Evolutionary-variational method in mathematical plasticity

The elastic–plastic infinitesimal deformation of a solid is considered within the framework of the incremental flow theory using the constitutive relation in the general rate form. The appropriate initial boundary value problem is formulated for the displacement in the form of the evolutionary-varia...

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Bibliographic Details
Published in:Acta mechanica Vol. 235; no. 11; pp. 6723 - 6738
Main Author: Brigadnov, Igor A.
Format: Journal Article
Language:English
Published: Vienna Springer Vienna 01.11.2024
Springer
Springer Nature B.V
Subjects:
Analysis
Boundary value problems
Cauchy problems
Classical and Continuum Physics
Constitutive relationships
Control
Differential equations
Dynamical Systems
Elastic deformation
Engineering
Engineering Fluid Dynamics
Engineering Thermodynamics
Equilibrium equations
Existence theorems
Flow theory
Heat and Mass Transfer
Hilbert space
Internal friction
Kinematics
Mathematical analysis
Methods
Nonlinear differential equations
Ordinary differential equations
Original Paper
Solid Mechanics
Theoretical and Applied Mechanics
Uniqueness theorems
Vibration
49M20
65K10
ISSN:0001-5970, 1619-6937
Online Access:Get full text
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Summary:The elastic–plastic infinitesimal deformation of a solid is considered within the framework of the incremental flow theory using the constitutive relation in the general rate form. The appropriate initial boundary value problem is formulated for the displacement in the form of the evolutionary-variational problem (EVP), i.e., as the abstract Cauchy problem in the Hilbert space which coincides with a weak form of the equilibrium equation, known as the principle of possible displacements in mechanics. The general existence and uniqueness theorem for the EVP is discussed. The main sufficient condition has a simple algebraic form and does not coincide with the classical Drucker and similar thermodynamical postulates; therefore, it must be independently verified. Its independence is illustrated for the non-associated plastic model of linear isotropic-kinematic hardening with dilatation and internal friction. The classical and endochronic models are analyzed too. The initial EVP is reduced by a spatial finite element approximation to the Cauchy problem for an implicit system of essentially nonlinear ordinary differential equations which can be stiff. Therefore, for the numerical solution the implicit Euler scheme is proposed. All theoretical results are illustrated by means of original numerical experiments.
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ISSN:0001-5970
1619-6937
DOI:10.1007/s00707-024-04064-0