Fractional Derivative Constitutive Models for Finite Deformation of Viscoelastic Materials

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Stress Tensor ◽  
Finite Deformation ◽  
Dynamical Behavior ◽  
Strain Tensor ◽  
Constitutive Models ◽  
Cauchy Stress ◽  
Viscoelastic Materials ◽  
Kirchhoff Stress Tensor ◽  
Biot Stress

A methodology to derive fractional derivative constitutive models for finite deformation of viscoelastic materials is proposed in a continuum mechanics treatment. Fractional derivative models are generalizations of the models given by the objective rates. The method of generalization is applied to the case in which the objective rate of the Cauchy stress is given by the Truesdell rate. Then, a fractional derivative model is obtained in terms of the second Piola–Kirchhoff stress tensor and the right Cauchy-Green strain tensor. Under the assumption that the dynamical behavior of the viscoelastic materials comes from a complex combination of elastic and viscous elements, it is shown that the strain energy of the elastic elements plays a fundamental role in determining the fractional derivative constitutive equation. As another example of the methodology, a fractional constitutive model is derived in terms of the Biot stress tensor. The constitutive models derived in this paper are compared and discussed with already existing models. From the above studies, it has been proved that the methodology proposed in this paper is fully applicable and effective.

Torsion of a Viscoelastic Cylinder

2000 ◽  
Vol 67 (2) ◽  
pp. 424-427 ◽  
Author(s):  
R. C. Batra ◽  
J. H. Yu
Keyword(s):  
Stress Tensor ◽  
Constitutive Relation ◽  
Linear Relation ◽  
Constitutive Relations ◽  
Strain Tensor ◽  
Time History ◽  
Cauchy Stress ◽  
Kirchhoff Stress Tensor ◽  
History Of ◽  
Energy Dissipated

Finite torsional deformations of an incompressible viscoelastic circular cylinder are studied with its material modeled by two constitutive relations. One of these is a linear relation between the determinate part of the second Piola-Kirchhoff stress tensor and the time history of the Green-St. Venant strain tensor, and the other a linear relation between the deviatoric Cauchy stress tensor and the left Cauchy-Green tensor, its inverse, and the time history of the relative Green-St. Venant strain tensor. It is shown that the response predicted by the latter constitutive relation is in better agreement with the test data, and this constitutive relation is used to compute energy dissipated during torsional oscillations of the cylinder. [S0021-8936(00)00502-X]

Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Entropy Elasticity and Generalized Maxwell Law

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Basic Assumption ◽  
Dynamical Behavior ◽  
Impact Dynamics ◽  
Viscoelastic Materials ◽  
Free Response ◽  
Scale Free ◽  
Collective Response ◽  
Elementary Constituent ◽  
Entropy Elasticity

Two types of models are proposed for describing nonlinear fractional derivative dynamical behavior of viscoelastic materials subject to impulse forces. The models are derived based on the thermodynamic elasticity in terms of entropy and on the “scale-free response of the material” under the basic assumption that the viscoelastic materials consist of stable coils of polymers, which we refer to as blobs. The blobs, which may be connected to each other by chemical bonds or physical bonds, are considered here as the elementary constituent of viscoelastic materials from which the nonlinear fractional derivative models are derived. Responses of individual blobs can determine the net collective response of the viscoelastic material to impulse forces. From the above consideration, two types of models are proposed in which the force elements or the stress elements are connected by the generalized Maxwell law.

scholarly journals Verification of Continuum Mechanics Predictions with Experimental Mechanics

Materials ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 77
Author(s):  
Cesar A. Sciammarella ◽  
Luciano Lamberti ◽  
Federico M. Sciammarella
Keyword(s):  
Stress Tensor ◽  
Continuum Mechanics ◽  
Strain Tensor ◽  
Scalar Fields ◽  
Identification Problem ◽  
Theoretical Concept ◽  
Cauchy Stress ◽  
Cauchy Stress Tensor ◽  
Mathematical Functions ◽  
Eulerian Derivative

The general goal of the study is to connect theoretical predictions of continuum mechanics with actual experimental observations that support these predictions. The representative volume element (RVE) bridges the theoretical concept of continuum with the actual discontinuous structure of matter. This paper presents an experimental verification of the RVE concept. Foundations of continuum kinematics as well as mathematical functions relating displacement vectorial fields to the recording of these fields by a light sensor in the form of gray-level scalar fields are reviewed. The Eulerian derivative field tensors are related to the deformation of the continuum: the Euler–Almansi tensor is extracted, and its properties are discussed. The compatibility between the Euler–Almansi tensor and the Cauchy stress tensor is analyzed. In order to verify the concept of the RVE, a multiscale analysis of an Al–SiC composite material is carried out. Furthermore, it is proven that the Euler–Almansi strain tensor and the Cauchy stress tensor are conjugate in the Hill–Mandel sense by solving an identification problem of the constitutive model of urethane rubber.

Finite Elastic Deformation Governed by Linear Equations

1986 ◽  
Vol 53 (4) ◽  
pp. 819-820 ◽  
Author(s):  
Joseph B. Keller
Keyword(s):  
Stress Tensor ◽  
Elastic Deformation ◽  
Finite Deformation ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Deformation Gradient ◽  
Kirchhoff Stress ◽  
Kirchhoff Stress Tensor ◽  
Three Components

It is observed that the equations of motion governing finite elastic deformation are linear if and only if the Piola-Kirchhoff stress tensor is linear in the deformation gradient. Then the three components of displacement satisfy uncoupled linear equations. These equations are used to solve some problems of finite deformation.

Kelvin-Voigt versus fractional derivative model as constitutive relations for viscoelastic materials

AIAA Journal ◽  
1995 ◽  
Vol 33 (3) ◽  
pp. 547-550 ◽  
Author(s):  
Lloyd B. Eldred ◽  
William P. Baker ◽  
Anthony N. Palazotto
Keyword(s):  
Fractional Derivative ◽  
Constitutive Relations ◽  
Viscoelastic Materials ◽  
Fractional Derivative Model

A Study on a Grade-One Type of Hypo-Elastic Models

2014 ◽  
Author(s):  
S. Alireza Momeni ◽  
Mohsen Asghari
Keyword(s):  
Constitutive Models ◽  
Specific Model ◽  
Elasticity Tensor ◽  
Elastic Model ◽  
Cauchy Stress ◽  
Spin Tensor ◽  
Positive Real ◽  
Corotational Rate ◽  
Deformation Rate Tensor ◽  
Grade One

In Hypo-elastic constitutive models an objective rate of the Cauchy stress tensor is expressed in terms of the current state of the stress and the deformation rate tensor D in a way that the dependency on the latter is a homogeneously linear one. In this work, a type of grade-one hypo-elastic models (i.e. models with linear dependency of the hypo-elasticity tensor on the stress) is considered for isotropic materials based on the objective corotational rates of stress. A positive real parameter denoted by n is involved in the considered type. Different values can be selected for this parameter, each selection leads to a specific model within the class of grade-one hypo-elasticity. The spin of the associated corotational rate is also dependent on the parameter n. In the special case of n=0, the corresponding hypo-elastic model reduces to a grade-zero one with the logarithmic rate of stress; noting that this rate is a corotational rate associated with the logarithmic spin tensor. Moreover, by choosing n=2, the model reduces to a grade-one hypo-elastic model with the Jaumann rate, i.e. the corotational rate associated with the vorticity spin tensor. As case studies, the simple shear problem is investigated with utilizing the considered type of hypo-elastic models with various values for parameter n, and the curves for the stress-shear response are depicted.

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