Example 4 edit


A monoclinic crystal has inversion symmetric about the  -  plane. Therefore, the material properties do not change for a mirror-reflection through this plane. The stress-strain relations must therefore remain unchanged under this transformation. The transformation matrix   for this for the mirror inversion is given by



If we apply this transformation to the stress and strain tensors, then the stiffness matrix of the material (in Voigt notation) is


Solution edit

In 3 3 matrix form, the strain tensor is given by


The transformation rule for a second order tensor   is


Applying this transformation to the strain tensor, we have


In engineering notation (Voigt notation),


Therefore, the transformed strain tensor can be written as


The expression for the strain energy density of a linear elastic material imposes a constraint on the components of the stiffness tensor in the presence of planes of material symmetry. This constraint is


where   are the components of the 6 6 matrix that contains the independent components of the stiffness tensor.



For a monoclinic material, replacing the transformed strain components by the equivalent original strain components, we get




Using the symmetry of the stiffness matrix, we have


Since the strains can be arbitrary, the above condition is satisfied only if


Therefore, the stiffness matrix is given by


Hence shown.