Journal of Applied Mechanics, Transactions ASME

Statistical models for predicting failure probability of brittle materials are investigated. A formula is derived from a physical consideration for the fracture of microcracks in materials based on the general forms of a fracture criterion and a statistical distribution function incorporating the weakest link principle. The relationships of this model and other statistical models in the literature are discussed; they were found to be equivalent for isotropic materials in which microcracks are randomly distributed in all directions. The statistical model is also used in a failure analysis of the round-notch four-point bending specimen made of an AISI 1008 steel. The grain boundary carbide particles are considered to be microcracks in the plastic zone near the notch tip. The distribution function in the statistical theory is derived from the density and size distribution of carbide particles in the steel. The statistical theory for a triaxial stress state is used to predict the failure probability for any given load on the specimen. The failure loads (loads corresponding to 50 percent of failure probability) are calculated for the specimen at different temperatures. The results are compared with experimental data; good agreement is obtained.

The solution for a point force applied at the interior of an infinite transversely isotropic solid is obtained by introducing three potential functions which govern the displacements. Unlike previous publications where the solutions are expressed in different forms depending on the conditions satisfied by the elastic constants, the present paper provides a systematic approach to obtain a unified solution which is applicable for all stable transversely isotropic materials. The expression obtained does not have the deficiency suffered by previous solutions, namely, each individual term in the present expression does not tend to infinity on the z-axis. Thus accurate numerical evaluation of the Green’s function can be directly performed without the need to resolve the singularity algebraically.

A class of one-dimensional models for the yielding behavior of materials and structures is presented. This class of models leads to stress-strain relations which exhibit a Bauschinger effect of the Massing type, and both the steady-state and nonsteady-state cyclic behavior are completely specified if the initial monotonic loading behavior is known. The concepts of the one-dimensional class of models are extended to three-dimensions and lead to a subsequent generalization of the customary concepts of the incremental theory of plasticity.

A generalized isotropic yield criterion of the form, (σ1−σ2)n+(σ2−σ3)n+(σ1−σ3)n21/n=Y where σ1 ≥ σ2 ≥ σ3 and 1 ≤ n ≤ ∞, is proposed. The corresponding flow rules, Lode variables, and effective strain functions are presented. Experimental and theoretical data on yielding under combined stresses can be described by a single parameter, n.

The role of plastic spin within the framework ofa phenomenological polycrystalline plasticity is examined. We show that if elastic and plastic strain rates are properly identified, partitioning of total spin and identification of its “plastic” part is not required in the elastoplastic constitutive analysis of elastically isotropic materials.

The plasticity of a polycrystalline aggregate is expressed in terms of the plasticity of the individual grains. It is assumed that the local deviation of stress from the average stress is proportional to the local deviation of strain from the average strain, and it is assumed that plastic flow begins when there is an average of three active slip systems per grain. The plastic states of a grain are mapped as a function of the orientation of the crystallographic axes. A coexistence of different states of strain with different rotations of the axes at the same state of stress can be correlated with the occurrence of deformation bands. A range of orientations is illustrated in which the axes tend to congregate quickly and then move more slowly toward a stable end orientation. The residual strain which would be observed by x-ray diffraction after the release of stress is calculated for three sets of diffracting planes. The applications of the theory have so far been limited to face-centered cubic polycrystals with intrinsic elastic isotropy.

Exact and variational methods of analysis are proposed which reduce a general elastic wave-propagation problem to a static problem plus an eigenvalue problem. The methods are shown to be immediately applicable to linear viscoelastic materials with constant Poisson’s ratio. The viscoelastic wave problem is thus reduced to an elastic wave problem plus an integral equation of the Volterra type involving time only.

Bounds and expressions for the effective elastic moduli of materials reinforced by parallel hollow circular fibers have been derived by a variational method. Exact results have been obtained for hexagonal arrays of identical fibers and approximate results for random array of fibers, which may have unequal cross sections. Typical numerical results have been obtained for technically important elastic moduli.

The application of press forging to parts with thin sections is desirable in many instances. It is generally recognized that in forging such parts, the pressure required may be very high. That this high pressure required is due to friction has been reported; however, very little specific information on the effects of friction, area, and thickness is available. The results of this analysis resemble, in some respects, those presented in an analysis of forces required in rolling by Nadai.