1.2.1 Formulation
Since the reference axis in Updated Lagrange formulation is regularly updated, the constitutive quantity should also be updated in large strain problems. Writing the deformed and undeformed length difference in indicial form:
\(\begin{aligned}
\left(ds\right)^2 – \left(dS\right)^2 &= dX \cdot 2E \cdot dX\\
&= 2\:{}_{0}^{}E_{pq} \: d_{}^{0}x_p \: d_{}^{0}x_q \:\:\rightarrow\:\: reference \:to\:C_0 \\
&= 2\:{}_{1}^{}E_{ij} \: d_{}^{1}x_i \: d_{}^{1}x_j \:\:\:\rightarrow\:\: reference \:to\:C_1
\end{aligned}\)
\(\displaystyle 2\:{}_{1}^{}E_{ij} \: d_{}^{1}x_i \: d_{}^{1}x_j = 2\:{}_{0}^{}E_{pq} \: d_{}^{0}x_p \: d_{}^{0}x_q \)
\(\boxed{_{1}^{}E_{ij} = \frac{d_{}^{0}x_p}{d_{}^{1}x_i}\:\frac{d_{}^{0}x_q}{d_{}^{1}x_j} \: {}_{0}^{}E_{pq}} \:\:(*)\)
Recall from conservation of mass and Jacobian transformation:
\(\displaystyle m = \int\limits_{} {}_{}^{0}\rho \: d_{}^{0}V = \int\limits_{} {}_{}^{t}\rho \: d_{}^{t}V = \int\limits_{} {}_{}^{t}\rho \: ({}_{0}^{t}J \: d_{}^{0}V) \)
\(\displaystyle {}_{}^{0}\rho = {}_{}^{t}\rho \: {}_{0}^{t}J \)
By using the definition of second Piola – Kirchoff stress tensor:
\(\displaystyle S = J\left(F^{- 1} \cdot \sigma \cdot F^{- T}\right)\)
\(\begin{aligned}
{}_{0}^{2}S_{ij} &= {}_{0}^{2}J \: F_{ik}^{-1} \: {}^{2}\sigma_{km} \: F_{mj}^{-T} \:\:\rightarrow\:\: reference \:to\:C_0\\
&= \frac{_{}^{0}\rho }{_{}^{2}\rho }\:\frac{d_{}^{0}x_i}{d_{}^{2}x_k}\:\frac{d_{}^{0}x_j}{d_{}^{2}x_m}\:{}_{}^{2}\sigma_{km}
\end{aligned}\)
\(\displaystyle _{}^2\sigma_{km} = \frac{_{}^2\rho }{_{}^0\rho }\:\frac{d_{}^2x_k}{d_{}^0x_i}\:\frac{d_{}^2x_m}{d_{}^0x_j}\:{}_{0}^{2}S_{ij} \:\:\rightarrow\:\: reference \:to\:C_0\)
\(\displaystyle _{}^2\sigma_{km} = \frac{_{}^2\rho }{_{}^1\rho }\:\frac{d_{}^2x_k}{d_{}^1x_p}\:\frac{d_{}^2x_m}{d_{}^1x_q}\:{}_{1}^{2}S_{pq} \:\:\rightarrow\:\: reference \:to\:C_1\)
Equating these two:
\(\displaystyle \frac{_{}^2\rho }{_{}^0\rho }\:\frac{d_{}^2x_k}{d_{}^0x_i}\:\frac{d_{}^2x_m}{d_{}^0x_j}\:{}_{0}^{2}S_{ij} = \frac{_{}^2\rho }{_{}^1\rho }\:\frac{d_{}^2x_k}{d_{}^1x_p}\:\frac{d_{}^2x_m}{d_{}^1x_q}\:{}_{1}^{2}S_{pq}\)
\(\boxed{_0^2S_{ij} = \frac{_{}^0\rho }{_{}^1\rho }\:\frac{d_{}^0x_i}{d_{}^1x_p}\:\frac{d_{}^0x_j}{d_{}^1x_q}\:{}_{1}^{2}S_{pq}} \:\:\rightarrow\:\: at\:C_2\)
\(\boxed{_0^1S_{ij} = \frac{_{}^0\rho }{_{}^1\rho }\:\frac{d_{}^0x_i}{d_{}^1x_p}\:\frac{d_{}^0x_j}{d_{}^1x_q}\:{}_{1}^{1}S_{pq}} \:\:\rightarrow\:\: at\:C_1\)
Find incremental stresses by subtracting \(_0^1S_{ij}\) from \(_0^2S_{ij}\):
\(\displaystyle (_0^2S_{ij} \:-\: _0^1S_{ij}) = \frac{_{}^0\rho }{_{}^1\rho }\:\frac{d_{}^0x_i}{d_{}^1x_p}\:\frac{d_{}^0x_j}{d_{}^1x_q}\:(_1^2S_{pq} \:-\: _1^1S_{pq}) \)
\(\boxed{_0S_{ij} = \frac{_{}^0\rho }{_{}^1\rho }\:\frac{d_{}^0x_i}{d_{}^1x_p}\:\frac{d_{}^0x_j}{d_{}^1x_q} \: _1S_{pq}} \:\:(**)\)
The relationship between incremental second Piola – Kirchoff stress tensor and Green – Lagrange strain tensor:
\(\displaystyle _0S_{ij} = {}_0C_{ijkl}\:_0E_{kl}\:\:\:\rightarrow\:\: reference \:to\:C_0\)
\(\displaystyle _1S_{pq} = {}_1C_{pqrs}\:_1E_{rs}\:\:\rightarrow\:\: reference \:to\:C_1\)
Substituting these relations into \((**)\):
\(\displaystyle {}_0C_{ijkl}\:_0E_{kl} = \frac{_{}^0\rho }{_{}^1\rho }\:\frac{d_{}^0x_i}{d_{}^1x_p}\:\frac{d_{}^0x_j}{d_{}^1x_q}\:{}_1C_{pqrs}\:_1E_{rs}\)
The insertion of \((*)\) yields:
\(\displaystyle {}_0C_{ijkl}\:_0E_{kl} = \frac{_{}^0\rho }{_{}^1\rho }\:\frac{d_{}^0x_i}{d_{}^1x_p}\:\frac{d_{}^0x_j}{d_{}^1x_q}\:{}_1C_{pqrs} \left(\frac{d_{}^{0}x_k}{d_{}^{1}x_r}\:\frac{d_{}^{0}x_l}{d_{}^{1}x_s} \: {}_{0}^{}E_{kl}\right) \)
\(\boxed{{}_1C_{pqrs} = \frac{_{}^1\rho }{_{}^0\rho }\:\frac{d_{}^1x_p}{d_{}^0x_i}\:\frac{d_{}^1x_q}{d_{}^0x_j}\:\frac{d_{}^1x_r}{d_{}^0x_k}\:\frac{d_{}^1x_s}{d_{}^0x_l}\:{}_0C_{ijkl}} \:\:(***)\)
Conservation of mass:
\(\displaystyle {}_{}^{0}\rho\:{}_{}^{0}V = {}_{}^{t}\rho\:{}_{}^{t}V\)
\(\displaystyle {}_{}^{0}\rho\:A\:{}_{}^{0}L = {}_{}^{t}\rho\:A\:{}_{}^{t}L \)
\(\displaystyle \frac{_{}^t\rho }{_{}^0\rho } = \frac{_{}^0L }{_{}^tL }\)
1.2.2 Finite Element Discretization
\(\displaystyle \left\{\begin{array}{cc} N_1 \\ N_2 \\ \end{array}\right\} = \left\{\begin{array}{cc} 1 – \frac{_{}^{0}x_1}{_{}^{0}L} \\ \frac{_{}^{0}x_1}{_{}^{0}L} \\ \end{array}\right\}\)
\(\displaystyle _{}^{t}x_i = \sum_{k = 1}^2N_k\:{}_{}^{t}x_i^k\)
\(\begin{aligned}
\frac{d_{}^{t}x_i}{d_{}^{0}x_j} &= \sum_{k = 1}^2\left(\frac{dN_k}{d_{}^{0}x_j}\right){}_{}^{t}x_i^k\\
&= \left(\frac{ – 1}{_{}^{0}L}\right){}_{}^{t}x_i^1 + \left(\frac{1}{_{}^{0}L}\right){}_{}^{t}x_i^2\\
&= \left(\frac{{}_{}^{t}x_i^2 \:-\: {}_{}^{t}x_i^1}{_{}^{0}L}\right)\\
&= \frac{_{}^{t}L}{_{}^{0}L}
\end{aligned}\)
Since truss element resists only in axial direction, we only focus on direction (1111). Finally, the equation \((***)\) becomes:
\(\displaystyle {}_tC_{1111} = \left(\frac{_{}^{t}L}{_{}^{0}L}\right)^3 {}_0C_{1111}\)
\(\boxed{{}_{}^{t}E = {}_{}^{0}E\left(\frac{_{}^{t}L}{_{}^{0}L}\right)^3}\)
References:
[1] Bathe, K.J. (2006) Finite Element Procedures.
[2] Yang YB, Kuo SR. (1994) Theory and analysis of nonlinear framed structures.
[3] Yang, Y.-B., & Leu, L.-J. (1991). Constitutive laws and force recovery procedures in nonlinear analysis of trusses. Computer Methods in Applied Mechanics and Engineering, 92(1), 121-131.
[4] Yang, Y.B. & Leu, Liang-Jenq. (1990). Postbuckling analysis of trusses with various Lagrangian formulations. Aiaa Journal – AIAA J. 28. 946-948. 10.2514/3.25146.