Problems of Nonlinear Deformation

Many problems in the mechanics of deformable solids reduce or can be reduced to the solution of systems of nonlinear algebraic, transcendental, differential or integral equations containing an explicit parameter. These are problems such as static nonlinear deformation, stability, optimization, and others. A parameter appearing in these nonlinear equations may be a load parameter, a temperature field parameter, a geometric or a structural parameter, etc.

This chapter deals with some forms of the continuation method that are based on the requirement for equivalent treatment of the unknowns X ₁, X ₂, …, X _m and the problem parameter P entering in the equations. This proposition has been previously advanced in [297, 390–392]. But its practical implementation involved the solution of linearized equations using elimination procedures. This, as will be shown later, amounts to rejection of the treatment of the unknowns and the parameter on an equal basis, giving preference to one of the unknowns or a combination of them. The actual equivalent treatment of the unknowns and the parameter can only be ensured using such methods for solving linearized systems that give no preference to any one of the unknowns or the parameter. One of these methods is the ortho-gonalization procedure. Its use allows us to avoid determining the continuation parameter and is equivalent to a continuation process with the continuation parameter being chosen as the arc length of the solution set K in R _{m +1}. Moreover, the continuation process ensures maximum conditioning of the solution of linearized systems and becomes unified for regular and limit points of the set of solutions. Consequently introducing the concept of a limit point is unnecessary.

The continuation methods developed in Chapter 1, in which the unknowns and the parameter are treated on an equal basis, have a unified continuation algorithm at regular and limit points in the solution set of nonlinear system equations. From a standpoint of these forms of the continuation algorithm, it is, therefore, unnecessary to introduce the concept of a limit point. Further to the discussion in the Introduction, primary attention is given to an analysis of the behaviour of the solution in the neighbourhood of essentially singular points, i.e., points where the augmented Jacobian matrix \( \bar{J} \) is singular. As a basic method of analysis we adopt a method of expansion of the solution in a Taylor series in the neighbourhood of a singular point. This enables us to construct the bifurcation equation, and, by its analysis, to find all branches of the solution. The complexity of the analysis depends on the degree of singularity of the Jacobian matrix \( \bar{J} \). We shall consider the case of a simple singularity of the matrix \( \bar{J} \) (rank \( (\bar{J}) = m - 1 \)), which is the most important case for practical applications, and also a more complicated case of its double singularity (rank \( (\bar{J}) = m - 2 \)).

Many problems on the mechanics of deformable solids reduce to nonlinear boundary value problems with a parameter for ordinary differential equations. Some examples of such problems will be considered in the next chapter. Here we shall formulate algorithms for the continuation method taking account of the specific features of these kind of boundary value problems.

In this chapter we consider the application of the methods of the previous chapter to nonlinear boundary value problems describing large deflections of arches and large axisymmetric deflections of shells of revolution. The principal aim is to demonstrate continuation algorithms, so we limit ourselves to the case of small elastic strains without any restrictions on the angles of rotation of the axis of an arch and the middle surface of a shell. In linear problems on the deformation of arches and shells it is customary to introduce normal and tangential displacements. Their introduction is justified by the similarity of the deformed to the undeformed states, and the different order of magnitude of normal and tangential displacements makes it possible to considerably simplify the governing system of equations. In the range of deformation where nonlinearity is still insignificant, say, for finite deflections, the introduction of normal and tangential displacements still offers these advantages for the most part. However, in the case of large displacements such an approach leads to cumbersome equations. The equations are comparatively simpler when the unknowns are taken to be the Cartesian coordinates of the deformed axis of an arch or the changes due to deformation in the Cartesian coordinates of the middle surface or shells of revolution. Such equations have been constructed. Large deflections of circular arches and toroidal shells have been considered on the basis of these equations.

Natural vibration and stability problems for membranes, plates and shells of non-canonical form in plan (parallelogram, trapezoid, ellipse, etc.) are often solved by the Rayleigh-Schrödinger perturbation method. A detailed survey of such solutions is given in [322, 265].