Anisotropy of Residual Stress Energy in Two-Component Plate Crystal Structures

Abstract

An analytical solution for residual stresses and their energy in an elastically anisotropic two-component plate structure, where the components have an identical type of elastic anisotropy, identical or proportional elastic constants and coinciding principal axes of elastic anisotropy is obtained. The obtained solution has been applied to analyze the anisotropy of the elastic energy of such crystalline structures as the raft structure γ/γ' of single-crystal nickel-base superalloys, multilayer erosion-resistant nanocoatings ZrN/CrN and single-layer coatings of various types. It has been shown that the factor of minimizing the elastic energy of residual stresses has a significant effect on the crystallographic orientation of the interface in multilayer structures and the direction of axis of the growth texture axis of coatings.

APPENDIX

Below there are the orientation dependences of the elastic compliances of a cubic crystal ${{s}_{{ij}}} = f\left( {{{S}_{{ij}}},{{\varphi }},\theta ,{{\psi }} = 0} \right)$, where φ, θ and ψ are the Euler angles. In this study, only those orientations are considered for which ψ = 0. The given formulas can be obtained using both the general elastic compliance transformation described in [23] and a special simplified transformation for cubic crystals proposed in [24]. The transformation matrix in Euler angles is presented in [25].

$${{s}_{{11}}} = {{S}_{{11}}} - \frac{S}{2}{\text{si}}{{{\text{n}}}^{2}}2{{\varphi ,}}$$

(1A)

$${{s}_{{22}}} = {{S}_{{11}}} - \frac{S}{2}({\text{si}}{{{\text{n}}}^{2}}2{{\varphi }}{{{\text{cos}}}^{4}}\theta + {\text{si}}{{{\text{n}}}^{2}}2\theta ),$$

(2A)

$${{s}_{{12}}} = {{S}_{{12}}} + \frac{S}{2}{\text{si}}{{{\text{n}}}^{2}}2{{\varphi }}{{{\text{cos}}}^{2}}\theta ,$$

(3A)

$${{s}_{{13}}} = {{S}_{{12}}} + \frac{S}{2}{\text{si}}{{{\text{n}}}^{2}}2{{\varphi }}{{{\text{sin}}}^{2}}\theta ,$$

(4A)

$${{s}_{{23}}} = {{S}_{{12}}} + \frac{S}{2}\left( {1 - \frac{1}{4}{\text{si}}{{{\text{n}}}^{2}}2{{\varphi }}} \right){\text{si}}{{{\text{n}}}^{2}}2\theta ,$$

(5A)

$${{s}_{{16}}} = - \frac{S}{2}{\text{sin}}4{{\varphi \cos}}\theta ,$$

(6A)

$${{s}_{{26}}} = \frac{S}{2}{\text{sin}}4{{\varphi }}{{{\text{cos}}}^{3}}\theta ,$$

(7A)

$${{s}_{{36}}} = \frac{S}{4}{\text{sin}}4{{\varphi \sin}}\theta {\text{sin}}2\theta ,$$

(8A)

$${{s}_{{66}}} = {{S}_{{44}}} + 2S{\text{si}}{{{\text{n}}}^{2}}2{{\varphi }}{{{\text{cos}}}^{2}}\theta ,$$

(9A)

where $S = {{S}_{{11}}} - {{S}_{{12}}} - {{S}_{{44}}}{\text{/}}2$.