M. L. Williams

Bio: M. L. Williams is an academic researcher from California Institute of Technology. The author has contributed to research in topics: Bending of plates & Stress concentration. The author has an hindex of 10, co-authored 14 publications receiving 5498 citations. Previous affiliations of M. L. Williams include Douglas Aircraft Company.

On the Stress Distribution at the Base of a Stationary Crack

Abstract: In an earlier paper it was suggested that a knowledge of the elastic-stress variation in the neighborhood of an angular corner of an infinite plate would perhaps be of value in analyzing the stress distribution at the base of a V-notch. As a part of a more general study, the specific case of a zero-angle notch, or crack, was carried out to supplement results obtained by other investigators. This paper includes remarks upon the antisymmetric, as well as symmetric, stress distribution, and the circumferential distribution of distortion strain-energy density. For the case of a symmetrical loading about the crack, it is shown that the energy density is not a maximum along the direction of the crack hut is one third higher at an angle ± cos^(-1) (1/3); i.e., approximately ±70 deg. It is shown that at the base of the crack in the direction of its prolongation, the principal stresses are equal, thus tending toward a state of (two-dimensional) hydrostatic tension. As the distance from the point of the crack increases, the distortion strain energy increases, suggesting the possibility of yielding ahead of the crack as well as ±70 deg to the sides. The maximum principal tension stress occurs on ±60 deg rays. For the antisymmetrical stress distribution the distortion strain energy is a relative maximum along the crack and 60 per cent lower ± 85 deg to the sides.

Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates in Extension

Abstract: As an analog to the bending case published in an earlier paper, the stress singularities in plates subjected to extension in their plane are discussed. Three sets of boundary conditions on the radial edges are investigated: free-free, clamped-clamped, and clamped-free. Providing the vertex angle is less than 180 degrees, it is found that unbounded stresses occur at the vertex only in the case of the mixed boundary condition with the strength of the singularity being somewhat stronger than for the similar bending case. For vertex angles between 180 and 360 degrees, all the cases considered may have stress singularities. In amplification of some work of Southwell, it is shown that there are certain analogies between the characteristic equations governing the stresses in extension and bending, respectively, if ν, Poisson's ratio, is replaced by -ν. Finally, the free-free extensional plate behaves locally at the origin exactly the same as a clamped-clamped plate in bending, independent of Poisson's ratio. In conclusion, it is noted that the free-free case analysis may be applied to stress concentrations in V-shaped notches.

Crack point stress singularities at a bi-material interface

Abstract: A continuing study of plane stress singularities at corners and cracks has been extended to the case of a crack in. a hard (soft) material ending normal to a continuous interface with a soft (hard) material. The increase (decrease) in stress singularity over the homogeneous material case. which is of the characteristic inverse square root of distance from the crack point, is given for all relative rigidities between zero and infinity. Associated changes in the principal stress and distortion strain energy density distribution are also discussed, along with indications of application to such situations as microcrack growth near grain boundaries and earth faults in layered strata.

The Bending Stress Distribution at the Base of a Stationary Crack

Abstract: Extending an earlier paper dealing with extensional stress distributions, this study considers the stresses around a crack point owing to bending loads. It is found first that the stresses possess the characteristic inverse square-root singularity in terms of distance from the crack point. Along the crack prolongation direction the symmetric principal stresses are of opposite sign and in the ratio (1 − ν)/(3 + ν), in contrast to the extensional situation where they are identically equal. This leads to the observation that more yielding might be expected as the percentage of bending to extension stress at the crack increases. In the antisymmetrical loading, the shear stress is a maximum ±90 deg to the side of the crack, where the distortion energy is also a maximum. Interesting reciprocity relationships are also shown to exist between the symmetrical and antisymmetrical loading conditions for the isochromatic fringe lines and the stress trajectories. Finally, the results are discussed in connection with the combined extensional and bending loading.

Combined Stresses in an Orthotropic Plate Having a Finite Crack

Abstract: Using a formulation in integral equations, a solution for the combined extension-classical bending stress and displacement solution is presented for the case of an infinite orthotropic flat plate containing a finite crack. While the solution can be expressed in closed form for the entire field, primary emphasis is placed upon the stresses near the crack point. Qualitatively, no major difference in behavior due to orthotropy was found although certain quantitative features are noted, mainly as a function of the characteristic rigidity ratio (Ex /Ey )1/2 . The inverse square-root character of the isotropic stress bending and extension is not changed by orthotropy, although amplitudes and distribution are affected. Account is taken of recent important work by Knowles and Wang dealing with Reissner bending of the plate which shows that the extensional and surface bending stresses are identical in singular character and circumferential distribution. A bending-extension interaction curve for fracture initiation is derived and shown to be linear when based upon the more exact bending theory.

A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks

Abstract: : An integral is exhibited which has the same value for all paths surrounding a class of notches in two-dimensional deformation fields of linear or non-linear elastic materials. The integral may be evaluated almost by inspection for a few notch configurations. Also, for materials of the elastic- plastic type (treated through a deformation rather than incremental formulation) , with a linear response to small stresses followed by non-linear yielding, the integral may be evaluated in terms of Irwin's stress intensity factor when yielding occurs on a scale small in comparison to notch size. On the other hand, the integral may be expressed in terms of the concentrated deformation field in the vicinity of the notch tip. This implies that some information on strain concentrations is obtainable without recourse to detailed non-linear analyses. Such an approach is exploited here. Applications are made to: Approximate estimates of strain concentrations at smooth ended notch tips in elastic and elastic-plastic materials, A general solution for crack tip separation in the Barenblatt-Dugdale crack model, leading to a proof of the identity of the Griffith theory and Barenblatt cohesive theory for elastic brittle fracture and to the inclusion of strain hardening behavior in the Dugdale model for plane stress yielding, and An approximate perfectly plastic plane strain analysis, based on the slip line theory, of contained plastic deformation at a crack tip and of crack blunting.

The mathematical theory of equilibrium cracks in brittle fracture

Abstract: Publisher Summary In recent years, the interest in the problem of brittle fracture and, in particular, in the theory of cracks has grown appreciably in connection with various technical applications. Numerous investigations have been carried out, enlarging in essential points the classical concepts of cracks and methods of analysis. The qualitative features of the problems of cracks, associated with their peculiar nonlinearity as revealed in these investigations, makes the theory of cracks stand out distinctly from the whole range of problems in terms of the theory of elasticity. The chapter presents a unified view of the way basic problems in the theory of equilibrium cracks are formulated and discusses the results obtained thereby. The object of the theory of equilibrium cracks is the study of the equilibrium of solids in the presence of cracks. However, there exists a fundamental distinction between these two problems, The form of a cavity undergoes only slight changes even under a considerable variation in the load acting on a body, while the cracks whose surface also constitutes a part of the body boundary can expand even with small increase of the load to which the body is subjected.

Mixed mode cracking in layered materials

Abstract: Publisher Summary This chapter describes the mixed mode cracking in layered materials. There is ample experimental evidence that cracks in brittle, isotropic, homogeneous materials propagate such that pure mode I conditions are maintained at the crack tip. An unloaded crack subsequently subject to a combination of modes I and II will initiate growth by kinking in such a direction that the advancing tip is in mode I. The chapter also elaborates some of the basic results on the characterization of crack tip fields and on the specification of interface toughness. The competition between crack advance within the interface and kinking out of the interface depends on the relative toughness of the interface to that of the adjoining material. The interface stress intensity factors play precisely the same role as their counterparts in elastic fracture mechanics for homogeneous, isotropic solids. When an interface between a bimaterial system is actually a very thin layer of a third phase, the details of the cracking morphology in the thin interface layer can also play a role in determining the mixed mode toughness. The elasticity solutions for cracks in multilayers are also elaborated.

Plane strain deformation near a crack tip in a power-law hardening material

Abstract: C rack-tip strain singularities are investigated with the aid of an energy line integral exhibiting path independence for all contours surrounding a crack tip in a two-dimensional deformation field of an elastic material (or elastic/plastic material treated by a deformation theory). It is argued that the product of stress and strain exhibits a singularity varying inversely with distance from the tip in all materials. Corresponding near crack tip stress and strain fields are obtained for the plane straining of an incompressible elastic/plastic material hardening according to a power law. A noteworthy feature of the solution is the rapid rise of triaxial stress concentration above the flow stress with increasing values of the hardening exponent. Results are presented graphically for a range of hardening exponents, and the interpretation of the solution is aided by a discussion of analogous results in the better understood anti-plane strain case.

Strain-energy-density factor applied to mixed mode crack problems

Abstract: This paper deals with the general problem of crack extension in a combined stress field where a crack can grow in any arbitrary direction with reference to its original position. In a situation, when both of the stress-intensity factors,k 1,k 2 are present along the crack front, the crack may spread in any direction in a plane normal to the crack edge depending on the loading conditions. Preliminary results indicate that the direction of crack growth and fracture toughness for the mixed problem of Mode I and Mode II are governed by the critical value of the strain-energy-density factor,S cr. The basic assumption is that crack initiation occurs when the interior minimum ofS reaches a critical value designatedS cr. The strain-energy-density factorS represents the strength of the elastic energy field in the vicinity of the crack tip which is singular of the order of 1/r where the radial distancer is measured from the crack front. In the special case of Mode I crack extensionS cr is related tok 1c alone asS cr = (κ − 1)k 1 2 /8μ. In general,S takes the quadratic forma 1 1 k 1 + 2a 1 2 k 1 k 2 +a 2 2 k 2 whose critical value is assumed to be a material constant. The analytical predictions are in good agreement with experimental data on the problem of an inclined crack in plexiglass and aluminum alloy specimens. The result of this investigation provides a convenient procedure for determining the critical crack size that a structure will tolerate under mixed mode conditions for a given applied stress.