Abstract
The maximum stresses in a plane-stressed component typically occur on the boundary. However, it is generally difficult to obtain reliable experimental data at an edge and thermoelastic stress analysis is no exception. The inability to measure reliable edge isopachic stresses has caused many previous thermoelastic stress analyses to be more qualitative than quantitative. This paper develops and implements an effective iterative least-squares method for calculating reliable edge isopachic stresses from measured interior values. The method is based upon the plane-stress isotropic compatibility equation. A regularization scheme is employed to minimize the sensitivity to measurement error and to improve the stability of the algorithm by controlling the rate of convergence. An illustrative example with actual measured thermoelastic data is included. The processes thermoelastically determined results compare well with those obtained using strain gages.
Similar content being viewed by others
Abbreviations
- l, m :
-
direction cosines on Γ
- n :
-
number of nodes per element
- nel :
-
number of elements
- N i :
-
shape functions
- S :
-
isopachic stress,S = σx + σy = σ1 + σ2
- \(\bar S\) :
-
approximating field for isopachics
- [A], [B]:
-
coefficient matrices composed of pseudo-stiffness terms
- {F}:
-
load vector resulting from imposing boundary conditions
- [k]:
-
element pseudo-stiffness matrix
- [K]:
-
global pseudo-stiffness matrix
- [K′]:
-
global pseudo-stiffness matrix with applied boundary conditions
- [N]:
-
shape function row vector
- [R]:
-
regularization matrix
- {S}:
-
global vector of nodal isopachic values
- {S e}:
-
vector of element nodal isopachics
- {S i}:
-
vector of interior nodal isopachic values
- {Ŝ i}:
-
vector of experimentally measured interior isopachic values
- {S u}, {S k}:
-
vector of unknown and known isopachic edge values, respectively
- [X]:
-
sensitivity matrix
- \(\varepsilon _1 ,\varepsilon _2 \) :
-
convergence criteria parameters
- σx, σy :
-
principal normal stresses
- σx, σy :
-
normal stresses in coordinate directions
- Γ:
-
boundary of a domain
- Ω:
-
interior of a domain
- \(\nabla ^2 \) :
-
Laplacian operator,\(\nabla ^2 = \partial ^2 0/\partial x^2 0/\partial y^2 \)
References
Rauch, B.J. and Rowlands, R.E., Thermoelastic Stress Analysis, SEM Handbook on Experimental Mechanics,2nd Ed., ed. A.S. Kobayashi, Chap. 14, 581–600 (1993).
Rauch, B.J., “Enhancement and Individual Stress Component Separation of Thermoelastically Measured Isopachic Data,” PhD Thesis, University of Wisconsin-Madison (1993).
Huang, Y.M. andRowlands, R.E., “Quantitative Stress Analysis Based on the Measured Trace of the Stress Tensor,”J. Strain Analysis,26,55–63 (1991).
Huang, Y.M., “Determination of Individual Stresses from Thermoelastically Measured Trace of Stress Tensor,” PhD Thesis, University of Wisconsin-Madison (1989).
Huang, Y.M., Rowlands, R.E., andLesniak, J.R., “Simultaneous Stress Separation, Smoothing of Measured Thermoelastic Information and Enhanced Boundary Data,”Experimental Mechanics,30 (4),398–304 (1990).
Feng, Z., Zhang, D., Rowlands, R.E., andSandor, B.I., “Thermoelastic Determination of Individual Stress Components in Loaded Composites,”Experimental Mechanics,32 (2),89–95 (1992).
Rauch, B.J., Lin, S.T. and Rowlands, R.E., “Quantitative Thermoelastic Stress Analysis,” Int. Symp. on Opt. Sci. and Eng. (SPIE), (1993).
Rauch, B.J. and Rowlands, R.E., “Recent Advances in Determining Individual Stresses From Thermoelastically Measured Isopachics,” Proc. 1993 SEM Spring Conf. on Exp. Mech. (1993).
Lesniak, J.R., “Stress Intensity Measurement Through Image Deconvolution,” Stress Photonics Thermoelastic Forum (1992).
Lin, S.T., “Quantitative Thermoelastic Stress Analysis of Orthotropic Composite Structures,” PhD Thesis, University of Wisconsin-Madison (1993).
Boyle, J.T. and Hamilton, R., “A Method of Thermographic Stress Separation,” 4th Conf. on Applied Solid Mechanics, ed. A.R.S. Porter, University of Leicester, Elsevier (1991).
Stanley, P., “Stress Separation from SPATE Data for a Rotationally Symmetrical Pressure Vessel,” Stress and Vibration: Recent Developments in Industrial Measurement and Analysis, ed. P. Stanley editor, SPIE 1084,72–83 (1987).
Rowlands, R.E., “Thermoelastic Stress Analysis: Smoothing of the Measured Information, Improved Edge Data and Stress Separation,” Proc. 1991 SEM Spring Conf. on Exp. Mech., 248–251 (1991).
Ryall, T.G., Cox, P.M. andEnke, N.F., “Determination of Dynamic and Static Stress Components from Experimental Thermoelastic Data,”Mech. of Mat.,14 47–58 (1992).
Ryall, T.G., Heller, M. andJones, R., “Determination of Stress Components From Thermoelastic Data Without Boundary Conditions,”J. Appl. Mech.,59,841–847 (1992).
Huiskamp, W.J.C. and Thompson, J.C., “Least-Squares Asymptotic Analysis of Finite Element Notch Stress Data,” Proc. 8th Canadian Cong. of Appl. Mech., 855–856 (1981).
Thompson, J.C. andNegus, K.J., “Least-Squares Asymptotic Analysis of Photoelastic Data from Stress Concentration Regions in Plane Problems,”Strain,19 179–184 (1983).
Thompson, J.C., “Analysis of Stress Concentration Data by Asymptotic Techniques,” Proc. 5th Int. Cong. on Exp. Methods, 758–764 (1984).
Thompson, J.C. andNegus, K.J., “Developments in a Least Squares Asymptotic Analysis of Isochromatic Data from Stress Concentration Regions in Plane Problems,”Strain,20,133–134 (1984).
Maniatty, A., Zabaras, N. andStetson, K., “A Finite Element Method Analysis of Some Inverse Elasticity Problems,”J. Eng. Mech.,115,1303–1317 (1989).
Das, S., “Numerical Solution of Inverse Problems in Mechanics Using the Boundary Element Method,” PhD Thesis, Iowa State Univ., (1991).
Cook, R.D., Malkus, D.S. andPlesha, M.E., Concepts and Application of Finite Element Analysis, 2nd Ed., John Wiley and Sons, New York (1989).
Harwood, N. and Cummings, W.M., Thermoelastic Stress Analysis, Adam Hilger (1991).
McKelvie, J., “Some Practical Limits to the Applicability of the Thermoelastic Effect,”Proc. VIII Int. Conf. on Exp. Stress Analysis, Martinus Nijhoff Publishers, Amsterdam, 507–518 (1986).
Peterson, R.E., Stress Concentration Factors, John Wiley & Sons (1974).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rauch, B.J., Rowlands, R.E. Determining reliable edge isopachic data from interior thermoelastic measurements. Experimental Mechanics 35, 174–181 (1995). https://doi.org/10.1007/BF02326477
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02326477