The von Karman plate equations are a singular perturbation of the Foppl membrane equation in the asymptotic limit of small thickness. We study the role of compressive membrane solutions in the small thickness asymptotic behavior of the plate solutions. In the case of the circular plate, we have three conditions arising from the requirement that the solution be smooth at the origin.
The components of the plotted solution are the radial stress at the edge v( 1) = A (abscissa) and the radial stress at the center v(O) (ordinate). For the case of a clamped circular plate, we show that the generalization of the stretch to the 0(1) edge load is not consistent in general. Chapter 4 investigates the cause of tensile failure in the case of a clamped circular plate.
In the case that g(x) is analytic, a series expansion valid for x sufficiently small can be obtained for a singular solution of the form. The components of the solution plotted are the radial stress at the edge v(l) = .X (ab,:cissa) and v'(a) (ordinate), which are related to the circumference. From the asymptotic expansion (2.5) we have. 2.6) (the branch section of the arctangent is chosen appropriately.) Differentiation with respect to a given.
Remember that the outer solution for the voltage is visually a tensile (negative) solution v0 of the Foppl membrane equation (2.1). Inspection of the asymptotic approximation in this limit shows that the expansion is not valid for the clamped plate unless the compressive load is self-equilibrating; i.e. g(1) = 0. We are forced to conclude that the assumption of smooth 0(1) compressive radial stress is not valid in the case of the clamped circular plate.
In this chapter, the cause of expansion failure in the case of clamped circular plate is investigated.
The Main Result
As discussed in Chapter 1, we consider circular and annular plates undergoing radially symmetric deformation; a schematic deformation is shown in figure (1.1). With respect to the dimensionless variables x, u and v, the original variables are evaluated at the midplane z = 0. We will use the formulas above to discuss the validity of the assumptions used to derive the von Karman plate theory for the asymptotic extensions derived in this thesis.
We must also check the validity of the assumption that the deformations remain approximately constant throughout the thickness. If these conditions are not met, we have to question the validity of the assumption that shear deformations are negligible. We should also note that the assumption hu ~ 0 is violated for x - O(h) in the construction (3.72), where the boundary layer appeared in the center of the plate.
For numerical calculations, the plate and membrane equations are replaced by finite difference equations. As a rule, the solutions of plate equations are calculated using the HOC continuation procedure; the exception is in Chapter 3, where the asymptotic expansion for the plate solution is used as the initial guess for the iteration procedure. The von Karman plate equations were found to be particularly suitable for high-order continuation due to the simple structure of the nonlinearity.
The difference scheme used here is essentially the box scheme with the slight difference that we include the factor x3 in the difference coefficient of the second and fourth equations. Also, we note that the accuracy of the finite difference approximation for the circular membrane equations increases with a finer mesh near the origin, regardless of the smoothness of the desired solution. To define a non-uniform network, a transformation of the independent variable is chosen to help solve the difficulties we expect to encounter.
We first discuss the case where we want to obtain a numerical solution of the circular membrane equations (2.1), which is satisfactory. For example, s can represent arc length or pseudo arc length.1·2 The continuation step gives an approximation ZA, >.A to a point. The high-order continuation method described in the previous section is particularly effective for the von Karman plate equations.
In the 1m-implementation of the HOC method for the plate problem, the FM functions could easily be expressed in closed form for any M. For the membrane problem, the formulas are not simple, so we decided to evaluate FM numerically. We could expand the drain in terms of the G particles with respect to Z and .>., but for numerical purposes it is better not to do so.
The choice of pseudo-arc length parameterization is found to be suitable for follow-up purposes; for other purposes, the true arc length may be a better choice for the parameter. VAN DYKEG uses an exact expansion of the sene m terms of the Dean number K for the problem of flow through a loosely coiled pipe to study the behavior of solutions such as K ---+ oo. A first step towards achieving such an expansion is to choose a parameterization such that there are no singularities on the real line between the origin of the expansion and the point of interest.
Parameterization in terms of true arc length along a smooth solution branch allows no singularities for real values of the parameter; thus, the true arc length may be an appropriate parameter for analytical studies. For such problems, the function FM can be calculated in a way similar to the calculation for the plate equations. Thus, we expect the number of operations for a competing continuation step of order N, including the iterations for the new solution, to be O(NP · nm + N · n2m- 1 -r n3m-2).
In our calculations it has been observed that the length of a continuation step of order N is typically a multiple R(N) 2:1 of the continuation step of the first order, with the factor R being essentially independent of the mesh size for a particular application. Shape = 1, the estimate (11.17) suggests that there is an optimal choice for N independent of the grid size for a given application. For the membrane problem it is observed to be approximately 2, while for the plate problem the optimal value is approximately 7.
We cannot predict the size of the factor R(N); however, for the plate problem R(7) was often around 4 or 5. Application of this method to equations with more complicated nonlinearities may be feasible if numerical differentiation as described for the membrane problem is used to calculate the functions F, , 1•. 34;On the existence of a solution to the membrane problem of the equilibrium of a circular membrane", Prik/.