Thermodynamic Topology Optimization for Elasto-Plastic Material Behavior

Core Concepts

A novel surrogate material model is developed to efficiently incorporate elasto-plastic material behavior into thermodynamic topology optimization, enabling the design of optimal structures under plastic deformation.

Abstract

The content presents an extension of the thermodynamic topology optimization (TTO) approach to account for non-linear elasto-plastic material behavior. The key contributions are:

Development of a novel surrogate plasticity model that computes the correct plastic strain tensor without considering dissipation-related hysteresis effects. This allows for efficient computation of the optimal structure under plastic deformation.

Formulation of the governing equations for the displacement field, plastic strains, and density variable (topology) within the TTO framework. The stationarity condition of an extended Hamilton functional yields these coupled differential-algebraic equations.

Detailed numerical implementation using a staggered scheme that combines the finite element method (FEM) for displacements and a finite difference method (FDM) for the density variable. This "neighbor element method" (NEM) approach reduces computational costs compared to a monolithic update.

Demonstration of the functionality of the proposed approach through topology optimization examples involving elasto-plastic material behavior, including the effects of hardening.

The surrogate plasticity model avoids the path-dependence and dissipation-related hysteresis of classical plasticity models, enabling efficient computation of the optimal structure under plastic deformation. The coupled TTO framework with the novel plasticity treatment allows finding optimal designs that account for the non-linear material behavior.

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Key Insights Distilled From

Thermodynamic topology optimization for hardening materials

by Miriam Kick,... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2103.03567.pdf

Deeper Inquiries

How can the proposed surrogate plasticity model be extended to account for more complex yield criteria beyond the von Mises criterion, such as the Drucker-Prager criterion

The proposed surrogate plasticity model can be extended to account for more complex yield criteria beyond the von Mises criterion, such as the Drucker-Prager criterion, by modifying the formulation of the yield function and the associated constraints.
For the Drucker-Prager criterion, the yield function is defined based on the stress components and the material parameters specific to this criterion. The plastic strains are then updated according to the new yield function, taking into account the stress state and the material behavior governed by the Drucker-Prager criterion. The derivative of the yield function with respect to the plastic strains is calculated accordingly to ensure the correct evolution of the plastic strains under this criterion.
Additionally, the constraints related to the Drucker-Prager criterion, such as the pressure-dependent yield behavior and the associated flow rule, need to be incorporated into the optimization framework. This involves adjusting the update equations for the plastic strains and the density variable to reflect the specific characteristics of the Drucker-Prager criterion. By considering these modifications, the surrogate plasticity model can effectively handle the Drucker-Prager criterion within the thermodynamic topology optimization framework.

What are the potential limitations of the assumption of volume-preserving plastic deformation, and how could this assumption be relaxed in the framework

The assumption of volume-preserving plastic deformation, while simplifying the formulation and numerical implementation, may have limitations in capturing certain material behaviors accurately. One potential limitation is that it restricts the plastic deformation to be purely deviatoric, neglecting any volumetric changes that may occur during plastic yielding. This can lead to inaccuracies in predicting the actual material response, especially in cases where volumetric changes play a significant role in the plastic behavior.
To relax this assumption within the framework, one approach could be to introduce a volumetric component to the plastic strains, allowing for both deviatoric and volumetric plastic deformation. This would involve modifying the plasticity model to include volumetric strains and considering the associated stress-strain relationships. By incorporating volumetric changes into the plastic deformation, the model can better capture the complete material response, especially in scenarios where volumetric effects are significant.

Can the thermodynamic topology optimization approach be further generalized to handle other non-linear material behaviors, such as damage or phase transformations, in addition to plasticity

The thermodynamic topology optimization approach can be further generalized to handle other non-linear material behaviors, such as damage or phase transformations, in addition to plasticity, by adapting the surrogate material model and constraints to accommodate these behaviors.
For damage modeling, the extension would involve introducing damage variables and evolving damage criteria into the optimization process. The surrogate model would need to incorporate the evolution of damage states and their influence on the material response. Constraints related to damage initiation and propagation would be included to ensure the structural integrity of the optimized design.
In the case of phase transformations, the optimization framework would need to account for the phase change kinetics and the associated energy considerations. The surrogate material model would be modified to capture the phase transformation behavior, and constraints related to phase stability and transformation kinetics would be integrated into the optimization process.
By extending the thermodynamic topology optimization approach to include these non-linear material behaviors, a more comprehensive and versatile framework can be developed to address a wider range of material responses and design requirements.