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Nonlinear Geometry: Cantilever beam

This benchmark assesses the nonlinear geometry (nlgeom) implementation in numgeo for both 2D (plane strain) and 3D analyses by direct comparison with reference solutions obtained using Abaqus. The primary objective is to validate the geometric formulation implemented in numgeo for material models formulated both with hyperelastic constitutive laws and with incremental (rate-type) stress updates.

The benchmark is based on a cantilever beam subjected either to prescribed displacement boundary conditions or to external loads that induce large strains and large rotations. For displacement-controlled cases, the comparison is performed in terms of nodal reaction forces, whereas for force-controlled cases the comparison is based on the resulting nodal displacements.

1. Scope and objective

The benchmark addresses two distinct classes of constitutive formulations:

  • Hyperelastic materials (Neo-Hookean):

    • stress is derived directly from a strain energy function,
    • no incremental stress update is required,
    • Hughes–Winget is not used.

  • Incremental (rate-type) constitutive models:

    • nonlinear geometry handled via the Hughes–Winget algorithm,
    • linear elasticity formulated and integrated incrementally,
    • used to validate the general nlgeom infrastructure for rate-form material models.

The purpose of this benchmark is therefore twofold:
to verify the finite-strain kinematic formulation for hyperelastic materials, and to validate the Hughes–Winget-based geometric update for incremental constitutive laws in boundary value problems that go beyond the single-element tests presented in nlgeom – single element.

2. Model specifications

In these simulations, a cantilever beam with the following properties is considered:

  • Beam length: \(L = 2\,\text{m}\)
  • Beam height: \(h = 0.2\,\text{m}\)
  • Beam width: \(d = 0.2\,\text{m}\)
  • Young’s modulus: \(E = 30000\,\text{kPa}\)
  • Poisson’s ratio: \(\nu = 0.3\)

The element edge length is set to 0.05 m, resulting in a total of 160 elements in 2D and 640 elements in 3D. To avoid locking effects, quadratically interpolated quadrilateral (2D) and hexahedral (3D) elements with reduced integration are employed. In Abaqus, these correspond to the CPE8R and C3D20R element types, while in numgeo the corresponding elements are u8-solid-red and u20-solid-red.

3. Investigated cases

Three load cases are investigated: two displacement-controlled cases and one force-controlled case.

The cases are defined as follows:

  1. Vertical displacement
    The top right-hand node(s) are displaced by \(-1.0\,\text{m}\) in the y-direction.

  2. Horizontal and vertical displacement
    The right-hand node(s) are displaced by \(-1.0\,\text{m}\) in the y-direction and by \(+1.0\,\text{m}\) in the x-direction.

  3. Point / line load
    The top right-hand node(s) are loaded by concentrated nodal forces acting in the y-direction.

Input files

The input files to reproduce this benchmark can be downloaded here.

4. Results and comparison

The combination of constitutive models and load cases introduced above results in a total of 12 benchmark simulations. The corresponding results are presented in the following subsections.

2D, Neo-Hooke

3D, Neo-Hooke

2D, Linear elastic

3D, Linear elastic

5. Conclusion

The comparisons presented above for a cantilever beam subjected to large displacements in both 2D and 3D confirm the correct implementation of the nlgeom algorithms in numgeo. This holds for both incrementally updated constitutive models and hyperelastic material formulations.