Stored Energy Function Research Articles

Constitutive Relation for Large Deformations of Fiber-Reinforced Rubberlike Materials with Different Response in Tension and Compression

ABSTRACT Fiber-reinforced rubberlike materials commonly used in tires undergo large deformations and exhibit different responses in tension and compression along the fiber direction. Assuming that the response of a fiber-reinforced rubberlike material can be modeled as transversely isotropic with the fiber direction as the axis of transverse isotropy, we express the stored energy function in terms of the five invariants of the right Cauchy-Green strain tensor and account for different response in tension and compression along the fiber direction. The constitutive relation accounts for both material and geometric nonlinearities and incorporates effects of the fifth strain invariant, I5. It has been shown by Merodio and Ogden that in shear dominated deformations, I5 makes a significant contribution to the stress-strain curve. We have implemented the proposed constitutive relation in the commercial software, LS-DYNA. The numerical solutions of a few boundary value problems studied here agree with their analytical solutions derived by using Ericksen's inverse approach, in which part of the solution is assumed and unknowns in the presumed solution are found by analyzing the pertinent boundary value problem. However, computed results have not been compared with experimental findings. When test data become available, one can modify the form of the strain energy density and replace the proposed constitutive relation by the new one in LS-DYNA.

A new set of biaxial and uniaxial experiments on vulcanized rubber and attempts at modeling it using classical hyperelastic models

Using a custom built experimental setup, biaxial and uniaxial extension tests are performed on the same specimen. In biaxial tests, the stretch ratio along y direction is held constant and the stretch ratio in the perpendicular x direction is increased gradually from 1 to 1.4. Stored energy function is obtained from seven different biaxial tests performed by varying the stretch ratio along the y direction at which it is held fixed. Stored energy functions are proposed for the tested rubber sheets by allowing the stored energy to be a function of the principal invariants of right Cauchy–Green deformation tensor or the invariants of the Hencky strain or the principal stretch ratios. To evaluate the performance of the developed stored energy functions, they are compared with 10 existing constitutive relations available in literature in terms of their ability to capture their biaxial response and predicting capability for the uniaxial experiments on the rubber specimens studied. This study reveals that the stored energy function developed in this study as a function of the principal invariants of the right Cauchy–Green deformation tensor predicts the uniaxial behavior the best and capture the biaxial response second best among the models studied for the tested material. However, even this model on an average has 25 percent error in the predicted stresses.

Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties

In this Note, we address the construction of a class of stochastic Ogden's stored energy functions associated with incompressible hyperelastic materials. The methodology relies on the maximum entropy principle, which is formulated under constraints arising in part from existence theorems in nonlinear elasticity. More specifically, constraints related to both polyconvexity and consistency with linearized elasticity are considered and potentially coupled with a constraint on the mean function. Two parametric probabilistic models are thus derived for the isotropic case and rely in part on a conditioning with respect to the random shear modulus. Monte Carlo simulations involving classical (e.g., Neo-Hookean or Mooney–Rivlin) stored energy functions are then performed in order to illustrate some capabilities of the probabilistic models. An inverse calibration involving experimental results is finally presented.

Macroscopic constitutive relations for elastomers reinforced with short aligned fibers: Instabilities and post-bifurcation response

This paper is concerned with the characterization of the macroscopic response and possible development of instabilities in a certain class of anisotropic composite materials consisting of random distributions of aligned rigid fibers of elliptical cross section in a soft elastomeric matrix, which are subjected to general plane strain loading conditions. For this purpose, use is made of an estimate for the stored-energy function that was derived by Lopez-Pamies and Ponte Castañeda (2006b) for this class of reinforced elastomers by means of the second-order linear comparison homogenization method. This homogenization estimate has been shown to lose strong ellipticity by the development of shear localization bands, when the composite is loaded in compression along the (in-plane) long axes of the fibers. The instability is produced by the sudden, collective rotation of a band of fibers to partially release the high stresses that develop in the elastomer matrix when the composite is compressed along the stiff, long-fiber direction. Consistent with the mode of the impending instability, a lower-energy, post-bifurcation solution is constructed where “striped domain” microstructures consisting of layers with alternating fiber orientations develop in the composite. The volume fractions of the layers and the fiber orientations within the layers adjust themselves to satisfy equilibrium and compatibility across the layers, while remaining compatible with the imposed overall deformation. Mathematically, this construction is shown to correspond to the rank-one convex envelope of the original estimate for the energy, and is further shown to be polyconvex and therefore quasiconvex. Thus, it corresponds to the “relaxation” of the stored-energy function of the composite, and can in turn be viewed as a stress-driven “phase transition,” where the symmetry of the fiber microstructures changes from nematic to smectic.

Material-symmetries congruency in transversely isotropic and orthotropic hyperelastic materials

In this work we present the material-symmetries congruency property for anisotropic hyperelastic materials. A transversely isotropic or orthotropic material model holding this property guarantees the recovery of the isotropic behavior when, for example, the volume of reinforcement in a composite material vanishes. A material presenting material-symmetries congruency must show it not only from an analytical, theoretical point of view, but also from a numerical, practical point of view. The former may be obtained from construction of the stored energy function, whereas the latter must be obtained guaranteeing that the parameter-fitting procedure yields material parameters which combined with the specific model results in an isotropic behavior prediction for isotropic materials. We show that some anisotropic models do not present either analytical or numerical material-symmetries congruency.

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.

On the Global Stability of Compressible Elastic Cylinders in Tension

Consider a three-dimensional, homogeneous, compressible, hyperelastic body that occupies a cylindrical domain in its reference configuration. We identify a variety of hypotheses on the structure of the stored-energy function under which there exists an axisymmetric, homogeneous deformation that globally minimizes the energy. For certain classes of energy functions the uniqueness of this minimizer is also established. The primary boundary condition considered is the extension of the cylinder via the prescription of its deformed axial length, but the biaxial extension of the curved surface is also briefly considered. In particular, the results contained in this paper give conditions on the stored-energy function under which material instabilities, such as necking or the formation of shear bands, are not energy favorable.

Constitutive Relations and Parameter Estimation for Finite Deformations of Viscoelastic Adhesives

We propose a constitutive relation for finite deformations of nearly incompressible isotropic viscoelastic rubbery adhesives assuming that the Cauchy stress tensor can be written as the sum of elastic and viscoelastic parts. The former is derived from a stored energy function and the latter from a hereditary type integral. Using Ogden’s expression for the strain energy density and the Prony series for the viscoelastic shear modulus, values of material parameters are estimated by using experimental data for uniaxial tensile and compressive cyclic deformations at different constant engineering axial strain rates. It is found that values of material parameters using the loading part of the first cycle, the complete first cycle, and the complete two loading cycles are quite different. Furthermore, the constitutive relation with values of material parameters determined from the monotonic loading during the first cycle of deformations cannot well predict even deformations during the unloading portion of the first cycle. The developed constitutive relation is used to study low-velocity impact of polymethylmethacrylate (PMMA)/adhesive/polycarbonate (PC) laminate. The three sets of values of material parameters for the adhesive seem to have a negligible effect on the overall deformations of the laminate. It is attributed to the fact that peak strain rates in the severely deforming regions are large, and the corresponding stresses are essentially unaffected by the long time response of the adhesive.

A combination of histological analyses and uniaxial tensile tests to determine the material coefficients of the healthy and atherosclerotic human coronary arteries.

Atherosclerosis is considered as the most severe form of cardiovascular diseases as it alters the structure of the elastin and collagen and, consequently, the mechanical properties of the artery wall. The role of collagen fibers orientations in the mechanical properties of the healthy and atherosclerotic human coronary arteries so far has not been well determined. In this study, a fiber family based constitutive equation was employed to address the mechanical behavior of healthy and atherosclerotic human coronary arteries using the combination of histostructural and uniaxial data. A group of six healthy and atherosclerotic human coronary arteries was excised at autopsy and histological analyses were performed on each artery to determine the mean angle of collagen fibers. The preconditioned arterial tissues were then subjected to a series of quasi-static axial and circumferential loadings. The key role of fiber orientation was explicitly added into a proposed strain energy density function. The constrained nonlinear optimization method was used to determine the material coefficients based on the axial and circumferential extension data of the arteries. The material coefficients of coronary arteries were given with R(2)≥0.991. The results regardless of loading direction revealed a significant load-bearing capacity and stiffness of atherosclerotic arteries compared to the healthy ones (p<0.005). The optimized fiber angles were in good agreement with the experimental histological data as only 2.52% and 10.10% differences were observed for the healthy and atherosclerotic arteries, respectively. The stored energy function of the healthy arteries was found to be higher than that of atherosclerotic ones. These findings help us to understand the directional mechanical properties of coronary arteries which may have implications for different types of interventions and surgeries, including bypass, stenting, and balloon-angioplasty.

Corneal hyper-viscoelastic model: derivations, experiments, and simulations.

The aim of this study is to propose a method to construct corneal biomechanical model which is the foundation for simulation of corneal microsurgery. Corneal material has two significant characteristics: hyperelastic and viscoelastic. Firstly, Mooney-Rivlin hyperelastic model of cornea obtained based on stored-energy function can be simplified as a linear equation with two unknown parameters. Then, modified Maxwell viscoelastic model of the cornea, whose analytical form is consistent with the generalized Prony-series model, is proposed from the perspective of material mechanics. Parameters of the model are determined by the uniaxial tensile tests and the stress-relaxation tests. Corneal material properties are simulated to verify the hyper-viscoelastic model and measure the effectiveness of the model in the finite element simulation. On this basis, an in vivo model of the corneal is built. And the simulation of extrusion in vivo cornea shows that the force is roughly nonlinearly increasing with displacement, and it is consistent with the results obtained by extrusion experiment of in vivo cornea. Conlusions: This paper derives a corneal hyper-viscoelastic model to describe the material properties more accurately, and explains the mathematical method for determination of the model parameters. The model is an effective biomechanical model, which can be directly used for simulation of trephine and suture in keratoplasty. Although the corneal hyper-viscoelastic model is taken as the object of study, the method has certain adaptability in biomechanical research of ophthalmology.

Joint Segmentation/Registration Model by Shape Alignment via Weighted Total Variation Minimization and Nonlinear Elasticity

This paper falls within the scope of joint segmentation-registration using nonlinear elasticity principles. Because Saint Venant--Kirchhoff materials are the simplest hyperelastic materials (hyperelasticity being a suitable framework when dealing with large and nonlinear deformations), we propose viewing the shapes to be matched as such materials. Then we introduce a variational model combining a measure of dissimilarity based on weighted total variation and a regularizer based on the stored energy function of a Saint Venant--Kirchhoff material. Adding a weighted total variation--based criterion enables us to align the edges of the objects even when the modalities are different. We derive a relaxed problem associated to the initial one for which we are able to provide a result of existence of minimizers. A description and analysis of a numerical method of resolution based on a decoupling principle is then provided including a theoretical result of $\Gamma$-convergence. Applications are illustrated in academic and biological images.

Quasistatic Nonlinear Viscoelasticity and Gradient Flows

We consider the equation of motion for one-dimensional nonlinear viscoelasticity of strain-rate type under the assumption that the stored-energy function is $\lambda$-convex, which allows for solid phase transformations. We formulate this problem as a gradient flow, leading to existence and uniqueness of solutions. By approximating general initial data by those in which the deformation gradient takes only finitely many values, we show that under suitable hypotheses on the stored-energy function the deformation gradient is instantaneously bounded and bounded away from zero. Finally, we discuss the open problem of showing that every solution converges to an equilibrium state as time $t \to \infty$ and prove convergence to equilibrium under a nondegeneracy condition. We show that this condition is satisfied in particular for any real analytic cubic-like stress-strain function.

Electromechanical instabilities in fiber-constrained, dielectric-elastomer composites subjected to all-around dead-loading

In this work we investigate the possible development of instabilities in a certain class of dielectric-elastomer composites (DECs) subjected to all-around dead electromechanical loading. The DECs consist of a dielectric elastomer matrix phase constrained to plane strain deformations by means of aligned, long, rigid dielectric fibers of elliptical cross-section that are also aligned but randomly distributed in the transverse plane. Two types of instabilities are considered: loss of positive definiteness (LPD), and loss of strong ellipticity (LE). LPD simply corresponds to the loss of local convexity of the homogenized electroelastic stored-energy function for the DECs and can be of two types depending on the resulting instability modes. When the modes are aligned with the ‘principal’ solution, the instability corresponds to a maximum in the nominal electric field, possibly followed by snapping behavior. Alternatively, when the modes are orthogonal to the principal solution, the instability corresponds to a bifurcation from the principal solution. The LE, on the other hand, corresponds to LPD of the electromechanical acoustic tensor and manifests itself by the onset of highly localized shear band instabilities. Our results show that the stability, as well as the type of instability (when stability is lost), of the DECs depends sensitively on the loading conditions and is also affected by the microstructure of the DECs (i.e. the volume fraction of the fibers and their aspect ratio). In particular, it is found that the non-aligned LPD instabilities typically precede aligned LPD and LE instabilities, especially for in-plane isotropic microstructures, but aligned (limit load) LPD and LE instabilities are also possible for composites with anisotropic microstructures. In this context, it should be noted that the possible development of non-aligned LPD bifurcation instabilities appears to have been ignored by prior work. Fortunately, however, such instabilities can usually be avoided by orienting the long (in-plane) axis of the fibers parallel to the tensile stress direction and orthogonal to the applied electric field.

On the identifiability of the stored energy function of hyperelastic materials from sensor data at the boundary

This article addresses the inverse problem of reconstructing the stored energy function of a certain class of hyperelastic materials from partial Cauchy data. It is motivated by so-called structural health monitoring systems, whose idea is to disclose defects in elastic, anisotropic structures from data which are recorded at sensors that are applied at the boundary of the object. Damage affects the spatially varying stored energy function and thus its reconstruction reveals defects of the structure. The dynamic behavior of hyperelastic materials is described by Cauchyʼs equation of motion, a second order, non-linear system of partial differential equations, and thus we get an identification problem for this PDE system. We prove conditions under which this identification problem is uniquely solvable and that the solution continuously depends on the measured data as well as on the initial conditions of the structure. An important assumption is that the stored energy function is a conic combination of finitely many given functions. Moreover, we show that in the linear case such a conic decomposition of the elasticity tensor exists if its entries are in a finite dimensional subspace of the space of continuous functions and some spectral conditions are satisfied. In the case of a homogeneous, isotropic medium this decomposition is explicitly known. As a consequence, we get the result that two sensors are sufficient to identify the two independent material parameters, the Lamé coefficients, of such a medium.

A Landau--de Gennes theory of liquid crystal elastomers

In this article, we study minimization of the energy of a Landau-de Gennes liquid crystal elastomer. The total energy consists of the sum of the Lagrangian elastic stored energy function of the elastomer and the Eulerian Landau-de Gennes energy of the liquid crystal. There are two related sources of anisotropy in the model, that of the rigid units represented by the traceless nematic order tensor $Q$, and the positive definite step-length tensor $L$ characterizing the anisotropy of the network. This work is motivated by the study of cytoskeletal networks which can be regarded as consisting of rigid rod units crosslinked into a polymeric-type network. Due to the mixed Eulerian-Lagrangian structure of the energy, it is essential that the deformation maps $\varphi$ be invertible. For this, we require sufficient regularity of the fields $(\varphi, Q)$ of the problem, and that the deformation map satisfies the Ciarlet-Necas injectivity condition. These, in turn, determine what boundary conditions are admissible, which include the case of Dirichlet conditions on both fields. Alternatively, the approach of including the Rapini-Papoular surface energy for the pull-back tensor $\tilde Q$ is also discussed. The regularity requirements also lead us to consider powers of the gradient of the order tensor $Q$ higher than quadratic in the energy. We assume polyconvexity of the stored energy function with respect to the effective deformation tensor and apply methods of calculus of variations from isotropic nonlinear elasticity. Recovery of minimizing sequences of deformation gradients from the corresponding sequences of effective deformation tensors requires invertibility of the anisotropic shape tensor $L$. We formulate a necessary and sufficient condition to guarantee this invertibility property in terms of the growth to infinity of the bulk liquid crystal energy $f(Q)$, as the minimum eigenvalue of $Q$ approaches the singular limit of $-\frac{1}{3}$. It turns out that $L$ becomes singular as the minimum eigenvalue of $Q$ reaches $-\frac{1}{3}$. Lower bounds on the eigenvalues of $Q$ are needed to ensure compatibility between the theories of Landau-de Gennes and Maier-Saupe of nematics [5].

Existence, uniqueness solution for nonlinear mixed problem in two dimensional elasticity

The aim of this article is to minimises the stored energy function, of two dimensional elasticity with mixed boundary condition, in order, that the Euler's equilibrium equations of the Saint-Venant-Kirchhoff problem, has one and only one solution. Keywords: Elasticity, Saint-Venant, Stored Energy.

Finite-horizon online transmission scheduling on an energy harvesting communication link with a discrete set of rates

As energy harvesting communication systems emerge, there is a need for transmission schemes that dynamically adapt to the energy harvesting process. In this paper, after exhibiting a finite-horizon online throughput-maximizing scheduling problem formulation and the structure of its optimal solution within a dynamic programming formulation, a low complexity online scheduling policy is proposed. The policy exploits the existence of thresholds for choosing rate and power levels as a function of stored energy, harvest state and time until the end of the horizon. The policy, which is based on computing an expected threshold, performs close to optimal on a wide range of example energy harvest patterns. Moreover, it achieves higher throughput values for a given delay, than throughput-optimal online policies developed based on infinite-horizon formulations in recent literature. The solution is extended to include ergodic time-varying (fading) channels, and a corresponding low complexity policy is proposed and evaluated for this case as well.

A Polyconvex Integrand; Euler–Lagrange Equations and Uniqueness of Equilibrium

In this manuscript we are interested in stored energy functionals W defined on the set of d × d matrices, which not only fail to be convex but satisfy \({{\rm lim}_{\det \xi \rightarrow 0^+} W(\xi)=\infty.}\) We initiate a study which we hope will lead to a theory for the existence and uniqueness of minimizers of functionals of the form \({E(\mathbf{u})=\int_\Omega (W(\nabla \mathbf{u}) -\mathbf{F} \cdot \mathbf{u}) {\rm d}x}\) , as well as their Euler–Lagrange equations. The techniques developed here can be applied to a class of functionals larger than those considered in this manuscript, although we keep our focus on polyconvex stored energy functionals of the form \({W(\xi)=f(\xi) +h( {\rm det} \xi)}\) – such that \({{\rm lim}_{t \rightarrow 0^+} h(t)=\infty}\) – which appear in the study of Ogden material. We present a collection of perturbed and relaxed problems for which we prove uniqueness results. Then, we characterize these minimizers by their Euler–Lagrange equations.

A thermodynamically compatible splitting procedure in hyperelasticity

A material is hyperelastic if the stress tensor is obtained by variation of the stored energy function. The corresponding 3D mathematical model of hyperelasticity written in the Eulerian coordinates represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. A classical approach for numerical solving of such a 3D system is a geometrical splitting: the 3D system is split into three 1D systems along each spatial direction and solved then by using a Godunov type scheme. Each 1D system has 7 independent eigenfields corresponding to contact discontinuity, longitudinal waves and shear waves. The construction of the corresponding Riemann solvers is not an easy task even in the case of isotropic solids. Indeed, for a given specific energy it is extremely difficult, if not impossible, to check its rank-one convexity which is a necessary and sufficient condition for hyperbolicity of the governing equations.In this paper, we consider a particular case where the specific energy is a sum of two terms. The first term is the hydrodynamic energy depending only on the density and the entropy, and the second term is the shear energy which is unaffected by the volume change. In this case a very simple criterion of hyperbolicity can be formulated. We propose then a new splitting procedure which allows us to find a numerical solution of each 1D system by solving successively three 1D sub-systems. Each sub-system is hyperbolic, if the full system is hyperbolic. Moreover, each sub-system has only three waves instead of seven, and the velocities of these waves are given in explicit form. The last property allows us to construct reliable Riemann solvers. Numerical 1D tests confirm the advantage of the new approach. A multi-dimensional extension of the splitting procedure is also proposed.

On Irreversibility and Dissipation in Hyperelasticity With Softening

Bulk and interface material failures are often modeled via hyperelastic stored energy functions incorporating softening behavior. The softening is reversible due to the hyperelastic nature of the constitutive law and material can “heal” under unloading. To prevent this healing, special numerical procedures (like finite element deletion) are usually used in computer simulations. In the present work, we suggest an alternative: very simple analytical formulation, which makes failure irreversible when a critical stored energy is reached. This new notion is directly incorporated into the constitutive equations, consequently, relieving the need for preliminary discretization of the boundary-value problem.