Incompressible Hyperelastic Materials Research Articles

Nonlinear incompressible shear wave models in hyperelasticity and viscoelasticity frameworks, with applications to Love waves

General equations describing shear displacements in incompressible hyperelastic materials, holding for an arbitrary form of strain energy density function, are presented and applied to the description of nonlinear Love-type waves propagating on an interface between materials with different mechanical properties. The model is valid for a broad class of hyper-viscoelastic materials. For the Murnaghan constitutive model, shear wave equations contain cubic and quintic differential polynomial terms, including viscoelasticity contributions in terms of dispersion terms that include mixed derivatives uxxt of the material displacement. Full (2+1)-dimensional numerical simulations of waves propagating in the bulk of a two-layered solid are undertaken and analysed with respect to the source position and mechanical properties of the layers. Interfacial nonlinear Love waves and free upper surface shear waves are tracked; it is demonstrated that in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies the linear Love wave existence condition c1<v<c2, while tending to the larger material wave speed c1 or c2 for large times.

Topology Optimization of Hard-Magnetic Soft Phononic Structures for Wide Magnetically Tunable Band Gaps

Abstract Hard-magnetic soft materials, which exhibit finite deformation under magnetic loading, have emerged as a promising class of soft active materials for the development of phononic structures with tunable elastic wave band gap characteristics. In this paper, we present a gradient-based topology optimization framework for designing the hard-magnetic soft materials-based two-phase phononic structures with wide and magnetically tunable anti-plane shear wave band gaps. The incompressible Gent hyperelastic material model, along with the ideal hard-magnetic soft material model, is used to characterize the constitutive behavior of the hard-magnetic soft phononic structure phases. To extract the dispersion curves, an in-house finite element model in conjunction with Bloch’s theorem is employed. The method of moving asymptotes is used to iteratively update the design variables and obtain the optimal distribution of the hard-magnetic soft phases within the phononic structure unit cell. Analytical sensitivity analysis is performed to evaluate the gradient of the band gap maximization function with respect to each one of the design variables. Numerical results show that the optimized phononic structures exhibit a wide band gap width in comparison to a standard hard-magnetic soft phononic structure with a central circular inclusion, demonstrating the effectiveness of the proposed numerical framework. The numerical framework presented in this study, along with the derived conclusions, can serve as a valuable guide for the design and development of futuristic tunable wave manipulators.

Rediscovering the Mullins effect with deep symbolic regression

The Mullins effect represents a softening phenomenon observed in rubber-like materials and soft biological tissues. It is usually accompanied by many other inelastic effects like for example residual strain and induced anisotropy. In spite of the long term research and many material models proposed in literature, accurate modeling and prediction of this complex phenomenon still remain a challenging task.In this work, we present a novel approach using deep symbolic regression (DSR) to generate material models describing the Mullins effect in the context of nearly incompressible hyperelastic materials. The two step framework first identifies a strain energy function describing the primary loading. Subsequently, a damage function characterizing the softening behavior under cyclic loading is identified. The efficiency of the proposed approach is demonstrated through benchmark tests using the generalized the Mooney–Rivlin and the Ogden–Roxburgh model. The generalizability and robustness of the presented framework are thoroughly studied. In addition, the proposed methodology is extensively validated on a temperature-dependent data set, which demonstrates its versatile and reliable performance.

Inverse Hessian by stochastic projection and application to system identification in nonlinear mechanics of solids

Parameter identification in nonlinear mechanics of solids, a class of non-convex optimization problems of significant interest, often has to balance fast convergence with enhanced exploration. Local optimization schemes based on Newton or quasi-Newton updates cannot tread this fine balance. A similar limitation also applies to many derivative-free global evolutionary searches that typically end up with slower convergence. Notwithstanding a relatively rapid convergence around a local optimum, schemes using classical gradients require that the objective function be smooth. Thus, applied to real-world scenarios, these algorithms face challenges such as non-smoothness including discontinuities in the objective function and costly evaluations of gradients or Hessians, whenever possible. Based on a novel stochastic projection to estimate inverse Hessians, this work aims at generalizing a fast Newton-type local search whilst retaining capabilities of noise-assisted exploration in non-convex optimization. In principle, such a meeting of the best of both worlds in optimization should provide for a powerful paradigm shift, offering a versatile and robust tool for solving complex identification and variational problems arising in nonlinear mechanics of solids. The approach is well-suited even for cases where the objective function is implicitly defined and not amenable to explicit evaluation. In a first implementation of our proposal, we assess its performance against a few benchmark non-convex problems. Finally, we solve a parameter estimation problem for an isotropic and incompressible hyperelastic material.

Stiffness and pre-stretching estimation from indentation test of hyperelastic membrane

Obtaining precise data on the mechanical properties of soft materials, often in the form of thin, pre-tensioned membranes, is crucial, especially when conventional testing methods have limited applicability. The article focuses on an innovative method for estimating the Young's modulus and pre-stretching the membranes using the indentation method, assuming an isotropic, incompressible hyperelastic material. The parameters were estimated for two models of indenter-membrane contact: with perfect slip and without slip. Experimental tests were performed for pre-stretched latex membranes glued to a stiffer ring to minimize the effect of attachment on membrane deformation. In order to validate the estimation method, the predictions of the model with the determined mechanical parameters were compared with the results of indentation tests with a liquid layer under the membrane. The discussion shows the consistency of the estimation results with the literature results for small latex deformations, and indicates the advantages and numerous limitations of the approach, especially related to the choice of the material model.

Development of the variation-of-elastic-energy-based virtual fields method for parameter identification of incompressible and compressible hyperelastic materials

The virtual fields method (VFM) has played a pivotal role in parameter identification of hyperelastic materials, spanning from rubbers to soft tissues. A key challenge in VFM lies in calculating the internal virtual work, which necessitates the computation of the stress and its conjugate virtual strain. In this study, we introduce a novel theoretical framework for VFM, which directly computes the internal virtual work based on the variation of elastic energy (VEE). The proposed method, namely VEE-VFM, eliminates the calculation of stress and virtual strain, making it remarkably user-friendly compared with the conventional approach. We extend VEE-VFM to both incompressible and compressible hyperelastic materials, including invariant-based and principal stretch-based models. In the end, we demonstrate the accuracy of VEE-VFM and its robustness against noise with numerical experiments.

A high‐order shell finite element for the large deformation analysis of soft material structures

SummaryThis work proposes a higher‐order unified shell finite element for the analysis of cylinders made of compressible and nearly incompressible hyperelastic materials. The nonlinear governing equations are derived employing the Carrera unified formulation (CUF), thanks to which it is possible to build shell elements with the capability to capture three‐dimensional (3D) transverse and out‐of‐plane effects. The material and geometric nonlinearities are expressed in an orthogonal curvilinear reference system and the coupled formulation of hyperelastic constitutive law is considered. The principle of virtual work and a total Lagrangian approach is used to derive the nonlinear governing equations, which are solved by a Newton–Raphson scheme. The numerical investigations deal with a curved arch and both thick and thin cylinders subjected to line and point loadings. The obtained results are validated by comparing them with those from the literature. They demonstrate the reliability of the proposed method to analyze compressible and incompressible hyperelastic shell structures.

Effect of the apron in the mechanical characterisation of hyperelastic materials by means of biaxial testing: A new method to improve accuracy

Biological soft tissues and polymers used in biomedical applications (e.g. in the cardiovascular area) are hyperelastic incompressible materials that commonly operate under multi-axial large deformation fields. Their characterisation requires biaxial tensile testing. Due to the typically small sample size, the gripping of the specimens commonly relies on rakes or sutures, where the specimen is punctured at the edges of the gauge area. This approach necessitates of an apron, excess of material around the gauge region.This work analyses the apron influence on the estimated mechanical response of biaxial tests performed by using a rakes gripping system, with the aim of verifying the test accuracy and propose improved solutions.In order to isolate the effect of the apron, avoiding the influence of anisotropy and inhomogeneity typical of most soft tissues, homogeneous and isotropic hyperplastic samples made from a uniform sheet of casted silicone were tested. The stress-strain response of specimens with different apron sizes/shapes was measured experimentally by means of biaxial testing and digital image correlation. Tests were replicated numerically, to interpret the experimental findings.The apron surrounding the gauge area acts as an additional annular constraint which stiffens the system, resulting in a significant overestimate in the stress values. This error can be avoided by introducing specific cuts in the apron.The study quantifies, for the first time, the correlation between the apron size/shape and the experimental stress overestimation, proposing a research protocol which, although identified on homogeneous hyperelastic materials, can be useful in providing more accurate characterisation of both, synthetic polymers and soft tissues.

Deformations of the Varga material II: Plane stress

The governing equations for plane stress deformations of isotropic incompressible hyperelastic materials are highly nonlinear and consequently very few exact solutions are known. The Varga material is the only material in which general results are known. A recent paper by the authors revealed a linearization for the governing equations for the plane strain case of the Varga material. In this paper we show similar results for the plane stress case where we show the governing equations are also linearizable.

Data-driven nonparametric identification of material behavior based on physics-informed neural network with full-field data

A Physics-Informed Neural Network (PINN) model is developed to extract material behavior from full-field displacement data. The PINN model consists of independent networks describing mechanical fields such as displacement, deformation gradient, stress, and strain energy density function. The displacement and deformation gradient networks learn from the deformation data of a specimen, while the stress and material networks predict the stress distribution within the specimen and learn the behavior of the target material, respectively. Each mechanical field should satisfy the physics laws of momentum balance, compatibility, constitutive relations, and boundary conditions. The constraints are wrapped up in the loss function, and the weights and biases of each hidden layer are determined by minimizing the loss function. This study introduces a two-stage learning strategy to deal with the minimization process, which is a multi-objective optimization problem. The accuracy of the proposed method is verified for linear elastic, power-law, and incompressible hyperelastic materials. The data-driven identification method successfully identifies the response of target materials in uniaxial tension, simple shear, and tension-shear-coupled loading cases. The robustness of the proposed method is verified on full-field data with noise and missing points. Data-driven identification can handle full-field data with missing points and maintain a reasonable accuracy level as the noise amplitude increases. Furthermore, data-driven identification results of an incompressible hyperelastic material are compared with conventional parametric identification results. The proposed method has advantages over the conventional method as it requires no predefined model. The results indicate that the data-driven identification method can be applied to new materials without a developed constitutive model or materials with manufacturing uncertainty.

A perturbation analysis of pure torsion of incompressible hyperelastic cylinders

The stretch-based formulation of basic problems for incompressible isotropic hyperelastic materials has been the subject of renewed recent attention largely motivated by application to modelling the mechanical response of soft tissues. Here we are concerned with the classical problem of simple torsion of a circular cylinder which has in the past been primarily investigated by employing the classical approach of Rivlin in terms of invariants of the Cauchy–Green deformation tensors. Here we propose a perturbation approach to the pure torsion problem for thin (but not necessarily infinitesimally thin) cylindrical specimens motivated by potential applications to rubber-like or biological filaments and for the infinitesimal twisting of thick cylinders. The advantage of this approach is that the results are valid for all strain–energy functions whether based in terms of the principal stretches or in terms of the classical deformation invariants. The asymptotic results obtained for the resultant force and moment are applied to the special case of the one-term stretch-based Ogden model. In particular, a prediction of a transition in the Poynting effect for values of the Ogden parameter α>6 is obtained. Other illustrative examples for both invariant and stretch-based strain energies are described. In view of the complexity of a general analysis of torsion for arbitrary strain energies, the perturbation approach provides a viable alternative for assessing the resultant force and moment.

Plane deformations of the Varga material

The governing equations for plane deformations of isotropic incompressible hyperelastic materials are highly nonlinear, and consequently, very few exact solutions are known. As far as we are aware, the Varga material is the only material in which results of any generality are known. In this paper, we show that for the Varga material, the full governing equations are in fact linearizable.

A semi-analytical inverse method to obtain the hyperelastic potential using experimental data

Determining the mathematical structure of the stored energy of hyperelastic materials that predicts the experimental data well can prove to be complicated. For instance, the mechanical response of the brain tissue exhibits a complex shear response in the combined tension/compression and shear loading. For such a loading that involves multiple components of stress, arriving at the stored energy can be difficult because the six components of stress need to be described by the scalar function. For such situations, a change of tack is necessary. We propose an inverse procedure for isotropic incompressible hyperelastic materials subjected to combined tension/compression and shear, a homogeneous deformation in which the measured shear stress is an input. Exploiting the advantages of the Lode invariants and using a combination of the universal relation and the existence of stored energy, we uniquely determine the mathematical form of the stored energy. The novel inverse procedure is used to derive the stored energy for human brain tissue using the data for shear stress obtained by Budday et al. (2017). The derived constitutive relation is implemented in Abaqus® using the UHYPER routine. A realistic three-dimensional simulation (inhomogeneous deformation) of Budday et al. (2017), i.e. shear superposed on tension and compression, corresponding to the inverse procedure and the Prasad-Kannan constitutive relations for the human brain tissue are compared. The level set of the stored energy show marked differences qualitatively and quantitatively. Consequently, the stress fields differ significantly as well.

Hyperelastic constitutive model parameters identification using optical-based techniques and hybrid optimisation

AbstractIn this paper we propose a new optical-based technique to identify the constitutive relation coefficients of the hyperelastic material using a hybrid optimisation approach. This technique can be used in place of traditional mechanical testing of elastomers for applications that involve inhomogeneous deformation. The purpose of the proposed method is to identify the incompressible hyperelastic material constitutive relation coefficients using a single experiment under different loading cases. The method comprises sample surface 3D reconstruction and uses finite element simulations to replicate the experiments, and uses a hybrid optimisation technique to minimise the error between actual 3D deformations and FE simulation results. The proposed hybrid technique predicts the hyperelastic constitutive relation coefficients more accurately than other optimisation methods. This study introduces a novel approach by employing a subpixel image registration algorithm for 3D reconstruction. The method requires a single experiment with diverse loading cases to accurately determine the coefficients of hyperelastic constitutive relations. The setup is portable and can be accommodated in a small suitcase. For this purpose, an apparatus was constructed comprising a stereoscopic system with eight cameras and a six-degree-of-freedom force-torque sensor to measure the induced forces and torques during the experiments. We identified the constitutive relation coefficients of Ogden N1, Ogden N3, Yeoh, and Arruda-Boyce relations which are commonly used models for silicone materials, using a traditional uniaxial test, optical uniaxial test (experiments performed using a constructed optical system), and inhomogeneous deformations tests. The study demonstrated that the coefficients obtained from inhomogeneous deformation tests provided the most accurate FE predictions. It was also shown that hyperelastic constitutive relation coefficients obtained from traditional uniaxial tests are insufficient to describe the material behaviour when the material undergoes inhomogeneous deformations.

Finite deformation analysis of the rotating cylindrical hollow disk composed of functionally-graded incompressible hyper-elastic material

Finite deformation analysis of the rotating cylindrical hollow disk composed of functionally-graded incompressible hyper-elastic material

High-performance unsymmetric 8-node hexahedral element in modeling nearly-incompressible soft tissues

In biomechanics problems, the biological soft tissues are usually treated as anisotropic nearly incompressible hyperelastic materials, but such complicated nonlinear material models often cause challenging problems of severe volumetric locking and instabilities in numerical simulations. In this paper, the recent unsymmetric 8-node, 24-DOF hexahedral solid element US-ATFH8 with different test and analytical trial functions (ATFs) is modified for the analysis of the anisotropic nearly-incompressible hyperelastic soft tissues. Unlike the original formulation, the linear analytical general solutions for anisotropic elasticity and the consistent tangent modulus are firstly employed for formulating the trial functions, and are used to construct the incremental displacement fields that result in the incremental deformation gradient. The total deformation gradient is obtained by multiplying the incremental deformation gradient by the deformation gradient, after which the Cauchy stresses can be directly calculated from a total-form constitutive equation relating to the deformation gradient. Numerical tests, including commonly used benchmarks and cardiac examples, demonstrate attractive properties of the proposed finite element formulation in modeling nearly-incompressible anisotropic hyperelastic materials. It is free of various locking and quite insensitive to mesh distortions, and provides high accuracy with faster convergence rates when compared with other existing models.

Unified plate finite elements for the large strain analysis of hyperelastic material structures

This paper proposes a high-order two-dimensional (2D) finite element model for the analysis of isotropic, nearly incompressible hyperelastic material structures based on a decoupled Neo–Hookean strain energy function. The model is based on the Carrera Unified Formulation (CUF) , which allows to automatically implement different kinematics by using an opportune recursive notation. The principle of virtual work and a finite element approximation are exploited to obtain the nonlinear governing equations. Considering the three-dimensional full Green–Lagrange strain components and given the material Jacobian tensor, the explicit forms of tangent stiffness matrices of unified plate elements are presented in terms of the fundamental nuclei, which are independent of the theory approximation order. Several problems of soft material plates under uniform pressure are investigated, including a silicone rubber clamped plate and a simply supported plate made of biological material. The proposed model is compared with literature results including those coming from experiments and numerical solutions. The numerical investigation demonstrated the validity and accuracy of the proposed methodology for the analysis of hyperelastic plates.

On homogeneous deformations for a new constitutive model for incompressible isotropic hyperelastic soft materials

On homogeneous deformations for a new constitutive model for incompressible isotropic hyperelastic soft materials

Pure torsion for incompressible hyperelastic materials of Valanis–Landel form

The stretch-based formulation of basic problems for incompressible isotropic hyperelastic materials has been the subject of renewed recent attention largely motivated by application to modelling the mechanical response of soft tissues. Here, we are concerned with this formulation of the classic problem of simple torsion of a circular cylinder that has been primarily in the past investigated by employing the classical approach of Rivlin in terms of invariants of the Cauchy–Green strain tensor. Attention is focused on examining the resultant axial force necessary to sustain pure torsion, thereby investigating the Poynting effect, and the resultant moment needed to generate the twist. General results for the resultant force and moment are obtained for all hyperelastic materials that can be modelled in the celebrated separable Valanis–Landel form. These expressions are specialized for the one-term Ogden model to provide explicit results valid for all strain-stiffening exponents in this model. Some particular examples are provided, mainly for even values of the exponent. A notable result obtained for the Ogden model is that for values of the hardening exponent n in that model for which n ≤ 6 , one obtains the classical Poynting effect, whereas for n &gt; 6 , a reverse Poynting effect always occurs for sufficiently small twist.

Locking treatment of penalty-based gradient-enhanced damage formulation for failure of compressible and nearly incompressible hyperelastic materials

Soft materials are of major interest for biomechanics applications due to their high deformability and susceptibility to experience damage events under different loading scenarios. The present study is concerned with modelling damage evolution processes in these nonlinear materials whose structural responses are prone to locking when low-order kinematic interpolation is employed in the context of nonlinear Finite Element schemes. For this reason, a pair of gradient-enhanced continuum damage schemes are proposed with the aim of tackling mechanical failure problems in applications that exhibit shear and volumetric locking. In particular, we present the consistent formulation and the assessment of the corresponding performance of (i) a mixed displacement-enhanced assumed strain Q1Q1E24 employing a total Lagrangian formulation, and (ii) a three-field mixed displacement-pressure-Jacobian Q1Q1P0 formulation. The novel Q1Q1E24 and Q1Q1P0 formulations are consistently derived and numerically implemented, providing a satisfactory agreement with respect to ABAQUS built-in elements handling the treatment of shear and volumetric locking, respectively, in conjunction to the modelling damage phenomena via the use of a penalty-based gradient-enhanced formulation. This performance is examined via several numerical applications. Furthermore, the final example justifies the need for a formulation combining both mixed FE approaches to simulate problems encompassing both locking issues (shear and volumetric locking), which can be performed using a combination of the Q1Q1E24 and Q1Q1P0 herein proposed.