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The problem explained in the title is formulated generally and given an explicit solution for tensile loadings opening a half-plane crack in an infinite body. For the half-plane crack, changes in the opening displacement between the crack surfaces and in the stress-intensity factor distribution along the crack front are calculated to first order in an arbitrary deviation of the crack-front position from a reference straight line. The deviations considered lie in the original crack plane. The results suggest that in the presence of loadings that would induce uniform conditions along the crack front, if it were straignt, small initial deviations from straightness should reduce in size during quasistatic crack growth if of small enough spatial wavelength but possibly enlarge in size if of longer wavelength. The solution methods rely on elastic reciprocity, in terms of a three-dimensional version of weight function theory for tensile cracks, and on direct solution of elastic crack problems. The weight function is derived for the half-plane crack by solving for the first-order variation in the elastic displacement field associated with arbitrary variations of the crack front from a straight reference line. Also, a new three-dimensional weight function theory is developed for planar cracks under general mixed-mode loading involving tension and shears relative to the crack, the connection between weight functions and the Green’s function for crack problems is shown, and some results are given for the half-plane crack on the variations of elastic fields for variation of crack-front location in the presence of general loadings including shear.

The shear resistance of slipping surfaces at fixed normal stress is given by τ = τ(V, state). Here V = slip velocity, dependence on “state” is equivalent to functional dependence with fading memory on prior V(t), and ∂τ(V, state)/∂V>0. We establish linear stability conditions for steady slip states (V(t), τ(t) constant). For single degree-of-freedom elastic or viscoelastic dynamical systems, instability occurs, if at all, by a flutter mode when the spring stiffness (or appropriate viscoelastic generalization) reduces to a critical value. Similar conclusions are reached for slipping continua with spatially periodic perturbations along their interface, and in this case the existence of propagating frictional creep waves is established at critical conditions. Increases in inertia of the slipping systems are found to be destabilizing, in that they increase the critical stiffness level required for stability.

The propagation and scattering of SH-waves of general form is investigated using the boundary element method. Attention is focused on semi-infinite regions containing subsurface cavities. The analysis is carried out within the context of linear elastodynamic theory formulated as an integral equation. Using a time-dependent Green’s function for the half space, the boundary integral equation is developed and is discretized into a finite set of algebraic relations. A time-stepping algorithm is constructed for the solution of general boundary value—initial value problems. Results of this solution technique are presented for the two-dimensional case of transient waves scattering off of buried stress-free cavities.

A crack paralleling a bonded plane interface between two dissimilar isotropic elastic solids is considered. When the distance of the crack from the interface is small compared to the crack length itself and to other length scales characterizing the geometry, a simple universal relation exists between the Mode I and Mode II stress intensity factors and the complex stress intensity factor associated with the corresponding problem for the crack lying on the interface. In other words, if the influence of external loading and geometry on the interface crack is known, then this information can immediately be used to generate the stress intensity factors for the sub-interface crack. Conditions for cracks to propagate near and parallel to, but not along, an interface are derived.

Elastic fracture mechanics concepts are reexamined for a crack on the interface between dissimilar solids. A derivation by function theory is given of the form of stress and displacement fields in the vicinity of the crack tip, equivalent to complete Williams expansions of both inner and outer (external to a nonlinear or contact zone) type. The complex stress intensity factor K associated with an elastic interface crack, for which contact is ignored, is discussed and, specifically, its validity as a crack tip characterizing parameter is noted for cases of small scale nonlinear material behavior and/or small scale contact zones at the crack tip. That is, similar values of K for two cracked bodies then imply similar states as the crack tip, so that conditions for crack growth can be phrased in terms of K reaching a critical failure locus in a complex plane. The maintenance of a similar state at a crack tip under change of crack length is shown to require alteration of both the magnitude and phase angle of a combined tension and shear loading. Some possible definitions of stress intensity factors KI and KII of classical type associated with interface cracks are discussed. Also, the scaling of interface crack tip plastic zone size with load under small scale yielding conditions is found to deviate from classical scaling, in proportion to the square of the applied load level, and dependences of the field on distance from the tip and on load phase angle are found to be linked together.

The paper is concerned with the structure of multiaxial stress-strain relations in time-dependent metal plasticity, as for transient creep and rate sensitive yielding. First, a general kinematical relation is developed between the macroscopic inelastic strain tensor and microstructural slip displacements, as modeled either by continuum shearing on crystallographic planes of individual grains or by the motion of discrete dislocation lines. It is assumed that at any given slipped state, the rate of slipping on a particular system is governed by the resolved shear stress on that system (or by the local “forces” on dislocation lines). This leads to the primary result of the paper: Components of the macroscopic inelastic strain rate tensor are derivable, at each instant in the course of deformation, from a potential function of stress. General features of the flow potential surfaces in stress space are discussed, and some specific functional forms are examined. Linear viscoelasticity and time-independent plasticity are developed as limiting cases of the flow potential formulation, and the appropriateness of a potential function for stationary creep is discussed.

The in-plane extension of two dissimilar materials with cracks or fault lines along their common interface is considered. A method is offered for solving such problems by the application of complex variables integrated with the eigenfunction-expansion technique presented in an earlier paper. The solution to any problem is resolved to finding a single complex potential resulting in a marked economy of effort as contrasted with the more laborious conventional methods which have not yielded satisfactory results. Boundary problems are formulated and solutions are given in closed form. The results of these evaluations also give stress-intensity factors (which determine the onset of rapid fracture in the theory of Griffith-Irwin) for plane problems.

A unique adaptation of the hydrogen-bubble flow visualization method was applied to measure velocity profiles and film thicknesses of very thin films on an inclined plane wall. Data were obtained in the three flow regions for a developing falling film with an initially uniform velocity profile and thickness ≤0.1 in. The measured profiles compared more favorably with parabolic profiles in the intermediate fully developed region than in the initial developing region. However, measured film thicknesses compared favorably with a simplified solution of the integral momentum equation based on parabolic velocity profiles. The results confirm the theoretical prediction that a relatively long distance may be required even for a thin film before nonaccelerating flow with a constant film thickness is obtained and Nusselt’s classical analysis applies. The experimental technique was shown to be a practical experimental method for obtaining data for the two-dimensional laminar flow of thin liquid films.

An elastic analysis is presented for the tensile stretching and bending of a plate containing a surface crack penetrating part-through the thickness, Fig. 1. The treatment is approximate, in that the two-dimensional generalized plane stress and Kirchhoff-Poisson plate bending theories are employed, with the part-through cracked section represented as a continuous line spring. The spring has both stretching and bending resistance, its compliance coefficients being chosen to match those of an edge cracked strip in plane strain. The mathematical formulation reduces finally to two-coupled integral equations for the thickness averaged force and moment per unit length along the cracked section. These are solved numerically for the case of a semi-elliptical part-through crack, with results compared to a simple but approximate closed-form solution. Extensive results are given for the stress intensity factor at the midpoint of the part-through crack for both remote tensile and bending loads on the plate. These results indicate that the stress-intensity factor is substantially lower, in general, than for a similarly loaded strip in plane strain with a crack of the same depth.

A line integral is exhibited which has the same value for all paths surrounding the tip of a notch in the two-dimensional strain field of an elastic or deformation-type elastic-plastic material. Appropriate integration path choices serve both to relate the integral to the near tip deformations and, in many cases, to permit its direct evaluation. This averaged measure of the near tip field leads to approximate solutions for several strain-concentration problems. Contained perfectly plastic deformation near a crack tip is analyzed for the plane-strain case with the aid of the slip-line theory. Near tip stresses are shown to be significantly elevated by hydrostatic tension, and a strain singularity results varying inversely with distance from the tip in centered fan regions above and below the tip. Approximate estimates are given for the strain intensity, plastic zone size, and crack tip opening displacement, and the important role of large geometry changes in crack blunting is noted. Another application leads to a general solution for crack tip separations in the Barenblatt-Dugdale crack model. A proof follows on the equivalence of the Griffith energy balance and cohesive force theories of elastic brittle fracture, and hardening behavior is included in a model for plane-stress yielding. A final application leads to approximate estimates of strain concentrations at smooth-ended notch tips in elastic and elastic-plastic materials.

A work-hardening elastic-plastic stress analysis is presented for a sharp notch or, as a limiting case, a crack perturbing a remotely applied uniform stress field. Mathematical complexities are reduced through considering the kinematically simple case of antiplane longitudinal shear deformations and by employing a deformation plasticity theory rather than the more appropriate incremental theory. Consequently, a general solution is available valid for any relation between stress and strain in the work-hardening range, so long as the remotely applied stress does not exceed the initial yield stress. When a power law relates stress to a strain in the work-hardening range, the deformation theory solution is also the correct incremental solution at low applied stress levels causing yielding on a scale small compared to notch depth. For cracks, the near crack tip strain field is shown to depend on loads and geometry only through the elastic stress intensity factor when yielding is on a small scale, and the elastic-plastic boundary and lines of constant strain magnitude are circles. Extensive numerical results are tabulated for a crack, 45 deg V-notch, and 90 deg V-notch in power-law-hardening materials, and exhibited graphically for a crack.

This paper considers the problem of bending of a plate composed of two plates of materials having dissimilar elastic properties, bonded together along a straight line which sustains a crack. Both materials are assumed to be isotropic and homogeneous. Upon obtaining stress solutions, it is found that the significant stresses are inversely proportional to the square root of the radial distance from the crack front and have an oscillatory character, which is shown to be confined to the immediate vicinity of the crack tip. A two-parameter set of equations expressing the general form of the stress distribution around the tip of such a crack is provided as it is of primary importance in predicting the strength of cracked plates. Some analogies are also observed between the characteristic equations occurring in the extension and bending of cracked plates composed of dissimilar materials.