ALBERT

Notizen: Abstract The fatigue process, viewed as a sequence of slow growth periods, is described by a non-linear differential equation of the first order which includes the “creep component” of crack growth in a visco-elastic solid. It is shown that fracture in a rate sensitive medium may extend even under sustained or decreasing loads. The total rate of growth consists therefore of two terms where the first term is a familiar power law, valid for a high-cycle range and limited plasticity effects, while the second one accounts for the time-dependent contribution. Here f denotes frequency, Y is the yield stress, Kcdenotes fracture toughness, ΔK stands for the stress intensity range while the crack closure correction α, the rate sensitivity C and the constant R * are defined in the text.

Notizen: Abstract Conditions under which fast fracture may occur in ductile solids are described in terms of a new quantity, named stability index, λ. This index, defined as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaabs% gacaWGsbGaai4laiaadsgacaWGHbGaeyOeI0IaeyOaIyRaamOuamaa% BaaaleaacaWGbbaabeaakiaac+cacqGHciITcaWGHbGaaiykaiaacI% cacqGHciITcaWGsbWaaSbaaSqaaiaadgeaaeqaaOGaai4laiabgkGi% 2kaadggacaGGPaWaaWbaaSqabeaacqGHsislcaaIXaaaaaaa!4C11!\[({\text{d}}R/da - \partial R_A /\partial a)(\partial R_A /\partial a)^{ - 1}\], encompasses the tearing modulus of Paris % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadw% eacaGGVaGaeq4Wdm3aa0baaSqaaiaaicdaaeaacaaIYaaaaOGaaiyk% aiaacIcacaqGKbGaamOsaiaac+cacaqGKbGaamyyaiaacMcadaWgaa% WcbaGaamyAaaqabaaaaa!42DC!\[(E/\sigma _0^2 )({\text{d}}J/{\text{d}}a)_i\]determined at the point of incipient crack extension, and other geometry dependent quantities. Stability index serves as a measure of the “distance” of any given state of the system considered (specimen containing an extending crack) from the critical state. Instability occurs at λ=0. This work suggests a means of computation of the quantities % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeizaiaabk% facaqGVaGaaeizaiaadggaaaa!3A28!\[{\text{dR/d}}a\]and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRaam% OuamaaBaaaleaacaWGbbaabeaakiaac+cacqGHciITcaWGHbaaaa!3C25!\[\partial R_A /\partial a\]necessary for evaluation of the stability index λ when a geometrical configuration of the test-piece is specified and under an assumption that the DBCS model employed here and extended to a moving crack situation, is adequate. Symbols R and R Adenote the extent of the plastic zone developed during a quasi-static crack growth and considered either as a measure of material toughness (R) or as an intensity of the applied field (R A). Symbol “a” denotes the current crack half-length. There are no restrictions imposed on the size of the plastic zone except that it should be much greater than the process zone size, Δ. Therefore, the solutions suggested here cover both the limiting cases, i.e., (a) the contained yielding case, when % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaac+% cacaWGHbaaaa!385D!\[R/a\]≪1, and (b) large scale (or unlimited) yielding case, when % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaac+% cacaWGHbaaaa!385D!\[R/a\]≫1. Interestingly, both these asymptotic cases have the closed-form solutions, as discussed in the text. For the intermediate range of the ratio % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaac+% cacaWGHbaaaa!385D!\[R/a\]the governing non-linear differential equation of the first order (valid for a center-cracked infinite width plate):% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% qGKbGaamOuaaqaaiaabsgacaWGHbaaaiabg2da9maalaaabaGaamyy% aiabgUcaRiaadkfaaeaacaWGHbaaamaacmaabaWaaSaaaeaaieGaca% WFufaabaGaeuiLdqeaaiabgkHiTKqzacWaaSaaaOqaaKqzacGaaGym% aaGcbaqcLbiacaaIYaaaaOGaaeiBaiaab+gacaqGNbWaamWaaeaada% WcaaqaaiaaikdacaWGLbGaamOuaiaacIcacaaIYaGaamyyaiabgUca% RiaadkfacaGGPaaabaGaamyyaiabfs5aebaaaiaawUfacaGLDbaaai% aawUhacaGL9baacqGHRaWkdaWcaaqaaiaadkfaaeaacaWGHbaaaiaa% cYcacaqGGaGaamOuaiabg2da9iaabccacaWGsbGaaiikaiaadggaca% GGPaaaaa!5FE9!\[\frac{{{\text{d}}R}}{{{\text{d}}a}} = \frac{{a + R}}{a}\left\{ {\frac{}{\Delta } - \frac{1}{2}{\text{log}}\left[ {\frac{{2eR(2a + R)}}{{a\Delta }}} \right]} \right\} + \frac{R}{a},{\text{ }}R = {\text{ }}R(a)\]can be integrated numerically for a given set of the initial data R 1 and a 0, thus generating an R-curve valid for any % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaac+% cacaWGHbaaaa!385D!\[R/a\]ratio which may be observed experimentally. It is noteworthy that the initial slope of an R-curve, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeikaiaabs% gacaWGsbGaai4laiaabsgacaWGHbGaaiykamaaBaaaleaacaWGPbaa% beaaaaa!3C9D!\[{\text{(d}}R/{\text{d}}a)_i\], is a material constant for the case of small scale yielding (R≪a), while for the other extreme case (R≫a) such slope becomes geometry dependent. Available data indicate that such dependence is not very strong. The closed form solutions for the two asymptotic cases are% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgk% HiTiaadggadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIYaGaamOu% amaaBaaaleaacaWGZbGaam4CaaqabaGcdaWadaqaaiaadweadaWgaa% WcbaGaaGymaaqabaGcdaqadaqaaiaabYgacaqGVbGaae4zamaalaaa% baGaamOuamaaBaaaleaacaWGZbGaam4CaaqabaaakeaacaWGsbaaaa% GaayjkaiaawMcaaiabgkHiTiaadweadaWgaaWcbaGaaGymaaqabaGc% daqadaqaaiaabYgacaqGVbGaae4zamaalaaabaGaamOuamaaBaaale% aacaWGZbGaam4CaaqabaaakeaacaWGsbWaaSbaaSqaaiaadMgaaeqa% aaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaiaacYcacaqGGaGaam% OuaiablQMi9iaabccacaWGHbaaaa!5B4B!\[a - a_0 = 2R_{ss} \left[ {E_1 \left( {{\text{log}}\frac{{R_{ss} }}{R}} \right) - E_1 \left( {{\text{log}}\frac{{R_{ss} }}{{R_i }}} \right)} \right],{\text{ }}R \ll {\text{ }}a\]% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabgk% HiTiaadkfadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamyyaiaac+ca% caWGHbWaaSbaaSqaaiaaicdaaeqaaOGaaiykam

Notizen: Abstract The model developed here links microstructural and continuum mechanics aspects of the early stages of the fracture process occurring in dissipative solids. A variable size damage zone, endowed with a structure of its own determined by the microstructural parameters related to the material ductility and grain size, is incorporated into a moving dominant crack. Prior to and during the course of crack extension, the energy dissipation mechanisms of diverse nature are activated within the volume of the localized damage zone, providing a substantial contribution to the effective material toughness. A criterion for quasi-static crack based on self-similarity of the crack-tip region is used to set up a governing equation of motion. The stress-transferring ability of the damage zone depends on the separation distance created between the two opposite boundaries of the fractures zone, and it determines the history of a quasi-static crack development including the attainment of the terminal instability point. Although detailed information regarding the distribution of stress prevailing within the nonlinear zone is lacking, it is shown that certain plausible governing equations may be constructed and employed to define material resistance curve in sufficiently large specimens (within the so-called ssy range), which is then used to predict the onset of catastrophic fracture. Six different specimen configurations are considered and the pertinent macro-mechanical stability analysis is presented in detail. The class of materials susceptible to this type of analysis is not restricted by the usual LEFM or EPFM constraint.

Notizen: Abstract A modified line-plasticity model, involving concept of structured nonlinear zone coupled with the final stretch criterion governing the crack propagation, is used to show the effects of material strain-hardening and the redistribution of strain caused by an advancing quasi-static crack, on the essential parameters pertinent to a mathematical description of elasto-plastic fracture process including its ductile limit. The model links micro-structural and continuum aspects of ductile fracture occurring in dissipative solids equipped with an ability to strain-harden. Attention has been focused on the crack tip opening displacement ( δ_t ) and the J-integral, both associated with either stationary of quasi-static crack contained in a power-hardening material of the Ramberg–Osgood type. The ratios of δ_t and J for a stationary and moving crack are represented via closed-form solutions and then compared against the earlier numerical results of Shih (1981), based on the finite element analyses. Expressions derived here, apart from having a theoretical merit, address an issue of significant interest to the researchers involved in the field of the Experimental Fracture Mechanics.

Notizen: Abstract A local energy criterion of Irwin's type is applied to a problem of a quasi-static extension of a crack embedded in an elastic-plastic or viscoelastic-plastic matrix. The derived nonlinear differential equation ℳ(σ, l, dσ/dl) + G(σ, l) = G c /Ψ(Δ/l) governs subcritical growth up to the point of gross instability. The creep rupture under sustained loads and fatigue crack propagation both described by the above equation, are shown to follow almost identical mathematical representations. The essential features such as the resistance curves, the dependence of growth rate on the current K-factor and the amount of growth vs. time relations are shown to be strikingly similar.

Notizen: Abstract The governing equation of motion, which describes the early stages of fracture in a rate-sensitive solid, has been integrated through the use of the analog computer type EAI 380. Three regimes of the tensile pulsating loading, for which there are no closed form solutions, are considered: (i) sinusoidal load, (ii) trapezoidal load, (iii) randomly oscillating load. Signals of these shapes superposed on a constant tensile stress produced three different patterns of crack growth and were photographed directly from the screen of a dual-beam oscilloscope. It appears that the rate of fatigue crack spreading in a time-dependent matrix is enhanced by the pulsating component of load in the degree coinciding with the sequence of the above listing.

Notizen: Abstract Matching the near tip stress and strain fields, resulting from the continuum mechanics models of the crack with the corresponding fields postulated within the small process region adjacent to the leading edge of the crack, allows one to gain a better understanding of the interrelations between the microstructure and the macroscopic response of a material undergoing extensive deformation and fracture process. In this work, attention is focused on the influence of the strain-hardening phenomenon of the Ramberg-Osgood type on the material fracture response. In particular, through detailed analysis of the relations pertinent to the onset of stable cracking process and to the attainment of the steady-state limit which corresponds to a fully developed energy absorption mechanism that a material is capable to provide, it has been found that even minimal amount of strain-hardening has a significant effect on the predicted value of the JM ss / Jc ratio. This ratio defines the margin of safety when dealing with fracture associated with a nonlinear elasto-plastic deformation process. The end results of this study can be applied to small scale yielding range and moderate amount of strain-hardening. Plane stress is usually implied, but for nonmetallic materials the division between plane stress and plane strain conditions is blurred (as exemplified by the localized craze which usually precedes fracture in polymer specimens regardless their width).

Notizen: Abstract Nonlinear differential equation governing mode I fracture under small scale yielding condition has been derived on the basis of the energy partition concept. This technique is associated with a cohesive crack model. The nonlinear zone which precedes a propagating crack has been assumed to have a structured nature. In addition to this microstructural assumption, it has been postulated that the energy dissipated within the process zone (Δ), embedded in a larger nonlinear zone (R), remains invariant to the extent of crack growth. Upper and lower bounds of the tearing modulus have been related to the material ductility via closed form expressions. It has been demonstrated that the energy screening, measured by the ratio of the true fracture energy (Ŵ) to the total work expended in the cohesive zone during the process of irreversible deformation, is a monotonic function of the crack growth increment, resembling a reciprocal of the apparent material resistance to cracking described by an R-curve.

Notizen: Abstract In has been shown that slow stable cracking process can be sustained by strain-softening materials such as cementitious composites, e.g., rock, concrete, mortars, ceramics and others. A mathematical model is proposed based on the theory of quasi-static crack extension governed by Wnuk's criterion of final stretch (equivalent to the CTOA condition). Requirement of self-similarity of the crack opening profile, maintained during the quasistatic growth phase, leads to a differential equation defining the material resistance to sustained crack extension, as a function of crack growth increment, material properties such as tearing modulus, strain-softening parameter, and the size of the process zone. The equations derived on the basis of the step-like distribution of the restraining stress which operates within a structured end zone associated with a moving crack, are compared against the Wnuk-Rice-Sorensen equation to ran R-curve in an ideal elasto-plastic material, and with the more recent results of Wnuk and Hunsacharoonroj [1] pertaining to strain-hardening materials which obey the Ramberg-Osgood power law. Present results suggest that the nature of constitutive equations substantially influence composite fracture resistance as measured by the R-curve. Ability to absorb energy and the ensuing resistance to crack growth can be significantly enhanced when the mechanisms of energy dissipation within the matrix are properly understood. This work establishes a “bridge” between the continuum mechanics and micromechanics of deformation and meture processes by providing a relation between material fracture toughness and its constitutive equations supplemented by certain microstructural characteristics.