4.2. ALTERNATIVE MEASURES OF STRESS
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Recall Nanson’s Equation (eq. (2.10)): dSn = (detF)N · F−1 dS0 0 −1 · σ dS dS 0 N · σ = (detF) 0N · F
(more formally: dS0 N · [σ 0 − (detF)F−1 · σ] = 0) σ 0 = (detF)F−1 · σ
(4.5)
Eq. (4.5) is the Nominal Stress Tensor Now, consider a “pseudo-force” acting in the initial frame of reference (Fig. 4.2).
Figure 4.2: 2nd Piola-Kirchhoff stress
note: The “pseudo-force” pictured in Fig. 4.2 will become more clear later. The following derivation of the “Second Piola-Kirchhoff” (P-K II) stress tensor is similar to the previous derivation of the “Nominal” stress tensor. The P-K II tensor will be quite useful later on to help us derive constitutive relationships. Recall dx = F · dX Similarly, dfn = F · dˆ fn ; dˆ fn = ˆ tn dS0 ; ˆ tn = N · σ ˆ ;σ ˆ = P-K II tensor We already know that dfn = tn ·dS = n·σdS So, dfn = F · dˆ fn = F · ˆ tn dSo = ˆ tn · FT dS0 = N · σ ˆ · FT dS0 Substituting for dfn , n·σdS = N·ˆ σ · FT dSo
(1)
Invoke Nanson’s equation: ndS = (detF)N · F−1 dS0
(2)