Incompressible Elastic Material Research Articles

Indentation of a thin incompressible layer with finite friction

If a thin layer of an incompressible elastic material is pressed between two plane surfaces, the effective stiffness is very sensitive to the presence of frictional slip. This effect is investigated using a low-order polynomial representation of the through-thickness displacement profile. Results show good agreement with finite element studies and also show that the stiffness is significantly affected by that part of the layer [if any] outside the loaded region. The same result is then used in convolution to approximate the load–displacement response for a convex indenter.

Transverse surface wave in non-planar layered nearly incompressible elastic structure having imperfect contact

The prime motive of the present work is to bring out the emphatic impacts of planar boundary and non-planar boundary as well as imperfectness of the interface on the displacement components and phase velocity of Love-type wave (LTW) propagating in the layered structure. The layered structure essentially incorporates two dissimilar nearly incompressible elastic materials bonded imperfectly to each other having non-planar boundaries at the free surface and at the common interface. A linear spring model has been adopted to describe the imperfect bonding at the interface. The solution methodology features the applicability of variable separable technique along with Taylor’s series expansion that leads to establish dispersion relation and displacement components of LTW. The displacement components and phase velocity of LTW have been numerically computed, and their dependence on wave number have been graphically manifested. As a special case of the study, the dispersion relations for incompressible and isotropic elastic layered structures have been deduced. A comparative analysis between compressible, nearly incompressible and incompressible layered structures comprised of Neo–Hookean material, and isotropic layered structure has been performed to trace out the impact of imperfect bonding parameter on the phase velocity of LTW. The established result of dispersion relation for the considered geometrical structure, as a particular deduction, concur well with the result present in the literature.

Exact solution of one problem of imposition of large deformations in a compound slab with pre-deformed layers of incompressible nonlinear elastic micropolar material

Exact solution of one problem of imposition of large deformations in a compound slab with pre-deformed layers of incompressible nonlinear elastic micropolar material

On the Extension-Torsion of Short Hyperelastic Tubes of Axially Functionally-Graded Materials

The mechanical behavior of thick-walled tubes made of axially functionally graded (FG) incompressible nonlinear elastic materials under combined extension-torsion loading is focused on in this study. To achieve more realistic modeling, a computation-efficient semi-analytical approach is proposed, which is subsequently validated through finite element simulations. Axially functional grading is addressed by considering finite length instead of the commonly assumed infinite length. Additionally, a comparative study is conducted to examine twelve different functional grading distributions based on silicon-rubber Neo-Hookean material constant. 3D numerical simulations are outperformed by our approach by up to 50 times in computational efficiency under identical hardware conditions. The results indicate the model's capability to evaluate the non-linear distribution of twisting along the length, also stress analysis suggests that decreased shear stress variation, reaction torque, and force can be achieved by increasing the material gradient in larger than one composition ratio. Overall, valuable insights for engineers are provided by our approach, ensuring structural integrity and safe operation of functionally graded cylindrical components in various engineering applications.

On the coupling of rectilinear shear and stretching in a slab made of incompressible elastic neo-Hookean materials: analytic formulation and finite element validation

On the coupling of rectilinear shear and stretching in a slab made of incompressible elastic neo-Hookean materials: analytic formulation and finite element validation

Gradient Robust Mixed Methods for Nearly Incompressible Elasticity

Within the last years pressure robust methods for the discretization of incompressible fluids have been developed. These methods allow the use of standard finite elements for the solution of the problem while simultaneously removing a spurious pressure influence in the approximation error of the velocity of the fluid, or the displacement of an incompressible solid. To this end, reconstruction operators are utilized mapping discretely divergence free functions to divergence free functions. This work shows that the modifications proposed for Stokes equation by Linke (Comput Methods Appl Mech Eng 268:782–800, 2014) also yield gradient robust methods for nearly incompressible elastic materials without the need to resort to discontinuous finite elements methods as proposed in Fu et al. (J Sci Comput 86(3):39–30, 2021).

Weakly nonlinear two dimensional wave scattering in incompressible pre-stressed materials: Far field approximation

We develop a series expansion approach to the calculation of the scattering coefficients of in-plane transverse elastic waves. The waves are scattered by a cylindrical obstacle as they propagate through an incompressible nonlinear material which is inhomogeneously pre-stressed. We deal with the calculation using a distorted wave Born approximation in the far field. A neo-Hookean strain energy function is considered for the incompressible elastic material, and a rigid cylinder is considered as an obstacle. From the static equilibrium configuration, a small-on-large approach is constructed via a series expansion of the wave equation which allows us to compute both the scattered wave and the induced pressure field, and their dependence on the applied pre-stressed. In the far field limit, the pre-stress effect on the scattering coefficients is calculated via partial waves. The formalism can be expanded directly to three-dimensional media with different obstacle symmetries.

A model for fibre–matrix interaction in non-linearly elastic incompressible orthotropic materials

The strain energy for incompressible anisotropic non-linearly elastic materials is decomposed into an isotropic part representing the mechanical response of an isotropic matrix and an anisotropic part representing the contribution to the mechanical response from the presence of fibres. It is the form of the anisotropic component that is of interest here. We note that the invariants can themselves be divided into two classes: the invariants that are homogeneous functions of degree two and those of degree four in the principal stretches. The approach adopted here is straightforward: assume that there is a linear proportional relationship between terms in the general stress–strain law that are of the same degree in the principal stretches. Setting these constants identically zero recovers many of the simplified strain energies commonly found in the literature. The proportionality constants are interpreted as being a measure of the fibre–matrix interaction and a measure of the interaction between fibres in anisotropic non-linear elasticity. An influential model of fibre dispersion is recovered as a special case. The results are illustrated using the homogeneous deformation of simple shear.

Geometric linearization of theories for incompressible elastic materials and applications

We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

Large deformation of hyperelastic thick-walled vessels under combined extension-torsion-pressure: analytical solution and FEM

The mechanical behavior of a thick-walled pressure vessel composed of an incompressible isotropic nonlinearly elastic material subjected to the combined pressure and extension-torsion loading is investigated. An analytical solution is proposed for the general form of the free strain energy density, and different models including well-known Neo-Hookean, Mooney-Rivlin, and recently presented free energy contained two exponential terms -exp-exp model- which is strongly stable and compatible with experimental data, are employed. Comparing different constitutive models involves compatible calibration for a specific material. Thus, the uniaxial and equi-biaxial experimental data of silicon-rubber are simultaneously considered to be used in a calibration optimization procedure. Regarding the application of extension-torsion problem, e.g., biological application in papillary muscles, it is required to inspect this loading on a hollow cylinder and considering the internal and external pressure make the problem more realistic to be employed as capillary muscle. After proposing a closed-form solution for this problem, the finite element analysis of the same problem is conducted to verify the proposed analytical solution. Comparing the results of non-dimensional stress, force and moment distribution of analytical and finite element methods illustrate excellent agreement which confirms the accuracy of the presented solution. Some designing factors including the value of applied twist, axial stretch and pressure, and radius ratio are studied. Results demonstrate that the exp-exp model is more conservative in a small amount of twist and axial stretch. However, it yields a higher value for the stress, force, and moment in larger twists and stretches.

The Effect of Rotation on Propagation of Rayleigh Wave in an Incompressible Monoclinic Elastic Solid

ABSTRACTIn this paper, the Rayleigh wave propagation is investigated in rotating half-space of incompressible monoclinic elastic materials which are subjected to the impedance boundary conditions. In particular, the explicit secular equation of the Rayleigh wave is obtained. The main objective of this paper is to illustrate the dependence of dimensionless speed of Rayleigh wave on rotation, anisotropy and impedance parameters. An algorithm in MATLAB software is developed to solve the secular equation of Rayleigh wave. The speed of Rayleigh wave is plotted against rotation, anisotropy and impedance parameters.

Mixed stress-displacement isogeometric collocation for nearly incompressible elasticity and elastoplasticity

We propose a mixed stress-displacement isogeometric collocation method for nearly incompressible elastic materials and for materials exhibiting von Mises plasticity. The discretization is based on isogeometric analysis (IGA) with non-uniform rational B-Splines (NURBS) as basis functions. As compared to conventional IGA Galerkin formulations, isogeometric collocation methods offer a high potential of computational cost reduction for higher-order discretizations as they eliminate the need for quadrature. In the proposed mixed formulation, both stress and displacement fields are approximated as primary variables with the aim of treating volumetric locking and instability issues, which occur in displacement-based isogeometric collocation for nearly incompressible elasticity and von Mises plasticity. The performance of the proposed approach is demonstrated by several numerical examples.

Influence of interface phenomena on the features of interaction between the probe of atomic force microscope and soft material

A new model of the interaction of AFM sharp probe with a soft incompressible elastic material is proposed. The model takes into account the finite deformations of the material. The novelty is the consideration of bending and increasing of the material-air interface during the process of indentation. The work performed by the probe is spent both on the elastic deformation of the sample and on changing the geometry of the material-air interface. The paper proposes an explanation of possibility to simulate the phase boundary by a thin elastic layer under the condition of its large shrinkage. The obtained approximation dependences allow quick computer calculations. The presented arguments justify the possibility of neglecting the probe jump to the sample surface. It is established that the Deryagin-Muller-Toporov model is a special case of the proposed model.

A higher order nonlocal operator method for solving partial differential equations

A higher order nonlocal operator method for the solution of boundary value problems is developed. The proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method (Ren et al., 2020) which only ensures first-order convergence. Furthermore, it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions. Only the functionals based on the nonlocal operators (termed as operator functional) are needed to obtain the final discrete system of equations, which significantly facilitates the implementation. Several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the Poisson equation in 2–5 dimensional space, Kirchhoff and von Kármán plate problems, incompressible elastic materials as well as phase field modeling of fracture.

Torsion stability of a cylinder with circular and elliptical section under finite perturbations

In this paper, the stability of a cylinder with circular and elliptical section made of compressible and incompressible nonlinear elastic materials under finite perturbations is considered. A set of computational experiments was performed. The permissible boundaries of the region with respect to final and initial perturbations for given parameters of loading and structures are established. Finite sequences of bifurcation points are constructed, confirming, in contrast to stability at small perturbations, the existence of a hierarchy of stable equilibrium states. Classical linearized stability theories are evaluated. New phenomena and characteristic effects are established.

The Unexpected Kinematics of Simple Extension and Contraction of Incompressible Materials

Simple extension and contraction of incompressible non-linearly elastic materials is considered here, with cuboid specimens stretched or contracted in one of the principal directions. Although the average stretch of infinitesimal line elements in such deformations is always extensile, the local response is much more nuanced. We examine the variation of the squared stretch with applied stretch and line element orientation in the undeformed configuration. The response of interest here is the percentage of directions at a point for which the material is in compression which we call the amount of material in compression. It is shown that the maximum amount of compression attainable is approximately $61\%$ which occurs for infinitesimal contractions. Further contraction results in a decrease in the amount of material in compression, with the material being extensile almost everywhere in the limit of zero axial stretch. On the other hand, for simple extension, the amount of material in compression is a monotonically decreasing function of axial stretch, with $39\%$ in compression for infinitesimal strains. The amount of material in compression therefore exhibits a discontinuity in the reference configuration. The amount of material in compression is still significant for moderate extensions, with, for example, $25\%$ of the material still in compression for $100\%$ stretch. Suspecting that this somewhat surprising response is a result of assuming perfect incompressibility, the effect of compressibility on the amount of material in compression is examined within the context of the linear theory for isotropic elastic materials.

Shear shock evolution in incompressible soft solids

Nonlinear evolution of shear waves into shocks in incompressible elastic materials is investigated using the framework of large deformation elastodynamics, for a family of loadings and commonly used hyperelastic material models. Closed form expressions for the shock formation distance are derived and used to construct non-dimensional phase maps that determine regimes in which a shock can be realized. These maps reveal the sensitivity of shock evolution to the amplitude, shape, and ramp time of the loading, and to the elastic material parameters. In light of a recent study (Espindola et al., 2017), which hypothesizes that shear shock formation could play a significant role in Traumatic Brain Injury (TBI), application to brain tissue is considered and it is shown that the size matters in TBI research. Namely, for realistic loadings, smaller brains are less susceptible to formation of shear shocks. Furthermore, given the observed sensitivity to the imparted waveform and the constitutive properties, it is suggested that the non-dimensional maps can guide the design of protective structures by determining the combination of loading parameters, material dimensions, and elastic properties that can avoid shock formation.

On problem of the stresses in a non-homogeneous composite of an infinite cylinder of orthotropic material

The prime objective of the present paper is to analyze the effect of inhomogeneity in an incompressible orthotropic elastic material in a cylinder. The dynamical problems of an orthotropic cylinder containing: (i) an isotropic core and (ii) a rigid core are considered. The radial displacement and stresses in different cases have been computed analytically. The numerical calculations arc carried out for the radial displacement and stresses of the composite elastic cylinder for various material inhomogeneity parameters are depicted graphically. The results indicate that the effect of inhomogeneity parameter on the components of displacement and the stresses is very pronounced. Several examples are considered. The approach is validated by comparing with previously known solutions. The MATHEMATICA software is applied to explicate the effect of these parameters graphically.

Large deformations of a cylindrical tube with prestressed coatings

The problem of large deformations in a combined nonlinear elastic hollow cylinder under internal and external pressures, loaded with a longitudinal force and torque at the end faces, is under consideration. The combined cylinder is a tube with the internal and external coatings in the form of prestressed hollow circular cylinders. An exact solution to the problem is found, which is valid for any model of isotropic incompressible elastic materials.

Large Deformations of a Cylindrical Tube with Prestressed Coatings

The problem of large deformations in a combined nonlinear elastic hollow cylinder under internal and external pressures, loaded with a longitudinal force and torque at the end faces, is under consideration. The combined cylinder is a tube with the internal and external coatings in the form of prestressed hollow circular cylinders. An exact solution to the problem is found, which is valid for any model of isotropic incompressible elastic materials.