Abstract Generated abstract
The paper examines plastic flow in a thin layer compressed between elastic bodies, where pressure in the layer is coupled to elastic displacements of the bounding surfaces through Green’s functions. It formulates the governing problem as a system consisting of a nonlinear first order pressure equation and an integral relation for the elastic displacement, then considers a special case with constant initial thickness and a Winkler-type elastic foundation. For this model an exact solution is obtained by the method of characteristics, and successive approximations are compared with it for a circular domain. The comparison indicates rapid convergence: under parameters chosen to resemble pressure-working conditions, the second approximation gives a displacement close to the exact value.
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THEORY OF ELASTICITY
I. A. KIIKO
EXACT SOLUTION OF A PROBLEM OF PLASTIC FLOW IN A THIN LAYER OVER ELASTIC SURFACES
(Presented by Academician A. Yu. Ishlinskii, 19 XI 1964)
Let us consider a layer of plastic material which occupies in the \(xy\)-plane a certain region \(S\) and is enclosed between the surfaces
\[ z_1=f_1(x,y)-\int_0^t V_0(t)\,dt \quad \text{and} \quad z_2=f_2(x,y), \]
bounding two elastic bodies; these bodies, approaching one another with velocity \(V_0(t)\), cause the layer to spread. We shall assume the functions \(f_1(x,y)\) and \(f_2(x,y)\) to be smooth, and their first derivatives with respect to each of the coordinates to be of order \(h_0/L \ll 1\), where \(h_0=z_1-z_2\), and \(L\) is the characteristic size of the region \(S\).
The boundary \(\Gamma\) of the region may be formed by places of abrupt change in the thickness of the layer or by grooves in the bodies, into which the material of the layer may flow, forming ribs; we shall regard \(\Gamma\) as a piecewise-smooth curve \(y_0=\varphi(x_0)\).
Under these conditions the pressure in the layer is determined from the equation (1)
\[ (\operatorname{grad} P)^2=\frac{4\tau_s^2}{h^2} \tag{1} \]
and the boundary condition
\[ \text{on } \Gamma \qquad P=2\sigma_s, \tag{2} \]
where \(P\) is the pressure; \(\sigma_s=\sqrt{3}\tau_s\) is the yield strength of the material of the layer; \(h\) is its thickness, which in the problem under consideration may be written in the form
\[ h=h_0+w_1+w_2; \]
in this expression \(w_1\) and \(w_2\) denote the normal displacements of the boundary points of the elastic bodies; they are taken as positive if directed into the bodies.
Assuming that the corresponding Green’s functions exist, we obtain
\[ w_1=\iint_S K_1(x,y,\xi,\eta)P(\xi,\eta)\,d\xi\,d\eta, \]
\[ w_2=\iint_S K_2(x,y,\xi,\eta)P(\xi,\eta)\,d\xi\,d\eta; \]
adding these equalities term by term and denoting \(w=w_1+w_2\), \(H(x,y,\xi,\eta)=2\tau_s[K_1(x,y,\xi,\eta)+K_2(x,y,\xi,\eta)]\), \(u=P/2\tau_s\), we find
\[ w=\iint_S H(x,y,\xi,\eta)u(\xi,\eta)\,d\xi\,d\eta; \tag{3} \]
equation (1) and boundary condition (2) take the form
\[ (\operatorname{grad} u)^2=\frac{1}{(h_0+w)^2}; \tag{4} \]
\[ \text{on } \Gamma \qquad u=\sqrt{3}. \tag{5} \]
The problem is reduced to the joint solution of the system (3), (4) under condition (5); in [2] it was proposed to solve this system by the method of successive approximations. We put \(w_0=0\) and from (4) find \(u_0\); substituting this into (3), we find \(w_1\), in accordance with which from (4) we determine \(u_1\), etc. Below an example is considered for which an exact solution of the system (3)—(5) is constructed and it is shown how rapidly the approximation process converges.
Put \(h_0=h_1=\mathrm{const}\) and \(H(x,y,\xi,\eta)=\lambda\delta(\xi-x,\eta-y)\); substituting this into (3), we find
\[ w(x,y)=\lambda \iint\limits_S \delta(\xi-x,\eta-y)u(\xi,\eta)\,d\xi\,d\eta=\lambda u(x,y); \]
hence it is seen that the adopted Green’s function describes the Winkler model of an elastic foundation, and from (4) we then obtain
\[ (\operatorname{grad}u)^2=1/(h_1+\lambda u)^2, \tag{6} \]
and thus the problem is reduced to the Cauchy problem for equation (6) under condition (5). The corresponding characteristic system in Monge’s notation has the form
\[ \frac{dx}{ds}=2p,\quad \frac{dy}{ds}=2q,\quad \frac{du}{ds}=2(p^2+q^2),\quad \frac{dp}{ds}=-\frac{2\lambda p}{(h_1+\lambda u)^3}, \]
\[ \frac{dq}{ds}=-\frac{2\lambda q}{(h_1+\lambda u)^3} \]
and the solution
\[ (h_1+\lambda u)^2=(h_1+\lambda\sqrt{3})^2 \pm (x_0-x)\frac{2\lambda\sqrt{1+y_0'^2}}{y_0'}, \tag{7} \]
\[ (x_0-x)=-(y_0-y)y_0',\qquad y=\varphi(x_0); \]
hence we find the final expression for the displacements
\[ \bar w=\delta u= \left[ (1+\delta\sqrt{3})^2 \pm \frac{x_0-x}{h_1}\, \frac{2\delta\sqrt{1+y_0'^2}}{y_0'} \right]^{1/2}-1, \tag{8} \]
where \(\bar w=w/h_1\), \(\delta=\lambda/h_1\). In what follows we shall assume \(\bar w_{\max}<1\); since \(u_{\max}\approx L/h_1\), the stated requirement will be satisfied if \(\delta L/h_1<1\).
We now use the method of approximations to solve the problem. Setting \(w_0=0\) and substituting this into (4), we obtain
\[ \bar w_1=\delta u_1=\delta\sqrt{3} \pm \delta\frac{x_0-x}{h_1}\frac{\sqrt{1+y_0'^2}}{y_0'}, \tag{9} \]
which coincides with the principal term in the formal expansion of (8) in the parameter \(\delta\). We construct the following approximations for a circular domain; from (7) for this case we obtain
\[ \bar w=\delta u=\left[(1+\delta\sqrt{3})^2+2\delta t\right]^{1/2}-1, \tag{10} \]
where \(t=(R-r)/h_1\), \(R\) is the radius of the circle, and from (9)
\[ \bar w_1=\delta u_1=\delta(\sqrt{3}+t). \]
Substituting this into (4), we find the second approximation
\[ \bar w_2=\delta u_2=\delta\sqrt{3}+\ln\left(1+\frac{\delta t}{1+\delta\sqrt{3}}\right); \]
analogously, the third approximation is obtained
\[ \bar{w}_3=\delta u_3=\delta\sqrt{3}+\int_1^{1+\beta t} \frac{dx}{1+\frac{1}{\alpha}\ln x} = \]
\[ =\delta\sqrt{3}+\sum_{m=0}^{\infty}(-1)^m \left[ \frac{x}{m+1} \sum_{k=0}^{m}(-1)^k(m+1)m(m-1)\ldots \right. \]
\[ \left. \ldots(m-k+1)\left(\frac{1}{\alpha}\ln x\right)^{m-k} \right]_{x=1}^{x=1+\beta t}, \]
where \(\alpha=1+\delta\sqrt{3}\), \(\beta=\delta/(1+\delta\sqrt{3})\), etc. Let, for example, \(\delta=1/30\), \(\delta R/h_1=0.7\); then for the center of the circle we successively find \(\bar{w}_1=0.758\), \(\bar{w}_2=0.556\), \(\bar{w}_3=0.614\), whereas the exact solution gives the result \(\bar{w}=0.59\). The first approximation is overestimated in comparison with the exact one by 28%, the second is underestimated by 6%, and the third exceeds the exact one by 4%. Hence it is clear that, in estimating the displacements \(w\) (in other words, the deviations from the prescribed value of the layer thickness due to elastic deformations of the tool), it is quite permissible to use the first approximation, while the second gives a practically exact result. The parameters have been chosen so as to bring the calculation closer to the real conditions of pressure-working technology \({}^{3}\).
Moscow State University
named after M. V. Lomonosov
Received
4 XI 1964
CITED LITERATURE
\({}^{1}\) A. A. Ilyushin, Prikl. matem. i mekh., No. 3 (1954). \({}^{2}\) I. A. Kiiko, DAN, 157, No. 3 (1964). \({}^{3}\) I. A. Kiiko, Inzh. zhurn. AN SSSR, 3, issue 1 (1963).