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## Über dieses Buch

The book provides a rigorous axiomatic approach to continuum mechanics under large deformation. In addition to the classical nonlinear continuum mechanics – kinematics, fundamental laws, the theory of functions having jump discontinuities across singular surfaces, etc. - the book presents the theory of co-rotational derivatives, dynamic deformation compatibility equations, and the principles of material indifference and symmetry, all in systematized form. The focus of the book is a new approach to the formulation of the constitutive equations for elastic and inelastic continua under large deformation. This new approach is based on using energetic and quasi-energetic couples of stress and deformation tensors. This approach leads to a unified treatment of large, anisotropic elastic, viscoelastic, and plastic deformations. The author analyses classical problems, including some involving nonlinear wave propagation, using different models for continua under large deformation, and shows how different models lead to different results. The analysis is accompanied by experimental data and detailed numerical results for rubber, the ground, alloys, etc. The book will be an invaluable text for graduate students and researchers in solid mechanics, mechanical engineering, applied mathematics, physics and crystallography, as also for scientists developing advanced materials.

## Inhaltsverzeichnis

### Chapter 1. Introduction: Fundamental Axioms of Continuum Mechanics

Abstract
Continuum mechanics, including nonlinear continuum mechanics, studies the behavior of material bodies or continua. We can mathematically define a body as follows: it is a set $$\mathcal{B}$$ consisting of elements $$\mathcal{M}$$ called material points. The concept of a material point in continuum mechanics is primary, i.e. axiomatic, as is the concept of a geometrical point in elementary geometry.
Yuriy I. Dimitrienko

### Chapter 2. Kinematics of Continua

Abstract
Let us consider a continuum $$\mathcal{B}$$. Due to Axiom2, at time t=0 there is a one-to-one correspondence between every material point $$\mathcal{M}\in \mathcal{B}$$and its radius-vector $$\mathop{\mathbf{x}}^{\circ } =\overrightarrow{ O\mathcal{M}}$$in a Cartesian coordinate system $$O\bar{{\mathbf{e}}}_{i}$$. Denote Cartesian coordinates of the radius-vector by $$\mathop{x}^ {\circ }{}^{i}$$($$\mathop{\mathbf{x}}^{\circ } =\mathop{ x}^ {\circ }{}^{i}\bar{{\mathbf{e}}}_{i}$$) and introduce curvilinear coordinates X i of the same material point $$\mathcal{M}$$in the form of some differentiable one-to-one functions
$$\mathop{x}^ {\circ }{}^{i} =\mathop{ x}^ {\circ }{}^{i}({X}^{k}).$$
(2.1)
Yuriy I. Dimitrienko

### Chapter 3. Balance Laws

Abstract
Let us consider fundamental laws of continuum mechanics, and complement the set of Axioms1–3 (see Introduction) with new axioms.
Yuriy I. Dimitrienko

### Chapter 4. Constitutive Equations

Abstract
The equation system (3.307) consists of 18 scalar equations (each of the vector equations in (3.307) is equivalent to three scalar equations, and each of the tensor ones is equivalent to nine scalar equations), but involves 29 scalar unknowns:
$$\rho,\ \ \mathbf{v},\ \ \mathbf{u},\ \ \mathbf{T},\ \ e,\ \ \eta,\ \ \theta,\ \ \mathbf{q},\ \ \mathbf{F},\ \ {q}^{{_\ast}}.$$
(4.1)
Yuriy I. Dimitrienko

### Chapter 5. Relations at Singular Surfaces

Abstract
Up to now we considered the case when all functions appearing in the balance laws: ρ, u, v, T, F, fetc. are continuously differentiable functions of coordinates X i (or of x i ) and time t. However, in practice one often meets with problems, where this condition is violated. For example, for the phenomena of impact, explosion, combustion etc., a part of the indicated functions in a domain Vconsidered can suffer a jump discontinuity across some surface
Yuriy I. Dimitrienko

### Chapter 6. Elastic Continua at Large Deformations

Abstract
Let us formulate the problem for the general case of elastic (i.e. ideal) solids at large deformations; their constitutive equations have been derived in Sects.4.5–4.9.
Yuriy I. Dimitrienko

### Chapter 7. Continua of the Differential Type

Abstract
Let us consider now nonideal continua. Practically all real bodies are nonideal media, and they can be considered as ideal ones in a certain approximation. According to the general theory of constitutive equations stated in Sect. 4.4.2, a continuum is nonideal if its operator constitutive equations (4.158) include the dissipation function w being nonzero.
Yuriy I. Dimitrienko

### Chapter 8. Viscoelastic Continua at Large Deformations

Abstract
Besides models of the differential type considered in Chap.7, in continuum mechanics there are other types of nonideal media. One widely uses models of viscoelastic materials, which are also called continua of the integral type, or hereditarily elastic continua. Models of viscoelastic continua most adequately describe the mechanical properties of polymer materials, composites based on polymers, different elastomers, rubbers and biomaterials, in particular, human muscular tissues.
Yuriy I. Dimitrienko

### Chapter 9. Plastic Continua at Large Deformations

Abstract
While models of viscoelastic continua most adequately describe a behavior of ‘soft’ materials (rubbers, polymers, biomaterials), for simulation of mechanical inelastic properties of ‘stiff’ materials (metals and alloys) one widely uses models of plastic media.
Yuriy I. Dimitrienko

### Backmatter

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