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Über dieses Buch

This book was written to serve as the standard textbook of elastoplasticity for students, engineers and researchers in the field of applied mechanics. The present second edition is improved thoroughly from the first edition by selecting the standard theories from various formulations and models, which are required to study the essentials of elastoplasticity steadily and effectively and will remain universally in the history of elastoplasticity. It opens with an explanation of vector-tensor analysis and continuum mechanics as a foundation to study elastoplasticity theory, extending over various strain and stress tensors and their rates. Subsequently, constitutive equations of elastoplastic and viscoplastic deformations for monotonic, cyclic and non-proportional loading behavior in a general rate and their applications to metals and soils are described in detail, and constitutive equations of friction behavior between solids and its application to the prediction of stick-slip phenomena are delineated. In addition, the return-mapping algorithm, the consistent tangent operators and the objective time-integration algorithm of rate tensor are explained in order to enforce the FEM analyses. All the derivation processes and formulations of equations are described in detail without an abbreviation throughout the book.

The distinguishable features and importance of this book is the comprehensive description of fundamental concepts and formulations including the objectivity of

tensor and constitutive equations, the objective time-derivative of tensor functions, the associated flow rule, the loading criterion, the continuity and smoothness conditions and their substantial physical interpretations in addition to the wide classes of reversible/irreversible constitutive equations of solids and friction behavior between solids.

Inhaltsverzeichnis

Frontmatter

Tensor Analysis

Abstract
Physical quantities appearing in continuum mechanics are mathematically expressed by tensors. Therefore, their relations are described by tensor equations. Before studying the main theme of this book, elastoplasticity theory, mathematical properties of tensors and mathematical rules on tensor operations are explained on the level necessary to understand elastoplasticity theory. The orthogonal Cartesian coordinate system is adopted throughout this book. A further advanced mathematics of tensors in the embedded curvilinear coordinate system is referred to Hashiguchi and Yamakawa (2012).
Koichi Hashiguchi

Motion and Strain (Rate)

Abstract
The tensor analysis providing the mathematical foundation for the continuum mechanics is described in Chapter 1. Basic concepts and quantities for continuum mechanics will be studied in the three chapters up to Chapter 4. The description of motion and deformation of a material body constitutes the basic introductory part of the continuum mechanics. Various expressions of motion and a variety of strain and strain rate measures are employed for the description of reversible and irreversible deformations of materials. Some selected basic expressions and measures will be explained in this chapter.
Koichi Hashiguchi

Conservation Laws and Stress Tensors

Abstract
Conservation laws must be fulfilled for mass, momentum, angular momentum, etc. during a deformation. These laws are described first in detail. Then, the Cauchy stress tensor is defined and further, based on it, various stress tensors are derived from the Cauchy stress tensor. Introducing the stress tensor, the equilibrium equations of force and moment are formulated from the conservation laws. The virtual work principle required for the analyses of boundary value problems are also described in this chapter.
Koichi Hashiguchi

Objectivity, and Objective and Corotational Rate Tensors

Abstract
Constitutive property of material is independent of observers. Therefore, constitutive equation has to be described by variables obeying the common objective transformation rule described in Section 1.3. State variables, e. g. stress, strain and back stress tensors in the same configuration obey the common coordinate transformation rule. However, the material-time derivatives of tensors in the current configuration do not obey the objective transformation rule, which is influenced by the rigid-body rotation. Then, instead of the material-time derivative of tensors, particular time-derivatives of tensors obeying the objective transformation rule have to be adopted in constitutive equations.
Koichi Hashiguchi

Elastic Constitutive Equations

Abstract
Elastic deformation is induced by the reversible change of distances between material particles without a mutual slip between them. They therefore exhibit high stiffness. Elastic constitutive equations are classifiable into the three types depending on their levels of reversibility, i.e. the hyperelasticity (or Green elasticity) possessing the strain energy function, the Cauchy elasticity possessing the one-to-one correspondence between stress and strain and the hyperelasticity possessing the linear relation between stress rate and strain rate. As preparation for the study of elastoplasticity in the subsequent chapters, they are explained in this chapter.
Koichi Hashiguchi

Basic Formulations for Elastoplastic Constitutive Equations

Abstract
Elastic deformation is induced microscopically by the elastic deformations of the material particles themselves, exhibiting a one-to-one correspondence to the stress. However, when the stress reaches an yield stress, slippages between material particles are induced, which do not disappear even if the stress is removed, leading to macroscopically to the plastic deformation. Then, the one-to-one correspondence between the stress and the strain, i.e. the stress-strain relation does not hold in the elastoplastic deformation process.
Koichi Hashiguchi

Unconventional Elastoplasticity Model: Subloading Surface Model

Abstract
Elastoplastic constitutive equations with the yield surface enclosing the elastic domain possess many limitations in the description of elastoplastic deformation, as explained in the last chapter. They are designated as the conventional model in Drucker’s (1988) classification of plasticity models. Various unconventional elastoplasticity models have been proposed, which are intended to describe the plastic strain rate induced by the rate of stress inside the yield surface. Among them, the subloading surface model is the only pertinent model fulfilling the mechanical requirements for elastoplastic constitutive equations. These mechanical requirements are first described and then the subloading surface model is explained in detail.
Koichi Hashiguchi

Cyclic Plasticity Models: Critical Reviews and Assessments

Abstract
Accurate description of plastic deformation induced during a cyclic loading process is required for the mechanical design of machinery subjected to vibration and buildings and soil structures subjected to earthquakes since the middle of the last century. Elastoplastic constitutive model formulated to this aim is called the cyclic plasticity model. Substantially, the key of the pertinence in cyclic plasticity model is how to describe appropriately a small plastic strain rate induced by the rate of stress inside the yield surface. Therefore, a quite delicate formulation of plastic strain rate developing gradually as the stress approaches the yield surface is required to this end. Here, needless to say, the continuity and the smoothness conditions described in Section 7.1 would have to be fulfilled in a cyclic plasticity model.
Koichi Hashiguchi

Extended Subloading Surface Model

Abstract
As was deliberated in Chapter 8, only the extended subloading surface model is capable of describing the cyclic loading behavior of materials pertinently. The explicit constitutive equation of this model is shown in this chapter. Then, this model will be applied to metals and soils, and their validities will be verified by comparisons with test data of metals in chapter 10 and soils in chapter 11.
Koichi Hashiguchi

Constitutive Equations of Metals

Abstract
The plasticity theory has highly developed through the prediction of deformation of metals up to date. The reason would be caused by the fact that, among various materials exhibiting plastic deformation, metals are used most widely as engineering materials and exhibit the simplest plastic deformation behavior without a pressure dependence, a plastic compressibility, an independence on the third invariant of deviatoric stress and a softening. Nevertheless, metals exhibit various particular aspects, e.g., the kinematic hardening and the stagnation of isotropic hardening in a cyclic loading. Explicit constitutive equations of metals will be delineated in this chapter, which are based on the general elastoplastic constitutive equations described in the preceding chapters.
Koichi Hashiguchi

Constitutive Equations of Soils

Abstract
The history of plasticity has begun to study the deformation behavior of soils by Coulomb (1773) when he has proposed the yield condition of soils by applying the friction law of himself. Thereafter, the soil plasticity has been superseded the leadership by the metal plasticity.
Koichi Hashiguchi

Viscoplastic Constitutive Equations

Abstract
Deformations of solids depend on the rate of loading or deformation, exhibiting the time-dependence or rate-dependence in general. Constitutive equations describing the rate-dependent plastic deformation is described in this chapter. The physical background of rate-dependent plastic deformation and the history of the development of rate-dependent plastic constitutive equation are reviewed first. Then, the pertinent formulation of the rate-dependent plastic constitutive equation is delineated, which is capable of describing the smooth elastic-plastic transition and the rate-dependence for the general rate ranging from the quasi-static deformation to the impact loading behavior.
Koichi Hashiguchi

Corotational Rate Tensor

Abstract
It was studied in Chapter 4 that the material-time derivatives of state variables, e.g. stress and internal variables in elastoplasticity do not possess the objectivity and thus, instead of them, we must use their objective time-derivatives. This chapter focuses on the responses of simple constitutive equations introducing corotational rates with various spins including the plastic spin.
Koichi Hashiguchi

Localization of Deformation

Abstract
Even if material is subjected to a homogeneous stress, the deformation concentrates in a quite narrow strip zone as the deformation becomes large and finally the material results in failure. Such a concentration of deformation is called the localization of deformation and the strip zone is called the shear band. The shear band thickness is the order of several microns in metals and ten and several times of particle diameter in soils. Therefore, a large shear deformation inside the shear band is hardly reflected in the change of external appearance of the whole body, although the stress is determined by the external load and the outer appearance of material. Therefore, a special care is required for the interpretation of element test data and the analysis taking account of the inception of shear band is indispensable when a large deformation is induced. The localization phenomenon of deformation and its pertinent analysis are addressed in this chapter.
Koichi Hashiguchi

Constitutive Equation for Friction: Subloading-Friction Model

Abstract
All bodies in the natural world are exposed to friction phenomena, contacting with other bodies, except for bodies floating in a vacuum. Therefore, it is indispensable to analyze friction phenomena rigorously in addition to the deformation behavior of bodies themselves in analyses of boundary value problems. The friction phenomenon can be formulated as a constitutive relation in a similar form to the elastoplastic constitutive equation of materials. A constitutive equation for friction with the transition from the static to the kinetic friction and vice versa and the orthotropic and rotational anisotropy is described in this chapter. The stick-slip phenomenon, which is an unstable and intermittent motion caused by the friction, is of importance for the prediction of earthquake and influences on the performance of machinery. It will be also delineated as the application of constitutive equation for friction.
Koichi Hashiguchi

Return-Mapping and Consistent Tangent Modulus Tensor

Abstract
Constitutive equations of irreversible deformation, e.g. elastoplastic, viscoelastic and viscoplastic deformations are described in rate forms in which the stress rate and the strain rate are related to each other through the tangent modulus. Therefore, numerical calculations are executed in their incremental forms by the input of load (stress) increment or displacement (deformation) increment, while the time increment is also input in rate-(or time-)dependent constitutive equations, e.g. viscoelastic and viscoplastic ones.
Koichi Hashiguchi

Backmatter

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