1.1.8 Uniaxial stretching of an elastic sheet with a circular hole

Product: ABAQUS/Standard  

This example considers the uniform large stretching of a thin, initially square sheet containing a centrally located circular hole. Plane stress conditions are assumed, and the results are compared with those provided in Oden (1972) for four different forms of the strain energy function using the experimental results of Treloar (1944). The example demonstrates the use and verifies the results of hyperelastic and viscoelastic materials in plane stress.

Problem description

Loading and controls

The sheet is stretched to a width of 1181 mm (46.5 in)—over seven  times its initial width—in the x-direction, while the edges parallel to the x-axis are restrained from stretching in the y-direction. The y-direction restraints are imposed directly with the *BOUNDARY option. The stretch in the x-direction is prescribed by imposing uniform normal displacement on the right-hand edge of the mesh. All the nodes on that edge are constrained to have the same x-displacement by using the *EQUATION option. The displacement of the retained node (node 1601) is then prescribed to stretch the sheet. This technique allows the total stretching force to be obtained directly as the reaction force at this node. The symmetry conditions at and at are also imposed with the *BOUNDARY option.

An initial increment of 5% of the final displacement is suggested. The size of subsequent increments is chosen by the automatic incrementation scheme.

In the viscoelastic case a second step is added, driven by the *VISCO procedure. The deformation is kept the same, and the stresses relax. The time period is 100 sec, which is much larger than the time constants of the material. As a result, the long-term behavior of the material should be obtained. Setting in the expression for the time-dependent moduli provides and Since the deformation is almost completely constrained during the relaxation step, we expect the stresses to be halved in this process. A CETOL value of 0.1 is specified, which enables automatic incrementation. CETOL controls the error in the integration of the viscoelastic model by limiting the difference in the strain increments defined by forward Euler and backward Euler integrations. The value of 10% strain error per increment used here is very large and suggests that no attempt is being made to limit this source of error: rather, we are allowing the automatic time incrementation to reach the long-term (steady-state) solution as quickly as possible.

Results and discussion

Input files

References

Figures

Figure 1.1.8–1 Rubber sheet and mesh.

Figure 1.1.8–2 Final displaced configuration, Biderman model.

Figure 1.1.8–3 Applied force versus overall nominal strain.

Figure 1.1.8–4 Load versus time, Biderman model, with a relaxation period of 100 secs.