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2021 | Buch

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Continuum Mechanics with Eulerian Formulations of Constitutive Equations

verfasst von: Prof. M.B. Rubin

Verlag: Springer International Publishing

Buchreihe : Solid Mechanics and Its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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Über dieses Buch

This book focuses on the need for an Eulerian formulation of constitutive equations. After introducing tensor analysis using both index and direct notation, nonlinear kinematics of continua is presented. The balance laws of the purely mechanical theory are discussed along with restrictions on constitutive equations due to superposed rigid body motion. The balance laws of the thermomechanical theory are discussed and specific constitutive equations are presented for: hyperelastic materials; elastic–inelastic materials; thermoelastic–inelastic materials with application to shock waves; thermoelastic–inelastic porous materials; and thermoelastic–inelastic growing biological tissues.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The objective of this introductory chapter is to present an overview of the contents of this book and to discuss the importance of Eulerian formulations of constitutive equations. Specifically, simple one-dimensional examples are used to identify unphysical arbitrariness in the classical Lagrangian formulations of constitutive equations that can and should be removed.

M. B. Rubin

Chapter 2. Basic Tensor Analysis

Abstract

Tensors are mathematical objects which ensure that mathematical equations characterizing physics are insensitive to arbitrary choices of a coordinate system. The objective of this chapter is to present a review of tensor analysis using both index and direct notations. To simplify the presentation of tensor calculus, attention is limited to tensors expressed relative to fixed rectangular Cartesian base vectors. (Some of the content in this chapter has been adapted from Rubin (Cosserat theories: shells, rods and points. Springer Science & Business Media, Berlin, 2000) with permission.)

M. B. Rubin

Chapter 3. Kinematics

Abstract

The objective of this chapter is to discuss nonlinear kinematics of deformable continua. Bodies, configurations and motion of continua are discussed along with a definition of the material time derivative, which is used to determine the velocity and acceleration of a material point. Deformation tensors and rate of deformation tensors are defined and analyzed. The notion of Superposed Rigid Body Motions (SRBM) is presented and the associated transformation relations of specific tensors are developed. In addition, an Eulerian formulation of evolution equations for elastic deformations is proposed and strongly objective, robust numerical integration algorithms for these evolution equations are developed.

M. B. Rubin

Chapter 4. Balance Laws for the Purely Mechanical Theory

Abstract

The objective of this chapter is to discuss the balance laws in the purely mechanical theory. Specifically, the conservation of mass and the balances of linear and angular momentum are presented in both global and local forms. The properties of the Cauchy stress tensor are derived and the rate of material dissipation is proposed. Invariance under Superposed Rigid Body Motions (SRBM) is discussed along with the development of the transformation relations for specific tensors. It is shown that the local forms of the balance laws can be derived by using invariance under SRBM of the rate of material dissipation and these transformation relations. Also, linearization of the kinematic quantities and balance laws are discussed.

M. B. Rubin

Chapter 5. Purely Mechanical Constitutive Equations

Abstract

The objective of this chapter is to discuss purely mechanical constitutive equations. After identifying unphysical arbitrariness of the classical Lagrangian formulation of constitutive equations, an Eulerian formulation for nonlinear elastic materials is developed using evolution equations for microstructural vectors \({\mathbf {m}}_i\). The influence of kinematic constraints on constitutive equations is discussed and specific nonlinear constitutive equations are presented for a number of materials including: elastic solids, viscous fluids and elastic–inelastic materials.

M. B. Rubin

Chapter 6. Thermomechanical Theory

Abstract

The objective of this chapter is to present the balance laws for the thermomechanical theory. Specifically, the balances of entropy and energy are presented and different forms of second law of thermodynamics are discussed. Invariance under Superposed Rigid Body Motions (SRBM) is considered for the new thermal quantities and thermal constraints on material response are discussed. In addition, specific nonlinear constitutive equations are presented for a number of materials modeling: thermoelastic, thermoelastic–inelastic and porous responses. Also, constitutive equations for growth of thermoelastic–inelastic biological tissues are presented.

M. B. Rubin

Backmatter

Titel: Continuum Mechanics with Eulerian Formulations of Constitutive Equations
verfasst von: Prof. M.B. Rubin
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-57776-6
Print ISBN: 978-3-030-57775-9
DOI: https://doi.org/10.1007/978-3-030-57776-6

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