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个人专栏—塑性力学
1.1 塑性力学基本概念 塑性力学基本概念
1.2 弹塑性材料的三杆桁架分析 弹塑性材料的三杆桁架分析
1.3 加载路径对桁架的影响 加载路径对桁架的影响
2.1 塑性力学——应力分析基本概念 应力分析基本概念
2.2 塑性力学——主应力、主方向、不变量 主应力、主方向、不变量
目录
- 个人专栏—塑性力学
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目录
- 个人专栏—塑性力学
应变分析 \color{blue}应变分析 应变分析
- 应变与位移的关系
如图所示，由几何方程得：
{ ε x = ∂ u ∂ x γ x y = ∂ v ∂ x + ∂ u ∂ y ε y = ∂ v ∂ y γ y z = ∂ w ∂ y + ∂ v ∂ z ε z = ∂ w ∂ z γ z x = ∂ u ∂ z + ∂ w ∂ x \begin{cases} \varepsilon_x=\frac{\partial u}{\partial x} & \gamma_{xy}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\\ \varepsilon_y=\frac{\partial v}{\partial y} & \gamma_{yz}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\\ \varepsilon_z=\frac{\partial w}{\partial z} & \gamma_{zx}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x} \end{cases} ⎩ ⎨ ⎧εx=∂x∂uεy=∂y∂vεz=∂z∂wγxy=∂x∂v+∂y∂uγyz=∂y∂w+∂z∂vγzx=∂z∂u+∂x∂w
剪应变张量表示为：
{ ε x y = 1 2 γ x y = 1 2 ∂ v ∂ x + ∂ u ∂ y ε y z = 1 2 γ y z = 1 2 ∂ w ∂ y + ∂ v ∂ z ε z x = 1 2 γ z x = 1 2 ∂ u ∂ z + ∂ w ∂ x \begin{cases} \varepsilon_{xy}=\frac{1}{2}\gamma_{xy}=\frac{1}{2}{\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}}\\ \varepsilon_{yz}=\frac{1}{2}\gamma_{yz}=\frac{1}{2}{\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}}\\ \varepsilon_{zx}=\frac{1}{2}\gamma_{zx}=\frac{1}{2}{\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}} \end{cases} ⎩ ⎨ ⎧εxy=21γxy=21∂x∂v+∂y∂uεyz=21γyz=21∂y∂w+∂z∂vεzx=21γzx=21∂z∂u+∂x∂w
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一点的应变状态：知道一点的6个独立的应变分量： ε x , ε y , ε z , γ x y , γ y z , γ z x \varepsilon_x,\varepsilon_y,\varepsilon_z,\gamma_{xy},\gamma_{yz},\gamma_{zx} εx,εy,εz,γxy,γyz,γzx,任一方向的应变即可确定，称该点的应变情况为应变状态。
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应变分量 $\varepsilon_x,\varepsilon_y,\varepsilon_z,\varepsilon_{xy},\varepsilon_{yz},\varepsilon_{zx} $构成应变张量
ε i j = [ ε x ε x y ε x z ε y x ε y ε y z ε z x ε z y ε z ] = [ ε x 1 2 γ x y 1 2 γ x z 1 2 γ y x ε y 1 2 γ y z 1 2 γ z x 1 2 γ z y ε z ] ε i j = 1 2 ( u i , j + u j , i ) \begin{gather*} \varepsilon_{ij}=\begin{bmatrix} \varepsilon_x & \varepsilon_{xy} & \varepsilon_{xz} \\ \varepsilon_{yx} & \varepsilon_y & \varepsilon_{yz} \\ \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_z \end{bmatrix}=\begin{bmatrix} \varepsilon_x & \frac{1}{2}\gamma_{xy} & \frac{1}{2}\gamma_{xz} \\ \frac{1}{2}\gamma_{yx} & \varepsilon_y & \frac{1}{2}\gamma_{yz} \\ \frac{1}{2}\gamma_{zx} & \frac{1}{2}\gamma_{zy} & \varepsilon_z \end{bmatrix}\\ \varepsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}) \end{gather*} εij= εxεyxεzxεxyεyεzyεxzεyzεz = εx21γyx21γzx21γxyεy21γzy21γxz21γyzεz εij=21(ui,j+uj,i)
- 应变张量的三个不变量
ε i j = [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] ε 3 − I 1 ε 2 + I 2 ε 2 − I 3 = 0 I 1 = ε x + ε y + ε z = ε 1 + ε 2 + ε 3 I 2 = ε x ε y + ε y ε z + ε z ε x − ε x y 2 − ε y z 2 − ε z x 2 = ε 1 ε 2 + ε 2 ε 3 + ε 3 ε 1 I 3 = ε x ε y ε z + 2 ε x y ε y z ε z x − ε x ε y z 2 − ε y ε z x 2 − ε z ε x y 2 = ε 1 ε 2 ε 3 \begin{gather*} \varepsilon_{ij}=\begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\ \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \end{bmatrix}\\ \varepsilon^3-I_1\varepsilon^2+I_2\varepsilon^2-I_3=0\\ I_1=\varepsilon_x+\varepsilon_y+\varepsilon_z=\varepsilon_1+\varepsilon_2+\varepsilon_3\\ I_2=\varepsilon_x\varepsilon_y+\varepsilon_y\varepsilon_z+\varepsilon_z\varepsilon_x-\varepsilon_{xy}^2-\varepsilon_{yz}^2-\varepsilon_{zx}^2=\varepsilon_1\varepsilon_2+\varepsilon_2\varepsilon_3+\varepsilon_3\varepsilon_1\\ I_3=\varepsilon_x\varepsilon_y\varepsilon_z+2\varepsilon_{xy}\varepsilon_{yz}\varepsilon_{zx}-\varepsilon_x\varepsilon_{yz}^2-\varepsilon_y\varepsilon_{zx}^2-\varepsilon_z\varepsilon_{xy}^2=\varepsilon_1\varepsilon_2\varepsilon_3 \end{gather*} εij= ε11ε21ε31ε12ε22ε32ε13ε23ε33 ε3−I1ε2+I2ε2−I3=0I1=εx+εy+εz=ε1+ε2+ε3I2=εxεy+εyεz+εzεx−εxy2−εyz2−εzx2=ε1ε2+ε2ε3+ε3ε1I3=εxεyεz+2εxyεyzεzx−εxεyz2−εyεzx2−εzεxy2=ε1ε2ε3
- 应变偏张量的三个不变量
ε i j = [ ε m 0 0 0 ε m 0 0 0 ε m ] + [ ε 11 − ε m ε 12 ε 13 ε 21 ε 22 − ε m ε 23 ε 31 ε 32 ε 33 − ε m ] ⏟ e i j = ε m δ i j + e i j ε m = ε x + ε y + ε z 3 = ε 1 + ε 2 + ε 3 3 ε x − ε m = 2 ε x − ε y − ε z 3 , ε y − ε m = 2 ε y − ε x − ε z 3 , ε z − ε m = 2 ε z − ε y − ε x 3 \begin{gather*} \varepsilon_{ij}=\begin{bmatrix} \varepsilon_m & 0 & 0 \\ 0 & \varepsilon_m & 0 \\ 0 & 0 & \varepsilon_m \end{bmatrix}+\underbrace{\begin{bmatrix} \varepsilon_{11}-\varepsilon_m & \varepsilon_{12} & \varepsilon_{13} \\ \varepsilon_{21} & \varepsilon_{22}-\varepsilon_m & \varepsilon_{23} \\ \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}-\varepsilon_m \end{bmatrix}}_{e_{ij}}=\varepsilon_m\delta_{ij}+e_{ij}\\ \varepsilon_m=\frac{\varepsilon_x+\varepsilon_y+\varepsilon_z}{3}=\frac{\varepsilon_1+\varepsilon_2+\varepsilon_3}{3}\\ \varepsilon_x-\varepsilon_m=\frac{2\varepsilon_x-\varepsilon_y-\varepsilon_z}{3}, \quad \varepsilon_y-\varepsilon_m=\frac{2\varepsilon_y-\varepsilon_x-\varepsilon_z}{3}, \quad \varepsilon_z-\varepsilon_m=\frac{2\varepsilon_z-\varepsilon_y-\varepsilon_x}{3} \end{gather*} εij= εm000εm000εm +eij ε11−εmε21ε31ε12ε22−εmε32ε13ε23ε33−εm =εmδij+eijεm=3εx+εy+εz=3ε1+ε2+ε3εx−εm=32εx−εy−εz,εy−εm=32εy−εx−εz,εz−εm=32εz−εy−εx
体积应变 $\theta=\varepsilon_x+\varepsilon_y+\varepsilon_z=3\varepsilon_m $,只引起单元体的体积改变；剪切应变： $e_{ij} $只产生形状改变。
I 1 ′ = e 11 + e 22 + e 33 = 0 I 3 ′ = ∣ e i j ∣ = e 1 e 2 e 3 I 2 ′ = 1 2 e i j e j i = 1 6 [ ( e 11 − e 22 ) 2 + ( e 22 − e 33 ) 2 + ( e 33 − e 11 ) 2 ] + e 12 2 + e 23 2 + e 31 2 = 1 6 [ ( e x − e y ) 2 + ( e y − e z ) 2 + ( e z − e x ) 2 ] + 1 4 ( γ x y 2 + γ y z 2 + γ z x 2 ) = 1 6 [ ( ε 1 − ε 2 ) 2 + ( ε 2 − ε 3 ) 2 + ( ε 3 − ε 1 ) 2 ] \begin{align*} I_1^{'}&=e_{11}+e_{22}+e_{33}=0\quad I_3^{'}=|e_{ij}|=e_1e_2e_3\\ I_2^{'}&=\frac{1}{2}e_{ij}e_{ji}=\frac{1}{6}[(e_{11}-e_{22})^2+(e_{22}-e_{33})^2+(e_{33}-e_{11})^2]+e_{12}^2+e_{23}^2+e_{31}^2\\ &=\frac{1}{6}[(e_x-e_y)^2+(e_y-e_z)^2+(e_z-e_x)^2]+\frac{1}{4}(\gamma_{xy}^2+\gamma_{yz}^2+\gamma_{zx}^2)\\ &=\frac{1}{6}[(\varepsilon_1-\varepsilon_2)^2+(\varepsilon_2-\varepsilon_3)^2+(\varepsilon_3-\varepsilon_1)^2] \end{align*} I1′I2′=e11+e22+e33=0I3′=∣eij∣=e1e2e3=21eijeji=61[(e11−e22)2+(e22−e33)2+(e33−e11)2]+e122+e232+e312=61[(ex−ey)2+(ey−ez)2+(ez−ex)2]+41(γxy2+γyz2+γzx2)=61[(ε1−ε2)2+(ε2−ε3)2+(ε3−ε1)2]
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八面体剪应变：与三个应变主轴方向具有相同倾角平面上的剪应变
{ ε 8 = 1 3 ( ε 1 + ε 2 + ε 3 ) γ 8 = 2 3 ( ε 1 − ε 2 ) 2 + ( ε 2 − ε 3 ) 2 + ( ε 3 − ε 1 ) 2 = 8 3 I 2 ′ \begin{cases} \varepsilon_8=\frac{1}{3}(\varepsilon_1+\varepsilon_2+\varepsilon_3)\\ \gamma_8=\frac{2}{3}\sqrt{(\varepsilon_1-\varepsilon_2)^2+(\varepsilon_2-\varepsilon_3)^2+(\varepsilon_3-\varepsilon_1)^2}=\sqrt{\frac{8}{3}I_2^{'}} \end{cases} {ε8=31(ε1+ε2+ε3)γ8=32(ε1−ε2)2+(ε2−ε3)2+(ε3−ε1)2=38I2′ -
Lode应变参数
μ ε = 2 ε 2 − ε 1 − ε 3 ε 1 − ε 3 = 2 ε 2 − ε 3 ε 1 − ε 3 − 1 \mu_{\varepsilon}=\frac{2\varepsilon_2-\varepsilon_1-\varepsilon_3}{\varepsilon_1-\varepsilon_3}=2\frac{\varepsilon_2-\varepsilon_3}{\varepsilon_1-\varepsilon_3}-1 με=ε1−ε32ε2−ε1−ε3=2ε1−ε3ε2−ε3−1
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单向拉伸 ε 1 = ε , ε 2 = ε 3 = − 0.5 ε , μ ε = − 1 \varepsilon_1=\varepsilon,\quad \varepsilon_2=\varepsilon_3=-0.5\varepsilon,\quad \mu_{\varepsilon}=-1 ε1=ε,ε2=ε3=−0.5ε,με=−1
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单向压缩 $\varepsilon_3=-\varepsilon,\quad \varepsilon_2=\varepsilon_1=0.5\varepsilon,\quad \mu_{\varepsilon}=1 $
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纯剪切 $\varepsilon_1=0.5\gamma,\quad \varepsilon_2=0,\quad \varepsilon_3=-0.5\gamma,\quad \mu_{\varepsilon}=0 $
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等效应变
ε ˉ = 4 3 I 2 ′ = 2 3 ( e 1 − e 2 ) 2 + ( e 2 − e 3 ) 2 + ( e 3 − e 1 ) 2 = 2 3 e 1 2 + e 2 2 + e 3 2 \begin{align*} \bar{\varepsilon}&=\sqrt{\frac{4}{3}I_2^{'}}\\ &=\frac{\sqrt{2}}{3}\sqrt{(e_1-e_2)^2+(e_2-e_3)^2+(e_3-e_1)^2}\\ &=\sqrt{\frac{2}{3}}\sqrt{e_1^2+e_2^2+e_3^2} \end{align*} εˉ=34I2′=32(e1−e2)2+(e2−e3)2+(e3−e1)2=32e12+e22+e32
应用示例 \color{blue}应用示例 应用示例
- 已知位移分量 $u=(2x+y)a,v=(2y+x)a,w=-az $，求：应变张量并分解应变强度。
ε x = ∂ u ∂ x = 2 a , ε y = ∂ v ∂ y = 2 a , ε z = ∂ w ∂ z = − a γ x y = ∂ u ∂ y + ∂ v ∂ x = 2 a , ε x y = 1 2 γ x y = a ε y z = 1 2 ( ∂ v ∂ z + ∂ w ∂ y ) , ε z x = 1 2 ( ∂ w ∂ x + ∂ u ∂ z ) ε m = ε x + ε y + ε z 3 = a ε i j = [ 2 a a 0 a 2 a 0 0 0 − a ] = [ a 0 0 0 a 0 0 0 a ] + [ a a 0 a a 0 0 0 − 2 a ] \begin{gather*} \varepsilon_x=\frac{\partial u}{\partial x}=2a, \quad \varepsilon_y=\frac{\partial v}{\partial y}=2a, \quad \varepsilon_z=\frac{\partial w}{\partial z}=-a\\ \gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=2a, \quad \varepsilon_{xy}=\frac{1}{2}\gamma_{xy}=a\\ \varepsilon_{yz}=\frac{1}{2}(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}), \quad \varepsilon_{zx}=\frac{1}{2}(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z})\\ \varepsilon_m=\frac{\varepsilon_x+\varepsilon_y+\varepsilon_z}{3}=a\\ \varepsilon_{ij}=\begin{bmatrix} 2a & a & 0\\ a & 2a & 0\\ 0 & 0 & -a \end{bmatrix}=\begin{bmatrix} a & 0 & 0\\ 0 & a & 0\\ 0 & 0 & a \end{bmatrix}+\begin{bmatrix} a & a & 0\\ a & a & 0\\ 0 & 0 & -2a \end{bmatrix} \end{gather*} εx=∂x∂u=2a,εy=∂y∂v=2a,εz=∂z∂w=−aγxy=∂y∂u+∂x∂v=2a,εxy=21γxy=aεyz=21(∂z∂v+∂y∂w),εzx=21(∂x∂w+∂z∂u)εm=3εx+εy+εz=aεij= 2aa0a2a000−a = a000a000a + aa0aa000−2a