## 【小呆的力学笔记】连续介质力学的知识点回顾二：应变度量

#### 文章目录

• 3. 格林应变与阿尔曼西应变

##### 3. 格林应变与阿尔曼西应变

∣ o a → ∣ 2 − ∣ O A → ∣ 2 = d x i d x i − d x i ′ d x i ′ = ∂ x i ∂ x j ′ d x j ′ ⋅ ∂ x i ∂ x k ′ d x k ′ − d x i ′ d x i ′ = ∂ x i ∂ x j ′ ∂ x i ∂ x k ′ d x j ′ d x k ′ − d x i ′ d x i ′ = ( ∂ x i ∂ x j ′ ∂ x i ∂ x k ′ − δ j k ) d x j ′ d x k ′ (3.1) \begin{aligned} |\overrightarrow{oa}|^2-|\overrightarrow{OA}|^2&=dx_idx_i-dx^{'}_idx^{'}_i\\ &=\frac{\partial x_i}{\partial x^{'}_j}dx^{'}_j\cdot\frac{\partial x_i}{\partial x^{'}_k}dx^{'}_k-dx^{'}_idx^{'}_i\\ &=\frac{\partial x_i}{\partial x^{'}_j}\frac{\partial x_i}{\partial x^{'}_k}dx^{'}_jdx^{'}_k-dx^{'}_idx^{'}_i\\ &=(\frac{\partial x_i}{\partial x^{'}_j}\frac{\partial x_i}{\partial x^{'}_k}-\delta_{jk}) dx^{'}_jdx^{'}_k \end{aligned}\tag{3.1}

E i j = 1 2 ( ∂ x k ∂ x i ′ ∂ x k ∂ x j ′ − δ i j ) (3.2) E_{ij}=\frac{1}{2}(\frac{\partial x_k}{\partial x^{'}_i}\frac{\partial x_k}{\partial x^{'}_j}-\delta_{ij}) \tag{3.2}

E = 1 2 ( F T F − I ) (3.3) E=\frac{1}{2}(F^TF-I) \tag{3.3}

∣ o a → ∣ 2 − ∣ O A → ∣ 2 = d x i d x i − d x i ′ d x i ′ = δ i j d x i d x j − ∂ x i ′ ∂ x j d x j ⋅ ∂ x i ′ ∂ x k d x k = δ i j d x i d x j − ∂ x i ′ ∂ x j ∂ x i ′ ∂ x k d x j d x k = ( δ j k − ∂ x i ′ ∂ x j ∂ x i ′ ∂ x k ) d x j d x k (3.4) \begin{aligned} |\overrightarrow{oa}|^2-|\overrightarrow{OA}|^2&=dx_idx_i-dx^{'}_idx^{'}_i\\ &=\delta_{ij} dx_idx_j-\frac{\partial x^{'}_i}{\partial x_j}dx_j\cdot\frac{\partial x^{'}_i}{\partial x_k}dx_k\\ &=\delta_{ij} dx_idx_j-\frac{\partial x^{'}_i}{\partial x_j}\frac{\partial x^{'}_i}{\partial x_k}dx_jdx_k\\ &=(\delta_{jk}-\frac{\partial x^{'}_i}{\partial x_j}\frac{\partial x^{'}_i}{\partial x_k}) dx_jdx_k \end{aligned}\tag{3.4}

e i j = 1 2 ( δ i j − ∂ x k ′ ∂ x i ∂ x k ′ ∂ x j ) (3.5) e_{ij}=\frac{1}{2}(\delta_{ij}-\frac{\partial x^{'}_k}{\partial x_i}\frac{\partial x^{'}_k}{\partial x_j}) \tag{3.5}

e = 1 2 ( I − F − T F − 1 ) (3.6) e=\frac{1}{2}(I-F^{-T}F^{-1}) \tag{3.6}

u i = x i ( x j ′ , t ) − x i ′ u i = x i − x i ′ ( x j , t ) (3.7) u_i=x_i(x^{'}_j,t)-x^{'}_i\\ u_i=x_i-x^{'}_i(x_j,t)\tag{3.7}

∂ u i ∂ x j ′ = ∂ x i ∂ x j ′ − δ i j ∂ u i ∂ x j = δ i j − ∂ x i ′ ∂ x j (3.8) \frac{\partial u_i}{\partial x^{'}_j}=\frac{\partial x_i}{\partial x^{'}_j}-\delta_{ij}\\ \frac{\partial u_i}{\partial x_j}=\delta_{ij}-\frac{\partial x^{'}_i}{\partial x_j}\tag{3.8}

E i j = 1 2 ( ∂ x k ∂ x i ′ ∂ x k ∂ x j ′ − δ i j ) = 1 2 [ ( ∂ u k ∂ x i ′ + δ k i ) ( ∂ u k ∂ x j ′ + δ k j ) − δ i j ] = 1 2 ( ∂ u k ∂ x i ′ ∂ u k ∂ x j ′ + δ k i ∂ u k ∂ x j ′ + δ k j ∂ u k ∂ x i ′ + δ k i δ k j − δ i j ) = 1 2 ( ∂ u k ∂ x i ′ ∂ u k ∂ x j ′ + ∂ u i ∂ x j ′ + ∂ u j ∂ x i ′ ) (3.9) \begin{aligned} E_{ij}&=\frac{1}{2}(\frac{\partial x_k}{\partial x^{'}_i}\frac{\partial x_k}{\partial x^{'}_j}-\delta_{ij}) \\ &=\frac{1}{2}[(\frac{\partial u_k}{\partial x^{'}_i}+\delta_{ki})(\frac{\partial u_k}{\partial x^{'}_j}+\delta_{kj})-\delta_{ij}]\\ &=\frac{1}{2}(\frac{\partial u_k}{\partial x^{'}_i}\frac{\partial u_k}{\partial x^{'}_j}+\delta_{ki}\frac{\partial u_k}{\partial x^{'}_j}+\delta_{kj}\frac{\partial u_k}{\partial x^{'}_i}+\delta_{ki}\delta_{kj}-\delta_{ij})\\ &=\frac{1}{2}(\frac{\partial u_k}{\partial x^{'}_i}\frac{\partial u_k}{\partial x^{'}_j}+\frac{\partial u_i}{\partial x^{'}_j}+\frac{\partial u_j}{\partial x^{'}_i})\tag{3.9} \end{aligned}

∇ u = [ ∂ ∂ x ∂ ∂ y ∂ ∂ z ] [ u x u y u z ] = [ ∂ u x ∂ x ∂ u y ∂ x ∂ u z ∂ x ∂ u x ∂ y ∂ u y ∂ y ∂ u z ∂ y ∂ u x ∂ z ∂ u y ∂ z ∂ u z ∂ z ] u ∇ = [ u x u y u z ] [ ∂ ∂ x ∂ ∂ y ∂ ∂ z ] = [ ∂ u x ∂ x ∂ u x ∂ y ∂ u x ∂ z ∂ u y ∂ x ∂ u y ∂ y ∂ u y ∂ z ∂ u z ∂ x ∂ u z ∂ y ∂ u z ∂ z ] (3.10) \nabla u=\begin{bmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z} \end{bmatrix}\begin{bmatrix} u_x& u_y& u_z \end{bmatrix}=\begin{bmatrix} \frac{\partial u_x}{\partial x}& \frac{\partial u_y}{\partial x}& \frac{\partial u_z}{\partial x}\\ \frac{\partial u_x}{\partial y}& \frac{\partial u_y}{\partial y}& \frac{\partial u_z}{\partial y}\\ \frac{\partial u_x}{\partial z}& \frac{\partial u_y}{\partial z}& \frac{\partial u_z}{\partial z} \end{bmatrix}\\\quad\\ u\nabla =\begin{bmatrix}u_x\\ u_y\\ u_z \end{bmatrix}\begin{bmatrix} \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z} \end{bmatrix}=\begin{bmatrix} \frac{\partial u_x}{\partial x}& \frac{\partial u_x}{\partial y}& \frac{\partial u_x}{\partial z}\\ \frac{\partial u_y}{\partial x}& \frac{\partial u_y}{\partial y}& \frac{\partial u_y}{\partial z}\\ \frac{\partial u_z}{\partial x}& \frac{\partial u_z}{\partial y}& \frac{\partial u_z}{\partial z} \end{bmatrix}\tag{3.10}

E = 1 2 ( ∇ 0 u + u ∇ 0 + ∇ 0 u ⋅ u ∇ 0 ) (3.11) E=\frac{1}{2}(\nabla_0 u+u\nabla_0 +\nabla_0 u\cdot u\nabla_0)\tag{3.11}

e i j = 1 2 ( δ i j − ∂ x k ′ ∂ x i ∂ x k ′ ∂ x j ) = 1 2 [ δ i j − ( δ k i − ∂ u k ∂ x i ) ( δ k j − ∂ u k ∂ x j ) ] = 1 2 ( δ i j − δ k i δ k j + δ k j ∂ u k ∂ x i + δ k i ∂ u k ∂ x j − ∂ u k ∂ x i ∂ u k ∂ x j ) = 1 2 ( ∂ u j ∂ x i + ∂ u i ∂ x j − ∂ u k ∂ x i ∂ u k ∂ x j ) (3.12) \begin{aligned} e_{ij}&=\frac{1}{2}(\delta_{ij}-\frac{\partial x^{'}_k}{\partial x_i}\frac{\partial x^{'}_k}{\partial x_j}) \\ &=\frac{1}{2}[\delta_{ij}-(\delta_{ki}-\frac{\partial u_k}{\partial x_i})(\delta_{kj}-\frac{\partial u_k}{\partial x_j})]\\ &=\frac{1}{2}(\delta_{ij}-\delta_{ki}\delta_{kj}+\delta_{kj}\frac{\partial u_k}{\partial x_i}+\delta_{ki}\frac{\partial u_k}{\partial x_j}-\frac{\partial u_k}{\partial x_i}\frac{\partial u_k}{\partial x_j})\\ &=\frac{1}{2}(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}-\frac{\partial u_k}{\partial x_i}\frac{\partial u_k}{\partial x_j}) \tag{3.12} \end{aligned}

e = 1 2 ( ∇ u + u ∇ − ∇ u ⋅ u ∇ ) (3.13) e=\frac{1}{2}(\nabla u+u\nabla -\nabla u\cdot u\nabla)\tag{3.13}

∣ o a → ∣ 2 − ∣ O A → ∣ 2 = 2 E i j d x i ′ d x j ′ (3.14) \begin{aligned} |\overrightarrow{oa}|^2-|\overrightarrow{OA}|^2&=2E_{ij}dx^{'}_idx^{'}_j \end{aligned}\tag{3.14}
d s = ∣ o a → ∣ ds=|\overrightarrow{oa}| d S = ∣ O A → ∣ dS=|\overrightarrow{OA}| ，那么
d s 2 − d S 2 = 2 E i j d x i ′ d x j ′ (3.15) ds^2-dS^2=2E_{ij}dx^{'}_idx^{'}_j\tag{3.15}

d s 2 d S 2 − 1 = 2 E i j d x i ′ d S d x j ′ d S = 2 E i j α i ′ α j ′ (3.16) \begin{aligned} \frac{ds^2}{dS^2}-1&=2E_{ij}\frac{dx^{'}_i}{dS}\frac{dx^{'}_j}{dS}\\ &=2E_{ij}\alpha_i^{'}\alpha_j^{'} \end{aligned}\tag{3.16}

( ϵ + 1 ) 2 = 1 + 2 E 11 → 2 ϵ + ϵ 2 = 2 E 11 (3.17) (\epsilon+1)^2=1+2E_{11}\rightarrow2\epsilon+\epsilon^2=2E_{11}\tag{3.17}

∣ o a → ∣ ⋅ ∣ o b → ∣ − ∣ O A → ∣ ⋅ ∣ O B → ∣ = d x i δ x i − d x i ′ δ x i ′ = ∂ x i ∂ x j ′ d x j ′ ⋅ ∂ x i ∂ x k ′ δ x k ′ − d x i ′ δ x i ′ = ∂ x m ∂ x i ′ ∂ x m ∂ x j ′ d x i ′ δ x j ′ − d x i ′ δ x i ′ = ( ∂ x m ∂ x i ′ ∂ x m ∂ x j ′ − δ i j ) d x i ′ δ x j ′ = 2 E i j d x i ′ δ x j ′ (3.1’) \begin{aligned} |\overrightarrow{oa}|\cdot |\overrightarrow{ob}|-|\overrightarrow{OA}|\cdot|\overrightarrow{OB}|&=dx_i\delta x_i-dx^{'}_i\delta x^{'}_i\\ &=\frac{\partial x_i}{\partial x^{'}_j}dx^{'}_j\cdot\frac{\partial x_i}{\partial x^{'}_k}\delta x^{'}_k-dx^{'}_i\delta x^{'}_i\\ &=\frac{\partial x_m}{\partial x^{'}_i}\frac{\partial x_m}{\partial x^{'}_j}dx^{'}_i\delta x^{'}_j-dx^{'}_i \delta x^{'}_i\\ &=(\frac{\partial x_m}{\partial x^{'}_i}\frac{\partial x_m}{\partial x^{'}_j}-\delta_{ij}) dx^{'}_i\delta x^{'}_j\\ &=2E_{ij}dx^{'}_i\delta x^{'}_j \end{aligned}\tag{3.1'}

∣ o a → ∣ ⋅ ∣ o b → ∣ − ∣ O A → ∣ ⋅ ∣ O B → ∣ = d s δ s cos ⁡ θ − d S δ S cos ⁡ θ 0 = 2 E i j d x i ′ δ x j ′ (3.18) \begin{aligned} |\overrightarrow{oa}|\cdot |\overrightarrow{ob}|-|\overrightarrow{OA}|\cdot|\overrightarrow{OB}| &=ds\delta s\cos\theta-dS\delta S\cos\theta_0\\ &=2E_{ij}dx^{'}_i\delta x^{'}_j \end{aligned}\tag{3.18}

d s d S δ s δ S cos ⁡ θ − cos ⁡ θ 0 = d s d S δ s δ S cos ⁡ θ = λ a λ b sin ⁡ θ a b = 2 E i j α i α j (3.19) \begin{aligned} \frac{ds}{dS} \frac{\delta s}{\delta S}\cos\theta-\cos\theta_0 &=\frac{ds}{dS} \frac{\delta s}{\delta S}\cos\theta\\ &=\lambda^{a}\lambda^{b}\sin\theta_{ab}\\ &=2E_{ij}\alpha_i\alpha_j \end{aligned}\tag{3.19}

sin ⁡ θ a b = 2 E i j α i α j λ a λ b (3.20) \sin\theta_{ab}=\frac{2E_{ij}\alpha_i\alpha_j}{\lambda^{a}\lambda^{b}}\tag{3.20}

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