9R14. Scaling of Structural Strength. - ZP Bazant (Dept of Civil Eng and Mat Sci, Northwestern University, Evanston IL 60201). Hermes Sci Publ, Paris. Distributed in USA by Taylor & Francis Publ, New York NY. 2002. 280 pp. ISBN 1-56032-984-X. $95.00.

Reviewed by G Lewis (Dept of Mech Eng, Univ of Memphis, 316 Eng Bldg, Memphis TN 38152).

It is widely acknowledged that the subject of scaling or size effects (particularly, in components and structures fabricated from quasi-brittle materials) is, arguably, one of the most important ones in solid mechanics. This is because, from a theoretical perspective, the analysis is very complex, even abstruse, in places. In practical terms, scaling has far-reaching implications for both the design and the cost of prototypes of these components and structures, ranging from reinforced concrete columns on highways and fine crystal wine glasses to silicon nitride machining tool inserts and ice sculptures. It is against this background that this book will be judged; that is, first, the extent to which the author succeeds in mapping out all the theoretical concepts that are relevant to scaling and, second, the extent to which the material covered in the book can aid structural designers. How the book fares on these scores will be discussed following the descriptions of its contents, which are organized into 11 chapters.

In Chapter 1 (Introduction), the author provides a sweeping summary of the field of scaling, beginning with two statements that are both catchy and tone-setting; namely, that scaling is central to all physical systems, and that an understanding of any theory describing any physical system exists only when scaling of that system is understood. The author then goes on to make comments about the nature of the problem and to present a brief review of the history of scaling (starting with Leonardo da Vinci’s observation about the relationship between the length and strength of cords, continuing with the development of the Weibull theory, and ending with some recent developments in quasi-brittle materials, as exemplified by work on concrete). In this chapter, the three basic theories of scaling in solid mechanics and the four indirect size effects are summarized, and a host of other topics are covered, namely, power scaling in the absence of a characteristic length, transitional size effect bridging power laws for different scales, and deductions of the size effect from dimensional analysis (by, for example, converting the mathematical formulation of the boundary value problem to dimensionless form), and stability of structures and the size effect (for example, in the case of beams, the difference in size effect under elastic buckling conditions compared to when they are on an elastic foundation).

Among the topics covered in Chapter 2 (Asymptotic analysis of size effect), are asymptotic analysis of size effect in structures with notches or large cracks (including a comparison of large- and small-size asymptotic expansions of the size effect), energistic size effect law and its asymptotic matching character (including the role of the brittleness number in linear elastic fracture mechanics, LEFM, scaling), use of Rice’s J-integral for asymptotic scaling analysis, identification of fracture parameters from size effect tests on concrete specimens, comparison of size effect law (as obtained from tests conducted on Indiana limestone, carbon-epoxy fiber composite, silicon oxide, and sea ice specimens), an examination of whether a universal size effect law exists (that is, the extent to which all expansions, such as the large-size expansion for short cracks, could be made to yield a single expression that matches all the asymptotic cases), interaction diagram (or failure envelope) for the case of many loads, and size effect on approach to zero size (including discussion of two physically meaningful boundary value problems of elasticity for a body with a crack).

As the title of Chapter 3 (Randomness and disorder) implies, relevant statistical concepts are described. The chapter opens with a summary of the tenets of the Weibull theory and a presentation of what the author calls “serious objections” (seven, in all) to the applicability of the Weibull theory to quasi-brittle structures. Following this are treatments of a number of relevant topics, such as nonlocal probabilistic theory of size effect, energistic-statistical formula for size effect for failures at crack initiation, and the size effect ensuing from J-integral for randomly located cracks. The chapter ends with detailed critical examinations of the roles of fracture fracticality and lacunar fracticality of microcracks in size effect.

In Chapter 4 (Energetic scaling for sea ice and concrete structures), a wide range of topics is covered, these being scaling of the fracture of floating sea ice plates (among the topics covered are thermal bending fracture, numerical simulation and approximate analytical solution of vertical penetration, and the force applied by moving ice on a fixed structure), size effect of softening in beams and plates, steel-concrete beams and the compound size effect, size effect provisions in Japan Concrete Institute, CEB (European), DIN (German), and ACI (United States) design codes, six reasons why use of an excessive dead load factor as a substitute for size effect, in a design code, is inadequate. The chapter ends with a treatment of no-tension design of concrete structures or rock from the perspective of size effect.

The topics covered in Chapter 5 (Energetic scaling of compression fracture and further applications to concrete, rock, and composites) are propagation of damage band in components and structures fabricated from these materials (under compressive stress), size effect in reinforced concrete columns, the fracturing truss (strut-and-tie) model for shear failure of reinforced concrete beams, experimental and analytical results for the breakout of boreholes in rocks, asymptotic equivalent LEFM analysis for cracks with residual bridging stress, and the applicability of compression kink bands and the effect of orthotropy, in the case of in fiber-reinforced composite materials.

In Chapter 6 (Scaling via J-integral, with application to kink bands in fiber composites), the author presents very useful summaries of the following topics: J-integral analysis of size effect on kink band failures (in, for example, single-edge notched carbon fiber-reinforced PEEK test specimens), calculations of the first and second parts of Rice’s J-integral, the derivation of an expression for the nominal strength of a specimen that contains a long kink band, failure of notched specimens containing kink bands, and comparison of the results of size effect tests of kink band failures in quasi-isotropic and orthotropic carbon fiber-reinforced PEEK laminates.

The material in Chapter 7 (Time dependence, repeated loads and energy absorption capacity) is presented in a summarized manner. The topics covered are the impact of the two causes of fracture growth in materials (the viscoelasticity of the material, in the case of polymers, and the time dependence of the bond ruptures that cause fracture, in, for example, rocks) on the scaling of fracture, the need to correct fatigue crack growth test results for specimen size effect, the viscosity-induced size effect, the relationship between ductility of a structure and its energy absorption capacity, and the influence of size effect on structural ductility.

Among the topics covered in Chapter 8 (Computational approaches to quasibrittle fracture and its scaling) are the use of eigenvalue analysis for calculating the size effect (when the cohesive, or fictitious, crack model is used), microplane constitutive model, basic features of the suite of numerical methods which may be used in the simulation of damage localization, fracture propagation size effect (for example, R-curve, finite element analysis, FEA, and element-free Galerkin models), nonlocal damage models, and two key steps to be taken, when performing FEA, to avoid the problems of spurious localization of the damage front into a band of vanishing width and spurious mesh sensitivity of solutions.

The focus in Chapter 9 (New asymptotic scaling analysis of cohesive crack model and smeared-tip method) is the extent to which analysis of asymptotic scaling properties of the cohesive crack model is informed by a new variant of the smeared-tip approach (the “K-version”). The chapter begins with a summarized account of the history of both the cohesive crack model and the smeared-tip method. Then, the limitations of the cohesive crack model are detailed. Following this, the application of the K version method to asymptotic scaling analysis for four cases is described. These cases are positive geometry with notch or stress-free initial crack, for fixed K-density (Case 1), fracture initiation from a smooth surface, for fixed K-density (Case 2), Cases 1 and 2 for standard cohesive crack model or first three terms of asymptotic expansion (Case 3), and negative-positive geometry transition (Case 4). Also covered in this chapter are small-size asymptotics of the cohesive crack model; scaling of cohesive fracture (using the nonlocal LEFM approach), the use of the Dirichlet series expansion in a broad-range size effect law, (for Case 1, as given above), and the size effect law in both small- and large-size asymptotics. The chapter ends with a very useful summary of the main points (11, in all) covered in the chapter.

In Chapter 10 (Size effect at continuum limit on approach to atomic lattice scale), a number of concepts on scaling that are based, essentially, on microscaling the role of plasticity in the theory of metal plasticity, are covered, among which are the definition of corresponding nominal stresses, an approximate asymptotic-matching formula for the dependence of nominal stress on size (from which the transitional size is estimated), and micro-torsion and Rockwell micro-hardness tests and results.

In Chapter 11 (Future perspectives), the author reminds the reader that, although much is now known about damage mechanics, there is a vast expanse of unknowns, and, in an effort to close this knowledge gap, he identifies 13 areas for future research, including the micromechanical basis of softening damage, scaling problems in geophysics, and the mixture of extreme value statistics and the scaling of loads, for a given extremely low probability of failure.

This book has a number of several attractive features. The synthesis of research findings from a large array of studies (the bibliography comprises some 600 titles!), spanning several decades, is impressive. However, one should point out that many of the more recent citations are articles by the author and his collaborators. The reading of the book is greatly facilitated by the fact that, in each chapter, the material is assembled into many sections and sub-sections. The equations presented were selected judiciously to enhance understanding of the underlying concepts. In all cases, modifications of the quoted equations are set out very carefully. The potential for the use of scaling effect in emerging areas, notably components and structures engineered using nanotechnology techniques, is acknowledged through a very impressive coverage, in Chapter 10, of continuum mechanics at the atomic lattice scale. Throughout the book, the writing is consistently lucid and well paced (expansive, where appropriate, and economical in other places). The physical layout of the book is aesthetic; in particular, the diagrams are well drawn and fully annotated.

Regrettably, the book does have three major flaws and one minor one. The first major flaw is that size effect in metallic structures is not covered at all. (The author’s rejoinder may be that the focus of the book is the analysis of size effect in components and structures fabricated from quasi-brittle materials, for which the effect is both acute and complicated. However, this path of defense, if it is taken, may be insufficient given the title of the book and the fact that the limit of the coverage to the aforementioned materials is not explained explicitly in the Introduction chapter. Furthermore, there are many cases of practical importance of ductile materials that undergo transformation to brittleness under certain environmental conditions; vide the ductile-brittle transition phenomenon in mild steel.) The second main flaw is that only in a few sections (such as that dealing with size effect tests of kink band failures in carbon fiber-reinforced epoxy laminates) does the author present and comment on comparisons between experimental and theoretical results. Other than in Chapter 9, there are no succinct summaries of the main points of the material covered in the chapter. Such summaries would have been very useful given the denseness of the material in nearly all chapters.

All things considered, the author is to be congratulated on having written a first-class text on a most important (but, sadly, either neglected or misunderstood) subject. Given the topics covered and the method of treatment, the audience for this book is more likely to be graduate students and researchers in structural mechanics than structural designers. The present reviewer most warmly recommends Scaling of Structural Strength to the former group.