Second Piola-Kirchhoff stress

Since the first Piola-Kirchhoff stress II is not symmetric as understood by (2.110), we introduce a symmetrized tensor T, called the second Piola-Kirchhoff stress, and the Euler stress t, which is the transformed tensor of T, into the deformed body using the rotation tensor R ... 

The specific first Piola-Kirchhoff stress 11, specific second Piola-Kirchhoff stress and specific EuIct stress t are defined by... 

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 = Lagrangian description (cf. Sect. 2.7). 

Po (X,t) = poT E - Divq + por where = T/pois the second Piola-Kirchhoff stress per unit mass. 

Second Piola-Kirchhoff stress tensor and related traction vector Lagrangian strain tensor and displacement vector... 

The second Piola-Kirchhoff stress tensor represents a contact force density measured in the reference configuration per unit of reference area. 

According to Eq. (31) and definitions (28)-(30), the most general form of the second Piola-Kirchhoff stress tensor for an isotropic and hyperelastic material is ... 

In the framework of nonlinear viscoelasticity, Fosdick and Yu [165] proposed their own constitutive equation. They assumed that the second Piola-Kirchhoff stress tensor is given by... 

A physical Lagrangian stress tensor is defined and established by applying vector transformation to the second Piola Kirchhoff stress tensor 11 components using equation (27), such that ... 

It is noted that within the Euler-Bemoulli theory context, it holds that 6 = v, 6y= — w (the cross section remains perpendicular to the deformed axis) hence, shear strain components Eqs. 3b and 3c vanish. Considering strains to be small and employing the second Piola-Kirchhoff stress tensor, the nonvanishing stress components are defined in terms of the strain ones as... 

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