Nonlinear continuum mechanics : an engineering approach / Carlos Agelet de Saracibar.
Agelet de Saracíbar Bosch, Carlos.
2023
QA808.2
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Title Nonlinear continuum mechanics : an engineering approach / Carlos Agelet de Saracibar.
Author Agelet de Saracíbar Bosch, Carlos.
ISBN 9783031152078
3031152077
3031152069
9783031152061
DOI https://doi.org/10.1007/978-3-031-15207-8
Publication Details Cham : Springer International Publishing AG, 2023.
Language English
Description 1 online resource (xx, 344 pages) : illustrations (chiefly color)
Item Number 10.1007/978-3-031-15207-8 doi
Call Number QA808.2
Dewey Decimal Classification 531/.7
Summary This textbook on Continuum Mechanics presents 9 chapters. Chapters 1 and 2 are devoted to Tensor Algebra and Tensor Analysis. Part I of the book includes the next 3 chapters. All the content here is valid for both solid and fluid materials. At the end of Part I, the reader should be able to set up in local spatial/material form, the fundamental governing equations and inequalities for a Continuum Mechanics problem. Part II of the book, Chapters 6 to 10, is devoted to presenting some nonlinear constitutive models for Nonlinear Solid Mechanics, including Finite Deformation Hyperelasticity, Finite Deformation Plasticity, Finite Deformation Coupled Thermoplasticity, and Finite Deformation Contact Mechanics. The constitutive equations are derived within a thermodynamically consistent framework. Finite deformation elastoplasticity models are based on a multiplicative decomposition of the deformation gradient and the notion of an intermediate configuration. Different formulations based on the intermediate configuration, the current or spatial configuration, and the material configuration are considered. The last chapter is devoted to Variational Methods in Solid Mechanics, a fundamental topic in Computational Mechanics. The book may be used as a textbook for an advanced Masters course on Nonlinear Continuum Mechanics for graduate students in Civil, Mechanical or Aerospace Engineering, Applied Mathematics, or Applied Physics, with an interest in Continuum Mechanics and Computational Mechanics.
Note Description based upon print version of record.
Access Note Access limited to authorized users.
Available in Other Form Nonlinear Continuum Mechanics
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Intro
Preface
Contents
About the Author
1 Tensor Algebra
1.1 Vector Algebra
1.1.1 Linear or Vector Spaces. Real Space
1.1.2 Euclidean Vector Spaces
1.1.3 Lie Algebra
1.2 Vector Algebra Geometry
1.2.1 Inner (or Dot, or Scalar) Product on mathbbR3
1.2.2 Cross (or Vector) Product on mathbbR3
1.2.3 Box (or Scalar Triple) Product on mathbbR3
1.3 Vector Algebra in Components
1.3.1 Standard Basis and Vector Components on mathbbR3
1.3.2 Inner (or Dot, or Scalar) Product on mathbbR3
1.3.3 Cross (or Vector) Product on mathbbR3

1.3.4 Box (or Scalar Triple) Product on mathbbR3
1.4 Tensor Algebra
1.4.1 Vector Space of Linear Transformations mathscrL(mathbbR3,mathbbR3)
1.4.2 Tensor Multiplication on mathscrL(mathbbR3,mathbbR3)
1.4.3 Tensor Product on mathbbR3
1.4.4 The Trace Function
1.4.5 The Determinant Function
1.4.6 Inner Product on mathscrL(mathbbR3,mathbbR3)
1.4.7 Inverse of a Second-Order Tensor
1.4.8 Lie Bracket Operation
1.4.9 Orthogonal Group O(3)
1.4.10 Special Orthogonal Group SO(3)
1.4.11 Spectral Decomposition of a Symmetric Second-Order Tensor

1.5 Tensor Algebra in Components
1.5.1 Standard Basis and Tensor Components
1.5.2 Tensor Multiplication on mathscrL(mathbbR3,mathbbR3)
1.5.3 Tensor Product on mathbbR3
1.5.4 The Trace Function
1.5.5 Inner (or Double Dot) Product on mathscrL(mathbbR3,mathbbR3)
1.5.6 The Determinant Function
1.5.7 Inverse of a Second-Order Tensors
1.5.8 Additive Split of a Second-Order Tensor into Its Symmetric and Skew-Symmetric Parts
1.5.9 Additive Split of a Second-Order Tensor into Its Volumetric and Deviatoric Parts
1.6 Fourth-Order Tensors
1.6.1 Fourth-Order Tensors

1.6.2 Tensor Multiplication on mathscrL(mathbbR3,mathbbR3)timesmathscrL(mathbbR3,mathbbR3)
1.6.3 Tensor Product on mathscrL(mathbbR3,mathbbR3)
1.6.4 Fourth-Order Identity Tensors
1.6.5 Fourth-Order Volumetric and Deviatoric Projection Operators
1.7 Fourth-Order Tensors in Components
1.7.1 Fourth-Order Tensors
1.7.2 Tensor Multiplication on mathscrL(mathbbR3,mathbbR3)timesmathscrL(mathbbR3,mathbbR3)
1.7.3 Tensor Product on mathscrL(mathbbR3,mathbbR3)
1.7.4 Fourth-Order Identity Tensors
1.7.5 Fourth-Order Volumetric and Deviatoric Projection Operators
1.8 Exercises

References
2 Tensor Analysis
2.1 Differential Analysis
2.1.1 The Frechet Derivative
2.1.2 The Directional Derivative
2.1.3 Tangent Spaces and Tangent Maps
2.1.4 Differentiable Manifolds and Charts
2.1.5 Tangent Spaces to a Manifold
2.2 Tangent Space to Lie Groups
2.2.1 Tangent Space to mathbbSmathbbL(3,mathbbR) at the Identity and Lie Algebra mathfraksmathfrakl(3,mathbbR)
2.2.2 Tangent Space to O(3) at the Identity and Lie Algebra mathfraksmathfrako(3)
2.3 Differential Operators in mathbbR3
2.3.1 Differential Operators in mathbbR3
2.3.2 Gradient Vector Differential Operator

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