which generalizes to the statement. This reduces the number of material constants from 81 to 54. In a similar fashion we can make use of the symmetry of the strain tensor This further reduces the number of material constants to 36. To further reduce the number of material constants consider the conclusion from the first law for elastic materials, equation(1) strain energy density per unit volume O∈ Oe 02U 02U O∈mOe 02U CH Assuming equivalence of the mixed partials 02U 0∈k1Oa;Oe;0e This further reduces the number of material constants to 21. The most general anisotropic linear elastic material therefore has 21 material constants We are going to adopt voigt's notation l「nC2C3C4C5Co「en1 2C23C24C25C26 C33C34C35C3 11) When the material has symmetries in its structure the number of material constants is reduced even further(see Unified treatment of this material) We are going to concentrate on the isotropic case:which generalizes to the statement. This reduces the number of material constants from 81 to 54. In a similar fashion we can make use of the symmetry of the strain tensor �ij = �ji ⇒ Cijlk = Cijkl (4) This further reduces the number of material constants to 36. To further reduce the number of material constants consider the conclusion from the first law for elastic materials, equation (1): ∂U� σ � ij = , U : strain energy density per unit volume (5) ∂�ij ∂U� Cijkl�kl = (6) ∂�ij ∂ � � ∂2U� Cijkl�kl = (7) ∂�mn ∂�mn∂�ij ∂2U� Cijklδkmδln = (8) ∂�mn∂�ij ∂2U� Cijmn = (9) ∂�mn∂�ij Assuming equivalence of the mixed partials: ∂2 � ∂ U 2U� Cijkl = = = Cklij (10) ∂�kl∂�ij ∂�ij∂�kl This further reduces the number of material constants to 21. The most general anisotropic linear elastic material therefore has 21 material constants. We are going to adopt Voigt’s notation:       σ11 C11 C12 C13 C14 C15 C16 �11  σ22   C22 C23 C24 C25 C26         �22         σ33   C33 C34 C35 C36   = = �33        (11)  σ23   C44 C45 C46   �23  σ13  symm C55 C56 �13 σ12 C66 �12 When the material has symmetries in its structure the number of material constants is reduced even further (see Unified treatment of this material). We are going to concentrate on the isotropic case: 2