{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/390514"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/390514","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Mathematical models of soft elasticity: elastomers, active solids and topological metamaterials","abstract":"The mechanics of soft solids, from everyday materials like rubber to exotic metamaterials for emerging technologies, is elegantly described by the theory of elasticity, which uses a continuum field approach to capture universal features in the macroscopic physics of solids. Using this approach, we study three types of soft material, ranging from conventional to exotic. We begin in Part I with elastomers, which are used in many technological applications that rely on hyperelastic constitutive models to predict the nonlinear stress–strain response of these materials. To obtain more accurate models, we use the Irving–Kirkwood–Noll procedure to construct a Cauchy stress tensor from forces along polymer chains. Combining this stress with previously reported experimental results on chain orientations in deformed polymer networks, we present two new chain stretch relations relating the molecular-level stretching of polymer chains to the macroscopic deformation. Our new models have only two fitting parameters, and fit experimental data more accurately than previously proposed two-parameter models. In Part II, we turn to active solids, which consist of energy-consuming constituents that enable work to be extracted using closed cycles of deformation. Previous work on linear odd elasticity captured their rich phenomenology, which occurs in both living systems and robotic metamaterials. We generalise odd elasticity to the nonlinear regime and identify a new odd modulus for quadratic terms that give rise to deformation-induced odd elasticity. Unlike linear odd elasticity, this odd response does not rely on broken chiral symmetry, so it can exist in three-dimensional isotropic materials, and enables work extraction from closed deformation cycles for which the linear theory predicts zero net work. We show that random networks of active fibres, in which each fibre's stiffness depends on the stretches of surrounding fibres, exhibit this novel form of odd elasticity. In Part III, we consider topological mechanical metamaterials that exhibit robust floppy modes, characterised by topological invariants. To capture universal features of these modes in the continuum, we augment linear elasticity with additional fields and define a continuum version of Maxwell counting, which expresses the balance between degrees of freedom and mechanical constraints. We define topological invariants for continuum floppy modes and show that one additional field is necessary to capture topological edge modes, whereas two are required for Weyl points, which are topological bulk modes. Although our results are independent of microscopic detail, we show that our theory arises naturally when taking the continuum limit of lattices, enabling efficient prediction of coarse-grained deformations.","abstract_html":"The mechanics of soft solids, from everyday materials like rubber to exotic metamaterials for emerging technologies, is elegantly described by the theory of elasticity, which uses a continuum field approach to capture universal features in the macroscopic physics of solids. Using this approach, we study three types of soft material, ranging from conventional to exotic. We begin in Part I with elastomers, which are used in many technological applications that rely on hyperelastic constitutive models to predict the nonlinear stress–strain response of these materials. To obtain more accurate models, we use the Irving–Kirkwood–Noll procedure to construct a Cauchy stress tensor from forces along polymer chains. Combining this stress with previously reported experimental results on chain orientations in deformed polymer networks, we present two new chain stretch relations relating the molecular-level stretching of polymer chains to the macroscopic deformation. Our new models have only two fitting parameters, and fit experimental data more accurately than previously proposed two-parameter models. In Part II, we turn to active solids, which consist of energy-consuming constituents that enable work to be extracted using closed cycles of deformation. Previous work on linear odd elasticity captured their rich phenomenology, which occurs in both living systems and robotic metamaterials. We generalise odd elasticity to the nonlinear regime and identify a new odd modulus for quadratic terms that give rise to deformation-induced odd elasticity. Unlike linear odd elasticity, this odd response does not rely on broken chiral symmetry, so it can exist in three-dimensional isotropic materials, and enables work extraction from closed deformation cycles for which the linear theory predicts zero net work. We show that random networks of active fibres, in which each fibre's stiffness depends on the stretches of surrounding fibres, exhibit this novel form of odd elasticity. In Part III, we consider topological mechanical metamaterials that exhibit robust floppy modes, characterised by topological invariants. To capture universal features of these modes in the continuum, we augment linear elasticity with additional fields and define a continuum version of Maxwell counting, which expresses the balance between degrees of freedom and mechanical constraints. We define topological invariants for continuum floppy modes and show that one additional field is necessary to capture topological edge modes, whereas two are required for Weyl points, which are topological bulk modes. Although our results are independent of microscopic detail, we show that our theory arises naturally when taking the continuum limit of lattices, enabling efficient prediction of coarse-grained deformations.","abstract_has_math":false,"creators":["Tan, Ian Roong Chern"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Savin, Thierry"],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-06-30","date_published":"2025-06-30","updated_at":"2026-07-22T22:24:24Z","subjects":["continuum mechanics","elasticity","elastomers","active matter","topological physics","floppy modes","coarse-graining","nonlinear response","soft solids","topological invariants","random networks"],"languages":["eng"],"rights":[],"rights_urls":["https://www.repository.cam.ac.uk/bitstreams/80d57295-49b4-46fc-9d0c-fbab86d8f856/download","http://purl.org/NET/rdflicense/allrightsreserved"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.122067","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Savin, Thierry"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["Cambridge Commonwealth European and International Trust"]},{"key":"dc:creator","label":"Author","values":["Tan, Ian Roong Chern"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2025-06-30"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/390514"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["continuum mechanics","elasticity","elastomers","active matter","topological physics","floppy modes","coarse-graining","nonlinear response","soft solids","topological invariants","random networks"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.repository.cam.ac.uk/bitstreams/80d57295-49b4-46fc-9d0c-fbab86d8f856/download","http://purl.org/NET/rdflicense/allrightsreserved"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.17863/CAM.122067"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/bc17bd44-7675-47d0-a047-edb8688626aa/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The mechanics of soft solids, from everyday materials like rubber to exotic metamaterials for emerging technologies, is elegantly described by the theory of elasticity, which uses a continuum field approach to capture universal features in the macroscopic physics of solids. Using this approach, we study three types of soft material, ranging from conventional to exotic. We begin in Part I with elastomers, which are used in many technological applications that rely on hyperelastic constitutive models to predict the nonlinear stress–strain response of these materials. To obtain more accurate models, we use the Irving–Kirkwood–Noll procedure to construct a Cauchy stress tensor from forces along polymer chains. Combining this stress with previously reported experimental results on chain orientations in deformed polymer networks, we present two new chain stretch relations relating the molecular-level stretching of polymer chains to the macroscopic deformation. Our new models have only two fitting parameters, and fit experimental data more accurately than previously proposed two-parameter models. In Part II, we turn to active solids, which consist of energy-consuming constituents that enable work to be extracted using closed cycles of deformation. Previous work on linear odd elasticity captured their rich phenomenology, which occurs in both living systems and robotic metamaterials. We generalise odd elasticity to the nonlinear regime and identify a new odd modulus for quadratic terms that give rise to deformation-induced odd elasticity. Unlike linear odd elasticity, this odd response does not rely on broken chiral symmetry, so it can exist in three-dimensional isotropic materials, and enables work extraction from closed deformation cycles for which the linear theory predicts zero net work. We show that random networks of active fibres, in which each fibre's stiffness depends on the stretches of surrounding fibres, exhibit this novel form of odd elasticity. In Part III, we consider topological mechanical metamaterials that exhibit robust floppy modes, characterised by topological invariants. To capture universal features of these modes in the continuum, we augment linear elasticity with additional fields and define a continuum version of Maxwell counting, which expresses the balance between degrees of freedom and mechanical constraints. We define topological invariants for continuum floppy modes and show that one additional field is necessary to capture topological edge modes, whereas two are required for Weyl points, which are topological bulk modes. Although our results are independent of microscopic detail, we show that our theory arises naturally when taking the continuum limit of lattices, enabling efficient prediction of coarse-grained deformations."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["809e3d6c46d919578aada5ac739c67af","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Mathematical models of soft elasticity: elastomers, active solids and topological metamaterials"]}]}],"canonical_facts":{"dc:contributor.advisor":["Savin, Thierry"],"dc:contributor.sponsor":["Cambridge Commonwealth European and International Trust"],"dc:creator":["Tan, Ian Roong Chern"],"dc:date.issued":["2025-06-30"],"dc:description.abstract":["The mechanics of soft solids, from everyday materials like rubber to exotic metamaterials for emerging technologies, is elegantly described by the theory of elasticity, which uses a continuum field approach to capture universal features in the macroscopic physics of solids. Using this approach, we study three types of soft material, ranging from conventional to exotic. We begin in Part I with elastomers, which are used in many technological applications that rely on hyperelastic constitutive models to predict the nonlinear stress–strain response of these materials. To obtain more accurate models, we use the Irving–Kirkwood–Noll procedure to construct a Cauchy stress tensor from forces along polymer chains. Combining this stress with previously reported experimental results on chain orientations in deformed polymer networks, we present two new chain stretch relations relating the molecular-level stretching of polymer chains to the macroscopic deformation. Our new models have only two fitting parameters, and fit experimental data more accurately than previously proposed two-parameter models. In Part II, we turn to active solids, which consist of energy-consuming constituents that enable work to be extracted using closed cycles of deformation. Previous work on linear odd elasticity captured their rich phenomenology, which occurs in both living systems and robotic metamaterials. We generalise odd elasticity to the nonlinear regime and identify a new odd modulus for quadratic terms that give rise to deformation-induced odd elasticity. Unlike linear odd elasticity, this odd response does not rely on broken chiral symmetry, so it can exist in three-dimensional isotropic materials, and enables work extraction from closed deformation cycles for which the linear theory predicts zero net work. We show that random networks of active fibres, in which each fibre's stiffness depends on the stretches of surrounding fibres, exhibit this novel form of odd elasticity. In Part III, we consider topological mechanical metamaterials that exhibit robust floppy modes, characterised by topological invariants. To capture universal features of these modes in the continuum, we augment linear elasticity with additional fields and define a continuum version of Maxwell counting, which expresses the balance between degrees of freedom and mechanical constraints. We define topological invariants for continuum floppy modes and show that one additional field is necessary to capture topological edge modes, whereas two are required for Weyl points, which are topological bulk modes. Although our results are independent of microscopic detail, we show that our theory arises naturally when taking the continuum limit of lattices, enabling efficient prediction of coarse-grained deformations."],"dc:format.checksum.md5":["809e3d6c46d919578aada5ac739c67af","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["https://doi.org/10.17863/CAM.122067"],"dc:identifier.uri":["https://www.repository.cam.ac.uk/bitstreams/bc17bd44-7675-47d0-a047-edb8688626aa/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/390514"],"dc:rights":["https://www.repository.cam.ac.uk/bitstreams/80d57295-49b4-46fc-9d0c-fbab86d8f856/download","http://purl.org/NET/rdflicense/allrightsreserved"],"dc:subject":["continuum mechanics","elasticity","elastomers","active matter","topological physics","floppy modes","coarse-graining","nonlinear response","soft solids","topological invariants","random networks"],"dc:title":["Mathematical models of soft elasticity: elastomers, active solids and topological metamaterials"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:24Z"}