Deformation Plasticity Based Topology Optimization of Anisotropic Elastoplastic Structures

Abstract

This thesis presents a topology optimization framework for anisotropic elastoplastic structures based on the deformation theory of plasticity. Traditional methods often assume linear elasticity and isotropic material behavior, which limits their ability to model real-world materials that exhibit anisotropy and undergo plastic deformation. Although incremental elastoplastic models can accurately capture such behavior, they are computationally expensive and memory-intensive due to their path-dependent nature and the need to store internal variables across multiple load steps.

To address these challenges, this work adopts a deformation plasticity approach under the assumption of monotonic proportional loading. This results in a path-independent total stress-strain relationship, allowing the load to be applied in a single step and enabling direct computation of the final equilibrium state. This significantly reduces both memory usage and computational cost compared to incremental approaches. In addition, it also incorporates anisotropic material behavior by implementing Hill’s yield criterion in the elastoplastic formulation, making the framework applicable to materials such as rolled metals and paperboard, or natural orthotropic materials such as timber.

The proposed method is integrated into a density-based topology optimization framework with stiffness maximization as the design objective. Numerical examples demonstrate the accuracy and efficiency of the approach, validating the assumption of proportional loading and confirming its applicability to practical structural design problems. The results show that this formulation offers a competitive and computationally efficient alternative to the cumbersome incremental approach, particularly when anisotropy and plastic deformation are critical design considerations.

Keywords: Deformation Theory of Plasticity, Elastoplastic Topology Optimization, Hill Orthotropic Plasticity, Nonlinear Elasticity

Popular Abstract

"Optimization - the action of making the best or most effective use of a situation or resource."

This definition, quoted from the Oxford English Dictionary, captures a concept that’s been around far longer than the word itself. From planning the fastest route to work to squeezing toothpaste from the very end of the tube, optimization is part of our daily lives.

In engineering, optimization often means using materials in the most efficient way possible. Less material often means lower cost, lighter structures, and reduced environmental impact. One powerful method used in structural engineering is topology optimization. This technique helps determine the most efficient way to distribute material within a design space to achieve a specific goal. Typically, that goal is to make a structure as stiff as possible while using as little material as necessary. Think of designing a bridge that can carry a heavy load but weighs very little.

Traditional methods usually assume materials are elastic, meaning they return to their original shape after being loaded. Many also assume materials behave the same in every direction. However, real-world materials are rarely that simple. A common example is wood: it’s much stiffer along the grain than across it. And in many cases, materials don’t fully bounce back after being deformed. Instead, they undergo plastic deformation, which results in permanent shape changes. Ignoring these behaviors can lead to inaccurate predictions in structural performance.

Most existing models that take plastic deformation into account rely on cumbersome iterative approaches, where loads are applied gradually in small steps to simulate the deformation. While this often produces accurate results, it can be very time-consuming. This thesis introduces a more direct and simplified alternative. If the load increases proportionally, as it often does in practice, it is possible to go directly to the final state where the worst-case scenario happens. After all, why take small steps if we can just take a big one?

The result is a method that balances realistic material behavior with computational efficiency, making it more practical for real-world applications. Industries such as automotive and packaging, where materials must perform while keeping costs and weight low, stand to benefit. Whether it's helping a car frame absorb impact more effectively or designing packaging that is both lighter and stronger, this approach contributes to smarter, faster, and more efficient structural design.

• author: N'Dao, Zacharia and Dar, Sobhan
• supervisor: Matti Ristinmaa
• organization: Solid Mechanics; Department of Construction Sciences
• alternative title: Topologioptimering av Anisotropa Elastoplastiska Strukturer Baserad på Deformationsplasticitet
• course: FHLM01 20251
• year: 2025
• type: H3 - Professional qualifications (4 Years - )
• subject: Technology and Engineering
• keywords: Deformation Theory of Plasticity, Elastoplastic Topology Optimization, Hill Orthotropic Plasticity, Nonlinear Elasticity
• publication/series: TFHF-5000
• report number: TFHF-5269
• language: English
• id: 9207983
• date added to LUP: 2025-07-04 11:19:25
• date last changed: 2025-07-04 11:19:25