On the state of pure shear (original) (raw)
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Journal of Elasticity, 2007
A result on pure shear provides the motivation for the determination of some new general results relating real second order Cartesian tensors.
PhysICal ComPonenTs oF Tensors PhysICal ComPonenTs oF Tensors PhysICal ComPonenTs oF Tensors
K24389 Illustrating the important aspects of tensor calculus, and highlighting its most practical features, Physical Components of Tensors presents an authoritative and complete explanation of tensor calculus that is based on transformations of bases of vector spaces rather than on transformations of coordinates. Written with graduate students, professors, and researchers in the areas of elasticity and shell theories in mind, this text focuses on the physical and nonholonomic components of tensors and applies them to the theories. It establishes a theory of physical and anholonomic components of tensors and applies the theory of dimensional analysis to tensors and (anholonomic) connections. This theory shows the relationship and compatibility among several existing definitions of physical components of tensors when referred to nonorthogonal coordinates. The book assumes a basic knowledge of linear algebra and elementary calculus, but revisits these subjects and introduces the mathematical backgrounds for the theory in the first three chapters. In addition, all field equations are also given in physical components as well. Comprised of five chapters, this noteworthy text: • Deals with the basic concepts of linear algebra, introducing the vector spaces and the further structures imposed on them by the notions of inner products, norms, and metrics • Focuses on the main algebraic operations for vectors and tensors and also on the notions of duality, tensor products, and component representation of tensors • Presents the classical tensor calculus that functions as the advanced prerequisite for the development of subsequent chapters • Provides the theory of physical and anholonomic components of tensors by associating them to the spaces of linear transformations and of tensor products and advances two applications of this theory Physical Components of Tensors contains a comprehensive account of tensor calculus , and is an essential reference for graduate students or engineers concerned with solid and structural mechanics.
Tensors The mathematics of Relativity Theory and Continuum Mechanics Anadijiban Das
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Tensor functions and constitutive relationships for two isotropic materials
Bulletin of the Polish Academy of Sciences, Technical Sciences
There are two aims of this contribution. Firstly, usilrg the representation theory of isotropic tensor functions, there is derived a general form of the constitutive reiationship for perfectly locking materials. The homogeneity condition ol degree zero imposed on the iocking behaviour permits the general form of the locking locus to be obtained. Some particular cases are a.lso studied. Secondly, regarding the dissipa,tion density dependent on hydrostatic pressure, the theory proposed by Sawczuk and Stutz [1] is generalized, Consequently, incompressible bełraviour of isotropic perfectly plastic materials obeyirrg plessule sensitive yield condition is described. 1. Introduction. The constitutive relations for perfectly 1ocking matelials have been first ploposed by Prager [2]. Counterpalts of the limit analysis theorems wele foTmulated in [3]. Next, more ligolous mathenatical studies of not necessalily perfectiy locking solids have been undertaken in [4-6]. Our aim here is to derive the general form of the constitutive equation (1) provided that (2) is satisfled. Condition (2) expresses the fact that Eq. (1) should not depend upon the time scale. Locking behaviour can strongly depend upon the density p of the material, cf. [7-9]. Therefore we assume that the tensor function C depends not only ". t, but also on p. The representation theory of tensor functions is a convenient tool for the study of trq. (1) satisfying condition (2). The derived locking law, is in general, not associated with the corresponding locking condition. The associated locking law is obtained under an additional condition. Several particular iocking ]oci are proposed.
Eigenvalue Problem for Tensors of Even Rank and its Applications in Mechanics
Journal of Mathematical Sciences, 2017
In this paper, we consider the eigenvalue problem for a tensor of arbitrary even rank. In this connection, we state definitions and theorems related to the tensors of moduli C2p(Ω) and R2p(Ω), where p is an arbitrary natural number and Ω is a domain of the n-dimensional Riemannian space R n. We introduce the notions of minor tensors and extended minor tensors of rank (2ps) and order s, the corresponding notions of cofactor tensors and extended cofactor tensors of rank (2ps) and order (N −s), and also the cofactor tensors and extended cofactor tensors of rank 2p(N −s) and order s for rank-(2p) tensor. We present formulas for calculation of these tensors through their components and prove the Laplace theorem on the expansion of the determinant of a rank-(2p) tensor by using the minor and cofactor tensors. We also obtain formulas for the classical invariants of a rank-(2p) tensor through minor and cofactor tensors and through first invariants of degrees of a rank-(2p) tensor and the inverse formulas. A complete orthonormal system of eigentensors for a rank-(2p) tensor is constructed. Canonical representations for the specific strain energy and determining relations are obtained. A classification of anisotropic linear micropolar media with a symmetry center is proposed. Eigenvalues and eigentensors for tensors of elastic moduli for micropolar isotropic and orthotropic materials are calculated.
On some problems of tensor calculus. I
Journal of Mathematical Sciences, 2009
Basic definitions of linear algebra and functional analysis are given. In particular, the definitions of a semigroup, group, ring, field, module, and linear space are given [1-3, 6]. A local theorem on the existence of homeomorphisms is stated. Definitions of the inner r-product, local inner product of tensors whose rank is not less than r, and of local norm of a tensor [22] are also given. Definitions are given and basic theorems and propositions are stated and proved concerning the linear dependence and independence of a system of tensors of any rank. Moreover, definitions and proofs of some theorems connected with orthogonal and biorthonormal tensor systems are given. The definition of a multiplicative basis (multibasis) is given and ways of construction bases of modules using bases of modules of smaller dimensions. In this connection, several theorems are stated and proved. Tensor modules of even orders and problems on finding eigenvalues and eigentensors of any even rank are studied in more detail than in [22]. Canonical representations of a tensor of any even rank are given. It is worth while to note that it was studied by the Soviet scientist I. N. Vekua, and an analogous problem for the elasticity modulus tensor was considered by the Polish scientist Ya. Rikhlevskii in 1983-1984.
On the representation theorem for linear, isotropic tensor functions
Journal of Elasticity, 1995
The well-know representation theorem for the elasticity tensor C of an isotropic body shows that C[E] = 2#E + ~ tr(E)I (1) for all symmetric tensors E, where tr(E) denotes the trace of E and I is the identity tensor. This theorem is actually a special case of a classical result (cf. e.g. [Je 31, Chapter 7]) on linear, tensor-valued mappings that are isotropic, i.e. C[QHQ T] = QC[HjQ T for all tensors H in the domain of C and all orthogonal tensors Q, where Q~ denotes the transpose of Q.
Restricted Invariants on the Space of Elasticity Tensors
Mathematics and Mechanics of Solids, 2005
Abstract: A linear function defined on the space of elasticity tensors is a restricted invariant under a group of rotations G if it has an invariant restriction to a proper subspace which is larger than the set left fixed by the action of G itself. A necessary and sufficient condition for a function to be a restricted invariant is given using concepts related with isotypic decomposition, Haar integration and G -dependence. The result is applied to characterize isotropic and transversely isotropic restricted invariants.