scholarly journals An effect of a purely dissipative process of microstresses on plane strain gradient plasticity problems

2018 ◽  
Vol 45 (2) ◽  
pp. 177-188
Author(s):  
Adebowale Borokinni ◽  
Odunayo Fadodun ◽  
Adegbola Akinola
Keyword(s):  
Plastic Strain ◽  
Strain Rate ◽  
Plane Strain ◽  
Strain Gradient ◽  
Dissipative Process ◽  
Strain Gradient Plasticity ◽  
Plastic Strain Rate ◽  
Flow Rule ◽  
Gradient Plasticity ◽  
Plane Strain Problem

This article considers a plane strain gradient plasticity theory of the Gurtin?Anand model [M. Gurtin, L. Anand, A theory of strain gradient plasticity for isotropic, plastically irrotational materials Part I: Small deformations, J. Mech. Phys. Solids 53 (2005), 1624?1649] for an isotropic material undergoing small deformation in the absence of plastic spin. It is assumed that the system of microstresses is purely dissipative, so that the free energy reduces to a function of the elastic strain, while the microstresses are only related to the plastic strain rate and gradient of the plastic strain rate via the constitutive relations. The plane strain problem of the Gurtin?Anand model for a purely dissipative process gives rise to elastic incompressibility. A weak formulation of the flow rule is derived, making the plane strain problem suitable for finite element implementation.

scholarly journals A theory of strain-gradient plasticity with effect of internal microforce

2017 ◽  
Vol 44 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Adebowale Borokinni ◽  
Adegbola Akinola ◽  
Olawanle Layeni
Keyword(s):  
Finite Element ◽  
Plastic Strain ◽  
Strain Gradient ◽  
Strain Gradient Plasticity ◽  
Finite Element Solution ◽  
Flow Rule ◽  
Length Scale ◽  
Element Solution ◽  
Gradient Plasticity ◽  
Microforce Balance

This paper develops a theory of strain gradient plasticity for isotropic bodies undergoing small deformation in the absence of plastic spin. The proposed theory is based on a system of microstresses which include a microstress vector consistent with microforce balance; the mechanical form of the second law of thermodynamics which includes work performed by the microstresses during plastic flow; and a constitutive theory that allows the free energy to depend on the elastic strain E??, divergence of plastic strain div E?? and the Burgers tensor G. Substitution of the constitutive relations into the microforce balance leads to a nonlinear partial differential equation in the plastic strain known as flow rule which captures the presence of an additional energetic length scale arising from the accounting of microstress vector. In addition to the flow rule, nonstandard boundary conditions are obtained, and as an aid to finite element solution a variational formulation of the flow rule is deduced. Finite element solution is obtained of one-dimensional problem of viscoplastic simple shearing under gravity force, where it is shown that for a fixed dissipative length scale, increase in the energetic length scales will result in decrease in the plastic strain.

A new strain-gradient theory for an isotropic plastically deformed polycrystalline solid body

2017 ◽  
Vol 23 (9) ◽  
pp. 1333-1344 ◽  
Author(s):  
AS Borokinni ◽  
AP Akinola ◽  
OP Layeni ◽  
OO Fadodun
Keyword(s):  
Plastic Strain ◽  
Solid Body ◽  
Strain Gradient ◽  
Plasticity Theory ◽  
Flow Rule ◽  
Gradient Plasticity ◽  
Polycrystalline Solid ◽  
Work Done ◽  
Strain Gradient Plasticity Theory ◽  
Internal Microstresses

This study considers strain-gradient plasticity theory in the context of small deformations for an isotropic solid body with a view to investigating the distortion effects associated with the divergence of plastic strain through the Burgers tensor. The principle of virtual power is employed and the constraint of irrotationality is imposed on the plastic component of the gradient of the displacement vector. It is obtained that the gradient, curl, and divergence of the plastic strain in the body are mutually related. This relation establishes the existence of work done through the divergence of plastic strain as distinct from the work done through the gradient of the plastic strain. Consequently, a polycrystalline solid body undergoing distortion associated with the divergence of plastic strain exhibits new internal microstresses; and the obtained model, consisting of the microforce balance, constitutive relations, and plastic flow rule, extends the known Gurtin–Anand model in a natural fashion. Furthermore, in the governing flow rule, it is revealed that the internal microstresses associated with the divergence of plastic strain act as opposing agents to the internal microstresses associated with the gradient of the plastic strain via the length scales Q, L, and the gradient of the divergence of the plastic strain. This work shows the distortion effects associated with the divergence of plastic strain which the Gurtin–Anand strain-gradient plasticity theory in literature does not apprehend.

NOTES ON STRAIN GRADIENT PLASTICITY: FINITE STRAIN COVARIANT MODELLING AND GLOBAL EXISTENCE IN THE INFINITESIMAL RATE-INDEPENDENT CASE

2009 ◽  
Vol 19 (02) ◽  
pp. 307-346 ◽  
Author(s):  
PATRIZIO NEFF ◽  
KRZYSZTOF CHEŁMIŃSKI ◽  
HANS-DIETER ALBER
Keyword(s):  
Strain Gradient ◽  
Finite Strain ◽  
Spatial Configuration ◽  
Initial Boundary ◽  
Strain Gradient Plasticity ◽  
Flow Rule ◽  
Gradient Plasticity ◽  
Mixed Variational Inequality ◽  
Dislocation Interaction ◽  
Plastic Distortion

We propose a model of finite strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition, a thermodynamically admissible flow rule for Fp is described involving as plastic gradient Curl Fp. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependence on the plastic metric [Formula: see text]. The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the nonsymmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: [Formula: see text] on the microscopically hard boundary ΓD ⊂ ∂Ω and [ Curl p] · τ = 0 on the microscopically free boundary ∂Ω\ΓD, where τ are the tangential vectors at the boundary ∂Ω. A weak reformulation of the infinitesimal model allows for a global in-time solution of the rate-independent initial boundary value problem. The method is based on a mixed variational inequality with symmetric and coercive bilinear form. We use a new Hilbert-space suitable for dislocation density dependent plasticity.

scholarly journals A gradient theory based on the Aifantis theory using the Gurtin-Anand strain gradient plasticity approach

2018 ◽  
Vol 27 (3-4) ◽  
Author(s):  
Adebowale S. Borokinni
Keyword(s):  
Strain Gradient ◽  
Flow Direction ◽  
Strain Gradient Plasticity ◽  
Flow Rule ◽  
Gradient Theory ◽  
Gradient Plasticity

AbstractThis article investigates the differences between the Aifantis and Gurtin-Anand strain gradient plasticity. The fact that Gurtin-Anand strain gradient plasticity is richer than the Aifantis strain gradient plasticity provides a basis for which we could derive equivalent Aifantis flow rule and extend it to accommodate scalar dissipative hardening. It was found that the major difference between the Gurtin-Anand and Aifantis theories lies on the use of the codirectionality hypothesis and constraint imposed on the gradient of the flow direction.

scholarly journals A Brief Note on the Nix–Gao Strain Gradient Plasticity Theory

Metals ◽  
2018 ◽  
Vol 8 (9) ◽  
pp. 708 ◽  
Author(s):  
A. Borokinni ◽  
Dabiao Liu
Keyword(s):  
Free Energy ◽  
Strain Gradient ◽  
Constitutive Equations ◽  
Plasticity Theory ◽  
Strain Gradient Plasticity ◽  
Flow Rule ◽  
Gradient Term ◽  
Gradient Plasticity ◽  
Virtual Power ◽  
Strain Gradient Plasticity Theory

The mathematical nature of the flow rule for the strain gradient plasticity theory proposed by Nix and Gao (W.D. Nix and H. Gao, J Mech Phys Solids 46(3), 411(1998)) is discussed based on the paradigm developed by Gurtin and Anand (M.E. Gurtin and L. Anand, J Mech Phys Solids 57 (3), 405 (2009)). It is shown that, when investigated on the basis of Gurtin–Anand theory, the Nix–Gao flow rule is a combination of constitutive equations for microstresses, balance law, and a constraint. As an accessory, we demonstrate that the strain gradient term introduced in the model is energetic. The results are obtained by combining a virtual-power principle of Fleck and Hutchinson, and the free-energy imbalance under isothermal conditions.

Experimentally Evaluating the Equilibrium Stress in Shear of Glassy Polycarbonate

2006 ◽  
Vol 128 (4) ◽  
pp. 537-542 ◽  
Author(s):  
Mehrdad Negahban ◽  
Ashwani Goel ◽  
Pierre Delabarre ◽  
Ruqiang Feng ◽  
Amy Dimick
Keyword(s):  
Plastic Strain ◽  
Strain Rate ◽  
Mechanical Response ◽  
Deformation Gradient ◽  
Back Stress ◽  
Glassy Polymers ◽  
Plastic Strain Rate ◽  
Flow Rule ◽  
Strain Rates ◽  
Equilibrium Stress

One group of models proposed for characterizing the mechanical response of glassy polymers is based on a structure that resembles finite plasticity. In most cases, a constitutive equation for stress is proposed, which depends on the elastic deformation gradient, supplemented by a flow rule for the plastic deformation, which depends on the “over stress.” The over stress is a properly invariant difference between the stress and the back stress (equilibrium stress). The back stress represents conditions under which relaxation events should stop and the material should be able to carry an applied load indefinitely without a need to change the strain. Questions that arise in using these models are whether such equilibrium stresses exist, how can they be evaluated, and what experiments can be used to characterize the flow rule. One challenge in accurately evaluating the locus of equilibrium conditions is the fact that the relaxation process substantially slow down around these points, and, therefore, a method that does not directly require being at the equilibrium is desirable. Focusing on shear, a thermodynamic theory for characterizing the response of glassy polymers, similar to models currently used for this purpose, is developed, and using this model it is shown that one can set up a method to calculate the plastic strain rate. This method is based on evaluating the slope of stress-strain response under conditions of similar elastic and plastic strain, but different strain rates. Since the equilibrium stress occurs when the plastic strain rate goes to zero, the evaluated plastic strain rates allow evaluation of the needed information for developing the flow rule and obtaining the back stress. This method is used to evaluate the plastic strain rate and back stress at room temperature for polycarbonate. The evaluated results match well with results obtained by direct probing of the equilibrium stress, in which one searches for points at which the stress remains constant at a constant strain over long durations. The method proposed looks promising in evaluating the back stress of glassy polymers. The added advantage of this method is that it also provides a map of plastic strain rate and tangent modulus over a large range of loading conditions.

Gradient plasticity in isotropic solids

2021 ◽  
pp. 108128652110502
Author(s):  
D. J. Steigmann
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Plastic Deformation ◽  
Plastic Strain ◽  
Plastic Strain Rate ◽  
Flow Rule ◽  
Order Partial Differential Equation ◽  
Riemannian Curvature ◽  
Gradient Plasticity ◽  
Isotropic Solids

We discuss a framework for the description of gradient plasticity in isotropic solids based on the Riemannian curvature derived from a metric induced by plastic deformation. This culminates in a flow rule in the form of a fourth-order partial differential equation for the plastic strain rate, in contrast to the second-order flow rules that have been proposed in alternative treatments of gradient plasticity in isotropic solids.

Evaluation of Associated and Non-Associated Flow Metal Plasticity; Application for DC06 Deep Drawing Steel

2012 ◽  
Vol 504-506 ◽  
pp. 661-666 ◽  
Author(s):  
Mohsen Safaei ◽  
Wim De Waele ◽  
Shun Lai Zang
Keyword(s):  
Finite Element ◽  
Plastic Strain ◽  
Strain Rate ◽  
Yield Stress ◽  
Deep Drawing ◽  
Kinematic Hardening ◽  
Plastic Strain Rate ◽  
Flow Rule ◽  
Strain Rate Tensor ◽  
Incremental Change

In this paper the capabilities of Associated Flow Rule (AFR) and non-AFR based finite element models for sheet metal forming simulations is investigated. In case of non-AFR, Hill’s quadratic function used as plastic potential function, makes use of plastic strain ratios to determine the direction of effective plastic strain rate. In addition, the yield function uses direction dependent yield stress data. Therefore more accurate predictions are expected in terms of both yield stress and strain ratios at different orientations. We implemented a modified version of the non-associative flow rule originally developed by Stoughton [1] into the commercial finite element code ABAQUS by means of a user material subroutine UMAT. The main algorithm developed includes combined effects of isotropic and kinematic hardening [2]. This paper assumes proportional loading cases and therefore only isotropic hardening effect is considered. In our model the incremental change of plastic strain rate tensor is not equal to the incremental change of the compliance factor. The validity of the model is demonstrated by comparing stresses and strain ratios obtained from finite element simulations with experimentally determined values for deep drawing steel DC06. A critical comparison is made between numerical results obtained from AFR and non-AFR based models

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