Anisotropic Linearly Elastic Materials

A completely general way of describing the constitutive equation of a linear elastic material (Hooke's law) is by the following equation:

T = L ^. E or T_ij = L_ijkl^. E_kl

Where:	T	is the stress tensor
	L	is the elasticity tensor with a total of 81 indices
	E	is the strain tensor

The assumptions implicit in this equation is that the reference configuration, i.e. E = 0 also is stress free, i.e. T = 0. For most materials the above equation is valid only for small strains and rotations.

Note that the strain tensor is also described by the differential equation:

E_ij = (u_i,j + u_j,i) e.g. E₁₂ = (dx/dy + dy/dx)

Before starting to simplify the strain tensor we take a look at the elasticity tensor complete with the indices:

L = [ L 1111 L 1112 L 1113 L 1121 L 1122 L 1123 L 1131 L 1132 L 1133 L 1211 L 1212 L 1213 L 1221 L 1222 L 1223 L 1231 L 1232 L 1233 L 1311 L 1312 L 1313 L 1321 L 1322 L 1323 L 1331 L 1332 L 1333 L 2111 L 2112 L 2113 L 2121 L 2122 L 2123 L 2131 L 2132 L 2133 L 2211 L 2212 L 2213 L 2221 L 2222 L 2223 L 2231 L 2232 L 2233 L 2311 L 2312 L 2313 L 2321 L 2322 L 2323 L 2331 L 2332 L 2333 L 3111 L 3112 L 3113 L 3121 L 3122 L 3123 L 3131 L 3132 L 3133 L 3211 L 3212 L 3213 L 3221 L 3222 L 3223 L 3231 L 3232 L 4233 L 3311 L 3312 L 3313 L 3321 L 3322 L 3323 L 3331 L 3332 L 3333 ]

The first step in reducing the elasticity tensor is to check equilibrium as this demands that the stress tensor is symmetric. Mathematically this can be written:

T_ij = T_ji         =>         L_ijkl = L_jikl

From the equation above we see that this implies that L_jikl= L_ijkl and the number of indices in the elasticity tensor can be reduced by 27 to 54.

L = [ L 1111 L 1112 L 1113 L 1121 L 1122 L 1123 L 1131 L 1132 L 1133 L 1211 L 1212 L 1213 L 1221 L 1222 L 1223 L 1231 L 1232 L 1233 L 1311 L 1312 L 1313 L 1321 L 1322 L 1323 L 1331 L 1332 L 1333 - - - - - - - - - L 2211 L 2212 L 2213 L 2221 L 2222 L 2223 L 2231 L 2232 L 2233 L 2311 L 2312 L 2313 L 2321 L 2322 L 2323 L 2331 L 2332 L 2333 - - - - - - - - - - - - - - - - - - L 3311 L 3312 L 3313 L 3321 L 3322 L 3323 L 3331 L 3332 L 3333 ]

Likewise the strain tensor is symmetric:

L_ijklE_kl = ½ L_ijklE_kl + ½ L_ijlkE_kl        =>      E_lk = E_kl

From the equation above we see that this implies that L_jikl= L_jilk and the number of indices can be reduced by 18 to36.

Thus the elasticity tensor for linearly elastic anisotropic materials include 36 independent elements:

L = [ L 1111 L 1112 L 1113 - L 1122 L 1123 - - L 1133 L 1211 L 1212 L 1213 - L 1222 L 1223 - - L 1233 L 1311 L 1312 L 1313 - L 1322 L 1323 - - L 1333 - - - - - - - - - L 2211 L 2212 L 2213 - L 2222 L 2223 - - L 2233 L 2311 L 2312 L 2313 - L 2322 L 2323 - - L 2333 - - - - - - - - - - - - - - - - - - L 3311 L 3312 L 3313 - L 3322 L 3323 - - L 3333 ]

The indices of the elasticity matrix can be simplified if we introduce the following equalities:

T₁=T₁₁

T₂=T₂₂

T₃=T₃₃

T₄=T₂₃

T₅=T₃₂

T₆=T₁₂

E₁=E₁₁

E₂=E₂₂

E₃=E₃₃

E₄=2E₂₃

E₅=2E₃₂

E₆=2E₁₂

We can then simply formulate the relationship for an anisotropic linear elastic material as:

T_K = L_KLE_L

In this equation the indices K and L run from 1 to 6, and L is thus a 6x6 matrix with the 36 independent elasticities as described above (Voigt matrix). Note that the new modified elasticity matrix L is no longer a tensor.

L = [ L 11 L 12 L 13 L 14 L 15 L 16 L 21 L 22 L 23 L 24 L 25 L 26 L 31 L 32 L 33 L 34 L 35 L 36 L 41 L 42 L 43 L 44 L 45 L 46 L 51 L 52 L 53 L 54 L 55 L 56 L 61 L 62 L 63 L 64 L 65 L 66 ]