scholarly journals The Poisson Function of Finite Elasticity

1986 ◽  
Vol 53 (4) ◽  
pp. 807-813 ◽  
Author(s):  
M. F. Beatty ◽  
D. O. Stalnaker
Keyword(s):  
Finite Elasticity ◽  
Constraint Equation ◽  
Deformation Tensor ◽  
Polyurethane Elastomers ◽  
Derived Relations ◽  
Hyperelastic Materials ◽  
Rubber Compounds ◽  
Poisson Function ◽  
Simple Tension ◽  
Constrained Materials

The Poisson function is introduced to study in a simple tension test the lateral contractive response of compressible and incompressible, isotropic elastic materials in finite strain. The relation of the Poisson function to the classical Poisson’s ratio and its behavior for certain constrained materials are discussed. Some experimental results for several elastomers, including two natural rubber compounds of the same kind studied in earlier basic experiments by Rivlin and Saunders, are compared with the derived relations. A special class of compressible materials is also considered. It is proved that the only class of compressible hyperelastic materials whose response functions depend on only the third principal invariant of the deformation tensor is the class first introduced in experiments by Blatz and Ko. Poisson functions for the Blatz-Ko polyurethane elastomers are derived; and our experimental data are reviewed in relation to a volume constraint equation used in their experiments.

scholarly journals How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity

2017 ◽  
Vol 473 (2207) ◽  
pp. 20170607 ◽  
Author(s):  
L. Angela Mihai ◽  
Alain Goriely
Keyword(s):  
Elastic Material ◽  
Mechanical Response ◽  
Finite Elasticity ◽  
Experimental Tests ◽  
Elastic Parameter ◽  
Hyperelastic Materials ◽  
Poisson Function ◽  
Elastic Responses ◽  
Nonlinear Elastic ◽  
Constitutive Parameters

The mechanical response of a homogeneous isotropic linearly elastic material can be fully characterized by two physical constants, the Young’s modulus and the Poisson’s ratio, which can be derived by simple tensile experiments. Any other linear elastic parameter can be obtained from these two constants. By contrast, the physical responses of nonlinear elastic materials are generally described by parameters which are scalar functions of the deformation, and their particular choice is not always clear. Here, we review in a unified theoretical framework several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests. These parameters represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. Universal relations between certain of these parameters are further established, and then used to quantify nonlinear elastic responses in several hyperelastic models for rubber, soft tissue and foams. The general parameters identified here can also be viewed as a flexible basis for coupling elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales.

Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues—With Examples

1987 ◽  
Vol 40 (12) ◽  
pp. 1699-1734 ◽  
Author(s):  
Millard F. Beatty
Keyword(s):  
Constitutive Equations ◽  
Mechanical Energy ◽  
Finite Elasticity ◽  
Physical Nature ◽  
Decomposition Theorem ◽  
Elastic Stability ◽  
Biological Tissues ◽  
Field Equations ◽  
Energy Principle ◽  
Hyperelastic Materials

This is an introductory survey of some selected topics in finite elasticity. Virtually no previous experience with the subject is assumed. The kinematics of finite deformation is characterized by the polar decomposition theorem. Euler’s laws of balance and the local field equations of continuum mechanics are described. The general constitutive equation of hyperelasticity theory is deduced from a mechanical energy principle; and the implications of frame invariance and of material symmetry are presented. This leads to constitutive equations for compressible and incompressible, isotropic hyperelastic materials. Constitutive equations studied in experiments by Rivlin and Saunders (1951) for incompressible rubber materials and by Blatz and Ko (1962) for certain compressible elastomers are derived; and an equation characteristic of a class of biological tissues studied in primary experiments by Fung (1967) is discussed. Sample applications are presented for these materials. A balloon inflation experiment is described, and the physical nature of the inflation phenomenon is examined analytically in detail. Results for the different materials are compared. Two major problems of finite elasticity theory are discussed. Some results concerning Ericksen’s problem on controllable deformations possible in every isotropic hyperelastic material are outlined; and examples are presented in illustration of Truesdell’s problem concerning analytical restrictions imposed on constitutive equations. Universal relations valid for all compressible and incompressible, isotropic materials are discussed. Some examples of non-uniqueness, including that of a neo-Hookean cube subject to uniform loads over its faces, are described. Elastic stability criteria and their connection with uniqueness in the theory of small deformations superimposed on large deformations are introduced, and a few applications are mentioned. Some previously unpublished results are presented throughout.

Implementing Stretch-Based Strain Energy Functions in Large Coupled Axial and Torsional Deformations of Functionally Graded Cylinder

2019 ◽  
Vol 11 (04) ◽  
pp. 1950039 ◽  
Author(s):  
Arash Valiollahi ◽  
Mohammad Shojaeifard ◽  
Mostafa Baghani
Keyword(s):  
Finite Element ◽  
Hoop Stress ◽  
Constitutive Models ◽  
Elastic Solid ◽  
Functionally Graded ◽  
Deformation Tensor ◽  
Element Analysis ◽  
Hyperelastic Materials ◽  
Ogden Model ◽  
Exponential Power

In this paper, coupled axial and torsional large deformation of an incompressible isotropic functionally graded nonlinearly elastic solid cylinder is investigated. Utilizing stretch-based constitutive models, where the deformation tensor is non-diagonal is complex. Hence, an analytical approach is presented for combined extension and torsion of functionally graded hyperelastic cylinder. Also, finite element analysis is carried out to verify the proposed analytical solutions. The Ogden model is employed to predict the mechanical behavior of hyperelastic materials whose material parameters are function of radius in an exponential fashion. Both finite element and analytical results are in good agreement and reveal that for positive values of exponential power in material variation function, stress decreases and the rate of stress variation intensifies near the outer surface. A transition point for the hoop stress is identified, where the distribution plots regardless of the value of stretch or twist, intersect and the hoop stress alters from compressive to tensile. For the Ogden model, the torsion induced force is always compressive which means the total axial force starts from being tensile and then eventually becomes compressive i.e., the cylinder always tends to elongate on twisting.

scholarly journals A Strut Finite Element for Exact Incompressible Isotropic Hyperelastic Analysis

2018 ◽  
Vol 26 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Vinicius F. Arcaro ◽  
Pietro C. Ferrazzo
Keyword(s):  
Mathematical Model ◽  
Finite Element ◽  
Total Potential Energy ◽  
Deformation Tensor ◽  
Analysis Software ◽  
Element Analysis ◽  
Hyperelastic Materials ◽  
Total Potential ◽  
Nodal Displacements ◽  
The Right

Abstract This text describes a mathematical model of a strut finite element for isotropic incompressible hyperelastic materials. The invariants of the Right Cauchy-Green deformation tensor are written in terms of nodal displacements. The equilibrium problem is formulated as an unconstrained nonlinear programming problem, where the objective function is the total potential energy of the structure and the nodal displacements are the unknowns. The constraint for incompressibility is satisfied exactly, thereby eliminating the need for a penalty function. The results of the examples calculated by the proposed mathematical model show five significant digits in agreement when compared with commercial finite element analysis software.

Influence of Boundary Conditions on Decay Rates in a Prestrained Plate1

2002 ◽  
Vol 69 (4) ◽  
pp. 515-520 ◽  
Author(s):  
B. Karp ◽  
D. Durban
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Boundary Data ◽  
Separation Of Variables ◽  
Finite Elasticity ◽  
Decay Rates ◽  
Constitutive Behavior ◽  
Hyperelastic Materials ◽  
Various Boundary Conditions ◽  
Decay Exponent

Decay of end perturbations imposed on a prestrained semi-infinite rectangular plate is investigated in the context of plane-strain incremental finite elasticity. A separation of variables eigenfunction formulation is used for the perturbed field within the plate. Numerical results for the leading decay exponent are given for three hyperelastic materials with various boundary conditions at the long faces of the plate. The study exposes a considerable sensitivity of axial decay rates to boundary data, to initial strain and to constitutive behavior. It is suggested that the results are relevant to the applicability of Saint-Venant’s principle even though the eigenfunctions are not always self-equilibrating.

A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part II: Applications

1996 ◽  
Vol 63 (4) ◽  
pp. 869-876 ◽  
Author(s):  
Jiun-Shyan Chen ◽  
Cheng-Tang Wu ◽  
Chunhui Pan
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Projection Method ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Finite Elasticity ◽  
Element Formulation ◽  
Density Functions ◽  
Hyperelastic Materials ◽  
Pressure Projection

In the first part of this paper a pressure projection method was presented for the nonlinear analysis of structures made of nearly incompressible hyperelastic materials. The main focus of the second part of the paper is to demonstrate the performance of the present method and to address some of the issues related to the analysis of engineering elastomers including the proper selection of strain energy density functions. The numerical procedures and the implementation to nonlinear finite element programs are presented. Mooney-Rivlin, Cubic, and Modified Cubic strain energy density functions are used in the numerical examples. Several classical finite elasticity problems as well as some practical engineering elastomer problems are analyzed. The need to account for the slight compressibility of rubber (finite bulk modulus) in the finite element formulation is demonstrated in the study of apparent Young’s modulus of bonded thin rubber units. The combined shear-bending deformation that commonly exists in rubber mounting systems is also analyzed and discussed.

Relationships between Stress and Strain

1997 ◽  
Author(s):  
Burak Erman ◽  
James E. Mark
Keyword(s):  
Free Energy ◽  
Closed System ◽  
Uniaxial Stress ◽  
Deformation Tensor ◽  
Stress And Strain ◽  
Stress Strain ◽  
Swelling Equilibrium ◽  
Simple Tension ◽  
Closed Systems ◽  
Link Model

In the first section of this chapter, the relationships between the Helmholtz free energy, the stress tensor, and the deformation tensor are given for uniaxial stress. These relations follow from the general discussion of stress and strain given in appendix C, and the notation and approach closely follow the classic treatment of Flory. The detailed forms of the stress-strain relations in simple tension (or compression) are given in the remaining sections of the chapter for the (1) phantom network, (2) affine network, (3) constrained-junction model, and (4) slip-link model. Results of theory are then compared with experiment. The effects of swelling on the stress-strain relations are also included in the discussion. It is to be noted that the stress-strain relations in this chapter are obtained by treating the swollen networks as closed systems. The conditions for such systems are fulfilled if solvent does not move in and out of the network during deformation. A network swollen with a nonvolatile solvent and subject to simple tension in air is an example of a closed system. The same network at swelling equilibrium and subjected to compression will exude some of the solvent under increased internal pressure, and is therefore not a closed system. For semiopen systems, such as those under compression, or, in general, networks stressed while immersed in solvent, a more general thermodynamic treatment is required. This situation will be taken up in the following chapter.

scholarly journals Stochastic isotropic hyperelastic materials: constitutive calibration and model selection

2018 ◽  
Vol 474 (2211) ◽  
pp. 20170858 ◽  
Author(s):  
L. Angela Mihai ◽  
Thomas E. Woolley ◽  
Alain Goriely
Keyword(s):  
Model Selection ◽  
Finite Elasticity ◽  
Experimental Tests ◽  
Mechanical Tests ◽  
Large Strains ◽  
Intrinsic Variability ◽  
Mean Values ◽  
Practical Applications ◽  
Hyperelastic Materials ◽  
Hyperelastic Models

Biological and synthetic materials often exhibit intrinsic variability in their elastic responses under large strains, owing to microstructural inhomogeneity or when elastic data are extracted from viscoelastic mechanical tests. For these materials, although hyperelastic models calibrated to mean data are useful, stochastic representations accounting also for data dispersion carry extra information about the variability of material properties found in practical applications. We combine finite elasticity and information theories to construct homogeneous isotropic hyperelastic models with random field parameters calibrated to discrete mean values and standard deviations of either the stress–strain function or the nonlinear shear modulus, which is a function of the deformation, estimated from experimental tests. These quantities can take on different values, corresponding to possible outcomes of the experiments. As multiple models can be derived that adequately represent the observed phenomena, we apply Occam’s razor by providing an explicit criterion for model selection based on Bayesian statistics. We then employ this criterion to select a model among competing models calibrated to experimental data for rubber and brain tissue under single or multiaxial loads.

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