Elastic constants mismatch

This range depends on the elastic constants of the layers, the layer number, the mismatch in CTEs, and A T. 

Background At elevated temperatures the rapid application of a sustained creep load to a fiber-reinforced ceramic typically produces an instantaneous elastic strain, followed by time-dependent creep deformation. Because the elastic constants, creep rates and stress-relaxation behavior of the fibers and matrix typically differ, a time-dependent redistribution in stress between the fibers and matrix will occur during creep. Even in the absence of an applied load, stress redistribution can occur if differences in the thermal expansion coefficients of the fibers and matrix generate residual stresses when a component is heated. For temperatures sufficient to cause the creep deformation of either constituent, this mismatch in creep resistance causes a progres-... 

Tills falls somewhat short of the measured value of about 30 MPa. We attribute this discrepancy to the difficulty of obtaining accurate elastic constants for the particle by the method of Chow, where we used a random spherical block componen t morphology that differs significantly from the randomly wavy rod morphology of the KRO-1 particles. We conclude nevertheless that the craze yield stress is high due to the minimal elastic and thermal expansion mismatch of the particle with its... 

Fig. 2.29. Plots of normalized radial curvature k in the directions 0 = 0, Kj 1 and 7T versus normalized mismatch strain Cm in a circular film-substrate system for V = 1/4. These results were obtained by finite element calculation, assuming identical elastic constants for the film and substrate materials. They involve no a priori assumptions on the deformed shape. Bifurcation occurs when em(bif) 1-54. The normalized curvature R and mismatch strain Cm are defined in (2.81). The results obtained for R/hg = 50, 100 and 200 are identical when expressed in terms of these parameters.

Suppose that a thin film of a cubic material is deposited on a substrate as a single crystal, oriented so that its (111) crystallographic plane is parallel to the substrate surface. The elastic constants of the material are c j- in a coordinate system aligned with the cube axes of the material. A global coordinate system is introduced with the X3—axis normal to the film-substrate interface, as in Section 3.5.1. In this frame, the known components of mismatch strain are = Cm, <... 

If do and the elastic constants of the film material would be known, it follows that the mismatch stress some direction in the material, say -00 defined by... 

If the thin film with an equibiaxial extensional mismatch strain and zero shear mismatch strain is an anisotropic crystal with (001) orientation, the strain given in (3.60) can be rewritten in terms of the single crystal elastic constants, using (3.22) and (3.41)-(3.44), as... 

The deposited material is assumed to have an equi-biaxial elastic mismatch m in the plane of the interface with respect to the substrate, and the elastic constants of the film are taken to be the same as those of the substrate. The total stress (strain) at any material point is the sum of the mismatch stress (strain), which is zero in the substrate, and a contribution due to change in shape of the film surface, that is,... 

First, the elastic stress distributions of the un-notched specimens are obtained from a finite element analysis. For the PI un-notched specimen, the discrepancy between the finite element and the analytical result is very small (about 0.01%), thus validating the finite element calculation in terms of accuracy through the meshing and the type of element used. Therefore a similar calculation is conducted on the G1 un-notched specimen where the span to height ratio is smaller. The mismatch on the maximum stresses at the bottom and at the top of the beam between the finite element calculations and the analytical solution is 0.74% in tension and 0.79% in compression (and remains constant upon further mesh refinement). This estimation of the stress distribution is then used for the following evaluation of the stress intensity factor. 

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