In this article we establish universal relations for a cube of a nonlinear elastic isotropic electroactive solid that undergoes a homogeneous deformation due to shear tractions but no normal tractions. When there is no electric field, in addition to a shearing deformation, the cube’s dimensions change because of the Poynting effect. In this work, we study the influence of the electric field vector on these dimensional changes using a previously developed constitutive equation for nonlinear electroactive solids. Expressions are obtained for these dimensional changes that depend the amount of shear for different directions of the electric field vector relative to the shearing direction. In addition, universal relations are obtained when there is no electric field and are extended for different electric field directions.

Elastomers and soft biological tissues can undergo large deformations and exhibit time-dependent behavior that is characteristic of nonlinear viscoelastic solids. An overview of this subject is contained herein, beginning with a review of pertinent topics from linear viscoelasticity. After stating the general constitutive assumption for nonlinear viscoelastic solids, and then imposing restrictions imposed by consideration of superposed rotations and material symmetry, a number of specific forms that have been proposed in the literature are discussed. The emphasis is then confined to nonlinear single integral constitutive equations, specific cases being finite linear viscoelasticity and the Pipkin–Rogers constitutive equations. The latter contains, as a special case, the quasi-linear viscoelastic model used in the biomechanics of soft tissue. Representations for the Pipkin–Rogers model are provided for isotropy, transverse isotropy, and orthotropy. Uniaxial stretch histories for isotropic materials are used to show the deviation from linear behavior as nonlinear effects become important. A number of examples involving non-homogeneous deformations that have appeared in the literature are summarized.

In his pioneering work in nonlinear elasticity, Rivlin studied the tension-torsion of a solid isotropic homogeneous elastic cylinder and obtained expressions for the twisting moment as well as the normal force for all bodies belonging to that class in terms of the angle of twist and the stretch. The deformation considered by Rivlin was a universal controllable deformation in that it can be engendered by just the application of the appropriate surface tractions. In this short note we consider the tension-torsion of a solid electroelastic cylinder under the action of an electric field along the axis of the cylinder, and determine expressions for the twisting moment and normal force that depend on the angle of twist, stretch and the applied electric field. We obtain an expression for the torsional rigidity of the electroelastic solid cylinder that depends on the electric field, thereby implying its torsional rigidity can be “tuned” by varying the electrical field. While within the context of classical Cauchy elasticity one observes the compression of the isotropic solid cylinder, namely the POYNTING effect, we show that the cylinder can be made shorter or longer by controlling the electrical field.

It has been observed in experiments that, for some materials, when a sheet is subjected to increasing equal biaxial tensile forces on its edges, its deformation can change from homogeneous equal biaxial extension to homogeneous unequal biaxial extension. Such response has been analyzed in the literature as a bifurcation from the base deformation to a new deformation. The analyses have used a constitutive equation that expresses the stress tensor as a function of a deformation tensor. The present work is concerned with analyzing this bifurcation using a new class of constitutive equations in which the deformation tensor is expressed as a function of the stress tensor. Using such a model for an incompressible isotropic nonlinear elastic material, a condition is derived for determining when during an equal biaxial extension of a sheet, there is a bifurcation into an unequal biaxial extension. An example is provided using a constitutive equation that has been fit to experimental data.

This work considers a cylindrical rod, acting as a nonlinear viscoelastic spring, with an attached mass that can displace along and rotate about the rod’s axis, thereby inducing an extension and twist in the rod. The coupling between them produces the Poynting effect in the body. The material is modeled by the nonlinear single integral Pipkin–Rogers constitutive equation. The mass has been assumed to be at rest in a long-time equilibrium state and then given a small axial or rotational disturbance and released (plucked). The subsequent motion is governed by a linear Volterra integro-differential equation. For a particular choice of material parameters, analytical expressions for the time-dependent decay of the disturbance are obtained in terms of the extension/torsion coupling, material stiffness, the amount and rate of stress relaxation, and the inertia of the mass. For other choices of material parameters, the results provide insight into how these quantities affect the time-dependent decay. A number of plucking scenarios are treated such as extensional or rotational disturbances from the undeformed state, rotational plucking after a finite axial stretch, and extensional plucking after a finite rotation. Simultaneous rotational and extensional plucking leads to a system of Volterra integro-differential equations whose treatment is deferred to later work.

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.

We study the response of a class of transversely elastic bodies, wherein the Green–Saint Venant strain tensor is a function of the second Piola–Kirchhoff stress tensor, when the body is residually stressed. The notion of such non-Cauchy elastic bodies being transversely isotropic is defined in Rajagopal (Mech. Res. Commun. 64, 2015, 38–41), and by a body being residually stressed, we mean the interior of the body is not in a stress-free state although the boundary is free of traction as considered by Coleman and Noll (Arch. Ration. Mech. Anal. 15, 1964, 87–111) and by Hoger (Arch. Ration. Mech. Anal. 88, 1985, 271–289).

This work considers the uniaxial compressive creep of a solid elastomeric circular cylinder under a fixed dead load as it undergoes time dependent and temperature dependent changes in its macromolecular structure. The changes occur at temperatures exceeding 100 °C and consist of chemical scission of macromolecular network junctions and their re-crosslinking to form new networks in new reference configurations. The scission process causes softening due to modulus reduction and the re-crosslinking causes stiffening due to deformation of newly formed networks. A condition is developed for the time when the homogeneous uniaxial creep deformation history develops an non-homogenous barrel shaped branch. Branching times and the corresponding compression ratio are calculated from this condition using mechanical and chemical scission properties from the experimental literature.

Intrinsic mechano-chemo-transduction (MCT) mechanisms at the single cardiomyocyte level are being studied via the Cell-in-Gel system, where isolated live cardiomyocytes are embedded in a constraining hydrogel. Knowing that normal heart function has a compensatory response to elevated systolic pressure (Anrep effect) and that excessive mechanical stress in myocardium can lead to arrhythmias and heart failure, our aim is to better understand MCT mechanisms at the single cell level subject to various mechanical scenarios.

This work considers the uniaxial compression of a solid circular cylinder of time-dependent material. Initially, the cylinder undergoes a homogeneous compression history. The purpose is to determine a time when this deformation history can form a new inhomogenous branch. Material time dependence is described by a nonlinear single-integral constitutive equation that relates the stress in the material to its deformation history. A criterion is developed for determining branching times using a Pipkin–Rogers constitutive equation for nonlinear incompressible isotropic viscoelastic solids. For the purpose of numerical examples, material parameters are chosen so that the material acts as a neo-Hookean material in its short-time response and a softer one in its long-time equilibrium response. Examples show that characteristic relaxation times and deformation histories influence the time to branch and the corresponding compressive stretch and compressive force.

We investigate the pure bending of an elastic prismatic beam, but unlike in the classical setting we assume that the material parameters are density-dependent. The corresponding boundary value problem admits a semi-analytical solution, and the derived formulae allow one to quickly assess the impact of density-dependent material parameters on the predicted deformation across various parameter regimes, and consequently make a decision on the importance of the density-dependent material parameters in the given setting.

This work considers a rubber cylinder under zero axial force that elongates in response to the normal stresses produced during torsion (the Poynting effect). The combined elongation and twisting deformation occurs at an elevated temperature at which the rubber undergoes time-dependent scission and re-crosslinking of its macromolecular network junctions. A constitutive theory accounting for this microstructural change is used in an analytical and numerical study of the interaction of the deformation and the scission or re-crosslinking process. Examples show the time-dependence of elongation for several twist histories.

We develop a viscoelastic generalization of the elastic Eshelby inclusion solution, where the inclusion and surrounding matrix are two different viscoelastic solids and the inclusion's eigenstrain is a time-periodic oscillatory input. The solution exploits the Correspondence Principle of Linear Viscoelasticity and a Discrete Fourier Transform to efficiently capture the steady-state oscillatory behavior of the 3-D mechanical fields. The approach is illustrated here in the context of the recently-developed in vitro Cell-in-Gel system, where an isolated live cardiomyocyte (the inclusion) is paced to contract periodically within a soft hydrogel (the matrix), for the purpose of studying the effect of mechanical load on biochemical signals that regulate contractility. The addition of viscoelasticity improves the fidelity of our previous elastic Eshelby inclusion analysis of the Cell-in-Gel system by accounting for the time-varying fields and the resulting hysteresis and dissipated mechanical energy. This mathematical model is used to study the parametric sensitivities of the relative stiffness of the inclusion, the inclusion's aspect ratio (slenderness), and the cross-link density of the hydrogel matrix.

In this note, we study the response of a viscoelastic body whose stress relaxation modulus and creep compliance depend on the density of the body in such a manner that the stress and strain appear linearly in the constitutive equation. Such models would be useful to study the response of porous viscoelastic bodies undergoing small deformations, as the moduli depend on the porosity, and hence the density. We study the problem of tension–torsion of cylinders of arbitrary cross-section within the context of this constitutive relation.

A fiber-reinforced material comprised of a soft polymeric matrix reinforced with polymeric filaments is often modeled as an equivalent anisotropic nonlinearly elastic solid. Although the response of a single constituent polymeric material can be modeled by nonlinear thermo-elasticity over a large range of deformations and temperatures, there can be conditions requiring a theory that extends the range of application to account for other features, such as nonlinear viscoelasticity and an evolving microstructure due to a combination of mechanical and nonmechanical factors. In a multi-constituent fiber-reinforced material these effects can be expected to occur with different initial triggering and ongoing potency in the separate polymer matrix and fiber constituents. This paper summarizes a number of constitutive models for fiber-reinforced materials that include these features, discusses the connection of these models to a nonlinearly elastic scaffold, provides a framework for the incorporation of these features into the constitutive theory for an equivalent general simple solid, and shows how certain terms in the mathematical structure can be associated with the matrix constituent while other terms can associated with the fibrous constituent.

Successful soft robot modeling approaches appearing in recent literature have been based on a variety of distinct theories, including traditional robotic theory, continuum mechanics, and machine learning. Though specific modeling techniques have been developed for and validated against already realized systems, their strengths and weaknesses have not been explicitly compared against each other. In this paper, we show how three distinct model structures —a lumped-parameter model, a continuum mechanical model, and a neural network— compare in capturing the gross trends and specific features of the force generation of soft robotic actuators. In particular, we study models for Fiber Reinforced Elastomeric Enclosures (FREEs), which are a popular choice of soft actuator and that are used in several soft articulated systems, including soft manipulators, exoskeletons, grippers, and locomoting soft robots. We generated benchmark data by testing eight FREE samples that spanned broad design and kinematic spaces and compared the models on their ability to predict the loading-deformation relationships of these samples. This comparison shows the predictive capabilities of each model on individual actuators and each model's generalizability across the design space. While the neural net achieved the highest peak performance, the first principles-based models generalized best across all actuator design parameters tested. The results highlight the essential roles of mathematical structure and experimental parameter determination in building high-performing, generalizable soft actuator models with varying effort invested in system identification.

When a rectangular block of a nonlinear material is subjected to a simple shearing deformation, specific normal tractions are required to ensure that the distances between the faces of the block, i.e. its dimensions, do not change. This work investigates the time-dependent dimensional changes during shear in the absence of these normal tractions (the Poynting effect) that occur in a block composed of an incompressible nonlinearly viscoelastic fiber-reinforced solid. The material is modeled using the Pipkin–Rogers nonlinear single integral constitutive equation for viscoelasticity. This constitutive equation is used because (1) it exhibits the essential features of nonlinear viscoelasticity; (2) it is straightforward to include the material symmetry restrictions due to the reinforcing fibers. A system of nonlinear Volterra integral equations is formulated for the dimensional changes in the block. Numerical solutions are presented for the case when the standard reinforcing model for nonlinearly elastic fiber-reinforced materials is incorporated in the Pipkin–Rogers constitutive framework. The results illustrate how the time-dependent dimensional changes depend on the fiber orientation and the viscoelastic properties of the fibers relative to those of the matrix.

In pulmonary hypertension and certain forms of congenital heart disease, ventricular pressure overload manifests at birth and is an obligate hemodynamic abnormality that stimulates myocardial fibrosis, which leads to ventricular dysfunction and poor clinical outcomes. Thus, an attractive strategy is to attenuate the myocardial fibrosis to help preserve ventricular function. Here, by analyzing RNA-sequencing databases and comparing the transcript and protein levels of fibrillar collagen in WT and global-knockout mice, we found that slit guidance ligand 3 (SLIT3) was present predominantly in fibrillar collagen-producing cells and that SLIT3 deficiency attenuated collagen production in the heart and other nonneuronal tissues. We then performed transverse aortic constriction or pulmonary artery banding to induce left and right ventricular pressure overload, respectively, in WT and knockout mice. We discovered that SLIT3 deficiency abrogated fibrotic and hypertrophic changes and promoted long-term ventricular function and overall survival in both left and right ventricular pressure overload. Furthermore, we found that SLIT3 stimulated fibroblast activity and fibrillar collagen production, which coincided with the transcription and nuclear localization of the mechanotransducer yes-associated protein 1. These results indicate that SLIT3 is important for regulating fibroblast activity and fibrillar collagen synthesis in an autocrine manner, making it a potential therapeutic target for fibrotic diseases, especially myocardial fibrosis and adverse remodeling induced by persistent afterload elevation.

Fibre-reinforced, fluid-filled structures are commonly found in nature and emulated in devices. Researchers in the field of soft robotics have used such structures to build lightweight, impact-resistant and safe robots. The polymers and biological materials in many soft actuators have these advantageous characteristics because of viscoelastic energy dissipation. Yet, the gross effects of these underlying viscoelastic properties have not been studied. We explore nonlinear viscoelasticity in soft, pressurized fibre-reinforced tubes, which are a popular type of soft actuation and a common biological architecture. Relative properties of the reinforcement and matrix materials lead to a rich parameter space connecting actuator inputs, loading response and energy dissipation. We solve a mechanical problem in which both the fibre and the matrix are nonlinearly viscoelastic, and the tube deforms into component materials' nonlinear response regimes. We show that stress relaxation of an actuator can cause the relationship between the working fluid input and the output force to reverse over time compared to the equivalent, non-dissipative case. We further show that differences in design parameter and viscoelastic material properties can affect energy dissipation throughout the use cycle. This approach bridges the gap between viscoelastic behaviour of fibre-reinforced materials and time-dependent soft robot actuation.