THEORY OF ELASTICITY

Corresponding Member of the Academy of Sciences of the USSR E. I. GRIGOLYUK

DYNAMICS OF ELASTIC-VISCOUS SHELLS AND PLATES

A general theory of vibrations of elastic-viscous shells is presented below. The shell material is assumed to be isotropic, homogeneous, and to obey a linear relation between three tensors—stress, stress rate, and strain rate. For the shell the Love–Kirchhoff hypotheses are valid. The shell is shallow; displacements of its middle surface are assumed small. A system of differential equations for the problem is obtained, which is solved for the case of a circular cylindrical shell streamlined by a supersonic gas flow along the generator.

1. Elastic-viscous shallow shells. The equilibrium equations for a small element of the shell have the form

\[ \frac{\partial N_1}{\partial x}+\frac{\partial T_2}{\partial y}+X=0,\qquad \frac{\partial N_2}{\partial y}+\frac{\partial T_1}{\partial x}+Y=0,\qquad \frac{N_1}{R_1}+\frac{N_2}{R_2}+\frac{\partial Q_1}{\partial x}+\frac{\partial Q_2}{\partial y}+Z=0, \]

\[ \frac{\partial H_1}{\partial x}+\frac{\partial M_2}{\partial y}-Q_2=0,\qquad \frac{\partial H_2}{\partial y}+\frac{\partial M_1}{\partial x}-Q_1=0; \tag{1,1} \]

here \(N_1, N_2, T_1=T_2=T,\ Q_1, Q_2,\ M_1, M_2, H_1=H_2=H\) are specific forces and moments; \(X,Y,Z\) are the components of the external surface load respectively along the orthogonal axes \(x,y,z\); \(R_1,R_2\) are the principal radii of curvature. The coordinate system coincides with the principal directions on the middle surface.

If \(\sigma_1(z),\ \sigma_2(z),\ \tau_{12}(z)\) are the stresses, then

\[ N_1=\int_{-h/2}^{h/2}\sigma_1(z)\,dz,\qquad N_2=\int_{-h/2}^{h/2}\sigma_2(z)\,dz,\qquad T=\int_{-h/2}^{h/2}\tau_{12}(z)\,dz, \]

\[ M_1=\int_{-h/2}^{h/2}\sigma_1(z)\,z\,dz,\qquad M_2=\int_{-h/2}^{h/2}\sigma_2(z)\,z\,dz,\qquad H=\int_{-h/2}^{h/2}\tau_{12}(z)\,dz. \tag{1,2} \]

For a linear homogeneous and isotropic Maxwell medium in the plane stress state we have

\[ \dot{\varepsilon}_1(z)= \frac{D_0}{2\mu}\sigma_1(z)+ \frac{1}{3}\left(\frac{D_1}{3K_\nu}-\frac{D_0}{2\mu}\right) \bigl(\sigma_1(z)+\sigma_2(z)\bigr) = \]

\[ = \frac{1}{3}\left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\sigma_1(z) - \frac{1}{3}\left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)\sigma_2(z), \]

\[ \dot{\varepsilon}_2(z)= \frac{D_0}{2\mu}\sigma_2(z)+ \frac{1}{3}\left(\frac{D_1}{3K_\nu}-\frac{D_0}{2\mu}\right) \bigl(\sigma_1(z)+\sigma_2(z)\bigr) = \]

\[ = \frac{1}{3}\left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\sigma_2(z) - \frac{1}{3}\left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)\sigma_1(z),\qquad \dot{\gamma}_{12}(z)=\frac{D_0}{\mu}\tau_{12}(z), \tag{1,3} \]

where \(\varepsilon_1(z),\varepsilon_2(z)\) are the relative strains of the surface \(z=\mathrm{const}\) along the axes \(x\) and \(y\); \(\gamma_{12}(z)\) is the shear angle of the surface \(z=\mathrm{const}\); \(K_\nu= \frac{2}{3}\mu+\lambda\) is the bulk viscosity; \(\mu,\lambda\) are rigidity coefficients; \(G\) is the shear modulus; \(K_e=2G(1+\nu)/3(1-2\nu)\); \(\nu\) is Poisson’s ratio; the dot denotes

differentiation with respect to time \(t\). Here the time operators are

\[ D_0=1+\frac{\mu}{G}\frac{\partial}{\partial t},\qquad D_1=1+\frac{K_\nu}{K_e}\frac{\partial}{\partial t}. \tag{1,4} \]

According to (1,3), the stresses are equal to

\[ \left(\frac{D_0}{2\mu}+\frac{2}{3}\frac{D_1}{K_\nu}\right)D_0\sigma_1(z) = 2\mu\left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\dot{\varepsilon}_1(z) + \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)\dot{\varepsilon}_2(z) \right], \]

\[ \left(\frac{D_0}{2\mu}+\frac{2}{3}\frac{D_1}{K_\nu}\right)D_0\sigma_2(z) = 2\mu\left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\dot{\varepsilon}_2(z) + \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)\dot{\varepsilon}_1(z) \right], \]

\[ D_0\tau_{12}(z)=\dot{\gamma}_{12}(z). \tag{1,5} \]

The Kirchhoff–Love hypotheses lead to the following expressions for the strains:

\[ \varepsilon_1(z)=\varepsilon_1-z\chi_1,\qquad \varepsilon_2(z)=\varepsilon_2-z\chi_2,\qquad \gamma_{12}(z)=\gamma_{12}-2z\chi_{12}, \tag{1,6} \]

where the relative strains \(\varepsilon_1\) and \(\varepsilon_2\) and the shear angle \(\gamma_{12}\) of the middle surface for a shallow shell with small displacements have the form

\[ \varepsilon_1=\frac{\partial u}{\partial x}-\frac{w}{R_1},\qquad \varepsilon_2=\frac{\partial v}{\partial y}-\frac{w}{R_2},\qquad \gamma_{12}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}, \tag{1,7} \]

\[ \chi_1=\frac{\partial^2 w}{\partial x^2},\qquad \chi_2=\frac{\partial^2 w}{\partial y^2},\qquad \chi_{12}=\frac{\partial^2 w}{\partial x\,\partial y}. \]

We substitute (1,5) into (1,2). Then, taking (1,6) into account, we obtain

\[ \left(\frac{D_0}{2\mu}+\frac{2}{3}\frac{D_1}{K_\nu}\right)D_0M_1 = -\frac{\mu h^3}{6} \left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\dot{\chi}_1 + \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)\dot{\chi}_2 \right], \]

\[ \left(\frac{D_0}{2\mu}+\frac{2}{3}\frac{D_1}{K_\nu}\right)D_0M_2 = -\frac{\mu h^3}{6} \left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\dot{\chi}_2 + \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)\dot{\chi}_1 \right], \]

\[ D_0H=-\frac{\mu h^3}{6}\dot{\chi}_{12},\qquad D_0T=\mu h\dot{\gamma}_{12}, \tag{1,8} \]

\[ \left(\frac{D_0}{2\mu}+\frac{2}{3}\frac{D_1}{K_\nu}\right)D_0N_1 = 2\mu h \left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\dot{\varepsilon}_1 + \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)\dot{\varepsilon}_2 \right], \]

\[ \left(\frac{D_0}{2\mu}+\frac{2}{3}\frac{D_1}{K_\nu}\right)D_0N_2 = 2\mu h \left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\dot{\varepsilon}_2 + \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)\dot{\varepsilon}_1 \right]. \]

We substitute (1,6) into (1,5), multiply both sides of the expression by \(dz\), and then by \(z\,dz\), and integrate the result over the thickness of the shell. We find

\[ \dot{\varepsilon}_1= \frac{1}{3h} \left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)N_1 - \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)N_2 \right], \]

\[ \dot{\varepsilon}_2= \frac{1}{3h} \left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)N_2 - \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)N_1 \right], \]

\[ \dot{\gamma}_{12}=\frac{1}{\mu h}D_0T,\qquad \dot{\chi}_{12}=-\frac{6}{\mu h^3}D_0H, \tag{1,9} \]

\[ \dot{\chi}_1= -\frac{4}{h^3} \left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)M_1 - \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)M_2 \right], \]

\[ \dot{\chi}_2= -\frac{4}{h^3} \left[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)M_2 - \left(\frac{D_0}{2\mu}-\frac{D_1}{3K_\nu}\right)M_1 \right]. \]

From the first three relations (1,7) we obtain the strain-compatibility equation

\[ \frac{\partial^2\varepsilon_1}{\partial y^2} + \frac{\partial^2\varepsilon_2}{\partial x^2} - \frac{\partial^2\gamma_{12}}{\partial x\,\partial y} = -\frac{1}{R_1}\frac{\partial^2 w}{\partial y^2} - \frac{1}{R_2}\frac{\partial^2 w}{\partial x^2}, \tag{1,10} \]

and after differentiation with respect to time, the compatibility equation for the strain rates,

\[ \frac{\partial^2\dot{\varepsilon}_1}{\partial y^2} + \frac{\partial^2\dot{\varepsilon}_2}{\partial x^2} - \frac{\partial^2\dot{\gamma}_{12}}{\partial x\,\partial y} = -\frac{1}{R_1}\frac{\partial^2\dot{w}}{\partial y^2} - \frac{1}{R_2}\frac{\partial^2\dot{w}}{\partial x^2}. \tag{1,11} \]

The first two equilibrium equations (1,1), for \(X=Y=0\), are satisfied by the stress function \(F\), defined as

\[ N_1=\frac{\partial^2F}{\partial y^2},\qquad N_2=\frac{\partial^2F}{\partial x^2},\qquad T=-\frac{\partial^2F}{\partial x\,\partial y}. \tag{1,12} \]

Substitute the fourth and fifth equations of (1.1) into the third, and substitute there, for \(M_1, M_2, H\), their values according to relations (1.8), and for \(N_1\) and \(N_2\), their expressions through the force function (1.12). We insert the first three dependences (1.9) into (1.11), taking (1.12) into account. As a result we obtain

\[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\nabla^4 w -\frac{6}{h^3}\left(\frac{D_0}{2\mu}+\frac{2}{3}\frac{D_1}{K_\nu}\right) \frac{D_0}{\mu} \left[ \frac{1}{R_1}\frac{\partial^2 F}{\partial y^2} +\frac{1}{R_2}\frac{\partial^2 F}{\partial x^2} +Z \right]=0, \tag{1.13} \]

\[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\nabla^4 F = -3h\left( \frac{1}{R_1}\frac{\partial^2 w}{\partial y^2} +\frac{1}{R_2}\frac{\partial^2 w}{\partial x^2} \right) \tag{1.14} \]

\[ \left(\nabla^4=\frac{\partial^4}{\partial x^4} +2\frac{\partial^4}{\partial x^2\partial y^2} +\frac{\partial^4}{\partial y^4}\right). \]

By \(Z\) in (1.13) one should understand the reduced transverse load, which in the case of a shell with initial forces \(N_1^0, N_2^0, T^0\) and the action of the external medium is equal to

\[ Z=N_1^0\frac{\partial^2 w}{\partial x^2} +N_2^0\frac{\partial^2 w}{\partial y^2} +2T^0\frac{\partial^2 w}{\partial x\partial y} +\frac{\gamma h}{g}\ddot w+p_c, \tag{1.15} \]

where \(\gamma\) is the specific weight of the shell material; \(g\) is the acceleration of gravity; \(p_c\) is the pressure of the medium on the shell surface.

From (1.13)—(1.14) one can obtain one differential equation with respect to the deflection

\[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)^2\nabla^8 w +\frac{6}{h^3}\left(\frac{D_0}{2\mu}+\frac{2}{3}\frac{D_1}{K_\nu}\right) \frac{D_0}{\mu} \left[ 3h\left( \frac{1}{R_1^2}\frac{\partial^4 w}{\partial x^4} +\frac{2}{R_1R_2}\frac{\partial^4 w}{\partial x^2\partial y^2} \right.\right. \]

\[ \left.\left. +\frac{1}{R_2^2}\frac{\partial^4 w}{\partial x^4} \right) -\left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right) \nabla^4\left( N_1^0\frac{\partial^2 w}{\partial x^2} +N_2^0\frac{\partial^2 w}{\partial y^2} +2T^0\frac{\partial^2 w}{\partial x\partial y} +\frac{\gamma h}{g}\ddot w+p_c \right) \right]=0. \tag{1.16} \]

If a new function \(\Phi\) is introduced by means of

\[ F=3h\left(\frac{1}{R_1}\frac{\partial^2\Phi}{\partial y^2} +\frac{1}{R_2}\frac{\partial^2\Phi}{\partial x^2}\right), \qquad w=-\left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\nabla^4\Phi, \tag{1.17} \]

then equation (1.14) is identically satisfied, and (1.13) takes the form

\[ \left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)^2\nabla^8\Phi +\frac{18}{h^2}\left(\frac{D_0}{2\mu}+\frac{2}{3}\frac{D_1}{K_\nu}\right) \frac{D_0}{\mu} \left\{ \frac{1}{R_1^2}\frac{\partial^4\Phi}{\partial y^4} +\frac{2}{R_1R_2}\frac{\partial^4\Phi}{\partial x^2\partial y^2} +\frac{1}{R_2^2}\frac{\partial^4\Phi}{\partial x^4} \right. \]

\[ \left. +\frac{1}{3h}\left[ N_1^0\frac{\partial^2 w}{\partial x^2} +N_2^0\frac{\partial^2 w}{\partial y^2} +2T^0\frac{\partial^2 w}{\partial x\partial y} -\frac{\gamma h}{g}\left(\frac{D_0}{\mu}+\frac{D_1}{3K_\nu}\right)\nabla^4\Phi +p_c \right] \right\}=0. \]

2. Oscillations of a cylindrical shell in a supersonic gas flow

Let a cylindrical shell be subjected to internal transverse pressure \(p\) and be externally washed by a supersonic gas flow. Then

\[ R_1=\infty,\qquad R_2=R,\qquad y=R\varphi,\qquad T^0=0,\qquad N_1^0=\frac{1}{2}pR,\qquad N_2^0=pR, \tag{2.1} \]

\[ -q=q_1\frac{\partial w}{\partial t}+q_2\frac{\partial w}{\partial x} \qquad \left(q_1=\frac{\rho U}{\sqrt{M^2-1}},\qquad q_2=\frac{\rho U^2}{\sqrt{M^2-1}}\right). \]

Here \(q\) is the additional pressure in the gas flow due to the deviation of the shell from the undisturbed cylindrical form during oscillations, corresponding to the theory of a stationary supersonic flow; \(U\) is the velocity of the undisturbed flow; \(M=U/c\); \(c\) is the speed of propagation of sound in the undisturbed flow; \(\rho\) is the density of the flow.

We substitute (2.1) into equations (1.13)—(1.14) and seek solutions of these equations in the form of traveling waves

\[ w=w_0e^{i(\omega t-kx)}\cos n\varphi,\qquad F=F_0e^{i(\omega t-kx)}\cos n\varphi, \tag{2.2} \]

where \(w_0, F, k\) are constants; \(n\) is the number of half-waves in the circumferential direction; \(\omega\) is the circular frequency of oscillation of the shell in the flow.

Substitution of (2.2) into (1.13)—(1.14) leads to the following equation for the reduced frequency \(\omega^*\) of oscillations of the shell in a flow:

\[ \omega^{*5}-i\omega^{*4}a_4-\omega^{*3}(a_{32}-ia_{31})+\omega^{*2}(a_{22}+ia_{21})+\omega^*(a_{12}-ia_{11})+ +a_{02}+ia_{01}=0. \tag{2.3} \]

Here

\[ a_4=\frac{1}{3(1-\nu^2)\pi}\left(\frac{3-\nu-\nu^2}{\mu^*}+\frac{3+2\nu-\nu^2}{3K_\nu^*}\right)+\frac{q_1^*}{\pi},\qquad a_{31}=\frac{q_2^*hk}{\pi^2}, \]

\[ \begin{aligned} a_{32}={}&\frac{\theta}{\pi^2}\left[\frac{h^2}{R^2}\frac{k^4}{(k^2+n^2/R^2)^4} +\frac{h^4}{12(1-\nu^2)}\left(k^2+\frac{n^2}{R^2}\right)^2\right] +N_1^*\frac{k^2R^2}{\pi^2}+N_2^*\frac{n^2}{\pi^2}+{}\\ &+\frac{1}{18(1-\nu^2)\pi^2} \left(\frac{11-4\nu}{2\mu^{*2}}+2\frac{7+\nu}{3K_\nu^*\mu^*} +\frac{4}{9}\frac{1+\nu}{K_\nu^{*2}}\right)+{}\\ &+\frac{1}{3(1-\nu^2)}\frac{q_1^*}{\pi^2} \left(\frac{3-\nu-\nu^2}{\mu^*}+\frac{3+2\nu-\nu^2}{3K_\nu^*}\right), \end{aligned} \]

\[ a_{22}=\frac{q_2^*hk}{3(1-\nu^2)\pi^3} \left(\frac{3-\nu-\nu^2}{\mu^*}+\frac{3+2\nu-\nu^2}{3K_\nu^*}\right), \]

\[ \begin{aligned} a_{21}={}&\frac{1}{3(1-\nu^2)\pi^3}\Bigg\{ \theta\Bigg[\frac{h^2}{R^2}\frac{k^4}{(k^2+n^2/R^2)^2} \left(\frac{2-\nu}{\mu^*}+2\frac{1+\nu}{3K_\nu^*}\right)+{}\\ &+\frac{1}{6}h^4\left(k^2+\frac{n^2}{R^2}\right)^2 \left(\frac{1}{\mu^*}+\frac{1}{3K_\nu^*}\right)\Bigg] +\left(\frac{3-\nu-\nu^2}{\mu^*}+\frac{3+2\nu-\nu^2}{3K_\nu^*}\right) \left(N_1^*R^2k^2+{} \right.\\ &\left. +N_2^*n^2\right) +\frac{1}{6}\frac{1}{\mu^*}\left(\frac{1}{2\mu^*}+\frac{2}{3K_\nu^*}\right) \left(\frac{1}{\mu^*}+\frac{1}{3K_\nu^*}\right) +\frac{1}{6}q_1^* \left(\frac{11-4\nu}{2\mu^{*2}}+2\frac{7+\nu}{3K_\nu^*\mu^*} +\frac{4}{9}\frac{1+\nu}{K_\nu^{*2}}\right)\Bigg\}, \end{aligned} \]

\[ \begin{aligned} a_{12}={}&\frac{1}{6(1-\nu^2)\pi^4}\Bigg\{ \theta\Bigg[\frac{h^2}{R^2}\frac{k^4}{(k^2+n^2/R^2)^2} \frac{1}{\mu^*}\left(\frac{1}{2\mu^*}+\frac{2}{3K_\nu^*}\right)+{}\\ &+\frac{1}{48(1-\nu^2)}h^4\left(k^2+\frac{n^2}{R^2}\right)^2 \left(\frac{1}{\mu^*}+\frac{1}{3K_\nu^*}\right)^2\Bigg]+{}\\ &+\frac{1}{3} \left(\frac{11-7\nu}{2\mu^{*2}}+2\frac{7+\nu}{3K_\nu^*\mu^*} +\frac{4}{9}\frac{1+\nu}{K_\nu^{*2}}\right) \left(N_1^*R^2k^2+N_2^*n^2\right)+{}\\ &+\frac{1}{3}\frac{q_1^*}{\mu^*} \left(\frac{1}{2\mu^*}+\frac{2}{3K_\nu^*}\right) \left(\frac{1}{\mu^*}+\frac{1}{3K_\nu^*}\right) \Bigg\}, \end{aligned} \]

\[ a_{11}=\frac{q_2^*hk}{18(1-\nu^2)\pi^4} \left(\frac{11-4\nu}{2\mu^{*2}}+2\frac{7+\nu}{3K_\nu^*\mu^*} +\frac{4}{9}\frac{1+\nu}{K_\nu^{*2}}\right), \]

\[ a_{02}=\frac{q_2^*hk}{18(1-\nu^2)\pi^5}\frac{1}{\mu^*} \left(\frac{1}{2\mu^*}+\frac{2}{3K_\nu^*}\right) \left(\frac{1}{\mu^*}+\frac{1}{3K_\nu^*}\right), \]

\[ a_{01}=\frac{1}{18(1-\nu^2)\pi^5}\frac{1}{\mu^*} \left(\frac{1}{2\mu^*}+\frac{2}{3K_\nu^*}\right) \left(\frac{1}{\mu^*}+\frac{1}{3K_\nu^*}\right) \left(N_1^*R^2k^2+N_2^*n^2\right). \]

\[ \omega^*=\frac{\omega h}{\pi c},\qquad \theta=\frac{gE}{\gamma c^2},\qquad \mu^*=\frac{\mu c}{Eh},\qquad K_\nu^*=\frac{K_\nu c}{Eh},\qquad q_1^*=\frac{q_1g}{\gamma c}=\frac{\rho g}{\gamma}\frac{M}{\sqrt{M^2-1}}, \]

\[ q_2^*=\frac{q_2g}{\gamma c^2}=\frac{\rho g}{\gamma}\frac{M^2}{\sqrt{M^2-1}},\qquad N_1^*=\frac{gN_1^0h}{\gamma c^2R^2},\qquad N_2^*=\frac{gN_2^0h}{\gamma c^2R^2}. \]

In (2.3) the reduced frequency is a complex quantity.

Institute of Hydrodynamics, Siberian Branch
of the Academy of Sciences of the USSR

Received
27 I 1961

CITED LITERATURE

\(^{1}\) C. Torre, Österreich. Ing. Arch., 8, H. 1, 55 (1954).
\(^{2}\) C. Torre, Koll. Zs., 138, H. 1, 11 (1954).
\(^{3}\) J. Martinek, G. C. K. Yeh, Actes IX Congr. Intern. Mec. appl. T. 5, Univ. Bruxelles, 1957, p. 340—351, 352—359.
\(^{4}\) V. Z. Vlasov, General Theory of Shells and Its Applications in Engineering, Moscow, 1949.
\(^{5}\) A. Ferri, Aerodynamics of Supersonic Flows, 1952.