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.

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.

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.

Description of Large Deformation Problem Using Means of Visual Strain

2013 ◽  
Vol 275-277 ◽  
pp. 16-22
Author(s):  
You Liang Xu
Keyword(s):  
Constitutive Equation ◽  
Large Deformation ◽  
Strain Tensor ◽  
Visual Image ◽  
Linear Strain ◽  
Deformation Problem ◽  
Practical Engineering ◽  
Large Deformation Problem ◽  
Strain Tensors ◽  
Linear Strain Tensor

The constitutive equation of large deformation problem is closely related to geometric description. Nowadays, linear strain tensor is no longer unsuitable to describe large deformation. However, the existing non-linear strain tensors have complicated forms as well as no apparent geometric or physical meaning. While, the increment method is used to solve, however, convergence and efficiency are low sometimes. Thus the idea of visual strain tensor is proposed, with distinct meaning and visual image. Beside, it is likely to be used in engineering measurement, and it can be connected with measured constitutive equation directly. Thus this research provides a new idea and method for solving large-deformation problems in practical engineering.

A Corotational Elastoplastic Formulation With Subloading Surface Model for Hardening Materials

Volume 1 ◽  
2004 ◽  
Author(s):  
Ali Reza Saidi ◽  
Koichi Hashiguchi
Keyword(s):  
Constitutive Model ◽  
Strain Rate ◽  
Simple Shear ◽  
Deformation Theory ◽  
Elastoplastic Deformation ◽  
Strain Tensor ◽  
Surface Model ◽  
Strain Rate Tensor ◽  
Hencky Strain ◽  
Stress Components

In this paper a corotational constitutive model for the large elastoplastic deformation of hardening materials using subloading surface model is formulated. This formulation is obtained by refining the large deformation theory of Naghdabadi and Saidi (2002) adopting the corotational logarithmic (Hencky) strain rate tensor and incorporating it into the subloading surface model of Hashiguchi (1980, 2003) falling within the framework of the unconventional plasticity. As an application of the proposed constitutive model, the large Elastoplastic deformation of simple shear example has been solved and the results have been compared with classical elasto-plastic model using the Hencky strain tensor. Also the effect of the choice of corotational rates on stress components has been studied.

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.

scholarly journals ON DIFFERENT DEFINITIONS OF STRAIN TENSORS IN GENERAL SHELL THEORIES OF VEKUA-AMOSOV TYPE

2021 ◽  
Vol 17 (1) ◽  
pp. 117-126
Author(s):  
Sergey Zhavoronok
Keyword(s):  
Shell Theory ◽  
Strain Tensor ◽  
Translation Vector ◽  
Dimensional Manifold ◽  
Vector Systems ◽  
Base Vector ◽  
Main Basis ◽  
Definition Of ◽  
Strain Tensors ◽  
Main Definition

Several possible definiions of strains in a general shell theory of I.N. Vekua – A.A. Amosov type are considered. The higher-order shell model is definedon a two-dimensional manifold within a set of fieldvariables of the firstkind determined by the expansion factors of the spatial vector fieldof the translation. Two base vector systems are introduced, the firs one so-called concomitant corresponds to the cotangent fibrtion of the modelling surface while the other is defind on a surface equidistant to the modelling one. The distortion appears as a two-point tensor referred to both base systems after covariant differentiationof the translation vector feld. Thus, two main definition of the strain tensor become possible, the firstone referred to the main basis whereas the second to the concomitant one. Some possible simplificationsof these tensors are considered, and the interrelation between the general theory of A.A. Amosov type and the classical ones is shown.

Easy-to-Compute Tensors With Symmetric Inverse Approximating Hencky Finite Strain and Its Rate

1998 ◽  
Vol 120 (2) ◽  
pp. 131-136 ◽  
Author(s):  
Zdeneˇk P. Bazˇant
Keyword(s):  
Finite Strain ◽  
Strain Tensor ◽  
Taylor Series Expansion ◽  
Inverse Transformation ◽  
Quadratic Term ◽  
Original Strain ◽  
Volume Changes ◽  
Hencky Strain ◽  
Deformation Rate Tensor ◽  
Deviatoric Part

It is shown that there exist approximations of the Hencky (logarithmic) finite strain tensor of various degrees of accuracy, having the following characteristics: (1) The tensors are close enough to the Hencky strain tensor for most practical purposes and coincide with it up to the quadratic term of the Taylor series expansion; (2) are easy to compute (the spectral representation being unnecessary); and (3) exhibit tension-compression symmetry (i.e., the strain tensor of the inverse transformation is minus the original strain tensor). Furthermore, an additive decomposition of the proposed strain tensor into volumetric and deviatoric (isochoric) parts is given. The deviatoric part depends on the volume change, but this dependence is negligible for materials that are incapable of large volume changes. A general relationship between the rate of the approximate Hencky strain tensor and the deformation rate tensor can be easily established.

scholarly journals The Exponentiated Hencky Strain Energy in Modeling Tire Derived Material for Moderately Large Deformations

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Giuseppe Montella ◽  
Sanjay Govindjee ◽  
Patrizio Neff
Keyword(s):  
Constitutive Model ◽  
Strain Energy ◽  
Large Deformations ◽  
Large Strain ◽  
Strain Tensor ◽  
Strain Energy Function ◽  
Cold Forging ◽  
Central Feature ◽  
Hencky Strain ◽  
Highly Nonlinear

This work presents a hyperviscoelastic model, based on the Hencky-logarithmic strain tensor, to model the response of a tire derived material (TDM) undergoing moderately large deformations. The TDM is a composite made by cold forging a mix of rubber fibers and grains, obtained by grinding scrap tires, and polyurethane binder. The mechanical properties are highly influenced by the presence of voids associated with the granular composition and low tensile strength due to the weak connection at the grain–matrix interface. For these reasons, TDM use is restricted to applications involving a limited range of deformations. Experimental tests show that a central feature of the response is connected to highly nonlinear behavior of the material under volumetric deformation which conventional hyperelastic models fail in predicting. The strain energy function presented here is a variant of the exponentiated Hencky strain energy, which for moderate strains is as good as the quadratic Hencky model and in the large strain region improves several important features from a mathematical point of view. The proposed form of the exponentiated Hencky energy possesses a set of parameters uniquely determined in the infinitesimal strain regime and an orthogonal set of parameters to determine the nonlinear response. The hyperelastic model is additionally incorporated in a finite deformation viscoelasticity framework that accounts for the two main dissipation mechanisms in TDMs, one at the microscale level and one at the macroscale level. The new model is capable of predicting different deformation modes in a certain range of frequency and amplitude with a unique set of parameters with most of them having a clear physical meaning. This translates into an important advantage with respect to overcoming the difficulties related to finding a unique set of optimal material parameters as are usually encountered fitting the polynomial forms of strain energies. Moreover, by comparing the predictions from the proposed constitutive model with experimental data we conclude that the new constitutive model gives accurate prediction.

scholarly journals BRITTLE DESTRUCTION OF INCOMPRESSIBLE MATERIAL DURING FLAT STRAIN

2019 ◽  
Vol 2 (5) ◽  
pp. 89-97
Author(s):  
Anvar Chanyshev ◽  
Olga Belousova ◽  
Olga Lukyashko
Keyword(s):  
Incompressible Material ◽  
Stress And Strain ◽  
System Of Differential Equations ◽  
Stress Strain ◽  
Uniform Compression ◽  
Strain Behavior ◽  
Destruction Zone ◽  
Strain Tensors ◽  
Brittle Destruction

In the paper stress-strain behavior of solid during flat strain in case of its volumetric incompressible behavior and ideally brittle destruction is studied. Parameters of the system of differential equations of balance and its correlations are obtained. In this case, condition of stress and strain tensors axiality is used. Boundary problem for determination of stress-strain behavior at destruction zone is formulated. As example, equations of ideally brittle out-of-limit deformation of solid in form of rectangular plate (pillar) during uniform compression are considered. It is shown that when displacement of side border of the plate is observed, it is possible to predict its destructions.

scholarly journals Hadamard’s variational formula in terms of stress and strain tensors

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Björn Gustafsson ◽  
Ahmed Sebbar
Keyword(s):  
Green Function ◽  
Stress Tensor ◽  
Strain Tensor ◽  
Variational Formula ◽  
Scalar Fields ◽  
Action Functional ◽  
Lagrangian Action ◽  
The Green Function ◽  
Strain Tensors ◽  
Hadamard Variational Formula

AbstractStarting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

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