Acoustoelasticity measurements in a sample of room dry Berea sandstone are conducted at various loading frequencies to explore the transition between the quasi‐static (
f→0) and dynamic (few ...kilohertz) nonlinear elastic response. We carry out these measurements at multiple confining pressures and perform a multivariate regression analysis to quantify the dependence of the harmonic content on strain amplitude, frequency, and pressure. The modulus softening (equivalent to the harmonic at 0f) increases by a factor 2–3 over 3 orders of magnitude increase in frequency. Harmonics at 2f, 4f, and 6f exhibit similar behaviors. In contrast, the harmonic at 1f appears frequency independent. This result corroborates previous studies showing that the nonlinear elasticity of rocks can be described with a minimum of two physical mechanisms. This study provides quantitative data that describes the rate dependency of nonlinear elasticity. These findings can be used to improve theories relating the macroscopic elastic response to microstructural features.
Key Points
We study the frequency dependence of nonlinear elasticity in Berea sandstone
We find different frequency dependence behavior among the harmonic content
These findings will ultimately help in better relating the nonlinear elastic behavior to rock microstructure
The mechanical response of a homogeneous isotropic linearly elastic material can be fully characterized by two physical constants, the Young’s modulus and the Poisson’s ratio, which can be derived by ...simple tensile experiments. Any other linear elastic parameter can be obtained from these two constants. By contrast, the physical responses of nonlinear elastic materials are generally described by parameters which are scalar functions of the deformation, and their particular choice is not always clear. Here, we review in a unified theoretical framework several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests. These parameters represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. Universal relations between certain of these parameters are further established, and then used to quantify nonlinear elastic responses in several hyperelastic models for rubber, soft tissue and foams. The general parameters identified here can also be viewed as a flexible basis for coupling elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales.
The von Mises truss has been widely studied in the literature because of its numerous applications in multistable and morphing structures. The static equilibrium of this structure was typically ...addresses by considering only geometric nonlinearities. However, Falope et al. (2021) presented an entirely nonlinear solution in finite elasticity and demonstrated that material nonlinearities play an important role in the prediction of both snap-through and Euler buckling. In such work, the von Mises truss was subjected to a vertical load and thus the system was symmetric and the deformations were relatively small. The present contribution extends the investigation to the case of a horizontal load, which is much more complex due to asymmetry and very large deformations. Since most rubbers employed in technological applications exhibit hardening under large stretches, we propose a new hyperelastic model capable of reproducing this behavior. The advantage of such model compared to the ones available in the literature is that the equilibrium solution maintains a straightforward mathematical form, even when considering compressibility of the material. In addition, in this work we present a new formulation in nonlinear elasticity to predict Euler buckling. The formulation takes into account shear deformation. The analytical prediction agrees well with both finite element (FE) and experimental results, thus demonstrating the accuracy of the proposed model.
•Fully nonlinear formulation for equilibrium of the von Mises truss.•New hyperelastic model that reproduces hardening of rubber-like materials.•Analytical model for the prediction of Euler buckling in nonlinear elasticity.•The proposed models are validated with accurate experiments and FE simulations.
Due to surface tension, a beading instability takes place in a long enough fluid column that results in the breakup of the column and the formation of smaller packets with the same overall volume but ...a smaller surface area. Similarly, a soft elastic cylinder under axial stretching can develop an instability if the surface tension is large enough. This instability occurs when the axial force reaches a maximum with fixed surface tension or the surface tension reaches a maximum with fixed axial force. However, unlike the situation in fluids where the instability develops with a finite wavelength, for a hyperelastic solid cylinder that is subjected to the combined action of surface tension and axial stretching, a linear bifurcation analysis predicts that the critical wavelength is infinite. We show, both theoretically and numerically, that a localized solution can bifurcate sub-critically from the uniform solution, but the character of the resulting bifurcation depends on the loading path. For fixed axial stretch and variable surface tension, the localized solution corresponds to a bulge or a depression, beading or necking, depending on whether the axial stretch is greater than a certain threshold value that is dependent on the material model and is equal to 23 when the material is neo-Hookean. At this single threshold value, localized solutions cease to exist and the bifurcation becomes exceptionally supercritical. For either fixed surface tension and variable axial force, or fixed axial force and variable surface tension, the localized solution corresponds to a depression or a bulge, respectively. We explain why the bifurcation diagrams in previous numerical and experimental studies look as if the bifurcation were supercritical although it was not meant to. Our analysis shows that beading in fluids and solids are fundamentally different. Fluid beading resulting from the Plateau–Rayleigh instability follows a supercritical linear instability whereas solid beading in general is a subcritical localized instability akin to phase transition.
•It is shown that beading in fluids and solids are fundamentally different phenomena.•It is demonstrated that localized necking and bulging are both possible depending on the loading path.•It is shown that there exists an exceptional case in which the bifurcation is super-critical.•Theoretical predictions are verified by numerical simulations.
In this work, we have explored growth-induced mechanical instability in an isotropic circular hyperelastic plate. Consistent two-dimensional governing equations for a plate under a general finite ...strain are derived using a variational approach. The derived plate equations are solved using the compound matrix method for two cases of axisymmetric growth conditions – purely radial, and combined radial and circumferential growth. The effect of growth on the buckling behaviour of the plate (in particular, the critical growth factor and the associated buckling mode shapes) is investigated for different thickness values. These results are applicable to model growth induced deformation in planar soft tissues such as skin.
We discuss the strong ellipticity (SE) conditions for strain gradient and micromorphic continua considering them as an enhancement of a simple nonlinearly elastic material called in the following ...primary material. Recently both models are widely used for description of material behavior of beam-lattice metamaterials which may possess various types of material instabilities. We analyze how a possible loss of SE results in the behavior of enhanced models. We shown that SE conditions for a micromorphic medium is more restrictive than for its gradient counterpart. On the other hand we see that a violation of SE for a primary material affects solutions within enhanced models even if the SE conditions are fulfilled for them.
•Strong ellipticity (SE) conditions are compared for nonlinear strain gradient (SG) and micromorphic (MM) elasticity.•Relations between SE of enhanced models and of simple nonlinear elastic (primary) material are clarified.•SE within SG approach is independent on SE of primary material, whereas SE of MM model elasticity inherits it partially.•Both models regularize primary material behavior, so non-existence of solutions is avoided.•SE conditions bring information on material instabilities within enhanced models of continua.
It is previously known that under inflation alone a spherical rubber membrane balloon may bifurcate into a pear shape when the tension in the membrane reaches a maximum, but the existence of such a ...maximum depends on the material model used: the maximum exists for the Ogden model, but does not exist for the neo-Hookean, Mooney–Rivlin or Gent model. This paper discusses how such a situation is changed when a pressurized dielectric elastomer balloon is subjected to additional electric actuation. A similar bifurcation condition is first deduced and then verified numerically by computing the bifurcated solutions explicitly. It is shown that when the material is an ideal dielectric elastomer, bifurcation into a pear shape is possible for all material models, and similar results are obtained when a typical non-ideal dielectric elastomer is considered. It is further shown that whenever a pear-shaped configuration is possible it has lower total energy than the co-existing spherical configuration.
On a uniformly-valid asymptotic plate theory Wang, Fan-Fan; Steigmann, David J.; Dai, Hui-Hui
International journal of non-linear mechanics,
June 2019, 2019-06-00, 20190601, Volume:
112
Journal Article
Peer reviewed
Open access
A uniformly-valid plate theory, independent of the magnitudes of applied loads, is derived based on the two-dimensional plate theory obtained from series expansions about the bottom surface of a ...plate. For five different magnitudes of surface loads, it is shown by using asymptotic expansions that this unified plate theory recovers five well-known plate models in the literature to leading-order. This demonstrates its uniform validity. More generally, it provides a uniformly-valid plate model provided that two asymptotic conditions are satisfied, which can be checked as a posteriori. The weak formulation of the uniformly-valid plate equations is furnished, which can be used for finite element implementation.
•A uniformly-valid plate theory, independent of the magnitudes of loads, is derived.•This plate theory recovers five well-known plate models based on a priori scalings.•The weak formulation of the uniformly-valid plate equations is furnished.
Soft biological tissues often exhibit notable strain stiffening under increasing stretch, and this can have significant effects on tissue growth and morphological development, such as causing ...symmetry breaking in growing airways and leading to mucosal folding and airway hyperresponsiveness. To investigate the role of strain stiffening and the multifactorial control in growth and remodeling, we consider a growing tubular structure with strain-stiffening effects caused by increased and tightened collagen. In addition, we employ the nonlinear hyperelastic Gent model and initial stress symmetry theory to include the coupling effects of differential growth and initial residual stress. Results show that for strain stiffening that takes place at higher strain (Jm>21), the maximum critical growth ratio matches that obtained using neo-Hookean model calculations. Meanwhile, for biological tissues that exhibit strain stiffening under moderate strain conditions (0.46<Jm<21), the strain-stiffening effect delays significantly the onset of growth instability. When strain stiffening takes place at very low strains (Jm<0.46), stiff biological tissues can prevent growth instability, resulting in a smooth hyperelastic cylindrical tubular structure, and the epithelial tissue remains stable at all growth stages without forming any unstable morphology. Our results suggest that strain stiffening can induce retardation instability during biological growth and remodeling, but airway remodeling can incorporate this effect by increasing wall stiffness and reducing obstruction. This highlights the importance of considering the impact of strain stiffening on biological growth and remodeling, which can inform the development of effective clinical interventions for chronic inflammatory airway diseases.
•Novel framework assesses strain stiffening’s impact on tissue growth with initially residual stress.•Nonlinear hyperelastic Gent model used to explore coupling effects on differential growth.•Strain stiffening impacts biological growth, induces retardation instability.•Insights offered into airway growth, potential implications for chronic diseases.
Revisiting the wrinkling of elastic bilayers I: linear analysis Alawiye, Hamza; Kuhl, Ellen; Goriely, Alain
Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences,
05/2019, Volume:
377, Issue:
2144
Journal Article
Peer reviewed
Open access
Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. It has been studied extensively for many different systems but a full description is still ...lacking. Here, we provide a systematic analysis of the wrinkling of a thin hyperelastic film over a substrate in plane strain using stream functions. For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire system. We perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects. Namely, we consider the effect of added pressure, surface tension, an upper substrate and fibres. We obtain analytical estimates of the instability in the two asymptotic regimes of long and short wavelengths. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.
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