Abstract
The boundary value problem of small static deformation range is formulated as the variational problem for the displacement. The general local conditions of convexity and rank-one-convexity are obtained for smooth elastic–plastic potentials depending on two first convex invariants of the Cauchy strain tensor. It is demonstrated that all types of convexity are equivalent for the Hencky elastic–plastic potential, but, for example, the strain energy function of continuum fracture can have every type of convexity for different relationships between material parameters. The classical regularization method is applied for the construction of the lower convex envelope of the Hencky strain energy potential for the experimental non-monotonic stress–strain relation with falling part. For this relation, the concept of the Maxwell line is used by analogy with the Van der Waals’ gas theory. The results of 1-D and 2-D numerical examples show that this regularization is principally necessary for the convergence of the finite-difference methods that are usually used for numerical computation within the framework of the small deformation models with non-monotonic stress–strain relations.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.