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In this work, the nonlinear forced vibration response of a hyperelastic combined cantilever plate is investigated. The plate consists of two irregular quadrilaterals and is under a uniformly distributed load perpendicular to the midplane. First, two irregular quadrilateral regions are projected into two rectangular regions by the iso-parametric element transformation. Then the nonlinear ordinary differential governing equations is obtained by the thin-plate theory, and Lagrange variational principle. In addition, the chaotic behavior is analyzed by the bifurcation diagram, Poincaré map, phase diagram, and maximum Lyapunov exponent. The results indicate that the amplitude and frequency of the external excitation have a significant effect on the nonlinear vibration, each of which can excite the chaotic response. Furthermore, it is found that the varying geometric parameters generate a larger chaotic region compared to the varying external excitation parameters. In summary, this study provides theoretical references for the excitation strategy and structural design of hyperelastic combined cantilever plates.

The investigation of hyperelastic responses of soft materials and structures is essential for understanding of the mechanical behaviors and for the design of soft systems. In this paper, by considering both the material and geometrical nonlinearities, a new neo-Hookean model for the hyperelastic beam is developed with focus on its nonlinear free vibration with large strain deformations. The neo-Hookean model is employed to capture the large strain deformation of the hyperelastic beam. The governing equations of the hyperelastic beam are derived by using Hamilton’s principle. To avoid expensive calculations for solving the nonlinear governing equations, a simplified Taylor-series expansion model is proposed. The effects of two key system parameters, i.e. the initial displacement amplitude and the slenderness ratio, on the nonlinear free vibrations of the hyperelastic beam are numerically analyzed. The bifurcation diagrams, displacement time traces, phase portraits and power spectral diagrams are presented for the nonlinear free vibrations of the hyperelastic beam. For small initial displacement amplitudes, it is found that the hyperelastic beam will undergo limit cycle oscillations, depending on the initial amplitude employed. For initial displacement amplitudes large enough, interestingly, the free vibration of the hyperelastic beam will become quasi-periodic or chaotic, which were rarely reported for the free vibration of linearly elastic beams. Also observed is the traveling wave feature of oscillating shapes of the hyperelastic beam, indicating that higher-order modes of the beam are excited even for free vibrations. All these new features in the nonlinear free vibrations of hyperelastic beams indicate that the material and geometric nonlinearities play a great role in the dynamic analysis of hyperelastic beams.

The nonlinear vibration of a hyperelastic moderately thick cylindrical shell with 2:1 internal resonance in a temperature field is investigated based on the third-order shear deformation theory. A radial harmonic excitation is applied to the shell. First, by employing the higher-order approximation for the curvature-related expansion, the displacement field of the moderately thick cylindrical shell with improved coefficients is derived. For the temperature field with gradient, the shear modulus of the shell is thickness dependent because of the temperature-dependent nature of the hyperelastic materials used. The graphical results manifest that the temperature gradient has a significant impact on the nonlinear vibration of the shell. In addition, the separation of the resonance peak caused by the variations of the structural parameter and temperature will result in a bubble shaped response curve for the shell.

Fast-moving soft robotics usually suffer large-strain deformation with a relatively high velocity. To improve their performance, it is necessary to study the underly dynamical mechanism. However, the inherent strong material nonlinearity brings considerable difficulties in modeling and calculating the soft structure’s nonlinear dynamics. This work focuses on the large-strain vibrations of the hyperelastic rod under an axial tensile force. Several typical hyperelastic models, including the neo-Hookean, Mooney, and Yeoh models, are used to derive the equations governing the rod’s dynamics. The equations based on these models are relatively complex, and their corresponding numerical calculation diverges in some cases. Therefore, a reference stretch ratio(s) polynomial fitting (RSRPF) model is proposed to derive the concise governing equation. Such a governing equation is effectively solved by the combination of the Galerkin discretization and the fourth-order Runge–Kutta integration algorithm. The correctness of the developed model and the employed algorithm is verified by comparing with the previous data of a hyperelastic membrane. In most cases, the rod periodically vibrates under a periodic tensile force. However, when the structural damping is small and the load frequency approaches the resonant frequency, the hyperelastic rod may quasi-periodically vibrate. The revealed large-strain vibration mechanism is supposed to be useful for guiding the dynamical design of soft structures.

Based on the modified third-order shear deformation theory, the harmonic balance method, and the pseudo-arclength continuation method with two-point prediction, the nonlinear forced vibration response of incompressible hyperelastic moderately thick cylindrical shells subjected to a concentrated harmonic load at mid-span and simply supported boundary conditions at both ends is investigated. The algorithmic procedure for solving steady-state periodic solutions of strongly nonlinear systems of differential equations is presented. The structural response characteristics of shells under different excitation amplitudes and structural parameters are analyzed. The numerical results indicate that the aspect ratio of moderately thick hyperelastic cylindrical shells has a significant effect on the natural frequency ratio. Different frequency ratios lead to varying nonlinear mode coupling effects. The coupling effects among modes result in complex nonlinear behavior in the vibration response of each mode, leading to abundant multi-valued phenomena in the response curve.

Collagen plays an extremely important role in carrying forces and maintaining the shape of the cornea. In keratoconus, the cornea shape can become distorted to the extent that normal vision is impossible, and the amount crosslinking between collagen fibrils are generally lower than in healthy eyes. In contrast, riboflavin-induced crosslinks can strengthen and stiffen the cornea. This article examined quantitatively how the extent of crosslinking in collagen fibrils influences the overall mechanical behavior of corneal tissue. Three models for the stress–strain behavior of the fibrils were examined, which is a function of the crosslink density within the fibrils. These models were then embedded in a matrix model, and tensile tests of cornea strips were examined using a finite element program. Results were compared with experiments from the literature for both normal and crosslinked corneas.

Finite element fingertip models are useful tools to assess product ergonomics. While “real geometry” approaches provide accurate results, developing models requires medical images. “Simpified geometry” approaches have to date not been tested to see whether they can provide equally accurate results in terms of mechanical response, i.e. force-displacement response and dimensions of fingertip contact area. Four fingertip models were built either from medical images (Visible Human project) or from simplified geometries. Simulations of fingertip flat contact compression at 20${}^{\circ }$ were performed. A 2nd order hyperelastic material property was used to effectively reproduce the mechanical behavior of the fingertip. Models based on simplified geometries such as conics proved as accurate as models reconstructed from medical images. However, accurate positioning of the bony phalanx is paramount if a biofidelic mechanical response is to be reproduced.

Finite element (FE) analysis is widely used for exploring the mechanical properties of layers and organs of the human body. The aim of this study was to construct and validate an FE model to monitor human abdominal fascia behavior under an uniaxial tensile test. In this study, our previous experimental results for abdominal fascia from three human cadavers where the age distance is about 20 years are chosen to validate FE model. The Yeoh hyperelastic constitutive model was used for the chosen abdominal fascia samples. The FE analysis was applied to simulate the experimental conditions and to calculate stress and strain distributions generated by the uniaxial tension test. The fitting accuracy of the experimental data to the theoretical one was evaluated. The analysis of the results showed a good correlation between the numerical and experimental data. The percentage error between experimental and FE results is in the interval 4.7–9.4%.

In this paper, we study the buckling of an incompressible hyperelastic rectangular layer due to compression with sliding end conditions. The combined series-asymptotic expansions method is used to derive two-coupled nonlinear ordinary differential equations (ODEs) governing the leading order of the axial strain W and shear strain G. Linear analysis yields the critical stress values of buckling. For the nonlinearly coupled system, by introducing a small parameter, the approximate analytical solutions for post-buckling deformations are obtained by using the method of multiple scales. The amplitude of buckling is expressed explicitly by the aspect ratio, the incremental dimensionless engineering stress and the mode of buckling. To the authors' best knowledge, it is the first time that such an analytical formula is obtained within the framework of two-dimensional field equations for nonlinearly elastic materials (including both geometric and material nonlinearity). Numerical computations of the coupled system are also carried out. Good agreements between the numerical and analytical solutions are found when the amplitudes of buckling are moderate. Finally, some energy analysis regarding material failure is made.

This work concerns mainly the finite element (FE) implementation of polyconvex incompressible hyperelastic models. A user material subroutine (UMAT) has been developed and can be utilized to define the aforementioned material behaviors in the FE system ABAQUS. The subroutine is written using a novel strategy in order to maximally simplify the relations for the analytical material Jacobian (MJ). The UMAT code is attached in the appendix. The developed subroutine allows to significantly decrease the time of computations and to avoid possible convergence difficulties. The structure of the code enables modifications which may lead to a rheological, damage or growth models, for instance.

In this paper, a hierarchic high-order three-dimensional finite element formulation is studied for hyperelastic and anisotropic elastoplastic problems at finite strains. The element formulation allows for anisotropic ansatz spaces supporting efficient discretizations of beam-, plate-, and shell-like structures. Several benchmark examples are investigated and the results of the high-order formulation are compared to analytic solutions and different mixed finite element formulations. Special emphasis will be placed on locking effects, robustness with respect to high aspect ratios and element distortion as well as anisotropies related to the material model. Furthermore, the interplay between the chosen ansatz space for the displacement field and mapping function in the context of geometrically nonlinear problems are studied.

This study proposes a method for calculating hyperelastic materials’ stress and tangent modulus using a generalized complex-step derivative approximation. This method is accurate and requires only the operational code of the strain energy density function. Moreover, because it uses only standard complex operations in Fortran, no prior preparation is required, making the method convenient and practical. The effectiveness of the method was confirmed by applying it to the implementation of isotropic and anisotropic hyperelastic models.

In this paper, a generalization of the standard continuum theory is proposed in order to describe materials with a more complex microstructure. The key idea consists in dividing the studied body onto a set of small but finite disjunct cells whose boundaries form what is called the grid. The state of the grid is described by macroscopically smooth functions. The interior of each cell, on the other hand, is described by an additional field that may be discontinuous or highly oscillating. Such an approach allows us to include non-trivial effects of the microstructure in resulting material models of macroscopic bodies. The theory is illustrated on two examples of two-scale hyperelastic models. The first one represents a crystalline material with a failure of the Cauchy-Born rule. The second one, motivated by an arrangement of soft tissues, includes prestress at the reference state as well as non-trivial response caused by a collapse of micro-constituents. Transparent physical meaning of several material parameters of the latter model provides for their direct identification instead of the least-squares fitting.

The scientific community has witnessed, lately, a tremendous progress in the fabrication and synthesis of nanomaterials. As a result, it is essential to develop new and efficient numerical techniques that are capable of modeling the behavior of materials at nanoscale with sufficient accuracy. In this work, a novel approach is presented for the multiscale analysis of brittle failure in nanostructures using the phase-field modeling. The specimen at microscale is discretized using finite elements (FEs), whose integration points lie in the representative volume elements (RVEs) at nanoscale. The displacement computed in upper scale for a microstructure that contains an evolving crack is imposed on the boundaries of the RVE in lower scale. On the other hand, the stresses and material properties obtained for the RVE in lower scale are transferred to upper scale to compute stiffness matrices and load vectors. The evolution of the phase-field variable indicates the initiation and propagation of cracks at microscale. In order to avoid time-consuming molecular dynamics (MD) simulations at nanoscale in each step of the analysis, the Mooney–Rivlin material model is used to simulate the behavior of Aluminum (AL) nanostructure at this scale. The approach that is utilized to compute the material constants and the formulation for the multiscale technique combined with the phase-field modeling in upper scale are described in detail. It is discussed how the phase-field variable in microstructure is evolved based on the properties of the RVE in nanostructure. Many numerical examples are presented to demonstrate the application of the proposed multiscale technique in the solution of engineering problems.

In this paper, a general model of elastic (non-dissipative) behavior is developed. This model belongs to a class of models, developed for the description of complex bodies, in which the local state is assumed to be determined not only by the deformation, but also by a family of additional material parameters. The latter, unlike some additional structures used in the mechanics of complex bodies (e.g., directors, order parameters, internal degrees of freedom), are not considered as interactions of microscopic nature; rather they are considered as variables of macroscopic nature that describe the internal structure of the material, while their rates describe the evolution of the internal structure in the course of deformation. Accordingly, these variables are assumed to evolve continuously with time in a manner that guaranties the reversibility of the applied dynamical process. A covariant theory for the continuum in question is derived by means of invariance properties of the global form of the spatial energy balance equation, under the superposition of arbitrary spatial diffeomorphisms. In particular, it is shown that the assumption of spatial covariance of the equation of balance of energy yields the standard conservation and balance laws of classical mechanics but it does not yield the standard Doyle–Ericksen formula. In fact, the "Doyle–Ericksen formula" derived in this work, has some extra terms in it, which are related directly to the internal structure of the material, as the latter is controlled by the additional parameters. In a similar manner, by assuming the absolute temperature as an additional state variable and by employing the invariance properties of the local form of the spatial balance of energy under superimposed spatial diffeomorphisms, which also include a temperature rescaling, a nonisothermal covariant constitutive theory is naturally obtained. A formal comparison of the proposed elastic material with the standard hyperelastic (Green elastic) solid is also presented.

In this paper, the compression of an isolated cell by two rigid indenters is analyzed. The neo-Hookean model is employed to characterize the hyperelastic behavior of biological cells. Owing to the greatly increased ratio between surface energy density and elastic modulus, surface energy plays important roles in the mechanical performance of biological cells. Using the dimensional analysis method and a finite element approach incorporating surface energy, we study the elastic compression of hyperelastic cells at finite deformation and give the explicit relations of contact radius and indent depth depending on compressive load. Our results reveal that surface energy obviously influences both the local deformation and the overall responses of hyperelastic cells at finite deformation. The obtained results are useful to determine the elastic properties of biological cells from indent-depth curves accurately.

Morphing wings covered with elastomeric skin have emerged as a promising technique for enhancing the performance and efficiency of unmanned aerial vehicles (UAVs). These morphing wings can change shape in flight, enabling UAVs to adapt to evolving aerodynamic conditions, fly more efficiently, maneuver more effectively and perform a broader range of missions. The durability of such elastomeric skins that cover the wings, on the other hand, is a critical issue that requires careful consideration. During the flight, elastomeric skins are subjected to a variety of mechanical stresses, including tear and fracture, which can significantly impact the performance and reliability of elastomeric morphing wings. To ensure the long-term durability of the morphing wings, a comprehensive understanding of the tear fracture of elastomeric skins is essential. This study employs a multi-faceted approach of experimental and computational research to investigate the tear fracture of elastomeric skins in morphing wings. Initially, the fracture properties of three materials — Latex, Oppo and Ecoflex — are evaluated experimentally for various cut positions. Subsequently, a continuum physics-based tear fracture model is derived to numerically simulate the mechanical behavior of elastomeric skins. The Griffith criterion, a well-established method, is adopted to investigate mode-III fracture tests, specifically the trousers test, which involves pulling two legs of a cut specimen horizontally apart. Finally, the derived tear fracture model is validated by comparing model solutions to tear test data obtained experimentally. The study suggests that adjusting the stretch ratio and cut position can significantly impact the stress distribution of elastomeric skins, the ability to resist fracture and the stretching behavior of elastomeric morphing wings.

This contribution aims to derive a nonlinear FE formulation for large elastic deformation of hyperelastic thin rubber-like elastomeric materials. The formulation accounts for isotropic as well as transversely isotropic rubber-like materials. Besides, with the developed formulation, deformation of incompressible as well as compressible materials can be investigated. The formulation is applicable for in-plane and out-of-plane deformation of thin membranes. Several problems are inspected including uniaxial and equibiaxial extension of rubber, axisymmetric deformation of annular sheet, stretching a perforated rubber, extension of an anisotropic rubber, and inflation of an initially-flat circular and a square membrane. The results show very good agreement compared with those reported in the literature. Moreover, this model is able to model the inflation of initially flat membranes which is not possible to be done by commercial software.

Conducting polymer (CP) is an electroactive polymer that displays specific electronic properties, including conductivity. The utility of CP-based soft actuators in various biomedical applications has recently been motivated due to their low voltage-driven specialty compared to the widely used high voltage-driven dielectric elastomers. In some biomedical applications, highly delicate CP actuators may be torn or damaged for unknown reasons. In this regard, this study develops a tear fracture model for fiber-reinforced CP actuators to investigate a specific fracture test of mode III, namely, the trousers test, which involves pulling two legs of a cut specimen horizontally apart. The development of the tear fracture model adopts a well-known Griffith criterion along with the thermodynamically consistent continuum mechanics approach. Additionally, prominent strain energy capturing elastomer strain-stiffening at a moderate strain range is used in conjunction with an empirically established correlation to couple the two internal phenomena, ion diffusion and mechanical deformation of the CP actuators. Later, the effects of various electrical and geometrical parameters on the tearing of the actuator are also addressed.

Most biological phenomena commonly involve growth and expansion mechanics. In this work, we propose an innovative model of cancerous growth which posits that an expandable tumor can be described as a poroelastic medium consisting of solid and fluid components. To verify the feasibility of the model, we utilized an established epithelial human breast cancer cell line (MDA-MB-231) to generate an in vitro tumorsphere system to observe tumor growth patterns in both constrained and unconstrained growth environments. The tumorspheres in both growth environments were grown with and without the FDA-approved anti-breast cancer anthracycline, Doxorubicin (Dox), in order to observe the influence small molecule drugs have on tumor-growth mechanics. In our biologically informed mechanical description of tumor growth dynamics, we derive the governing equations of the tumor’s growth and incorporate them with large deformation to improve the accuracy and efficiency of our simulation. Meanwhile, the dynamic finite element equations (DFE) for coupled displacement field and pressure field are formulated. Moreover, the porosity and growth tensor are generalized to be functions of displacement and pressure fields. We also introduce a specific porosity and growth tensor. In both cases, the formalism of continuum mechanics and DFE are accompanied by accurate numerical simulations.