Form Of Strain-energy Function Research Articles

Implementation of Particle Swarm Optimization Algorithm in Matlab Code for Hyperelastic Characterization

The purpose of this paper is to demonstrate the applicability of Particle Swarm Optimization algorithm to determine material parameters in incompressible isotropic elastic strain-energy functions using combined tension and torsion loading. Simulation of rubber behavior was conducted from the governing equations of the deformation of a cylinder composed of isotropic hyperelastic incompressible materials. Four different forms of strain-energy function were considered based respectively on polynomial, exponential and logarithmic terms to reproduce load force (N) and torque (M) trends using natural rubber experimental data. After highlighting the minimization of the objective function generated in the fitting process, the study revealed that a particle swarm optimization algorithm could be successfully used to identify the best material parameters and characterize the behavior of rubber-like hyperelastic materials.

Residually Stressed Fiber Reinforced Solids: A Spectral Approach.

We use a spectral approach to model residually stressed elastic solids that can be applied to carbon fiber reinforced solids with a preferred direction; since the spectral formulation is more general than the classical-invariant formulation, it facilitates the search for an adequate constitutive equation for these solids. The constitutive equation is governed by spectral invariants, where each of them has a direct meaning, and are functions of the preferred direction, the residual stress tensor and the right stretch tensor. Invariants that have a transparent interpretation are useful in assisting the construction of a stringent experiment to seek a specific form of strain energy function. A separable nonlinear (finite strain) strain energy function containing single-variable functions is postulated and the associated infinitesimal strain energy function is straightforwardly obtained from its finite strain counterpart. We prove that only 11 invariants are independent. Some illustrative boundary value calculations are given. The proposed strain energy function can be simply transformed to admit the mechanical influence of compressed fibers to be partially or fully excluded.

Closed form solutions for large deformation of cylinders under combined extension-torsion

In this paper, the mechanical response of an incompressible isotropic nonlinear elastic solid circular cylinder is investigated under combined extension and torsion. Since the deformation tensor in extension-torsion is non-diagonal, implementing stretch based energy functions are complex. Hence, in this study, an analytical solution is proposed for both invariant- and stretch-based models. Moreover, finite element analysis of extension-torsion of hyperelastic materials is carried out using both UHYPER and VUMAT user-defined subroutine in ABAQUS to verify the presented analytical methods. Both stretch and invariant-based exponential forms of strain energy function are employed, and its corresponded material parameters are calibrated for silicon-rubber. The finite element results for stress distribution show a good agreement with analytical findings which confirms the validity and accuracy of the proposed method. Results show that the both invariant- and stretch-based exp-exp models are less conservative for a relatively small stretch and twist while for large stretches they yield higher stresses than the other models and the rate of stress variation increases significantly, especially for a stretch-based exp-exp model. For both Invariant- and stretch-based exp-exp models, an alteration axial stretch is identified where the torsion induced axial force alters from tensile to compressive. For stretches smaller than this alteration point, the cylinder always elongates while for the larger stretches, the cylinder tends to shorten for small twists and then elongates on further twisting. Also, as the axial stretch increases the alteration point appears in larger twists.

Biomechanical Properties of Human Ascending Thoracic Aortic Dissections.

Thoracic aortic dissections are associated with a significant risk of morbidity and mortality, and currently challenge our understanding of the biomechanical factors leading to their initiation and propagation. We quantified the biaxial mechanical properties of human type A dissections (n = 16) and modeled the stress-strain data using a microstructurally motivated form of strain energy function. Our results show significantly higher stiffness for dissected tissues as compared to control aorta without arterial disease. Higher stiffness of dissected tissues did not, however, correlate with greater aortic diameter measured prior to surgery nor were there any age dependent differences in the tissue properties.

The effect of different forms of strain energy functions in hyperelasticity‐based crystal plasticity models on texture evolution and mechanical response of face‐centered cubic crystals

SUMMARYMicro‐mechanical and macro‐mechanical behavior of face‐centered cubic (FCC) crystals is investigated by using different forms of strain energy functions in hyperelastic material models in crystal plasticity finite element framework. A quadratic strain energy function with anisotropic elastic constants, a polyconvex strain energy function with invariants associated with the cubic symmetry, and a strain energy function from an inter‐atomic potential are considered in hyperelastic material models to describe the elastic deformation of FCC crystals. In our numerical experiments, the trajectories of {111} poles in the pole figure and the accumulated plastic slips of FCC coppers under uniaxial tension and simple shear depend on the choice of strain energy functions when the slip resistance of the slip systems is high. The ability of strain energy functions in this study to represent elastic lattice distortions in crystals varies with the amount of elastic deformation and the shape of deformed lattice. However, numerical results show that the change of macroscopic mechanical behavior of FCC coppers is not significant for the choice of strain energy functions, compared with the change of crystallographic texture evolution. Copyright © 2014 John Wiley & Sons, Ltd.

Propagation of surface wave in a fluid layer overlying a slightly compressible, finitely deformed elastic medium

In this paper the dispersion relation of surface wave for a fluid layer overlying a slightly compressible, finitely deformed half-space is discussed. It has been observed that as the wave number tends to zero the dispersion relation transformed to the speed equation of surface waves. For graphical representation we have considered Neo-Hookean form of strain energy function. Phase speed curves have been plotted and different discussions have been made.

생체 유사재료를 설명하는 물질 상수 추정

It is important to have a well developed strain energy function in order to understand the mechanical behavior of biomaterial like the blood vessel of artery. However, since it is not possible to have a complete form of strain energy function of artery, theoretical framework describing the behaviour of Rubber-like material which is similar to blood vessel is applied to infer useful forms of strain energy function of biomaterial. Based on Chuong-Fung model and Mooney-Rivlin model, material parameters are estimated based on experimental data. From the results, it can be inferred that the estimated parameters can be used to explain the difference of mechanical characteristics between normal vessel and vessel with stent.

Experiment Which is Done in Rubber Mechanics

Literaturde, kaucuk turu malzemelerin elastik davranisini modelleyebilmek icin bir cok farkli formda sekil degistirme enerji fonksiyonlari onerilmistir. Teorik olarak onerilen bu sekil degistirme enerji fonksiyonlarinin dogrulugu ise cekme, basma ve kayma deneyleri ile kontrol edilmektedir. Hazirlanan bu calismada kaucuk mekaniginde yapilan deneyler tanimlanmis ve siniflandirilmistir. Ayrica, calismada literaturde kaucuk turu malzemeler icin yapilan deneysel calismalar incelenmistir.

An Improved Non-Affine Arruda-Boyce Type Constitutive Model for Collagen Networks

This work investigates, by means of computational modeling, the mechanical properties of soft collagen tissues on the basis of elasticity theory. Bio-polymer networks are structurally disordered and thus compelled to deform non-affine. To capture that in our computational modeling, we supplement the well-known affine Arruda-Boyce model with positional disorder and compute the resultant changes in mechanical response. We characterize this mechanical behavior as a response to various homogeneous deformations in 3D networks, assuming different constitutive behavior for the individual fibers (in the small deformations linear regime, hookean springs under the entropic elasticity assumption, and in the nonlinear regime freely-jointed and worm-like chains), as well as different coordination numbers (4, 6 or 8 chains connecting at each cross linking point) of the resulting fiber networks. Furthermore we compare the moduli of the simulated networks with their affine deformed counterparts. Previous work has clearly demonstrated that non-affine deformation modes in elastic (bio)polymer networks greatly affect their mechanics. As the original Arruda-Boyce model can be represented with a particular form of strain-energy function that is micro-mechanically motivated, incorporation of the non-affinity yields amended predictions of the macroscopic mechanical behavior of soft fibrous networks, based on an improved representation of microscopic network structure and deformations. We show that shear and bulk moduli in the Arruda-Boyce model can be as off as 30% when compared with the shear and bulk moduli in the non-affine model. This entire evaluation of the ways non-affinity enhances the well known Arruda-Boyce model sets the groundwork for developing accurate constitutive relations for fibrous biological materials, for use in finite element analysis of soft tissues.

Fluid pressure loading of a hyperelastic membrane

This paper examines the problem of the fluid pressure loading of a hyperelastic membrane made of a natural gum rubber, that is fixed along a circular boundary. The fluid pressure loading of hyperelastic membranes have several important technological applications particularly in the area of bio-medical engineering and the response of the membrane to fluid pressure loading can be useful in identifying the applicability of the various forms of strain energy functions that are proposed in the literature for describing hyperelastic behaviour. The results of the fluid pressure induced deflected profiles of the circular membrane together with computational modelling of the experiments are used to identify the range of applicability of several forms of strain energy functions for the hyperelastic materials.

Studies of biaxial mechanical properties and nonlinear finite element modeling of skin.

The objective of this research is to conduct mechanical property studies of skin from two individual but potentially connected aspects. One is to determine the mechanical properties of the skin experimentally by biaxial tests, and the other is to use the finite element method to model the skin properties. Dynamic biaxial tests were performed on 16 pieces of abdominal skin specimen from rats. Typical biaxial stress-strain responses show that skin possesses anisotropy, nonlinearity and hysteresis. To describe the stress-strain relationship in forms of strain energy function, the material constants of each specimen were obtained and the results show a high correlation between theory and experiments. Based on the experimental results, a finite element model of skin was built to model the skin's special properties including anisotropy and nonlinearity. This model was based on Arruda and Boyce's eight-chain model and Bischoff et al.'s finite element model of skin. The simulation results show that the isotropic, nonlinear eight-chain model could predict the skin's anisotropic and nonlinear responses to biaxial loading by the presence of an anisotropic prestress state.

Nonlinear transversely isotropic elastic solids: an alternative representation

A strain energy function which depends on five independent variables that have immediate physical interpretation is proposed for finite strain deformations of transversely isotropic elastic solids. Three of the five variables (invariants) are the principal stretch ratios and the other two are squares of the dot product between the preferred direction and two principal directions of the right stretch tensor. The set of these five invariants is a minimal integrity basis. A strain energy function, expressed in terms of these invariants, has a symmetry property similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress-strain relations are given. The formulation is applied to several types of deformations, and in these applications, a mathematical simplicity is highlighted. The proposed model is attractive if principal axes techniques are used in solving boundary-value problems. Experimental advantage is demonstrated by showing that a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed. A specific form of strain energy function can be easily obtained from the general form via a triaxial test. Using series expansions and symmetry, the proposed general strain energy function is refined to some particular forms. Since the principal stretches are the invariants of the strain energy function, the Valanis-Landel form can be easily incorporated into the constitutive equation. The sensitivity of response functions to Cauchy stress data is discussed for both isotropic and transversely isotropic materials. Explicit expressions for the weighted Cauchy response functions are easily obtained since the response function basis is almost mutually orthogonal.

Large deformation analysis of gellan gels

Gellan gel, a typical polysaccharide gel, is ruptured with different deformation behaviors from gelatin gel or rubber. It exhibits both strain hardening and softening; hardening is observed for moderate strain and softening occurs for larger strain. From the analyses of stress–strain curves of gellan gels, we propose forms of strain energy function. The fit with the proposed equation was excellent, while the existing models fail because they consider only one of hardening or softening effect. Furthermore, these equations are shown to be capable of extracting the hardening and softening effects separately from the observed stress–strain curves. By using these fitting equations, the concentration dependences of hardening and softening are investigated. It is shown that the degrees of hardening and softening both increase with increasing gellan concentration.

The Nonlinear Material Properties of Liver Tissue Determined From No-Slip Uniaxial Compression Experiments

The mechanical response of soft tissue is commonly characterized from unconfined uniaxial compression experiments on cylindrical samples. However, friction between the sample and the compression platens is inevitable and hard to quantify. One alternative is to adhere the sample to the platens, which leads to a known no-slip boundary condition, but the resulting nonuniform state of stress in the sample makes it difficult to determine its material parameters. This paper presents an approach to extract the nonlinear material properties of soft tissue (such as liver) directly from no-slip experiments using a set of computationally determined correction factors. We assume that liver tissue is an isotropic, incompressible hyperelastic material characterized by the exponential form of strain energy function. The proposed approach is applied to data from experiments on bovine liver tissue. Results show that the apparent material properties, i.e., those determined from no-slip experiments ignoring the no-slip conditions, can differ from the true material properties by as much as 50% for the exponential material model. The proposed correction approach allows one to determine the true material parameters directly from no-slip experiments and can be easily extended to other forms of hyperelastic material models.

Constitutive Equations for Isotropic Rubber-Like Materials Using Phenomenological Approach: A Bibliography (1930–2003)

Abstract To describe the elastic behavior of rubber-like materials, numerous specific forms of strain energy functions have been proposed in the literature. This bibliography provides a list of references on the strain energy functions for rubber-like materials on isothermal condition using the phenomenological approach. The published works, either containing the strain energy function proposals or the discussions on such proposals, based upon the phenomenological approach, are classified.

Dynamic extension of a compressible nonlinearly elastic membrane tube

The dynamic response of an isotropic compressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. The equations governing dynamic axially symmetric deformations of the membrane tube are presented for a general form of compressible isotropic elastic strain-energy function. Numerical results, obtained using a Godunov-type finite volume method and valid up to the time at which reflections occur at the fixed end of the tube, are given for two specific forms of the strain-energy function that characterizes a class of compressible elastomers (the Blatz-Ko model). The question of how the numerical results are related to the exact solution obtained for a limiting case is discussed.

Phenomenological Modeling of the Stress-Stretch Behavior of EPDM Rubber with Loading-rate and Damage Effects

This paper formulates a phenomenological continuum model, which includes the idealized Mullins effect and loading-rate effects, for the stress-stretch behavior of ethylene-propylene-diene terpolymer (EPDM). This proposed model is based on Ogden and Roxburgh’s pseudoelasticity theory and a hyperelastic form of strain-energy function describing the Mullins effect. Based on experimental results, the loading-rate effects are introduced into this model by replacing the material parameters with an exponential formation as a function of loading rate. The loading rate has effects only on the material parameters and the degree of damage in a specimen is only dependent on the maximum stretch ratio it experiences. It is demonstrated that this model provides a good agreement with the experimental results on EPDM stress-stretch behavior over a wide range of stretching rates.

Rayleigh Waves and Surface Stability for Bell Materials in Compression; Comparison with Rubber

The stability of a Bell-constrained half-space in compression is studied. To this end, the propagation of Rayleigh waves on the surface of the material when it is maintained in a static state of triaxial prestrain is considered. The prestrain is such that the free surface of the half-space is a principal plane of deformation. The exact secular equation is established for surface waves traveling in a principal direction of strain with attenuation along the principal direction normal to the free plane. As the half-space is put under increasing compressive loads, the speed of the wave eventually tends to zero and the bifurcation criterion, or stability equation, is reached. Then the analysis is specialized to specific forms of strain energy functions and prestrain, and comparisons are made with results previously obtained in the case of incompressible neo-Hookean or Mooney-Rivlin materials. It is found that these rubber-like incompressible materials may be compressed more than "Bell empirical model" materials, but not as much as "Bell simple hyperelastic" materials, before the critical stretches, solutions to the bifurcation criterion, are reached. In passing, some classes of incompressible materials which possess a relative-universal bifurcation criterion are presented.

Strain Energy Function for Filled and Unfilled Rubberlike Material

Abstract A general separable form of strain energy function with linear parameters is developed so that if a standard curve fitting method is used only a handful of linear equations need to be solved; the proposed function is formulated in such a way that only a few parameters are required to characterized the rubberlike materials. One of the parameters is the Young's modulus and this could facilitate a curve fitting algorithm if the value of the Young's modulus is known. A single constant, strain energy function is also proposed. The predicted curves agree well with numerous experimental data.

Application of variational principles to the axial extension of a circular cylindical nonlinearly elastic membrane

In this paper stationary potential-energy and complementary-energy principles are formulated for boundary-value problems for compressible or incompressible nonlinearly elastic membranes, and full justification for adoption of the complementary principle is provided. The stationary principles are then extended to extremum principles, which provide upper and lower bounds on the energy functional associated with the solution of a given problem. The principles are then illustrated by their application to the nonlinear problem of the axially symmetric static deformation of an isotropic elastic membrane. In its undeformed natural configuration the membrane has the form of a circular cylindrical surface. The cylinder is subject to a prescribed (tensile) axial force with the ends of the cylinder constrained so that their radii remain constant. The alternative boundary condition in which the axial displacement of the ends is prescribed instead of the axial force is also considered. The extremum principles are applied first without restriction on the form of strain-energy function in order to obtain primitive bounds on the energy of Voigt and Reuss type commonly used in composite-material mechanics. Then, for particular forms of strain-energy function, specific bounds are obtained by selecting suitable trial deformation and stress fields and the bounds are optimized using a numerical procedure (which is readily adapted for other forms of strain-energy function). It is found that these bounds are very close and hence give a good estimate of the actual energy. The associated deformed geometry of the membrane is described together with the resulting principal stresses.