Nonlinearity

1. Material nonlinearity: constitutive relation \(\sigma(\varepsilon)\) is nonlinear
2. Geometric nonlinearity: strain-displacement relation is nonlinear (large deformation), equilibrium equations are nonlinear
3. Boundary condition nonlinearity: contact-impact

• Linear

  \[ \mathbf{K} \mathbf{u}(t) = \mathbf{P}(t) \]

  In linear analysis, results can be scaled linearly (proportionally) with the load.

• Nonlinear

  \[ \mathbf{K}(\mathbf{u}(t), t) \mathbf{u}(t) = \mathbf{P}(\mathbf{u}(t), t) \]

  If the deformation is large, one needs to account the shape change.

Nonlinear Analysis

• Linear: small deformation

  \[ \varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \]

• Nonlinear: large deformation

  \[ \varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} + \bbox[5px,border:2px dashed orange]{\frac{\partial u_k}{\partial x_i} \frac{\partial u_k}{\partial x_j}} \right) \]

Nonlinear terms can account for large deformation accurately, but its formulation is much more complicated, and the computational cost is much higher.

Contact-Impact

Contact-impact problem is one of the most difficult nonlinear problems.

• The response in contact-impact problems is non-smooth.
• When impact happens, the normal speed at the interface is incontinuous. The tangential speed may also be incontinuous if friction is involved.

Refine slave surface mesh to avoid intrusion.

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ABAQUS Nonlinear 实操

• Step
  • Nlgeom 打开
  • Automatic stabilization（增加阻尼，帮助收敛。不得已的时候用）
    • energy traction
    • damping factor
  • 修改时间步长 Incrementation（关系到收敛）
    • 初始步长调小
    • 固定步长利于画图
    • 要反复尝试
• Interaction
  • Contact

如果每一步都输出，大的模型可能输出文件很大。可以在 Field Output 中