strain. For a total strain of 15%, the delayed elastic strain was found to be about 0.15 or 25% of the elastic strain. A possible relationship has been proposed between the delayed elastic strain measured at 15% strain and area fraction of γ′ facet‐sized cracks for the onset of ultimate fracture. Nonetheless, delayed elastic effect in the single‐crystal material could not be detected during the primary creep period of small strain of engineering importance.
The lack of a reliable, simple – with a limited number of material constants or parameters – microstructurally sensitive, three‐dimensional constitutive model of polycrystalline materials at high homologous temperatures is the stumbling block against the use of sophisticated computer modeling and numerical simulations available today. A successful constitutive equation should have a physical basis and be able to describe the features that can be readily seen in the real field conditions. Such a model for inelastic deformation must be able to predict the usual empirically derived relations and be adequate for extrapolation to account for the effects of microstructural altercations or loading conditions. Moreover, such a model must be capable of predicting delayed elastic phenomena that can be quantified on the removal of the external driving forces. This means, predictability of the rebound of the structure on partial or full unloading should be built in the model. To date, most constitutive equations are highly empirical in nature and are useful for interpretation in the range investigated, but inadequate for extrapolation.
The chronology of the development of the three‐term rheological model, called EDEV model, is described in Chapter 5. This model relies on the incorporation of grain‐boundary shearing processes to relate with the delayed elastic phenomenon. A review of the phenomenological aspects of the failure processes at elevated temperatures is presented in Chapter 6. The crucial roles played by the phenomenon of delayed elasticity are examined in depth. Delayed elasticity can be linked to the predictability of the conditions necessary for the onset of grain‐facet‐sized crack formation and the kinetics of microcracking (Chapter 7). This leads to the development of the crack‐enhanced EDEV model that can handle, for example, the strain‐rate sensitivity of strength of polycrystalline solids at high temperatures (Chapter 8). EDEV is a nonlinear constitutive equation for high‐temperature applications. It consists of elastic, delayed elastic, and viscous components corresponding to three micromechanisms: lattice deformation, intergranular shearing/sliding, and intragranular dislocation motion. The model incorporates the predictability of the onset of cracking activity and damage accumulation due to the mechanism(s) of high‐temperature grain‐boundary embrittlement.
Grain facet long cracks develop when a critical grain‐boundary sliding or shearing (gbs) displacement or an equivalent delayed elastic strain is reached. Further damage is given in terms of the excess gbs, displacement over its critical value. As cracks form, they enhance the deformation matrix affecting the overall creep rate, leading to a minimum rate and then tertiary creep. Formulations have been developed in Chapter 8, using this model, for predicting the deformation and cracking activity for conditions of constant strain‐rate strength tests. The theory was tested with published experimental data on the strain‐rate sensitivity of the compressive strength of transversely isotropic, columnar‐grained, pure polycrystalline ice with a load applied in the plane of isotropy. Calculations using material constants were obtained from constant‐stress creep and recovery (SRRT) experiments totally independent of the strength tests. One‐to‐one correspondence of theory and experiments was noted for the dependence of strength, failure strain, and failure time on strain rate. The theory has the capabilities for predicting the popular empirically obtained relations between these quantities, such as the MG relationship. The EDEV theory also points out the limitations of the empirical relations, for example the inapplicability of the empirical power‐law relation between t f and σ f or between ε f and strain rate. It is shown that strain‐rate sensitivity of volumetric dilatation can also be reasonably predicted.
The EDEV model has been extended to stress relaxation tests (SRTs) at elevated temperatures in Chapter 9 and applied successfully to nickel‐base superalloys, like In‐738LC, titanium‐base alloys, like Ti‐6246, and polycrystalline ice (especially the grain‐size effects on stress relaxation). Theoretical predictions on total strain as well as the components of elastic, delayed elastic, and viscous strain at any time during creep and SRTs can be examined and compared with experimental observations. EDEV model is based on observations that the viscous strain rate (for dislocation creep) is constant throughout the primary creep under constant stress (for conditions of no microcracking activities) and the shape of the primary creep is governed by delayed elastic deformation that is recoverable on unloading. The acid tests of the developed model are provided by the comparisons between the predicted permanent strain and the recovered elastic and delayed elastic strain components with the corresponding experimental observations of both SRRTs and SRTs. Since delayed elastic response in single‐crystal materials may be neglected, the EDEV model reduces to a simple nonlinear elasto–viscous relation, as presented in Chapter 5, for most of the deformation of engineering design purposes.
But why does the viscous strain rate, associated with dislocation creep, appear to be constant during normal primary or transient creep? This SRRT‐based experimental observation in single‐phase crystalline materials and complex crystalline materials, including at least one rock type, ice, and complex alloys, goes against the dislocation theories developed over many decades. Theoreticians have to modify some of the classical assumptions and start working on the development of some new ideas.
1.8 Paradigm Shifts
The Oxford English Dictionary defines paradigm (pær&ip.schwa;da&ip.iscp;m) as “example or pattern, esp., of inflexions of noun, verb, etc.” (Simpson and Weiner 1989). In science and philosophy, a paradigm may be considered as a distinct set of concepts or thought patterns that have developed to guide workers in a specific area. In his book, The Structure of Scientific Revolutions, Kuhn (1996; first published in 1962) defines a scientific paradigm as: “universally recognized scientific achievements that, for a time, provide model problems and solutions for a community of practitioners.”
1.8.1 Paradigm Shift in Experimental Approach
The history of cultures and nations, including various economic and political aspects of the inhabitants, provides opportunities to look back and make judgments that can, eventually, influence and improve our understanding of the global society. Looking back is always healthy as long as the approach is rational and forward looking. This approach has been the key to success for the development of science and technology and building bonds between diverse societies and linguistic groups of the world using a more‐or‐less common multidisciplinary scientific and technical language and jargon. A thorough and critical, but unbiased (hopefully), review of literature is therefore essential for embarking on any scientific work. It is said, “Hindsight 20‐20.” Why not apply this approach with a fresh outlook to high‐temperature materials science? But then, what would be that approach?
Materials exhibit elastic and inelastic deformation on application of a load. Inelastic deformation is commonly known as plastic. The paradigm of plasticity theories was developed on the basis of engineering experience with materials at low homologous temperatures. Plastic deformation is thought to occur when stress exceeds a specific range. The thoughts of practitioners in several engineering disciplines are molded by theories of plasticity proven to be very successful in explaining failures. Plastic deformation is traditionally assumed to be independent of time and hence independent of strain rate or stress rate. As a consequence, failure processes of geological materials have continued to be presented/discussed