On Volumetric Locking

It has been established that one of the main disadvantages of implementing the Finite

Element Method in solving problems in quasi-static linear elasticity. When the case of

incompressible elasticity (with =0.5) or even nearly incompressible elasticity (with

>0.4) is studied, standard finite element solutions are not satisfactory.

In a standard Finite Element Method approach to nearly incompressible, quasi-static, linear

elasticity, the weak from of the problem is defined in terms of the strong form (boundary

value problem) governing equations and the Dirichlet and Neumann boundary conditions.

In the traditional formulation of the quasi-static linear elasticity problem, the only

independent variable of interest is the displacement field, u.

The main problem in using pure displacement formulations lies in the presence of severe

stiffening near the incompressible limit. Using very refined meshes can be helpful in

arriving to an acceptable solution, but is associated with an increased computational effort.

Moreover, there is an intrinsic problem with this formulation in the near incompressible

limit. Let us remember the strong form of the problem:

div[] + ()() = ()

= tr[l] + 2l

l =1

2(grad[] + grad[]T)

While:

=E

(1 2)(1 + )

And it is noted that, in this formulation, an accurate representation of the strain tensor El

will not be achieved, as will trend towards infinity as the incompressible limit is reached.

The formulation is the, through the strain tensor, inherently associated with the volumetric

variations as the body is deformed. If these variations stop being permitted ( > 0.4),

solutions for u will be rendered inaccurate. This phenomenon is referred to as volumetric

locking, and it occurs because the formulation is not capable of representing a volume-

preserving strain field.

To avoid volumetric locking while maintaining the original formulation, one useful

technique is reduced integration. In this case, the idea is to reduce the order (accuracy) of

integration in computing the element stiffness to compensate the effects of elevated values

of . In this sense, reduced integration is a way (or trick) to stabilize the model and

continue obtaining acceptable solution on the same formulation.

However, there are methods to change the initial formulation itself. In this sense, the

problem shall be formulated with two variables of interest, the displacement field, u and

the traction t. These alternative formulations, or mixed methods, also serve as ways of

stabilizing the solution when nearing the incompressible limit.