Residual stresses in a finite cylinder. Direct and inverse problems and their solving using the variational method of homogeneous solutions

2018;
: 119-133
https://doi.org/10.23939/mmc2018.02.119
Received: September 20, 2018
1
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine; Kuyawy and Pomorze University in Bydgoszcz
2
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine

Mathematical models and methods for determination of axisymmetric residual stresses in a finite cylinder are considered. The model of residual stresses is built using the conception of incompatible eigenstrain tensor. Within the frame of this model, a direct problem for residual stresses determination is formulated. A method based on the variational method of homogeneous solutions is developed for solving the direct problem. Using the obtained solution, features of residual stresses, caused by continuous and piece-wise homogeneous distributions of eigenstrain components are studied. A variational formulation of the inverse problem for residual stresses determination on the base of empirical data obtained by a photoelasticity method is suggested. The inverse problem is solved numerically with the use of iterative calculations of values of the criterion functional. The results presented in the paper can be used for the development of methods and means for nondestructive testing and engineering characterization of materials and structural elements.

  1. Schajer G. S. Practical residual stress measurement methods. John Wiley & Sons, Ltd (2013).
  2. Dally J. W., Riley W. F. Experimental stress analysis. McGraw-Hill, New York; 3rd edition (1991).
  3. Chekurin V. F. A variational method for solving of the problems of tomography of the stressed state of solids. Materials Science. 35 (5), 623–633 (1999).
  4. Chekurin V. F. An approach to solving of stress state tomography problems of elastic solids with incompatibility strains. Mechanics of Solids. 35 (6), 29–37 (2000).
  5. Aben H. Integrated Photoelasticity. McGraw-Hill, New York (1979).
  6. Chekurin V. F. Integral photoelasticity relations for inhomogeneously strained dielectrics. Mathematical Modeling and Computing. 1 (2), 144–155 (2014).
  7. Mura T. Micromechanics of Defects in Solids. Martinus Nijhof Publishers, Dordrecht, The Netherlands; 2nd edition (1987).
  8. Chekurin V. F., Postolaki L. I. A variational method of homogeneous solutions for axisymmetric elasticity problems for cylinder. Mathematical Modeling and Computing. 2 (2), 128–139 (2015).
  9. Chekurin V. F., Postolaki L. I. Application of variational method of homogeneous solutions for optimal control of axisymmetric thermoelastic state of cylinder. Mathematical methods and physico-mechanical fields. 60 (2), 105–116 (2017).
  10. Saad H. M. Elasticity. Theory, Applications and Numerics. Elsevier Academic Press (2005).
  11. Chekurin V., Postolaki L. Application of the least square method in axisymmetric biharmonic problems. Mathematical Problems in Engineering. 2016, Article ID 3457649, 9 pages (2016).
  12. Kantorovich L. V., Krylov V. I. Approximate methods of higher analysis. Translated from the 3rd Russian Edition by C. D. Benster. Interscience Publ., New York (1958).
  13. Dennis J. E., Schnabel R. B. Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall Inc., New Jersey (1983).
Math. Model. Comput. Vol. 5, No. 2, pp. 119-133 (2018)