L. B. Freund

Crack propagation in an elastic solid subjected to general loading—I. Constant rate of extension

Abstract A particular motion of a half-plane crack in an elastic solid subjected to general loading is considered. The loading is prescribed so that the mode of deformation of the crack is the plane-strain opening mode. The crack is initially at rest and then, at a certain instant, begins to move with a constant velocity which is less than the Rayleigh wave speed. A fundamental solution is derived for particular loading on the body which makes it possible to obtain the solution for the complete dynamic stress field due to crack extension by linear superposition. The details of the solution are worked out for the dynamic stress intensity factor for the moving crack. It is found that the stress intensity factor varies almost linearly with the crack speed, from the static value at zero speed to zero at the Rayleigh wave speed. Some numerical results are also presented for the shape of the crack tip as a function of crack speed. Finally, the energy-release rate of the moving crack tip is plotted as a function of crack-tip speed.

CRACK PROPAGATION IN AN ELASTIC SOLID SUBJECTED TO GENERAL LOADING-II. NON-UNIFORM RATE OF EXTENSION

Abstract The stress intensity factor of a half-plane crack extending non-uniformly in an isotropic elastic solid subjected to general loading is determined. The loading is applied in such a way that a state of plane strain exists and that crack extension takes place in Mode I. The crack tip is initially at rest and then moves in an arbitrary way in the plane of the crack. In the process of obtaining the stress intensity factor, the complete elastic field is determined for a crack which starts from some initial position, extends at a constant rate for some time, and then suddenly stops. Once the stress intensity factor is known for arbitrary motion of the crack tip, the Griffith fracture criterion is applied to obtain an equation of motion for the crack tip which is consistent with the assumptions of this criterion. Numerical results are included for the stress intensity factor and for the velocity-dependent term in the equation of motion.

Extensions of the Stoney Formula for Substrate Curvature to Configurations with Thin Substrates or Large Deformations

Two main assumptions which underlie the Stoney formula relating substrate curvature to mis-match strain in a bonded thin film are that the film is very thin compared to the substrate, and the deformations are infinitesimally small. Expressions for the curvature-strain relastionship are derived for cases in which thses assumptions are relaxed, thereby providing a biasis for interpretation of experimental observations for a broader class of film-substrate configurations.

A criterion for arrest of a threading dislocation in a strained epitaxial layer due to an interface misfit dislocation in its path

In a strained layer grown epitaxially on a substrate, the motion of a dislocation on any particular glide plane in the layer can be influenced by the presence of dislocations on other glide planes. The focus here is on the glide of a dislocation extending from the free surface of the layer to the layer‐substrate interface, the so‐called threading dislocation. A general definition of driving force for glide of a threading dislocation in a nonuniform stress field is adopted to calculate the driving force on a threading dislocation due to an encounter with an interface misfit dislocation on an intersecting glide plane. The result is examined in detail for the case of cubic materials, taking into account different combinations of Burgers vectors. The analysis makes it clear that the misfit dislocation forces the threading dislocation to glide through a channel of width less than the full layer thickness. A blocking criterion is proposed, based on the presumption that blocking will occur if the channel width i...

Crack propagation in an elastic solid subjected to general loading—III. Stress wave loading

Abstract T he stress intensity factor of a half-plane crack extending non-uniformly in an isotropic elastic solid subjected to stress wave loading is determined. A plane stress pulse strikes the crack at time t = 0, the wavefront being parallel to the plane of the crack. At some arbitrary later time t = τ, the crack begins to extend at a non-uniform rate. It is found that the stress intensity factor is a universal function of instantaneous crack-tip velocity times the stress intensity factor for an equivalent stationary crack. An energy rate balance fracture criterion is applied to obtain an equation of motion for the crack tip. The delay time between the arrival of the incident pulse and the onset of fracture is also calculated for this fracture criterion.

A CRITICAL THICKNESS CONDITION FOR A STRAINED COMPLIANT SUBSTRATE/EPITAXIAL FILM SYSTEM

The physical system under study is a single crystal film grown epitaxially on a substrate of comparable thickness which is constrained to remain flat. In general, the layers are strained due to a mismatch in lattice parameter between the film and substrate materials. The free energy change of the system due to formation of strain‐relaxing interface misfit dislocations is estimated, and the discriminating case of zero energy change leads to a critical thickness condition on mismatch strain, film thickness, substrate thickness, and crystallographic slip orientation which is necessary for the spontaneous formation of such dislocations. The condition obtained generalizes the Matthews–Blakeslee (MB) criterion for a thin film on a thick substrate to the case of a complaint substrate/epitaxial film system, and it reduces to the MB criterion when either the film or substrate is relatively thick.

Model for stress generated upon contact of neighboring islands on the surface of a substrate

In the early stage of growth of a metal film on a substrate by the Volmer–Weber mechanism, a tensile stress in the film is observed to arise at about the point in the process when islands of deposited material begin to coalesce. The mechanism commonly proposed as the origin of this tensile stress is that the coalescing islands deform in order to form a relatively low energy grain boundary, at the expense of some surface energy by surface area reduction, and that this proceeds until a stress is generated that has magnitude sufficient to prevent further area reduction. Several models have been proposed for this process, but the inferred tensile stress estimates have been much larger than observed stress magnitudes in many cases. The purpose here is to introduce a model for the process based on the theory of contact of elastic solids with cohesion. A description of the process is developed on this premise for one-dimensional, two-dimensional, and three-dimensional states of deformation of coalescing islands....

Substrate curvature due to thin film mismatch strain in the nonlinear deformation range

The physical system considered is a thin film bonded to the surface of an initially flat circular substrate, in the case when a residual stress exists due to an incompatible mismatch strain in the film. The magnitude of the mismatch strain is often inferred from a measurement of the curvature it induces in the substrate. This discussion is focused on the limit of the linear range of the relationship between the mismatch strain and the substrate curvature, on the degree to which the substrate curvature becomes spatially nonuniform in the range of geometrically nonlinear deformation, and on the bifurcation of deformation mode from axial symmetry to asymmetry with increasing mismatch strain. Results are obtained on the basis of both simple models and more detailed finite element simulations.

MECHANICS OF COHERENT AND DISLOCATED ISLAND MORPHOLOGIES IN STRAINED EPITAXIAL MATERIAL SYSTEMS

A combined analytical and computational model is developed to study the mechanics of strained epitaxial island growth in typical semiconductor systems. Under certain growth conditions in systems with a film/substrate lattice mismatch, deposited material is known to aggregate into islandlike shapes with geometries having arc shaped cross-sections. A two-dimensional model assuming linear elastic behavior is used to analyze an isolated arc shaped island with elastic properties similar to those of the substrate. The substrate is assumed to be much larger than the island. Finite element analysis shows that in order to minimize the total energy, which consists of strain energy, surface energy, and film/substrate interface energy, a coherent island will adopt a particular height-to-width aspect ratio that is a function of only the island volume. It is then shown that for an island with volume greater than a certain critical size, the inclusion of a mismatch strain relieving edge dislocation is favorable. The cri...

Stress intensity factor calculations based on a conservation integral

Abstract It is observed that one of the integral conservation laws of elastostatics, the so-called M-integral conservation law, has certain special features which make it possible to apply this conservation law for a class of plane elastic crack problems in order to calculate the elastic stress intensity factor in each case without solving the corresponding boundary value problem. The main characteristics which a problem must have in order for the approach to be useful are (1) for points very near to the origin of coordinates, the known elastic stresses are 0(r−r) where r is the radial coordinate and γ ⩽ 1, (2) for points very far from the origin, the known elastic stresses are 0(r−r) where γ ⩾ 1, and (3) the boundary of the body is made up of radial lines on which certain traction and/or displacement conditions are satisfied. The approach is demonstrated by determining the stress intensity factors for four familiar elastic crack problems directly from the conservation law, and then four similar additional applications of the M-integral conservation law are discussed.