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## 5.1.1 Elastic Model

In this model the materials are treated as elastic solids parameterized by their Young Modulus E and Poisson ratio . The stress tensor is calculated uniquely from the strain tensor which is solved from the Navier Stokes equations [Zie91] together with the displacement boundary conditions.

In theory of linear elasticity with small displacements the strain tensor can be defined as

 = = . = L . (5.1)

with the displacement field:
 (x, y, z) = (5.2)

Assuming a linear material law, the stress tensor can now be calculated using the equation

 = D . ( - ) + (5.3)

where are prestrains and are prestresses due to change of temperature, crystal growth or for, e.g., volumetric expansions as in case of oxidation. Assuming an isotropic material (5.3) can be written as:
 = . (5.4)

Introducing the distributed body forces

 f (x, y, z) = (5.5)

the system of differential equations to be solved is finally found to:
 + + = (5.6) + + = (5.7) + + = (5.8)

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