Crack singularity

Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

In this case the crack is said to have a zeroth opening. The cracks of a zeroth opening prove to possess a remarkable property which is the main result of the present section. Namely, the solution % is infinitely differentiable in a vicinity of T, dT provided that / is infinitely differentiable. This statement is interpreted as a removable singularity property. In what follows this assertion is proved. Let x G T dT and w > (f in O(x ), where O(x ) is a neighbourhood of x. For convenience, the boundary of the domain O(x ) ia assumed to be smooth. 

The condition [%] = 0 is shown to provide the infinite differentiability of the solution only for > 0. For the problem (2.153), corresponding to = 0, one cannot state that w G H 0 x j) provided that [%] = 0 on O(x ) n F,, since, in general, in this case dw/dv 0 on O(x ) n F,. The result of Theorem 2.17 on C °°-regularity actually shows that the condition [x] = 0 provides the disappearance of singularity which takes place in view of the presence of a crack. It means that under the condition mentioned, we can forget about the crack since the behaviour of the plate is the same as that without the crack. 

In this section cracks of minimal opening are considered for thermoelastic plates. It is proved that the cracks of minimal opening provide an equilibrium state of the plate, which corresponds to the state without the crack. This means that such cracks do not introduce any singularity for the solution, and actually we have to solve a boundary value problem without the crack. 

The crack is said to have a zero opening in this case. As it turned out there is no singularity of the solution provided the crack has a zero opening. What this means is the solution of (3.144), (3.147), (3.148) coincides with the solution of (3.140)-(3.142) found in the domain Q with the initial and boundary conditions (3.144), (3.145) (and without (3.143)). In the last case the equations (3.141), (3.142) hold in Q. This removable singularity property is of local character. Namely, if O(x ) is a neighbourhood of the point and... 

The decrease in with crack depth for fracture of IG-11 graphite presents an interesting dilemma. The utihty of fracture mechanics is that equivalent values of K should represent an equivalent crack tip mechanical state and a singular critical value of K should define the failure criterion. Recall Eq. 2 where K is defined as the first term of the series solution for the crack tip stress field, Oy, normal to the crack plane. It was noted that this solution must be modified at the crack tip and at the far field. The maximum value of a. should be limited to and that the far... 

Fig. 18. Adhesive contact of elastic spheres. pH(r) and pa(r) are the Hertz pressure and adhesive tension distributions, (a) JKR model uses a Griffith crack with a stress singularity at the edge of contact (r = a) (b) Maugis model uses a Dugdale crack with a constant tension aa in a < r < c [1111.

If no laminae have failed, the load must be determined at which the first lamina fails (so-called first-ply failure), that is, violates the lamina failure criterion. In the process of this determination, the laminae stresses must be found as a function of the unknown magnitude of loads first in the laminate coordinates and then in the principal material directions. The proportions of load (i.e., the ratios of to Ny, to My,/ etc.) are, of course, specified at the beginning of the analysik The loaa parameter is increased until some individual lamina fails. The properties, of the failed lamina are then degraded in one of two ways (1) totally to zero if the fibers in the lamina fail or (2) to fiber-direction properties if the failure is by cracking parallel to the fibers (matrix failure). Actually, because of the matrix manipulations involved in the analysis, the failed lamina properties must not be zero, but rather effectively zero values in order to avoid a singular matrix that could not be inverted in the structural analysis problem. The laminate strains are calculated from the known load and the stiffnesses prior to failure of a lamina. The laminate deformations just after failure of a lamina are discussed later. 

S. G. Larsson and A. J. Carlsson, Influence of Non-Singular Stress Terms and Specimen Geometry on Small Scale Yielding at Crack Tips in Elastic-Plastic Material, J. Mech. Phys. Solids, 21, 263-278 (1973). 

According to LEFM the stress singularity at the tip of a (sharp) crack of length a can be conveniently described using a stress intensity factor K(a) in the crack opening mode K(a) is given by ... 

Fig. 10.2 Effective stress versus distance from crack tip based on matching of the singular crack tip stress fields of Eqs. (3), (4), and (7).

Another mathematical approach to modeling cohesive zones is to consider the crack tip fully shielded, that is Keff = 0, rather than partially shielded as in the case considered above. In this case, a cohesive zone lies in front of a traction free crack (Zone 1). This is the classical Barenblatt-Dugdale model in which the stress-intensity factor at the end of the cohesive zone is now zero that is, stress singularities are completely removed by the cohesive forces.29 The requirement of complete shielding results in a cusp-shaped cohesive zone or bridging zone profile. This approach has advantages, particularly for the elevated temperature case, in that the cohesive zone can... 

As surface corrugations grow, they will eventually touch each other and then folded parts will form on the surface. If the gel is sufficiently soft, the free energy can be lowered below the value in the homogeneous state by formation of a periodic pattern [20,21,89], The folded parts are singular surfaces like cracks and we encounter great mathematical difficulty. Nevertheless, we may analytically calculate the folded patterns in the highly compressible case 1... 

After cooling and by virtue of their shape, these hillocks form singularities where stress concentrations will encourage cracking and then flaking of any brittle thin passivation film after it has been deposited on the aluminum (Figure 5). 

By requiring that the stress singularity at the stationary crack tip must vanish the level of applied stress can be determined to be... 

On the contrary to the basic premise of the conventionnal LEFM approach, a polymer is never completely elastic. It doesn t describe the distribution of stress field of equation (3), sketched in Fig. 3(i) and characterised by the presence of an infinite stress in front of the crack tip, consequence of its 1/Vr singularity. 

To and the crack faces. The curve T is shrunk onto the crack tip to give J. ni is a unit vector that is normal to T or To and that points away from the crack tip. ni is the component of n, into xi direction. U is the strain energy density and u is the displacement vector. The material density is p. The advantage of a far-field integral such as Equation (4) is that J can be evaluated along any curve surrounding the crack tip and the singularity at the crack tip, which is problematic in numerical models, can be avoided. 

When the sound wave generated from the branched microcrack or reflected from the boundary of the plate interact with the crack, the singular stress fields at the tip of the crack are modulated in such form as given by Eqs. (25), which are proportional to cos((wr - P x). On the other hand the 1-st order fields consist of two terms which are proportional either to cos(o>r- p x) or sin(o>r-/5 jX). The latter term, for example, comes from the real part of such term. 

If we confine our attention to the re on very near to the crack tip, then clearly the first term dominates since it is singular and for this region we have ... 

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