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shape-programming in Hyperelasticity Through Differential Growth

APPLIED MATHEMATICS AND OPTIMIZATION 2024年第2期89卷 49-49页

作者： Ortigosa-Martinez, Rogelio Martinez-Frutos, Jesus Mora-Corral, Carlos Pedregal, Pablo Periago, Francisco Tech Univ Cartagena Dept Appl Math & Stat Campus Muralla Mar Cartagena 30202 Murcia Spain Tech Univ Cartagena Multiphys Simulat & Optimizat Lab Campus Muralla Mar Cartagena 30202 Murcia Spain Univ Autonoma Madrid Dept Math Madrid 28049 Spain Inst Ciencias Matemat CSIC UAM UC3M UCM Madrid 28049 Spain Univ Castilla La Mancha Dept Math INEI Camilo Jose Cela2 Ciudad Real 13071 Castilla La Man Spain

This paper is concerned with the growth-driven shape-programming problem, which involves determining a growth tensor that can produce a deformation on a hyperelastic body reaching a given target shape. We consider the two cases of globally compatible growth, where the growth tensor is a deformation gradient over the undeformed domain, and the incompatible one, which discards such hypothesis. We formulate the problem within the framework of optimal control theory in hyperelasticity. The Hausdorff distance is used to quantify dissimilarities between shapes;the complexity of the actuation is incorporated in the cost functional as well. Boundary conditions and external loads are allowed in the state law, thus extending previous works where the stress-free hypothesis turns out to be essential. A rigorous mathematical analysis is then carried out to prove the well-posedness of the problem. The numerical approximation is performed using gradient-based optimisation algorithms. Our main goal in this part is to show the possibility to apply inverse techniques for the numerical approximation of this problem, which allows us to address more generic situations than those covered by analytical approaches. Several numerical experiments for beam-like and shell-type geometries illustrate the performance of the proposed numerical scheme.

关键词： Soft robotics Differential growth Hyperelasticity shape-programming Optimal control Numerical simulation methods

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A theoretical scheme for shape-programming of thin hyperelastic plates through differential growth

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MATHEMATICS AND MECHANICS OF SOLIDS 2022年第8期27卷 1412-1428页

作者： Wang, Jiong Li, Zhanfeng Jin, Zili South China Univ Technol Sch Civil Engn & Transportat Guangzhou 510640 Guangdong Peoples R China South China Univ Technol State Key Lab Subtrop Bldg Sci Guangzhou Peoples R China

In this paper, a theoretical scheme is proposed for shape-programming of thin hyperelastic plates through differential growth. First, starting from the 3D governing system of a hyperelastic (neo-Hookean) plate, a consistent finite-strain plate equation system is formulated through a series expansion and truncation approach. Based on the plate equation system, the problem of shape-programming is studied under the stress-free assumption. By equating the stress components in the plate equations to be zero, the explicit relations between the growth functions and the geometrical quantities of target shape of the plate are derived. Then, a theoretical scheme of shape-programming is proposed, which can be used to identify the growth fields corresponding to arbitrary 3D shapes of the plate. To demonstrate the efficiency of the scheme, some typical examples are studied. The predicted growth functions in these examples are adopted in the numerical simulations, from which the target shapes of the plate can be recovered completely. The scheme of shape-programming proposed in the current work is applicable for manufacturing intelligent soft devices.

关键词： Hyperelastic plate differential growth finite-strain plate theory analytical results shape-programming

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A general theoretical scheme for shape-programming of incompressible hyperelastic shells through differential growth

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INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES 2023年第1期265卷

作者： Li, Zhanfeng Wang, Jiong Hossain, Mokarram Kadapa, Chennakesava South China Univ Technol Sch Civil Engn & Transportat Guangzhou 510640 Peoples R China South China Univ Technol State Key Lab Subtrop Bldg Sci Guangzhou 510640 Peoples R China Swansea Univ Zienkiewicz Ctr Computat Engn ZCCE Swansea SA1 8EN Wales Edinburgh Napier Univ Sch Engn & Built Environm Edinburgh EH10 5DT Scotland

In this paper, we study the problem of shape-programming of incompressible hyperelastic shells through differential growth. The aim of the current work is to determine one of the possible growth tensors (or growth functions) that can produce the deformation of a shell to the desired shape. First, a consistent finite-strain shell theory is introduced. The shell equation system is established from the 3D governing system through a series expansion and truncation approach. Based on the shell theory, the problem of shape-programming is studied under the stress-free assumption. For a special case in which the parametric coordinate curves generate a net of curvature lines on the target surface, the sufficient condition to ensure the vanishing of the stress components is analyzed, from which the explicit expression of the growth tensor can be derived. In the general case, we conduct the variable changes and derive the total growth tensor by considering a two-step deformation of the shell. With these obtained results, a general theoretical scheme for shape-programming of thin hyperelastic shells through differential growth is proposed. To demonstrate the feasibility and efficiency of the proposed scheme, several typical examples are studied. The derived growth tensors in these examples have also been implemented in the numerical simulations to verify their correctness and accuracy. The simulation results show that the target shapes of the shell samples can be recovered completely. The scheme for shape-programming proposed in the current work is helpful in designing and manufacturing intelligent soft devices.

关键词： Hyperelastic shell Differential growth shape-programming Theoretical scheme Numerical simulations

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General shape transformations of thin hyperelastic shells through stress-free differential growth

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INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE 2025年 213卷

作者： Li, Zhanfeng Wang, Jiong South China Univ Technol Sch Civil Engn & Transportat Guangzhou 510641 Peoples R China South China Univ Technol State Key Lab Subtrop Bldg Sci Guangzhou 510641 Peoples R China

To address the needs of engineering applications, researchers often wish that the shapes of samples can be precisely controlled. The current work aims to propose a promising approach, i.e., through stress-free differential growth, to realize general shape transformations of thin hyperelastic shells. First, within the finite-strain regime, we formulate the 3D governing equations system for modeling the growth behavior of hyperelastic shells. To facilitate the derivations, it is assumed that the shell sample attains the stress-free state in its current configuration. Subsequently, through series expansions of the unknown variables, we derive the explicit analytical formulas that elucidate the intricate relationships between growth functions and the geometric quantities of general 3D target surfaces. Based on these analytical formulas, we propose a theoretical framework for controlling the shape changes of the shell sample from the reference configuration to a desired target configuration. Notably, our framework accommodates a wide array of geometric mappings, including topology transformation, conformal mapping, and isometry mapping. To promote applications of the theoretical framework, a numerical scheme is further proposed to achieve shape transformations of shell samples between complex surfaces without explicit parametric equations. Both the theoretical framework and the numerical scheme are validated through 3D finite element simulations. The results of the current work can be applied for the design of novel intelligent soft devices, which also reveal the connections between solid mechanics and differential geometry.

关键词： Hyperelastic shell Differential growth shape-programming Theoretical framework Numerical simulation

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A general multi-layered hyperelastic plate theory for growth-induced deformations in soft material samples

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APPLIED MATHEMATICAL MODELLING 2023年 115卷 300-336页

作者： Du, Ping Li, Zhanfeng Chen, Xiaoyi Wang, Jiong South China Univ Technol Sch Civil Engn & Transportat Guangzhou 510640 Peoples R China South China Univ Technol State Key Lab Subtrop Bldg Sci Guangzhou 510640 Peoples R China BNU HKBU United Int Coll Div Sci & Technol 2000 Jintong Rd Zhuhai 519087 Guangdong Peoples R China

In this paper, we propose a general multi-layered hyperelastic plate theory to study the growth-induced deformations of soft material samples. First, by considering a multilayered plate sample with an arbitrary geometrical shape and adopting a general hyperelastic constitutive model, the 3D governing system is established, which incorporates the growth effects of the different layers in the plate. Then, a series expansion-truncation approach is adopted to eliminate the thickness variables in the 3D governing system. An elaborate calculation scheme is applied to derive the iteration relations of the coefficient functions in the series expansions. Through some further manipulations, a 2D vector plate equation system with the associated boundary conditions is established. To show the efficiency of the plate theory, three typical examples regarding the growth-induced deformations and instabilities of multi-layered plate samples are studied, where the incompressible neo-Hookean constitutive model is adopted. Some analytical and numerical solutions to the plate equation system are obtained, which can provide accurate predictions on the growth behaviors of the plates. Furthermore, the problem of `shape-programming' of multi-layered plates through differential growth is studied and the explicit formulas for some typical examples are derived. By using these formulas, the shape evolutions of the plates during the growing processes can be controlled accurately. The results obtained in the current work are applicable for the design of intelligent soft devices with multi-layered plate structures. (c) 2022 Elsevier Inc. All rights reserved.

关键词： Multi-layered plate theory Soft material Growth-induced deformation Analytical solution shape-programming

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Realization of planar and surface conformal mappings through stress-free growth of hyperelastic plates: Analytical formulas and numerical calculations

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JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 2024年 190卷

作者： Wang, Jiong Jin, Zili Li, Zhanfeng South China Univ Technol Sch Civil Engn & Transportat Guangzhou 510640 Guangdong Peoples R China South China Univ Technol State Key Lab Subtrop Bldg & Urban Sci Guangzhou 510640 Guangdong Peoples R China

Conformal mapping is a well-known concept and has a long research history in mathematics. Accompanying the developments of computational science and 3D digital scanning technology, conformal mapping has also found wide applications in geometric modeling, computer graphics, medical imaging and other fields. In the virtual image or animation world, conformal mapping offers a convenient approach to achieve various shape changes of planar regions and 3D surfaces. In this work, we propose a promising approach, i.e., through the stress-free growth of hyperelastic plates, to realize arbitrary planar and surface conformal mappings in the physical world. The growth field in a hyperelastic plate can be represented by a symmetric tensor field. For any given planar or surface conformal mappings with explicit analytical expressions, the formulas of growth functions in the growth tensor are derived. In the case that the target surfaces of conformal mappings have no explicit analytical expressions, we further propose a numerical scheme to determine the growth data in the hyperelastic plates. The efficiency of the approach proposed in the current work is validated through several typical examples, where the 3D finite element simulations are conducted. The results demonstrate that, by incorporating the obtained growth functions or growth data, the target surfaces can be generated accurately through growth deformations of hyperelastic plates. As an application of the current work, we further propose an approach of shape-control for double-layer grid structures with plate forms. According to any given target shapes, the values of elongation or contraction of the bar elements are determined, then the desired deformations of the grid structures can be realized.

关键词： Conformal mapping Hyperelastic plate Stress-free growth shape-programming Numerical calculation

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