scholarly journals A new type of constitutive equation for nonlinear elastic bodies. Fitting with experimental data for rubber-like materials

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210330
Author(s):  
Roger Bustamante ◽  
Kumbakonam R. Rajagopal
Keyword(s):  
Experimental Data ◽  
Constitutive Equation ◽  
Constitutive Relation ◽  
Cauchy Stress ◽  
Gibbs Potential ◽  
Hencky Strain ◽  
Biaxial Stretching ◽  
Elastic Bodies ◽  
New Type ◽  
Nonlinear Elastic

In this article, we develop a new implicit constitutive relation, which is based on a thermodynamic foundation that relates the Hencky strain to the Cauchy stress, by assuming a structure for the Gibbs potential based on the Cauchy stress. We study the tension/compression of a cylinder, biaxial stretching of a thin plate and simple shear within the context of our constitutive relation. We then compare the predictions of the constitutive relation that we develop and that of Ogden’s constitutive relation with the experiments of Treloar concerning tension/compression of a cylinder, and we show that the predictions of our constitutive relation provide a better description than Ogden’s model, with fewer material moduli.

Some Universal Solutions for a Class of Incompressible Elastic Body that is Not Green Elastic: The Case of Large Elastic Deformations

2020 ◽  
Vol 73 (2) ◽  
pp. 177-199
Author(s):  
R Bustamante
Keyword(s):  
Constitutive Equation ◽  
Strain Tensor ◽  
Elastic Deformations ◽  
Hencky Strain ◽  
New Class ◽  
Elastic Bodies ◽  
Kirchhoff Stress Tensor ◽  
New Type ◽  
Incompressible Elastic Body ◽  
Universal Solutions

Summary Some universal solutions are studied for a new class of elastic bodies, wherein the Hencky strain tensor is assumed to be a function of the Kirchhoff stress tensor, considering in particular the case of assuming the bodies to be isotropic and incompressible. It is shown that the families of universal solutions found in the classical theory of nonlinear elasticity, are also universal solutions for this new type of constitutive equation.

scholarly journals A Modified Fractional Maxwell Numerical Model for Constitutive Equation of Mn-Cu Damping Alloy

Materials ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 2020
Author(s):  
Baoquan Mao ◽  
Rui Zhu ◽  
Zhiqian Wang ◽  
Yuying Yang ◽  
Xiaoping Han ◽  
...  
Keyword(s):  
Constitutive Equation ◽  
Strain Rate ◽  
Constitutive Relation ◽  
Constant Strain Rate ◽  
Strain Curve ◽  
Maxwell Model ◽  
Nonlinear Viscoelastic ◽  
Fractional Maxwell Model ◽  
Hysteretic Characteristics ◽  
Nonlinear Elastic

To better describe its constitutive relation, we need a new constitutive equation for an important nonlinear elastic material, Mn-Cu damping alloy. In this work, we studied the nonlinear and hysteretic characteristics of the stress-strain curve of the M2052 alloy with the uniaxial cyclic tensile test with constant strain rate. The strain rate and amplitude correlations of M2052 resembled those of nonlinear viscoelastic material. Therefore, we created a new constitutive equation for the M2052 damping alloy by modifying the fractional Maxwell model, and we used the genetic algorithm to carry out numerical fitting with MATLAB. By comparing with the experimental data, we confirmed that the new constitutive equation could accurately depict the nonlinear constitutive relation and hysteretic property of the damping alloy. Taken together, this new constitutive equation for Mn-Cu damping alloy based on the fractional Maxwell model can serve as an effective tool for further studies of the constitutive relation of the Mn-Cu damping alloys.

Some topics on a new class of elastic bodies

2009 ◽  
Vol 465 (2105) ◽  
pp. 1377-1392 ◽  
Author(s):  
Roger Bustamante
Keyword(s):  
Constitutive Equation ◽  
Stress Tensor ◽  
Cauchy Stress ◽  
Cauchy Stress Tensor ◽  
Weak Formulation ◽  
New Class ◽  
Elastic Bodies ◽  
Small Deformations

In this paper, we study the problem of prescribing deformation as a function of stresses. For the particular case of small deformations, we find a weak formulation, from which we define the constitutive equation of a Green-like material, where an energy function that depends on the Cauchy stress tensor is proposed. Constraints on the deformation are studied for this new class of elastic bodies.

An implicit constitutive relation for describing the small strain response of porous elastic solids whose material moduli are dependent on the density

2021 ◽  
pp. 108128652110214
Author(s):  
KR Rajagopal
Keyword(s):  
Constitutive Relation ◽  
Constitutive Relations ◽  
Deformation Gradient ◽  
Short Note ◽  
Small Strain ◽  
Cauchy Stress ◽  
Porous Metals ◽  
Elastic Solids ◽  
Elastic Bodies ◽  
Implicit Equations

In this short note, we develop a constitutive relation that is linear in both the Cauchy stress and the linearized strain, by linearizing implicit constitutive relations between the stress and the deformation gradient that have been put into place to describe the response of elastic bodies (Rajagopal, KR. On implicit constitutive theories. Applications of Mathematics 2003; 28: 279–319), by assuming that the displacement gradient is small. These implicit equations include the classical linearized elastic constitutive approximation as well as some classes of constitutive relations that imply limiting strain in tension, as special subclasses. Moreover, the constitutive relations that are developed allow the material moduli to depend on the density; thus, they can be used to describe the response of porous materials, such as porous metals, bone, rocks, and concrete undergoing small deformations.

scholarly journals On the behaviour of spherical inclusions in a cylinder under tension loads

Ingenius ◽  
2018 ◽  
pp. 69-78
Author(s):  
Sebastian Montero Guarda ◽  
Roger Bustamante Plaza ◽  
Alejandro Ortiz Bernardin
Keyword(s):  
Large Strains ◽  
Nonlinear Functions ◽  
Surrounding Matrix ◽  
Elastic Bodies ◽  
Spherical Inclusions ◽  
Linearized Strain ◽  
New Type ◽  
Material Constitutive Relation ◽  
Nonlinear Elastic ◽  
External Loads

In the present paper the behaviour of a hyperelastic body is studied, considering the presence of one, two and more spherical inclusions, under the effect of an external tension load. The inclusions are modeled as nonlinear elastic bodies that undergo small strains. For the material constitutive relation, a relatively new type of model is used, wherein the strains (linearized strain) are assumed to be nonlinear functions of the stresses. In particular, it is used a function such that the strains are always small, independently of the magnitude of the external loads. In order to simplify the problem, the hyperelastic medium and the inclusions are modelled as axial-symmetric bodies. The finite element method is used to obtain results for these boundary value problems. The objective of using these new models for elastic bodies in the case of the inclusions is to study the behaviour of such bodies in the case of concentration of stresses, which happens near the interface with the surrounding matrix. From the results presented in this paper, it is possible to observe that despite the relatively large magnitude for the stresses, the strains for the inclusions remain small, which would be closer to the actual behaviour of real inclusions made of brittle materials, which cannot show large strains.

Viscosity Models Based on Network Theory for Polymer Melts

2010 ◽  
Vol 129-131 ◽  
pp. 1244-1247
Author(s):  
Hai Hang Xu ◽  
Lei Zhong
Keyword(s):  
Experimental Data ◽  
Kinetic Equation ◽  
Constitutive Equation ◽  
Polymer Composites ◽  
Network Theory ◽  
Polymer Melts ◽  
Structure Parameter ◽  
Viscosity Models ◽  
Model Predictions ◽  
Good Agreement

New shear and extensional viscosity models based on Fredrickson kinetic equation coupled with Dewitt constitutive equation were established to predict viscosities of polymer melts. The experimental data of 125°C LDPE and LDPE filled with 35% glass beads reported from references were compared with the model predictions. The predictions showed good agreement with the measurements. The models are simple and easy to use. Because they contain no structure parameter, they are capable to describe the viscosities of pure polymer and polymer composites.

Stury of Post-Failure Constitutive Relation Based on SMP Criterion

2013 ◽  
Vol 790 ◽  
pp. 405-409
Author(s):  
Jian Ming Zhu ◽  
Ze Xiang Wu ◽  
Qi Zhao ◽  
Chong Yang
Keyword(s):  
Residual Stress ◽  
Elastic Modulus ◽  
Constitutive Model ◽  
Constitutive Relation ◽  
Strain Softening ◽  
Friction Angle ◽  
Peak Friction Angle ◽  
Confining Pressures ◽  
Intermediate Variables ◽  
Nonlinear Elastic

In this paper, based on SMP criteria, combination of strain softening of rock material mechanics theory, the after peak friction angleφfor the intermediate variables, the residual strainεto express the after peak nonlinear elastic modulusE, and finally establish a unified non-linear constitutive model of the rock peak residual stress. Combination Xiao Guanzhuang Eastern Mine typical breakdown rock of diorite triaxial test , get stress-strain curves for different confining pressures by this model. It shows that peak constitutive relation of this study can simulate the experimental results, prove the rationality of the model.

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