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THEORY OF ELASTICITY
Ya. S. Ufliand
A MIXED PROBLEM FOR AN ELASTIC LAYER
(Presented by Academician N. I. Muskhelishvili, 4 VIII 1958)
In the present paper an exact solution is given of a spatial problem of the theory of elasticity for an unbounded layer \((-\infty < x, y < \infty,\ 0 \leq z \leq h)\), on one boundary plane \((z = 0)\) of which elastic displacements \((u, v, w)\) are prescribed, and on the other, stresses \((\sigma_z, \tau_{zx}, \tau_{yz})\).
To solve the problem posed, we represent the displacements \(u, v, w\) in terms of the four harmonic functions \((\Phi_0, \Phi_1, \Phi_2, \Phi_3)\) of Papkovich–Neuber \((^{1,2})\).
\[ \begin{gathered} 2\mu u = -\frac{\partial F}{\partial x} + 4(1-\nu)\Phi_1,\qquad 2\mu v = -\frac{\partial F}{\partial y} + 4(1-\nu)\Phi_2,\\ 2\mu w = -\frac{\partial F}{\partial z} + 4(1-\nu)\Phi_3,\qquad F = \Phi_0 + x\Phi_1 + y\Phi_2 + z\Phi_3 \end{gathered} \tag{1} \]
\((\mu\) is the shear modulus, \(\nu\) is Poisson’s ratio).
We shall also give the expressions for the stresses entering the boundary conditions:
\[ \sigma_z = 2(1-\nu)\frac{\partial \Phi_3}{\partial z} - \frac{\partial^2 \Phi_0}{\partial z^2} + 2\nu\left( \frac{\partial \Phi_1}{\partial x} + \frac{\partial \Phi_2}{\partial y} \right) - \left( x\frac{\partial^2 \Phi_1}{\partial z^2} + y\frac{\partial^2 \Phi_2}{\partial z^2} + z\frac{\partial^2 \Phi_3}{\partial z^2} \right); \]
\[ \tau_{zx} = \frac{\partial \Phi}{\partial x} + 2(1-\nu)\frac{\partial \Phi_1}{\partial z}, \qquad \tau_{yz} = \frac{\partial \Phi}{\partial y} + 2(1-\nu)\frac{\partial \Phi_2}{\partial z}, \tag{2} \]
\[ \Phi = (1-2\nu)\Phi_3 - \frac{\partial \Phi_0}{\partial z} - \left( x\frac{\partial \Phi_1}{\partial z} + y\frac{\partial \Phi_2}{\partial z} + z\frac{\partial \Phi_3}{\partial z} \right). \]
Using the arbitrariness of one of the harmonic functions entering the solution, we supplement the boundary conditions of the problem
\[ \begin{gathered} u\big|_{z=0}=u_0(r,\varphi),\qquad v\big|_{z=0}=v_0(r,\varphi),\qquad w\big|_{z=0}=w_0(r,\varphi),\\ \sigma_z\big|_{z=h}=\sigma(r,\varphi),\qquad \tau_{zx}\big|_{z=h}=\tau_x(r,\varphi),\qquad \tau_{yz}\big|_{z=h}=\tau_y(r,\varphi) \end{gathered} \tag{3} \]
\((r,\varphi,z\) are cylindrical coordinates) by the following two additional conditions:
\[ F\big|_{z=0}=0,\qquad \Phi\big|_{z=h}=0. \tag{4} \]
In this case, for the functions \(\Phi_1\) and \(\Phi_2\) one immediately obtains separate boundary conditions:
\[ \begin{gathered} 2(1-\nu)\Phi_1\big|_{z=0}=\mu u_0(r,\varphi),\qquad 2(1-\nu)\frac{\partial \Phi_1}{\partial z}\bigg|_{z=h}=\tau_x(r,\varphi),\\ 2(1-\nu)\Phi_2\big|_{z=0}=\mu v_0(r,\varphi),\qquad 2(1-\nu)\frac{\partial \Phi_2}{\partial z}\bigg|_{z=h}=\tau_y(r,\varphi). \end{gathered} \tag{5} \]
If the functions \(\Phi_1\) and \(\Phi_2\) are regarded as found, then we arrive at a boundary-value problem for the harmonic functions \(\Phi_0\) and \(\Phi_3\) with mixed boundary conditions of the form\(^*\)
\[ \begin{gathered} \left.\Phi_0\right|_{z=0} = -\frac{\mu}{2(1-\nu)}(x u_0+y v_0) = F_1(r,\varphi), \\ \left[(3-4\nu)\Phi_3-\frac{\partial \Phi_0}{\partial z}\right]_{z=0} = 2\mu w_0+ \left(x\frac{\partial \Phi_1}{\partial z} + y\frac{\partial \Phi_2}{\partial z}\right)_{z=0} = F_2(r,\varphi), \\ \left[ 2(1-\nu)\frac{\partial \Phi_3}{\partial z} - z\frac{\partial^2\Phi_3}{\partial z^2} - \frac{\partial^2\Phi_0}{\partial z^2} \right]_{z=h} = \sigma+ \left(x\frac{\partial^2\Phi_1}{\partial z^2} + y\frac{\partial^2\Phi_2}{\partial z^2}\right)_{z=h} \\ \qquad - 2\nu\left( \frac{\partial \Phi_1}{\partial x} + \frac{\partial \Phi_2}{\partial y} \right)_{z=h} = F_3(r,\varphi), \\ \left[ (1-2\nu)\Phi_3 - z\frac{\partial \Phi_3}{\partial z} - \frac{\partial \Phi_0}{\partial z} \right]_{z=h} = \frac{x\tau_x+y\tau_y}{2(1-\nu)} = F_4(r,\varphi). \end{gathered} \tag{6} \]
The exact solution of the mixed problem of potential theory thus obtained can be found by means of the Hankel integral transform.
We shall seek the harmonic functions \(\Phi_0\) and \(\Phi_3\) in the form of the following expansions in Bessel functions:
\[ \begin{gathered} \Phi_0 = \sum_{n=-\infty}^{\infty} e^{in\varphi} \int_0^\infty \left(A_0^n \operatorname{ch}\lambda z+B_0^n \operatorname{sh}\lambda z\right) J_n(\lambda r)\,d\lambda, \\ \Phi_3 = \sum_{n=-\infty}^{\infty} e^{in\varphi} \int_0^\infty \left(A_3^n \operatorname{ch}\lambda z+B_3^n \operatorname{sh}\lambda z\right) J_n(\lambda r)\,d\lambda. \end{gathered} \tag{7} \]
Substituting (7) into (6) and representing the functions \(F_k(r,\varphi)\) \((k=1,2,3,4)\) in the form of the corresponding expansions \((^{3,4})\)
\[ \begin{gathered} F_k(r,\varphi) = \sum_{n=-\infty}^{\infty} e^{in\varphi} \int_0^\infty f_k^n(\lambda)J_n(\lambda r)\lambda\,d\lambda, \\ f_n^k(\lambda) = \frac{1}{2\pi} \int_0^\infty J_n(\lambda r)r\,dr \int_0^{2\pi} F_k(r,\varphi)e^{-in\varphi}\,d\varphi, \end{gathered} \tag{8} \]
we immediately obtain, for the unknown quantities \(A_0^n,\ B_0^n,\ A_3^n,\ B_3^n\), a system of linear algebraic equations.
Since from the first equation (6) the quantity \(B_0^n=\lambda f_1^n(\lambda)\) is found at once, and from the second equation (6) we have the relation \((3-4\nu)A_3^n-A_0^n=f_2^n(\lambda)\), the matter is in fact reduced to the solution of a system of two equations
\[ -\left[(1-2\nu)\operatorname{sh}\gamma+\gamma\operatorname{ch}\gamma\right]A_3^n + \left[2(1-\nu)\operatorname{ch}\gamma-\gamma\operatorname{sh}\gamma\right]B_3^n = \psi_1^n(\gamma), \]
\[ -\left[2(1-\nu)\operatorname{ch}\gamma+\gamma\operatorname{sh}\gamma\right]A_3^n + \left[(1-2\nu)\operatorname{sh}\gamma-\gamma\operatorname{ch}\gamma\right]B_3^n = \psi_2^n(\gamma) \tag{9} \]
with determinant
\[ D(\gamma)=(3-4\nu)\operatorname{sh}^2\gamma+\gamma^2+4(1-\nu)^2 \qquad (\gamma=\lambda h), \tag{10} \]
where the notation has been introduced
\[ \psi_1^n = \lambda f_1^n\operatorname{ch}\gamma - f_2^n\operatorname{sh}\gamma + \frac{1}{\lambda}f_3^n, \qquad \psi_2^n = \lambda f_1^n\operatorname{sh}\gamma - f_2^n\operatorname{ch}\gamma + f_4^n. \tag{11} \]
\[ \rule{0.18\textwidth}{0.4pt} \]
\(^*\) It is also required that, as \(r\to\infty\), the Papkovich–Neuber functions be of order \(1/r\), and their derivatives of order \(1/r^2\), which ensures the proper behavior of displacements and stresses at infinity. The functions prescribed in the right-hand sides of (3) must also satisfy the corresponding conditions at infinity.
As an example, let us consider a layer with a fixed base \((z=0)\), deformed by a tangential force \(T\) applied at the point \((0,0,h)\) in the direction of the \(Ox\) axis.
Since in the case under consideration \(u_0=v_0=w_0=\sigma=\tau_y=0\), we have \(\Phi_2=0\), while for \(\Phi_1\) we have the boundary conditions
\[ \left.\Phi_1\right|_{z=0}=0,\qquad \left.2(1-\nu)\frac{\partial \Phi_1}{\partial z}\right|_{z=h} =\tau_x(r,\varphi). \tag{12} \]
Distributing the force \(T\) over a circle of radius \(\varepsilon\), applying the Hankel transform and passing in the obtained solution to the limit as \(\varepsilon\to 0\), we find
\[ \Phi_1=\frac{T}{4\pi(1-\nu)} \int_0^\infty \frac{\operatorname{sh}\lambda z}{\operatorname{ch}\lambda h} J_0(\lambda r)\,d\lambda . \tag{13} \]
Substituting the expression found for \(\Phi_1\) into the right-hand sides of the basic system (6) and expanding the resulting functions in Hankel integrals, we find the following expressions for the functions \(\Phi_0\) and \(\Phi_3\):
\[ \Phi_0=\cos\varphi\int_0^\infty A_0^1\operatorname{ch}\lambda z\,J_1'(\lambda r)\,d\lambda, \]
\[ \Phi_3=\cos\varphi\int_0^\infty \left(A_3^1\operatorname{ch}\lambda z+B_3^1\operatorname{sh}\lambda z\right) J_1(\lambda r)\lambda\,d\lambda, \tag{14} \]
where
\[ A_0^1=(3-4\nu)A_3^1-\frac{Th\,\operatorname{sh}\gamma}{4\pi(1-\nu)\operatorname{ch}^2\gamma}, \]
and the quantities \(A_3^1\) and \(B_3^1\) must be found from system (9) with the following values of its right-hand sides:
\[ \psi_1^1=-\frac{Th}{4\pi(1-\nu)} \left[1+(1-2\nu)\frac{\operatorname{th}\gamma}{\gamma}\right], \qquad \psi_2^1=-\frac{Th}{4\pi(1-\nu)}\operatorname{th}\gamma . \tag{15} \]
Let us give the expression for the tangential stress \(\tau_0\equiv \tau_{zx}|_{z=0}\) in the fixed base:
\[ \tau_0=\frac{Th}{2\pi} \left\{ \frac{1}{r}\int_0^\infty \frac{\operatorname{sh}\gamma}{\operatorname{ch}^2\gamma} J_1(\lambda r)\lambda\,d\lambda + \int_0^\infty \frac{D_1(\gamma)}{D(\gamma)} \left[ \frac{J_1(\lambda r)}{r}\cos 2\varphi -\lambda J_0(\lambda r)\cos^2\varphi \right]\lambda\,d\lambda \right\}, \tag{16} \]
where the notation has been introduced
\[ D_1(\gamma)=2(1-\nu)\operatorname{sh}\gamma +\frac{\gamma}{\operatorname{ch}\gamma} -\frac{(1-2\nu)^2}{\gamma}\operatorname{th}\gamma\,\operatorname{sh}\gamma . \tag{17} \]
Physico-Technical Institute
Academy of Sciences of the USSR
Received
8 VII 1958
CITED LITERATURE
- P. F. Papkovich, Theory of Elasticity, 1939.
- G. Neuber, Stress Concentration, 1947.
- G. N. Watson, Bessel Functions, IL, 1949.
- I. Sneddon, Fourier Transforms, IL, 1955.