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Variational formulation for dissipative continua and an incremental J-integral

Rahaman, Md Masiur and Dhas, Bensingh and Roy, D and Reddy, J N (2018) Variational formulation for dissipative continua and an incremental J-integral. In: PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 474 (2209).

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Official URL: http://dx.doi.org/10.1098/rspa.2017.0674

Abstract

Our aim is to rationally formulate a proper variational principle for dissipative (viscoplastic) solids in the presence of inertia forces. As a first step, a consistent linearization of the governing nonlinear partial differential equations (PDEs) is carried out. An additional set of complementary (adjoint) equations is then formed to recover an underlying variational structure for the augmented system of linearized balance laws. This makes it possible to introduce an incremental Lagrangian such that the linearized PDEs, including the complementary equations, become the Euler-Lagrange equations. Continuous groups of symmetries of the linearized PDEs are computed and an analysis is undertaken to identify the variational groups of symmetries of the linearized dissipative system. Application of Noether's theorem leads to the conservation laws (conserved currents) of motion corresponding to the variational symmetries. As a specific outcome, we exploit translational symmetries of the functional in the material space and recover, via Noether's theorem, an incremental J-integral for viscoplastic solids in the presence of inertia forces. Numerical demonstrations are provided through a two-dimensional plane strain numerical simulation of a compact tension specimen of annealed mild steel under dynamic loading.

Item Type: Journal Article
Additional Information: Copy right for this article belong to the ROYAL SOC, 6-9 CARLTON HOUSE TERRACE, LONDON SW1Y 5AG, ENGLAND
Department/Centre: Division of Mechanical Sciences > Civil Engineering
Depositing User: Id for Latest eprints
Date Deposited: 02 Mar 2018 14:55
Last Modified: 02 Mar 2018 14:55
URI: http://eprints.iisc.ac.in/id/eprint/59060

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