Strain Energy Function Research Articles

Effect of Skeletal Muscle Immobilization on Regional Anisotropic Viscohyperelastic Properties Change in a Rat Model

Passive mechanical properties in three different zones (distal, proximal, and medial) of biceps brachii immobilized in short position and its free contralateral were investigated. For that, in vitro equibiaxial relaxation tests were performed with samples harvested from skeletal muscles of immobilized rats during one or two weeks. From data obtained in two plane axes of loading, a viscohyperelastic anisotropic model described by a strain energy function coupled with second order Maxwell's model, was used to identify the material parameters. It has been shown that the zone influences the material parameters of the hyperelasticity behavior while the immobilization acts rather on the viscoelasticity response. Based on measurements of histological parameters from flexor carpi ulnaris muscles, we observed that immobilization produced contractile tissue atrophy and connective tissue thickening. The muscle atrophy caused by immobilization could lead to a more linear mechanical behavior along the axis of the muscle fibers. Furthermore, fibrosis, quantified by histological analysis on flexor carpi ulnaris muscles, could result in highly non-linear behavior along the other axis. These structural changes also contribute to an increase in relaxation following immobilization along both axes (+11.7% and +15.5% on average with p < 0.001 and p < 0.01 respectively for each axis). Statement of significanceTo our best knowledge, this work is the first to investigate the anisotropic, viscohyperelastic and heterogeneous properties within a passive skeletal muscle immobilized in a short position. For that, from a rat model, equibiaxial relaxation tests were performed and material parameters were found from a strain energy function coupled with a Maxwell's model. We showed that excising zone of samples greatly influences the parameters of the hyperelastic behavior while the immobilization influences the viscoelasticity response instead. Furthermore, histological analysis permitted to quantify fibrosis in the skeletal muscle and bear out, at least in part, changes in relaxation behavior following immobilization.

Dynamic Mechanical Properties and Constitutive Modeling of Polyurethane Microporous Elastomers

The present study fabricated samples of polyurethane elastomers (PUEs) with three distinct densities and assessed their mechanical responses using split Hopkinson pressure bar (SHPB) tests. The findings reveal a significant increase in PUE stress with increasing strain rate and density. To further investigate the influence of strain rate sensitivity on PUEs, a strain rate sensitivity coefficient was employed to quantify the impact of strain rate on the mechanical properties of PUEs. Separate quantifications were performed for collapse stress, plateau stress, and densification strain as indicators of the strain rate sensitivity coefficient. The results demonstrate that the collapse stress sensitivity coefficient was notably affected by the applied strain rate. Additionally, both collapse and plateau stresses exhibited an increase with increasing density, which could be described by a power function relationship. Based on the theory of strain energy function, a constitutive model considering density and strain rate effects was developed to describe the stress–strain behavior of PUEs under various densities and strain rates. A comparison between this constitutive relationship and experimental results showed good agreement, highlighting its potential in describing dynamic mechanical behavior.

Research on bridge safety early warning method based on strain energy theory and health monitoring data

Bridges are technology-intensive and heavily invested permanent infrastructure. After completion and opening to traffic, bridge structures are easy to be affected by factors such as traffic load and atmospheric environment. Therefore, it is necessary to do safety warning and evaluation of bridges, especially the abnormal behavior in the early stages of bridge operation. In this research paper, a large-span continuous rigid frame bridge installed with the health monitoring system (HMS), of which a large amount of health monitoring data are collected by the HMS, is used as an example, and then a bridge safety early warning method is proposed when the bridge is during early operating period. First of all, the research finding that the internal stress of the prototype bridge obeys normal distribution through statistical analysis is used; next, we deduce that the strain energy inside the prototype bridge is subject to the Non-central Chi-square distribution combined with the strain energy and statistical theories; in the end, the key probability density distribution function of strain energy and its parameters are derived by using the key stress distribution function of the high performance concrete C50 strength grade used in the prototype bridge. The method recommended in this paper is conducive to the formulation of bridge preventive maintenance strategies.

Nonlinear vibrations and static bifurcation of a hyperelastic pipe conveying fluid

This study investigates the free and forced nonlinear vibrations, along with the static bifurcation behavior, of a fluid transmission hyperelastic pipe supported by clamped ends. Both geometrical and material nonlinearities are taken into account. The mathematical model is grounded in Euler–Bernoulli beam theory, incorporating von Kármán nonlinearity to capture the complexities of the system. For the hyperelastic materials, the strain energy function proposed by Yeoh is employed. Considering that the lowest critical velocity of the fluid flow is associated with the first mode, after deriving the governing equation using Hamilton’s principle, Galerkin discretization is applied to obtain the reduced order model in the first mode. Analysis of static bifurcation and stability is performed and critical velocity values of fluid flow are determined. The analysis of free and forced vibrations of the hyperelastic pipe is conducted using the method of multiple time scales, specifically focusing on the sub-critical velocity range of fluid flow, which corresponds to the unbuckled state of the pipe. This approach allows for a comprehensive examination of how fluid flow velocity influences the frequency characteristics of free vibrations, as well as the frequency response in the vicinity of the main resonance. The results obtained from this study can significantly enhance our understanding of the design and application of hyperelastic pipes in various fluid transfer scenarios. By elucidating the dynamic behavior and critical parameters associated with these pipes, the findings can inform engineering practices and contribute to the optimization of systems that utilize hyperelastic materials for fluid conveyance.

A covariant formulation for finite strain modelling of orthotropic elasticity and orthotropic plasticity with plasticity-induced evolution of orthotropy: Application to natural fibres

We introduce a rate-independent model for orthotropic elastic and orthotropic plastic material behaviour in a hyper-elasto-plastic framework at finite strains. The model is based on the postulate of covariance and does not rely on a multiplicative decomposition of the deformation gradient. Furthermore, a plastic-deformation-induced evolution of orthotropy is considered, similar to the notion of plastic spin. We propose that the orthotropic strain energy function and the orthotropic yield criterion are guided by identical structural tensors that evolve with plasticity. The modelled material behaviour is significant for natural fibres such as flax, hemp, or pulp fibres. Our formulation has three findings. Firstly, the covariant formulation of plasticity provides rate equations for the plastic variables and the structural tensors suitable for reproducing stress–strain diagrams of natural fibres. Secondly, the introduction of plastic-deformation-induced evolution of orthotropy in the proposed covariant setting results in a non-associative plasticity algorithm. Thirdly, the covariant setting allows the incorporation of suitable constitutive equations for the structural tensors to evolve orthotropy. The latter successfully models the stiffness increase in stress–strain diagrams of cyclic tensile tests of natural fibres.

Padé acoustoporoelasticity finite-difference simulation of elastic wave propagation in prestressed porous media

Abstract Acoustoelasticity has significant implications in seismic exploration, enabling the inference of geological structures and the properties of oil and gas reservoirs through rock physical acoustoelastic parameters, as well as elucidating the propagation of elastic waves in subsurface rocks. In traditional acoustoporoelastic numerical simulations, third-order elastic constants are derived from the Taylor expansion of the strain energy function. However, under conditions of high effective stress, the Taylor approximation can lead to divergent elastic wave velocities. To address this limitation, we propose employing the Padé approximation for the expansion of the strain energy function, as it provides superior accuracy compared to the Taylor approximation. By calibrating the Padécoefficients using experimental data for various rock types, the resultant acoustoelastic constants exhibit a reasonable theoretical limit on elastic wave velocities as effective stress increases. This method allows for a more precise characterization of velocity variations with stress, especially under high-pressure conditions. The acoustoporoelasticity numerical simulation is mainly limited to the 2D cases, so we consider confining prestressed mode in the 3D cases.

A model for capturing the rate-dependent mechanical behaviour of liquid crystal elastomers

A modelling framework is adopted and adapted here to model the rate-dependent deformation behaviour of liquid crystal elastomers (LCEs). On using an isotropic hyperelastic strain energy function WI1,I2 of binomial form, previously employed for capturing the (quasi-static) deformation of polydomain LCEs, an extension is presented for application to the rate-dependent behaviour of LCEs by incorporating the deformation rate into the model parameters. That is, the model parameters of the core hyperelastic strain energy function W are considered to evolve with, i.e., be a function of, the deformation rate. A specific measure of deformation rate ɛ̇ is utilised, and a particular functional dependency of the hyperelastic model parameters on ɛ̇, of a skewed exponential form, is devised and presented. The ensuing rate-dependent model Wr is then applied to a range of extant experimental datasets, encompassing various LCE specimens and across different deformation rates, under uniaxial loading and including both tension and compression deformations. The model is shown to capture and predict the data favourably. Given the simplic of devising and implementing the presented modelling framework, the flexibility it provides for incorporating different choices of strain energy W function as well as the functional form(s) of the dependency of the model parameters on the deformation rate, and the accuracy of the modelling results, the application of this modelling approach to capturing the rate-dependent deformation behaviour of LCEs is hereby presented.

An unconditionally stable scheme for the immersed boundary method with application in cardiac mechanics

The stability of the immersed boundary (IB) method is a challenge in simulating fluid–structure interaction problems, where time step constraints are significantly stricter than in pure fluid simulations. We propose a novel unconditionally stable scheme for the immersed boundary finite element (IBFE) method. The structure is handled implicitly and characterized by strain energy functions, rather than being modeled as fibers or membranes. Through energy estimate, we prove the unconditional stability of the fully discrete approximation in the absence of the convective term. In real simulations of cardiac mechanics problems, the time step is much larger, only limited by the Courant–Friedrichs–Lewy condition of the fluid. The novelty of this work lies in the combination of dual interpolation and distribution operators, the Jacobian-free Newton–Krylov method for solving nonlinear algebraic systems, and the semi-Lagrangian method for handling the convective term. To validate the effectiveness and accuracy of our approach, we present various benchmarks and conduct a quasi-static simulation of a three-dimensional real left ventricular model. We have shown that the numerical stability of our scheme is very robust even with much larger time step compared to conventional explicit IB methods. Our work paves the way for further works on efficient solvers of the IBFE method.

Versatile data-adaptive hyperelastic energy functions for soft materials

Applications of soft materials are customarily linked to complex deformation scenarios and material nonlinearities. In the bioengineering field, soft materials typically mimic the low stiffness of biological matter subjected to extreme deformations. Computational frameworks surge as a versatile tool to assist the design of functional applications. The constitutive model lies at the core of such frameworks. In this regard, the customary extreme non-linear behavior of elastomers poses an additional challenge to thoroughly capture the material behavior. Here, data-driven methodologies hold considerable promise for enhancing constitutive modeling when contrasted with phenomenological approaches. In this investigation, we introduce a versatile data-adaptive method tailored to the modeling of hyperelastic soft materials at finite strains. Specifically, our method substitutes an a priori chosen model for the strain energy function by a flexible interpolant defined on a discretized invariant space. Within this framework, the interpolation values assume the role of material parameters and are determined through finite element model updating to conform to measured experimental data — comprising full-field displacements coming from Digital-Image-Correlation and global reaction forces. We validate the method on uniaxial experimental tests of soft elastomers, encompassing ELASTOSILTM, DOWSILTM, and V HBTM. Overall, we aim to establish a new route for the construction of hyperelastic energy functions, untethered from any predefined existing models or assumptions regarding the shape of the energy.

Non-linear analysis of anisotropic coated fabric utilizing Variational Asymptotic Method

The dimensional reduction of a hyperelastic coated fabric from 3D to 2D is accomplished asymptotically using the Variational Asymptotic Method (VAM). This method integrates constrained calculus of variations and asymptotics. The VAM bifurcates the analysis into two: a 1D through-the-thickness analysis and a 2D reference surface analysis. The 1D analysis leads to the derivation of an asymptotically correct 3D warping functions and a 2D non-linear constitutive law. The 2D non-linear reference surface analysis utilizes the derived 2D non-linear constitutive law to obtain 2D displacements and strains through the 2D non-linear FEA. The classification of 3D strain energy density into distinct orders is enabled by introducing two intrinsic small parameters: 1) a geometric small parameter denoted by the ratio of thickness to characteristic length (h/l≪1), and 2) a physical small parameter that ensures the largest component of 3D strain is restricted to 20 percent, which is less than 1. The model takes into account both geometric and material nonlinearities. The strain energy function, which describes the anisotropic characteristics of the coated fabric has contributions from the strain energies of fiber, matrix, and fiber-fiber interaction. The findings of the study include analytically derived 3D warping functions, a 2D nonlinear constitutive law, and the prediction of warp and weft stresses and strain for a biaxial loaded tensile specimen. These findings align with the experimental outcomes.

A discrete-to-continuum model for the human cornea with application to keratoconus.

We introduce a discrete mathematical model for the mechanical behaviour of a planar slice of human corneal tissue, in equilibrium under the action of physiological intraocular pressure (IOP). The model considers a regular (two-dimensional) network of structural elements mimicking a discrete number of parallel collagen lamellae connected by proteoglycan-based chemical bonds (crosslinks). Since the thickness of each collagen lamella is small compared to the overall corneal thickness, we upscale the discrete force balance into a continuum system of partial differential equations and deduce the corresponding macroscopic stress tensor and strain energy function for the micro-structured corneal tissue. We demonstrate that, for physiological values of the IOP, the predictions of the discrete model converge to those of the continuum model. We use the continuum model to simulate the progression of the degenerative disease known as keratoconus, characterized by a localized bulging of the corneal shell. We assign a spatial distribution of damage (i.e. reduction of the stiffness) to the mechanical properties of the structural elements and predict the resulting macroscopic shape of the cornea, showing that a large reduction in the element stiffness results in substantial corneal thinning and a significant increase in the curvature of both the anterior and posterior surfaces.

A hyperelastic strain energy function for isotropic rubberlike materials

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.

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.

Biomechanics of annulus fibrosus: Elastic fiber simplification and degenerative impact on damage initiation and propagation

This study addresses three primary objectives related to lumbar intervertebral disc (IVD) biomechanics under ramping quasi-static loading conditions. First, we explore the conditions justifying the simplification of axisymmetric elastic fiber families into single fiber bundles through discretized strain energy functions. Simulations reveal that a concentration factor exceeding 10 allows for a consistent deviation below 10% between simplified and non-simplified responses. Second, we investigate the impact of elastic fibers on the physiological stiffness in IVDs, revealing minimal influence on biological motions but significant effects on degeneration. Lastly, we examine the initiation and progression of annulus fibrosus (AF) damage. Our findings confirm the validity of simplifying elastic fiber families and underscore the necessity of considering elastic fiber damage in biomechanical studies of AF tissues. Elastic fibers contribute to increased biaxial stretch stiffness, and their damage significantly affects the loading capacity of the inner AF. Additionally, degeneration significantly alters the susceptibility to damage in the AF, with specific regions exhibiting higher vulnerability. Damage tends to extend circumferentially and radially, emphasizing the regional variations in collagen and elastic fiber properties. This study offers useful insights for refining biomechanical models, paving the way for a more comprehensive understanding of IVD responses and potential clinical implications.

Anisotropic Hyperelastic Strain Energy Function for Carbon Fiber Woven Fabrics.

The present paper introduces an innovative strain energy function (SEF) for incompressible anisotropic fiber-reinforced materials. This SEF is specifically designed to understand the mechanical behavior of carbon fiber-woven fabric. The considered model combines polyconvex invariants forming an integrity basisin polynomial form, which is inspired by the application of Noether's theorem. A single solution can be obtained during the identification because of the relationship between the SEF we have constructed and the material parameters, which are linearly dependent. The six material parameters were precisely determined through a comparison between the closed-form solutions from our model and the corresponding tensile experimental data with different stretching ratios, with determination coefficients consistently reaching a remarkable value of 0.99. When considering only uniaxial tensile tests, our model can be simplified from a quadratic polynomial to a linear polynomial, thereby reducing the number of material parameters required from six to four, while the fidelity of the model's predictive accuracy remains unaltered. The comparison between the results of numerical calculations and experiments proves the efficiency and accuracy of the method.

Investigation of Macroscopic Mechanical Behavior of Magnetorheological Elastomers under Shear Deformation Using Microscale Representative Volume Element Approach.

Magnetorheological elastomers (MREs) are a class of smart materials with rubber-like qualities, demonstrating revertible magnetic field-dependent viscoelastic properties, which makes them an ideal candidate for development of the next generation of adaptive vibration absorbers. This research study aims at the development of a finite element model using microscale representative volume element (RVE) approach to predict the field-dependent shear behavior of MREs. MREs with different elastomeric matrices, including silicone rubber Ecoflex 30 and Ecoflex 50, and carbonyl iron particles (CIPs) have been considered as magnetic particles. The stress-strain characteristic of the pure silicon rubbers was evaluated experimentally to formulate the nonlinear Ogden strain energy function to describe hyper-elastic behavior of the rubbery matrix. The obtained mechanical and magnetic properties of the matrix and inclusions were integrated into COMSOL Multiphysics to develop the RVE for the MREs, in 2D and 3D configurations, with CIP volume fraction varying from 5% to 40%. Periodic boundary condition (PBC) was imposed on the RVE boundaries, while undergoing shear deformation subjected to magnetic flux densities of 0-0.4 T. Comparing the results from 2D and 3D modeling of isotropic MRE-RVE with the experimental results from the literature suggests that the 3D MRE-RVE can be effectively used to accurately predict the influence of varying factors including matrix type, volume fraction of magnetic particles, and applied magnetic field on the mechanical behavior of MREs.

I too [formula omitted]: A new class of hyperelastic isotropic incompressible models based solely on the second invariant

In contemporary elasticity theory, the strain–energy function predominantly relies on the first invariant I1 of the deformation tensor; a practice that has been influenced by models derived from rubber elasticity. However, this approach may not fully capture the complexities of materials exhibiting pronounced shear deformations, such as very soft biological tissues. Here, we explore the implications and potential benefits of constitutive models where the strain–energy function is exclusively a function of the second invariant, I2. By shifting the focus towards I2, we aim to address the limitations of current models in accurately describing shear-dominated behaviors and to provide a more comprehensive understanding of material responses, particularly for materials that do not conform to the assumptions underlying I1-centric theories. Through theoretical musings, data analysis, and automated model discovery, we investigate the feasibility of this approach and its consequences for predicting material behavior under various loading conditions. We show that the so-called “second-invariant materials” conforming to I2-only have interesting properties that are found in biological tissues and are fundamentally different from the traditional “first-invariant materials”.

An anisotropic hyperelastic strain energy function based on 21 icosahedron fiber distributions

The microscopic Cauchy strain energy for linear elasticity based on the sum of quadratic strain energies due to pair potentials has only 15 material rari-constants. It is shown that the six vectors connecting opposing vertices of a regular icosahedron can be used to develop a strain energy function for general linear elastic anisotropic response with 21 material constants. Specifically, the six strains of material fibers characterized by these vectors are enhanced by 15 fiber distribution strains due to all combinations of distinct pairs of these vectors. These two-vector fiber distributions introduce coupling that is essential to obtaining general anisotropy. The model is generalized for large deformations by replacing the strains with stretches and by using a Fung-type exponential strain energy which couples the responses of the 21 stretches. The resulting nonlinear hyperelastic strain energy function can be used to model the anisotropic hyperelastic response of fibrous tissues.

Experimental and numerical investigations on the acoustoelastic effect in hyperelastic waveguides

Guided ultrasonic wave based structural health monitoring has been of interest over decades. However, the influence of prestress states on the propagation of Lamb waves in thin-walled structures is not fully covered yet. So far experimental work presented in the literature only focuses on a few individual frequencies, which does not allow a comprehensive verification of the numerous numerical investigations. Furthermore, most work is based on the strain-energy density function by Murnaghan. To validate the common modeling approach and to investigate the suitability of other nonlinear strain-energy density functions, an extensive experimental and numerical investigation covering a large frequency range is presented here. The numerical simulation comprises the use of the Neo-Hooke as well as the Murnaghan material model. It is found that these two material models show qualitatively similar results. Furthermore, the comparison with the experimental results reveals that the Neo-Hooke material model reproduces the effect of prestress on the difference in the Lamb wave phase velocity very well in most cases. For the [Formula: see text] wave mode at higher frequencies, however, the sign of this difference is only correctly predicted by the Murnaghan model. In contrast to this, the Murnaghan material model fails to predict the sign change for the [Formula: see text] wave mode.

Connecting weakly nonlinear elasticity theories of isotropic hyperelastic materials

Soft materials exhibit significant nonlinear geometric deformations and stress–strain relationships under external forces. This paper explores weakly nonlinear elasticity theories, including Landau’s and Murnaghan’s formulations, advancing understanding beyond linear elasticity. We establish connections between these methods and extend strain-energy functions to the third and fourth orders in power of ε , where ε = H · H and 0 < ε ≤ 1 , and H is the perturbation to the deformation gradient tensor F = I + H . Furthermore, we address simplified strain-energy functions applicable to incompressible materials. Through this work, we contribute to a comprehensive understanding of nonlinear elasticity and its relationship to weakly nonlinear elasticity, facilitating the study of moderate deformations in soft material behavior and its practical applications.