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We would of course like to have σ = E e in matrix notation. 11) E=  E 44 E 45 E 46     E 55 E 56 symm E 66 The components E pq of E are related to the components E i jk of E through an appropriate mapping that preserves the product relation. For example: σ11 = E 1111 e11 + E 1122 e22 + E 1133 e33 + E 1112 e12 + E 1121 e21 + E 1113 e13 + E 1131 e31 + E 1123 e23 + E 1132 e32 maps to σ11 = E 11 e11 + E 12 e22 + E 13 e33 + E 14 2e23 + E 15 2e31 + E 16 2e12 , whence E 11 = E 1111 , E 14 = E 1123 + E 1132 , etc.

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The body is referred to a three dimensional, rectangular, right-handed Cartesian coordinate system xi ≡ {x1 , x2 , x3 }. The body is in static equilibrium under the action of body forces bi in V , prescribed surface tractions tˆi on St and prescribed displacements uˆ i on Su , where St ∪ Su ≡ S are two complementary portions of the boundary S. 2. The three unknown internal fields are displacements u i , strains ei j = e ji and stresses σi j = σ ji . All of them are defined in V . 2 At internal interfaces (for example a change in material) certain strain and stress components may jump, but such “jump conditions” are ignored in the present treatment.

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