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1. Universal deformations and inhomogeneities in isotropic Cauchy elasticity

   https://doi.org/10.1098/rspa.2024.0229  

   Yavari, Arash (July 2024, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   For a given class of materials, universal deformations are those deformations that can be maintained in the absence of body forces and by applying solely boundary tractions. For inhomogeneous bodies, in addition to the universality constraints that determine the universal deformations, there are extra constraints on the form of the material inhomogeneities—universal inhomogeneity constraints. Those inhomogeneities compatible with the universal inhomogeneity constraints are called universal inhomogeneities. In a Cauchy elastic solid, stress at a given point and at an instance of time is a function of strain at that point and that exact moment in time, without any dependence on prior history. A Cauchy elastic solid does not necessarily have an energy function, i.e. Cauchy elastic solids are, in general, non-hyperelastic (or non-Green elastic). In this paper, we characterize universal deformations in both compressible and incompressible inhomogeneous isotropic Cauchy elasticity. As Cauchy elasticity includes hyperelasticity, one expects the universal deformations of Cauchy elasticity to be a subset of those of hyperelasticity both in compressible and incompressible cases. It is also expected that the universal inhomogeneity constraints to be more stringent than those of hyperelasticity, and hence, the set of universal inhomogeneities to be smaller than that of hyperelasticity. We prove the somewhat unexpected result that the sets of universal deformations of isotropic Cauchy elasticity and isotropic hyperelasticity are identical, in both the compressible and incompressible cases. We also prove that their corresponding universal inhomogeneities are identical as well. 

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2. The universal program of linear elasticity

   https://doi.org/10.1177/10812865221091305  

   Yavari, Arash; Goriely, Alain (January 2022, Mathematics and Mechanics of Solids)

   Universal displacements are those displacements that can be maintained, in the absence of body forces, by applying only boundary tractions for any material in a given class of materials. Therefore, equilibrium equations must be satisfied for arbitrary elastic moduli for a given anisotropy class. These conditions can be expressed as a set of partial differential equations for the displacement field that we call universality constraints. The classification of universal displacements in homogeneous linear elasticity has been completed for all the eight anisotropy classes. Here, we extend our previous work by studying universal displacements in inhomogeneous anisotropic linear elasticity assuming that the directions of anisotropy are known. We show that universality constraints of inhomogeneous linear elasticity include those of homogeneous linear elasticity. For each class and for its known universal displacements, we find the most general inhomogeneous elastic moduli that are consistent with the universality constrains. It is known that the larger the symmetry group, the larger the space of universal displacements. We show that the larger the symmetry group, the more severe the universality constraints are on the inhomogeneities of the elastic moduli. In particular, we show that inhomogeneous isotropic and inhomogeneous cubic linear elastic solids do not admit universal displacements and we completely characterize the universal inhomogeneities for the other six anisotropy classes. 

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3. Controllable deformations in compressible isotropic implicit elasticity

   https://doi.org/10.1007/s00033-024-02305-9  

   Yavari, Arash; Goriely, Alain (August 2024, Zeitschrift für angewandte Mathematik und Physik)

   Abstract For a given material,controllable deformationsare those deformations that can be maintained in the absence of body forces and by applying only boundary tractions. For a given class of materials,universal deformationsare those deformations that are controllable for any material within the class. In this paper, we characterize the universal deformations in compressible isotropic implicit elasticity defined by solids whose constitutive equations, in terms of the Cauchy stress$$\varvec{\sigma }$$ $\sigma$and the left Cauchy-Green strain$$\textbf{b}$$ $b$, have the implicit form$$\varvec{\textsf{f}}(\varvec{\sigma },\textbf{b})=\textbf{0}$$ $f(\sigma ,b)=0$. We prove that universal deformations are homogeneous. However, an important observation is that, unlike Cauchy (and Green) elasticity, not every homogeneous deformation is constitutively admissible for a given implicit-elastic solid. In other words, the set of universal deformations is material-dependent, yet it remains a subset of homogeneous deformations. 

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4. Universal displacements in inextensible fiber-reinforced linear elastic solids

   https://doi.org/10.1177/10812865231181924  

   Yavari, Arash (July 2023, Mathematics and Mechanics of Solids)

   For a given class of materials, universal displacements are those displacements that can be maintained for any member of the class by applying only boundary tractions. In this paper, we study universal displacements in compressible anisotropic linear elastic solids reinforced by a family of inextensible fibers. For each symmetry class and for a uniform distribution of straight fibers respecting the corresponding symmetry, we characterize the respective universal displacements. A goal of this paper is to investigate how an internal constraint affects the set of universal displacements. We have observed that other than the triclinic and cubic solids in the other five classes (a fiber-reinforced solid with straight fibers cannot be isotropic), the presence of inextensible fibers enlarges the set of universal displacements. 

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5. Normal stress anisotropy and marginal stability in athermal elastic networks

   https://doi.org/10.1039/c8sm02192a  

   Shivers, Jordan L.; Feng, Jingchen; Sharma, Abhinav; MacKintosh, F. C. (February 2019, Soft Matter)

   Hydrogels of semiflexible biopolymers such as collagen have been shown to contract axially under shear strain, in contrast to the axial dilation observed for most elastic materials. Recent work has shown that this behavior can be understood in terms of the porous, two-component nature and consequent time-dependent compressibility of hydrogels. The apparent normal stress measured by a torsional rheometer reflects only the tensile contribution of the axial component σ zz on long (compressible) timescales, crossing over to the first normal stress difference, N 1 = σ xx − σ zz at short (incompressible) times. While the behavior of N 1 is well understood for isotropic viscoelastic materials undergoing affine shear deformation, biopolymer networks are often anisotropic and deform nonaffinely. Here, we numerically study the normal stresses that arise under shear in subisostatic, athermal semiflexible polymer networks. We show that such systems exhibit strong deviations from affine behavior and that these anomalies are controlled by a rigidity transition as a function of strain. 

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