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.

scholarly journals A Method of Determining the Strain-Rate Tensor at the Surface of a Glacier

1959 ◽  
Vol 3 (25) ◽  
pp. 409-419 ◽  
Author(s):  
J. F. Nye
Keyword(s):  
Strain Rate ◽  
Least Squares Method ◽  
Strain Tensor ◽  
Rate Of Change ◽  
Principal Strain ◽  
Strain Rate Tensor ◽  
Strain Rates ◽  
Principal Stresses ◽  
Independent Components ◽  
Good Agreement

AbstractThe rate of strain tensor at a point on the surface of a glacier may be determined by setting up a number of stakes in a pattern and measuring the rate of change of the distances between them. A suitable pattern consists of four stakes at the corners of a square with one stake at the center. Five such patterns were used on Austerdalsbreen, Norway, in August 1956. The problem is to deduce the best values of the 3 independent components of the strain-rate tensor from the 8 measured quantities, and, for this purpose, a least-squares method, invented by Bond for the analogous problem in crystal physics, is used. The principal strain-rates are found to within about ±0.005 yr.−1and their directions relative to the stake system to within about ±0.5°. The directions and magnitudes of the principal stresses are then deduced from Glen’s flow law and a suitable general theory. The directions of the principal strain-rates are in good agreement with the directions of the crevasses, but the experiment is inconclusive on the question of the magnitude of the stress needed to form a crevasse.

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 A Method of Determining the Strain-Rate Tensor at the Surface of a Glacier

1959 ◽  
Vol 3 (25) ◽  
pp. 409-419 ◽  
Author(s):  
J. F. Nye
Keyword(s):  
Strain Rate ◽  
Least Squares Method ◽  
Strain Tensor ◽  
Rate Of Change ◽  
Principal Strain ◽  
Strain Rate Tensor ◽  
Strain Rates ◽  
Principal Stresses ◽  
Independent Components ◽  
Good Agreement

AbstractThe rate of strain tensor at a point on the surface of a glacier may be determined by setting up a number of stakes in a pattern and measuring the rate of change of the distances between them. A suitable pattern consists of four stakes at the corners of a square with one stake at the center. Five such patterns were used on Austerdalsbreen, Norway, in August 1956. The problem is to deduce the best values of the 3 independent components of the strain-rate tensor from the 8 measured quantities, and, for this purpose, a least-squares method, invented by Bond for the analogous problem in crystal physics, is used. The principal strain-rates are found to within about ±0.005 yr.−1 and their directions relative to the stake system to within about ±0.5°. The directions and magnitudes of the principal stresses are then deduced from Glen’s flow law and a suitable general theory. The directions of the principal strain-rates are in good agreement with the directions of the crevasses, but the experiment is inconclusive on the question of the magnitude of the stress needed to form a crevasse.

Nonlinear Plastic Modeling of Materials Based on the Generalized Strain Rate Tensor

2008 ◽  
Author(s):  
Kamyar Ghavam ◽  
Reza Naghdabadi
Keyword(s):  
Strain Rate ◽  
Simple Shear ◽  
Large Deformations ◽  
Kinematic Hardening ◽  
Flow Rule ◽  
Strain Rate Tensor ◽  
Governing Equations ◽  
Elastic Plastic ◽  
Shear Problem ◽  
Plastic Hardening

In this paper, a method for modeling of elastic-plastic hardening materials under large deformations is proposed. In this model the generalized strain rate tensor is used. Such a tensor is obtained on the basis of the method which was introduced by the authors. Based on the generalized strain rate tensor, a flow rule, a Prager-type kinematic hardening equation and a kinematic decomposition is proposed and the governing equations for such materials are obtained. As an application, the governing equations for the simple shear problem are solved and some results are compared with those in the literature.

scholarly journals 3D Muscle Deformation Mapping at Submaximal Isometric Contractions: Applications to Aging Muscle

2020 ◽  
Vol 11 ◽  
Author(s):  
Vadim Malis ◽  
Usha Sinha ◽  
Shantanu Sinha
Keyword(s):  
Extracellular Matrix ◽  
Strain Rate ◽  
Shear Strain ◽  
Compressive Strain ◽  
Strain Tensor ◽  
Medial Gastrocnemius ◽  
Strain Rate Tensor ◽  
Shear Strain Rate ◽  
Velocity Mapping ◽  
Lateral Transmission

3D strain or strain rate tensor mapping comprehensively captures regional muscle deformation. While compressive strain along the muscle fiber is a potential measure of the force generated, radial strains in the fiber cross-section may provide information on the material properties of the extracellular matrix. Additionally, shear strain may potentially inform on the shearing of the extracellular matrix; the latter has been hypothesized as the mechanism of lateral transmission of force. Here, we implement a novel fast MR method for velocity mapping to acquire multi-slice images at different % maximum voluntary contraction (MVC) for 3D strain mapping to explore deformation in the plantar-flexors under isometric contraction in a cohort of young and senior subjects. 3D strain rate and strain tensors were computed and eigenvalues and two invariants (maximum shear and volumetric strain) were extracted. Strain and strain rate indices (contractile and in-plane strain/strain rate, shear strain/strain rate) changed significantly with %MVC (30 and 60% MVC) and contractile and shear strain with age in the medial gastrocnemius. In the soleus, significant differences with age in contractile and shear strain were seen. Univariate regression revealed weak but significant correlation of in-plane and shear strain and shear strain rate indices to %MVC and correlation of contractile and shear strain indices to force. The ability to map strain tensor components provides unique insights into muscle physiology: with contractile strain providing an index of the force generated by the muscle fibers while the shear strain could potentially be a marker of lateral transmission of force.

The influence of strain rate on the cavitation time and deformation of an underwater impulsively rectangular plate

2021 ◽  
pp. 030932472110208
Author(s):  
Habib Ramezannejad Azarboni ◽  
Abolfazl Darvizeh
Keyword(s):  
Constitutive Equation ◽  
Strain Rate ◽  
Rectangular Plate ◽  
Total Pressure ◽  
Elastoplastic Deformation ◽  
Underwater Explosion ◽  
Strain Rate Effect ◽  
Linear Solution ◽  
Explosion Load ◽  
Explosive Forming

The effect of strain rate on the cavitation time and elastoplastic deformation of steel rectangular plate subjected to underwater explosion load is analytically and numerically investigated in this study. At the cavitation time, the total pressure of the explosion is eliminated so that the cavitation time plays a significant role in the elastoplastic deformation of underwater explosive forming of plate. Taking into account the strain rate effect, the Cowper-Symond constitutive equation of mild steel is employed. Exact linear solution using the Eigen function and numerical linear and nonlinear solution using finite difference method (FDM) of dynamic response of impulsively plate is obtained. Implementing the linear work hardening, the stress, strain, displacement, and velocity in any steps of loading are calculated. The time of cavitation can be recognized in elastic or plastic regimes by applying the Cowper-Symond constitutive equation. Considering the strain rate influence, the effects of charge mass and standoff are investigated to occur of cavitation and time dependent deflection and velocity of a rectangular plate.

Constitutive Modeling for Elevated Temperature Flow Behavior of ZHMn34-2-2-1 Manganese Brass

2017 ◽  
Vol 872 ◽  
pp. 30-37
Author(s):  
Meng Han Wang ◽  
Kang Wei ◽  
Xiao Juan Li
Keyword(s):  
Constitutive Model ◽  
Strain Rate ◽  
Flow Behavior ◽  
Peak Stress ◽  
Strain Rates ◽  
Hollomon Parameter ◽  
Prediction Ability ◽  
Absolute Relative Error ◽  
Deformation Behaviors ◽  
State Stress

The hot compressive deformation behaviors of ZHMn34-2-2-1 manganese brass are investigated on Thermecmastor-Z thermal simulator over wide processing domain of temperatures (923K-1073K) and strain rates (0.01s-1-10s-1). The true stress-strain curves exhibit a single peak stress, after which the stress monotonously decreases until a steady state stress occurs, indicating a typical dynamic recrystallization. A revised constitutive model coupling flow stress with strain, strain rate and deformation temperature is established with the material constants expressed by polynomial fitting of strain. Moreover, better prediction ability of the constitutive model is achieved by implementation of a simple approach for modified the Zener-Hollomon parameter considering the compensation of strain rate and temperature increment. By comparing the predicted and experimented values, the correlation coefficient and mean absolute relative error are 0.997 and 2.363%, respectively. The quantitative statistical results indicate that the proposed constitutive model can precisely characterize the hot deformation behavior of ZHMn34-2-2-1 manganese brass.

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.

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