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2.8.3 Mixture Free Energy and Stress
The referential free energy density of the mixture is obtained as where is the specific free energy of constituent and is the specific free energy of the mixture. An important function of state which arises later in our treatment is the chemical potential of constituent , given by The mixture Cauchy stress is given by where is the right Cauchy-Green tensor. The spatial elasticity tensor may be evaluated from
In reactive frameworks where evolves according to eq.(2.8.2-2), it may be convenient to define the mass fraction in which case we may rewrite eq.(2.8.3-1) as where is the referential strain energy density of solid under the assumption that it is the sole mixture constituent, normalized by the referential volume of the master constituent . Based on eq.(2.8.2-3) the mass fractions satisfy . In this case, when and is most conveniently expressed as a function of , we may use eq.(2.8.1-2) to evaluate and calculate the mixture stress using the alternative form This expression shows that the mixture stress may evolve not only due to temporal changes in the state of strain but also due to reactive changes in the mass fractions .
In FEBio the referential strain energy density for any solid mixture constituent is evaluated from the same library of constitutive models used in single-constituent solids. In this library the calculation of the referential strain energy density is based on the assumption that the reference configuration corresponds to the configuration when the deformation gradient passed to those functions is equal to the identity tensor. Thus, passing to those functions returns the correct . However, when passing as an argument to those functions, the referential volume is based on the configuration . Let the referential free energy density returned by FEBio for an argument be denoted by . We may similarly denote the corresponding Cauchy stress and spatial elasticity tensors by and . Here, the subscript has two meanings: First it emphasizes that the corresponding function is evaluated using for the mixture constituent ; second it emphasizes that the calculation returns the corresponding measure under the assumption that the mixture consists entirely of that constituent, so that its multiplication by the scale factor returns the actual contribution of that measure to the entire mixture. Based on eq.(2.8.1-3) we find that so that the mixture free energy density may be evaluated from whereas the mixture Cauchy stress and spatial elasticity are given by where and These relations show that the Cauchy stress and spatial elasticity tensor of constituent in eq.(2.8.3-8) may be evaluated using existing FEBio functions without needing to adjust for the choice of reference configuration or , as evidenced by eq.(2.8.3-9) in the case of the stress. However the referential strain energy density needs to be properly scaled by as shown in eq.(2.8.3-7).
The calculation of the 2nd Piola-Kirchhoff stress for each generation is more elaborate. When using the master reference configuration , this stress is given by When using the reference configuration , the stress is evaluated from a similar relation It can be shown that these stresses are related according to FEBio does not use this calculation for solid mixtures, as all internal calculations employ the Cauchy stress.