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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.

 

In the field of soft robotics, flexibility, adaptability, and functionality define a new era of robotic systems that can safely deform, reach, and grasp. To optimize the design of soft robotic systems, it is critical to understand their configuration space and full range of motion across a wide variety of design parameters. Here we integrate extreme mechanics and soft robotics to provide quantitative insights into the design of bio-inspired soft slender manipulators using the concept of reachability clouds. For a minimal three-actuator design inspired by the elephant trunk, we establish an efficient and robust reduced-order method to generate reachability clouds of almost half a million points each to visualize the accessible workspace of a wide variety of manipulator designs. We generate an atlas of 256 reachability clouds by systematically varying the key design parameters including the fiber count, revolution, tapering angle, and activation magnitude. Our results demonstrate that reachability clouds not only offer an immediately clear perspective into the inverse problem of control, but also introduce powerful metrics to characterize reachable volumes, unreachable regions, and actuator redundancy to quantify the performance of soft slender robots. 

One of the key problems in active materials is the control of shape through actuation. A fascinating example of such control is the elephant trunk, a long, muscular, and extremely dexterous organ with multiple vital functions. The elephant trunk is an object of fascination for biologists, physicists, and children alike. Its versatility relies on the intricate interplay of multiple unique physical mechanisms and biological design principles. Here, we explore these principles using the theory of active filaments and build, theoretically, computationally, and experimentally, a minimal model that explains and accomplishes some of the spectacular features of the elephant trunk. 

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 theuniversality constraints(equilibrium equations and arbitrariness of the elastic constants) completely specify theuniversal elastic strainsfor 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.

 

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. 

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.