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A study of bending waves in infinite and anisotropic plates. (English) Zbl 0898.35019
The authors obtain integral representation formulas describing the propagation of bending waves in an infinite anisotropic plate. The latter is governed by (Kirchhoff’s) equation \[ \frac{\partial ^4 w}{\partial x^4}+ A_1\frac{\partial^4 w}{\partial x^3\partial y}+ A_2\frac{\partial ^4 w}{\partial x^2\partial y^2}+ A_3\frac{\partial ^4 w}{\partial x \partial y^3}+ \frac{\partial ^4 w}{\partial y^4}+ \frac {1}{a^2} \frac{\partial ^2 w}{\partial t^2}= \frac 1D q(x,y,t), \] with the initial conditions \(w{|}_{t=0}=(\partial w/\partial t){|}_{t=0}=0\). Here \(w=w(x,y,t)\), \((x,y)\in \mathbb R^2\), is the deflection of the plate and \(t\) represents time; \(A_1,A_2,A_3,a\) and \(D\) are constants related to the stiffness, density and thickness of the plate, and \(q\) is the perpendicular load per unit area, assumed to be of the form \(q(x,y,t)=f(t)p(x,y)\). The resulting formula represents the solution \(w\) as an integral involving the Fourier transform \(\hat p\) of \(p\); in the special case of an isotropic plate (\(A_1=A_3=0,A_2=2\)) this formula can be simplified further to only contain \(p\) instead of \(\hat p\). A number of both new and well-known formulas are obtained as special cases – e.g. for \(f(t)=\delta (t)\) and \(p(x,y)=\delta (x,y)\) (the delta functions) one gets a generalization of the classical Boussinesq formula. A comparison of the results with experimental data is also included.
MSC:
35C15 Integral representations of solutions to PDEs
74K20 Plates
35Q72 Other PDE from mechanics (MSC2000)
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References:
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