Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model

Abstract

The Boussinesq problem, which describes quasi-static indentation of a rigid punch into a deformable body, is studied within the context of nonlinear constitutive equations. By this, the material response expresses the linearized strain in terms of the stress and cannot be inverted in general. A contact area between the punch and the body is unknown a priori , whereas the total contact force is prescribed and yields a non-local integral condition. Consequently, the unilateral indentation problem is stated as a quasi-variational inequality for unknown variables of displacement, stress and indentation depth. The Lagrange multiplier approach is applied in order to establish well-posedness to the underlying physically and geometrically nonlinear problem based on augmented penalty regularization and applying the minimax theorem of Ekeland and Témam. A sufficient solvability condition implies response functions that are bounded, hemi-continuous, coercive and obey a convex potential. A typical example is power-law hardening models for titanium alloys, Norton–Hoff and Ramberg–Osgood materials. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Forward and inverse problems for creep models in viscoelasticity

“…In the particular case when N = 1 and B 0 = 0, the differentiation of equation (1.13) with respect to t yields the Kelvin-Voigt model. The constitutive relation (1.13) was applied to the Boussinesq indentation problem by Itou et al [17,18] and Kovtunenko [19]. In this study, we consider all representations (1.9)- (1.13).…”

Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle

“…Open questions in the contact mechanics concern non-smooth behaviour, e.g. owing to non-coercive [2,3] and non-convex functions [4], nonlinear constitutive equations [5][6][7], time-discontinuous evolution [8] and geometry singularities [9].…”

Non-smooth variational problems and applications

“…The variational problems under consideration are subjected to gradient constraint [ 1 ] and unilateral constraints [ 2 , 3 ], they obey non-differentiable objectives and may lose the property of coercivity [ 4 ]. These features result in non-smooth optimization, quasi-variational inequalities [ 5 ] and hemi-variational inequalities [ 6 ]. The cases of stochastic optimal control [ 7 ] and coefficient identification from measured data at the boundary [ 8 ] belong to the field of inverse problems, which are known to be ill-posed.…”

“…The different types of models for solids describe elastic junctions [ 2 ], thermo-elastic composites under mechanical vibration [ 12 ], dynamic behaviour of the Euler–Bernoulli beams [ 8 ] and thermo-elastic Kirchhoff–Love plates [ 3 ]. There are considered bodies that exhibit power-law hardening like the Norton–Hoff and Ramberg–Osgood materials [ 5 ], and ideal elasto-plastic behaviour [ 1 ]. By this, nonlinear boundary conditions are of the first importance for physically consistent modelling.…”

“…The rate-independent evolutionary systems driven by non-convex energies have been suggested [ 10 ], which are successful to model properly jump discontinuities in time during quasi-brittle crack propagation. The elaborated mathematical and mechanical description gives an impact to practise testing methodologies by rigid punch indentation [ 5 ], in fracture mechanics and seismology [ 13 ].…”