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Hybrid stress analysis by digitized photoelastic data and numerical methods

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Abstract

A method is presented that determines photoelastic isochromatic values at the nodal points of a grid mesh which in turn is generated by a computer program that accepts digitized input. Values of σ1 - σ2 are computed from the digitized fringe orders. The Laplace equation is solved to separate the principal stresses at each nodal point. The method is extended to digitize isoclinics. Subsequently, σ x - σ y and τ xy are calculated to be used as starting values for the solution of the pertaining partial differential equations to enhance convergence. For further accelerating the rate of convergence, superfluous boundary conditions are added from the digitized data; significant improvement is demonstrated. Estimated values of σ x - σ y from the digitized data are further used in conjunction with the solution of the Laplace equation to determine the state of stress without solving the boundary value problems.

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Abbreviations

A :

coefficient matrix

b :

column vector of known functions

C :

weighting functions

D :

strictly diagonal matrix

D :

σ x - σ y , difference of two normal stresses

k :

number of iterations

L :

strictly lower triangular matrix

l 2 :

Euclidean norm

T G :

Gauss-Seidel iteration matrix

T J :

Jacobi iteration matrix

U :

strictly upper triangular matrix

x :

column vector of unknown function

α:

fractional part for horizontal unequal grid spacing

β:

fractional part for vertical unequal grid spacing

ϱ(T J ):

spectral radius ofT J

σ x - σ y :

normal stresses

σ1 - σ2 :

principal stresses

τ xy :

shear stress

ϕ:

σ x + σ y , first stress invariant

▽2 :

Laplacian operator

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Tuskegee University, 36088, Tuskegee, AL

    H. Mahfuz (Assistant Professor)

  2. Department of Mechanical Engineering, Florida Atlantic University, 33431, Boca Raton, FL

    R. O. Case (Professor) & T. L. Wong (Assistant Professor)

Authors
  1. H. Mahfuz
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  2. R. O. Case
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  3. T. L. Wong
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Mahfuz, H., Case, R.O. & Wong, T.L. Hybrid stress analysis by digitized photoelastic data and numerical methods. Experimental Mechanics 30, 190–194 (1990). https://doi.org/10.1007/BF02410247

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  • DOI: https://doi.org/10.1007/BF02410247

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