##
**Mathematical foundations of elasticity.**
*(English)*
Zbl 0545.73031

Prentice-Hall Civil Engineering and Engineering Mechanics Series. Englewood Cliffs, New Jersey: Prentice-Hall, Inc. XVIII, 556 p. $ 58.00 (1983).

This book, written in the language of contemporary geometry and functional analysis, contains a preliminary chapter while the main text is presented in seven chapters. In the preliminary chapter, entitled ”A point of departure”, basic notions of continuum mechanics are reviewed, yet in a clear manner. Particularly the Eulerian and Lagrangian description are discussed and balance laws are formulated. The definition of a nonhomogeneous elastic material is introduced and hyperelastic materials are defined in the usual way. The reader interested in constitutive equations for rubber-like materials, only briefly mentioned by the authors, is advised by the reviewer to consult the synthetic paper of R. W. Ogden, ”Mechanics of solids”, Rodney Hill 60th Anniv. Vol., 499-537 (1982; Zbl 0491.73045).

The first chapter concerns geometry and kinematics of bodies, their motions and their configurations. All concepts are carefully defined and discussed. Starting from the definition of a simple body as an open set of the Euclidean space \({\mathbb{R}}^ 3\) and its configuration, the fundamental kinematical geometrical objects such as velocity and acceleration are introduced in a general coordinate system. Then the passage to manifolds is straightforward. Particularly various deformation tensors, well-known in continuum mechanics, are formulated in both the absolute and component forms. Relative to rate-of deformation tensors I would like to indicate the recent results by Zh.-H. Guo [Rates of stretch tensors, J. Elasticity, 14, 263-268 (1984)] who offered explicit expressions for the rates of the left and right stretch tensors. Two- point tensors are introduced as multilinear mappings, together with their basic algebraic properties. Covariant differentiation of both tensor fields and two-point tensor fields is concisely discussed. I would like to indicate that in the theory of geometric objects two-point tensors are called ”product tensors” [cf. the two papers by the reviewer, Ann. Polon. Math., 27, 67-72 (1972; Zbl 0244.53013) and 34, 197-200 (1977; Zbl 0403.53002)]. Assuming the existence of a mass density function, the conservation of mass for simple bodies and thin shells is examined.

Having introduced the notion of the Lie derivative it is shown that all objective rates of second-order tensors can be expressed as Lie derivatives. Since the subject of objective rates is still controversial, the authors define what they mean by objective. This new proposal seems to be interesting. Yet it would be advisable to know its consequences for finite deformation theory, including finite plasticity. The considerations about differential forms and the so called Piola transformation close chapter 1. The Piola transformation is an operation relating the material and spatial descriptions of a continuum. I think that the comprehensive paper by R. Hill [Adv. Appl. Mech. 18, 1-75 (1978; Zbl 0475.73026)] can be advised as completion to chapters 1-3.

In the second chapter, devoted to balance principle, primarily the transport theorem and the so-called master balance law and master balance inequality are introduced for both the material and spatial descriptions. Discontinuities (singular surfaces) are also examined. These master laws interrelate two scalar fields and a vector field. Localization leads to well-known local forms of balance equations, or inequalities from thermodynamics. The master balance law and inequality are abstract forms of the basic laws of continuum mechanics. Rigorous discussion of the first and second laws of thermodynamics follows next. In this chapter special attention is paid to classical space-times, the principle of virtual work and covariance of balance of energy. Covariance can be identified with invariance known from the theory of tensor functions. The fundamental laws of continuum mechanics, including the principle of virtual work, formulated on the manifold representative of a continuum are also studied in a paper by P. Rougée [J. Méc. 19, 7-32 (1980; Zbl 0429.73002)]. Balance laws for shells are also briefly discussed by the authors.

The two chapters that follow deal with constitutive relations: chapter 3 with the general constitutive theory and chapter 4 with linearized equations. Constitutive elastic and thermoelastic equations are defined as appropriate mappings. The principle theorems of the constitutive theory of nonlinear elasticity and thermoelasticity are obtained by two approaches, which differ in that the underlying sets of axioms are different. The second set is called ”covariant constitutive axioms” and is given for elasticity and thermoelasticity. After the formulation of constitutive equations, the typical boundary and initial boundary value problems are formulated. Simplifications of constitutive functions result when materials have additional symmetries, for instance isotropy. Linearization of nonlinear elasticity is accomplished by using the implicit function theorem for Banach spaces. For a completion, see Zh.-H. Guo, Ing.-Archiv. 52, 121-129 (1982; Zbl 0487.73041). The cases when the linearization procedure fails are briefly mentioned and the specific case when the linearized operator fails to be surjective is examined abstractly (linearization stability).

Some variational aspects, including the reciprocal theorems, of linear and nonlinear elasticity are examined in chapter 5 by using the contemporary geometrical methods. Thus, the classical linear elastodynamics with homogeneous boundary conditions is treated as linear Hamiltonian system. For the purpose the generalization of the concepts of Hamiltonian mechanics to infinite dimensions is preliminarily presented. To study nonlinear elasticity, the notion of an abstract Hamiltonian system (on Banach spaces) is introduced and examined. As in the finite- dimensional case, a Lagrangian approach, suitably generalized, is also available. Thus constraints, such as incompressibility, can be taken into account. The contemporary approach to Noether’s theorem and its consequences for conservation laws of elasticity are also examined. In my opinion, as an easily readable introduction to Noether’s theorem and conservation laws the following book can be recommended: J. D. Logan, Invariant variational principles, Academic Press, New York (1977). Basic concepts of relativistic elasticity close this rich in informations chapter. The following synthetic paper can here be cited as completion: G. A. Maugin, Acta Mech. 35, 1-70 (1980; Zbl 0428.73095).

During the last decade important progress has been achieved in the field of a qualitative study of boundary value problems of nonlinear elasticity. This progress is mainly due to the application of the methods of functional analysis. Chapter 6 presents the fundamental results obtained by using the methods of linear and nonlinear functional analysis to the boundary value problems of linear and nonlinear elastostatics and elastodynamics. For nonlinear elasticity the energy criterion, which states that minima of the potential energy are dynamically stable, is discussed and the difficulties concerning this criterion, not yet well settled, are revealed. The results concerning the abstract problem of controlling a semilinear evolution equation are used for the case of a vibrating beam.

The last chapter is intended as an introduction to the static and dynamic bifurcation theory. Nontrivial examples from both the elastostatics and elastodynamics are given.

I think, that to the reader of this book, the No.3, of Vol. 86(1984) of Arch. Ration. Mech. Anal. will be of exceptional value.

Some misprints in the formulas given in the book will easily be perceived by the attentive reader. The book is ingeniously written and very carefully edited. Open problems are also suggested. The necessary completions of concepts from functional analysis and geometry are appended to a pertinent chapter, in the proper place. The authors have successfully combined an up-to-date and advanced approach with accessibility. This impressive book is strongly recommended to researchers and post-graduate students in all branches of continuum mechanics.

The first chapter concerns geometry and kinematics of bodies, their motions and their configurations. All concepts are carefully defined and discussed. Starting from the definition of a simple body as an open set of the Euclidean space \({\mathbb{R}}^ 3\) and its configuration, the fundamental kinematical geometrical objects such as velocity and acceleration are introduced in a general coordinate system. Then the passage to manifolds is straightforward. Particularly various deformation tensors, well-known in continuum mechanics, are formulated in both the absolute and component forms. Relative to rate-of deformation tensors I would like to indicate the recent results by Zh.-H. Guo [Rates of stretch tensors, J. Elasticity, 14, 263-268 (1984)] who offered explicit expressions for the rates of the left and right stretch tensors. Two- point tensors are introduced as multilinear mappings, together with their basic algebraic properties. Covariant differentiation of both tensor fields and two-point tensor fields is concisely discussed. I would like to indicate that in the theory of geometric objects two-point tensors are called ”product tensors” [cf. the two papers by the reviewer, Ann. Polon. Math., 27, 67-72 (1972; Zbl 0244.53013) and 34, 197-200 (1977; Zbl 0403.53002)]. Assuming the existence of a mass density function, the conservation of mass for simple bodies and thin shells is examined.

Having introduced the notion of the Lie derivative it is shown that all objective rates of second-order tensors can be expressed as Lie derivatives. Since the subject of objective rates is still controversial, the authors define what they mean by objective. This new proposal seems to be interesting. Yet it would be advisable to know its consequences for finite deformation theory, including finite plasticity. The considerations about differential forms and the so called Piola transformation close chapter 1. The Piola transformation is an operation relating the material and spatial descriptions of a continuum. I think that the comprehensive paper by R. Hill [Adv. Appl. Mech. 18, 1-75 (1978; Zbl 0475.73026)] can be advised as completion to chapters 1-3.

In the second chapter, devoted to balance principle, primarily the transport theorem and the so-called master balance law and master balance inequality are introduced for both the material and spatial descriptions. Discontinuities (singular surfaces) are also examined. These master laws interrelate two scalar fields and a vector field. Localization leads to well-known local forms of balance equations, or inequalities from thermodynamics. The master balance law and inequality are abstract forms of the basic laws of continuum mechanics. Rigorous discussion of the first and second laws of thermodynamics follows next. In this chapter special attention is paid to classical space-times, the principle of virtual work and covariance of balance of energy. Covariance can be identified with invariance known from the theory of tensor functions. The fundamental laws of continuum mechanics, including the principle of virtual work, formulated on the manifold representative of a continuum are also studied in a paper by P. Rougée [J. Méc. 19, 7-32 (1980; Zbl 0429.73002)]. Balance laws for shells are also briefly discussed by the authors.

The two chapters that follow deal with constitutive relations: chapter 3 with the general constitutive theory and chapter 4 with linearized equations. Constitutive elastic and thermoelastic equations are defined as appropriate mappings. The principle theorems of the constitutive theory of nonlinear elasticity and thermoelasticity are obtained by two approaches, which differ in that the underlying sets of axioms are different. The second set is called ”covariant constitutive axioms” and is given for elasticity and thermoelasticity. After the formulation of constitutive equations, the typical boundary and initial boundary value problems are formulated. Simplifications of constitutive functions result when materials have additional symmetries, for instance isotropy. Linearization of nonlinear elasticity is accomplished by using the implicit function theorem for Banach spaces. For a completion, see Zh.-H. Guo, Ing.-Archiv. 52, 121-129 (1982; Zbl 0487.73041). The cases when the linearization procedure fails are briefly mentioned and the specific case when the linearized operator fails to be surjective is examined abstractly (linearization stability).

Some variational aspects, including the reciprocal theorems, of linear and nonlinear elasticity are examined in chapter 5 by using the contemporary geometrical methods. Thus, the classical linear elastodynamics with homogeneous boundary conditions is treated as linear Hamiltonian system. For the purpose the generalization of the concepts of Hamiltonian mechanics to infinite dimensions is preliminarily presented. To study nonlinear elasticity, the notion of an abstract Hamiltonian system (on Banach spaces) is introduced and examined. As in the finite- dimensional case, a Lagrangian approach, suitably generalized, is also available. Thus constraints, such as incompressibility, can be taken into account. The contemporary approach to Noether’s theorem and its consequences for conservation laws of elasticity are also examined. In my opinion, as an easily readable introduction to Noether’s theorem and conservation laws the following book can be recommended: J. D. Logan, Invariant variational principles, Academic Press, New York (1977). Basic concepts of relativistic elasticity close this rich in informations chapter. The following synthetic paper can here be cited as completion: G. A. Maugin, Acta Mech. 35, 1-70 (1980; Zbl 0428.73095).

During the last decade important progress has been achieved in the field of a qualitative study of boundary value problems of nonlinear elasticity. This progress is mainly due to the application of the methods of functional analysis. Chapter 6 presents the fundamental results obtained by using the methods of linear and nonlinear functional analysis to the boundary value problems of linear and nonlinear elastostatics and elastodynamics. For nonlinear elasticity the energy criterion, which states that minima of the potential energy are dynamically stable, is discussed and the difficulties concerning this criterion, not yet well settled, are revealed. The results concerning the abstract problem of controlling a semilinear evolution equation are used for the case of a vibrating beam.

The last chapter is intended as an introduction to the static and dynamic bifurcation theory. Nontrivial examples from both the elastostatics and elastodynamics are given.

I think, that to the reader of this book, the No.3, of Vol. 86(1984) of Arch. Ration. Mech. Anal. will be of exceptional value.

Some misprints in the formulas given in the book will easily be perceived by the attentive reader. The book is ingeniously written and very carefully edited. Open problems are also suggested. The necessary completions of concepts from functional analysis and geometry are appended to a pertinent chapter, in the proper place. The authors have successfully combined an up-to-date and advanced approach with accessibility. This impressive book is strongly recommended to researchers and post-graduate students in all branches of continuum mechanics.

Reviewer: J.J.Telega

### MSC:

74B20 | Nonlinear elasticity |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74Axx | Generalities, axiomatics, foundations of continuum mechanics of solids |

53A45 | Differential geometric aspects in vector and tensor analysis |

53B50 | Applications of local differential geometry to the sciences |

74F05 | Thermal effects in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |