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.

Proposed experimental tests of a theory of fine microstructure and the two-well problem

1992 ◽  
Vol 338 (1650) ◽  
pp. 389-450 ◽  
Keyword(s):  
Free Energy ◽  
Mechanical Response ◽  
Experimental Tests ◽  
Martensitic Transformations ◽  
Free Energy Function ◽  
Minimizing Sequences ◽  
Nonlinear Elastic Material ◽  
Theoretical Predictions ◽  
Fine Microstructure ◽  
Nonlinear Elastic

Predictions are made based on an analysis of a new nonlinear theory of martensitic transformations introduced by the authors. The crystal is modelled as a nonlinear elastic material, with a free-energy function that is invariant with respect to both rigid-body rotations and the appropriate crystallographic symmetries. The predictions concern primarily the two-well problem , that of determining all possible energy-minimizing deformations that can be obtained with two coherent and macroscopically unstressed variants of martensite. The set of possible macroscopic deformations obtained is completely determined by the lattice parameters of the material. For certain boundary conditions the total free energy does not attain a minimum , and the finer and finer oscillations of minimizing sequences are interpreted as corresponding to microstructure. The predictions are am enable to experimental tests. The proposed tests involve the comparison of the theoretical predictions with the mechanical response of properly oriented plates subject to simple shear.

Effect of Constitutive Parameters on Formation of a Spherical Void in a Compressible Nonlinear Elastic Material

1994 ◽  
Vol 61 (2) ◽  
pp. 395-401 ◽  
Author(s):  
Shiro Biwa ◽  
Eiji Matsumoto ◽  
Toshinobu Shibata
Keyword(s):  
Strain Energy ◽  
Finite Elasticity ◽  
Void Formation ◽  
Strain Energy Function ◽  
Elastic Materials ◽  
Spherical Void ◽  
Nonlinear Elastic Material ◽  
Nonlinear Elastic ◽  
Constitutive Parameters ◽  
Bifurcation Behavior

Void formation in materials under isotropic tension is considered within the theory of finite elasticity. This phenomenon is described as the bifurcation of the solution containing a spherical cavity from the state of homogeneous deformation of a solid hyperelastic sphere. The equation giving the bifurcation point is derived, and the critical stretch and stress are numerically calculated for a special class of compressible nonlinear elastic materials for which the strain energy function was proposed by Hill. The effects of constitutive parameters on the post-bifurcation behavior as well as on the critical stretch and stress are discussed.

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.

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.

Brain tissue elastic behavior and experimental brain compression

1988 ◽  
Vol 255 (5) ◽  
pp. R799-R805 ◽  
Author(s):  
A. Schettini ◽  
E. K. Walsh
Keyword(s):  
Brain Tissue ◽  
Mechanical Response ◽  
Fluid Pressure ◽  
Elastic Parameter ◽  
Elastic Behavior ◽  
Present Observation ◽  
Tissue Elasticity ◽  
Tissue Pressure ◽  
Brain Compression ◽  
Nonlinear Elastic

This study was designed to test the hypothesis that the progressive expansion of an extradural mass causes detectable changes in brain mechanical response properties, in particular the nonlinear elastic behavior, before any significant changes in intracranial cerebrospinal fluid pressure can be detected. In 10 chronically prepared and anesthetized dogs, incremental inflation (0.07 ml/s) of an extradural balloon caused 1) a progressive fall in the brain nonlinear elastic parameter (G0, mmHg/mm2), 2) nonsignificant changes in brain tissue elasticity (G0, mmHg/mm), 3) a disproportionate progressive rise in subpial tension, and 4) a progressive fall in local cerebral blood flow (H2 clearance), despite a modest decrease in cerebral perfusion pressure (extracranial). In previous brain compression experiments (Brain Res. 305: 141-143, 1984) we have shown that the compression site becomes compacted and stiffer (increased G0) and its nonlinear elastic parameter (G0) increases markedly. These earlier findings, coupled with the present observation of a loss in tissue nonlinearity distally to the compression site, are most likely the major mechanisms by which, with a rapidly expanding intracranial mass, tissue pressure gradients and brain displacement, including transtentorial herniation, develop.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

An Alternative Technique to Evaluate Crack Propagation Path in Hyperelastic Materials

2012 ◽  
Vol 40 (1) ◽  
pp. 42-58 ◽  
Author(s):  
R. R. M. Ozelo ◽  
P. Sollero ◽  
A. L. A. Costa
Keyword(s):  
Constitutive Model ◽  
Crack Propagation ◽  
Experimental Tests ◽  
Growth Direction ◽  
Propagation Path ◽  
J Integral ◽  
Alternative Technique ◽  
Crack Propagation Path ◽  
Hyperelastic Materials ◽  
Material Constitutive Model

Abstract REFERENCE: R. R. M. Ozelo, P. Sollero, and A. L. A. Costa, “An Alternative Technique to Evaluate Crack Propagation Path in Hyperelastic Materials,” Tire Science and Technology, TSTCA, Vol. 40, No. 1, January–March 2012, pp. 42–58. ABSTRACT: The analysis of crack propagation in tires aims to provide safety and reliable life prediction. Tire materials are usually nonlinear and present a hyperelastic behavior. Therefore, the use of nonlinear fracture mechanics theory and a hyperelastic material constitutive model are necessary. The material constitutive model used in this work is the Mooney–Rivlin. There are many techniques available to evaluate the crack propagation path in linear elastic materials and estimate the growth direction. However, most of these techniques are not applicable to hyperelastic materials. This paper presents an alternative technique for modeling crack propagation in hyperelastic materials, based in the J-Integral, to evaluate the crack path. The J-Integral is an energy-based parameter and is applicable to nonlinear materials. The technique was applied using abaqus software and compared to experimental tests.

Numerical Investigation of Ductile Fracture in Pipelines Under Complex Loading Using a Phenomenological Damage Model

2021 ◽  
Author(s):  
Iago S. Santos ◽  
Diego F. B. Sarzosa
Keyword(s):  
Ductile Fracture ◽  
Mechanical Response ◽  
Numerical Study ◽  
Damage Model ◽  
Stress Triaxiality ◽  
Numerical Procedure ◽  
Complex Loading ◽  
Experimental Tests ◽  
Compact Tension ◽  
Axial Tensile

Abstract This paper presents a numerical study on pipes ductile fracture mechanical response using a phenomenological computational damage model. The damage is controlled by an initiation criterion dependent on the stress triaxiality and the Lode angle parameter, and a post-initiation damage law to eliminate each finite element from the mesh. Experimental tests were carried out to calibrate the elastoplastic response, damage parameters and validate the FEM models. The tested geometries were round bars having smooth and notched cross-section, flat notched specimens under axial tensile loads, and fracture toughness tests in deeply cracked bending specimens SE(B) and compact tension samples C(T). The calibrated numerical procedure was applied to execute a parametric study in pipes with circumferential surface cracks subjected to tensile and internal pressure loads simultaneously. The effects of the variation of geometric parameters and the load applications on the pipes strain capacity were investigated. The influence of longitudinal misalignment between adjacent pipes was also investigated.

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.

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