Elastic deformation under multiaxial loads

We already saw in section 2.2.2 that a load that causes a normal strain in its direction also causes transversal normal strains. For example, a stress in xi direction, cth, causes the following strains, according to equations (2.13) and (2.14) n = a /E, 22 = 33 = —vawjE. One component of the stress tensor a thus acts on several components of the strain tensor e. Similarly, a prescribed strain in one direction may change the stresses in other directions. If we restrict ourselves to small deformations, the relation between stress and strain is linear. Mathematically, an arbitrary linear relation between two tensors of second order can be described using a double contraction  

The elasticity tensor (7 is a tensor of fourth order. It can be considered as a four-dimensional matrix with three components in each of its 4 directions. Its 3 = 81 components Cjjfc are the material parameters that completely describe the (linear) elastic behaviour. 

Because the stress and the strain tensor contain only 6 independent components each, due to their symmetry, the elasticity tensor C needs only 6 = 36 independent parameters. 

The storage and dissipation of energy is also discussed in exercise 26. 

That not all 81 components of the elasticity tensor are needed can be most easily understood using an example. For 712, we find from equation (2.20) 

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