3R2. Mechanics in Material Space: With Applications to Defect and Fracture Mechanics. - R Kienzler (Univ Bremen, Postfach 330440, Bremen, D-28334, Germany) and G Herrmann (Stanford Univ, Ortstrasse 7, Davos Platz, CH-7270, Switzerland). Springer-Verlag, Berlin. 2000. 298 pp. ISBN 3-540-66965-5. $54.00.
Reviewed by HW Haslach Jr (Dept of Mech Eng, Univ of Maryland, College Park MD 20742-3035).
The mechanics of bodies with defects is typically described by material forces which are negative gradients of an energy or by related path independent integrals. The goal of this book is to draw a parallel between classical linear elastic mechanics and the mechanics of bodies with defects, called here Mechanics in Material Space, which describes a mathematical model for the behavior of material defects such as inclusions, voids, cracks, or dislocations. For example, the Cauchy stress and the Eshelby tensor are viewed as corresponding concepts, as are Newtonian force and derivatives of an energy such as the energy release rate.
To construct the model for mechanics in material space, the authors first introduce the mathematical ideas needed to describe conservation laws. Because their theory is applied only to linear elastic materials, a short review of linear elasticity is given in which the conservation laws are derived from a Lagrangian. This analysis is based on infinitesimal transformations of the independent and dependent variables of state space and the Noether theorem that if the action is invariant under an infinitesimal transformation, then a conservation law exists. The possible transformations depend on the Lagrangian through the Euler-Lagrange equation. A conservation law is the statement that the divergence of a vector valued function of several variables is zero. A major emphasis of this work is to show explicitly how to compute conservation laws from a given Lagrangian. Alternatively, their idea of a neutral action may be used in cases in which the mechanics is described by a system of differential equations, perhaps representing dissipation. The Euler-Lagrange equations are applied to an action with zero variation constructed from the system of differential equations.
The path independent integrals of defect analysis are obtained from conservation laws, each based on a particular infinitesimal transformation. The construction produces the J, L, and M integrals viewed as objects describing material mechanics. The authors then give a physical interpretation of the Eshelby tensor, as well as its eigenvalues and eigenvectors.
The J integral is also established for inhomogeneous functionally gradient (graded) materials (FGM), those whose properties change smoothly in a given direction, by recourse to a conservation law involving translational symmetries. The computation is analogous to that for a homogeneous material.
Linear elastodynamics fracture theory is developed by constructing conservation laws by three alternative methods. Here, a conservation law in material space and the duality with classical mechanics implies that the stress equations of motion are related to conservation of mass. Application of linear elastodynamics is made to wave motion.
Dissipative systems, those without Lagrangians, are discussed by example, including diffusion, the nonlinear wave equation, linear viscoelasticity based on the archaic spring and dashpot models. The creep C* integral is obtained from a potential for constant strain rate. The primary tool for the construction of conservation laws is the neutral action method.
Conservation laws for coupled systems, such as piezoelectric materials, linear thermoelasticity, and porous materials by analogy with thermoelasticity, are constructed from actions which superpose the subsystems. The neutral action method is used in the case of time-dependent thermoelasticity.
Finally the techniques are applied to the strength of materials description of bars, shafts, beams, and plates and shells with defects. In these cases, the construction begins with a Lagrangian appropriate to strength of materials, rather than elasticity. Shells are more difficult due to the curvature and are not fully developed.
This book, in a clear presentation, succeeds in establishing a foundation for the mechanics of linear elastic bodies with defects, or mechanics in material space, and in clarifying the parallels with classical linear elastic mechanics. A concise table listing the parallels is given in the Introduction. This fundamental approach in terms of conservation laws is an improvement over dealing with fracture and defects in the classical almost ad hoc manner of defining the path-independent integrals and forces on the defects. Mechanics in Material Space: With Applications to Defect and Fracture Mechanics should be useful to those engaged in research on bodies with defects. It could also be a text for a graduate course on this topic or at least used as a supplement. Although no problems are given, many examples are provided of constructing the conservation laws. Engineering research libraries should own this book.
