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Orthotropic material constants

Most elastic, engineering problems carried out using the LUSAS software have isotropic material properties with two independent elastic constants which are used in the relationship between stress and strain.  This material has an infinite number of planes of material property symmetry.

A fully anisotropic material (also known as a triclinic material) will have 21 independent elastic constants and will have no planes of symmetry for the material properties.  A material with one plane of material property symmetry is termed monoclinic and has 13 independent elastic constants.

An orthotropic material has nine elastic constants when modelling a 3D continuum and has two orthogonal planes of material property symmetry.  The engineering constants which must be provided are as follows:

E1, E2, E3        =  Young's moduli in 1, 2 and 3 directions respectively
nij                    =  Poisson's ratio for transverse strain in the j-direction when stressed in the i-direction, that is:

nij = -ej/ei

For si = s  and all other stresses are zero (a uni-directional stress state condition)

G23, G31, G12  =  shear moduli in the 2-3, 3-1 and 1-2 planes, respectively.

These material parameters are used to form the Compliance Matrix, [Sij], which relates strain to stress, as follows:

[e] = [Sij] [s]

There are nine independent constants because:

Sij = Sji (the matrix is symmetric) and therefore:

nij/Ei = nji/Ej

The inverse of the compliance matrix is termed the Stiffness Matrix, [Cij], which relates stress to strain and this is also a symmetric matrix.

The orthotropic material constants above are related in such a way that the laws of physics are obeyed and energy is not created.  Both the compliance matrix and the stiffness matrix which use these material parameters must be positive-definite which, in turn, leads to the fact that all leading diagonal terms of both matrices are positive.

Other conditions which must be satisfied and which arise from the relationship between the compliance matrix and the stiffness matrix are as follows:




A spreadsheet can be used to check the above conditions are satisfied and also carries out other checks on the material data.  The spreadsheet ensures that a proposed set of orthotropic elastic material constants is valid and does not violate any physical law.  The nine parameters are entered in the unprotected data-cells at the top of the spreadsheet and various checks are then carried out on the data.  The stiffness matrix is produced and should any messages such as "bad value" or "violation of orthotropic law" arise, then the set of parameters is invalid.

Please refer to the following reference for further information on orthotropic elastic material constants:

JONES, ROBERT, M.  (1975).  "Mechanics of composite materials", Taylor and Francis.

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