C'est la Vie – comme ci comme ça

April 24, 2012

(Wave-01)纳维方程

Filed under: Study and Work — hitzws @ 10:18 PM

(\lambda + \mu)\nabla\theta + \mu\nabla^2\mathbf{u} + \rho\mathbf{f} = \rho\dfrac{\partial^2u}{\partial t^2}

位移矢量场方程,也称之为纳维方程,包含了弹性力学所依据的力学、几何学和物理学三方面的条件,分别为:

1、表示物体内每个微分体的运动方程;
2、变形的几何要素\mathbf{u}\theta
3、确定物体的弹性和密度的参数\lambda, \mu, \rho

式中\theta = \nabla\cdot\mathbf{u} = \dfrac{\partial u_j}{\partial x_j},即体积应变。对上式两边取散度,可将方程最终化为:

v_p^2\nabla^2\theta + \nabla\cdot\mathbf{f} = \dfrac{\partial^2\theta}{\partial t^2}

该式为传播速度为v_p的无旋场的非齐次波动方程;胀缩扰动以速度v_p传播,称之为胀缩波或弹性纵波。

采用哑指标的方式,纳维方程可以写为:

(\lambda + \mu)\dfrac{\partial^2u_j}{\partial x_j\partial x_i} + \mu\dfrac{\partial^2u_i}{\partial x_j\partial x_j} + \rho f_i = \rho\dfrac{\partial^2u_i}{\partial t^2}

取其中一种情况,如i = 1时,

(\lambda + \mu)(\dfrac{\partial^2 u_1}{\partial x_1 \partial x_1} + \dfrac{\partial^2 u_2}{\partial x_2 \partial x_1} + \dfrac{\partial^2 u_3}{\partial x_3 \partial x_1}) + \mu(\dfrac{\partial^2 u_1}{\partial x_1 \partial x_1} + \dfrac{\partial^2 u_1}{\partial x_2 \partial x_2} + \dfrac{\partial^2 u_1}{\partial x_3 \partial x_3}) + \rho f_1 = \rho\dfrac{\partial^2u_1}{\partial t^2}

合并后得到:

(\lambda + 2\mu)\dfrac{\partial^2 u_1}{\partial x_1 \partial x_1} + (\lambda + \mu)(\dfrac{\partial^2 u_2}{\partial x_2 \partial x_1} + \dfrac{\partial^2 u_3}{\partial x_3 \partial x_1}) + \mu(\dfrac{\partial^2 u_1}{\partial x_2 \partial x_2} + \dfrac{\partial^2 u_1}{\partial x_3 \partial x_3}) + \rho f_1 = \rho\dfrac{\partial^2u_1}{\partial t^2}

对于二维问题,忽略u2:

(\lambda + 2\mu)\dfrac{\partial^2 u_1}{\partial x_1 \partial x_1} + (\lambda + \mu)(\dfrac{\partial^2 u_3}{\partial x_3 \partial x_1}) + \mu(\dfrac{\partial^2 u_1}{\partial x_3 \partial x_3}) + \rho f_1 = \rho\dfrac{\partial^2u_1}{\partial t^2}

如果分别用u, v, w, x, y, z来表示u_1, u_2, u_3, x_1, x_2, x_3上式也可以写为:

(\lambda + 2\mu)\dfrac{\partial^2 u}{\partial x^2} + (\lambda + \mu)(\dfrac{\partial^2 w}{\partial z \partial x}) + \mu(\dfrac{\partial^2 u}{\partial z^2}) + \rho f_x = \rho\dfrac{\partial^2u}{\partial t^2}

当i = 2时

(\lambda + 2\mu)\dfrac{\partial^2 w}{\partial z^2} + (\lambda + \mu)(\dfrac{\partial^2 u}{\partial z \partial x}) + \mu(\dfrac{\partial^2 w}{\partial x^2}) + \rho f_z = \rho\dfrac{\partial^2w}{\partial t^2}

April 15, 2012

Protected: (Piezo-13)高频情况下的剪力滞解

Filed under: Study and Work — hitzws @ 10:44 PM

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