Multimaterial topology optimization for finite strain elastoplasticity: theory, methods, and applications

TL;DR

This work develops a multimaterial topology optimization framework for structures undergoing finite-strain elastoplasticity, enabling simultaneous optimization of geometry and material phases under large deformations with isochoric plastic flow. It integrates a mechanics-based elastoplastic theory, a reversed adjoint sensitivity approach, and automatic differentiation to handle history-dependent design updates, delivering accurate global responses and gradients. The method is demonstrated across 2D and 3D applications—dampers, hyperelastic–elastoplastic beams, crashworthy bumpers, and multi-stage profiled sheets—revealing mechanisms such as the shift from kinematic to isotropic hardening and the formation of stress-concentrating regions that enhance energy dissipation. Practically, the framework tailors stiffness, strength, and structural toughness by combining multiple candidate materials, and it extends to real-world constraints like cost, weight, and CO$_2$ footprint, offering a pathway to next-generation multimaterial elastoplastic structures.

Abstract

Plasticity is inherent to many engineering materials such as metals. While it can degrade the load-carrying capacity of structures via material yielding, it can also protect structures through plastic energy dissipation. To fully harness plasticity, here we present the theory, method, and application of a topology optimization framework that simultaneously optimizes structural geometries and material phases to customize the stiffness, strength, and structural toughness of designs experiencing finite strain elastoplasticity. The framework accurately predicts structural responses by employing a rigorous, mechanics-based elastoplasticity theory that ensures isochoric plastic flow. It also effectively identifies optimal material phase distributions using a gradient-based optimizer, where gradient information is obtained via a reversed adjoint method to address history dependence, along with automatic differentiation to compute the complex partial derivatives. We demonstrate the framework by optimizing a range of 2D and 3D elastoplastic structures, including energy-dissipating dampers, load-carrying beams, impact-resisting bumpers, and cold working profiled sheets. These optimized multimaterial structures reveal important mechanisms for improving design performance under large deformation, such as the transition from kinematic to isotropic hardening with increasing displacement amplitudes and the formation of twisted regions that concentrate stress, enhancing plastic energy dissipation. Through the superior performance of these optimized designs, we demonstrate the framework's effectiveness in tailoring elastoplastic responses across various spatial configurations, material types, hardening behaviors, and combinations of candidate materials. This work offers a systematic approach for optimizing next-generation multimaterial structures with elastoplastic behaviors under large deformations.

Figures (16)

• Figure 1: Multimaterial and multiobjective topology optimization for finite strain elastoplasticity. (a) Optimization setups. (b) Multimaterial topology optimization framework. The variable $J^\texttt{p}$ is the determinant of the plastic part of the deformation gradient. (c) Optimized elastoplastic designs with real-world applications.
• Figure 2: Relationships among the deformation tensors used in the finite strain elastoplasticity theory.
• Figure 3: Relationships between the primary state variables at load steps $n$ and $n+1$. The dashed arrows signify that $\widehat{\gamma}_{n+1}$ generally does not admit to an explicit expression and needs to be numerically solved from \ref{['Function of gamma']} with \ref{['Nonlinear Solution of gamma']}.
• Figure 4: Design and optimization of metallic yielding dampers. (a) Design setups: the design domain, boundary conditions, and candidate materials (bronze, steel, and void). The variable $u$ is the applied displacement. (b) Uniaxial Kirchhoff stress--Lagrangian strain ($\tau_{11}$--$E_{11}$) curves of candidate materials. (c) Optimization objective: maximizing the total energy of dampers. The variable $F$ is the reaction force. (d) Intuitive and optimized dampers. The total energy ($\Pi$) is in kN$\cdot$m. The percentages are the total energy increments compared to the intuitive design. (e) Normalized equivalent plastic strain ($\alpha/\max (\alpha)$). (f) Force--displacement ($F$--$u$) curves. (g) Energy--displacement ($\Pi$--$u$) curves.
• Figure 5: Optimized dampers under increasing applied displacements. (a) Optimized dampers. The total energy ($\Pi$) is in kN$\cdot$m, and the percentages are the total energy increments compared to the intuitive design in Fig. \ref{['Fig: Damper-Part 1']}(d). (b) Force--displacement ($F$--$u$) curves. (c) Energy--displacement ($\Pi$--$u$) curves. The insets are the normalized total energy density ($W/\max(W)$) at the final load step.

Theorems & Definitions (8)

• Remark 1
• Proposition 1
• proof
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6