2.5 Problematic Element Behavior

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Some of the elements discussed earlier exhibit anomalies under certain conditions. This

invariably results from the approximations in the finite element formulation. These are

twofold: the real displacement field is approximated by the shape functions and the continuous

integration is replaced by a sum in discrete points. The most important anomalies that

LINEAR MECHANICAL APPLICATIONS 87

can occur in the volume elements considered here are shear locking, volumetric locking

and hourglassing.

2.5.1 Shear locking

Shear locking predominantly occurs in linear elements with full integration (8-node brick).

It results in a deformation behavior that is too stiff, that is, the displacements are too small.

This can best be explained by looking at a two-dimensional view of a beam subjected to

pure bending in Figure 2.14.

The shear force is zero and the shearing strain should everywhere be zero. However,

the linear brick element cannot model the curvature appropriately and will approximate the

deformed shape by a piecewise-linear curve (recall that the edges of an 8-node brick

element are straight). Whereas in the real deformation cross sections remain perpendicular

to the beam axis (Figure 2.14(b)), this is not necessarily the case in the finite

element approximation (Figure 2.14(c)). Figures 2.14(d) and 2.14(e) show just one element

from Figure 2.14(c). If full integration is used (2 × 2 × 2 integration points) as shown

in Figure 2.14(d), the shear strain at the integration points is not zero and a considerable

amount of energy is absorbed by the fake shearing phenomenon, not leaving enough

energy for bending: the displacements are too small. The problem can be alleviated by

using reduced integration (1 × 1 × 1 integration point) as shown in Figure 2.14(e): the

shear strain at the integration point is zero and the correct displacements result.

2.5.2 Volumetric locking

The problem of volumetric locking occurs for incompressible or nearly incompressible

behavior. It can be explained using the example in Figure 2.15. Element 1 is fixed alongside

1–2 and 1–4. It is a standard two-dimensional quadrilateral element. The material

is assumed to be incompressible. Accordingly, J = 1 everywhere. Since u1 = u2 =

u4 = v1 = v2 = v4 = 0, the displacements in the element amount to (reduce the threedimensional

shape function in Equation (2.39) to the present two-dimensional case)

u = 1

4

(1 + ξ)(1 + η)u3

v = 1

4

(1 + ξ)(1 + η)v3.

(2.134)

For simplicity, the local and global coordinates are assumed to coincide, that is, x,ξ =

y,η = 1, x,η = y,ξ = 0. Hence,

u,ξ = 1

4

(1 + η)u3

u,η = 1

4

(1 + ξ)u3

v,ξ = 1

4

(1 + η)v3

v,η = 1

4

(1 + ξ)v3.

(2.135)

88 LINEAR MECHANICAL APPLICATIONS

90◦

_= 90◦

(a)

(b)

(c)

(d)

(e)

M

Figure 2.14 The shear locking phenomenon

LINEAR MECHANICAL APPLICATIONS 89

ξ

η

1 2

4 3

Element1

Figure 2.15 Locking behavior in corner elements

Since

xk

,K

= (XLδk

L

+ uk),K = δk

K

+ uk

,K (2.136)

and

J = det(xk

,K ) (2.137)

one finds

J = 1 + 1

4

(1 + η)u3 + 1

4

(1 + ξ)v3. (2.138)

The requirement J = 1 (incompressibility) amounts to

1

4

(1 + η)u3 + 1

4

(1 + ξ)v3 = 0. (2.139)

If we take full integration, Equation (2.139) has to be satisfied at ξ, η = ±0.57:

0.39u3 + 0.39v3 = 0

0.11u3 + 0.39v3 = 0

0.39u3 + 0.11v3 = 0

0.11u3 + 0.11v3 = 0

(2.140)

which can only be satisfied if u3 = v3 = 0. This results in a zero-deformation field for

the complete element: the element locks. The same argument can be repeated for the

90 LINEAR MECHANICAL APPLICATIONS

neighboring elements (dashed line in Figure 2.15). If, on the other hand, reduced integration

is applied, there is only one integration point at ξ = η = 0. Now, there is only one equation

to satisfy:

u3 + v3 = 0 (2.141)

and no locking occurs. Therefore, it is often advantageous to use reduced integration to

avoid locking.

Another option is to use hybrid elements (Zienkiewicz and Taylor 1989), in which the

pressure is considered as an additional independent variable. Indeed, the problem is that for

truly incompressible behavior the pressure cannot be derived from the displacement field,

since increasing the hydrostatic pressure does not lead to a change in the displacement field.

The resulting hybrid elements are a special case of what are now called assumed stress

and assumed strain elements. In these elements, the stresses and/or strains are interpolated

independent of the displacements. They require the application of multifield variational principles

such as the Hu–Washizu weak form. Interested readers are referred to (Belytschko

et al. 2000).

2.5.3 Hourglassing

Hourglassing implies the existence of zero-energy modes: these are displacement modes

that do not lead to any strain or stress at the integration points. Since the field values at the

integration points are the only ones entering the integration scheme (Equation (2.106)),

hourglass modes can be added at will without disturbing the equilibrium condition

(Equation (2.1)). This usually results in wildly varying displacement fields but correct

stress and strain fields. Reducing the number of integration points naturally increases the

number of hourglassing modes. A linear brick element with reduced integration has one

integration point where six strain components prevail. However, the same element has 8

nodes leading to 24 degrees of freedom. Accordingly, 18 undetermined modes are left of

which 6 are rigid body modes leading to 12 hourglass modes in total. Figure 2.16 shows

how hourglassing for a beam under bending might look like.

To get rid of hourglassing, several stabilization methods such as the introduction of artificial

stiffness and the enhanced strain method have been proposed. For details, the reader

is referred to (Belytschko et al. 2000), (Belytschko and Bindeman 1993), (Puso 2000),

(Reese and Wriggers 2000) and (Reese 2003a). Notice that the solution in Figure 2.14(e)

to eliminate shear locking works because the shear deformation is an hourglass mode for

this one integration point. This shows that the locking phenomena and hourglassing are

intimately related, (see also (Reese 2002) for more information).

For the 20-node brick element with reduced integration, there are only (3 degrees of

freedom) × (20 nodes) − (8 integration points) × (6 strain components) − (6 rigid body

modes) = 6 hourglass modes. Because of the quadratic shape of the element sides, these

modes cannot propagate through the mesh. Consequently, hourglassing in these elements

is rare. In fully integrated brick elements, in tetrahedral elements and wedge elements,

hourglassing cannot occur.

LINEAR MECHANICAL APPLICATIONS 91

Figure 2.16 Hourglassing in a cantilever beam

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