by R. M. Christensen, Oxford University, New York, 2013


The formulation of a physically based mathematical failure characterization of materials is a long-standing problem of great complexity. The author takes the point of view that failure is governed by a constitutive theory, which should be formulated next to the theories of elasticity and plasticity. In this book, such a constitutive theory is derived for a homogeneous isotropic material in terms of a paraboloid in principal stress space. The theory covers the whole range from perfectly brittle to perfectly ductile material. The theory is defined by two independent constitutive properties: the one-dimensional strengths for tensile and compression, T and C, respectively. Failure is defined as the crossing of the effective limit of linearly elastic behavior. The theory is supplemented in later chapters by discussions of fracture mechanics, effective failure after plastic flow, and limited extensions to anisotropic and inhomogeneous materials, as well as probabilistic failure and probabilistic life prediction.

Before a detailed discussion of the derivation of the new theory, the author examines the difficulties and limitations of the four most often used current failure criteria, those of von Mises, Tresca, Coulomb–Mohr, and Drucker–Prager. In his view, only the Mises criterion is of lasting importance.

Chapter 4, where the author derives his failure theory, is the most important chapter of the book. The general isotropic failure behavior is determined by the spectrum of T/C values, which captures the entire range from brittle to ductile behavior between T/C = 0 and T/C = 1. The theory is based on an expansion of the elastic energy in terms of the first and second stress invariants. The relevant constants are determined in terms of T/C by considering appropriate limit cases. An elegant constitutive equation emerges in terms of the principal stresses. In principal stress space, the geometric form of this fracture criterion is that of a paraboloid. In the limit of perfect ductility, at T/C = 1, the expression reduces to the Mises criterion. On the brittle side (i.e., T/C < 0.5), additional considerations have been introduced, related to failure due to fracture. Here, it is assumed that, in a homogeneous material, cracks generate and propagate normal to the direction of maximum normal stress. The proposed additional condition is that the largest principal stress should be smaller than T. Whichever of the two failure criteria—the polynomial invariants form or the fracture mode form—is the most limiting is then the controlling failure condition at that stress state. For T/C < 0.5, the planes defining the fracture condition cut slices out of the paraboloid. Cases corresponding to the failure paraboloid sliced by the fracture failure planes are shown and discussed for widely different classes of materials in Chapter 5. Chapter 6 presents comparisons of the new theory with three sets of experimental data. It is not surprising that good agreement is found with the data for ductile metals obtained by Taylor and Quinney, since these data already showed good agreement with the Mises criterion but not with the Coulomb–Mohr and the Tresca criteria. Good agreement is also obtained for biaxial failure data on iron and for triaxial failure data for Blair dolomite. The Coulomb–Mohr criterion gives poor comparisons for the latter two sets of experimental data. The reviewer hopes that further experimental validation of the new theory will be forthcoming.

Chapter 7 provides further insight by discussion of applications of the theory to very ductile polymers, brittle polymers, glasses, ceramics, minerals, and geomaterials. Chapter 8 is concerned with the important question of the ductile/brittle transition. The definition of the yield stress and the failure stress in a one-dimensional test, which is a crucial matter for the new theory, is discussed in Chapter 9. The author would prefer to define the yield stress in a rational manner at the point of the stress-strain curve where the second order derivative of the stress with respect to strain is a maximum. After some reflection, he concludes, however, that this point would be difficult to determine from experimental data, and he settles for the stress where the strain is 5% higher than that of the linear elastic projection. The effective failure stress is obtained on the basis of an energy consideration.

The field of fracture mechanics is briefly discussed in Chapter 10 as a separate field parallel to failure theory. As the author sees it, fracture mechanics is relevant when a material contains cracks and it is concerned with the growth of cracks, while failure theory applies when a material may contain very small cracks but can be considered as basically homogeneous, and fracture takes place in a field of high normal stress. The difference is illustrated by an example for fracture mechanics (an edge crack in a tension field) and one for failure theory (the stress concentration for a rigid spherical inclusion in a one-dimensional stress field). It is concluded that both approaches are needed to determine the performance of any material in any application.

Chapters 13 and 14 are concerned with micromechanics and nanomechanics failure analysis, respectively. The author points out that an understanding of failure mechanism on these small scales is vital to understanding of failure at the macroscale but that it is very difficult to quantitatively predict macroscale failure from smaller scale results. Chapter 15 deals with damage, cumulative damage, creep, and fatigue failure. Residual strength, life prediction, and residual life are briefly touched upon. Chapter 16, the last chapter, discusses probabilistic failure and probabilistic life prediction. The Weibull distribution and a power law representation are used for static and dynamic creep rupture under constant stress and constant stress rate. An interesting feature of this chapter is probabilistic failure results based on a long-term program of creep rupture testing carried out at Lawrence Livermore National Laboratory.

This is a very interesting book. The reviewer believes that it will generate discussion and controversy, particularly from readers with a materials science orientation but also from structural engineers. The book is very well written with a relatively large ratio of text to equations. The author often repeats his main points and equations, which makes the book easier to read. Every chapter is concluded with a set of problem areas for further study. The author has not shied away from stating his opinion of earlier work. He states some unusual ideas, such as a comparison of failure in solids to turbulence in fluids, and a ductility scale for the elements, inspired by the periodic scale of elements, which relates the Poisson’s ratio to a measure of ductility from perfect ductility to perfect brittleness (Chapter 14). He also speculates on the correlation of the ductility rank ordering of the elements to the numbers of shells containing electrons in the atomic configurations.

This stimulating book was a pleasure to read. It will be of great interest to mature graduate students and researchers in solid mechanics.


Departments of Civil and Environmental Engineering and Mechanical Engineering, Northwestern University, Evanston, IL 60208.