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and of the current time t

(2.55)

To keep this mapping continuous and bijective at all times, the determinant of the Jacobian of this transformation must not equal zero and must be strictly positive, since it is equal to the determinant of the deformation gradient tensor F^π

      (2.56)

      where U^π is the right stretch tensor, V^π the left stretch tensor, and the skew‐symmetric tensor R^π gives the rigid body rotation. Differentiation with respect to the appropriate coordinates of the reference or actual configuration is respectively denoted by comma or slash, i.e.

      (2.57)

Because of the non‐singularity of the Lagrangian relationship (2.55), its inverse can be written and the Eulerian or spatial description of motion follows

      (2.58)

      The material time derivative of any differentiable function f^π(X, t) given in its spatial description and referred to a moving particle of the π phase is

(2.59)

      If superscript α is used for , the time derivative is taken moving with the α phase.

      2.5.2 Microscopic Balance Equations

      In the hybrid mixture theories, the microscopic situation of any π phase is first described by the classical equations of continuum mechanics. At the interfaces to other constituents, the material properties and thermodynamic quantities may present step discontinuities. As throughout the book, the effects of the interfaces are here not taken into account explicitly. These are introduced, e.g. in Schrefler (2002) and Gray and Schrefler (2001, 2007).

      For a thermodynamic property ψ, the conservation equation within the π phase may be written as

(2.60)

where is the local value of the velocity field of the π phase in a fixed point in space, i is the flux vector associated with Ψ, g the external supply of Ψ, and G is the net production of Ψ. The relevant thermodynamic properties Ψ are mass, momentum, energy, and entropy. The values assumed by i, g, and G are given in Table 2.2 (Hassanizadeh and Gray 1980, 1990; Schrefler 1995). The constituents are assumed to be microscopically nonpolar; hence, the angular momentum balance equation has been omitted. This equation shows, however, that the stress tensor is symmetric.

Table 2.2 Thermodynamic properties for the microscopic mass balance equations.

      Sources: Adapted from Hassanizadeh and Gray (1980, 1990), and Schrefler (1995).

Quantity	ψ	i	g	G
Mass	1	0	0	0
Momentum		t m	g	0
Energy				0
Entropy	Λ	Φ	S	φ

      2.5.3 Macroscopic Balance Equations

For isothermal conditions, as here assumed, the macroscopic

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