Investigation of implicit constitutive relations in which both the stress and strain appear linearly, adjacent to non-penetrating cracks

Author:

Itou Hiromichi1,Kovtunenko Victor A.23,Rajagopal Kumbakonam R.4

Affiliation:

1. Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

2. Institute for Mathematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstr.36, 8010 Graz, Austria

3. Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk, Russia

4. Department of Mechanical Engineering, Texas A&M University, College Station, Texas 77843, USA

Abstract

A novel class of implicit constitutive relations is studied, wherein the stress and the linearized strain appear linearly, that describe material response in elastic porous bodies like rocks, ceramics, concrete, cement, bones and metals. The constitutive relation is applied to a body with a crack subjected to non-penetration conditions between the opposite crack faces. To treat well-posedness of a corresponding variational inequality, we rely on a new approximation by thresholding dilatation and apply the Lions existence theorem on pseudo-monotone variational inequalities. An analytical solution to a specific example (without crack) under uniform triaxial loading is constructed, wherein blow-up can take place at a finite load, and this difficulty is overcome within a thresholding approximation so that blow-up does not occur.

Funder

Scientific Research

Austrian Science Fund (FWF) Project

European Research Council

JSPS and Russian Foundation for Basic Research

RFBR Project

Publisher

World Scientific Pub Co Pte Ltd

Subject

Applied Mathematics,Modeling and Simulation

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