Spherical elastic shells commonly appear both in nature and man-made devices. Often, their functionality is governed by an incoming- or outgoing flux of fluid. The transient traction that the fluid ...exerts in the process causes the shell to depart from sphericity. Here, we develop a framework for determining non-spherical axisymmetric deformations, by combining tools from nonlinear continuum mechanics, structural mechanics, and asymptotic analysis. We apply our framework to analyze an exemplary problem of a Mooney–Rivlin shell that is filled by viscous fluid. Collectively, our framework and the insights gained from its application, promote the understanding of the mechanics of such fluid-filled deformable membranes and shells.
•We develop a framework for determining non-spherical deformations of elastic shells.•We apply our framework to analyze an exemplary problem of shell that is filled by viscous fluid.•The insights gained from its application, promote the understanding of the mechanics of such fluid-filled deformable membranes and shells.
We prove upper and lower bounds for a variational functional for convex functions satisfying certain boundary conditions on a sector of the unit ball in two dimensions. The functional contains two ...terms: The full Hessian and its determinant, where the former is treated as a small perturbation in the space L2 and the latter as the leading-order term, in the negative Sobolev space W−2,2. We point out how this setting is motivated by problems in nonlinear elasticity, and obtain a corollary for a variational problem based on the so-called Föppl-von Kármán energy.
A three parameter novel hyperelastic strain energy function is introduced in this paper for soft and rubber-like materials. The function integrates a non-separable exponential component with a single ...term Ogden-type polynomial-like function, resulting in an exponential-polynomial based strain energy function. This helps in capturing both small and large deformation (stretch) behaviours of hyperelastic materials. The structure of the model is simple and validated against several experimental datasets including rubbers, hydrogel, and soft tissues. The model is reported to capture key material behaviors, including strain stiffening and various deformation paths. Through comparative studies with well-known models like the Ogden (six parameters) and Yeoh (three parameters), the model’s effectiveness is established. Furthermore, the model successfully addresses pressure-inflation instability in thin spherical balloons. It’s applicability extends to biological materials, as evidenced by its effectiveness in characterizing porcine brain tissue and a monkey’s bladder.
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•A three parameter strain energy function for soft and rubber-like materials has been developed.•Combined an exponential component with an Ogden-type polynomial term.•Experimental data of rubbers, hydrogel, and soft tissue are used to validate the model.•The model effectively addresses pressure-inflation instability in thin spherical balloons.•Advantages of the model are illustrated through comparisons with Ogden and Yeoh models.
•The mass sensitivity of hyperelastic plates is examined.•Experimental vibration tests are performed for different forces and masses.•A theoretical model is presented and validated through different ...tests.•Different internal resonance phenomena are investigated.
This paper presents a joint experimental and theoretical approach to the dynamics and mass sensitivity of hyperelastic plates including cases with modal interactions. For the theoretical approach, the plate structure is assumed to undergo large strains and deformations using the Mooney-Rivlin hyperelastic strain energy density model and the von-Kármán geometric nonlinearity, respectively. The plate is modelled using the continuum mechanics definitions and the Kirchhoff–Love plate theory. The coupled in-plane and out-of-plane equations of motion are obtained using Hamilton's equation and solved afterwards using a combination of Galerkin's procedure together with a dynamic equilibrium technique. For the experimental analysis, the properties of the material are first obtained by performing a set of stress-strain tests on the samples following the ASTM D-412 standard, and then from the same rubber material, a plate structure is fabricated, and an externally actuated vibration test is performed. The nonlinear frequency response of the structure both with and without a concentrated mass is investigated and the capability of the structure for mass sensing is discussed. By obtaining the amplitude-frequency response of the structure due to the applied external periodic load, using both theoretical and experimental approaches, it is shown that the given model has good accuracy in simulating the nonlinear dynamics of the hyperelastic plate structure under different conditions. Furthermore, a set of analyses on the nonlinear forced vibration behaviour of hyperelastic plates at different internal resonances, using concentrated mass and the length-to-width ratio is presented. The results of this study are useful in designing systems involving hyperelastic plates, such as soft robots and soft functional devices.
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Elastic buckling of slender structures is of high interest in engineering since the buckling could bring many unintended and disastrous consequences. The critical load of buckling in nonlinear ...elasticity can be highly affected by either the structure’s geometry or the material properties. This paper studies the buckling behaviors of an elastic block of soft materials with graded modulus at finite deformation. We give the total potential energy of the conservative system and formulate a boundary-value problem and its incremental forms. We solve the buckling problem analytically for an elastic block of neo-Hookean materials. Though our problem is different from Euler’s buckling, our results agree well with the Euler’s formula in the case of slender columns. In addition, our analysis can be used to predict the buckling load of short columns. Since our loading device controls the displacement of the block, the critical load is computed by using the critical stretch at which the loaded block begins to buckle. Depending on the varied forms of the moduli, the graded modulus can tune the critical load. The forms of the linear and exponential functions of the Young moduli of the material can only reduce the critical load. However, the quadratic form can either reduce or increase the critical load by about 50%, making a weaker or stronger structure without changing its geometry. This paper contributes to our understanding of the buckling behaviors of elastic structures with graded moduli.
•A closed-form analysis of the buckling of a finite block of soft materials.•The energy formulation and the complete linear bifurcation analysis are given.•A system of PDEs and ODEs are solved analytically and numerically.•The graded modulus can either reduce or increase the critical load by about 50%.
The Winkler foundation model is often used to analyze the wrinkling of a film/substrate bilayer under compression, and it can be rigorously justified when both the film and substrate are homogeneous ...and the film is much stiffer than the substrate. We assess the validity of this model when the substrate is still homogeneous but the film has periodic material properties in the direction parallel to the interface. More precisely, we assume that each unit cell is piecewise homogeneous, and each piece can be described by the Euler–Bernoulli beam theory. We provide analytical results for the critical compression when the substrate is viewed as a Winkler foundation with stiffness modeled either approximately (as in some previous studies) or exactly (using the Floquet theory). The analytical results are then compared with those from Abaqus simulations based on the three-dimensional nonlinear elasticity theory in order to assess the validity of the Euler–Bernoulli beam theory and the Winkler foundation model in the current context.
•Buckling of a periodic film/substrate bilayer is analyzed using a beam model.•The response of the substrate is described analytically using the Floquet theory.•A semi-analytical bifurcation condition is derived using the PWE method.•A formula is given that determines whether a whole wave consists of one or two unit cells.•Validity of various assumptions is assessed by comparing with Abaqus simulations.
The equations of nonlinear elastodynamics have attracted the attention of pure and applied mathematicians since the second half of the last century. Nevertheless, many aspects of this theory are ...still obscure or need a more systematic analysis. In this paper we study the wave propagation in an isotropic elastic solid by taking into account the effects due to dispersion. Specifically, we determine a class of global (in both time and space) solutions, find exact solutions for longitudinal travelling waves within the fourth-order theory of elasticity, and derive two asymptotic reductions of the governing equations for waves of small amplitudes.
•We study the wave propagation in a nonlinear isotropic elastic solid by taking into account the effects due to dispersion. We determine a class of global (in both time and space) solutions.•We find exact solutions for longitudinal travelling waves within the fourth-order theory of elasticity.•We derive two asymptotic reductions of the governing equations for waves of small but finite amplitudes.
In the present work, two machine learning based constitutive models for finite deformations are proposed. Using input convex neural networks, the models are hyperelastic, anisotropic and fulfill the ...polyconvexity condition, which implies ellipticity and thus ensures material stability. The first constitutive model is based on a set of polyconvex, anisotropic and objective invariants. The second approach is formulated in terms of the deformation gradient, its cofactor and determinant, uses group symmetrization to fulfill the material symmetry condition, and data augmentation to fulfill objectivity approximately. The extension of the dataset for the data augmentation approach is based on mechanical considerations and does not require additional experimental or simulation data. The models are calibrated with highly challenging simulation data of cubic lattice metamaterials, including finite deformations and lattice instabilities. A moderate amount of calibration data is used, based on deformations which are commonly applied in experimental investigations. While the invariant-based model shows drawbacks for several deformation modes, the model based on the deformation gradient alone is able to reproduce and predict the effective material behavior very well and exhibits excellent generalization capabilities. In addition, the models are calibrated with transversely isotropic data, generated with an analytical polyconvex potential. For this case, both models show excellent results, demonstrating the straightforward applicability of the polyconvex neural network constitutive models to other symmetry groups.
For geometric nonlinear elastic and thermoelastostatic analysis of the nylon springs (artificial muscles) with negative thermal expansion an innovative geometrically nonlinear nylon spring finite ...element is established. This straight two-noded finite element is based on the Updated Lagrange Formulation (ULF) at finite displacements. The stiffness matrix of the spring finite element containing a linear and nonlinear part is established without any linearization. The required and measured initial parameters of the nylon spring finite element are its length, the linear spring constant, and the compressive prestress force. The additionally measured negative thermal spring expansion coefficient is included into the thermomechanical load. Numerical experiments using the herein proposed algorithm are performed based on nonlinear elastic and thermoelastic analysis of the Nylon Twisted Coiled Spring (NTCS) which is also known as an artificial muscle. Additionally, a geometrically nonlinear elastostatic analysis of the passive nylon damper is performed. The established nonlinear finite element equations of the nylon spring systems are solved using Newton’s incremental method. Selected numerical results are verified experimentally using a physical model of the nylon spring system. These measurements indicate perfect accuracy of the proposed numerical approach, and it turns out that large extension of the nylon spring can only be captured by nonlinear calculations. This innovative procedure can be used in modelling and dimensioning of artificial muscles, special thermoelastic actuators as well as in the design of flexible storage of small stationary or mobile systems.
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•Nylon Twisted Coiled Springs (NTCS) with negative thermal expansion were studied.•Nonlinear elasticity and thermoelasticity of the NTCS were observed.•NTCS nonlinear finite element was established for numerical deformation analysis.•Artificial muscles and nylon damper were numerically analysed.•Results of the nonlinear numerical analyses were confirmed by measurement.
•Nonlinear modeulation equation for acoustic metamaterial.•Mutual influence of nonlinearities on the wave behavior.•Correspondence between discrete and continuum modeling of metamaterial.
Two kinds ...of nonlinearity are studied in the framework of a model of a discrete diatomic model of an acoustic metamaterial. It is shown that the continuum limit of a discrete model is similar to the equations obtained using a model of a reduced continuum with a microstructure. An asymptotic approach is developed to obtain a modulation nonlinear governing equation for dynamical processes in an acoustic metamaterial. New metamaterial features caused by nonlinearity are found.
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