Analytical Solution for Mode Ⅱ Dynamic Rupture of Standard Linear Viscoelastic Solid With Nonlinear Damping
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摘要: 把非线性Rayleigh阻尼引入标准线性固体粘弹性介质的Ⅲ型破裂的控制方程中,此方程是一个偏微分积分方程;首先设法消去积分项,得到一个三阶非线性偏微分方程,然后用小参数摄动法,得出线性化的各阶渐近控制方程;把每一个具有变系数的三阶线性控制方程分解为弹性部分及剩余部份,而前者的解析解为已知,后者是一个二阶变系数线性偏微分方程;它化不成Mathieu方程,也化不成Hill方程,故采用WKBJ的方法得出其渐近的解析解。
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关键词:
- Rayleigh阻尼 /
- Ⅲ型破裂 /
- 粘弹性标准线性固体
Abstract: Introducing the nonlinear Rayleigh damping into the governing equation of the Mode Ⅲ dynamic rupture for standard viscoelastic solid,this equation is a partial differential and integral equation.First,eliminating the integral term,a PDE of order three is obtained.Then,applying the small parameter expansion method,linearized asymptotic governing equation for each order of the small parameter is obtained.Dividing the order three PDE into an elastic part with known solution,the rest part pertains to viscous effect which is neither a Mathieu equation nor a Hill one.The WKBJ method is still adopted to solve it analytically.-
Key words:
- Rayleigh damping /
- mode Ⅲ rupture /
- standard linear viscoelastic solid
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