Material inhomogeneities in elasticity.

*(English)*Zbl 0797.73001
Applied Mathematics and Mathematical Computation. 3. London: Chapman & Hall. xii, 276 p. (1993).

The central purpose of the book is the derivation of mechanical balance laws suitable for the analysis of the dynamics of deformation of hyperelastic materials containing inhomogeneities such as point defects, dislocations and cracks. The presentation is based on the fact that the classical mechanics of these materials may be formulated using variational methods: virtual variations of the displacement vector of a particle give rise to physical force. On the other hand material forces are generated by virtual displacements of the particle position vector in the body manifold, thus providing the basis for the study of inhomogeneities, which are space variations in material properties.

In chapter 4, taking Ericksen’s fully material covariant derivation of the equations for static balance of a hyperelastic material subject to conservative body force as a starting point, a corresponding equation for pseudomomentum balance is derived which involves a tensor, named the Eshelby stress tensor, and an inhomogeneous force vector accounting for inhomogeneities in the material density and material properties. The Eshelby stress tensor is composed of the strain and kinetic energies and a term involving the second Piola-Kirchoff stress tensor. Earlier in chapter 2, the linearized forms of the Eshelby tensor and inhomogeneous force vector are introduced successively to account for the ‘force’ on a singular defect and material inhomogeneity in that order. Pseudomomentum may be interpreted in terms of a material velocity the physical significance of which is illustrated for a deforming string, and in its linearised form is shown to equate to the crystal momentum of crystal physics. Based on a 4-dimensional space-time formulation of the pseudomomentum balance equation, two other balance laws are derived by taking the trace and skew parts of its tensor product with the position vector in this space. Equations of Eischen and Hermann (loc. cit.) derived for small strains are shown to approximate these.

As indicated above, the earlier chapters are used to motivate while chapter 3 lays down the foundations of the standard nonlinear theory of hyperelasticity. In some respects these chapters contain unnecessary philosophical diversions and generality. For example, there is no need for the discussion on the ‘principle of objectivity’ to derive the dependence of the elastic strain energy in chapter 3; obviously it should be independent of rigid body rotations.

Chapter 5 presents variational formulations of the theory in a 4- dimensional space-time setting – electro-magnetic effects are considered in Chapter 8 – and also enlarges the discussion to include ‘second gradient’ theories of elastic response. Chapter 6 concerns itself with the differential geometric properties of the material manifold. Chapter 7 deals with material inhomogeneities in relation to brittle fracture, i.e. the study of cracks and criterion for their growth. To this end generalized Reynolds and Green-Gauss theorems are constructed, and results of classical small strain theory are extended to the exact theory. chapter 8 considers material forces associated with inhomogeneities in electro-magnetic elastic materials. Chapter 9 deals with the pseudomomentum of quasi-particles such as photons and phonons, and includes conservation laws for soliton theory. Chapter 10 tentatively extends the ideas to dissipative materials using a constitutive theory depending on internal (state) variables and a local form of Clausius- Duhem (dissipation) inequality.

The whole is based heavily on past researches of the author and represents a considerable body of work, being very broad in its ramifications. The first seven chapters are the most basic and in themselves provide sufficient material for a most stimulating discourse. Occasionally the language becomes convoluted and there are lapses in the use of the required font for mathematical symbols.

In chapter 4, taking Ericksen’s fully material covariant derivation of the equations for static balance of a hyperelastic material subject to conservative body force as a starting point, a corresponding equation for pseudomomentum balance is derived which involves a tensor, named the Eshelby stress tensor, and an inhomogeneous force vector accounting for inhomogeneities in the material density and material properties. The Eshelby stress tensor is composed of the strain and kinetic energies and a term involving the second Piola-Kirchoff stress tensor. Earlier in chapter 2, the linearized forms of the Eshelby tensor and inhomogeneous force vector are introduced successively to account for the ‘force’ on a singular defect and material inhomogeneity in that order. Pseudomomentum may be interpreted in terms of a material velocity the physical significance of which is illustrated for a deforming string, and in its linearised form is shown to equate to the crystal momentum of crystal physics. Based on a 4-dimensional space-time formulation of the pseudomomentum balance equation, two other balance laws are derived by taking the trace and skew parts of its tensor product with the position vector in this space. Equations of Eischen and Hermann (loc. cit.) derived for small strains are shown to approximate these.

As indicated above, the earlier chapters are used to motivate while chapter 3 lays down the foundations of the standard nonlinear theory of hyperelasticity. In some respects these chapters contain unnecessary philosophical diversions and generality. For example, there is no need for the discussion on the ‘principle of objectivity’ to derive the dependence of the elastic strain energy in chapter 3; obviously it should be independent of rigid body rotations.

Chapter 5 presents variational formulations of the theory in a 4- dimensional space-time setting – electro-magnetic effects are considered in Chapter 8 – and also enlarges the discussion to include ‘second gradient’ theories of elastic response. Chapter 6 concerns itself with the differential geometric properties of the material manifold. Chapter 7 deals with material inhomogeneities in relation to brittle fracture, i.e. the study of cracks and criterion for their growth. To this end generalized Reynolds and Green-Gauss theorems are constructed, and results of classical small strain theory are extended to the exact theory. chapter 8 considers material forces associated with inhomogeneities in electro-magnetic elastic materials. Chapter 9 deals with the pseudomomentum of quasi-particles such as photons and phonons, and includes conservation laws for soliton theory. Chapter 10 tentatively extends the ideas to dissipative materials using a constitutive theory depending on internal (state) variables and a local form of Clausius- Duhem (dissipation) inequality.

The whole is based heavily on past researches of the author and represents a considerable body of work, being very broad in its ramifications. The first seven chapters are the most basic and in themselves provide sufficient material for a most stimulating discourse. Occasionally the language becomes convoluted and there are lapses in the use of the required font for mathematical symbols.

Reviewer: J.Dunwoody (Belfast)

##### MSC:

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

74B20 | Nonlinear elasticity |

74R99 | Fracture and damage |

74A60 | Micromechanical theories |

74M25 | Micromechanics of solids |