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18.104.22.168 Energy Balance
For an elastic solid, in the absence of heat exchanges (i.e., in elastodynamics), the equation of energy balance reduces where is the specific internal energy and is the rate of deformation tensor. Recall that , where is the specific free energy, is the absolute temperature and is the specific entropy. Since in elasticity (due to the temperature remaining constant), the above energy balance may be combined with the mass balance (6.2.1-2) as or where is the free energy density (per volume of the material in the reference configuration).
In our time integration scheme, to satisfy energy balance, this equation needs to be evaluated at , thus However, the solution for obtained from the momentum balance may not necessarity satisfy this equation. Thus, to satisfy energy balance over consecutive time steps, we want to evaluate an effective stress such that To find a solution for , we follow the procedure of Gonzalez  and let where is some scalar function to be determined. Substituting this relation, (22.214.171.124-6), into the previous equation, (126.96.36.199-5), produces Hence, the equation for an effective stress needed to satisfy energy balance between consecutive time steps is In the limit when , we use . Recall that this scheme produces conservation of linear and angular momentum and total energy only with , or equivalently, , and . Therefore, this effective stress calculation is only applied when the user employs .