scholarly journals A fourth-order gauge-invariant gradient plasticity model for polycrystals based on Kröner’s incompatibility tensor

2019 ◽  
Vol 25 (2) ◽  
pp. 129-159 ◽  
Author(s):  
François Ebobisse ◽  
Patrizio Neff
Keyword(s):  
Plastic Strain ◽  
Fourth Order ◽  
Strain Tensor ◽  
Reference Configuration ◽  
Isotropic Hardening ◽  
Gradient Plasticity ◽  
Gauge Invariant ◽  
Tensor Formula ◽  
Model Features ◽  
Plastic Strain Tensor

In this paper we derive a novel fourth-order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kröner’s incompatibility tensor [Formula: see text], where [Formula: see text] is the symmetric plastic strain tensor. Here, gauge-invariance denotes invariance under diffeomorphic reparametrizations of the reference configuration, suitably adapted to the geometrically linear setting. The model features a defect energy contribution that is quadratic in the tensor [Formula: see text] and it contains isotropic hardening based on the rate of the plastic strain tensor [Formula: see text]. We motivate the new model by introducing a novel rotational invariance requirement in gradient plasticity, which we call micro-randomness, suitable for the description of polycrystalline aggregates on a mesoscopic scale and not coinciding with classical isotropy requirements. This new condition effectively reduces the increments of the non-symmetric plastic distortion [Formula: see text] to their symmetric counterpart [Formula: see text]. In the polycrystalline case, this condition is a statement about insensitivity to arbitrary superposed grain rotations. We formulate a mathematical existence result for a suitably regularized non-gauge-invariant model. The regularized model is rather invariant under reparametrizations of the reference configuration including infinitesimal conformal mappings.

scholarly journals On the Plastic and Viscoplastic Constitutive Equations—Part I: Rules Developed With Internal Variable Concept

1983 ◽  
Vol 105 (2) ◽  
pp. 153-158 ◽  
Author(s):  
J. L. Chaboche ◽  
G. Rousselier
Keyword(s):  
Plastic Strain ◽  
Kinematic Hardening ◽  
Strain Tensor ◽  
Internal Variable ◽  
Cyclic Behavior ◽  
Internal Variables ◽  
Plastic Behavior ◽  
Hardening Rules ◽  
Plastic Strain Tensor ◽  
Natural Description

The description of monotonic and cyclic behavior of material is possible by generalizing the internal stress concept by means of a set of internal variables. In this paper the classical isotropic and kinematic hardening rules are briefly discussed, using present plastic strain tensor and cumulated plastic strain as hardening variables. Some additional internal variables are then proposed, giving rise to many possibilities. What is called the “nonlinear kinematic hardening” leads to a natural description of the nonlinear plastic behavior under cyclic loading, but is connected to other concepts such as the Mroz’s model, limited to only two surfaces, and similarities with other approaches are pointed out in the context of a generalization of this rule to viscoplasticity.

An improved monolithic Newton-Raphson scheme for solving plastic flow with nonlinear flow laws

2020 ◽  
Author(s):  
Casper Pranger ◽  
Dave May ◽  
Laetitia Le Pourhiet
Keyword(s):  
Plastic Strain ◽  
Yield Strength ◽  
Strain Tensor ◽  
Effective Plastic Strain ◽  
Non Linear ◽  
Elasto Plasticity ◽  
Consistent Linearization ◽  
Newton Raphson ◽  
Objective Variable ◽  
Plastic Strain Tensor

<p>Brittle-plastic flows where the yield strength is a decreasing, non-linear function of plastic strain are thought to be commonplace in the Earth, and responsible for some of its most catastrophic events. Recent work [1] has highlighted again the computational benefit of an iterative Newton-Raphson scheme that contains a linearization of the plastic flow problem that is consistent with its time discretization. However, such a consistent linearization requires a nested set of iterations to converge on a yield strength if it is governed by a law that is non-linear in strain (or strain rate).</p><p>Eckert and co-authors [2] have shown that the construction of a consistent linearization can be avoided altogether, including these inner iterations, though at the considerable cost of including the full plastic strain tensor as an objective variable alongside the displacement vector. The resulting system is therefore larger, but as it can be expressed directly, posesses the quality that it may be linearized automatically, cheaply, and accurately by finite-differencing the non-linear residual with respect to the solution variables. Their algorithm naturally incorporates predictor and corrector polynomials that are second-order accurate in time, contrasting with traditional methods that are often derived using a Backward Euler time integrator. We present a modification to this algorithm that suppresses the cost of operating it significantly by replacing the symmetric second-order plastic strain tensor with a single effective plastic strain scalar objective variable, cutting the number of unknowns by 40% (2D) and 55% (3D) This makes it computationally more on par with existing schemes that employ a consistent tangent modulus.</p><p>We demonstrate this improved algorithm with test cases of non-linear strain softening laws relevant to Earth scientists, that include regularization by both Kelvin visco-plasticity [3] and non-local measures of effective plastic strain [4]. In addition, we analyse performance of this scheme with respect to existing algorithms.</p><p><em>References</em><br>[1] Duretz et al. (2018). “The benefits of using a consistent tangent operator for viscoelastoplastic computations in geodynamics.” <em>Geochemistry, Geophysics, Geosystems</em>, 19, 4904–4924.</p><p>[2] Eckert et al. (2004). “A BDF2 integration method with step size control for elasto-plasticity.” <em>Computational Mechanics</em> 34.5, 377–386.</p><p>[3] Duretz et al. (2019). “Finite Thickness of Shear Bands in Frictional Viscoplasticity and Implications for Lithosphere Dynamics.” <em>Geochemistry, Geophysics, Geosystems</em>, 20, 5598–5616.</p><p>[4] Engelen et al. (2003). “Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour.” <em>International Journal of Plasticity</em><br>19.4, 403–433.</p>

Continuum Foundations of Endochronic Plasticity

1984 ◽  
Vol 106 (4) ◽  
pp. 367-375 ◽  
Author(s):  
K. C. Valanis
Keyword(s):  
Plastic Strain ◽  
Experimental Determination ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Strain History ◽  
Periodic Stress ◽  
Strain Space ◽  
Plastic Strain Tensor ◽  
Standard Tests

In this paper a number of issues are addressed. The proper measure of intrinsic time is shown to be the norm of the increment of the plastic strain tensor, if the memory path is a line in the cojoint stress-strain space and elasticity at reversal points is stipulated. The convexity of the kernel function suffices to show that a periodic plastic strain history will give rise to a periodic stress history—as is observed. Standard tests are then established for the unequivocal experimental determination of the kernel and hardening function, valid for use in general three-dimensional histories.

scholarly journals Учет влияния полей остаточных деформаций в современных физико-механических технологиях обработки конструкционных материалов

2019 ◽  
Vol 45 (1) ◽  
pp. 27
Author(s):  
Ю.Н. Кульчин ◽  
В.Е. Рагозина ◽  
О.В. Дудко
Keyword(s):  
Plastic Strain ◽  
Elastic Waves ◽  
Strain Tensor ◽  
Elastic Field ◽  
Reversible Strain ◽  
Elastic Plastic ◽  
Elastic Process ◽  
Elastic Shock ◽  
Plastic Strain Tensor

AbstractA strict determination of the mechanisms of redistribution of previously accumulated irreversible strains as a result of additional elastic shock actions on the material is given for a nonlinear gradient model of large elastic–plastic strain. It is shown that this redistribution is limited by rigid transport and rotation of the plastic strain tensor. Formulas for a change in the initial components of the plastic strain tensor in elastic waves are derived. It is shown that the preliminary plastic field affects the dynamics of further reversible strain as one of the factors of formation of the initial quasi-static elastic field, which cannot be obtained in a purely elastic process.

A new approach to curvature measures in linear shell theories

2020 ◽  
pp. 108128652097275
Author(s):  
Miroslav Šilhavý
Keyword(s):  
Strain Tensor ◽  
Middle Surface ◽  
Surface Strain ◽  
Affine Function ◽  
Shear Deformations ◽  
Strain Measure ◽  
Bending Strain ◽  
Curvature Measures ◽  
Tensor Formula ◽  
Strain Measures

The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor [Formula: see text], which is different from the widely used Naghdi’s bending strain tensor [Formula: see text]. In the particular case of Kirchhoff–Love deformations, the tensor [Formula: see text] reduces to a tensor [Formula: see text] introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, [Formula: see text] is different from Koiter’s bending strain tensor [Formula: see text] (frequently used in this context). AMS 2010 classification: 74B99

Predictions of microtorsional experiments by micropolar plasticity

2005 ◽  
Vol 461 (2053) ◽  
pp. 189-205 ◽  
Author(s):  
Paschalis Grammenoudis ◽  
Charalampos Tsakmakis
Keyword(s):  
Size Effects ◽  
Bauschinger Effect ◽  
Kinematic Hardening ◽  
Circular Cylinders ◽  
Isotropic Hardening ◽  
Gradient Plasticity ◽  
Material Response ◽  
Micropolar Plasticity ◽  
Classical Plasticity ◽  
Hardening Rules

Kinematic hardening rules are employed in classical plasticity to capture the so–called Bauschinger effect. They are important when describing the material response during reloading. In the framework of thermodynamically consistent gradient plasticity theories, kinematic hardening effects were first incorporated into a micropolar plasticity model by Grammenoudis and Tsakmakis. The aim of the present paper is to investigate this model by predicting size effects in torsional loading of circular cylinders. It is shown that kinematic hardening rules compared with isotropic hardening rules, as adopted in the paper, provide more possibilities for modelling size effects in the material response, even if only monotonous loading conditions are considered.

Tensor representation of magnetostriction for all crystal classes

2018 ◽  
Vol 24 (9) ◽  
pp. 2814-2843 ◽  
Author(s):  
Salvatore Federico ◽  
Giancarlo Consolo ◽  
Giovanna Valenti
Keyword(s):  
Fourth Order ◽  
Strain Tensor ◽  
Tensor Representation ◽  
Magnetostrictive Strain

A systematic representation of the fourth-order magnetostriction tensor for all crystal classes is presented, based on the formalism proposed by Walpole ( Proc R Soc Lond Ser A 1984; 391: 149–179). This representation allows the general, unconstrained case, as well as the case in which the magnetostrictive strain is assumed to be isochoric, to be studied. The knowledge of the fourth-order magnetostriction tensor enables the stress-free magnetostrictive strain tensor as well as the scalar strain in a given direction to be calculated.

Strain-based fatigue life analysis of pipelines with external defects under cyclic internal pressure

2020 ◽  
pp. 030932472095756
Author(s):  
Hojjat Gholami ◽  
Shahram Shahrooi ◽  
Mohammad shishehsaz
Keyword(s):  
Fatigue Life ◽  
Stress Concentration ◽  
Plastic Strain ◽  
Internal Pressure ◽  
Oil And Gas ◽  
Isotropic Hardening ◽  
Element Simulation ◽  
Life Analysis ◽  
Fatigue Life Analysis ◽  
Pipe Geometry

Gouge and dent are common mechanical defects in oil and gas pipelines. These defects with plastic strain cause stress concentration in the pipelines. Plastic strain is dependent on initial deformation and spring-back behavior of materials. Therefore, they reduce the fatigue life of pipelines. In this paper, the strain-base fatigue life analysis is investigated in pipelines with smooth dent or combination smooth dent and gouge defects under cyclic internal pressure. For this purpose, elastic-plastic multilinear isotropic hardening finite element simulation was used to investigate the effects of various factors, such as residual stress of dent, amplitude internal pressure, pipe geometry, gouge geometry, and smooth dent geometry on stress concentration factor (SCF). Finally, a new method is proposed for predicting the fatigue life of pipelines with uniform dent and uniform dent and gouge combination defects. The model is presented based on the Smith-Watson-Topper (SWT) criterion. A set of fatigue life test specimens with various pipe materials, size and geometry were prepared and tested. The specimens carried a smooth dent, as well as a combination of smooth dent and gouge defects, results of which were collected to validate those obtained based on the proposed model. The results of the predicted tests using the developed formula showed a good correlation to practical experiments.

The cyclic boundary conditions and crystal vibrations

1993 ◽  
Vol 442 (1915) ◽  
pp. 373-395 ◽  
Keyword(s):  
Boundary Conditions ◽  
Degrees Of Freedom ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Normal Modes ◽  
Inertial Mass ◽  
Mean Value ◽  
Reference Configuration ◽  
Large Crystal ◽  
Homogeneous Coordinates

In considering the vibrational properties of a crystal, a rigorous finite transformation of the particle displacements from their reference configuration is introduced. This transformation shows that an arbitrary set of such displacements may be regarded as made up of a rotation, a translation, a homogeneous deformation of the reference configuration, and a set of inhomogeneous deformational orthogonal modes. For a three-dimensional crystal, there are 3 N – 12 such inhomogeneous modes, which, in the limit of a large crystal can be considered wave-like. In the usual treatment beginning with the cyclic boundary conditions, 3 N wave-like modes are assumed and rotational displacements, for example, must be ignored. The present treatment accounts satisfactorily for all degrees of freedom, including rotational. Because of the non-singular nature of the above transformation, the transformation of the above modes to the normal modes proves that some normal modes are admixtures of inhomogeneous and homogeneous modes and therefore cannot possibly satisfy the Born cyclic boundary conditions. The vibrational hamiltonian is shown to contain the elastic energy and the elastic–phonon interaction terms as well as the usual wave energies. In the limit of a large crystal, it is shown that, for all processes involving phonons, the homogeneous coordinates may be regarded as effectively static, in much the same way as, in a simple theory of the Earth–Sun motion, the Sun, because of its large inertial mass, is considered stationary and its position coordinates static. The above transformation enables the case of a crystal, free or confined in a container, to be satisfactorily discussed. It is proved that the quantum mean value of the tensor whose independent elements define the homogeneous coordinates is, in the limit of a large crystal, equal to the strain tensor of the container, when it is being used to deform the crystal by being itself homogeneously deformed. A rigorous quantum treatment of crystal elastic constants may then be developed. For practical use, the 3 N – 12 inhomogeneous modes may be assumed to obey the cyclic boundary conditions. Thus a satisfactory complete basic treatment of lattice dynamics may be given which accounts for all degrees of freedom including rotation.

Sign in / Sign up

Export Citation Format

Share Document