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Applications of tensor functions in solid mechanics. (English) Zbl 0657.73001

This book contains lectures by J.-P. Boehler, A. J. M. Spencer and J. Betten on the theory of scalar invariants and tensor functions from the viewpoint of solid mechanics, including some applications.
The lectures by Boehler comprise Chapters 1-7 and summarize his results concerning the representation of tensor functions. Namely by introducing a structural tensor (or tensors), the representations of anisotropic scalar, vector and tensor functions result from the corresponding isotropic representations. Application of such a rational approach to the formulation of constitutive relations is general, elegant and sometimes reveals new effects. For instance, more general models of isotropic and anisotropic hardening are obtained by using the tensor function approach. As regarding practical aspects, Boehler’s lectures concentrate on applications of tensor functions to elasticity and plasticity.
The next three chapters (8-10) are by A. J. M. Spencer, who largely contributed to the theory of invariants suitable for applications in continuum mechanics. Chapter 8 presents basic results concerning derivation of (mainly isotropic) polynomial scalar invariants and form- invariant tensor functions. Once one knows how to generate scalar invariants, the tensor functions are derived from the corresponding invariant scalar functions. The next chapter reviews the problem of generating of polynomial scalar invariants and tensor functions in the case of orthotropy and transverse isotropy. Comments on crystal symmetries are also included. The process of reducing tensors to their simplest forms leads naturally to traceless tensors (deviatoric, for second-order tensors). This problem is briefly discussed. Formulas for the determination of the number of linearly independent invariants are adduced without derivation. Chapter 10 discusses the problem of the formulation of the constitutive relations for elastic and plastic solids in the presence of internal constraints such as incompressibility and inextensibility.
The last four chapters by J. Betten deal with scalar invariants and tensor functions involving fourth order tensors. Particularly, the integrity basis is constructed for the forth order tensor by using methods known for second order tensors. To describe damage phenomena in initially anisotropic materials the following tensor relation is discussed \(d_{ij}=f_{ij}(\sigma_{kl},\omega_{kl},A_{klpq})\). Certainly, such a general relation is practically unmanageable, therefore simplified representations are also investigated. The whole Chapter 13 is concerned with an extension of the polynomial interpolation method to tensor-valued functions. As an application a tensorial generalization of Norton’s creep law is studied. The chapter on potential flow rules for isotropic and anisotropic materials ends up the book.
The reviewer hopes that the book will be very useful for solid mechanicians wanting to get acquainted quickly with the fundamentals of the theory of scalar invariants and tensor functions. The material presented is mainly of algebraic character and analytical problems such as regularity and convexity deserve further studies. For instance, more extensive investigations in the vein of the paper by M. D. P. M. Marques and J. J. Moreau [Isotropie et convexité dans l’espace des tenseurs symétriques, Sémin. Anal. Convexe, Univ. Sci. Tech. Languedoc 12, No.1, Exp. No.6, 9 p. (1982)] would be very desirable. The list of references is not exhaustive and at least the following book is worth of being mentioned: T. A. Springer, Invariant Theory (1977; Zbl 0346.20020).
Reviewer: J.J.Telega

MSC:

74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74-06 Proceedings, conferences, collections, etc. pertaining to mechanics of deformable solids
15A72 Vector and tensor algebra, theory of invariants
74C99 Plastic materials, materials of stress-rate and internal-variable type
74E10 Anisotropy in solid mechanics
74A20 Theory of constitutive functions in solid mechanics
20G45 Applications of linear algebraic groups to the sciences

Citations:

Zbl 0346.20020