Nonlinear Elastic Material Research Articles

Plane Stress Problems for Isotropic Incompressible Hyperelastic Materials

The analysis of plane stress problems has long been a topic of interest in linear elasticity. The corresponding problem for non-linearly elastic materials is considered here within the context of homogeneous incompressible isotropic elasticity. It is shown that when the problem is posed in terms of the Cauchy stress, a semi-inverse approach must be employed to obtain the displacement of a typical particle. If however the general plane stress problem is formulated in terms of the Piola-Kirchhoff stress, the deformation of a particle requires the solution of a non-linear partial differential equation for both simple tension and simple shear, the trivial solution of which yields a homogeneous deformation. It is also shown that the general plane stress problem can be solved for the special case of the neo-Hookean 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

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

Experimental testing combined with inverse-FE for mechanical characterisation of penile tissues

Erectile dysfunction (ED) predominantly affects men in their 40–70s and can lead to poor quality of life. One option for ED treatment is surgical implantation of an inflatable penile prosthesis (IPP). However, they can be associated with negative outcomes including infection, migration or fibrosis. To improve outcomes, the interaction between the IPP device and surrounding tissues needs further investigation and this could be achieved using pre-clinical testbeds, but they need to be informed by extensive tissue testing.In this study, an experimental approach is adopted to characterise the mechanics of horse penile tissue and establish a testing protocol for penile tissue. The whole penis segments were tested in plate compression tests to obtain whole penis behaviour which is necessary for validation of a pre-clinical testbed, whilst tensile and compression tests were performed on individual penile tissues, namely corpus cavernosa and tunica albuginea. The second part of the paper deals with the development of a computational model employing an inverse finite element approach to estimate the material parameters of each tissue layer. These material parameters are in good agreement with the experimental results obtained from the individual tissue layers and whole organ tissue tests. This paper presents the first study proposing realistic nonlinear elastic material parameters for penile tissues and offers a validated testbed for IPPs. Statement of significanceErectile Dysfunction (ED) affects over half the male population aged 40–70 potentially leading to poor quality of life. Patients not responding to conventional treatments of ED, are advised to use penile prostheses which can create an erection using implanted inflatable cylinders. A significant drawback of such prostheses, however, is the substantial tissue damage they can induce during their usage. Preclinical testbeds, including computational and bench-top models, could offer an efficient means of improving device designs to mitigate this damage but such testbeds require extensive knowledge of penile tissue properties. In this study, the authors determine penile tissue mechanics and apply an inverse FE approach to characterise the penile material properties required to validate preclinical models of the penis.

An Analytical Model for Nonlinear-Elastic Compliant Mechanisms with Tension-Compression Asymmetry

Abstract While nonlinear-elastic materials demonstrate potential in enhancing the performance of compliant mechanisms, their behavior still needs to be captured in a generalized mechanical model. To inform new designs and functionality of compliant mechanisms, a better understanding of nonlinear-elastic materials is necessary and, in particular, their mechanical properties that often differ in tension and compression. In the current work, a beam-based analytical model incorporating nonlinear-elastic material behavior is defined for a folding compliant mechanism geometry. Exact equations are derived capturing the nonlinear curvature profile and shift in the neutral axis due to the material asymmetry. The deflection and curvature profile are compared with finite element analysis along with stress-distribution across the beam thickness. The analytical model is shown to be a good approximation of the behavior of nonlinear-elastic materials with tension-compression asymmetry under the assumptions of the von Kármán strain theory. Through a segmentation approach, the geometries of a semicircular arc and folding compliant mechanism design are defined. The deflection of the folding compliant mechanism due to an applied tip load is then evaluated against finite element analysis and experimental results. The generalized methods presented highlight the utility of the model for designing and predicting the behavior of other compliant mechanism geometries and different nonlinear-elastic materials.

Quantitative stochastic homogenization of nonlinearly elastic, random laminates

In this paper we study quantitative stochastic homogenization of a nonlinearly elastic composite material with a laminate microstructure. We prove that for deformations close to the set of rotations, the homogenized stored energy function W_{\hom} is C^{3} and that W_{\hom} , the stress tensor DW_{\hom} , and the tangent-moduli D^{2}W_{\hom} can be represented with the help of stochastic correctors. Furthermore, we study the error of an approximation of these quantities via representative volume elements. More precisely, we consider periodic representative volume elements (RVEs) obtained by periodizing the distribution of the random material. For materials with a fast decay of correlations on scales larger than a unit scale, we establish error estimates on the random and systematic error of the RVE with optimal scaling in the size of the RVE and with a multiplicative random constant that has exponential moments.

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.

Transient analysis of trusses considering nonlinear elastic and viscoelastic material models

Transient analysis of trusses considering nonlinear elastic and viscoelastic material models

Fracture resistance of polyacrylamide-alginate hydrogels

Polyacrylamide-alginate hydrogels are biphasic materials with double-polymeric structure and considerable percentage of water inside. They exhibit complex mechanical behaviour in which finite deformations, non-linearity and time dependence are mixed. Fracture behaviour is less known, and this work analyses the fracture resistance obtained via methodologies of non-linear elastic materials such as that developed by Rivlin and Thomas and that attained with Non-Linear Fracture Mechanics procedures based on the use of J-integral. Special attention is paid to the fracture parameters on both crack growth initiation and crack propagation stages. The validity of the methodologies was evaluated, and conclusions were drawn about the effect of the alginate content on the fracture resistance of polyacrylamide.

Approaches to the Solution of the Lamé–Gadolin Problem for a Composite Hollow Ball Made of Nonlinear Elastic and Elasto-Plastic Materials Under Superimposed Finite Deformations

Approaches to the Solution of the Lamé–Gadolin Problem for a Composite Hollow Ball Made of Nonlinear Elastic and Elasto-Plastic Materials Under Superimposed Finite Deformations

Phase-field simulation of fracture in Polymethyl Methacrylate

Significant interest and developments in phase-field fracture modeling have emerged. It approximates fracture continuously using a length-scale parameter, accurately simulating fracture initiation and crack propagation in brittle materials like Polymethyl Methacrylate (PMMA) with nonlinear elasticity. This study proposes a phase-field approach for simulating PMMA fracture behavior using ABAQUS with a nonlinear elastic material model (UMAT) and a three-layered element composition (UEL). The staggered approach sequentially solves displacements and phase-field variables. Numerical results are validated against experiments considering various geometries, stress concentrators, and pre-cracks. The computational approach accurately predicts fracture initiation and crack growth in complex patterns and diverse loading conditions.

A mixed least‐squares finite element method for a geometrically nonlinear TPM formulation

AbstractIn the present work, a mixed least‐squares finite element method (LSFEM) is used, which is applied on the partial differential equations arising from the Theory of Porous Media (TPM), see for example [1, 2, 3]. Since the LSFEM is not limited to the LBB condition, the method has some theoretical advantages over the well‐known (mixed) Galerkin method, see [4]. In particular, the LSFEM leads to positive definite and symmetric system matrices, even for differential equations with non‐self‐adjoint operators. In detail, we study an incompressible binary model with solid and liquid phases. The modeling of saturated porous structures serves as the basis for the presented approach. The resulting finite element is a four‐field formulation in terms of solid displacements, fluid pressure, mixture stresses, and a new variable associated with the pressure gradient. Vector‐valued Raviart‐Thomas and conventional Lagrangian functions are used to achieve a conforming discretization of the unknowns in the spaces H(div) and H1. The applicability of the LSFEM approach is demonstrated with numerical examples for fluid‐saturated porous structures, which consider incompressible, geometrically nonlinear elastic material behavior.

Data-driven enhanced FDEM for simulating the rock mechanical behavior

In this paper, a data-driven enhanced combined finite-discrete element method (DDFDEM) is proposed to simulate the rock mechanical behavior by directly assigning the rock experimental data into mechanical simulations, thus bypassing the empirical constitutive model of rock. To achieve this, a novel data-driven iterative solver with transient dynamic system equilibrium constraint is first proposed and embedded into the FDEM computing framework. Then, the bonding force update algorithm for crack elements is improved to mitigate the error accumulation during the iteration process. After that, a recommended formation for the proper size of the experimental data set is proposed to balance the computational accuracy and efficiency of DDFDEM, and an efficient nearest neighbor search algorithm is introduced to significantly improve the computational efficiency of DDFDEM. After the above improvements, the computational accuracy and efficiency of DDFDEM in simulating the rock deformation and complex failure process are validated by conducting several examples, including the thin plate tension test, uniaxial compression test, and Brazilian disk test. Finally, the advantage of DDFDEM over conventional FDEM in comprehensively presenting the information hidden behind the experimental data sets is validated by reproducing the nonlinear mechanical behaviors of rock samples based on a data set following the nonlinear elastic material law.

Advancements in Phase-Field Modeling for Fracture in Nonlinear Elastic Solids under Finite Deformations

The prediction of failure mechanisms in nonlinear elastic materials holds significant importance in engineering applications. In recent years, the phase-field model has emerged as an effective approach for addressing fracture problems. Compared with other discontinuous fracture methods, the phase-field method allows for the easy simulation of complex fracture paths, including crack initiation, propagation, coalescence, and branching phenomena, through a scalar field known as the phase field. This method offers distinct advantages in tackling complex fracture problems in nonlinear elastic materials and exhibits substantial potential in material design and manufacturing. The current research has indicated that the energy distribution method employed in phase-field approaches significantly influences the simulated results of material fracture, such as crack initiation load, crack propagation path, crack branching, and so forth. This impact is particularly pronounced when simulating the fracture of nonlinear materials under finite deformation. Therefore, this review outlines various strain energy decomposition methods proposed by researchers for phase-field models of fracture in tension–compression symmetric nonlinear elastic materials. Additionally, the energy decomposition model for tension–compression asymmetric nonlinear elastic materials is also presented. Moreover, the fracture behavior of hydrogels is investigated through the application of the phase-field model with energy decomposition. In addition to summarizing the research on these types of nonlinear elastic body fractures, this review presents numerical benchmark examples from relevant studies to assess and validate the accuracy and effectiveness of the methods presented.

Uniform in-plane stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity

We establish the uniformity of the internal in-plane elastic field of stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity embedded in an infinite linear isotropic elastic matrix subjected to uniform remote in-plane stresses. The nonlinear elastic material occupying the inhomogeneity can incorporate a power-law hardening material. The original boundary value problem is ultimately reduced to a single non-linear equation. It is proved rigorously that the non-linear equation has a unique solution. Once the non-linear equation is solved numerically, the uniform elastic field within the nonlinear elastic elliptical inhomogeneity and the non-uniform elastic field in the linear elastic matrix can be determined.

Conditions of strong ellipticity and calculations of M-eigenvalues for a partially symmetric tensor

Strong ellipticity condition (SE-condition) of the equilibrium equations for general nonlinear elastic materials can be equivalently transformed into the SE-condition of a partially symmetric tensor. Qi et al. in 2009 proved that the SE-condition of a partially symmetric tensor holds if and only if its smallest M-eigenvalue is positive. In this paper, a criterion for the SE-condition of a partially symmetric tensor is first given. And then, an alternative method to compute all M-eigentriples of a partially symmetric tensor is presented. Finally, two numerical examples show the effectiveness of the proposed criterion and the calculation of M-eigenvalues in judging the SE-condition of the equilibrium equations.

A nonlinear bending theory for nematic LCE plates

In this paper, we study an elastic bilayer plate composed of a nematic liquid crystal elastomer in the top layer and a nonlinearly elastic material in the bottom layer. While the bottom layer is assumed to be stress-free in the flat reference configuration, the top layer features an eigenstrain that depends on the local liquid crystal orientation. As a consequence, the plate shows non-flat deformations in equilibrium with a geometry that non-trivially depends on the relative thickness and shape of the plate, material parameters, boundary conditions for the deformation, and anchorings of the liquid crystal orientation. We focus on thin plates in the bending regime and derive a two-dimensional bending model that combines a nonlinear bending energy for the deformation, with a surface Oseen–Frank energy for the director field that describes the local orientation of the liquid crystal elastomer. Both energies are nonlinearly coupled by means of a spontaneous curvature term that effectively describes the nematic-elastic coupling. We rigorously derive this model as a [Formula: see text]-limit from three-dimensional, nonlinear elasticity. We also devise a new numerical algorithm to compute stationary points of the two-dimensional model. We conduct numerical experiments and present simulation results that illustrate the practical properties of the proposed scheme as well as the rich mechanical behavior of the system.

Equilibrium shapes of liquid drops on pre-stretched nonlinear elastic membranes

We investigate the equilibrium of an axisymmetric system consisting of sessile and pendent drops on pre-stretched nonlinear elastic membranes. The membrane experiences large deformations due to the drop's weight and interfacial interactions. We first show that force balance alone leads to non-unique equilibrium solutions. Identifying the system's equilibrium with the minimum of its free energy, we then demonstrate that the equilibrium solution is made unique by requiring the continuity of meridional stretches across the three-phase contact circle. For a special class of nonlinear elastic materials – $I_2$ materials – we then compute the equilibrium configurations of the drop–membrane system for a range of drop volumes and membrane pre-tensions. Finally, the present work facilitates two important applications: (a) the membrane's pre-tension and current tension are related exactly to help in utilizing the system as an elastocapillary probe for membrane pre-tension and (b) we suggest an experimental protocol for measuring the membrane's surface properties.

Breakdown of smooth solutions in one dimensional nonlinear nonlocal elasticity

We write the equations of a nonlinear nonlocal elastic material in the system of Riemann invariants and view them as the perturbation of a simplified system under specific conditions that utilize a finite influence distance. For the perturbed system we study formation of singularities, a quality that passes on to the solutions of the original system.

Large-strain Elastic and Elasto-Plastic Formulations for Host-Inclusion Systems and Their Applications in Thermobarometry and Geodynamics

Mineral inclusions are trapped in a variety of geological environments and physical conditions. If brought to conditions different than their entrapment, mineral inclusions will generally experience different stress conditions than their hosts due to differences in their thermo-elastic properties and the associated deformation. These stress differences develop both in prograde and retrograde metamorphic conditions. The currently available analytical solutions consider isotropic materials and employ either fully linear-elastic behavior or they account for the non-linear-elastic volumetric deformation of minerals. Here we show that, by taking into account the finite volumetric deformation, we are able to explain the systematic differences amongst the available linear and non-linear elastic solutions. Furthermore, we employ a newly derived analytical solution for fully non-linear elastic materials (generalized Varga materials) to the host-inclusion problem. This solution considers both the geometric non-linearity and the material non-linearity by employing a Murnaghan equation of state. Our results show that the complete non-linear, hyperelastic behavior is not needed to explain the pressure differences that develop in common, unreacting, host-inclusion systems. The effects of plastic yielding are also investigated for the case of large finite deformations that can be relevant for the cases of phase transitions and mineral reactions that induce significant volume changes. Our results show that in the case of very large volumetric deformations the incorporation of finite strain effects may become important. Moreover, depending on the yield stress of the materials, the effects of plasticity may be dominant. In the latter case, significant pressure gradients will be developed as a consequence of stress balance. These results are general and they can also be used for elastic-barometry/volcanology applications and for benchmarking compressible Navier-Stokes geodynamic models. Accurate stress predictions in mechanical problems with large volumetric deformation can be significant in modeling the effects of mineral reactions that are generally non-isochoric.

Unit cell of three nonlinear layers for the design of tunable nonlinear metamaterials

Tang et al. (2012) and Kube (2017–2018) discussed the harmonic scattering of elastic waves from nonlinear inclusions through a theoretical perspective modeling early-stage damages as quadratic and cubic nonlinear elastic material models. Achenbach and Wang (2017–2018) studied the harmonic scattering of a wave from a single nonlinear layer embedded in a linear elastic material using the reciprocity theorem in elastodynamics and the concept of compensatory waves. They demonstrated sensitivity of the back and forward scattered waves towards the dimension and the intensity. Based on this understanding, a new unit cell is proposed in this study that can act as a tunable nonlinear metamaterial, where we can tune the amplitudes of the back and forward-scattered nonlinear waves. The unit cell contains three nonlinear layers of equal widths; the intensity of the central layer, along with the total width of the unit cell, is varied to tune the harmonic responses of scattered waves. Computational studies are carried out to demonstrate the presence and absence of harmonically backscattered waves from a unit cell. In some case studies, parameters are tuned so that the amplitudes of 2nd harmonics (2f) of forward, backscattered waves and waves recorded at the interfaces of the three nonlinear layers.