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Testing for Nonlinear Material Models for CAE

In many engineering applications, materials experience permanent plastic deformation before failure. Accurately simulating such behavior in FEA (finite element analysis) requires the use of nonlinear material models, especially for plastics and some metals that do not conform well to simple linear assumptions. However, modeling these materials is complex due to their nonlinear, time-dependent, and often poorly defined mechanical behavior.

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Assumptions of Nonlinear Material Models

Nonlinear behavior in FEA is typically modeled using an elastic-plastic decomposition of the stress-strain curve:

ε tot = ε e + ε p

where:

ε tot = total strain ε e = elastic (recoverable) strain ε p = plastic (irrecoverable) strain

This model assumes:

  • Elastic behavior is linear and fully recoverable up to a defined yield point.
  • Beyond yield, plastic deformation accumulates.
  • The elastic modulus remains constant in the elastic region.

This approach is generally accurate for metals, which show clear yield points and linear elasticity. For non-metals, especially plastics, these assumptions often introduce significant artifacts that reduce simulation accuracy.

Limitations of Nonlinear Material Models for Plastics

Plastics behave very differently than metals:

  • Significant strain (often 5–7%) occurs before yield.
  • Much of this early strain is plastic, meaning permanent deformation.
  • Nonlinear elasticity dominates before the onset of necking or failure.
  • The elastic modulus decreases continuously with strain.
  • Plasticity begins before the yield point, contrary to metal-based assumptions.

As illustrated in research by Courtney, the plastic stress-strain behavior of plastics can be divided into three regions:

  1. Viscoelastic region (recoverable strain).
  2. Nonlinear elastic limit, beyond which plastic flow (shear bands, crazing) begins.
  3. Necking region, characterized by localized deformation and material instability.

These behaviors make it difficult to:

  • Distinguish between elastic and plastic strains.
  • Accurately define the true onset of plasticity.
  • Simulate post-loading recovery or damage accumulation.

Furthermore, plastics exhibit viscoelasticity — strain recovery occurs over time, not instantly. Many FEA programs only support linear viscoelastic models, which become invalid once the material exceeds the nonlinear elastic limit.

Challenges with Classic Elastic-Plastic Models

The traditional metal-based elastic-plastic model assumes a clear division between linear elasticity and post-yield plasticity. For plastics, this model oversimplifies:

  • The elastic region isn’t linear.
  • The plastic region may begin gradually.
  • Recovery may be partial and time-dependent (viscoelastic).

Thus, two fundamental compromises arise:

  1. Fidelity to plastic onset → underestimates stiffness at low stress.
  2. Fidelity to elastic stiffness → overpredicts plastic deformation.

These compromises lead to inaccurate results, particularly when using FEA for stress predictions, failure assessments, or simulations involving unloading, cyclic loading, or creep.

Approaches to Modeling Nonlinear Elastic-Plastic Behavior

To bridge the gap between theory and practice, two main modeling strategies are used to implement plasticity in FEA:

Secant Modulus-Based Elastoplasticity

This method approximates the entire elastic region (up to an arbitrary point) with a secant modulus, effectively averaging the nonlinear elasticity into a single slope. An assumed "yield point" defines when plasticity begins, and beyond this, the model treats strain as plastic.

Advantages:

  • Simpler implementation.
  • Acceptable accuracy for many monotonic loading scenarios.

Disadvantages:

  • Underestimates stiffness at low strains.
  • Cannot accurately simulate unloading or recoverable behavior.
  • Plastic strain results are not trustworthy for detailed failure or damage analysis.

This approach is a compromise, often used when high-fidelity data or advanced material models are unavailable.

Tangent Modulus-Based Elastoplasticity

In this method, the initial elastic modulus (i.e., the tangent modulus at zero strain) is used to describe the elastic portion of the stress-strain curve. The yield point is chosen — often at or near the nonlinear elastic limit — and plasticity is computed from that point onward.

A chord or tangent line is drawn to the yield point, and the rest of the curve is decomposed into plastic strain using the conventional elastic-plastic model.

Advantages:

  • Better represents initial stiffness.
  • Results in a more accurate loading response.
  • Preferred when true stress-strain data is available.

Disadvantages:

  • Still not ideal for simulating unloading or cyclic behavior.
  • Requires careful selection of yield point to avoid artificial plastic strain predictions.

This method is often favored for polymers when more precision is required in the elastic response without completely adopting advanced viscoelastic or damage models.

Conclusion

Nonlinear material modeling is essential for accurate FEA of plastics and nonlinear metals, especially where permanent deformation or post-yield behavior is important. However, the traditional elastic-plastic framework developed for metals often falls short for plastics due to:

  • Nonlinear elasticity
  • Early onset of plasticity
  • Time-dependent viscoelastic recovery

Two main modeling strategies — secant and tangent modulus-based approaches — are commonly used to approximate this complex behavior in FEA tools. While they can provide reasonable accuracy for loading simulations, they are less reliable for unloading, cyclic loading, or failure prediction.

To advance simulation accuracy, analysts must understand the material-specific stress-strain behavior, be cautious in defining yield points, and choose modeling approaches that balance complexity, data availability, and simulation goals.

To explore the topics discussed on this page further, see Hubert Lobo (Founder, DatapointLabs) and Brian Croop (CEO, DatapointLabs), Determination and Use of Material Properties for Finite Element Analysis (NAFEMS, 2016), Ch.5.

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