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1. 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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2. Universality in Anisotropic Linear Anelasticity

   https://doi.org/10.1007/s10659-022-09910-7  

   Yavari, Arash; Goriely, Alain (August 2022, Journal of Elasticity)

   Abstract In linear elasticity, universal displacements for a given symmetry class are those displacements that can be maintained by only applying boundary tractions (no body forces) and for arbitrary elastic constants in the symmetry class. In a previous work, we showed that the larger the symmetry group, the larger the space of universal displacements. Here, we generalize these ideas to the case of anelasticity. In linear anelasticity, the total strain is additively decomposed into elastic strain and anelastic strain, often referred to as an eigenstrain. We show that the universality constraints (equilibrium equations and arbitrariness of the elastic constants) completely specify the universal elastic strains for each of the eight anisotropy symmetry classes. The corresponding universal eigenstrains are the set of solutions to a system of second-order linear PDEs that ensure compatibility of the total strains. We show that for three symmetry classes, namely triclinic, monoclinic, and trigonal, only compatible (impotent) eigenstrains are universal. For the remaining five classes universal eigenstrains (up to the impotent ones) are the set of solutions to a system of linear second-order PDEs with certain arbitrary forcing terms that depend on the symmetry class. 

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3. The Universal Program of Nonlinear Hyperelasticity

   https://doi.org/10.1007/s10659-022-09906-3  

   Yavari, Arash; Goriely, Alain (January 2022, Journal of Elasticity)

   Abstract For a given class of materials, universal deformations are those that can be maintained in the absence of body forces by applying only boundary tractions. Universal deformations play a crucial role in nonlinear elasticity. To date, their classification has been accomplished for homogeneous isotropic solids following Ericksen’s seminal work, and homogeneous anisotropic solids and inhomogeneous isotropic solids in our recent works. In this paper we study universal deformations for inhomogeneous anisotropic solids defined as materials whose energy function depends on position. We consider both compressible and incompressible transversely isotropic, orthotropic, and monoclinic solids. We show that the universality constraints —the constraints that are dictated by the equilibrium equations and the arbitrariness of the energy function—for inhomogeneous anisotropic solids include those of inhomogeneous isotropic and homogeneous anisotropic solids. For compressible solids, universal deformations are homogeneous and the material preferred directions are uniform. For each of the three classes of anisotropic solids we find the corresponding universal inhomogeneities —those inhomogeneities that are consistent with the universality constraints. For incompressible anisotropic solids we find the universal inhomogeneities for each of the six known families of universal deformations. This work provides a systematic approach to study analytically functionally-graded fiber-reinforced elastic solids. 

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4. Residual stress and material symmetry

   https://doi.org/10.1016/j.ijengsci.2023.104013  

   Rajagopal, KR; Wineman, A (April 2024, International Journal of Engineering Science)

   There seems to be a basic misconception in several recent papers concerning the material symmetry of bodies in configurations that are pre-stressed. In this short paper we point to the source of the error and show that the material symmetry that is possible depends on the nature of the pre-stress. We also extend the results for material symmetry which have been well known within the context of simple elastic solids to the general class of simple materials. This generalization has relevance to the material symmetry of biological solids that are viscoelastic. 

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5. Statistics and topology of fluctuating ribbons

   https://doi.org/10.1073/pnas.2122907119  

   Yong, Ee Hou; Dary, Farisan; Giomi, Luca; Mahadevan, L. (August 2022, Proceedings of the National Academy of Sciences)

   Ribbons are a class of slender structures whose length, width, and thickness are widely separated from each other. This scale separation gives a ribbon unusual mechanical properties in athermal macroscopic settings, for example, it can bend without twisting, but cannot twist without bending. Given the ubiquity of ribbon-like biopolymers in biology and chemistry, here we study the statistical mechanics of microscopic inextensible, fluctuating ribbons loaded by forces and torques. We show that these ribbons exhibit a range of topologically and geometrically complex morphologies exemplified by three phases—a twist-dominated helical phase (HT), a writhe-dominated helical phase (HW), and an entangled phase—that arise as the applied torque and force are varied. Furthermore, the transition from HW to HT phases is characterized by the spontaneous breaking of parity symmetry and the disappearance of perversions (that correspond to chirality-reversing localized defects). This leads to a universal response curve of a topological quantity, the link, as a function of the applied torque that is similar to magnetization curves in second-order phase transitions. 

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