# Limit Analysis of Collapse States in Cellular Solids

Lead Research Organisation:
Cardiff University

Department Name: Sch of Mathematics

### Abstract

Cellular solids are two or three dimensional bodies divided into cells, the walls of which are made of a solid material capable of undertaking (large) elastic deformations without plastic failure or fracture. Due to their exceptional mechanical efficiency, they are ubiquitous in nature and industry, yet they are less well understood than almost any other class of materials. Among the best known mechanical qualities of these structures are their high strength-to-weight ratio and their energy absorption capacity, which are due the inextricable relation between the geometric architecture and the constitutive properties of the underlying solid mater. The aim of this project is to gain insight into the properties of cellular structures of nonlinear elastic material and their overall response under loads by providing both a new framework to understand the mechanical properties of these structures and rigorous mathematical techniques for the analysis of their large elastic deformations caused by the application of external loads. Taking into account the interplay between the geometry and the mechanical qualities of the elastic cell walls, novel numerical methods will be devised to compute effective lower and upper bounds for the critical load causing densification by cell closure in various cellular structures, and the bounds gap will be used as an indicator for the computational error. In this context, the proposed investigation and non-standard numerical procedures are novel and have many potential applications. For example, the development of new flexible stents and scaffolds for soft tissue re-growth in biomedical applications is a rapidly growing multidisciplinary area of biomaterials and tissue engineering, and many foams and sponges designed for cushioning and re-usability can also be found in everyday life as well as in several industrial areas, e.g. microelectronics, aerospace, pharmaceutical and food processes. For these complex materials to be understood and optimised with respect to their mechanical response, reliable computer models supported by rigorous mathematical mechanical analysis are needed, and may also open the way for new applications.

### Planned Impact

In addition to the academic beneficiaries identified in this case for support, the potential impact of the proposed research may be broken down into the following areas.

KNOWLEDGE AND PEOPLE. The limit analysis of elastic cellular structures is an important issue which has received little attention from the mathematical community to date. The long-term ambition is to make the new results useful for modern practical applications in bio-medical and material engineering, and also to help to identify new applications. From an applied mathematics point of view, the immediate goal is to inform the engineering research community who would then be able to test the mathematical results on a wide range of man-made materials of engineering interest.

LAM is an effective and welcome communicator within both the mathematical and engineering communities, as demonstrated by her track record, and her presentations at conferences and research seminars during the project will enhance the visibility of cellular materials and also encourage young researchers from both these communities into the area.

As a direct outcome, the project will train a young postdoctoral researcher in the solid mechanics underlying the physical model of cellular materials and the mathematical and numerical techniques needed to understand and analyse these materials.

In order to validate the computational results on real physical structures, Prof Sam Evans (Cardiff, Engineering) has agreed to carry out experimental tests on cellular structures made of a nonlinear elastic material. This will offer opportunities for feed-back on the computational models and feed-forward for the design of new model structures of engineering interest for further testing and validation.

Mathematical findings resulting from the project will also be included in the Finite Elasticity course for the fourth year MMath programme at Cardiff University, established by LAM in 2013-14. A new postgraduate-level course in Nonlinear Elasticity will also be introduced by LAM as part of the MAGIC (Mathematics Access Grid Instruction and Collaboration) programme funded by EPSRC. The MAGIC group contains 20 Universities across England and Wales, including Cardiff.

ECONOMY AND SOCIETY. Cellular materials can be found in most things where light-weight, shock-absorbing, multi-functional materials are required, and although it is hard to make a precise estimate of how much they cost or what the value of those protected by them is, it is not difficult to appreciate their overwhelming importance for the economy and society. For example, many foams and sponges designed for cushioning and re-usability can be found in everyday life as well as in several industrial areas, e.g. microelectronics, aerospace, pharmaceutical and food processes.

In the UK, the development of new flexible scaffolds for tissue re-growth in biomedical applications is a rapidly growing multidisciplinary area of Biomaterials and Tissue Engineering, which in turn underpins two of the current EPSRC Challenge Themes, namely Manufacturing the Future and Healthcare Technologies. In this context, the results of the proposed research will contribute to the development of associated research areas in material engineering, and may ultimately be used to optimize the design of cellular materials for highly flexible stents and soft tissue scaffolds, which are under continuing research and development.

For these complex materials to be understood and optimised with respect to their mechanical response, reliable computer models based on rigorous mathematical mechanical analysis are needed and may open the way for new applications.

KNOWLEDGE AND PEOPLE. The limit analysis of elastic cellular structures is an important issue which has received little attention from the mathematical community to date. The long-term ambition is to make the new results useful for modern practical applications in bio-medical and material engineering, and also to help to identify new applications. From an applied mathematics point of view, the immediate goal is to inform the engineering research community who would then be able to test the mathematical results on a wide range of man-made materials of engineering interest.

LAM is an effective and welcome communicator within both the mathematical and engineering communities, as demonstrated by her track record, and her presentations at conferences and research seminars during the project will enhance the visibility of cellular materials and also encourage young researchers from both these communities into the area.

As a direct outcome, the project will train a young postdoctoral researcher in the solid mechanics underlying the physical model of cellular materials and the mathematical and numerical techniques needed to understand and analyse these materials.

In order to validate the computational results on real physical structures, Prof Sam Evans (Cardiff, Engineering) has agreed to carry out experimental tests on cellular structures made of a nonlinear elastic material. This will offer opportunities for feed-back on the computational models and feed-forward for the design of new model structures of engineering interest for further testing and validation.

Mathematical findings resulting from the project will also be included in the Finite Elasticity course for the fourth year MMath programme at Cardiff University, established by LAM in 2013-14. A new postgraduate-level course in Nonlinear Elasticity will also be introduced by LAM as part of the MAGIC (Mathematics Access Grid Instruction and Collaboration) programme funded by EPSRC. The MAGIC group contains 20 Universities across England and Wales, including Cardiff.

ECONOMY AND SOCIETY. Cellular materials can be found in most things where light-weight, shock-absorbing, multi-functional materials are required, and although it is hard to make a precise estimate of how much they cost or what the value of those protected by them is, it is not difficult to appreciate their overwhelming importance for the economy and society. For example, many foams and sponges designed for cushioning and re-usability can be found in everyday life as well as in several industrial areas, e.g. microelectronics, aerospace, pharmaceutical and food processes.

In the UK, the development of new flexible scaffolds for tissue re-growth in biomedical applications is a rapidly growing multidisciplinary area of Biomaterials and Tissue Engineering, which in turn underpins two of the current EPSRC Challenge Themes, namely Manufacturing the Future and Healthcare Technologies. In this context, the results of the proposed research will contribute to the development of associated research areas in material engineering, and may ultimately be used to optimize the design of cellular materials for highly flexible stents and soft tissue scaffolds, which are under continuing research and development.

For these complex materials to be understood and optimised with respect to their mechanical response, reliable computer models based on rigorous mathematical mechanical analysis are needed and may open the way for new applications.

### Publications

Lee C
(2017)

*Strain smoothing for compressible and nearly-incompressible finite elasticity*in Computers & Structures
Mihai L
(2017)

*A Microstructure-Based Hyperelastic Model for Open-Cell Solids*in SIAM Journal on Applied Mathematics
Mihai L
(2016)

*Guaranteed Upper and Lower Bounds on the Uniform Load of Contact Problems in Elasticity*in SIAM Journal on Applied Mathematics
Mihai L
(2017)

*Hyperelastic bodies under homogeneous Cauchy stress induced by non-homogeneous finite deformations*in International Journal of Non-Linear Mechanics
Mihai L
(2017)

*Hyperelastic bodies under homogeneous Cauchy stress induced by three-dimensional non-homogeneous deformations*in Mathematics and Mechanics of Solids
Mihai L
(2017)

*A family of hyperelastic models for human brain tissue*in Journal of the Mechanics and Physics of Solids
Mihai L
(2015)

*Paws, pads and plants: the enhanced elasticity of cell-filled load-bearing structures*in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Mihai LA
(2015)

*A comparison of hyperelastic constitutive models applicable to brain and fat tissues.*in Journal of the Royal Society, Interface
Mihai LA
(2018)

*Stochastic isotropic hyperelastic materials: constitutive calibration and model selection.*in Proceedings. Mathematical, physical, and engineering sciences
Mihai LA
(2017)

*How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity.*in Proceedings. Mathematical, physical, and engineering sciencesDescription | OVERVIEW: Cellular solids, or foams, are two or three dimensional bodies divided into cells, which may be open or closed, and can be filled with a fluid or solid core. In open-celled solids, the cell walls consist of the cell edges which form an interconnected network, while in closed-cell foams, the cell walls contain both the cell edges and the cell faces forming disconnected cell compartments, although some foams may contain both open and closed cells. Even though it is at a cellular level that the structural basis of foams is best addressed, both theoretically and computationally, due to the inherent complexity and diversity of cellular bodies, the mathematical modelling of every individual cell in a structure with a large number of cells is generally unfeasible, and a combined study spanning the micro-, meso-, and macroscopic scales is required. Since, in many natural and engineered cellular materials, plastic damage or fracture rarely occurs during functional performance, and the material recovers completely after large deformations, these materials can be reasonably treated within the theoretical framework of finite elasticity, which in principle provides a complete description of the elastic responses of a solid material under loading. In particular, the analysis of soft cellular structures operating in large strain deformation involves nonlinear hyperelastic models for which the mathematical and numerical treatment poses many physical, theoretical, and computational challenges. For these materials, boundary value problems can be formulated equivalently as variational problems, which provide powerful methods for obtaining approximate solutions and can also be used to generate finite element methods. Cellular structures are the subject of significant research efforts in biomedical applications, where the development of new flexible scaffolds for tissue re-growth is a rapidly growing multidisciplinary area of biomaterials and tissue engineering. Cellular bodies are also ubiquitous in nature and can be found in most things where light-weight, shock-absorbing, multi-functional materials are required, in everyday life as well as in several industrial areas (e.g., microelectronics, aerospace, pharmaceutical and food processes). Although, in many cases, it is hard to estimate their precise cost or the exact value of those protected by them, it is not difficult to appreciate their overwhelming significance for the economy and society. For these complex materials to be understood and optimised with respect to their mechanical responses, reliable models that take into account the interplay between the geometry and the mechanical qualities of the constituents across the scales are needed and may also open the way to new applications. (I) PAWS, HEEL-PADS AND PLANT STEMS: Paws, plantar pads, and plant stems are some of the most remarkable load bearing biological structures. They generally rely on closed cells filled with fluids or adipose tissue to cushion large distortions. Physical evidence suggests that several main factors determine the magnitude of the enhancement of stress level in their cells, involving the individual cell geometry, the cell wall thickness, and the presence of cell inclusions. These competing factors play significant roles in the manner in which such structures achieve their superior mechanical performance. Histologically, paw and foot cushions are soft cellular bodies built from closed compartments or cells separated by collagen reinforced elastic septa and filled with fat (adipose tissue). As part of the locomotor system, they are designed to absorb mechanical shock, redistribute excessive local stress, and store and return elastic strain energy. When the septa walls become thicker or more fibrous, they become more difficult to stretch and deform under loading, while if the walls break down, then the structure becomes more easily deformed resulting in an atrophic pad. The mechanical properties of collagenous septa also depend on osmotic pressure developed within the aqueous cell core. In plant stems, the mechanical support system is usually formed through a combination of increase in the cell number or size and sustained sclerification (thickening and lignification) of the cell walls, while turgor pressure induces stretching in the parenchyma cell walls. Some monocotyledon plants attain tree stature (e.g., palm trees with maximum heights of 20-40 meters) comparable to that of arborescent dicotyledons and conifers (e.g., magnolia, maple, oak, sycamore), but their stems are relatively slender. By contrast, tall dicot trees have bigger stem diameters relative to their height than small trees. The intriguing behaviours of dicot and monocot stems inspired La Fontaine's fable ``The oak and the reed'', which was illustrated also in a painting by A. E. Michallon, now at Fitzwilliam Museum. Although the wood density representing the relative quantity of the cell wall in a given volume of wood tissue vary significantly among wood species, the composition and strength of the cell wall is generally the same for all woods. A key challenge is to understand how the function of these structures is enhanced by their geometric and material design. To do so, we compared different elastic models capable of large strain deformations when the cells are empty or filled with an incompressible liquid or solid core. For cellular structures with uniform cell size, shape, and distribution, we demonstrated that the stiffness in cell walls made from an isotropic hyperelastic material increases when the wall thickness increases or if the wall is multi-layered, as well as when the number of cells increases while the volume of solid material and the ratio between the thickness and the length of the wall remain fixed. While different factors may contribute to this behaviour, in our view, this is due to the enhanced elasticity of the cell walls when more material is added or when the elastic material is distributed more uniformly throughout the structure. Our analysis further indicates that the cell inclusion in a geometrically closed cell plays a two-fold role: on the one hand, it maintains the integrity of the deforming structure by preventing densification or compaction, and on the other hand, it enhances the stiffness of the cell walls through additional local constraints. Since a reduction in the energy absorbing capacity of a cellular structure is typical when the cell wall stiffness increases, such changes in the material properties play significant roles in the manner in which these structures perform their roles. Even though natural structures are irregular, and their large deformations are typically more complex and diverse, idealised mathematical models have the advantage that they can be studied systematically to identify the independent influence of different mechanical characteristics. In particular, our analysis offers reasonable characterisation of the increase in stiffness of atrophied heel-pads with thicker septa or reduced fat tissue, and may be relevant in understanding pathologies or impairments related to their weight bearing and protective ability. (II) FRUIT-CELL DEBONDING: Many natural structures are cellular solids at millimetre scale and fibre-reinforced composites at micrometer scale. For these structures, mechanical properties are associated with cell strength, and phenomena such as cells separation through debonding of the middle lamella in cell-walls is key in explaining some important characteristics or behaviour. Cell debonding is a spontaneous mechanism for damage initiation in many natural structures. Cells separation through debonding of the middle lamella in cell walls is key in explaining the property or behaviour of fruit and legumes during storage or cooking, and are decisive for the perceived quality of many food products. Fruit tissue is a hydrostatic structure in which individual fluid-filled cells provide resistance to compressive forces, and the fluid pressure may also influence the elastic properties of the cell walls. During pre-harvest ripening, the firmness of fruit (apple, pear, tomato) decreases and the cell walls of the fruit tissue become softer, then continues to decrease in post-harvest storage due to the loss of cell-to-cell contact, even though the stiffness of the cell walls increases. Cellular pressure also decreases after harvest causing cell wall relaxation, accelerating the process of loss of adhesion. However, both the cell wall strength and the inter-cellular adhesion decline as fruit enter the over-ripe stage. The relevant scales at which such phenomena occur, though beyond the capacity of the human eye, can be followed by mechanical analysis and mathematical models based on micro-structural evidence. However, obtaining suitable models that are, at the same time, physically plausible, mathematically tractable, and computationally feasible raises many theoretical and numerical challenges. To explore such phenomena, we modelled cellular structures with nonlinear hyperelastic cell walls under large shear deformation, and incorporated cell wall material anisotropy and unilateral contact between neighbouring cells in our models. The theoretical and computational challenges raised by these structures range from the non-linear deformation of the individual elastic walls to the detection of contact and openings between adjacent cells. For example, in systems formed from linear elastic bodies (e.g., ceramics, metals), surface roughness can impede active contact when a surface is pressed against another, and contact forces cannot be transmitted where surface separation occur. By contrast, nonlinear elastic bodies (e.g., rubber, soft-tissue) are more pliable and thus capable of attaining more active support through which contact forces can be transmitted. Finally, the fundamental question arises: what is the critical external force or internal cell pressure that will cause failure by cell separation? In material analysis and design, limit analysis supplements elastic analysis in order to determine the limit value of the load such that a material acted upon by loads smaller than the limit load will not collapse (static principle), and predict a possible collapse mode of the material under the limit load (kinematic principle). Limit analysis principles are central to the modelling and assessment of many structural materials, and involve: mathematical analysis of partial differential equations, numerical analysis based on finite element methods, numerical optimization methods to solve the discrete mathematical programming problems, and extensive computational experience to help choose between possible combinations of these methods for various applications. To obtain an effective numerical solution, we devised a successive deformation decomposition procedure as follows: (i) first, we deform the entire structure as a seamless assembly of cells subject to the given external boundary conditions; (ii) then, for the pre-deformed structure, we take into account the contact conditions at cell interfaces to allow for possible cell separation. The two-step procedure proves significantly faster and more accurate than if the external boundary conditions and contact constraints were imposed simultaneously in a single step. From the numerical results, we infer that separation is less likely between cells with high internal cell pressure (e.g., in fresh and growing fruit and vegetables) than between cells where the internal pressure is low (e.g., in cooked or ageing plants). This is in agreement with physical observations that, under applied forces, tissue from high-maturity fruit breaks down into small clumps of undamaged cells, while cell-walls from less mature fruit, which are relatively strongly attached to each other, will rupture. (III) MICRO-STRUCTURE BASED CONTINUUM MODELS: Bridging the micro-structural responses of individual cells with the apparent macro-structural behaviour is a challenging modelling problem in materials sciences. To date, there is no established continuum model for cellular structures of nonlinear elastic material, even though this should stand on the shoulders of nonlinear elasticity theory. Moreover, under uniform stresses, many isotropic hyperelastic materials deform uniformly, whereas in cellular structures, the deformation generally concentrates in regions that are more easily deformable. For stretch-dominated architectures, which are structurally more efficient due to a higher stiffness-to-weight ratio than the bending-dominated ones, assuming that the cells are either open or closed cells and the cell walls are arbitrarily oriented and made from a general isotropic hyperelastic material, we derived explicitly continuum hyperelastic models at a mesoscopic level where the number of cells is finite and the size of the structure is comparable to the size of the cells. These models can be useful as an initial approximation in multi-level approaches, whereby a cellular structure is represented first as an elastic material deforming uniformly, and after the loading is increased, the areas where the stress field reaches critical values are re-modelled as individual cells to predict local effects. In computer simulations, multi-scale models reduce computational cost and may even mean the difference between finding a solution or not. (IV) NONLINEAR ELASTIC CONSTITUTIVE PARAMETERS: Many natural and man-made cellular bodies are light-weight, shock-absorbing, multi-functional materials, capable of undertaking large elastic deformations. These properties are due to a complex system of local deformations which can lead to changes in the material properties as the deformation progresses, but their study is non-trivial since the corresponding stresses are non-trivial functions of volume fraction, micro-geometry, and material properties of the components. For cellular bodies of hyperelastic material, several main factors determine the magnitude of the stress level, including the cell geometry, the cell wall thickness, and the presence of cell inclusions. In our work, nonlinear elastic parameters have been identified, namely a nonlinear shear modulus, a nonlinear stretch modulus, and a Poisson function, which are defined in terms of the large stresses and strains in the elastic cell walls, and their utility in estimating how different competing factors may contribute to the complex mechanical behaviour of cellular structures has been investigated. These parameters represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. Universal relations between certain of these parameters have also been established, and then used to quantify nonlinear elastic responses in several hyperelastic models for rubber, soft tissue, and foams. |

Exploitation Route | Constant material parameters are standard in engineering applications where linear elastic models are commonly used. In nonlinear elasticity, similar constitutive parameters can be defined that are functions of the deformation, including the nonlinear stretch modulus, the nonlinear shear modulus, and the Poisson function, which are measurable under axial or shear experimental tests. These functions can play significant roles in both the fundamental understanding and the application of many elastic materials under large elastic strains, as they represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. As shown by our microstructure-based models, they can provide also a flexible basis for the coupling of elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales. |

Sectors | Aerospace, Defence and Marine,Agriculture, Food and Drink,Construction,Electronics,Healthcare,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology,Other |

URL | https://www.cardiff.ac.uk/people/view/98660-mihai-angela |

Description | Fast Track to Fellowships awarded to PDRA, Dr Hayley Wyatt, by the College of Physical Sciences and Engineering, Cardiff University, for experimental testing. |

Amount | £288 (GBP) |

Organisation | Cardiff University |

Sector | Academic/University |

Country | United Kingdom |

Start | 03/2016 |

End | 03/2016 |

Title | Numerical data for the enhanced elasticity of cell-filled load-bearing structures |

Description | This is the complete data set for the research article ''Paws, pads, and plants: The enhanced elasticity of cell-filled load-bearing structures'' by LA Mihai, K Alayyash & A Goriely, accepted for publication by the Proceedings of the Royal Society of London A on April 30, 2015. It comprises (i) files for the computational models of periodic cellular structures of nonlinear elastic material with or without inclusions subject to axial tension or compression - produced within the open-source software Finite Elements for Biomechanics (FEBio) environment; and (ii) tables with the corresponding numerical results for the mean effective Cauchy stress and apparent elastic modulus plotted in Figures 10, 11, 12 and 14 - presented in the numerical examples section (section 3) of the paper. ----------- Mihai LA, Alayyash K, Goriely A. 2015 Paws, pads and plants: The enhanced elasticity of cell-filled load-bearing structures. Proc. R. Soc. A 20150107 (http://dx.doi.org/10.1098/rspa.2015.0107) |

Type Of Material | Computer model/algorithm |

Year Produced | 2015 |

Provided To Others? | Yes |

Impact | The associated research article Mihai LA, Alayyash K, Goriely A. 2015 Paws, pads and plants: The enhanced elasticity of cell-filled load-bearing structures. Proc. R. Soc. A 20150107 (http://dx.doi.org/10.1098/rspa.2015.0107) has been noticed by researchers in Zoology and Ecology. |

URL | http://research.cardiff.ac.uk/converis/portal/Dataset/1572490?auxfun=&lang=en_GB |

Description | Soapbox Science |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Public/other audiences |

Results and Impact | Soapbox Science is a grass-roots science outreach event that brings cutting edge research onto urban streets whilst also promoting the visibility of women in science. Speakers who take part in the event simply stand on their soapbox to deliver scientific talks to the general public, aiming to make science fun, encouraging people from all backgrounds to take an interest in science, and helping to break down gender stereotypes and provide positive female role models. At the first Soapbox Science event organised in Cardiff, on June 4, 2016, Dr Hayley Wyatt delivered a talk entitled 'Structures within nature and modern engineering applications'. This talk was based on the research undertaken by Dr Angela Mihai (PI) and Dr Hayley Wyatt (PDRA) on the EPSRC funded project EP/M011992/1. |

Year(s) Of Engagement Activity | 2016 |

URL | https://www.cardiff.ac.uk/news/view/356718-soapbox-science-in-cardiff |

Description | Structures within Nature and Modern Engineering Applications, Western Mail (Cardiff, Wales), June 6, 2016. |

Form Of Engagement Activity | A press release, press conference or response to a media enquiry/interview |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Public/other audiences |

Results and Impact | Public outreach and media coverage In this newspaper article, Dr Hayley Wyatt, research associate on EPSRC project EP/M011992/1 led by Dr Angela Mihai, explains the significance of fundamental research in the mathematics and mechanics of cellular structures, following her presentation at the first Soapbox Science public outreach event in Cardiff, on June 4, 2016. |

Year(s) Of Engagement Activity | 2016 |

URL | https://www.cardiff.ac.uk/news/view/356718-soapbox-science-in-cardiff |