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47) Note that the AMB, eq. 5), was exploited in the derivation of eq. 28), and earlier of eq. 24), but its weak form of eq. 12) is not present in the integrand of eq. 46). Three-field potential. On the basis of eq. 47), by using the CL of eq. 32), we can define the three-field potential . 48) where Fext is defined in eq. 42). This also proves that the use of δskew(QT P) in eq. 45) was indeed correct. Remark 1. The right stretch U can also be eliminated from the fourfield formulation in another way.

Besides, we must take det F > 0 to exclude annihilation of line elements and negative volumes, see [159], pp. 85 and 87. Hence, (det F)/(det U) > 0 and, therefore, det R = +1. 2 Rotation Constraint equation Instead of calculating R as FU−1 , we can find a tensor Q ∈ SO(3), by solving the RC equation . C = skew(QT F) = 0. 8) This it permitted because the equations QT F = U and skew(QT F) = 0, are equivalent, which is shown below. 1. skew(QT F) = 0 ⇒ QT F = U. e. QT F is symmetric. Using this symmetry, we have (QT F)2 = (QT F)(QT F) = (QT F)T (QT F) = FT QQT F = U2 .

If we use in eq. 47) the CL sym(QT P) = ∂U˜ W of eq. 31), then the nominal stress P remains only in the term . skew(QT P). Hence, we can define a skew-symmetric tensor Ta = skew(QT P) with only three components and abandon using P with nine components. That is the basic motivation behind using the Biot stress in the next section. 4 3-F and 2-F formulations for Biot stress In this section, we describe a three-field formulation in terms of {χ, Q, Ta }, developed in [42, 191]. This formulation can be obtained from the threefield formulation for the nominal stress tensor, which is described in the previous section, just by introducing the definition of the Biot stress.

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