ALBERT

Description: The published traditional crack problem solutions usually consider cracks located in the planes parallel to the plane of isotropy, which is usually denoted as z =  0. The case of a crack located in the plane x =  0 and subjected to arbitrary normal or tangential loading was solved recently in Fabrikant (Eur J Mech A 30:902–912, 2011 ). This article may be considered as its logical continuation. We consider here a transversely isotropic body, related to the system of axes ( x 1 , x 2 , x 3 ), weakened in the plane x 1 =  0 by a flat crack of arbitrary shape. The plane x 1 =  0 is perpendicular to the planes of isotropy of the transversely isotropic body. The axis Ox 1 coincides with the major axis Ox , and the axes Ox 2 and Ox 3 are obtained by rotation about the axis Ox by arbitrary angle φ . The governing integral equation is derived for such a general case. The case of an elliptical crack is considered in detail. The complete solution for the fields of displacements and stresses is presented as single contour integrals of elementary integrands. Stress intensity factors are computed explicitly.

Description: The so-called Stokes’ hypothesis for a Newtonian fluid is reconsidered, and a possible explanation is given of the fact that, in spite of its evidently weak physical justification, it permits to obtain good results for the description of most compressible flows. The explanation stands upon a closer analysis of the effect of the terms of the complete stress tensor in which the viscosity coefficients appear. An alternative formulation of the hypothesis is proposed, which also permits to clearly identify the very particular flow conditions in which it cannot justify the experimental evidence.

Description: Considering the chemical reaction during metal oxidation at high temperature, a coupled diffusion–reaction–mechanics model of the metallic oxidation is developed, which is different from the existing mechanochemistry coupling model. Seen from the coupled model, the hydrostatic pressure depending on the diffusion and reaction is a harmonic function if the body force is ignored. Analytical solutions of the concentration and hydrostatic pressure in the steady state are obtained. Then a numerical example in the unsteady state is performed by means of the finite difference method. Numerical results show that the distribution of the concentration in the thin plate is nonlinear due to the interactions among diffusion, reaction and stress.

Description: A nonlinear continuum model of van der Waals interactions is established to relate the size-dependent nonlinear properties of crystalline polyethylene. The explicit formulas are derived from the Lennard-Jones potential function for the van der Waals force between any two polymer chains. The present results are in reasonable agreement with those from present united-atom molecular dynamics simulations. This work is a new effort to establish nonlinear analytical models of molecular mechanics for crystalline polymers, and it should be helpful to provide an efficient route for mechanical characterization of crystalline polymers and related materials.

Description: The suppression of large vibrations of a smart thin elastic rectangular von Kármán’s plate is considered. The plate is subjected to external disturbances and generalized control forces produced by electromechanical feedback. The considered nonlinear initial-boundary value problem is spatially discretized by means of the time spectral method. The implicit Newmark- β iterative method is employed for the time integration of the obtained system of nonlinear equations of motion. Nonlinear controllers are designed, based on a fuzzy inference system. Two numerical algorithms involving a general control of displacement/velocity and a direct control of the Fourier coefficients are proposed. The techniques have been implemented within MATLAB environment with the use of the fuzzy logic toolbox. Numerical examples are presented.

Description: The flow induced above an impermeable membrane undergoing orthogonal linear stretching and orthogonal linear shearing is investigated. For an exact solution of the Navier–Stokes equations, the orthogonal shearing motions must be related through the constant σ = γ δ , where γ and δ are the dimensionless streamwise and transverse shear rates, respectively. The resulting similarity reduction leads to three nonlinearly coupled ordinary differential equations governed by σ and the ratio of membrane stretch rates β . All possible solutions of these equations are found either numerically or, in special cases, analytically. Features of the σ = 0 solutions at β = 0 and asymptotically as β → ∞ are found to be in excellent agreement with numerical calculations. An aside calculation shows that orthogonal shearing in the absence of any plate stretching cannot exist. However, shearing in one coordinate direction is possible as long as the membrane stretches in at least one direction with the caveat that there exists uniform suction through a porous membrane.

Description: A two-dimensional problem for an infinite thermoelastic half-space with a permeating substance in contact with the bounding plane is developed. The formulation is applied to the generalized thermoelastic diffusion based on Lord–Shulman theory. The bounding surface is traction free and subjected to a known axisymmetric temperature distribution, and the chemical potential is assumed to be a known function of time. Integral transform technique is used to find the analytic solution in the transform domain by using a direct approach. Inversion of transforms is done employing a numerical scheme. The mathematical model is prepared for copper material, and numerical results for temperature, stress, displacement, chemical potential and concentration are obtained and illustrated graphically.

Description: The study on reliability of an aging structure requires taking into account the influence of time. Typical approaches for performing time-variant reliability assessment are always based upon the random process model, where the dynamic distributions of uncertain parameters are determined by a substantial number of samples. In this paper, a new time-variant reliability measurement based on a non-probabilistic interval process model is proposed, in which we describe the time-varying uncertain variables at any time as intervals and define the corresponding auto-covariance function and correlation coefficient function to characterize the correlation between limit states at different times. By combining with the set-theory approach and the classical first-passage theory, a new non-probabilistic model of safety evaluation for time-dependent structures is established, and its measure index is then analytically calculated. The proposed model of time-variant reliability is suitable for both the cases of stationary process and non-stationary process. Moreover, the Monte Carlo method is also presented as a means of verification. The comparison between the presented model and the Monte Carlo-based model is eventually carried out on two application examples; the usage, efficiency and accuracy of the developed approach can be demonstrated.

Description: In this work, the concept of a linearized damage variable is introduced within the framework of continuum damage mechanics. Instead of considering one single fictitious undamaged configuration, a number n of smaller fictitious undamaged configurations are utilized. Thus, a smaller and linearized damage variable can be defined for each individual fictitious undamaged configuration. Additionally, the equations of damage evolution are formulated with respect to each individual fictitious undamaged configuration. Some interesting and surprising results are obtained. In this regard, a new result is obtained for the strain energies with respect to the n fictitious undamaged configurations. The linearized damage variable can be used for states of damage where the damage is small. The formulation is linear elastic based on linear superposition and should be applicable to many high-cycle fatigue problems.

Description: By extending the pseudo-Stroh formalism to multilayered one-dimensional orthorhombic quasicrystal plates, we derive an exact closed-form solution for simply supported plates under surface loadings. The propagator matrix method is introduced to efficiently and accurately treat the multilayered cases. As a numerical example, a sandwich plate made of quasicrystals and crystals with two different stacking sequences is investigated. The displacement and stress fields for these two stacking sequences are presented, which clearly demonstrate the importance of the stacking sequences on the induced physical quantities. Our exact closed-form solution should be of particular interest to the design of one-dimensional quasicrystal laminated plates. The numerical results can be further used as benchmarks to various numerical methods, such as the finite element and finite difference methods, on the analysis of laminated composites made of one-dimensional quasicrystals.

Description: In the present article, a new two-dimensional integrable system containing 17 free parameters is introduced. For giving certain values for these parameters, new integrable problems can be constructed, which generalize some known previous problems, and in some cases, we can restore some previous integrable problems. Two new integrable problems are announced, describing the motion in an Euclidean plane and on a pseudo-sphere. In the irreversible case, a new integrable problem in rigid body dynamics, which generalizes Goriachev–Chaplygin’s case (Varshav Univ Izvest 3:1–13, 1916 ), Yehia’s case (Mech Res Commun 23:423–427, 1996 ) and Elmandouh’s case (Acta Mech 226:2461–2472, 2015 ), is announced.

Description: Response variability of Timoshenko beams with uncertain Young’s modulus subjected to deterministic static loads is analyzed. The uncertain material property is idealized within a non-probabilistic context by using an interval field model recently proposed by the first two authors. Such a model is able to quantify the dependency between adjacent values of an interval uncertainty by means of a real, deterministic, symmetric, nonnegative, bounded function conceived as the non-probabilistic counterpart of the autocorrelation function characterizing random fields. In order to analyze the effects of Young’s modulus uncertainty on the static response of the Timoshenko beam, a finite difference discretization of the coupled interval ordinary differential equations of equilibrium is performed. Then, approximate explicit expressions of the lower bound and upper bound of the interval response are derived. Numerical results showing the effects of interval material uncertainty on the static response of a simply supported beam under uniformly distributed load are presented.

Description: This paper presents a simple method for finding the analytical solution for nonlinear boundary problems. The shifting function method is applied to developing the static deflection of an out-of-plane curved Timoshenko beam with nonlinear boundary conditions. Considering the out-of-plane motion of a uniform curved Timoshenko beam of constant radius R and a doubly symmetric cross section, three coupled governing differential equations are derived via Hamilton’s principle. After some simple arithmetic operations, the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. An example is given to illustrate the analysis and show that the proposed method performs very well for problems with strong nonlinearity.

Description: One of the significant issues in the nanotube research community is buckling behavior. In the present work, the buckling of single-walled nanotubes (SWNTs), double-walled nanotubes (DWNTs) and multi-walled nanotubes (MWNTs) under axial compression is investigated. Buckling analysis for nanotube composite structures is performed by using layer-wise theory based on the nonlocal constitutive relations of Eringen. The governing equations of SWNTs, DWNTs and MWNTs are developed. Then analytical solutions are obtained using the state-space method. The effects of nanotube length, diameter and nonlocal parameter on the buckling loads are studied. The numerical results indicate that the nonlocal parameter is important for the buckling analysis of nanotube composite structures.

Description: This paper presents the experimental results concerning the vortex-induced vibration phenomena of a flexibly mounted enclosed smooth rigid pipe. The experiment was newly designed and fabricated, utilizing the circulating water tunnel concept. New results on the dynamic response of the vibrating system are shown. The stream flow was generated by using a large displacement volume submersible water tank, controlled through the ABB inverter. The rotation frequency controlled based was implemented in the ABB inverter, and the measurement of stream flow was taken by using a flow rate sensor at the respective pump rotation frequency. The Reynolds number of the experiments ranges from \({\sim4900}\) to \({\sim15,000}\) , and its corresponding reduced velocities, based on the natural frequency in still water, vary up to \({\sim7}\) . The riser modeled pipe was positioned in vertical direction, and it is flexibly mounted at both ends. The system has low mass ratio and damping coefficient, with the values 1.184 and 0.08, respectively. The mass–damping ratio \({({m}^{*}\zeta)}\) is 0.09472. The results for the bare pipe cylinder in this experimental setup are in good agreementwith othermeasurements found in the literature.

Description: The influence of local thermal nonequilibrium with Cattaneo effects in the solid on the onset of thermal-convective instability in a horizontal layer of Darcy porous medium saturated by a ferrofluid in the presence of a uniform vertical magnetic field is investigated. The presence of the Cattaneo effect is to instill instability via oscillatory motion as well which is not reminiscent of the observed instability phenomenon in its absence. Increase in the value of the solid thermal relaxation time parameter τ is found to advance the onset of oscillatory ferroconvection. The onset of stationary ferroconvection is delayed, while the onset of oscillatory convection is accelerated with an increase in the value of the interphase heat transfer coefficient H t . The threshold value of H t , at which the transition from stationary to oscillatory convection takes place, decreases with increasing τ noticeably and marginally with increasing magnetic parameter M 3 , while it increases with increasing ratio of conductivities α and magnetic number M 1 . The critical wave number for stationary convection is found to be higher than for oscillatory convection.

Description: In existing literature, it remains an unexplored question whether any inclusion shape can achieve a uniform internal strain field in an elastic half-plane under either given uniform remote loadings or given uniform eigenstrains imposed on the inclusion. This paper examines the existence and construction of such single or multiple non-elliptical inclusions that achieve prescribed uniform internal strain fields in an elastic half-plane under given uniform anti-plane shear eigenstrains imposed on the inclusions. Such non-elliptical inclusion shapes in a half-plane can be determined by solving the original problem of an unknown holomorphic function in a multiply connected half-plane, which is transferred to an equivalent problem of an unknown holomorphic function in a multiply connected whole plane based on analytic continuation techniques. Extensive numerical examples are shown for single inclusion, multiple inclusions and two geometrically symmetrical inclusions, respectively. It is found that the inclusion shapes which achieve uniform internal strain fields depend on the given uniform eigenstrains, and the inclusion shapes that achieve uniform internal strain fields for arbitrarily given uniform eigenstrains do not exist. Moreover, specific conditions are derived on the given uniform eigenstrains and prescribed uniform internal strain fields for the existence of two geometrically symmetrical inclusions that achieve uniform internal strain fields.

Description: The nonlinear dynamics of a thin axisymmetric liquid film subjected to axial forcing of a horizontal cylindrical surface for initial conditions obtained from various states of a corresponding unforced system on its path to rupture, including near-rupture film configurations, is investigated in this paper. We exploit a nonlinear evolution equation describing the long-wave spatiotemporal dynamics of the film thickness derived in our earlier work and studied there for small-amplitude initial conditions. In this case, the existence of a critical curve separating between the domains of ruptured and contiguous states in the forcing amplitude–frequency space was shown. We find that near-rupture configurations of the film may be healed by applying harmonic forcing of the substrate with a sufficiently large forcing amplitude. We also reveal that for large forcing delay times the critical forcing amplitude is determined from a constant value of a linear combination of the maximal velocity and acceleration imparted by the forcing on the system when the rest of the problem parameters remains fixed. In the long-range limit, the dynamics of healed films reproduces the long-time dynamics of the films evolving from small-amplitude initial conditions for the same parameter set. It is found that local axial movement of the satellite drop in the near-rupture zone leads to a substantial slow-down of an increase in the critical amplitude in the intermediate range of the forcing delay time when the forcing frequency is fixed.

Description: While the response of a damped harmonic oscillator to random excitation offers the basic model in mechanics, stochastic dynamics, and stochastic fatigue of structures, the response due to random forcings with fractal and Hurst characteristics is studied for the first time. We investigate two types of such forcings: Cauchy and Dagum. Given the fact that their covariance functions lack explicit Fourier transforms, the spectral analyses are not possible, and, hence, the studies have to be conducted directly in the time domain—mathematically, a much more challenging task. In comparison with conventional models, we also give responses of the oscillator under random excitations of the white noise, Ornstein–Uhlenbeck (OU), and Matérn type. In contradistinction to the latter which has some limited ability of modeling multiscale phenomena, the Cauchy and Dagum forcings allow decoupling of fractal and Hurst effects. On the basis of a second-order stochastic differential equation, we calculate the transient second-order characteristics of the response under any given type of excitation. Overall, we find that, given the same variance on input, the variance on output is strongest for Matérn, then Cauchy, then OU, then white noise, and finally Dagum forcing. Also, if the excitation correlation function is Matérn, the correlation function of response is approximately Matérn, but this is not the case with the Cauchy excitation.

Description: This study deals with forced vibration analysis of a microplate subjected to a moving load. The formulation is developed based on the modified couple stress theory in conjunction with Kirchhoff–Love plate theory. The equations of motion of the problem are derived using Lagrange’s equations. In order to obtain the response of the microplate, the trial function for the dynamic deflection is expressed in the polynomial form. The equations of motion are solved by using the implicit time integration Newmark-β method, and then displacements, velocities and accelerations of the microplate at the considered point and time are determined. Five different sets of boundary condition are considered. For this purpose, boundary conditions are satisfied by adding some auxiliary functions to the trial functions. A parametric study is conducted to study the effects of the material length scale parameter, plate aspect ratio, boundary conditions and the moving load velocity on the dynamic response of the microplate. Also, in order to validate the present formulation and solution method, some comparisons with those available in the literature are performed. Good agreement is found. The results show that the dynamic deflections are significantly affected by the scale parameter and the load velocity.

Description: The oscillation of rigid rods is a simple and classical problem. The oscillation theory of rigid rods is widely applicable in various fields. However, the deformation effect on the oscillation periods of slender rods has not yet been studied. Herein we employ the energy method to derive the expression for the oscillation periods of deformable and slender rods. The derivation of the expression is straightforward. We present the deformation effect on the oscillation periods of slender iron wires. As the diameter of wires decreases and the length increases, the deformation effect becomes more significant and the percentage error of the oscillation period without including deformation effect increases. The expression for the oscillation periods of deformable and slender rods is verified by oscillation experiments.

Description: Molecular dynamics simulation of nano-indentation of single-crystal and bicrystal FCC aluminum is performed. The role of crystallographic orientation during nano-indentation of single-crystal aluminum is assessed. Then, the influence of the presence of a grain boundary is analyzed by adding high-angle symmetric tilt boundaries of Σ5〈210〉/(100) and Σ5〈310〉/(100) parallel to the surface on which the indentation is performed. Furthermore, in both cases, the size of the indenter is changed to investigate how the surface curvature of the indenter affects the nano-indentation process. The results suggest that in each crystallographic orientation, the presence of a grain boundary increases the required force for indentation, while the distance of a grain boundary from the indentation surface could affect the increase in the required force. Simulations prove that the grain boundary acts as a source of generation and emission of dislocations and restricts the penetration of the indenter by limiting the slip band formation and plastic deformation. The dislocation emission from the grain boundary restricts the penetration of the indenter and limits the formation of the octahedral slip systems of type {111}〈110〉 and consequently increases the required force for indentation in bicrystals.

Description: To extend the restricted applicability of the traditional perturbation method with small uncertainty level, this paper presents a first-order subinterval parameter perturbation method (FSPPM) and a modified subinterval parameter perturbation method (MSPPM) to solve the heat convection-diffusion problem with large interval parameters in material properties, external loads and boundary conditions. Based on the subinterval theory, the original uncertain-but-bounded parameters with limited information are divided into several small subintervals. The eventual response interval is assembled by the interval union operation. In both methods, the Taylor series is used to approximate the interval matrix and vector. The inversion of interval matrix in FSPPM is evaluated by the first-order Neumann series, while the modified Neumann series with higher-order terms is proposed to calculate the interval matrix inverse in MSPPM. By comparing the results with the traditional Monte Carlo simulation, two numerical examples evidence the remarkable accuracy and effectiveness of the proposed methods at predicting an uncertain temperature field in engineering.

Description: Applying the operator method, general solutions for dynamic governing equations involving anisotropic piezoelectric materials are derived. Based on the general solutions, a rigid solid with a sinusoidal surface moving along the surface of anisotropic piezoelectric materials is concerned. Dual series equations are obtained and solved analytically. Then, the electro-elastic fields are determined in series forms. A full contact problem leads to more simple forms. Numerical analysis is performed for the anisotropic piezoelectric material \({0^{\circ }}\) Graphite-epoxy, in which an illustrative example concerning the vertical mechanical displacement on the surface is displayed to validate the present approach. Results demonstrate that the contact behavior is considerably affected by both the velocity and material properties.

Description: A simplified analysis is carried out to derive the analytical solution for the large deflection of a fully clamped metal sandwich beam under impulsive loading. Using the classical yield criterion for sandwich structures, the moment–axial force interaction induced by large deflection is uncoupled. Then, plastic bending and stretching deformations are separated into two distinct phases, i.e., a bending-only phase and a plastic string phase. The simplified analytical solutions for the maximum deflection and structural response time have been obtained and are in good agreement with the previous theoretical and numerical results. Moreover, the analytical formulae are employed to determine the optimal geometries of a sandwich beam that maximize the resistance to impulsive loading at a given mass.

Description: In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Nambu system with fractional derivatives (the fractional Nambu dynamics), and study its applications to mechanical systems. By using the definition of combined fractional derivative, we construct unified fractional Nambu equations and, respectively, give four new kinds of fractional Nambu equations based on the different definitions of fractional derivative. Further, we study the relations between the fractional Nambu system, the fractional generalized Hamiltonian system, the fractional Birkhoffian system, the fractional Hamiltonian system, the fractional Lagrangian system and a series of integer-order dynamical systems and present the transformation conditions. And furthermore, as applications of the fractional Nambu method, we construct two kinds of fractional dynamical models, which include the fractional Euler–Poinsot model of rigid body which rotates with respect to a fixed point and the fractional Yamaleev oscillator model. This work provides a general method for studying a fractional dynamical problem which is related to science and engineering.

Description: A closed solution for one-dimensional heat conduction in a slab with nonhomogenous time-dependent boundary condition at one end and homogenous boundary condition with time-dependent heat transfer coefficient at the other end is proposed. The shifting function method developed by Lee and his colleagues is used to derive the solution of the temperature distribution of the slab. By splitting the Biot function into a constant plus a function and introducing two particularly chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions only. Consequently, the transformed system can be solved by a series expansion method. Two limiting cases, including time-independent boundary condition and constant heat transfer coefficient, are proved to be identical to those in the literature. Three-term approximation used in numerical examples can always result in an error 〈1 % in the present study, rendering the proposed methodology efficient and accurate. Finally, the influence of parameters of heat flux function or Biot function on the temperature distribution is presented.

Description: This research work addresses questions on the vibration characteristics of single-walled carbon nanotubes (SWCNTs) using multi-scale analysis. Atomistic finite element method (AFEM) is one such multi-scale technique where sequential mode is used to transfer information between two length scales to model and simulate the nanostructures at continuum level. This method is used to investigate the vibration characteristics of SWCNTs. Open- and capped-end armchair and zigzag nanotubes are considered with clamped-free and clamped–clamped boundary conditions. The dependence of vibration characteristic of SWCNTs on their length, diameter and atomic structure is also demonstrated. The body interatomic Tersoff–Brenner (TB) potential is used to represent the energy between two carbon atoms. Based on the TB potential, a new set of force constant parameters is established for carbon nanotubes and presented in this paper. Molecular and structural mechanics analogy is used to find the equivalent geometric and elastic properties of the space frame element to represent the carbon–carbon bond. To validate the vibration results of AFEM incorporating the proposed new set of force constants, molecular dynamics simulation is also carried out on the same structure of carbon nanotube, and it is found that they are in good agreement with each other.

Description: It is well known that the main approaches of the analytical solving of the elasticity mixed plane problems for a semi-strip are based on the different representations of the equilibrium equations’ solutions: the representations through the harmonic and by harmonic functions, through the stress function, Fadle–Papkovich functions and so on. The main shortcoming of these approaches is connected with the fact that to obtain the expression for the real mechanical characteristics, one should execute additional operations, not always simple ones. The approach that is proposed in this paper allows the direct solution of the equilibrium equations. With the help of the matrix integral transformation method applied directly to the equilibrium equations, the initial boundary problem is reduced to a vector boundary problem in the transformation’s domain. The use of matrix differential calculations and Green’s matrix function leads to the exact vector solution of the problem. Green’s matrix function is constructed in the form of a bilinear representation which simplifies the calculations. The method is demonstrated by the solving of the thermoelastic problem for the semi-strip. The zones and conditions of the strain stress occurrence on the semi-strip’s lateral sides, important to engineering applications, are investigated.

Description: Modal analysis free vibration response only (MAFVRO) is one of the output-only modal analysis methods which is able to determine the modal parameters of a vibration system by the free vibration responses. However, most of the vibrating systems are subjected to random excitations, and in such cases, the traditional MAFVRO cannot extract the modal parameters. In this study, the traditional MAFVRO method has been developed to randomly excited vibration systems by using some modifications. In order to verify the accuracy of modified MAFVRO, this method is applied to vibration analysis of a cantilever beam and a vehicle suspension system which are subjected to the random white noise excitation. The estimated results are compared with the exact solution obtained from the structural eigenvalue problem. Comparing the results reveals that the developed method can estimate the natural frequencies and mode shapes of a randomly excited system with a good accuracy. In addition, the effect of the measurement noise on the estimated modal parameters is investigated.

Description: Following their discovery in the early 1960s, there has been a continuous quest for ways to take advantage of the extraordinary properties of shape memory alloys (SMAs). These intermetallic alloys can be extremely compliant while retaining the strength of metals and can convert thermal energy to mechanical work. The unique properties of SMAs result from a reversible diffussionless solid-to-solid phase transformation from austenite to martensite. The integration of SMAs into composite structures has resulted in many benefits, which include actuation, vibration control, damping, sensing, and self-healing. However, despite substantial research in this area, a comparable adoption of SMA composites by industry has not yet been realized. This discrepancy between academic research and commercial interest is largely associated with the material complexity that includes strong thermomechanical coupling, large inelastic deformations, and variable thermoelastic properties. Nonetheless, as SMAs are becoming increasingly accepted in engineering applications, a similar trend for SMA composites is expected in aerospace, automotive, and energy conversion and storage-related applications. In an effort to aid in this endeavor, a comprehensive overview of advances with regard to SMA composites and devices utilizing them is pursued in this paper. Emphasis is placed on identifying the characteristic responses and properties of these material systems as well as on comparing the various modeling methodologies for describing their response. Furthermore, the paper concludes with a discussion of future research efforts that may have the greatest impact on promoting the development of SMA composites and their implementation in multifunctional structures.

Description: Visco-elastic material models with fractional characteristics have been used for several decades. This paper provides a simple methodology for Finite-Element-based dynamic analysis of structural systems with viscosity characterized by fractional derivatives of the strains. In particular, a re-formulation of the well-known Newmark method taking into account fractional derivatives discretized via the Grünwald–Letnikov summation allows the analysis of structural systems using standard Finite Element technology.

Description: This paper presents a novel three-dimensional hybrid smoothed finite element method (H-SFEM) for solid mechanics problems. In 3D H-SFEM, the strain field is assumed to be the weighted average between compatible strains from the finite element method (FEM) and smoothed strains from the node-based smoothed FEM with a parameter α equipped into H-SFEM. By adjusting α, the upper and lower bound solutions in the strain energy norm and eigenfrequencies can always be obtained. The optimized α value in 3D H-SFEM using a tetrahedron mesh possesses a close-to-exact stiffness of the continuous system, and produces ultra-accurate solutions in terms of displacement, strain energy and eigenfrequencies in the linear and nonlinear problems. The novel domain-based selective scheme is proposed leading to a combined selective H-SFEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed 3D H-SFEM is an innovative and unique numerical method with its distinct features, which has great potential in the successful application for solid mechanics problems.

Description: In this work, we study the transformation properties of the local form of the material (referential) balance of energy equation under the superposition of arbitrary material diffeomorphisms. For this purpose, the tensor analysis on manifolds is utilized. We show that the material balance of energy equation, in general, cannot be invariant; in fact an extra term appears in the transformed balance of energy equation, which is directly related to the work performed by the configurational stresses. By making the fundamental assumption that the body and the ambient space manifolds are always related in the course of deformation and by utilizing the metric concept, we determine this extra term. Building on this, we derive several constitutive equations for the material stress tensor. The compatibility of these constitutive equations with the second law of thermodynamics is evaluated. Finally, we postulate that the material balance of energy equation is covariant, and we study this case in detail, as well.

Description: Multi-axial hyper-elastic models for large strain rubberlike elasticity are established in a broad sense free of the commonly assumed constraint of incompressibility. Results are derived directly from uniaxial stress–strain relations by means of certain explicit procedures. Novelties in four respects are incorporated in the new models: (i) constitutive parameters of direct physical meanings may be introduced to represent features of rubberlike elasticity; (ii) the usual incompressibility constraint is no longer assumed and relevant issues may be rendered irrelevant; (iii) the incompressibility case may be derived as a natural limit; and (iv) accurate agreement with benchmark data for several deformation modes may be achieved, including uniaxial extension, equi-biaxial extension as well as plane-strain extension and others.

Description: Hypersingular integral equations for two intersecting three-dimensional cracks are derived by using Somigliana formula, in which the unknown functions are the displacement discontinuities on the crack-surfaces. The singularity of the unknowns at the crack-front is analyzed by the analytical asymptotic method, and the analytical solutions of the singular stresses near the crack-front are given. Numerical solutions of the stress intensity factors for some examples are presented and discussed.

Description: This article presents new constructive formulas to steady-state thermoelastic Green’s functions for a plane generalized boundary value problem of thermoelasticity for a generalized rectangle. The constructive formulas are expressed in terms of Green’s functions for Poisson’s equation. These results are formulated in a special theorem, which is proved using the author’s developed harmonic integral representations method. On the base of derived constructive formulas, it is possible to obtain many analytical expressions for Green’s functions for thermoelastic displacements and stresses to 28 concrete boundary value problems for: rectangle-16, half-strip-8, strip-4. An example of such kind is presented for a concrete plane boundary value problem for a rectangle, Green’s functions of which are presented in the form of a sum of elementary functions and ordinary series. These results are presented in another theorem, which is proved on the base of derived general constructive formulas. In the particular case for a half-strip and strip, ordinary series vanish and Green’s functions are presented in elementary functions. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle subjected to a constant boundary temperature gradient are also derived. Numerical investigation has shown that the infinite series are convergent. All solutions obtained for thermal stresses, caused by a constant temperature gradient and by a unit heat source are validated by checking the respective equilibrium equation and continuity of deformation equations (Beltrami–Michel equations), written in the terms of thermal stresses. The graphics for thermal stresses and their infinite series also are presented.

Description: This paper aims to study the vibration characteristics of viscoelastic double-walled carbon nanotubes embedded in a viscoelastic medium. In doing this, the governing equations of the system are derived by combining the Euler–Bernoulli beam theory, nonlocal viscoelastic model and Kelvin viscoelastic foundation model. Subsequently, the transfer function method is employed to solve the governing equations, which enables one to obtain the natural frequencies and the corresponding mode shapes in closed form for the DWCNTs with arbitrary boundary conditions. Here, the developed mechanics model is first compared with the existing techniques available in the literature, where excellent agreement is achieved. Also, a detailed parametric study is conducted to examine the effect of boundary conditions, nonlocal parameter, relaxation time, slenderness ratio, stiffness coefficient and damping coefficient on the vibration response of the DWCNTs.

Description: In this paper, Reddy’s third-order shear deformable plate theory is employed for the analysis of centrosymmetric anisotropic plate structures within strain gradient elasticity. The general three-dimensional anisotropic gradient theory is reduced to a two-dimensional formulation for the analysis of thick plate structures. The third-order shear deformation theory (TSDT) takes into account quadratic variation of the transverse shear strains of the plate and does not require shear correction factors. In order to investigate the case of small strains but moderate rotations, the von Kármán strains are considered. The TSDT is also simplified to anisotropic Kirchhoff plate theory within gradient elasticity. To study specific material properties in more detail, the (Kirchhoff and TSDT) gradient plate theory of general anisotropy is simplified to the more practical case of orthotropic plates. It is observed that the gradient theory provides the capability to capture the size effects in anisotropic plate structures. As case studies, the bending and buckling behaviors of the simply supported orthotropic (Kirchhoff and TSDT) plates are studied. Variationally consistent boundary conditions are also discussed. Finally, analytical solutions are presented for the bending and buckling of simply supported orthotropic Kirchhoff plates. The effects of internal length scales on deflections and buckling loads are presented.

Description: The deflection of thin, elastic plates with different curvatures is experimentally investigated under pulsatile cross-flow conditions. Average and maximum deflections are reported in terms of the plate’s curvature, flow frequency, and stroke fraction. The laboratory results indicate that the maximum deflections can be reduced significantly when the curvature is increased (around 50 % for low-to-intermediate curvatures and up to 80 % for high curvatures). Moreover, the comparative analysis shows that the deflection is inversely proportional to the rigidity of the plates, which depends nonlinearly on the curvature. Based on the observed behavior, a simplified Euler–Bernoulli beam model is derived to qualitatively explain the effects of geometrical factors such as the curvature, the thickness, and the length of the plates. Some traits of the flow evolving around the plate and the need to have accurate measurements of the pressure are discussed.

Description: An analytical approach is used to investigate the SH-wave scattering problems of the bi-material half-space including a circular inclusion embedded and an interfacial crack. Considering the multiple scattering phenomena between the boundaries and the circular inclusion, the “image” principle is used to establish the expressions of the wave field. With the aid of interfacial conjunction, the interfacial crack is constructed. According to the displacement and stress continuity conditions, the Fredholm integral equations of the first kind are obtained to determine the unknown forces attached to the linking sections. Finally, analytical expressions for dynamic stress intensity factors are obtained. The numerical results for dynamic stress intensity factors are presented graphically considering the factors of the crack length, the incident wave number, the inclusion defect and the different media parameters.

Description: This paper studies the reflection and transmission of a plane wave through a fluid layer of finite width sandwiched between two dissimilar monoclinic elastic half-spaces. Closed-form expressions for the phase velocity of quasi-waves (qP and qSV) have been obtained. The reflection/transmission coefficients and energy divisions have been procured for all the reflected and transmitted waves in terms of phase velocity, propagation vector, elastic constants and width of the layer. It has been noticed that these epitomes depend not only upon the incident angle and width of the layer, but also on the character of incident wave. Energy proportions have been calculated numerically to validate the rule of energy conservation at different angles of incidence. Graphical representation has been performed to demonstrate the analytical findings.

Description: Due to their superior mechanical characteristics, nanoporous materials and structures have attracted increasing interest in the last decade. The mechanical and physical properties of the surface layer are different from the bulk material when the structural size is at nanoscale. In this work, a nanoporous model with periodically arranged cuboid-shaped pores is presented. Based on the surface elasticity, the expression of the effective Young’s modulus is obtained theoretically. For a given relative density, the influences of the surface effect on the nanoporous structure are examined. It is shown that the effective Young’s modulus grows with increasing surface Young’s modulus or decreasing inclination angle for the ligament. Moreover, it is suggested from the results that a negative effective Young’s modulus could be achieved when the residual surface stress is less than zero. In addition, the impact of the relative density on the proposed nanoporous material is also investigated.

Description: Appropriate representation of the displacement field is important to establish proper stress distribution including shear-free condition at free surfaces for a laminated composite shell used in real engineering applications. The present study attempted to develop a more accurate higher-order displacement field for the analysis of a doubly curved laminated composite shell with \(\hbox {C}^{0 }\) finite element model based on higher-order shear deformation theory. A new displacement function is proposed for static and free vibration analysis of such a shell. The accurate strain displacement relationship is applied in the analysis of a shell structure with exactly zero shear-free condition at top and bottom surfaces. The proposed model is capable of determining the accurate shear stress distribution across the thickness of the laminate. Moderately deep and thick shells can be analyzed very accurately as the ratio of thickness coordinate to radius of curvature is incorporated in the formulation. An eight-noded isoparametric shell element with seven degrees of freedom at each node is used to formulate the finite element model. The numerical results in terms of deflection, stresses and natural frequencies obtained by the present formulations are compared with those available in the published literature to validate the accuracy of the proposed model.

Description: The paper addresses the frequency analysis of bars with an arbitrary number of dampers, subjected to harmonically varying loads. Multiple external/internal dampers occurring at the same position along the bar, modelling external damping devices and internal damping due to damage or imperfect connections, are considered. In this context, the challenge is to handle simultaneous discontinuities of the response variables, i.e. axial force/displacement discontinuities at the location of external/internal dampers. Based on the theory of generalized functions, the paper will present exact closed-form expressions of the frequency response under point/polynomial loads, which hold regardless of the number of dampers. In addition, closed-form expressions will be derived for the exact dynamic stiffness matrix and load vector of the bar, to be used in a standard assemblage procedure for an exact frequency response analysis of 2D truss structures. Changes to consider a single damper at a given position are straightforward. Numerical applications show the advantages of the proposed method.

Description: The present work aims to investigate the effect of the gravitational field on a two-dimensional thermoelastic medium influenced by thermal loading due to a laser pulse. The bounding plane surface is heated by a non-Gaussian laser beam. The problem is discussed under Green–Naghdi theory with and without energy dissipation. The normal mode analysis method is used to get the expressions for the physical quantities. The results are illustrated graphically.

Description: A novel modified edge-based smoothed finite element method (modified ES-FEM) is developed to compute the band gap of acoustic metamaterials. The stiffness in the modified ES-FEM is created by the edge-based smoothed finite element method (ES-FEM) which is aimed at softening the overly stiffness of standard finite element method (FEM). On the other hand, the mass matrix is constructed by mass-redistributed method to tune the balance between the smoothed stiffness and mass matrix. The present modified ES-FEM adopts linear triangular elements generated automatically, which enables automation in computation and saving computational cost in mesh generation. Two numerical examples are presented to verify the computational efficiency of the modified ES-FEM. The numerical results have clearly demonstrated that the modified ES-FEM is very effective to minimize the dispersion errors in the simulation of band gap of acoustic metamaterials.

Description: In this paper, equilibrium bifurcations of an axially moving Timoshenko beam are studied in the supercritical region. For the first time, Timoshenko beam theory is applied to investigate nonlinear dynamics of high-speed axially moving structures. The static equilibrium equation is deduced from the governing equation of transverse vibration of the axially moving Timoshenko beam. Two kinds of boundary conditions are considered. The non-trivial equilibrium solutions are analytically determined. Moreover, the equilibrium equations are discretized by using the finite difference method. Therefore, equilibrium configurations are numerically verified by proposing an iterative scheme. This investigation shows that non-trivial equilibrium solutions of Timoshenko beams bifurcate with axially moving speed. By comparing with Euler–Bernoulli beam theory, this study finds that the critical speed, determined by the Timoshenko beam, is remarkably smaller. Nevertheless, the equilibrium deformation of the moving Timoshenko beam is obviously larger. Furthermore, the present work derives the critical speed of the axially moving Timoshenko beam. At last, the effects of the system parameters on the equilibrium bifurcation and the critical speed are presented.

Description: In this paper, we extend the modeling effort of our earlier work concerning the development of an implicit elastic model for describing the response of biological fibers. We put into place a fractional-order viscoelastic (FOV) solid model that can quantify the properties of biological fibers comprised of collagen fibrils and elastic filaments. The compliance version of the FOV memory function is regularized, thereby removing the singularity from the viscoelastic kernel. The ensuing model is used to describe stress relaxation in mitral-valve chordæ tendineæ. Numerical solutions for the convolution integral are acquired via a midpoint quadrature rule with a Laplace endpoint correction.

Description: The aim of this paper is to study the problem of stress concentration in an infinite plate with an elliptic hole reinforced by a functionally graded layer based on the complex variable method combined with the technique of conformal mapping. With using the method of piece-wise homogeneous layers, the general solution for the functionally graded layer having normal arbitrary elastic properties is derived when the plate is subjected to arbitrary constant loads at infinity, and then numerical results are presented for several special examples. It is found that the existence of the functionally graded layer can influence the stress distribution in the plate, and thus choosing proper variations of the normal elastic properties and proper thicknesses of the layer can effectively reduce the stress concentration around the elliptic hole.

Description: The Noether symmetries and conserved quantities of the Birkhoffian systems in terms of fractional derivatives of variable order are studied. Firstly, the Pfaff–Birkhoff–d’Alembert principle within fractional derivatives of variable order is obtained, and corresponding variable order fractional Birkhoff’s equations are deduced. Secondly, the invariance of the fractional Pfaff action of variable order is studied under the one-parameter group of infinitesimal transformations, and the definition of the variable order fractional conserved quantity is given. Finally, the Noether’s theorem for the fractional Birkhoffian system of variable order is established. At the end of this paper, an example is given to illustrate the application of the results.

Description: Experimental pulse transmission in impulsively loaded, homogeneous, and dimer granular chains, optionally embedded in a viscoelastic matrix, is studied. All tested chains are composed of spherical elastic beads of common radius. Homogeneous chains were composed of granules with equal mass, whereas dimer chains had alternating ‘heavy’ and ‘light’ granules with different masses. These media are strongly nonlinear due to Hertzian interactions between adjacent beads under compressive loads, and separations and collisions between them in the absence of compression. A series of experimental tests was performed to study primary pulse transmission in the non-embedded chains, and assess the effect of the surrounding viscoelastic matrix on pulse transmission in the embedded ones. For the case of dimer chains, the effect of mass inhomogeneity on pulse attenuation caused by scattering at the interfaces between adjacent beads is studied. In total, two embedded dimer chains, as well as an embedded homogeneous one, were manufactured and tested, and a previous theoretical model is used to compare theoretical predictions to experimental measurements. Whereas one of the non-embedded dimer chains differs from the others in that its light beads are hollow and so its experimental responses are not captured well by our theoretical model, for the other embedded and non-embedded chains, the theoretical predictions match remarkably well with the experimental measurements, despite the complexity in the acoustics induced by the surrounding matrix, and the conceptual simplicity of the theoretical model. The results of this work contribute to the development of practical acoustic metamaterials incorporating embedded granular media.

Description: The purpose of this research is to present the wave propagation analysis of a functionally graded nano-rod made of magneto-electro-elastic material subjected to an electric and magnetic potential. The unified nonlocal elasticity theory and Love's rod model are used in this study. All mechanical, electrical and magnetic properties are assumed to be variable along the thickness direction based on a power law distribution. Two-dimensional electric and magnetic potential distributions due to an applied potential and a magnet at the top of the rod are considered. The governing equations of motion are obtained using equilibrium and nonlocal theory of elasticity in conjunction with the Hamilton principle. The effect of important parameters of the functionally graded magneto-electro-elastic nano-rod such as nonlocal parameters, power index, wave number, applied magnetic and electric potentials on the wave propagation characteristics is studied.

Description: We study the generalized plane strain deformations of an elastically isotropic circular inhomogeneity partially bonded to an unbounded generally anisotropic elastic matrix. The two-phase composite is subjected to a uniform loading at infinity, and meanwhile a line force and a line dislocation are applied both in the inhomogeneity and in the matrix. An elegant closed-form solution is obtained by reducing the original boundary value problem to a non-homogeneous Riemann–Hilbert problem of vector form which can be analytically solved by using a decoupling method and evaluating the Cauchy integrals. Surface traction on the bonded part of the interface, displacement jump across the debonded part of the interface, and the complex and real stress intensity factors at the crack tips are explicitly derived when the composite is only subjected to a remote uniform loading.

Description: We examine the torsional wave propagation along a micro-/nanowire with consideration of the surface elasticity effect. The surface of the wire is modeled by a two-dimensional “membrane” which is described by the surface/interface mechanics of Gurtin and Murdoch. The wave dispersion diagram is numerically presented with the surface/size effect which is characterized by two surface/size factors. They are “dynamic size/surface factor” and “static size/surface factor.”

Description: Piezoelectric transducers with high directional-dependent response are of interest in several applications because of their ability to sense and actuate along specific directions where they can distinguish individual principal strain components. This paper presents an improved unidirectional sensor obtained using a multi-laminate piezocomposite that maximizes the electromechanical coupling factor or the piezoelectric strain constant in one direction with respect to the other directions. Furthermore, multi-laminate structures provide significant design potential by the variation of the orientation and stacking sequence of fibers to obtain the desired properties. The effective properties depend on the number of layers, the fibers orientation as well as the thickness of each layer, and they are estimated by the variational asymptotic method for unit cell homogenization. A design that guarantees maximum directional dependence in terms of piezoelectric strain constant is determined by a global optimization technique using a genetic algorithm. Layered transducers composed of several orthotropic passive layers and a single active layer are considered.

Description: Surface stresses can significantly affect the mechanical behavior of structures when they are scaled down to deep submicron dimensions. The Gurtin–Murdoch surface elasticity theory has the capability to capture the size-dependent behavior of nanostructures due to the surface stress effect in a continuum manner. The present work is concerned with the application of Gurtin–Murdoch theory to the nonlinear free vibration analysis of circular cylindrical nanoshells with considering surface stress and shear deformation effects. The nonlinear governing equations of motion together with the corresponding boundary conditions are firstly derived using Hamilton’s principle, the first-order shear deformation shell theory and von Kármán’s assumption. An analytical approach is then presented to solve the nonlinear free vibration problem. Selected numerical results are given to illustrate the effects of surface energy on the nonlinear free vibration behavior of shear deformable nanoshells with different material and geometrical parameters. It is shown that there is a large difference between the results of Gurtin–Murdoch theory and those of its classical counterpart for very thin nanoshells.

Description: In previous works, a model was developed for the transition of an epoxy liquid to a cured solid as a result of the continuous formation of solid epoxy networks by crosslinking. The model accounts for volume decrease during crosslinking, thermal expansion due to heat generation, deformation and stress generation. The networks were assumed to be linear thermo-elastic isotropic solids, material properties were deduced from the literature, and the model was used to determine the stresses generated by curing in a number of examples. The purpose of the present work is to revisit the ideas on which this model is based and incorporate them into a general framework within the context of continuum mechanics. This will show the assumptions that have been made in the existing model because of the lack of experimental results and also provide a proper framework for extending the model as new experimental results are obtained. Each network is now assumed to be a nonlinear thermo-viscoelastic solid with its own properties. The assumption of small deformations is introduced to lead to a simplified model that is linear in the strains.

Description: The impact of a rigid-rod into a low-strength target is an ubiquitous research problem in many scientific and engineering fields. Typically, this impact has been modeled by the expansion of a pressurized spherical cavity in an unbounded elasto-plastic medium. In this paper, an engineering penetration model was developed using the dynamic spherical cavity expansion theory for an elasto-plastic, compressible, strain-dependent and strain-rate-sensitive material. The material plastic flow behavior was modeled using the Cowper–Symonds strength model. The engineering model was numerically solved, and its predictions were compared with results of computational finite-elements simulations performed in ANSYS/AUTODYN. Additionally, engineering and computational models were validated with experimental data using spherical-nosed, M300 steel projectiles impacting a semi-infinite Al 6061-T651 plate. Comparisons showed good agreement among engineering model results, computational model results and experimental data.

Description: A coupled analytical layer-element solution is presented to study the three-dimensional thermo-mechanical behavior of layered material with anisotropic thermal diffusivity in Cartesian coordinates. From the governing equations of three-dimensional thermo-elastic material, the coupled analytical layer elements expressing the relation between generalized displacements and stresses of a single finite layer and the underlying half-space are derived by the Laplace transform and the double Fourier transform. Considering the continuity conditions between adjacent layers and the boundary conditions, the global stiffness matrix of the multilayered half-space is assembled and solved in transformed domain. The real solutions in the physical domain are obtained by applying numerical quadrature schemes for the Laplace–Fourier transform inverse. Finally, numerical computations are carried out to investigate the time-dependent thermo-mechanical response of the material system.

Description: Peeling of a thin film adhesively bonded to a rigid substrate is analytically studied using a Bernoulli–Euler beam theory for viscoelastic materials. The film (adherend) is modeled as a viscoelastic Bernoulli–Euler beam, and the normal and shear stresses on the film-adhesive interface are assumed to satisfy constant traction laws. Closed-form solutions are derived for the following two cases: (i) only the interfacial normal stress is present (mode I loading) and (ii) the interfacial shear stress is acting alone (mode II loading). The Boltzmann superposition integral is used to obtain the constitutive relations for the viscoelastic beam, and the methods of separation of variables and Laplace transforms are employed in the formulation. To illustrate the newly derived analytical solutions, sample cases are quantitatively studied. The three-parameter Kohlrausch–Williams–Watts model is adopted to compute the compliance. The numerical results show that both the vertical displacement under mode I loading and the horizontal displacement under mode II loading increase with time and/or temperature.

Description: Electroelastic behavior induced by a penny-shaped dielectric crack in a piezoelectric cylinder (i.e., a finite piezoelectric body in the radius direction) is considered in this paper. Two coupled Fredholm integral equations are derived by employing the Hankel transform technique with the introduction of certain auxiliary functions. The intensity factors of stress, electrical displacement and crack opening displacement (COD) are obtained in closed forms. The effects of the radius of the piezoelectric cylinder, the applied electrical field and permittivity of the crack interior on the COD intensity factor are illustrated numerically. The results of both the electrically permeable and impermeable boundary conditions are simultaneously displayed as special cases.

Description: The scattering problem of the SH wave on a limited permeable crack in a functionally graded piezoelectric substrate bonded to a homogeneous piezoelectric strip is investigated. We adopted the limited permeable crack surface boundary condition. By using the Fourier cosine transform, this mixed boundary value problem is reduced to two pairs of dual integral equations, which are solved numerically by the Copson method. Numerical results showed the effects of gradient parameter, electric loading, electric boundary conditions, incident angle, thickness of PM strip, the distance from the crack to the interface, and wave number on the dynamic stress intensity factor.

Description: A two-dimensional generalized plane strain micromechanical model is developed to study the electro-elastic behavior of piezoelectric fiber-reinforced composite (PFRC) systems. The composite system consists of long parallel piezoelectric fibers with orthotropic and/or transversely isotropic properties and perfectly bounded to the isotropic matrix in a square array arrangement. In addition, the constituents are assumed to have both linear elastic and electrical behavior, whereas the matrix is piezoelectrically passive. The element-free Galerkin (EFG) method is employed to obtain the solution for the governing system of partial differential equations. The performance of the model is examined for both axial and transverse polarizations and various fiber cross sections. Comparison of the presented results with other techniques available in the literature reveals good agreement. It is demonstrated that the piezoelectric coefficient e 31 in the transverse polarization is considerably improved in comparison with the corresponding coefficient for pure piezoelectric material. Furthermore, results also show that elliptical fibers may enhance the electrical sensitivity of PFRCs for a specific direction, which is related to the elliptical fiber orientation, in both polarization directions.

Description: A maximum entropy-based stochastic micromechanical framework considering the inter-particle interaction effect is proposed to characterize the probabilistic behavior of the effective properties of two-phase composite materials. Based on our previous work, the deterministic micromechanical model of the two-phase composites is derived by introducing the strain concentration tensors considering the inter-particle interaction effect. By modeling the volume fractions and properties of constituents as stochastic, we extend the deterministic framework to stochastics, to incorporate the inherent randomness of effective properties among different specimens. A distribution-free method is employed to get the unbiased probability density function based on the maximum entropy principle. Further, the normalization procedures are utilized to make the probability density functions more stable. Numerical examples including limited experimental validations, comparisons with existing micromechanical models, commonly used probability density functions and the direct Monte Carlo simulations indicate that the proposed models provide an accurate and computationally efficient framework in characterizing the effective properties of two-phase composites.

Description: The coupled three-dimensional flexural vibrations of a micro-rotating shaft–disk system, as a basic model for micro-engines, are investigated in this paper by considering small-scale effects utilizing the modified couple stress theory. Governing equations of motion are derived by the use of Hamilton’s principle. Then, implementing the Galerkin approach, an infinite set of ordinary differential equations is obtained for the system. With truncated two-term equations, expressions for the first two natural frequencies are written, and for the two corresponding modes, the maximum rotational speed up to which the system will be stable is analytically determined. Parametric studies on the results for different responses illustrate that the length-scale value has a significant effect on the natural frequencies of the shaft and the threshold of instability of the system.

Description: Flexoelectricity, representing the coupling between electrical polarizations and strain gradients, should be taken into account in the analysis of electromechanical responses of nanostructures where large strain gradients are expected. In this paper, we will explore the influence of flexoelectricity on the electromechanical coupling behavior of a simply supported piezoelectric nanoplate by using the Kirchhoff plate theory. The governing equations and corresponding boundary conditions are deduced from Hamilton’s principle, and the analytical solutions are obtained for the deflection and natural frequency. The results indicate that the deflections predicted by the present model are smaller than those calculated by the classical one which only considers piezoelectricity, while the frequencies exhibit the opposite trend. In addition, the flexoelectric effect is more prominent for thinner plates; the differences of the deflections or frequencies between the two models are gradually diminishing with an increase in the plate thickness. The current work may contribute to the understanding of the higher-order electromechanical coupling mechanism. Moreover, the modified plate model can be utilized to accurately design novel piezoelectric nanoplate-based sensors in nanoelectromechanical systems.

Description: This manuscript provides a theoretical stability analysis for the configuration of two-layer immiscible flow down an inclined plane with velocity slip along the incline in the limit of zero Reynolds number. Surfactants may be present at the air–liquid interface, liquid–liquid interface, or both. In addition to an Orr–Sommerfeld analysis (again at zero Reynolds number), a long wavelength stability analysis is performed and the results are shown to be consistent. The interface mode, namely the mode of instability that arises because of viscosity stratification, is examined. Stability results (growth rates as a function of slip parameter and neutral stability boundaries) for various configurations of viscosity, surfactant placement, and layer thickness are compared with those of the previous literature and found to agree. It is found that velocity slip along the inclined plane reduces the maximum growth rate of instabilities in configurations where they occur, and the range of unstable wave numbers shrinks as well, indicating that slip has a promise for stabilization. This suggests that there is a possibility of using this favourably as a control option for two-layer flows in the absence or presence of surfactants, in relevant applications by designing the substrate to be a porous substrate with small permeability or a slippery substrate or a rough substrate or a hydrophobic substrate which can be modelled as substrates with velocity slip.

Description: This paper investigates the probability density function (PDF) of multi-degree-of-freedom nonlinear systems excited by nonzero mean Poisson impulses. The PDF solution is governed by the Kolmogorov–Feller equation which is also called the generalized Fokker–Planck–Kolmogorov (FPK) equation. First the high-dimensional generalized FPK equation is reduced to a low-dimensional equation by a state-space-split method. The reduced FPK equation is further solved by an exponential-polynomial closure method. In numerical analysis, a ten-degree-of-freedom Duffing system is further investigated under nonzero mean Poisson impulses. The PDF distribution of impulse amplitudes is adopted with either a nonzero mean Gaussian distribution or a nonzero mean Rayleigh distribution. Comparison with the simulated results shows that the proposed solution procedure is effective to obtain a satisfactory PDF solution, especially in the tail region. The nonzero mean PDF of displacement is formulated due to nonzero mean Poisson impulses. The obtained PDF of displacement is not symmetrically distributed about its mean.

Description: In the present work, the active control of vibration of annular plates is presented by the design of a cylindrically orthotropic short/continuous piezoelectric fiber-reinforced composite (SPFRC/CPFRC) actuator. The unidirectional piezoelectric fibers of the smart composite are oriented along the radial direction within a reference cylindrical coordinate frame and poled in the same direction. First, a finite element analysis of the effective electro-elastic properties of the smart composite is presented, and the optimal geometry of its unit cell is determined with an objective of improved magnitude of an effective piezoelectric coefficient ( e 11 , 1 for radial direction) for both short (SPFRC) and continuous (CPFRC) forms of piezoelectric fibers. Next, an arrangement of surface electrodes is presented for its effectual utilization as an actuator based on the coefficient e 11 . Subsequently, the smart actuator is attached to the surface of a host annular plate in the form of actuator patches for substantiating its control performance by the numerical evaluation of controlled frequency responses of the overall smart annular plate. The actuator patches act as smart dampers by means of supplying voltage according to the velocity feedback control strategy. The numerical results reveal more control power of the SPFRC actuator than that of a CPFRC actuator even though the magnitude of the major effective coefficient ( e 11 ) for SPFRC is lesser than that for CPFRC. The overall analysis shows a meaningful control power of present cylindrically orthotropic SPFRC/CPFRC actuators in control of vibration of annular plates and suggests short piezoelectric fibers instead of continuous fibers within it (smart actuator) for achieving its larger control power, flexibility and conformability.

Description: A stable and accurate Smoothed Particle Hydrodynamics (SPH) method is proposed for solving elastodynamics in solid mechanics. The SPH method is mesh-free, and it promises to overcome most of disadvantages of the traditional finite element techniques. The absence of a mesh makes the SPH method very attractive for those problems involving large deformations, moving boundaries and crack propagation. However, the conventional SPH method still has significant limitations that prevent its acceptance among researchers and engineers, namely the stability and computational costs. In approximating unsteady problems using the SPH method, attention should be given to the choice of time integration schemes as accuracy and efficiency of the SPH solution may be limited by the timesteps used in the simulation. This study presents an attempt to reconstruct an unconditionally stable SPH method for elastodynamics. To achieve this objective we implement an explicit Runge–Kutta Chebyshev scheme with extended stages in the SPH method. This time stepping scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. Numerical results are shown for several test problems in elastodynamics. For the considered elastic regimes, the obtained results demonstrate the ability of our new algorithm to better maintain the shape of the solution in the presence of shocks.

Description: This paper proposes that Kane’s equations for a simple nonholonomic system are the first-order form of generalized speeds. When the first-order form of Kane’s equations is put in matrix form, the element of the mass matrix is identical to the inertia coefficient. Because the orthogonal set of partial velocities will decouple the first-order equations, one can use the orthogonal criterion to generate efficient equations of motion. With the presented first-order form, Kane’s equations are different from Maggi’s or Gibbs–Appell’s equations. Moreover, in order to clarify the relationship between Kane’s approach and classical approaches, we start from Kane’s equations and introduce kinetic energy or acceleration energy functions to derive Lagrange’s or Gibbs–Appell’s forms of Kane’s equations for the system. We found that the Lagrange’s forms of Kane’s equations can be used to solve nonholonomic systems without introducing Lagrangian multipliers. At last, the first-order form, Lagrange’s forms, and Gibbs–Appell’s forms of Kane’s equations are, respectively, used to depict the derivation of first-order equations of motion for a rolling coin.

Description: A fiber-reinforced composite in which aligned fibers of uniform size are randomly embedded in a continuous matrix is considered. Bounds on the transverse effective transport property of the composite which were statistically evaluated by Monte Carlo simulations are discussed and analyzed with reference to the third-order perturbation bounds. Three-point geometric parameters for the transverse cross section of the composite are inferred from one of the previously reported results, and comparisons are made with those from other models. It is concluded that the three-point parameters can be used to quantify the geometrical arrangement of the fibers in the matrix. With the extracted three-point parameters, the fourth-order bounds can be exploited to sharpen the numerical predictions.

Description: We analytically investigate the contribution of surface elasticity to the Saint-Venant torsion problem of a circular cylinder containing a non-radial crack. The surface elasticity for the crack faces is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. Both internal and edge cracks are studied. By employing the Green’s function method, we reduce the original boundary value problem to two coupled first-order Cauchy singular integro-differential equations which can be numerically solved by the Chebyshev polynomials and an adapted collocation method. The analysis indicates that in general the stresses at the crack tips exhibit both the weak logarithmic and the strong square root singularities. The jump in the warping function across the crack faces and the size-dependent torsional rigidity are calculated.

Description: Studies on the fracture criterions of structural materials would be of great significance to the service security of engineering structures. This note proposes a novel criterion on the basis of the uniformity of plastic work under various stress states. In order to realize abundant stress states, modified Arcan fixtures were designed and integrated with a self-made in situ tensile device. Accordingly, by changing the stress ratio of tensile to shear components, true stress–strain relationships of a typical polycrystalline ductile material (Gr-4 titanium alloy) specimens on the basis of various stress states were obtained and piecewisely fitted. By, respectively, adopting linear and polynomial fitting in the elastic and plastic deformation stages, the plastic work, namely the envelope areas of true \({\sigma_{\rm t}}\) – \({\varepsilon_{\rm t}}\) curves, was quantitatively calculated. Approximate uniformity of plastic work was verified as no correlation between plastic work and stress state was observed. Moreover, the evolution behavior of microvoids inside a single grain and the equivalent average slip distance of polycrystalline ductile materials during trans-granular fracture process were also analyzed theoretically to explain the uniformity. The trans-granular slip and fracture behavior of Gr-4 titanium alloy specimen and orientations of crack propagation of extruded AZ61B magnesium alloy specimens were also examined experimentally.

Description: Strain and stress concentrations are studied for elastomers at finite deformations. Effects of strain-induced crystallization, filler reinforcement and deformation rate are also investigated, and micromechanical descriptions are provided for the observed results. A simple problem is subjected to finite element simulations to show the results evidently. Material parameters are obtained from experimental tests conducted on standard tensile samples of filled and unfilled natural rubber (NR) as well as styrene–butadiene rubber (SBR) as crystallizing and non-crystallizing rubbers, respectively. In all simulations, the strain concentration factor \(K_E\) is shown to decrease monotonically where the reduction is more apparent as the filler content increases. At enough large stretches, \(K_E\) is higher for filled NRs compared to the unfilled NR which is not the case for SBR. The stress concentration factor \(K_S\) rises sharply by deformation of the samples. At large stretches, in the case of SBR, filler reinforcement only shifts the maximum value of \(K_S\) to a lower level of strain, while in the case of NR, it reduces \(K_S\) significantly. It is concluded that \(K_S\) can rise from its theoretical value remarkably which should be noticed in design purposes particularly for crystallizing elastomers. Furthermore, the effect of deformation rate is investigated employing a visco-hyperelastic constitutive law along with an associated VUMAT in ABAQUS/Explicit. It is observed that, at high deformation rates, \(K_E\) decreases. Despite the reduction in strain concentration, \(K_S\) would be higher which is not desired in design of mechanical parts.

Description: In the present paper, the effect of parameterization on the results of isogeometric analysis of free-form approximated curved beams is investigated. An Euler–Bernoulli beam element for an initially curved beam with variable curvature is developed. The model is applied to four different examples. The effect of three parameterization strategies (the equally spaced method, the chord length method and the centripetal method) in the curve approximation process is considered. Also, the effect of least square approximation error is taken into consideration. The results strongly suggest avoiding the equally spaced method. Among the chord length and centripetal methods, the method which leads to a less least square error is recommended.

Description: The thermally induced vibration of a simply supported single-walled carbon nanotube (SWCNT) subject to thermal stress is investigated by using the models of planar and non-planar nonlinear beams with initial stress, respectively. The dynamic equations of nonlinear stochastic vibration of the SWCNT are established, with the geometric nonlinearity of the large deformation taken into account. The thermal vibration of SWCNT is predicted by numerically integrating both the dynamic equations of the nonlinear beam models via the Runge–Kutta algorithm of fourth order. The root-mean-square (RMS) amplitude and the stationary probability density of the thermal vibration of the SWCNT are obtained via the planar and non-planar nonlinear beam models with simply supported boundary conditions for both pre-buckling case and post-buckling case. The RMS amplitude of the thermal vibration of the SWCNT is given by using the numerical integration of probability density function. The RMS amplitude of thermal vibration of a SWCNT predicted via the non-planar nonlinear model with thermal stress is lower than that predicted via the planar nonlinear beam model, but higher than that predicted via the planar linear model.

Description: The dynamic analysis of two-dimensionalmultilayered anisotropic soilwith rigid bedrock is studied. An efficient numerical approach named the modified scaled boundary finite element method (SBFEM) is proposed in the time domain. Based on introducing the continued fraction method and auxiliary variables, the time domain solution is obtained. This solution can be applied to the transversely isotropic medium without any difficulty. For the modified SBFEM, the original scaling center is replaced by a scaling line. These characteristics enable the modified SBFEM to model the horizontal layered medium. Three significant technologies have been introduced in the formula derivation and solving process. First, the dual system is used to derive the displacement equation of the modified SBFEM, which is built on a Hamilton system. According to the principle of virtual work, the displacement equation is transformed to the dynamic stiffness equation. Second, the new continued fraction method for the unbounded domain resting on rigid bedrock is proposed. By introducing auxiliary variables, the displacement equation of motion of an unbounded domain is built. Third, it is an extremely important point that the accurate precise time-integration method is first employed to solve the global equation of motion of the modified SBFEM. This numerical integral method can achieve the machine precision. By using this method in solving the equation of motion of the modified SBFEM, an extremely accurate solution can be achieved. Finally, numerical examples validate the accuracy of the new proposed method, especially for the complex inclined model with anisotropic soil.

Description: Considering initial axial loads, dynamics and stability of an inner functionally graded cylindrical shell conveying swirling fluid (i.e., water) in the annulus between the flexible inner shell and the identical rigid outer shell are investigated by the traveling wave approach. Shell motions are described by Donnell’s thin shell equations. The fluid forces associated with shell motions are treated in the frame of the potential flow theory. The theoretical analysis is conducted by the zero-level contour method. The critical velocities of losing stability are determined. The influences of angular flow on the critical axial velocity and axial flow on the critical annular velocity are studied. Moreover, effects of the magnitude and the direction of initial axial loads on the critical velocities are fully discussed.

Description: Heterogeneity and anisotropy are two important features of biological materials, such as bone in animals, trunk of wood, and culm of bamboo. Both material heterogeneity and anisotropy have their particular biological functions and are formed through a complex and systematic biochemical process. In this paper, we intend to use a simple mechanical theory—the elastic growth theory—to predict these two features that are widely observed in bamboo. The theory assumes that the deformation gradient tensor is composed of two parts: one related to volumetric growth and the other corresponding to deformation. The analysis is carried out for a two-layered hollow cylinder model, which captures the main geometric characteristic of the cross-section of bamboo culm. Though the problem is nonlinear, we are able to derive exact and explicit expressions for the stresses and displacements in both layers. It is shown that, due to the volumetric growth, a residual stress field and elastic deformation are induced in the structure, which in turn leads to an equivalent and macroscopic material gradient and anisotropy. Numerical examples are considered to confirm the validity of the theoretical model and to perform a parametric study.

Description: We consider in this study the frictional sliding contact problem between a functionally graded magneto-electro-elastic material and a perfectly conducting rigid punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the unknowns are the normal contact stress, the electric displacement and the magnetic induction. These integral equations are then solved numerically to obtain the distributions of the normal contact stress, electric displacement and magnetic induction at the surface of the graded medium. The main objective of this paper is to study the effect of the non-homogeneity parameter, the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat and circular punch profiles.

Description: Phase field simulations are conducted to investigate the frequency-dependent behavior of a ferroelectric single crystal with and without dislocation arrays. For the dislocation-free ferroelectric, both coercive field and remnant polarization increase with the increase in the applied electric field frequency. For the ferroelectric with dislocation arrays, however, the variation of remnant polarization with frequency depends on the amplitude of the applied alternating electric field. When the applied electric amplitude ranges from 0.3 to 0.7, called the low field, the remnant polarization increases first and then decreases. On the other hand, if the applied field amplitude is higher than 0.8, the remnant polarization does not change too much at low frequency (3.13 × 10 −4 –1.56 × 10 −2 ), while it decreases sharply at high frequency (1.56 × 10 −2 –0.156). For various applied electric amplitudes ranging from 0.3 to 1.5, the overall trend of the coercive field is to increase in the low-frequency range, while it varies in the high-frequency range. The frequency-dependent properties are attributed to the generalized pinning and depinning of the dislocation arrays to polarizations, which is endorsed by the corresponding domain structures.

Description: We propose a method to characterize an undeformed double-layer strand with residual stress by considering the deviation from the self-equilibrium configurationwhere torque is balanced without any external forces.We first derive the residual twist defined as the deviation of internal twisting moment in the strand from the self-equilibrium configuration. The residual twist gives a measure of the effective torsional rigidity of the strand, determined by the geometrical, material, and mechanical parameters from the microstructural viewpoint. Then, we correlate the residual twist with the fatigue life of (3+8)-glass strands embedded in a rubber matrix under cyclic bending. We show that the residual twist correlates with the number of cycles to failure for specific (3+8)-glass strands having the identical twist direction.

Description: An industrially applicable nozzle is the subject of this study. The nozzle is an auxiliary equipment of a pneumatic pulsator system for unclogging outlets of silos which store loose materials. The aim is to determine the amount of heat which is generated during one work cycle of the system. Investigation in this field has not been carried out so far, and the present-day designing process is significantly based on heuristic knowledge. The heat is calculated by using results of a numerical simulation. The Finite Volume Method has been used with a thermodynamically ideal gas model. The airflow is assumed to be transient, compressible and supersonic, and it is driven by a time-varying pressure difference. There is an estimation of discretization error of the numerical results carried out in order to confirm the reliability of the solution. The error estimation shows that the results lie in the vicinity of the exact solution of the governing equations. Instantaneous results of the simulation indicate a locally flow which intensifies flow parameters in a similar way as the converging-diverging nozzles do. The value of the total heat generated during gas conversion within the nozzle is negative; thus, the nozzle could be cooled during its functioning.

Description: The boundary effect on the asymmetrical motion of a porous spherical particle in an eccentric spherical cavity is investigated in the quasi-steady limit under creeping flow conditions. The porous particle translates and rotates in the viscous fluid, located within the spherical cavity, normal to the line connecting their centers. The fluid inside the porous particle is governed by the Brinkman equation. A tangential stress jump condition at the interface between the fluid and the porous particle is applied. A semi-analytical approach based on a collocation technique is used. Due to the linearity of the present problem, the flow variables for the clear fluid region are constructed by superposing basic solutions of two problems: the first one is the regular solution inside the cavity region in the absence of the porous particle where a first system of coordinates has its origin at the center of the cavity, while the second problem is the regular solution in the infinite region outside the spherical porous particle where a second coordinate system with its origin at the center of the porous particle is used. Numerical results displaying the resistance coefficients acting on the particle are obtained with good convergence for various values of the physical parameters of the problem. The results are tabulated and represented graphically. The findings demonstrate that the collocation results of the resistance coefficients are in good agreement with the corresponding results for the impermeable solid particle.

Description: In the present investigation, the buckling of generally laminated conical shells with various boundary conditions subjected to axial pressure is studied using an analytical approach. The governing equations are obtained using classical shell theory with Donnell assumptions in strain–deformation relations and the principle of minimum potential energy. The differential equations are solved using trigonometric functions in circumferential and power series in longitudinal directions. All types of boundary conditions can be applied in this method. The results are compared and validated with the results available in the literature, and good agreement is observed. Finally, the effects of the length, semi-vertex angle, and lamination sequences on the buckling load and mode shapes of generally laminated conical shells are presented.

Description: Rotating elements supported on journal bearings are widely encountered structures in engineering practice. Most commonly, these are asymmetrically manufactured and loaded rigid rotors transmitting torque and carrying transverse as well as axial forces. Nowadays, despite high operational demands and high rotational velocities, such systems are still expected to exhibit stable working, even in the presence of small assembly deviations, light unbalance or external disturbances. The surrounding environment of a rotating machine may interact with it by kinematic excitation from vibrating foundation. This, in turn, may lead to hazardous response and the onset of irregular and chaotic motion of the rotor. The subject of the study is to find and analyze regions of the occurrence of such vibrations in the system of a rigid rotor supported in journal bearings. The bearings themselves are assumed to be non-perfectly mounted in the housing, i.e., their sleeves are inserted in rings possessing some viscoelastic properties. These properties are treated as variable parameters, and the aim is to move the regions of irregular and chaotic vibration outside the operational regime (angular velocity). The adjustability of the viscoelastic parameters may be realized by incorporation of smart materials such as piezoelectric or magnetorheological ones. The considered system is an asymmetric rigid rotor supported on two journal bearings subject to a steady kinematic excitation. The system is described by eight coupled nonlinear ordinary differential equations of motion. Results of the examinations prove that by selecting an appropriate magnitude of damping and stiffness of the bearing mountings, it is possible to enlarge the region of stable operation of the rotating system and thus secure its safety. This, however, does not mean the elimination of chaotic response at all, but only a shift of it outside the range of operational rotation speed.

Description: The hydrogen-containing solid is assumed to consist of an elastic matrix with voids filled by hydrogen. This model is used for modeling the process of diffusion and trapping the hydrogen in an elastic rod subjected to high-frequency excitation. The differential equation for the trapped hydrogen concentration is obtained in the one-dimensional case. The method of direct separation of vibrational processes allows one to introduce the “fast” and “slow” components in the process of hydrogen redistribution in the material. The governing equation for the evolution of the trapped hydrogen concentration in the vibrating elastic rod is derived, and it reflects the considerable impact of high-frequency vibration on the evolution of trapped hydrogen. The resulting equation is shown to differ significantly from the original equation for the hydrogen concentration.

Description: In this study, micro-machining of f.c.c. single-crystal materials was investigated based on a hybrid modelling approach combining smoothed particle hydrodynamics and continuum finite element analysis. The numerical modelling was implemented in the commercial software ABAQUS/Explicit by employing a user-defined subroutine VUMAT for a crystal plasticity formulation to gain insight into the underlying mechanisms that drive a plastic response of materials in high deformation processes. The numerical studies demonstrate that cutting force variations in different cutting directions are similar for different f.c.c. crystals even though the magnitudes of the cutting forces are different.

Description: This paper shows an approach to computing the effective properties of multi-field composite materials and their first-order sensitivities. The approach is based on the application of the complex variable step method for the micromechanical Mori–Tanaka scheme; hence, the first-order sensitivities can be computed in the same analysis. Numerical results are presented for magnetoelectroelastic properties of piezoelectric–piezomagnetic composite materials. A comparison of the results to those obtained by other methods shows that the presented sensitivity analysis gives highly accurate and stable results, but the values of the results are dependent on the applied micromechanical model. The presented approach may be used to solve ill-posed problems of optimal design or identification in coupled fields micromechanics.

Description: The present investigation enquires the role of the backup plate mechanical properties in the brittle failure of a ceramic tile. It provides a full-field solution for the elastostatic problem of an infinite Kirchhoff plate containing a semi-infinite rectilinear crack (the tile) resting on a two-parameter elastic foundation (the backup plate) and subjected to general transverse loading condition. The backup plate is modeled as a weakly non-local (Pasternak type) foundation, which reduces to the familiar local (Winkler) model once the Pasternak modulus is set to zero. The same governing equations are obtained for a curved plate (shell) subjected to in-plane equi-biaxial loading. Fourier transforms and the Wiener–Hopf technique are employed. The solution is obtained for the case when the Pasternak modulus is greater than the Winkler modulus. Superposition and a two-step procedure are employed: First, an infinite uncracked plate subjected to general loading is considered; then, the bending moment and shearing force distribution acting along the crack line are adopted as the (continuous) loading condition to be fed in the solution for the cracked plate. Results are obtained as a function of the ratio of the Pasternak over the Winkler foundation stiffness times the tile flexural rigidity. It is established that the elastic foundation significantly affects the mechanical behavior of the elastic plate. In particular, the Winkler model substantially underestimates the stress state near the crack tip. Stress-intensity factors are determined, and they are employed as a guideline for increasing the composite toughness. The analytical solution presented in this paper may serve as a benchmark for a more refined numerical analysis.

Description: This work focuses on investigation of structural (phase) transformations in crystal lattices from continuum and discrete points of view. Namely, the continuum, which is equivalent to a simple lattice in the sense of the Cauchy–Born energy, is constructed using long-wave approximation, and its strong ellipticity domains in finite strain space are obtained. It is shown that various domains correspond to variants of triangular and square lattices, and the number of the domains depends on the interaction potential parameters. Non-convex energy profiles and stress–strain diagrams, which are typical for materials allowing twinning and phase transformations, are obtained on the straining paths which connect the domains and cross non-ellipticity zones. The procedures of the lattice stability examinations and estimation of energy relaxation by means of molecular dynamical (MD) simulation are developed, and experimental construction of the envelope of the energy profiles, corresponding to the energy minimizer, is done on several straining paths. The MD experiment also allows to observe the energy minimizing microstructures, such as twins and two-phase structures.

Description: The evaluation of crack initiation and growth at microscopic scale is a crucial issue for the safety assessment of macroscopical fractures. In the present paper, the crack propagation in 17Mn1Si steel macroscale is investigated by taking into account the microstructural damage accumulation in polycrystalline solids. The revealed regularities are in good agreement with the concept of physical mesomechanics, which allows obtaining a generalized view of the material deformation and the failure process in the vicinity of the concentrator, which fulfills the limiting, initial and physical conditions and allows obtaining the generalized regularities in deformation and failure of 17Mn1Si steel.

Description: The boundary element method has been applied with success to linear elastic fracture mechanic problems, involving static and dynamic cases. In order to solve body force problems (e.g., gravitational forces and transient problems with velocities and accelerations), Nardini and Brebbia presented, in 1982, the dual reciprocity formulation. Originally with the intention of solving transient problems using fundamental solutions of the static formulation, the procedure was found to be very efficient in the solution of body force problems as well. Also, a Green’s function corresponding to an embedded crack within the infinite medium can be introduced into the boundary element formulation as the fundamental solution. This yields accurate means of calculating only the external boundary unknown displacements and tractions and, in a post-processing scheme, determining the crack opening displacements. This paper introduces an approach that involves the numerical Green’s function procedure, of Telles and coworkers, and the dual reciprocity formulation. It compares beam solutions with the simulated effect of the total weight applied as a concentrated boundary force, the actual self-weight as a body force and a frequency- and time-dependent transient Heaviside load applied to a plate with a central crack.

Description: In incompressible isotropic elasticity, the Valanis and Landel strain energy function has certain attractive features from both the mathematical and physical view points. This separable form of strain energy has been widely and successfully used in predicting isotropic elastic deformations. We prove that the Valanis–Landel hypothesis is part of a general form of the isotropic strain energy function. The Valanis–Landel form is extended to take anisotropy into account and used to construct constitutive equations for anisotropic problems including stress-softening Mullins materials. The anisotropic separable forms are expressed in terms of spectral invariants that have clear physical meanings. The elegance and attractive features of the extended form are demonstrated, and its simplicity in analysing anisotropic and stress-softening materials is expressed. The extended anisotropic separable form is able to predict, and compares well with, numerous experimental data available in the literature for different types of materials, such as soft tissues, magneto-sensitive materials and (stress-softening) Mullins materials. The simplicity in handling some constitutive inequalities is demonstrated. The work here sets an alternative direction in formulating anisotropic solids in the sense that it does not explicitly use the standard classical invariants (or their variants) in the governing equations.

Description: This paper represents a modified formulation of the wave finite element (WFE) method for propagating analysis of thermoelastic waves in a hollow cylinder without energy dissipation. The 2D-high-order spectral element with the Gauss–Legendre–Lobatto integration is applied into the WFE method, which produces the diagonal mass matrix. Based on the assumption of harmonic displacement fields by Fourier series expansion, the general discretization wave equation is simplified from the 3D problem to 2D. Dispersion properties of elastic wave propagation in the hollow cylinder are computed considering the choice of the spectral element orders, and the results indicate the high efficiency and high accuracy of the modified formulation compared with that of the software Disperse . Then, using the modified formulation, the thermoelastic dynamic equation of the cylinder is derived from the generalized thermoelasticity theory. The propagation of the thermoelastic wave (including two kinds of wave modes) in the cylinder without energy dissipation is discussed in different cases. Finally, wave structures along the radial direction of thermoelastic wave modes are shown at the nondimensional frequency 1.25, which can be used for the recognition of different modes.

Description: Steady solutions of an inverse problem relevant to gravity-driven film flows over an undulated slippery bottom are considered. Given a target free surface shape, the goal is to obtain the corresponding bottom topography of a slippery substrate which causes the specified free surface shape for a film flowing over it. The approaches followed by Sellier (Phys Fluids 20:062106, 2008 ) for creeping films and by Heining and Aksel (Phys Fluids 21:083605, 2009 ) for inertial films for reconstructing a rigid bottom topography for a target free surface profile are extended to the reconstruction of a slippery bottom topography. The model equations for film thickness above the bottom topography are derived for creeping flows under lubrication approximation and for inertial films using the weighted-residual integral boundary layer method and are solved numerically. The influence of inertia, slip parameter and surface tension on the shape of the reconstructed bottom topography is analyzed for different prescribed free surface shapes (sinusoidal, trench and bell-shaped). It is observed that the nonlinearities that appear in the reconstructed rigid bottom substrate with no slip at the substrate are suppressed by seeking the bottom substrate to be reconstructed as a slippery substrate. A spatial linear stability analysis of the corresponding direct problem is examined using Floquet theory, and the results reveal that the slip parameter and surface tension have a high influence on the critical Reynolds number. The results provide a strategy for controlling surface defects in a gravity-driven film over a substrate; namely, in order to achieve a target free surface profile, one can design the bottom substrate to be an undulated rough/textured/grooved or a superhydrophobic surface which can be modelled as an undulated smooth substrate with velocity slip at the substrate.