Cracks In Brittle Solids Research Articles

Minimum Spacing of Thermally Induced Cracks in Brittle Solids: Addendum

Sharper lower bounds for the minimum spacing of thermally induced tension cracks in brittle half-planes are presented, and four previous [1] numerical estimates are improved.

Experimental investigation of thermally induced interacting cracks in brittle solids

The results of an experimental investigation of thermally induced parallel edge cracks in a half-plane consisting of brittle material are reported. A free edge of a glass plate heated to a uniform temperature, was brought in contact with a liquid bath cooled by dry ice. The thermal contraction of the boundary layer produced interacting tension cracks. When no initial cracks existed, a large amount of potential energy was released in the formation of a few cracks which extended in a dynamic manner beyond the thermal layer. To prevent this type of cracking, initial cracks were formed before thermally inducing crack growth. This resulted in stable crack growth which qualitatively agreed well with the previously developed theory. The quantitative comparison was also reasonably good.

On stability and posteritical behavior of interactive tension cracks in brittle solids

The growth of interactive tension cracks in brittle solids is investigated in general. Cases are considered where the growth regime of several cracks, which are extending simultaneously in a stable manner, may become unstable. The fracture process is assumed to be governed by the Mode I conditions, and only straight-crack extension is examined. First, a variational formulation is presented, and a parametric description of the corresponding potential energy is given. Conditions under which unique crack growth rates correspond to given rates of loading are established from the stationary character of the potential energy, and critical states at which uniqueness, fails, are identified. Four types of critical states are identified, three of which stem from the constraint imposed by the physical conditions associated with crack growth phenomena, whereas the fourth emerges from the vanishing of the Hessian of the second variation of potential energy. The postcritical behavior is determined by minimization of the parametric form of the potential energy. Then a perturbation method is used, and the effect of material inhomogeneity and imperfection sensitivity is examined. Explicit results are obtained for two interacting cracks where imperfection arises from small differences in the fracture toughness of the cracks. It is shown in particular, that if the toughness of the cracks differs by a quantity of the order e, then the strain level at which instability takes place, drops from its value for the case with no imperfection by a quantity of the order e1/2.

Atomically sharp cracks in brittle solids: an electron microscopy study

The issue of bond rupture versus microplasticity as an essential mechanism of crack propagation in brittle solids is addressed. A detailed survey of existing theoretical and experimental evidence relating to this issue highlights the need for direct observations of events within the crack-tip “process zone”, at a level approaching 10 nm. Transmission electron microscopy is accordingly used to study arrested cracks about sharp-contact (Vickers indentation and particle impact) sites in Si, Ge, SiC and Al2O3. The nature of the deformation which accommodates the irreversible contact impression is first investigated, in the light of Marsh's proposal of an “equivalence” between indentation and crack-tip zone processes. Interfacial and tip regions of the surrounding cracks are then examined for any trace of a plasticity-controlled fracture process. Dislocation-like images are indeed evident at the crack planes, but these are shown to be totally inconsistent with any conventional slip mechanism. The close connection between the dislocation patterns and moire fringe systems along the cracks points to “lattice mismatch” contrast in association with a partial closure and healing operation at the interface. Analysis of all other details in the crack patterns, e.g. the presence of a crack-front contrast band indicative of a residual strain field and the disposition of interfacial fracture steps relative to the dislocation/moire system, reinforces this interpretation. It is concluded that the concept of an atomically sharp crack provides a sound basis for the theory of fracture of brittle solids.

A new combined analytical and finite-element solution method for stability analysis of the growth of interacting tension cracks in brittle solids

The analysis of stable and unstable growth regimes of a system of tension cracks in brittle solids, requires accurate estimates of the values of the stress intensity factors at various crack tips, and their derivatives with respect to the crack lengths. A new combined analytical and finite-element solution method is developed which is specially suited to problems of this kind. For illustration the method is applied to estimate the results corresponding to: (1) a single Griffith crack in an infinite space; and (2) possible unstable growth regimes corresponding to a system of edge cracks in a strip of finite width. Comparison of the results with those obtained by other methods, reveals that the present method is indeed very effective and efficient, and therefore provides a useful tool for the stability analysis of interacting cracks.

Unstable growth of tension cracks in brittle solids: Stable and unstable bifurcations, snap-through, and imperfection sensitivity

The growth pattern of a system of tension cracks in a linearly elastic brittle solid, may undergo abrupt changes, e.g. some cracks may stop or actually close, as others snap to finitely longer lengths. For the growth regime of a system of straight edge cracks in plane strain, the concepts of fundamental equilibrium path, stable and unstable bifurcation points and snap-through critical point are introduced and the corresponding behaviors are studied. In particular, it is shown that instabilities of this kind are highly imperfection-sensitive in the sense that, for example, a small material inhomogeneity can decrease by a large amount the critical value of the load parameter at which a growth regime abruptly changes to a new one. For thermally induced tension edge cracks in an infinite strip of finite width, numerical results are presented and various aspects of the theory are illustrated. For the calculation the new combined analytic and finite-element solution method recently given by the authors is employed.

Minimum Spacing of Thermally Induced Cracks in Brittle Solids

When the temperature at the free surface of a linearly elastic brittle half-plane, which is initially uniform throughout the solid, is suddenly reduced by a large amount and then kept constant thereafter, a thermal boundary layer whose thickness increases with time, forms close to the free surface. Because of the consequent thermal contraction, edge cracks may form within the thermal boundary layer. For a system of equally spaced straight-edge cracks, growing collinearly with increasing thickness of the thermal boundary layer, the average minimum crack spacing is estimated on the basis of: (a) energy consideration, (b) stress consideration, and (c) consideration of stability of the growth of interacting cracks. It is shown that for a given temperature profile, one can develop a general stability chart which, in particular, gives a complete growth regime of the interacting cracks (no crack branching is included).

Unstable growth of thermally induced interacting cracks in brittle solids: Further results

This paper is complementary to a previous work[1] in which the growth and stability of a system of thermally induced equally spaced parallel edge cracks in a half-plane consisting of a homogeneous isotropic linearly elastic brittle solid has been studied. Initially the half-plane has a uniform temperature, and the edge cracks are introduced by continuous cooling of its free surface. The cracks grow into the solid at an equal rate with the increasing thickness of the thermal boundary layer which forms close to its free surface. However, because of the interaction between adjacent cracks, a critical state may be reached after which some of the cracks stop growing while the remaining ones grow at a faster rate. This new growth regime may again be interrupted at a new critical state where either the cracks which had stopped growing would then suddenly snap closed while the remaining cracks jump to a finitely longer length, or a different growth regime takes place, depending on the nature of the temperature profile. The present work is concerned with a careful examination of the growth regime at and after the above-mentioned second critical state. This examination requires consideration of three interacting unequal cracks which involves crack extension in both Modes I and II. As in [1] two different temperature profiles, relevant to the problem of heat extraction from hot dry rock masses, are considered. It is shown that when the temperature profile in the solid is in the form of an error function, the inclusion of the third interacting crack changes the previously obtained results qualitatively (i.e. no crack closure is attained in this case), whereas for the second temperature profile our new (more complicated) calculations only confirm the previously obtained results.

Unstable growth of thermally induced interacting cracks in brittle solids

The growth and stability of thermally induced equally spaced parallel cracks in a half-plane consisting of a homogeneous isotropic linearly elastic brittle material are studied. At the initial time, the uniform temperature of the half-plane is reduced at its surface by a large increment, T0, and then kept constant (at the surface). Because of heat conduction and possible heat convection due to fluid flow, a temperature gradient forms close to the surface and penetrates into the half-plane. Thermal contraction results in the formation of cracks perpendicular to the free surface. It is shown that if the cracks are initially parallel and equally spaced, and if the possibility of branching is excluded, then they grow in time until a critical state is reached. At this state alternate cracks stop growing, while the others begin to grow at a much faster rate. This process continues until another critical state is attained, where the cracks which had stopped growing (together with some other cracks, depending on the temperature profile), suddenly close, while the cracks which have continued growing, suddenly “snap” into a finitely longer length. At this state the crack spacing is doubled (or quadrupled, depending on the temperature profile). The whole process then repeats itself. Applications to geothermal energy extraction from hot dry rock masses is mentioned.