NINI
81 lines
27 KiB
Markdown
81 lines
27 KiB
Markdown
$$
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+ \left[ \frac{\xi^3}{2} \, h_1 \phi_1 \quad \frac{\xi^3}{2} \, h_2 \phi_2 \quad \frac{\xi^3}{2} \, h_3 \phi_3 \quad \frac{\xi^3}{2} \, h_4 \phi_4 \right] \begin{bmatrix} \Delta^0 \mathbf{V}_n^1 \\ \Delta^0 \mathbf{V}_n^2 \\ \Delta^0 \mathbf{V}_n^3 \\ \Delta^0 \mathbf{V}_n^4 \\ \Delta^0 \mathbf{V}_n^4 \end{bmatrix}
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$$
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## 3. FE Formulation
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현재 형상 $(t + \Delta t)$ 에서의 평형방정식은 다음과 같다.
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$$
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\nabla_{x} \cdot^{t+\Delta t} \mathbf{\sigma} + \rho^{t+\Delta t} \mathbf{f} = \rho^{t+\Delta t} \ddot{\mathbf{u}}
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$$
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가상일 원리를 적용하면
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$$
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\begin{split} &\int_{V} \delta^{i+\Delta t} \mathbf{u} \cdot \left( \nabla_{X} \cdot^{i+\Delta t} \mathbf{\sigma} + \rho^{i+\Delta t} \mathbf{f} \right) dV = \int_{V} \delta^{i+\Delta t} \mathbf{u} \cdot \rho^{i+\Delta t} \mathbf{u} dV \\ &\Rightarrow \int_{V} \delta u_{i} \left( \frac{\partial \sigma_{ij}}{\partial x_{j}} + \rho f_{i} \right) dV = \int_{V} \delta u_{i} \cdot \rho \ddot{u}_{i} dV \\ &\Rightarrow \int_{V} \left\{ \frac{\partial \left( \delta u_{i} \sigma_{ij} \right)}{\partial x_{j}} - \frac{\partial \delta u_{i}}{\partial x_{j}} \sigma_{ij} + \delta u_{i} \rho f_{i} \right\} dV = \int_{V} \delta u_{i} \cdot \rho \ddot{u}_{i} dV \\ &\Rightarrow \int_{\partial V} \left\{ \frac{\partial \left( \delta u_{i} \sigma_{ij} \right)}{\partial x_{j}} - \frac{\partial \delta u_{i}}{\partial x_{j}} \sigma_{ij} + \delta u_{i} \rho f_{i} \right\} dV = \int_{V} \delta u_{i} \cdot \rho \ddot{u}_{i} dV \\ &\Rightarrow \int_{\partial V} \delta u_{i} \sigma_{ij} n_{j} dA + \int_{V} \left\{ -\frac{\partial \delta u_{i}}{\partial x_{j}} \sigma_{ij} + \delta u_{i} \rho f_{i} \right\} dV = \int_{V} \delta u_{i} \cdot \rho \ddot{u}_{i} dV \quad \left( \text{geometric } B.C \right) \\ &\Rightarrow \int_{\partial V_{x}} \delta u_{i} \sigma_{ij} n_{j} dA + \int_{\partial V_{m}} \delta u_{i} \overline{t}_{i} dA + \int_{V} \left\{ -\frac{\partial \delta u_{i}}{\partial x_{j}} \sigma_{ij} + \delta u_{i} \rho f_{i} \right\} dV = \int_{V} \delta u_{i} \cdot \rho \ddot{u}_{i} dV \\ &\left( \int_{\partial V_{x}} \delta u_{i} \sigma_{ij} n_{j} dA = 0, \ 7 \right) \vec{\sigma} \vec{\sigma} \vec{\sigma} \vec{\sigma} \vec{\sigma} \vec{\sigma} \vec{\sigma} \vec{\sigma}
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$$
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비선형 해석을 위해 하중을 조금씩 증가시켜 물체의 변형을 순차적으로 구해 나가며 이러한 반복계산에 있어 기준이 되는 물체의 형상을 설정하는 방법에는 크게 Total Lagrange formulation과 Updated Lagrange formulation이 있다. Total Lagrange formulation은 변형과 관련된 변수들을 초기형상을 이용하여 정의하고 Updated Lagrange formulation은 변수들을 현재 변형된 형상으로 정의한다.
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$$
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\begin{split} &\int_{V} \delta u_{i} \cdot \rho \ddot{u}_{i} dV + \int_{V} \delta \varepsilon_{ij} \sigma_{ij} dV = \int_{\partial V_{m}} \delta u_{i} \overline{t}_{i} dA + \int_{V} \delta u_{i} \rho f_{i} dV \\ &\left( \int_{V} (\bullet) dV = \int_{V_{0}} (\bullet) \det (\mathbf{F}) dV_{0} = \int_{V_{0}} (\bullet) J dV_{0} \right. \\ &\left. \int_{\partial V} (\bullet) \mathbf{n} dA = \int_{\partial V} (\bullet) J \mathbf{F}^{-T} \tilde{\mathbf{n}} dA \left( Nanson's \ formula \right) \right. \\ &\left. \int_{V_{0}} \delta \varepsilon_{ij} \sigma_{ij} dV_{0} = \int_{V_{0}} \delta F_{ij} P_{ij} dV_{0} = \int_{V_{0}} \delta E_{ij} S_{ij} dV_{0} \right. \\ &\Rightarrow \int_{V_{0}} \delta u_{i} \cdot \rho J \ddot{u}_{i} dV_{0} + \int_{V_{0}} \delta E_{ij} S_{ij} dV_{0} = \int_{\partial V_{0m}} \delta u_{i} \sigma_{ij} J \left[ F^{-T} \right]_{kj} \tilde{n}_{k} dA_{0} + \int_{V_{0}} \delta u_{i} \rho J f_{i} dV_{0} \\ &\Rightarrow \int_{V_{0}} \delta u_{i} \cdot \rho_{0} \ddot{u}_{i} dV_{0} + \int_{V_{0}} \delta E_{ij} S_{ij} dV_{0} = \int_{\partial V_{0m}} \delta u_{i} \sigma_{ij} J \left[ F^{-T} \right]_{kj} \tilde{n}_{k} dA_{0} + \int_{V_{0}} \delta u_{i} \rho_{0} f_{i} dV_{0} \\ &\Rightarrow \int_{V_{0}} \delta u_{i} \cdot \rho_{0} \ddot{u}_{i} dV_{0} + \int_{V_{0}} \delta E_{ij} S_{ij} dV_{0} = \int_{\partial V_{0m}} \delta u_{i} \sigma_{ij} J \left[ F^{-T} \right]_{kj} \tilde{n}_{k} dA_{0} + \int_{V_{0}} \delta u_{i} \rho_{0} f_{i} dV_{0} \\ &\Rightarrow \int_{V_{0}} \delta v_{i} \cdot \rho_{0} \ddot{u}_{i} dV_{0} + \int_{V_{0}} \delta E_{ij} S_{ij} dV_{0} = \int_{\partial V_{0m}} \delta u_{i} \sigma_{ij} J \left[ F^{-T} \right]_{kj} \tilde{n}_{k} dA_{0} + \int_{V_{0}} \delta u_{i} \rho_{0} f_{i} dV_{0} \\ &\Rightarrow \int_{V_{0}} \delta v_{i} \cdot \rho_{0} \ddot{u}_{i} dV_{0} + \int_{V_{0}} \delta E_{ij} S_{ij} dV_{0} = \int_{\partial V_{0m}} \delta u_{i} \sigma_{ij} J \left[ F^{-T} \right]_{kj} \tilde{n}_{k} dA_{0} + \int_{V_{0}} \delta u_{i} \rho_{0} f_{i} dV_{0} \\ &\Rightarrow \int_{V_{0}} \delta v_{i} \cdot \rho_{0} \ddot{u}_{i} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u} dV_{0} + \int_{V_{0}} \delta v_{i} \dot{u
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$$
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위 식을 정리하면 다음과 같다.
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$$
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\begin{split} &\int_{V_0} \delta^{t+\Delta t} \mathbf{u} \cdot \boldsymbol{\rho_0}^{t+\Delta t} \ddot{\mathbf{u}} dV_0 + \int_{V_0} \delta^{t+\Delta t} \mathbf{E} \stackrel{t+\Delta t}{\circ} \mathbf{S} dV_0 = \int_{\partial V_{0m}} \delta^{t+\Delta t} \mathbf{u}^{t+\Delta t} \mathbf{f} \mathbf{f} dV_0 \mathbf{F}^{t+\Delta t} \tilde{\mathbf{n}} dA_0 + \int_{V_0} \delta^{t+\Delta t} \mathbf{u} \boldsymbol{\rho_0}^{t+\Delta t} \mathbf{f} dV_0 \\ & \Leftrightarrow \int_{V_0} \delta^{t+\Delta t} \mathbf{u} \cdot \boldsymbol{\rho_0}^{t+\Delta t} \ddot{\mathbf{u}} dV_0 + \int_{V_0} \delta^{t+\Delta t} \mathbf{F} \stackrel{t+\Delta t}{\circ} \mathbf{P} dV_0 = \int_{\partial V_{0m}} \delta^{t+\Delta t} \mathbf{u}^{t+\Delta t} \mathbf{f} \mathbf{f} dA_0 + \int_{V_0} \delta^{t+\Delta t} \mathbf{u} \boldsymbol{\rho_0}^{t+\Delta t} \mathbf{f} dV_0 \end{split}
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$$
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$\mathbf{E}$ 는 Green-Lagrange strain, $\mathbf{S}$ 는 $2^{nd}$ PK stress, $\mathbf{P}$ 는 $1^{st}$ PK stress, $\mathbf{F}$ 는 deformation gradient를 나타낸다. 여기에서는 Green-Lagrange strain과 $2^{nd}$ PK stress를 사용한다.
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$$
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\int_{V_0} \delta^{t+\Delta t} \mathbf{u} \cdot \rho_0^{t+\Delta t} \ddot{\mathbf{u}} dV_0 + \int_{V_0} \delta^{t+\Delta t} \mathbf{E} :_0^{t+\Delta t} \mathbf{S} dV_0 = \int_{\partial V_{0m}} \delta^{t+\Delta t} \mathbf{u}_0^{t+\Delta t} \mathbf{F}_0^{t+\Delta t} \mathbf{S}^{t+\Delta t} \tilde{\mathbf{n}} dA_0 + \int_{V_0} \delta^{t+\Delta t} \mathbf{u} \rho_0^{t+\Delta t} \mathbf{f} dV_0
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$$
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먼저 좌변을 살펴보면 첫 번째 항을 다음과 같이 정리 할 수 있다.
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$$
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\begin{split} &\int_{V_0} \delta^{t+\Delta t} \mathbf{u} \cdot \rho^{t+\Delta t} \ddot{\mathbf{u}} J dV_0 = \int_{V_0} \delta \left( {}^t \mathbf{N}^{t+\Delta t} \mathbf{u}_n + {}^0 \tilde{\mathbf{N}} \Delta^0 \tilde{\mathbf{X}}_n \right) \cdot \rho^t \mathbf{N}^{t+\Delta t} \ddot{\mathbf{u}} J dV_0 \\ &= \int_{V_0} \delta \left( {}^t \mathbf{N}^{t+\Delta t} \mathbf{u}_n \right) \cdot \rho \left( {}^t \mathbf{N}^{t+\Delta t} \ddot{\mathbf{u}} \right) J dV_0 \\ &= \left[ \delta^{t+\Delta t} \mathbf{u}_n \right]^T \int_{V_0} \rho \left[ {}^t \mathbf{N} \right]^T \left[ {}^t \mathbf{N} \right] J dV_0 \left[ {}^{t+\Delta t} \ddot{\mathbf{u}}_n \right] \end{split}
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$$
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두 번째 항도 마찬가지로 다음과 같이 정리 할 수 있다. 먼저 Green-Lagrange strain은 다음과 같이 나타낼수 있다.
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$$
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\begin{split} & = \frac{1}{2} \left( \frac{t^{+\Delta l}}{{}_{0}} \mathbf{g}_{i} \cdot \frac{t^{+\Delta l}}{{}_{0}} \mathbf{g}_{j} - {}_{0} \mathbf{G}_{i} \cdot {}_{0} \mathbf{G}_{j} \right) \left( {}_{0} \mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \\ & = \frac{1}{2} \left( \frac{\partial^{t+\Delta l}}{\partial \xi^{i}} \mathbf{X} \cdot \frac{\partial^{t+\Delta l}}{\partial \xi^{j}} - \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \right) \left( {}_{0} \mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \\ & = \frac{1}{2} \left( \frac{\partial \left( {}^{0} \mathbf{X} + \frac{t+\Delta l}}{\partial \xi^{i}} \mathbf{u} \right)}{\partial \xi^{j}} \cdot \frac{\partial \left( {}^{0} \mathbf{X} + \frac{t+\Delta l}}{\partial \xi^{j}} \mathbf{u} \right)}{\partial \xi^{j}} - \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \right) \left( {}_{0} \mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \\ & = \frac{1}{2} \left( \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} + \frac{\partial^{0} \mathbf{X}}{\partial \xi^{i}} \cdot \frac{\partial^{t+\Delta l}}{\partial \xi^{j}} \mathbf{u} + \frac{\partial^{t+\Delta l}}{\partial \xi^{j}} \mathbf{u} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} + \frac{\partial^{t+\Delta l}}{\partial \xi^{i}} \mathbf{u} \cdot \frac{\partial^{t+\Delta l}}{\partial \xi^{j}} \mathbf{u} \cdot \frac{\partial^{t+\Delta l}}{\partial \xi^{j}} \mathbf{u} - \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \right) \left( {}_{0} \mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \\ & = \frac{1}{2} \left( \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \cdot \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} + \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} + \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \right) \left( {}_{0} \mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \\ & = \frac{1}{2} \left( \frac{\partial^{0} \mathbf{X}}{\partial \xi^{i}} \cdot \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} + \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} + \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} \right) \left( {}_{0} \mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \\ & = \frac{1}{2} \left( \frac{\partial^{0} \mathbf{X}}{\partial \xi^{i}} \cdot \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} + \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \right) \left( {}_{0} \mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \\ & = \frac{1}{2} \left( \frac{\partial^{0} \mathbf{X}}{\partial \xi^{i}} \cdot \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} + \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} \cdot \frac{\partial^{0} \mathbf{X}}{\partial \xi^{j}} \right) \left( {}_{0} \mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \right) \\ & = \frac{1}{2} \left( \frac{\partial^{0} \mathbf{X}}{\partial \xi^{i}} \cdot \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} + \frac{\partial \left( {}^{t} \mathbf{u} + \Delta^{t} \mathbf{u} \right)}{\partial \xi^{j}} \right) \left( {}_{0} \mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \right)
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$$
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정리하면 Green-Lagrange strain을 $\Delta \mathbf{u}$ 에 대한 상수 term, 선형 term, 비선형 term으로 나눌 수 있다.
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$$
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\begin{split} & \overset{t+\Delta t}{{}_{0}}\mathbf{E} = \overset{t+\Delta t}{{}_{0}}\mathbf{E}_{0} + \overset{t+\Delta t}{{}_{0}}\mathbf{E}_{C} + \overset{t+\Delta t}{{}_{0}}\mathbf{E}_{L} \\ & \left( \overset{t+\Delta t}{{}_{0}}\mathbf{E}_{0} = \frac{1}{2} \left( \frac{\partial^{0}\mathbf{X}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} + \frac{\partial^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial^{0}\mathbf{X}}{\partial \boldsymbol{\xi}^{j}} + \frac{\partial^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} \right) \left( {}_{0}\mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \\ & \overset{t+\Delta t}{{}_{0}}\mathbf{E}_{C} = \frac{1}{2} \left( \frac{\partial^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial \Delta^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} + \frac{\partial^{0}\mathbf{X}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial \Delta^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} + \frac{\partial \Delta^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial^{0}\mathbf{X}}{\partial \boldsymbol{\xi}^{j}} + \frac{\partial \Delta^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} \cdot \frac{\partial^{0}\mathbf{X}}{\partial \boldsymbol{\xi}^{j}} \cdot \frac{\partial^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} \right) \left( {}_{0}\mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \\ & \overset{t+\Delta t}{{}_{0}}\mathbf{E}_{L} = \frac{1}{2} \left( \frac{\partial \Delta^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial \Delta^{t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} \right) \left( {}_{0}\mathbf{G}^{i} \otimes_{0} \mathbf{G}^{j} \right) \end{split}
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$$
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와 같다. 두 번째 항을 위의 표현으로 나타내면
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$$
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\begin{split} &\int_{V_0} \mathcal{S}^{t+\Delta t} \mathbf{E} \stackrel{t+\Delta t}{\circ} \mathbf{S} dV_0 = \int_{V_0} \mathcal{S}^{t+\Delta t} \mathbf{E} \stackrel{t+\Delta t}{\circ} \mathbf{C} \stackrel{t+\Delta t}{\circ} \mathbf{E} dV_0 \\ &= \int_{V_0} \underbrace{\left( \underbrace{\mathcal{S}^{t+\Delta t}}_{0} \mathbf{E}_0 + \mathcal{S}^{t+\Delta t}_{0} \mathbf{E}_C + \mathcal{S}^{t+\Delta t}_{0} \mathbf{E}_L \right)}_{:t+\Delta t} \stackrel{t+\Delta t}{\circ} \mathbf{C} \stackrel{t+\Delta t}{\circ} \mathbf{E}_0 + \underbrace{t+\Delta t}_{0} \mathbf{E}_C + \underbrace{t+\Delta t}_{0} \mathbf{E}_L \right) dV_0 \\ &= \int_{V_0} \underbrace{\underbrace{\mathcal{S}^{t+\Delta t}}_{0} \mathbf{E}_C \stackrel{t+\Delta t}{\circ} \mathbf{C} \stackrel{t+\Delta t}{\circ} \mathbf{E}_0}_{\text{constant term}} \mathbf{E}_0 dV_0 + \int_{V_0} \underbrace{\underbrace{\mathcal{S}^{t+\Delta t}}_{0} \mathbf{E}_C \stackrel{t+\Delta t}{\circ} \mathbf{C} \stackrel{t+\Delta t}{\circ} \mathbf{E}_C + \mathcal{S}^{t+\Delta t}_{0} \mathbf{E}_L \stackrel{t+\Delta t}{\circ} \mathbf{C} \stackrel{t+\Delta t}{\circ} \mathbf{E}_D}_{\text{linear term}} dV_0 \\ &+ \int_{V_0} \underbrace{\underbrace{\mathcal{S}^{t+\Delta t}}_{0} \mathbf{E}_C \stackrel{t+\Delta t}{\circ} \mathbf{C} \stackrel{t+\Delta t}{\circ} \mathbf{E}_L + \mathcal{S}^{t+\Delta t}_{0} \mathbf{E}_L \stackrel{t+\Delta t}{\circ} \mathbf{C} \stackrel{t+\Delta t}{\circ} \mathbf{E}_C + \mathcal{S}^{t+\Delta t}_{0} \mathbf{E}_L \stackrel{t+\Delta t}{\circ} \mathbf{C} \stackrel{t+\Delta t}{\circ} \mathbf{E}_L}_{0} dV_0}_{\text{high order term}} \end{split}
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$$
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와 같다. 이후 선형화 시키고(High order term 무시) component형태로 나타내면 아래와 같다.
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$$
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\int_{V_{0}} \underbrace{\int_{V_{0}}^{t+\Delta t} \mathbf{E}_{C} :_{0}^{t+\Delta t} \mathbf{E}_{C} :_{0}^{t+\Delta t} \mathbf{E}_{0}}_{\text{constant term}} dV_{0} + \int_{V_{0}} \underbrace{\int_{0}^{t+\Delta t} \mathbf{E}_{C} :_{0}^{t+\Delta t} \mathbf{E}_{C} :_{0}^{t+\Delta t} \mathbf{E}_{C} + \int_{0}^{t+\Delta t} \mathbf{E}_{L} :_{0}^{t+\Delta t} \mathbf{E}_{L} :_{0}^{t+\Delta t} \mathbf{E}_{0}}_{\text{linear term}} dV_{0}
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= \int_{V_{0}} \underbrace{\left[\int_{0}^{t+\Delta t} \mathbf{E}_{C}\right]_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{C} \end{bmatrix}_{ij}^{ijkl} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{kl} dV_{0}}_{\text{constant term}}
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+ \int_{V_{0}} \underbrace{\left[\int_{0}^{t+\Delta t} \mathbf{E}_{C}\right]_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{C} \end{bmatrix}_{ij}^{ijkl} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{kl} + \begin{bmatrix} \int_{0}^{t+\Delta t} \mathbf{E}_{L} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{L} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{L} \end{bmatrix}_{ij} dV_{0}}_{\text{linear term}}
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= \int_{V_{0}} \underbrace{\left[\int_{0}^{t+\Delta t} \mathbf{E}_{C}\right]_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta t \\ 0 \mathbf{E}_{C} \end{bmatrix}_{ij} \begin{bmatrix} t+\Delta
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$$
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Green-Lagrange strain의 변분을 구해보면 다음과 같다. 변분은 incremental displacement에만 적용되는데 이는 incremental displacement가 정의되지 않았기 때문이다. 따라서 $\delta^{t+\Delta t}\mathbf{u} = \delta \begin{pmatrix} t \mathbf{u} + \Delta^t \mathbf{u} \end{pmatrix} = \delta \Delta^t \mathbf{u}$ 이고 $\delta^{t+\Delta t}\mathbf{e} = 0$ 이 성립한다.
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$$
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\begin{split} & \mathcal{S}^{t+\Delta t}_{\phantom{t}0}\mathbf{E}_{C} = \frac{1}{2} \Bigg( \frac{\partial^{0}\mathbf{X}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} + \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial^{0}\mathbf{X}}{\partial \boldsymbol{\xi}^{j}} + \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial^{0}\mathbf{X}}{\partial \boldsymbol{\xi}^{j}} + \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial^{t+\Delta t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} - \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} \Bigg) \Big( {}_{0}\mathbf{G}^{i} \otimes_{0}\mathbf{G}^{j} \Big) \\ & \mathcal{S}^{t+\Delta t}_{\phantom{t}0}\mathbf{E}_{L} = \frac{1}{2} \Bigg( \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial \Delta_{0}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} + \frac{\partial \Delta_{0}\mathbf{u}}{\partial \boldsymbol{\xi}^{i}} \cdot \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \boldsymbol{\xi}^{j}} \Bigg) \Big( {}_{0}\mathbf{G}^{i} \otimes_{0}\mathbf{G}^{j} \Big) \end{split}
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$$
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Component form으로 나타내면
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$$
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\begin{split} & \begin{bmatrix} \iota^{+\Delta t} & \mathbf{E}_0 \end{bmatrix}_{ij} = \frac{1}{2} \left( \frac{\partial^0 \mathbf{X}}{\partial \xi^i} \cdot \frac{\partial^t \mathbf{u}}{\partial \xi^j} + \frac{\partial^t \mathbf{u}}{\partial \xi^j} \cdot \frac{\partial^0 \mathbf{X}}{\partial \xi^j} + \frac{\partial^t \mathbf{u}}{\partial \xi^j} \cdot \frac{\partial^t \mathbf{u}}{\partial \xi^j} \right) \\ & = \begin{bmatrix} {}^0 \mathbf{X}_n \end{bmatrix}^T \frac{1}{2} \left\{ \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^i} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \right\} \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix} \\ & + \frac{1}{2} \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix} \\ & + \frac{1}{2} \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \right\} \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix} \\ & + \frac{1}{2} \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix}^T \frac{1}{2} \left\{ \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \right\} \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix} \\ & + \frac{1}{2} \begin{bmatrix} t \mathbf{X}_n + \mathbf{0} \mathbf{X}_n \end{bmatrix}^T \frac{1}{2} \left\{ \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \right\} \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix} \\ & = \frac{1}{2} \begin{bmatrix} t \mathbf{X}_n + \mathbf{0} \mathbf{X}_n \end{bmatrix}^T \frac{1}{2} \left\{ \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \\ \frac{\partial \xi^j}{\partial \xi^j} \end{bmatrix} + \frac{\partial^0 \mathbf{N}}{\partial \xi^j} + \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \right\} + \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \cdot \frac{\partial^0 \mathbf{N}}{\partial \xi^j} + \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \cdot \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \right\} \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix} \\ & = \begin{bmatrix} t \mathbf{X}_n \end{bmatrix}^T \frac{1}{2} \left\{ \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \right\} \begin{bmatrix} \Delta^t \mathbf{u}_n \end{bmatrix} \\ & + \begin{bmatrix} t \mathbf{X}_n - \mathbf{0} \mathbf{X}_n \end{bmatrix}^T \frac{1}{2} \left\{ \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \right\} \begin{bmatrix} \Delta^t \mathbf{u}_n \end{bmatrix} \\ & = \begin{bmatrix} t \mathbf{X}_n \end{bmatrix}^T \frac{1}{2} \left\{ \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix}^T \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \right\} \begin{bmatrix} \Delta^t \mathbf{u}_n \end{bmatrix} \\ & = \begin{bmatrix} t \mathbf{N}_n \end{bmatrix}_n \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} + \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \end{bmatrix} \begin{bmatrix} \Delta^t \mathbf{u}_n \end{bmatrix} \\ & = \begin{bmatrix} t \mathbf{N}_n \end{bmatrix}_n \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^0 \mathbf{N}}{\partial \xi^j} \end{bmatrix} \begin{bmatrix} \frac{\partial^
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$$
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