Generalized Strain Measure and its Conjugate Stress in a Continuum

2 Generalized Strain Measure and its Conjugate Stress in a Continuum  

As shown in (2.46) the deformation gradient is decomposed by a polar decomposition as F = RU = VR. Since U is positive definite, the characteristic form is given by 

Note that Green s strain corresponds to the case E (1) as mentioned previously E = E )). 

The inner product of second order tensors A, B is given hy A B= x A B) thus (3.34) ctm be proved. 

It is understood that the density function of the second Piola-Kirchhoff stress = TjjEI 0 is the energy-conjugate or dual stress to E under the definition of the increment of internal energy dua = r D. Note that the r.h.s. term of (3.34)i, T E, is denoted by a Lagrangian description (cf. Sect. 2.7). 

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