This book provides readers with a deep understanding of the use of objective algorithms for integration of constitutive relations (CRs) for Hooke-like hypoelasticity based on the use of corotational stress rates. The purpose of objective algorithms is to perform the step-by-step integration of CRs using fairly large time steps that provide high accuracy of this integration in combination with the exact reproduction of superimposed rigid body motions. Since Hooke-like hypoelasticity is included as a component in CRs for elastic-inelastic materials (e.g., in CRs for elastic-plastic materials), the scope of these algorithms is not limited to hypoelastic materials, but extends to many other materials subjected to large deformations. The authors performed a comparative analysis of the performance of most currently available objective algorithms, provided some recommendations for improving the existing formulations of these algorithms, and presented new formulations of the so-called absolutely objective algorithms. The proposed book will be useful for beginner researchers in the development of economical methods for integrating elastic-inelastic CRs, as well as for experienced researchers, by providing a compact overview of existing objective algorithms and new formulations of these algorithms. The book will also be useful for developers of computer codes for implementing objective algorithms in FE systems. In addition, this book will also be useful for users of commercial FE codes, since often these codes are so-called black boxes and this book shows how to test accuracy of the algorithms of these codes for integrating elastic-inelastic CRs in modeling large rotations superimposed on the uniform deformation of any sample.
Author(s): Sergey Korobeynikov, Alexey Larichkin
Series: SpringerBriefs in Applied Sciences and Technology: Continuum Mechanics
Publisher: Springer
Year: 2023
Language: English
Pages: 113
City: Cham
Preface
Acknowledgements
Contents
Acronyms
1 Introduction
1.1 State of the Art
1.2 Book Writing Goals
References
2 Preliminaries
2.1 Local Body Deformations and Basic Kinematics
2.2 Objective Tensor Rates
2.2.1 Objective Rates of Eulerian Tensors
2.2.2 Objective Rates of Lagrangian Tensors
2.2.3 Relationship Between Lagrangian and Eulerian Tensors and Their Rates
2.3 Hooke-Like Isotropic Hypoelasticity Models Based on Corotational Stress Rates
References
3 Incremental Tensors and the Incremental Objectivity of Tensors
3.1 Definition of the Incremental Objectivity of Tensors
3.2 Incrementally Objective Kinematic Tensors
References
4 Incrementally Objective Algorithms for Integrating CRs for Hooke-Like Hypoelastic Models in the Eulerian Form
4.1 Generalized Midpoint Approximations
4.2 Weak Incrementally Objective Algorithms
4.2.1 Hughes–Winget Algorithm
4.2.2 Rubinstein–Atluri Algorithm and Its Generalization
4.2.3 Simo–Hughes Algorithm and Its Generalization
4.3 Strong Incrementally Objective Algorithms
4.3.1 Necessary Conditions for the Strong Incremental Objectivity of the S–H Algorithm
4.3.2 Expressions for Approximate Incremental Strains for Strong Incrementally Objective Algorithms
4.3.3 Expressions for Approximate Incremental Rotations for Strong Incrementally Objective Algorithms
4.3.4 Expressions for Determining the Kirchhoff Stress Tensors
References
5 Absolutely Objective Algorithms for Integrating CRs for Hooke-Like Hypoelastic Models
5.1 Absolutely Lagrangian-Objective Algorithm
5.1.1 Absolutely Lagrangian-Objective Approximate Incremental Strain Tensors
5.1.2 Absolutely Lagrangian-Objective Approximate Incremental Rotation Tensor
5.1.3 Expression for Determining the Absolutely Lagrangian-Objective Rotated Kirchhoff Stress Tensor
5.2 Absolutely Eulerian-Objective Algorithm
5.2.1 Absolutely Eulerian-Objective Approximate Incremental Strain Tensors
5.2.2 Absolutely Eulerian–Lagrangian-Objective Approximate Incremental Rotation Tensor
5.2.3 Expression for Determining the Absolutely Eulerian-Objective Kirchhoff Stress Tensor
5.3 Discussion of Absolutely Objective Algorithms
References
6 Comparative Analysis and Verification of Objective Algorithms
6.1 Problems of Uniform Deformation of Hypoelastic Bodies: Exact Solutions and Estimates of the Accuracy of Approximate Incremental Strains
6.2 Background for Computer Simulations
6.3 Simulations Using Weak Incrementally Objective Algorithms
6.3.1 Solutions of Problems without Superimposed Rotations
6.3.2 Solutions of Problems with Superimposed Rotations
6.4 Simulations Using Strong Incrementally Objective Algorithms
6.5 Simulations Using Absolutely Objective Algorithms
6.6 Discussion of the Results of Computer Simulations
References
7 Concluding Remarks
Reference
Appendix A Flowcharts of Objective Algorithms for Integrating Hypoelastic Constitutive Relations
Appendix B Algorithms for Determining the Polar Decomposition
B.1 Algorithms for Determining the Polar Decomposition in Lagrangian Variables
B.2 Algorithms for Determining the Polar Decomposition in Eulerian Variables
Appendix C The Rodrigues Formula for Determining an Incremental Rotation Tensor
Reference
Index
