1.1 Ideal Strength of Solids 19 The standard Voigt notation for strain nj=n(1+)is used here.The corresponding energy expansion is then 1 U=Uo(V)+Voon+VCoanana +(n). (1.7) Note that the energy expansion at Equation 1.7 contains a double sum instead of a quadruple sum in Equation 1.6. 1.1.2.1 Elastic Moduli The 6x6 matrix of elastic moduli generally contains 21 independent elements that do not transform like the second-rank tensor components.According to the number of point group symmetry operations,the amount of independent elastic moduli can be lower.With respect to the energy expansion,the elastic moduli can be defined as 82U ongonB or 1/82U Ca8= VdEadEB From now on,the Cj will denote the elastic moduli defined on the basis of the finite strain and cij will stand for the elastic moduli based on the small strain. When,for example,the crystal is subjected to small isotropic deformation, the lattice parameter a is related to the reference parameter ar as a=ar(1+ e).Here e is a small stretch that represents diagonal components of the small strain tensor.Then,the finite strain is related to e according to m=n2= n3-n=e+,and the deformation gradient can be expressed as V1+2万 0 0 1+2万 0 0 √1+2 The corresponding energy expansion at Equation 1.7 gives U=U6()+3V,m+23C12+6C12)+ from which the following combination of elastic moduli: 1 d2U C11+2C12= 3V,d71.1 Ideal Strength of Solids 19 The standard Voigt notation for strain ηij = 1 2 ηα(1 + δij ) is used here. The corresponding energy expansion is then U = U0(V ) + V σαηα + 1 2 V Cαβηαηβ + O(η3). (1.7) Note that the energy expansion at Equation 1.7 contains a double sum instead of a quadruple sum in Equation 1.6. 1.1.2.1 Elastic Moduli The 6×6 matrix of elastic moduli generally contains 21 independent elements that do not transform like the second-rank tensor components. According to the number of point group symmetry operations, the amount of independent elastic moduli can be lower. With respect to the energy expansion, the elastic moduli can be defined as Cαβ = 1 V  ∂2U ∂ηα∂ηβ  or cαβ = 1 V  ∂2U ∂εα∂εβ  . From now on, the Cij will denote the elastic moduli defined on the basis of the finite strain and cij will stand for the elastic moduli based on the small strain. When, for example, the crystal is subjected to small isotropic deformation, the lattice parameter a is related to the reference parameter ar as a = ar(1 + e). Here e is a small stretch that represents diagonal components of the small strain tensor. Then, the finite strain is related to e according to η1 = η2 = η3 = η = e + e2 2 , and the deformation gradient can be expressed as Jiso = ⎛ ⎝ 1 + e 0 0 0 1+ e 0 0 0 1+ e ⎞ ⎠ or Jiso = ⎛ ⎝ √1+2η 0 0 0 √1+2η 0 0 0 √1+2η ⎞ ⎠ . The corresponding energy expansion at Equation 1.7 gives U = U0(Vr)+3Vrση + 1 2 Vr(3C11η2 + 6C12η2) + ... from which the following combination of elastic moduli: C11 + 2C12 = 1 3Vr d2U dη2