Application of Finite Elastic Theory to the Behavior of Rubberlike Materials

1963 ◽  
Vol 36 (5) ◽  
pp. 1459-1496 ◽  
Author(s):  
Paul J. Blatz
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Finite Elasticity ◽  
Strain Energy Function ◽  
Elastic Theory ◽  
Mechanical Parameters ◽  
Molecular Interpretation ◽  
Rubberlike Materials

Abstract A brief review of the theory of finite elasticity is presented. The theory is applied to the characterization of the mechanical response parameters of a polyurethan foam. The incorporation of compressibility and anisotropy effects into the strain energy function are discussed. An example of the behavior of a composite or filled foam is presented. Finally some of the problems associated with the molecular interpretation of mechanical parameters are discussed.

Large Deformation Isotropic Elasticity—On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids

1973 ◽  
Vol 46 (2) ◽  
pp. 398-416 ◽  
Author(s):  
R. W. Ogden
Keyword(s):  
Mathematical Analysis ◽  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Strain Energy Function ◽  
Usual Practice ◽  
Principal Axes ◽  
Simple Tension ◽  
Isotropic Elasticity ◽  
Strain Invariants

Abstract Many attempts have been made to reproduce theoretically the stress-strain curves obtained from experiments on the isothermal deformation of highly elastic ‘rubberlike’ materials. The existence of a strain-energy function has usually been postulated, and the simplifications appropriate to the assumptions of isotropy and incompressibility have been exploited. However, the usual practice of writing the strain energy as a function of two independent strain invariants has, in general, the effect of complicating the associated mathematical analysis (this is particularly evident in relation to the calculation of instantaneous moduli of elasticity) and, consequently, the basic elegance and simplicity of isotropic elasticity is sacrificed. Furthermore, recently proposed special forms of the strain-energy function are rather complicated functions of two invariants. The purpose of this paper is, while making full use of the inherent simplicity of isotropic elasticity, to construct a strain-energy function which: (i) provides an adequate representation of the mechanical response of rubberlike solids, and (ii) is simple enough to be amenable to mathematical analysis. A strain-energy function which is a linear combination of strain invariants defined by ϕ(α)=(α1α+α2α+α3α)/α is proposed; and the principal stretches α1, α2, and α3 are used as independent variables subject to the incompressibility constraint α1α2α3=1. Principal axes techniques are used where appropriate. An excellent agreement between this theory and the experimental data from simple tension, pure shear and equibiaxial tension tests is demonstrated. It is also shown that the present theory has certain repercussions in respect of the constitutive inequality proposed by Hill.

scholarly journals Evolution of anisotropy in soft tissue

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130548 ◽  
Author(s):  
J. G. Murphy
Keyword(s):  
Linear Theory ◽  
Energy Function ◽  
Anisotropic Material ◽  
Strain Energy ◽  
Mechanical Response ◽  
Essential Feature ◽  
Sufficient Conditions ◽  
Strain Energy Function ◽  
Phenomenological Approach ◽  
Reduced Form

The phenomenological approach to the modelling of the mechanical response of arteries usually assumes a reduced form of the strain-energy function in order to reduce the mathematical complexity of the model. A common approach eschews the full basis of seven invariants for the strain-energy function in favour of a reduced set of only three invariants. It is shown that this reduced form is not consistent with the corresponding full linear theory based on infinitesimal strains. It is proposed that compatibility with the linear theory is an essential feature of any nonlinear model of arterial response. Two approaches towards ensuring such compatibility are proposed. The first is that the nonlinear theory reduces to the full six-constant linear theory, without any restrictions being imposed on the constants. An alternative modelling strategy whereby an anisotropic material is compatible with a simpler material in the linear limit is also proposed. In particular, necessary and sufficient conditions are obtained for a nonlinear anisotropic material to be compatible with an isotropic material for infinitesimal deformations. Materials that satisfy these conditions should be useful in the modelling of the crimped collagen fibres in the undeformed configuration.

scholarly journals A Bimodular Polyconvex Anisotropic Strain Energy Function for Articular Cartilage

2006 ◽  
Vol 129 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Stephen M. Klisch
Keyword(s):  
Articular Cartilage ◽  
Strong Interaction ◽  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Deformation Gradient ◽  
Strain Energy Function ◽  
Finite Deformations ◽  
Orthotropic Materials ◽  
Interaction Terms

A strain energy function for finite deformations is developed that has the capability to describe the nonlinear, anisotropic, and asymmetric mechanical response that is typical of articular cartilage. In particular, the bimodular feature is employed by including strain energy terms that are only mechanically active when the corresponding fiber directions are in tension. Furthermore, the strain energy function is a polyconvex function of the deformation gradient tensor so that it meets material stability criteria. A novel feature of the model is the use of bimodular and polyconvex “strong interaction terms” for the strain invariants of orthotropic materials. Several regression analyses are performed using a hypothetical experimental dataset that captures the anisotropic and asymmetric behavior of articular cartilage. The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson’s ratios, as well as axial stress–axial strain responses, in tension and compression for finite deformations.

MULTIAXIAL STRAIN ENERGY FUNCTIONS OF RUBBERLIKE MATERIALS: AN EXPLICIT APPROACH BASED ON POLYNOMIAL INTERPOLATION

2014 ◽  
Vol 87 (1) ◽  
pp. 168-183 ◽  
Author(s):  
Xiao-Ming Wang ◽  
Hao Li ◽  
Zheng-Nan Yin ◽  
Heng Xiao
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Polynomial Interpolation ◽  
Strain Energy Function ◽  
Shear Plane ◽  
Unknown Parameters ◽  
Energy Functions ◽  
Explicit Approach ◽  
Strain Energy Functions ◽  
Rubberlike Materials

ABSTRACT We propose an explicit approach to obtaining multiaxial strain energy functions for incompressible, isotropic rubberlike materials undergoing large deformations. Via polynomial interpolation, we first obtain two one-dimensional strain energy functions separately from uniaxial data and shear data, and then, from these two, we obtain a multiaxial strain energy function by means of direct procedures based on well-designed logarithmic invariants. This multiaxial strain energy function exactly fits any given data from four benchmark tests, including uniaxial and equibiaxial extension, simple shear, plane–strain extension, and so forth. Furthermore, its predictions for biaxial stretch tests provide good accord with test data. The proposed approach is explicit in a sense without involving the usual procedures both in deriving forms of the multiaxial strain energy function and in estimating a number of unknown parameters.

Characterization of Elastic Properties of Carbon-Black-Filled Rubber Vulcanizates

1990 ◽  
Vol 63 (5) ◽  
pp. 792-805 ◽  
Author(s):  
O. H. Yeoh
Keyword(s):  
Carbon Black ◽  
Shear Modulus ◽  
Elastic Properties ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Stress Strain ◽  
Filled Rubber ◽  
Filled Rubbers

Abstract A novel strain-energy function which is a simple cubic equation in the invariant (I1−3) is proposed for the characterization of the elastic properties of carbon-black-filled rubber vulcanizates. Conceptually, the proposed function is a material model with a shear modulus which varies with deformation. This contrasts with the neo-Hookean and Mooney-Rivlin models which have a constant shear modulus. The variation of shear modulus with deformation is commonly observed with filled rubbers. Initially, the modulus falls with increasing deformation, leading to a flattening of the shear stress/strain curve. At large deformations, the modulus rises again due to finite extensibility of the network, accentuated by the strain amplication effect of the filler. This characteristic behavior of filled rubbers may be described approximately by the proposed strain-energy function by requiring the coefficient C20 to be negative, while the coefficients C10 and C30 are positive. The use of the proposed strain-energy function has been shown to permit the prediction of stress/strain behavior in different deformation modes from data obtained in one simple deformation mode. This circumvents the need for a rather difficult experiment in general biaxial extension. The simple form of the proposed function also simplifies the regression analysis. This strain-energy function is consistent with the general Rivlin strain-energy function and is easily obtained from the popular third-order deformation approximation. Thus, it is already available in many existing finite-element analysis programs.

A Theoretical Framework to Analyze Bend Testing of Soft Tissue

2006 ◽  
Vol 129 (1) ◽  
pp. 117-120 ◽  
Author(s):  
Mark A. Nicosia
Keyword(s):  
Soft Tissue ◽  
Energy Function ◽  
Strain Energy ◽  
Heart Valves ◽  
Finite Elasticity ◽  
Strain Energy Function ◽  
Element Model ◽  
Theoretical Framework ◽  
Radius Of Curvature ◽  
Bend Testing

It has been hypothesized that repetitive flexural stresses contribute to the fatigue-induced failure of bioprosthetic heart valves. Although experimental apparatuses capable of measuring the bending properties of biomaterials have been described, a theoretical framework to analyze the resulting data is lacking. Given the large displacements present in these bending experiments and the nonlinear constitutive behavior of most biomaterials, such a formulation must be based on finite elasticity theory. We present such a theory in this work, which is capable of fitting bending moment versus radius of curvature experimental data to an arbitrary strain energy function. A simple finite element model was constructed to study the validity of the proposed method. To demonstrate the application of the proposed approach, bend testing data from the literature for gluteraldehyde-fixed bovine pericardium were fit to a nonlinear strain energy function, which showed good agreement to the data. This method may be used to integrate bending behavior in constitutive models for soft tissue.

A Strain Energy Function for the Annulus Fibrosus Specified Using Biaxial Experimental Data

1999 ◽  
Author(s):  
Elisa C. Bass ◽  
Jeffrey C. Lotz
Keyword(s):  
Energy Function ◽  
Annulus Fibrosus ◽  
Strain Energy ◽  
Mechanical Response ◽  
Biaxial Tension ◽  
Elastic Strain Energy ◽  
Strain Energy Function ◽  
Constitutive Formulation

Abstract The mechanical behavior of the annulus fibrosus has typically been characterized through the use of uniaxial tests. In contrast, its in vivo constraints are multiaxial and likely result in a mechanical response very different from that observed to date in vitro. The goal of this study was to test the annulus in biaxial tension and use these data to determine an elastic strain energy function for the annulus. Our results demonstrate that the mechanical response of the annulus is dramatically influenced by a biaxial constraint, and that these experiments provide important data for the determination of the constitutive formulation for this strongly anisotropic and nonlinear tissue.

Mesoscale bounds in finite elasticity and thermoelasticity of random composites

2006 ◽  
Vol 462 (2068) ◽  
pp. 1167-1180 ◽  
Author(s):  
Z.F Khisaeva ◽  
M Ostoja-Starzewski
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Variational Principles ◽  
Finite Elasticity ◽  
Effective Strain ◽  
Strain Energy Function ◽  
Free Energy Function ◽  
Thermoelastic Effects ◽  
Thermoelastic Response ◽  
Strain Energy Functions

Under consideration is the problem of size and response of the Representative Volume Element (RVE) in the setting of finite elasticity of statistically homogeneous and ergodic random microstructures. Through the application of variational principles, a scale dependent hierarchy of strain energy functions (i.e. mesoscale bounds) is derived for the effective strain energy function of the RVE. In order to account for thermoelastic effects, the variational principles are first generalized, and then analogous bounds on the effective thermoelastic response are derived. It is also shown that, in contradiction to results obtained for random linear composites, the hierarchy on the effective strain energy function in nonlinear elasticity cannot be split into volumetric and isochoric terms, while the hierarchy on the effective free energy function cannot be divided into purely mechanical and thermal contributions.

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