Test Large Deformation in Structural Analysis

Introduction

Large deformation refers to a state in which a material or structure undergoes changes in shape or size that are too significant to be ignored, due to external loads, temperature variations, or internal stresses.
In such cases, displacement gradients and rotations are no longer small, and linear approximations cease to be valid. Therefore, the traditional small strain assumption cannot accurately describe the behavior, and geometric nonlinearity must be taken into account.

Theory

Deformation Gradient

F (Deformation Gradient) represents how the body is deformed. It contains all information: stretching, compression, rotation, and shear.

C (Right Cauchy–Green Tensor) is obtained from F after removing the rotational part. It represents only the change of shape, such as changes in lengths and angles.

\mathbf{F} = \mathbf{I} + \nabla \mathbf{u},
\quad
\mathbf{C} = \mathbf{F}^\mathsf{T} \mathbf{F},
\quad
J = \sqrt{\det \mathbf{C}}

Green–Lagrange Strain

E (Green–Lagrange Strain) is a strain measure derived from C. It correctly describes strain even under large deformation or rotation. For small deformations, it becomes the same as the usual linear strain.

\mathbf{E}
= \frac{1}{2}\bigl(\mathbf{C} - \mathbf{I}\bigr)

Variation of the Green–Lagrange Strain

δE (Variation of the Green–Lagrange Strain) represents how the strain E changes when the displacement field is perturbed slightly. This small change is introduced by the virtual displacement δu.

\delta \mathbf{E}(\delta\mathbf{u},\mathbf{u})
= \frac{1}{2}\Bigl(
  (\nabla \delta\mathbf{u})\,\mathbf{F}
  + \bigl((\nabla \delta\mathbf{u})\,\mathbf{F}\bigr)^\mathsf{T}
\Bigr)

Second Piola–Kirchhoff Stress

S(u) (Second Piola–Kirchhoff Stress) is a stress measure defined in the reference configuration. It describes how much the material is stretched or compressed, based on the strain E. The parameters μ and λ are material constants (Lamé parameters). The formula uses the volume change J and the inverse tensor C^{-1} to represent nonlinear stress behavior.

\mathbf{S}(\mathbf{u})
= \mu\bigl(\mathbf{I} - \mathbf{C}^{-1}\bigr)
  + \lambda \,\ln J\, \mathbf{C}^{-1}

Weak Form Residual

R(u; v) (Weak Form Residual) represents the equilibrium of forces in the weak form. It balances the internal strain energy with the external loads.

R(\mathbf{u};\mathbf{v})
=
\int_{\Omega}
  \delta \mathbf{E}(\mathbf{v},\mathbf{u}) : \mathbf{S}(\mathbf{u}) \,\mathrm{d}\Omega
-
\ell_f
\int_{\Gamma_\text{right}}
  \mathbf{t} \cdot \mathbf{v} \,\mathrm{d}\Gamma
= 0

Practical Implementation

In general, textbooks state that nonlinear structural problems are solved using the Newton–Raphson method, and the explanation usually ends there. However, in practice this is often not sufficient. In fact, even a seemingly simple cantilever example failed to converge.

Why does this happen?

It is likely because nonlinear equations frequently fall into local solutions or otherwise fail to converge depending on the initial guess and loading conditions.

To address this, I adopted a method in which the Neumann boundary load is increased gradually. This is so-called load-scaling. Although this is a fairly heuristic approach, it worked quite effectively. There are certainly more sophisticated methods available, arc-length for example, but in terms of ease of implementation, this strategy offers significant value.

Result

The result of cantilever is shown below, numerically converged enough as well.