Lagrange multiplier approach to unilateral indentation problems: Well-posedness and application to linearized viscoelasticity with non-invertible constitutive response

Lagrange multiplier approach to unilateral indentation problems: Well-posedness and application to linearized viscoelasticity with non-invertible constitutive response Full article

Journal

Mathematical Models and Methods in Applied Sciences
ISSN: 0218-2025

Output data

Year: 2021, Volume: 31, Number: 03, Pages: 649-674 Pages count : 26 DOI: 10.1142/s0218202521500159

Tags

augmented Lagrangian; Boussinesq problem; cone indenter; indentation testing; non-local constraint; quasi-variational inequality; variational solution; viscoelastic model; Volterra convolution operator

Authors

Itou Hiromichi ¹ , Kovtunenko Victor A. ^2,3 , Rajagopal Kumbakonam R. ⁴

Affiliations

1	Department of Mathematics, Tokyo University of Science
2	Institute for Mathematics and Scientific Computing, University of Graz, NAWI Graz,
3	Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences
4	Department of Mechanical Engineering, Texas A&M University, College Station, Texas

The Boussinesq problem describing indentation of a rigid punch of arbitrary shape into a deformable solid body is studied within the context of a linear viscoelastic model. Due to the presence of a non-local integral constraint prescribing the total contact force, the unilateral indentation problem is formulated in the general form as a quasi-variational inequality with unknown indentation depth, and the Lagrange multiplier approach is applied to establish its well-posedness. The linear viscoelastic model that is considered assumes that the linearized strain is expressed by a material response function of the stress involving a Volterra convolution operator, thus the constitutive relation is not invertible. Since viscoelastic indentation problems may not be solvable in general, under the assumption of monotonically non-increasing contact area, the solution for linear viscoelasticity is constructed using the convolution for an increment of solutions from linearized elasticity. For the axisymmetric indentation of the viscoelastic half-space by a cone, based on the Papkovich-Neuber representation and Fourier-Bessel transform, a closed form analytical solution is constructed, which describes indentation testing within the holding-unloading phase.

Itou H. , Kovtunenko V.A. , Rajagopal K.R.
Lagrange multiplier approach to unilateral indentation problems: Well-posedness and application to linearized viscoelasticity with non-invertible constitutive response
Mathematical Models and Methods in Applied Sciences. 2021. V.31. N03. P.649-674. DOI: 10.1142/s0218202521500159 WOS Scopus РИНЦ OpenAlex

Web of science:	WOS:000651438800006
Scopus:	2-s2.0-85101743497
Elibrary:	46757568
OpenAlex:	W3134088252

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