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 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 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.

A note on the wave equation for a class of constitutive relations for nonlinear elastic bodies that are not Green elastic

2016 ◽  
Vol 23 (2) ◽  
pp. 148-158 ◽  
Author(s):  
R Meneses ◽  
O Orellana ◽  
R Bustamante
Keyword(s):  
Wave Equation ◽  
Stress Tensor ◽  
Constitutive Relations ◽  
Strain Tensor ◽  
Nonlinear Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Travelling Wave ◽  
Elastic Bodies ◽  
New Type

A class of constitutive relations for elastic bodies has been proposed recently, where the linearized strain tensor is expressed as a nonlinear function of the stress tensor. Considering this new type of constitutive equation, the initial boundary value problem for such elastic bodies has been expressed only in terms of the stress tensor. In this communication, this new type of nonlinear wave equation is studied for the case of a one-dimensional straight bar. Conditions for the existence of the travelling wave solutions are given and some self-similar solutions are obtained.

Corrigendum: Direct determination of stresses from the stress equations of motion and wave propagation for a new class of elastic bodies

2017 ◽  
Vol 25 (3) ◽  
pp. 866-868 ◽  
Author(s):  
Roger Bustamante
Keyword(s):  
Exact Solution ◽  
Wave Propagation ◽  
Equations Of Motion ◽  
Direct Determination ◽  
One Dimensional ◽  
New Class ◽  
Elastic Bodies ◽  
Wave Propagation Problem ◽  
New Type

In this note it is shown there is an error in the analysis used to obtain an exact solution for the wave propagation problem, for the case of a one-dimensional body considering a new type of constitutive equation.

The Physically Nonlinear Model of an Elastic Material and Its Identification

2019 ◽  
Vol 11 (07) ◽  
pp. 1950064
Author(s):  
Alexey Markin ◽  
Marina Sokolova ◽  
Dmitrii Khristich ◽  
Yuri Astapov
Keyword(s):  
Linear Part ◽  
Strain Tensor ◽  
Volumetric Strain ◽  
Nonlinear Effects ◽  
Incompressible Material ◽  
Material Constants ◽  
Hencky Strain ◽  
New Variant ◽  
Kirchhoff Stress Tensor ◽  
Strain Tensors

This work is devoted to the new variant of relations between the energetically conjugated Hencky strain tensor and corotational Kirchhoff stress tensor. The elastic energy is represented as a third-order polynomial of the Hencky tensor containing five material constants. Unlike the Almansi tensor in the Murnaghan model, the Hencky tensor allows assigning a clear physical meaning to material constants. Linear part of the constitutive relation represents the Hencky model and contains the bulk modulus and the shear modulus. The two extra constants express nonlinear effects at a purely volumetric strain and a purely isochoric strain, whereas the third constant takes into account the possible deviation from the similarity of the deviators of the Hencky stress and strain tensors. The resulting relations are naturally generalized for incompressible materials. In this case, the overall number of constants decreases from five to two. The designed test unit was used for a compression test of prismatic specimens made of incompressible material. The proposed version of the relations is in good agreement with the experimental data on the compression of rubber samples.

Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
Wooseok Ji ◽  
Anthony M. Waas ◽  
Zdeněk P. Bažant
Keyword(s):  
Finite Element ◽  
Fiber Composite ◽  
Strain Tensor ◽  
Stress And Strain ◽  
Hencky Strain ◽  
Tangential Stiffness ◽  
Jaumann Rate ◽  
Lagrangian Strain ◽  
Strain Formulation ◽  
Accurate Expression

Many finite element programs including standard commercial software such as ABAQUS use an incremental finite strain formulation that is not fully work-conjugate, i.e., the work of stress increments on the strain increments does not give a second-order accurate expression for work. In particular, the stress increments based on the Jaumann rate of Kirchhoff stress are work-conjugate with the increments of the Hencky (logarithmic) strain tensor but are paired in many finite element programs with the increments of Green’s Lagrangian strain tensor. Although this problem was pointed out as early 1971, a demonstration of its significance in realistic situations has been lacking. Here it is shown that, in buckling of compressed highly orthotropic columns or sandwich columns that are very “soft” in shear, the use of such nonconjugate stress and strain increments can cause large errors, as high as 100% of the critical load, even if the strains are small. A similar situation may arise when severe damage such as distributed cracking leads to a highly anisotropic tangential stiffness matrix, or when axial cracks between fibers severely weaken a uniaxial fiber composite or wood. A revision of these finite element programs is advisable, and will in fact be easy—it will suffice to replace the Jaumann rate with the Truesdell rate. Alternatively, the Green’s Lagrangian strain could be replaced with the Hencky strain.

Theoretical Studies of the Physical Properties of Rodlike and Related Polymers

1988 ◽  
Vol 134 ◽  
Author(s):  
W. J. Welsh ◽  
J. E. Mark ◽  
Y. Yang ◽  
G. P. Das
Keyword(s):  
High Performance ◽  
Liquid Crystalline ◽  
Theoretical Work ◽  
Environmental Resistance ◽  
New Class ◽  
Protonated Forms ◽  
New Type ◽  
Electrooptical Properties ◽  
Cis And Trans ◽  
Strong Acids

ABSTRACTThis review focuses on a new type of para-catenated aromatic polymer being used in the preparation of high-performance films and fibers of exceptional strength, thermal stability, and environmental resistance, including inertness to essentially all common solvents. Polymers of this type include the cis- and trans-poly(p-phenylene benzobisoxazole) (PBO), the cis- and trans-forms of the corresponding poly(pphenylene benzobisthiazole) (PBT), and the structurally similar poly(5,5ʹ-bibenzoxazole-2.2ʹ-diyl-l,3-phenylene) (AAPBO) and poly(2,5-benzoxazole) (ABPBO) and their sulfurcontaining analogues. Because of their rigidity, these polymers become highly oriented in solution and some display liquid crystalline behavior. The purpose of this paper is to summarize the authorsʹ theoretical work on the structures, conformational energies, intermolecular interactions, electronic properties, electrical conductivity, and electrooptical properties of these chains, including, in some cases, the so-called articulated forms and the protonated forms known to exist in strong acids. The emphasis is on how such studies provide a molecular understanding of the unusual properties and processing characteristics of this new class of materials.

A New Class of Oxide Superconductor (Nd, Ce, Sr)2CuO4

1989 ◽  
Vol 156 ◽  
Author(s):  
E. Takayama-Muromachi
Keyword(s):  
Crystal Structure ◽  
Crystal Chemistry ◽  
Superconducting Phase ◽  
Type Structure ◽  
Oxide Superconductor ◽  
New Class ◽  
System 2 ◽  
New Type ◽  
System 1 ◽  
Tetragonal Cell

ABSTRACTSince the discovery of the high-Tc superconductor in the La-Ba-Cu-O system [1], a great deal of experimental and theoretical effort have been made to clarify the nature of the Cu-based oxides. In order to elucidate mechanism of the high-Tc superconductivity, discovery of a new type of superconductor is no doubt of great importance. Recently, Akimitsu et al. found a new oxide superconductor in the Nd-Ce-Sr-Cu-O system [2]. Soon after their discovery, the superconducting phase was isolated and identified [3]. It has a tetragonal cell with space group P4/nmm and has a structure closely related to but different from the K2NiF4− or T'-Nd2CuO4− -type structure. Although, Tc of the Nd-Ce-Sr-Cu oxide is not so high (ca. 20 K) compared with the 1–2–3 or Bi(Tl)-based superconductors, it has aroused interest widely due to a very simple crystal structure. In this article, I will discuss superconductivity and crystal chemistry of the Nd-Ce-Sr-Cu oxide. Also, various compounds isostructural to it will be presented.

Allee’s Effect Bifurcation in Generalized Logistic Maps

2019 ◽  
Vol 29 (03) ◽  
pp. 1950039
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha
Keyword(s):  
Allee Effect ◽  
Dynamical Behavior ◽  
Complete Classification ◽  
Unimodal Maps ◽  
New Class ◽  
New Type ◽  
Local And Global Bifurcations ◽  
One Dimensional Maps ◽  
Necessary And Sufficient

This paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee’s effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.

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