Eulerian Framework for Inelasticity Based on the Jaumann Rate and a Hyperelastic Constitutive Relation—Part I: Rate-Form Hyperelasticity

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Amin Eshraghi ◽  
Katerina D. Papoulia ◽  
Hamid Jahed
Keyword(s):  
Material Model ◽  
Elasticity Tensor ◽  
Constitutive Relationship ◽  
Deformation Tensor ◽  
Isotropic Case ◽  
Jaumann Rate ◽  
Principal Coordinates ◽  
Corotational Rate ◽  
Hypoelastic Model ◽  
Rate Formulation

An integrable Eulerian rate formulation of finite deformation elasticity is developed, which relates the Jaumann or other objective corotational rate of the Kirchhoff stress with material spin to the same rate of the left Cauchy–Green deformation measure through a deformation dependent constitutive tensor. The proposed constitutive relationship can be written in terms of the rate of deformation tensor in the form of a hypoelastic material model. Integrability conditions, under which the proposed formulation yields (a) a Cauchy elastic and (b) a Green elastic material model are derived for the isotropic case. These determine the deformation dependent instantaneous elasticity tensor of the material. In particular, when the Cauchy integrability criterion is applied to the stress-strain relationship of a hyperelastic material model, an Eulerian rate formulation of hyperelasticity is obtained. This formulation proves crucial for the Eulerian finite strain elastoplastic model developed in part II of this work. The proposed model is formulated and integrated in the fixed background and extends the notion of an integrable hypoelastic model to arbitrary corotational objective rates and coordinates. Integrability was previously shown for the grade-zero hypoelastic model with use of the logarithmic (D) rate, the spin of which is formulated in principal coordinates. Uniform deformation examples of rectilinear shear, closed path four-step loading, and cyclic elliptical loading are presented. Contrary to classical grade-zero hypoelasticity, no shear oscillation, elastic dissipation, or ratcheting under cyclic load is observed when the simple Zaremba–Jaumann rate of stress is employed.

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.

Study on Formability about ME20M Magnesium Alloy Sheet

2010 ◽  
Vol 136 ◽  
pp. 23-27
Author(s):  
Ting Fang Zhang ◽  
Shi Kun Xie
Keyword(s):  
Numerical Simulation ◽  
Magnesium Alloy ◽  
Deep Drawing ◽  
Material Model ◽  
Alloy Sheet ◽  
Simulation Software ◽  
Constitutive Relationship ◽  
Magnesium Alloy Sheet ◽  
Forming Temperature ◽  
Made In

Warm forming of magnesium alloy sheet has attracted more and more attention in recent years. Mechanics tension test has been made in this paper in order to study the constitutive relationship of ME20M magnesium alloy sheet at different temperatures and strain rates. And a constitutive relationship which includes a softening factor has been put forward. Warm deep drawing experiment and numerical simulation on ME20M magnesium alloy sheet have been made in which the attention was focused on the forming temperature. The results showed that the limit deep drawing height of ME20M magnesium alloy sheet can be dramatically improved as the temperature goes up, especially when the temperature was over about 250°C. Simultaneity, it is feasible and effective to add a material model into numerical simulation software by user subroutine.

Reduced Material Model of Composite Laminates for 3D Finite Element Analysis

2014 ◽  
Author(s):  
Goldy Kumar ◽  
Vadim Shapiro
Keyword(s):  
Finite Element ◽  
Structural Analysis ◽  
Composite Laminates ◽  
Substantial Reduction ◽  
Computational Cost ◽  
Material Model ◽  
Basis Functions ◽  
Constitutive Relationship ◽  
Element Analysis ◽  
3D Fea

Laminate composites are widely used in automotive, aerospace, medical, and increasingly in consumer industries, due to their reduced weight, superior structural properties and cost-effectiveness. However, structural analysis of complex laminate structures remains challenging. 2D finite element methods based on plate and shell theories may be accurate and efficient, but they generally do not apply to the whole structure, and require identification and preprocessing (dimensional reduction) of the regions where the underlying assumptions hold. Differences in and limitations of theories for thin/thick plates and shells further complicate modeling and simulation of composites. Fully automated structural analysis using 3D elements with sufficiently high order basis functions is possible in principle, but is rarely practiced due to the significant increase in computational integration cost in the presence of a large number of laminate plies. We propose to replace the actual layup of the laminate structure by a simplified material model, allowing for a substantial reduction of the computational cost of 3D FEA. The reduced model, under the usual assumptions made in lamination theory, has the same constitutive relationship as the corresponding 2D plate model of the original laminate, but requires only a small fraction of computational integration costs in 3D FEA. We describe implementation of 3D FEA using the reduced material model in a meshfree system using second order B-spline basis functions. Finally, we demonstrate its validity by showing agreement between computed and known results for standard problems.

scholarly journals Experimental and Numerical Investigations of High-Speed Projectile Impacts on 7075-T651 Aluminum Plates

Materials ◽  
2019 ◽  
Vol 12 (17) ◽  
pp. 2736 ◽  
Author(s):  
Jae-Wook Jung ◽  
Sang Eon Lee ◽  
Jung-Wuk Hong
Keyword(s):  
High Speed ◽  
High Strain ◽  
Damage Model ◽  
Computational Cost ◽  
Material Model ◽  
Constitutive Relationship ◽  
Mesh Dependency ◽  
Element Size ◽  
Aluminum Plates ◽  
The Relationship

Simulation of the material failure under high strain rate conditions is one of the most difficult problems in the finite element analyses, and many researchers have tried to understand and reproduce dynamic material fracture. In this study, we investigate a failure criterion that minimizes the mesh dependency at high strain rates and incorporates the criterion into the Johnson-Cook constitutive relationship by developing a user-defined material model. Impact tests were performed using a gas-gun system in order to investigate the response of the 7075-T651 aluminum plate in high-speed collision. On the other hand, numerical simulations are carried out by considering various element sizes and the relationship between element size and failure strain is inversely obtained using numerical results. By accommodating the relationship into the damage model and implementing in the user-defined material model, mesh dependency is significantly reduced, and sufficient accuracy is achieved with alleviated computational cost than the existing damage model. This study suggests an element size-dependent damage criterion that is applicable for impact simulation and it is expected that the criterion is useful to obtain accurate impact responses with a small computational cost.

Numerical simulation of cracking in concrete using damage mechanics

2021 ◽  
Author(s):  
Todor Zhelyazov ◽  
Radan Ivanov
Keyword(s):  
Finite Element ◽  
Damage Mechanics ◽  
Elasticity Tensor ◽  
General Purpose ◽  
Damage Variable ◽  
Constitutive Relationship ◽  
Self Healing ◽  
Structural Element ◽  
Healing Agent ◽  
Strain Relationship

<p>Damage Mechanics is employed to simulate the crack initiation and propagation in concrete structural elements. To this end, the stress-strain relationship for concrete is modified by introducing a damage variable which affects the elasticity tensor. The damage-based constitutive relationship defined for concrete is integrated into a general-purpose finite element code. The damage accumulated in each finite element is quantified throughout the loading history. Finite elements in which a critical value of the damage variable is reached are deactivated. The volume of cracks can also be approximately evaluated. The relative amount of cracks is considered by the authors to be an important characteristic of the material in the context of smart, self-healing concrete. It provides valuable information for the design of a smart structural element, namely in optimizing the amount and pattern of placement of the healing agent.</p>

Numerical Simulation on Drop Test of the Conical Shell

2010 ◽  
Vol 44-47 ◽  
pp. 2341-2345
Author(s):  
Yi Xia Yan ◽  
Wei Fang Xu ◽  
Xi Cheng Huang ◽  
Gang Chen ◽  
Zhi Ming Hao
Keyword(s):  
Numerical Simulation ◽  
Failure Mode ◽  
Conical Shell ◽  
Material Model ◽  
Drop Test ◽  
Constitutive Relationship ◽  
Plastic Behavior ◽  
Deformation And Failure ◽  
Conical Shells ◽  
Nonlinear Transient

The drop test for the thin 2A12 conical shells was developed on a drop hammer. The dynamical responses, typical deformation histories and failure mode of the shells were presented. The drop impact response of the thin conical shells were numerically simulated and analyzed in detail by using the explicit, nonlinear transient dynamic code, LS-DYNA. In the calculation, the material plastic behavior of the conical shells was described by Johnson-Cook constitutive relationship, which includes the effects of the strain rate, strains harden and temperatures soften. The deformation and failure model of the conical shell obtained from the numerical simulation were consistent well with the experiment. It was shown that the calculation method, material model and the failure criterion were available. The test and numerical simulation results were all shown that the failure mode was different because of the different drop height.

Viscoelastic modelling of rotor—shaft systems using an operator-based approach

2010 ◽  
Vol 225 (1) ◽  
pp. 73-87 ◽  
Author(s):  
J K Dutt ◽  
H Roy
Keyword(s):  
Equations Of Motion ◽  
Broad Band ◽  
Viscoelastic Model ◽  
Stability Limit ◽  
Material Model ◽  
Time Response ◽  
Constitutive Relationship ◽  
Rotor Shaft ◽  
Forcing Function ◽  
Shaft System

Damping exists in every material in varying degrees, so materials in general are viscoelastic in nature. Energy storage, as well as dissipation in varying degrees, accompanies every time-varying deformation, with the effect that stress and strain in a material get out of phase. This work presents the development of equations of motion of a rotor—shaft system with a viscoelastic rotor after discretizing the system into finite elements. Subsequently, these equations are used to study the dynamics of the rotor—shaft system in terms of stability limit of spin speed and time response of a disc as a result of unbalance. The primary inspiration for a viscoelastic model arises from the need to capture the influence of broad band spectral behaviour of rotor—shaft materials, primarily polymers and polymer composites, which are principally the materials of light rotors, on the dynamics of rotor—shaft system. For this, the material constitutive relationship has been represented by a differential time operator. Use of operators enables one to consider general linear viscoelastic behaviours, represented in the time domain by multi-element (three, four, or higher elements) spring—dashpot models or internal variable models, for which, in general, instantaneous stress and its derivatives are proportional to instantaneous strain and its derivatives. Again such representation is fairly generic, in a sense that the operator may be suitably chosen according to the material model to obtain the equations of motion of a rotor—shaft system. The equations so developed may be easily used to find the stability limit speed of a rotor—shaft system as well as the time response when the rotor—shaft system is subjected to any dynamic forcing function.

Effect of damage on the free radial oscillations of an incompressible isotropic tube

2017 ◽  
Vol 23 (8) ◽  
pp. 1177-1205 ◽  
Author(s):  
Kriti Arya ◽  
Somnath Sarangi
Keyword(s):  
Surgical Procedure ◽  
Arterial Wall ◽  
Inflammatory Diseases ◽  
Cylindrical Tube ◽  
Material Model ◽  
Finite Amplitude ◽  
Damage Function ◽  
Parent Material ◽  
Deformation Tensor ◽  
Thin Walled

The effect of damage on the finite-amplitude, free radial oscillations of an arbitrary incompressible, isotropic, homogeneous cylindrical tube is investigated. Pressure is applied to the inner and outer surfaces of the damaged cylinder and constrained from both ends. A purely radial motion is observed when the pressure is removed. The corresponding equation of motion is obtained, incorporating the effect of damage. A simple exponential front factor damage function is introduced in the tube problem. The damage function is a function of the first invariant of the left Cauchy–Green deformation tensor and is dependent on its maximum previous ever value. It is found that the period of oscillation for a thin-walled neo-Hookean membrane varies with the damage function. In contrast, the respective period for an undamaged neo-Hookean membrane is a constant. The described study may help in the surgical procedure of angioplasty, performed during inflammatory diseases such as atherosclerosis. During angioplasty, owing to inflation of the balloon, the arterial wall is damaged; this study may help to gain more insight on the surgical procedure. Both thick- and thin-walled analysis of the damaged cylindrical tube are performed and compared with the undamaged case. Several results are inferred and illustrated graphically for two types of parent material model, namely Gent and neo-Hookean.

scholarly journals Consistent numerical implementation of hypoelastic constitutive models

2020 ◽  
Vol 71 (5) ◽  
Author(s):  
Mehrdad Palizi ◽  
Salvatore Federico ◽  
Samer Adeeb
Keyword(s):  
Constitutive Models ◽  
Elasticity Tensor ◽  
Element Analysis ◽  
Kirchhoff Stress ◽  
User Subroutine ◽  
Jaumann Rate ◽  
The Finite Element Analysis ◽  
Kirchhoff Stress Tensor ◽  
Objective Stress Rate ◽  
Objective Stress

Abstract In hypoelastic constitutive models, an objective stress rate is related to the rate of deformation through an elasticity tensor. The Truesdell, Jaumann, and Green–Naghdi rates of the Cauchy and Kirchhoff stress tensors are examples of the objective stress rates. The finite element analysis software ABAQUS uses a co-rotational frame which is based on the Jaumann rate for solid elements and on the Green–Naghdi rate for shell and membrane elements. The user subroutine UMAT is the platform to implement a general constitutive model into ABAQUS, but, in order to update the Jacobian matrix in UMAT, the model must be expressed in terms of the Jaumann rate of the Kirchhoff stress tensor. This study aims to formulate and implement various hypoelastic constitutive models into the ABAQUS UMAT subroutine. The developed UMAT subroutine codes are validated using available solutions, and the consequence of using wrong Jacobian matrices is elucidated. The UMAT subroutine codes are provided in the “Electronic Supplementary Material” repository for the user’s consideration.

The Nonlinear Viscoelastic Behavior for Asphalt Mixture Materials

2011 ◽  
Vol 255-260 ◽  
pp. 3268-3271 ◽  
Author(s):  
Yong Ye ◽  
Yi Zhou Cai
Keyword(s):  
Asphalt Mixture ◽  
Viscoelastic Behavior ◽  
Least Square Method ◽  
Material Model ◽  
Least Square ◽  
Constitutive Relationship ◽  
Nonlinear Fitting ◽  
Fitting Procedure ◽  
Nonlinear Viscoelastic ◽  
Stress Dependent

Compressive behavior of asphalt mixture is studied in creep and strain recovery tests observing large nonlinear viscoelastic strains. The nonlinear viscoelastic material model for asphalt mixture is presented, based on a modified version of Schapery’s constitutive relationship. For the description of the nonlinear viscoelastic response of the material, simple creep and recovery tests for different stress levels were executed. An analytical method and a nonlinear fitting procedure by the least square method are developed to determine nonlinear viscoelastic stress dependent parameters. Constant stress creep testing were also performed to validate the developed material model. The model successfully describes the main features for asphalt mixture and shows good agreement with test data within the considered stress range.

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