Continuum Mechanics and Theory of Materials

Abstract

3R1. Continuum Mechanics and Theory of Materials. - P Haupt (Inst of Mech, Univ of Kassel, Monchebergstr 7, Kassel, 34109, Germany). Springer-Verlag, Berlin. 2000. 583 pp. ISBN 3-540-66114-X. $82.00.Reviewed by JL Wegner (Dept of Mech Eng, Univ of Victoria, Eng Office Wing, Room 537, Victoria BC, V8W 3P6, Canada).The author attempts to portray the ideas and general principles of the theory of materials within the framework of phenomenological continuum mechanics. It is a well-written rigorous mathematical treatment of classical continuum mechanics and deals with such concepts such as elasticity, plasticity, viscoelasticity, and viscoplasticity in nonlinear materials. The volume consists of 13 chapters. Chapter 1 covers kinematics, that is the geometry of motion and the deformation of material bodies. The outline follows the script of most texts on classical continuum mechanics—the concepts of material bodies and the material derivative are introduced in Euclidean space. The deformation gradient tensor is introduced, and its physical meaning is described by the transformation of material line, surface, and volume elements. Similar to most treatments on classical continuum mechanics, the appropriate strain and stretch tensors are described. However, the introduction of convective coordinates in this volume is a departure from most treatises on classical continuum mechanics. In this volume, a treatment of strain rates in convective coordinates is provided with the argument that the choice of convective coordinates not only affords a deeper understanding of the strain tensors but also of their strain rates. Chapter 1 finishes with a section on incompatible configurations, that is when a material body can identify a configuration with a non-Euclidean space. Chapter 2 develops the classical balance relations of mechanics, for example: balance of mass in spatial and referential form, conservation of linear and rotational momentum in spatial and referential form. Here, the Cauchy, first Piola-Kirchhoff, weighted Cauchy, and the second Piola-Kirchhoff stress tensors are introduced, as well as their physical significance. This treatise finishes with the balance of mechanical energy and the balance of virtual work.A comprehensive theory of phenomenological material behavior, based on the general principles of thermomechanics, is developed and presented in this volume. This general theory is expounded in the remaining chapters, beginning in Chapter 3 where the classical balance relations of thermodynamics are presented. In Chapter 4, the terms frame of reference, change of frame, and objectivity are clarified, in preparation for the discussion of the constitutive equations in chapter five. Classical constitutive relations are presented in Chapter 5, that is, equations defining the perfect fluid, the linear-viscous fluid and the linear-elastic isotropic solid. Two extensions to the model of linear elasticity are also included in this treatise, namely the theories of linear viscoelasticity and plasticity. Chapter 6 contains the results of experimental testing of different materials such as steel and elastomers. The experimental results provide invaluable insight to the reader when compared to the classical theories of continuum mechanics. The classical constitutive models do reflect significant aspects of the material behavior observed. However, there are considerable discrepancies which cannot be resolved within the context of classical theories presented thus far in the volume. Hence, the motivation for a comprehensive theory of phenomenological material behavior based on the general principles of thermomechanics. The aim of material theory is to provide general principles and systematic methods for constructing mathematical models suitably representing the individual properties of material bodies. The general theory of material behavior, as it is developed in Chapter 7, is mainly due to W Noll. In Chapter 9, the constitutive relations for isotropic elastic and isotropic hyperelastic (compressible, and incompressible) solids are derived. Of interest, the one-dimensional stress-strain curves for the Mooney-Rivlin and Neo-Hookean models are plotted, along with a discussion of their limitations for applications to large deformations. What separates this volume from most on continuum mechanics is the treatise in Chapter 9 on constitutive relations for anisotropic hyperelastic solids. Chapter 10 considers nonlinear viscoelasticity, and Chapter 11 covers plasticity theory. Chapter 12 covers viscoplastisticity, which depicts rate-dependent material behavior with equilibrium hysteresis phenomena. Constitutive models, for all of these types of materials, in thermomechanics is discussed in Chapter 13. The author achieves his goals of presenting, in a rigorous manner, the ideas and general principles of the theory of materials within the framework of phenomenological continuum mechanics, providing the reader general theories of material behavior from which a reader can select the constitutive model that applies best. Continum Mechanics and Theory of Materials will be invaluable to advanced graduate students of materials science in engineering and in physics.