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Analytical solution for mode III dynamic rupture of standard linear viscoelastic solid with nonlinear damping

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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 third-order is obtained. Then, applying the small parameter expansion method, linearized asymptotic governing equation for each order of the small parameter is obtained. Dividing the third-order 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.

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Abbreviations

\(\bar x,\bar y\) :

dimensional abscissa and ordinate respectively (m)

l :

half length of the crack (m)

δ:

half width of the crack (m)

\(\bar t\) :

dimensional time variable (s)

\(\bar w\) :

dimensional vertical displacement (m)

\(\bar A/\rho\) :

one of the dimensional parameter in Rayleigh damping formula (s−1)

\(\bar B/\rho\) :

the other dimensional parameter in Rayleigh damping formula (s/m2)

c p :

dimensional p-wave velocity (m/s)

c a :

dimensional s-wave velocity (m/s)

k=c a/c p :

the ratio of wave velocities,k<1

\(x = \bar x/l, y = \bar y/l\) :

nondimensional abscissa and ordinate respectively

\(t = \bar tc_p /l\) :

nondimensional time variable

\(A = \bar Al/c_p \rho\) :

one dimensionless coefficient in Rayleigh damping formula

ρ:

dimensional density of mass (kg/m3)

\(B = \bar Bc_p l/\rho\) :

the other nondimensional coefficient in Rayleigh damping formula

η:

the viscous coefficient of the standard linear viscoelastic solid (N·s/m2)

E 1,E 2 :

two elastic coefficients in the standard

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Communicated by Wu Ruifeng

Foundation item: the Science Research Foundation of Yunnan Provincial Education Committee (9712063)

Biography: Fan Jiashen (1929-)

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Jiashen, F. Analytical solution for mode III dynamic rupture of standard linear viscoelastic solid with nonlinear damping. Appl Math Mech 21, 461–470 (2000). https://doi.org/10.1007/BF02463769

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  • DOI: https://doi.org/10.1007/BF02463769

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