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.
Keywords
Boundary Condition Differential Equation Mechanical Engineer Partial Differential Equation Fluid DynamicsList of Symbols
- 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
- l2
Euclidean norm
- TG
Gauss-Seidel iteration matrix
- TJ
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(TJ)
spectral radius ofT J
- σx - σy
normal stresses
- σ1 - σ2
principal stresses
- τxy
shear stress
- ϕ
σ x + σ y , first stress invariant
- ▽2
Laplacian operator
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References
- 1.Segerlind, L.J., “Stress-difference Elasticity and Its Application to Photomechanics,”Experimental Mechanics,11,440–445 (1971).CrossRefGoogle Scholar
- 2.Chandrashekhara, K. andJacob, K.A., “An Experimental-Numerical Hybrid Technique for Two-Dimensional Stress,”Strain,13 (4),25–31 (1977).Google Scholar
- 3.Dally, J.W. andRiley, W.F., Experimental Stress Analysis, McGraw-Hill Book Co., New York (1978).Google Scholar
- 4.James, M.L., Smith, G.M. andWolford, J.C., Applied Numerical Methods for Digital Computation, Harper and Row Publishers, New York (1977).Google Scholar
- 5.Volterra, A. andGaines, J.H., Advanced Strength of Materials, Prentice-Hall, Inc., Englewood Cliffs, NJ (1971).Google Scholar
- 6.Ortega, J.M. Numerical Analysis — A Second Course, Academic Press, New York (1972).Google Scholar
- 7.Burden, R.L. andFaires, J.D., Numerical Analysis, 3rd Ed., Prindle, Weber and Schmidt, Boston (1985).Google Scholar
- 8.Garlach, H.D., “Electronic Aids for the Automation of Photoelastic Measurements,”PhD Thesis, Univ. of Karlsruhe, FRG (1968).Google Scholar
- 9.Muller, R.K. andSaackel, L.R., “Complete Automatic Analysis of Photoelastic Fringes,”Experimental Mechanics,19,245–251 (1979).Google Scholar
- 10.Seguchi, Y., Tomita, Y. andWatanabe, M., “Computer-aided Fringe-pattern Analyzer—A Case of Photoelastic Fringe,”Experimental Mechanics,19,362–370 (1979).CrossRefGoogle Scholar