Using Triangular and Tetrahedral Elements in the Practical Analysis of Elastomeric Components

This article was written by Alan R. Leewood of AC Engineering, Inc., the ABAQUS representative in the U.S. Midwest.

We are advised to use quadrilateral or brick elements when analyzing elastomeric components. The use of triangular or tetrahedral elements is strongly discouraged by various warning messages in the User's manual. What, exactly, is the reason for these warnings?

The basic issue is that solid rubber is almost incompressible: its bulk modulus is typically 200­2000 times its shear modulus, with the lower ratio resulting when filler is added to the rubber. Thus, the volume of a rubber component must remain almost constant as it is loaded. Standard, displacement-based elements cannot respond accurately to this incompressibility constraint: they "lock." Plane stress cases (including membranes and shells) are an exception because the through-thickness strain is available to keep the volume constant. ABAQUS/Standard includes "hybrid" elements to handle incompressibility. In these elements the pressure stress is treated as an additional variable, used to impose the almost constant volume constraint in an average sense over each element. This augmented formulation generally works well with quadrilateral and brick elements. But it can result in overconstrained elements if the mesh is made from triangles or tetrahedra. The resulting response is far too stiff.

Unfortunately, the geometry of many problems is too complicated to be modeled easily with brick elements. The analyst is expected to respond to the demands of concurrent engineering. He is provided with a relatively complex CAD representation of his component, usually in the form of a solid model, and is expected to provide design guidance in a short time frame. This usually involves many iterations involving changes to the geometry.

Engineers in this environment often find that they do not have the time to use the mapped meshing procedures necessary to create models with brick elements. CAD/MCAE programs, such as I-DEAS, MSC/PATRAN, and Pro/Engineer, have robust capabilities for free meshing of 3-D solid models using tetrahedra. The analyst responsible for a rubber component thus faces a dilemma: must he take the time to generate a mostly brick mesh, or can he accept the results from a mesh made of tetrahedra?

We will use the keyboard dome (used as the "spring" under the keys of a computer keyboard) illustrated in the ABAQUS brochure to demonstrate this issue. Keyboard domes are usually made from a synthetic rubber, whose behavior is nonlinear elastic and essentially incompressible. This material is modeled here with the fully incompressible Mooney-Rivlin *HYPERELASTIC model. A mesh using CAX4H (hybrid quadrilateral) elements is shown below. The keyboard dome is pushed downward by applying concentrated nodal loads to the top surface. Since the mechanics of the problem necessitate a "snap-through" behavior, the modified RIKS technique is used. The deformed geometry is also shown in the figure.

We next analyze the dome using hybrid, linear triangular elements (CAX3H). The deformed geometry is illustrated opposite. We see some unexpected local deformation: ABAQUS is clearly having a difficult time allowing the structure to deform normally while enforcing the incompressible material constraint.

The load-deflection responses for these cases are also shown opposite. As expected, the response predicted by the CAX3H elements is much "stiffer" than that of the recommended CAX4H elements. We have graphic confirmation of the warnings in the User's manual.

We next try modeling the dome with the higher-order hybrid axisymmetric triangle (CAX6H). The load-deflection response for this element essentially matches the response of the CAX4H model. At least in this particular case, the second-order triangular element seems capable of modeling fully incompressible material behavior without "locking."

Unfortunately this observation does not necessarily carry over to general problems, especially in three-dimensional cases. While the higher-order tetrahedral elements are the proper choice for some problems, they perform poorly in contact problems. Further, ABAQUS/Explicit does not contain any higher-order elements. We need an alternative.

Fortunately, there is an engineering solution to this problem in cases where the rubber volume change is not a critical parameter. The introduction of enough compressibility can improve the behavior of the tetrahedral elements so as to allow them to be used. To illustrate this point, the material model used for the dome problem is modified by the addition ofthe material's bulk flexibility, in the *HYPERELASTIC material definition. While the ratio of the material's bulk modulus, K, to its shear modulus, G, may be greater than 200, setting K/G = 10 means that the volume change will still be an order of magnitude smaller than the shape change if the maximum shear and pressure stresses are equal. The keyboard dome's load-deflection response predicted with 3-node, hybrid, triangular elements (CAX3H) with this value of K/G is shown below. It matches the behavior predicted by the preferred quadrilateral (CAX4H) element fairly well. The load-deflection response for an intermediate value of compressibility (K/G = 100) is also shown. It is too stiff.

Our experience at ACE is that results of sufficient accuracy for preliminary design work are provided with K/G ratios of order 10, unless the rubber component is well confined by stiffer components, as occurs in O-rings, elastomeric bearings, or reinforced sections of tires. As the rubber becomes more confined it becomes more important to model its relatively incompressible behavior accurately: we must use hybrid quadrilaterals or bricks. But, in cases of relatively unconfined rubber components, we can take advantage of the convenience of tetrahedron meshing and, so long as we define the rubber to have a rather low bulk modulus compared to its real behavior, we can obtain useful results.

This should not be interpreted as a license to disregard the warnings in the ABAQUS User's manual. There are counter examples where this technique will not provide reasonable results. Quadrilaterals or bricks are always the preferred selection. But, when the analyst is faced with severe time constraints and so has the choice of free meshing or no analysis at all, the preceding approach may be of practical use. As always, the analyst is responsible for extracting useful information from a less than perfect analysis approach.