Elasticity Mismatch

A partial answer to the first question has been provided by a theoretical treatment (1,2) that examines the conditions under which a matrix crack will deflect along the iaterface betweea the matrix and the reinforcement. This fracture—mechanics analysis links the condition for crack deflection to both the relative fracture resistance of the iaterface and the bridge and to the relative elastic mismatch between the reinforcement and the matrix. The calculations iadicate that, for any elastic mismatch, iaterface failure will occur whea the fracture resistance of the bridge is at least four times greater than that of the iaterface. For specific degrees of elastic mismatch, this coaditioa can be a conservative lower estimate. This condition provides a guide for iaterfacial desiga of ceramic matrix composites. 

The variation in hardness theoretically predicted for the TiN/NbN multilayers compared with data for the isostructural nitrides shown in Fig. 8.1(a). The predictions are based on the ideas of an increment of hardness arising from the elastic mismatch across the layers, equation (8.6), shown as the horizontal dashed line, and for the lateral flow of material within the interlayers, equations (8.10-8.12), shown as the dashed line increasing as the wavelength decreases. In the isostructural multilayers, the upper limit to the increase in hardness should occur when the stress is high enough to drive dislocation motion across the layers. In the nonisostructural, this condition does not apply, or is substantially modified. The lines shown assume that the hardness of monolithic TiN and NbN is 25 GPa and that the layer thicknesses are equal. Other data are given in Table 8.1. 

The SC solutions appear to run into difficulties when there is a large elastic mismatch between the constituent phases, for example, at high concentrations of a rigid phase in a compliant matrix or of a porous phase in a stiff matrix. The latter situation will be discussed in Section 3.6. One approach to this problem is known as the Generalized Self-Consistent Approach, the concept behind which is illustrated in Fig. 3.14. Instead of a single inclusion in an effective medium, a composite sphere is introduced into the medium. As in the composite sphere assemblage discussed in the last section, the relative size of the spheres reflects the volume fraction, i.e., V = a bf as before. Interestingly, this approach leads to the HS bounds for the bulk modulus. The solution for the shear modulus is complex but can be written in a closed form. 

Another important problem in elastic stress distributions, that relates to microstructures, is the disturbance of a stress field by an inclusion with different elastic properties, i.e., there is an elastic mismatch. Extreme examples occur when the inclusion is a pore or a rigid particle. Consider a plate under uniaxial tension that contains a circular hole. Fig. 4.20. As the hole surface is free from applied stresses, o rr-o re distances from the hole, the disturbance in... 

Among the most common mechanisms of relaxation of the elastic mismatch strain in epitaxial films is the formation of glide dislocations at the film substrate interface, the so-called misfit dislocations. An illustration of... 

Fig. 1.34. A model configuration of a distribution of islands on the substrate. The islands are subject to an elastic mismatch strain with respect to the substrate, which gives rise to substrate curvature.

Suppose that a thin film is bonded to one surface of a substrate of uniform thickness hs- It will be assumed that the substrate has the shape of a circular disk of radius R, although the principal results of this section are independent of the actual shape of the outer boundary of the substrate. A cylindrical r, 0, z—coordinate system is introduced with its origin at the center of the substrate midplane and with its z—axis perpendicular to the faces of the substrate the midplane is then at z = 0 and the film is bonded to the face at z = hs/2. The substrate is thin so that hs R, and the film is very thin in comparison to the substrate. The film has an incompatible elastic mismatch strain with respect to the substrate this strain might be due to thermal expansion effects, epitaxial mismatch, phase transformation, chemical reaction, moisture absorption or other physical effect. Whatever the origin of the strain, the goal here is to estimate the curvature of the substrate, within the range of elastic response, induced by the stress associated with this incompatible strain. For the time being, the mismatch strain is assumed to be an isotropic extension or compression in the plane of the interface, and the substrate is taken to be an isotropic elastic solid with elastic modulus Es and Poisson ratio Vs the subscript s is used to denote properties of the substrate material. The elastic shear modulus /Xg is related to the elastic modulus and Poisson ratio by /ig = Es/ 1 + t s). 

The expression for curvature in (2.7) is the famous Stoney formula relating curvature to stress in the film (Stoney 1909). Stoney s original analysis of the stress in a thin film deposited on a rectangular substrate was based on a uniaxial state of stress. Consequently, his expression for curvature did not involve use of the substrate biaxial modulus Mg. Consequently, (2.7) can be applied in situations in which mismatch derives from inelastic effects. However, the relationship (2.7) is based on Stoney s concept as outline in this section, and it has become known as the Stoney formula. It has the important property that the relationship between curvature k and membrane force / does not involve the properties of the film material. The elastic mismatch strain Cm corresponding to the stress <7 given in (2.8) is... 

The issue of film thickness effects on substrate curvature evolution is pursued now by recourse to the energy minimization method which was introduced in Section 2.1 for the derivation of the Stoney formula. All other features of the system introduced in that section are retained in this discussion, which follows the work of Freund et al. (1999). It is assumed that the film material carries an elastic mismatch strain in the form of an isotropic extension em (or contraction if Cm is negative) in the plane of the interface the physical origin of the mismatch strain is immaterial. The mismatch strain is spatially uniform throughout the film material. In this case, em is... 

Fig. 2.3. In the upper diagram, the elastic mismatch is maintained by externally applied traction of magnitude (7 there is no interaction between the film and substrate in this condition and the substrate is unrestrained. If the externally applied traction is relaxed, the mismatch strain in the film induces a curvature in the substrate as shown in the lower diagram.

The description of deformation in which the development in Section 2.1 is based is retained. However, if the thickness of the film is to be taken into account, then the strain energy of the film material must be included in the calculation of total potential energy. This is accomplished by adopting the strain expression (2.2) for the film as well as the substrate, but augmenting it by the elastic mismatch strain Cm in the former case. The strain energy density throughout the system is then... 

Fig. 2.13. An elastic mismatch strain, which varies in an arbitrary way through the thickness of the film, is maintained by an externally applied traction. Release of this traction induces curvature in the substrate.

Fig. 3.1. Illustrations of two epitaxial multilayers comprising alternating layers of materials identified as film f and substrate s materials for convenience. The multilayers are chosen to be symmetric to avoid overall bending of the structures and to each contain film material of total thickness and substrate material of total thickness hg. Otherwise, the arrangement of layers is arbitrary. For lattice mismatch the elastic mismatch strains are given in (3.28).

Fig. 3.6. Schematic diagram of a thin film bonded to the snrface of a snbstrate. The film thickness hi is much less than the thickness hg of the substrate. Due to an elastic mismatch strain in the film, a membrane force per unit length must be imposed around the periphery of the film to balance the internal stress due to mismatch without deforming the substrate. Relaxation of this artificial membrane force resultant to render the edge free of applied moment induces curvature in the substrate.

A rough estimate of the thickness ratio that results in = 0 can be obtained on the basis of the following reasoning. For the case considered by Suo and Hutchinson (1989), the elastic mismatch strain across the interface prior to delamination is zero, and the substrate is unstrained. Therefore, the conditions that the extensional strain of the film surface z = 0 after delamination must still be zero may correspond to a relatively small value of shear stress on the interface at the delamination front a nonsingular value of shear stress axz = 0 corresponds to the condition that ip = 0, provided that the normal stress azz is still singular and positive. The methods of Section 2.2 can be used to show that this condition on strain implies that / f// sl must have the value 2. While the estimate is crude, it may provide a useful indicator of behavior for arbitrary combinations of material parameters and film thicknesses in cases for which other indicators are not available. 

If a thin film is bonded to a substrate, and if that film is subject to a residual tensile stress as a result of elastic mismatch with the substrate, then the stress can be partially relaxed by formation of cracks in the film. In this section, the behavior of through-the-thickness cracks within the film is considered. First, the behavior of an isolated, fuUy formed crack is examined and, subsequently, the mechanics of formation of an array of cracks is considered. 

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