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Problem I The elasticity tensor C has at most 21 distinct material coefficients (properties) such that the equivalent condensed matrix form of T = C : E is T22 V@T23 (h 111 Cl 122 ("2211 ('2222 C2233 C.B.322 C3333 V'QC2322 V'QC3122 vQC3133 Q 223 (A.T23 2 C2.VX3 2C3123 122.3 2C3131 1231 C2212 12 2 ('2312 2C3112 1212 Ell E'22 E33 The 6 x 6 coefficient matrix appearing above is known as the stiffness matri:r in the Voigt (or Mandel) notation. (a) Derive the Voigt forms(6 x G matrix) of the fourtlH)rder tensors I I and IEI. (b) Starting from T = A(trE)I + 2/1E, evaluate trT and use the resulting expression to solve for trE. Then, using the relations between (A, 11) and (EY, v) derived in class, show that you can invert the stress- strain relation to produce (c) Write the indicial form of this inverted relation and use it to evaluate the Cartesian components of the compliance, tensor S=dE / dT. Express your results in indicial notation and tensorial notation. (d) Write the matrix of S in Voigt (Mandel) notation. Evaluate the eigenvalues of this 6 x 6 compliance matrix (you may Ilse Mathematica, Matlab, or other software, or solve by hand). What conditions rnllst be satisfied for stability?