θ A B x y z r free edge interface edge interface corner faces 0 φ Figure 1: Three-dimensional bimaterial interface corner geometry showing coordinate axes. The faces (planes) and edges (lines) are arbitrarily oriented. It is reasonably well known that a stress and a displacement field of the form [1-5] (1) ∑ ∑ ∞ =−∞ ∞ =−∞ − = = m Mm i D m Mi m Mm ij D m Mij u K r g f K r m m ( , ) ( , ) 3 1 3 φ θ φ θ σ λ λ exist in the region surrounding the tip of the three-dimensional bimaterial interface corner of Figure 1. In general, there are an infinite number of terms in the series, each corresponding to a specific deformation mode, m. Here λm-1 are the orders of the stress singularities, and describe the angular variation of the stress and displacement fields in each material (M = A, B) and are the corresponding stress intensities. λ ( , )φ θ m ij f ( , )φ θ mig K ( , )φ θ m ij f D m 3 m-1, and depend on the elastic mismatch and asymptotic interface corner geometry, and depend on the far-field loading and geometry. This expansion of the stress state is a natural generalization of the classical mode I, II and III fields in homogeneous isotropic cracked solids; however, the deformation modes generally do not possess the simple symmetry of cracks in homogeneous media. The deformation modes strongly depend on the elastic mismatch and the nature of the asymptotic corner geometry. In Eqn. 1, λ ( , )φ θ m ij f D mK3 ) K ( , )φ θ mig m, and can be determined from an asymptotic analysis of the stress state near the three-dimensional interface corner [2-5]. Note that may be singular along reentrant free edges and interface edges (see Figure 1). Only the stress intensities cannot be determined from the asymptotic analysis. They depend on the far-field geometry and loading of the solid. ( , )φ θ mig ( , φ θ m ij f D m 3 Terms that give rise to both singular and nonsingular stresses exist in the series. Finite strain-energy at the three-dimensional bimaterial interface corner requires Re{ λ} > -1/2 (for two-dimensional corners Re{ λ} > 0) and finite displacements require Re{ λ} > 0; however, as previously discussed [6,7] these are not completely satisfactory reasons for limiting the range of Re{ λ} since they are based on assessment of the elastic solution in a region where it is not valid. Very close to the interface corner the actual solution is usually perturbed by material nonlinearity and/or geometric perturbations from the ideal interface corner, and far from the interface corner, it is perturbed by the far-field boundaries and loads. The terms with eigenvalues in the range 0 < Re{ λ} < 1 may dominate the other terms in the series expansion of Eqn. 1 in an annular region surrounding the interface corner. In this annulus, these terms dominate higher-order singular terms because these have amplitudes that are small, and they dominate the nonsingular terms because the annulus is sufficiently close to the tip of the interface corner. Furthermore, in this elastic annulus the elastic fields for a particular loading mode exhibit a universal structure; their magnitude is simply scaled by a single parameter, the stress intensity . As such, this annulus is termed the K-annulus. The region where the higher-order singular terms are significant compared to the -term is embedded within D mK3 D mK3
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