Civil Engineering Reference
In-Depth Information
6
Material Non-linearity
6.1
Introduction
Non-linear processes pose very much greater analytical problems than do the linear pro-
cesses so far considered in this topic. The non-linearity may be found in the dependence of
the equation coefficients on the solution itself or in the appearance of powers and products
of the unknowns or their derivatives.
Two main types of non-linearity can manifest themselves in finite element analysis
of solids: material non-linearity, in which the relationship between stresses and strains (or
other material properties) are complicated functions, which result in the equation coefficients
depending on the solution, and geometric non-linearity (otherwise known as “large strain”
or “large displacement” analysis), which leads to products of the unknowns in the equations.
In order to keep the present topic to a manageable size, the 11 programs described in
this chapter deal only with material non-linearity. As far as the organisation of computer
programs is concerned, material non-linearity is simpler to implement than geometric non-
linearity. However, readers will appreciate how programs could be adapted to cope with
geometric non-linearity as well (see e.g. Smith, 1997).
In practical finite element analysis two main types of solution procedure can be
adopted to model material non-linearity. The first approach, which has already been seen
in Program 4.5, involves “constant stiffness” iterations in which non-linearity is introduced
by iteratively modifying the right hand side “loads” vector. The (usually elastic) global
stiffness matrix in such an analysis is formed once only. Each iteration thus represents
an elastic analysis of the type described in Chapter 5. Convergence is said to occur when
stresses generated by the loads satisfy some stress-strain law or yield or failure criterion
within prescribed tolerances. The loads vector at each iteration consists of externally applied
loads and self-equilibrating “body loads”. The body loads have the effect of redistributing
stresses (or moments) within the system, but as they are self-equilibrating, they do not
alter the net loading on the system. The “constant stiffness” method is shown diagram-
matically in Figure 6.1. For load-controlled problems, many iterations may be required as
