4. Measuring the stiffness of elastic packings. просмотров: 587
To calculate the efficient elastic constants for rubber pads, which are used to absorb shocks in joints of rigid elements, we use loading diagrams of the different kinds of accuracy order. Such problems are difficult because of geometric parameters, i. e. complex shapes of these elements, unknown in advance size range of contact areas, as well as mechanical or nonlinear deformation parameters of rubbers, material nonhomogenity, which is caused both by the material production peculiarities and the loaded static state. As a matter of fact, these rubber elements function as usual in dynamic regimes, being under considerable static load. Finally, rubber is almost incompressible because the Poisson ratio v for rubber materials is approximately equal to 0.5, that is why there takes place the volume-loaded state in a rubber element even under the simple deformation conditions.
Now let us consider the compression rate of a flexible material layer between the rigid surfaces. The accurate solution for this problem can be found in [Johnson, 1982]. Infinite large loads take place in the angular points, as in case there is a die block with angular points and the flat base. The infinity order of these loads is proportional to Δ–a, where Δ is the distance from the angular point, a is the fractional exponent, which takes the value from -0.43 to -0.29, depending on the friction conditions. In addition, there is a load drop near the free edge of the material.
To perform a numerical analysis we will use a somewhat different technique than in [Diveyev, Mykytiuk, 2000]. If authors considered the contact layer surface pressure as a free term in the equation, then we consider a certain homogeneous distribution of the vertical layer compression instead of the nonhomogeneous term (fig. 2).
Fig. 2. The loading diagram of a packing
Let us consider the displacement as follows
u = u0 + uijsin ((2i – 1) πz / H·y2j-1, (7)
w = w0 + wijsin((2i πz / H) ·y2j-2
Here u0, w0 are the displacements, which correspond to the homogeneous compression along z axis. According to the kinematic hypotheses (7) we consider the continuous contact between the layer and the compressing surfaces here. When inserting (7) into the variation principle given in [Diveyev, Mykytiuk, 2000] we obtain the equation
Other summands on the right-hand side are missing according to (7).м