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2.8.1 Mixture Kinematics
Consider a constrained mixture of multiple solid constituents . The motion of each constituent is given by , where denotes a material point in the reference configuration of that constituent. At the current time , various constituents which occupy an elemental region with a spatial position may have originated from distinct referential positions . We often label a convenient constituent as the master constituent (e.g., the oldest constituent in a reactive mixture with evolving composition) and call the reference configuration the master reference configuration. All the referential mass densities and mass density supplies (see below) are evaluated relative to the master reference configuration . The kinematics of each constituent may be related to the kinematics of the master constituent through Taking the material time derivative of this relation in the material frame and recognizing that this relation must hold for all in the case of a constrained mixture establishes that all constituents share the same velocity . However, as detailed previously [8, 77], constituents may have distinct deformation gradients . When a reaction converts a reactant into a product , these constituents may have distinct reference configurations. The deformation gradient of the master constituent , which also serves as the total deformation gradient, may be related to the relative deformation gradient of constituent by applying the chain rule to eq.(2.8.1-1), producing In this expression, is the deformation gradient of relative to , which must be postulated by constitutive assumption. The relationship between and is time-invariant; consequently, is a time-invariant spatial mapping. It follows that only one deformation gradient represents an independent state variable in a constrained mixture framework, whereas all others are related to it via eq.(2.8.1-2); any of the 's may be selected, based on convenience. In eq.(2.8.1-2) the spatio-temporal arguments have been written explicitly for clarity. These dependencies are implied in the forthcoming sections and henceforth those arguments may be selectively suppressed. Taking the determinant of eq.(2.8.1-2) produces a relation between the volume ratios and , where .