UDC 532.4

THEORY OF ELASTICITY

B. A. DRUYANOV

ON THE INTEGRATION OF THE EQUATIONS OF PLANE FLOW OF IDEALLY PLASTIC BODIES

(Presented by Academician Yu. N. Rabotnov, June 24, 1965)

In plastic regions, where both families of characteristics are curvilinear, the following functions of the curvilinear coordinates \(\alpha,\beta\) are to be determined: \(u, v\)—the projections of the velocity on the directions of the characteristics; \(R, S\)—the radii of curvature of the characteristics; the quantities \(\bar{x}, \bar{y}\), related to the Cartesian coordinates \(x, y\) by the formulas \(\bar{x}=x\cos\varphi+y\sin\varphi\), \(\bar{y}=-x\sin\varphi+y\cos\varphi\) (\(\varphi\) is the angle of inclination of the characteristics of the family \(\alpha\) to the axis of abscissas, \(\varphi=\alpha+\beta\)). The functions \(u, v, R, S, \bar{x}, \bar{y}\) satisfy the equation \(\partial^2 f/\partial\alpha\,\partial\beta+f=0\), for which the Riemann function is the Bessel function of the first kind of zero order

\[ J_0\!\left[2\sqrt{(\alpha-a)(\beta-b)}\right] \tag{1} \]

(\(a,b\) are parameters).

The application of analytical methods, however, is complicated by the necessity of solving a chain of boundary-value problems, since usually the plastic region has a “patchwork” form. A successive solution of boundary-value problems leads to lengthy calculations. The method presented below by way of an example considerably simplifies the matter.

Consider the problem of the initial flow of a strip under indentation by a convex smooth punch. Let the punch move translationally. The field of characteristics, generalizing the known fields \((^2)\), is shown in Fig. 1. In the region \(A_{34}A_{04}A_{15}A_{25}\) the characteristics of the family \(\beta\) are rectilinear; in the region \(A_{43}A_{40}A_{51}A_{52}\) the characteristics \(\alpha\) are rectilinear. In the remaining plastic regions the characteristics are curvilinear. The velocity distribution can be found after the actual construction of the net of characteristics in the physical plane \((^2)\). The impermeability condition makes it possible to determine the distributed pressures along the contact arc \(A_{52}A_{25}\) and the forming straight characteristics \(A_{04}A_{15}\) and \(A_{40}A_{51}\). The opening angle of the sector \(A_{24}A_{25}A_{15}A_{14}\), as well as of the sector \(A_{42}A_{52}A_{51}A_{41}\), is determined only roughly. Therefore one may assume that along the lines \(A_{40}A_{43}\), \(A_{43}A_{34}\), \(A_{34}A_{04}\) the quantities \(\bar{x}, \bar{y}\) and their derivatives are known as functions of the coordinates \(\alpha,\beta\). This makes it possible to determine \(\bar{x}, \bar{y}\), and consequently the net of characteristics in the whole plastic region.

Fig. 1

Let us determine \(\bar{x}, \bar{y}\) at the point \(M\) with coordinates \(\alpha=a,\ \beta=b\). For this it is necessary successively to determine \(\bar{x}, \bar{y}\) in the regions \(A_{43}A_{33}A_{34}\), \(A_{33}A_{34}A_{04}A_{03}\), \(A_{33}A_{43}A_{40}A_{30}\), \(A_{33}A_{03}A_{00}A_{30}\). Another possibility consists in the following. Let

\[ dU_x=\left(G\,\frac{\partial \bar{x}}{\partial\alpha}-\bar{x}\,\frac{\partial G}{\partial\alpha}\right)d\alpha+ \left(\bar{x}\,\frac{\partial G}{\partial\beta}-G\,\frac{\partial \bar{x}}{\partial\beta}\right)d\beta. \tag{1} \]

Integrating (1) along \(A_{33}A_{34}A_{43}A_{33}\), \(A_{33}A_{43}ABA_{33}\), \(A_{33}CDA_{34}A_{33}\), \(A_{33}BMCA_{33}\), we shall have

\[ \int_{A_{43}A_{33}} dU_{\bar{x}}+ \int_{A_{33}A_{34}} dU_{\bar{x}}+ \int_{A_{34}A_{43}} dU_{\bar{x}}=0, \tag{2} \]

\[ \int_{A_{33}A_{43}} dU_{\bar{x}}+ \int_{A_{43}N} dU_{\bar{x}}+ \int_{NK} dU_{\bar{x}}+ \int_{KA_{33}} dU_{\bar{x}}=0 \]

and so on.

Adding these equalities, we obtain an equality containing the integral of \(dU_{\bar{x}}\) along \(MDA_{34}A_{43}AM\) and integrals along the internal boundaries of these plastic regions, containing jumps of the functions \(\bar{x}\), \(\partial \bar{x}/\partial \alpha\), \(\partial \bar{x}/\partial \beta\).

Since \(x,y\) are continuous, \(\bar{x},\bar{y}\) are also continuous. In passing across the boundary of two regions, only the following quantities can undergo a discontinuity: \(\partial \bar{x}/\partial \alpha\) when crossing \(\beta\), and \(\partial \bar{y}/\partial \beta\) when crossing \(\alpha\). However, from (1) it is seen that \(dU_{\bar{x}}\) does not contain \(\partial \bar{x}/\partial \alpha\) on the lines \(\beta\). Similarly, \(dU_{\bar{y}}\) does not contain \(\partial \bar{y}/\partial \beta\) on the lines \(\alpha\). Consequently, after adding the equalities (2) we obtain

\[ \int_{MD} dU_{\bar{x}}+ \int_{DA_{34}} dU_{\bar{x}}+ \int_{A_{34}A_{43}} dU_{\bar{x}}+ \int_{A_{43}A} dU_{\bar{x}}+ \int_{AM} dU_{\bar{x}}=0. \tag{3} \]

Hence

\[ \bar{x}_M=\frac{1}{2}\left(\bar{x}_A+\bar{x}_D\right)+ \frac{1}{2}\int_{DA_{34}A_{43}A} dU_{\bar{x}}. \tag{4} \]

An analogous formula is obtained for \(\bar{y}_M\).

For \(a=b=0\), at the point \(A_{00}\), \(\bar{x}=x,\bar{y}=y\). Then

\[ \eta_{00}=\frac{1}{\sqrt{2}}(y_{00}-x_{00}) =\frac{1}{2\sqrt{2}}\left(\bar{y}_{40}-\bar{x}_{40}+\bar{y}_{04}-\bar{x}_{04}\right) +\frac{1}{2\sqrt{2}}\int_{A_{04}A_{34}A_{43}A_{40}} dU_{(\bar{y}-\bar{x})/\sqrt{2}}. \]

Since it must be that \(\eta_{00}=0\), we obtain an equation imposing a condition on the parameters of the problem, for example, on the position of the points \(A_{43}, A_{34}\).

Received
17 V 1965

REFERENCES

¹ R. Hill, Mathematical Theory of Plasticity, 1956. ² B. Druyanov, Journal of Applied Mechanics and Technical Physics, No. 6 (1961).