Classical Continuum Plasticity Research Articles

Computational Modeling of Dislocation Slip Mechanisms in Crystal Plasticity: A Short Review

The bridge between classical continuum plasticity and crystal plasticity is becoming narrower with continuously improved computational power and with engineers’ desire to obtain more information and better accuracy from their simulations, incorporating at the same time more effects about the microstructure of the material. This paper presents a short overview of the main current techniques employed in crystal plasticity formulations for finite element analysis, as to serve as a point of departure for researchers willing to incorporate microstructure effects in elastoplastic simulations. We include both classical and novel crystal plasticity formulations, as well as the different approaches to model dislocations in crystals.

Strain gradient plasticity for amorphous and crystalline polymers with application to micro- and nano-scale deformation analysis

Microscale mechanical responses of semicrystalline glassy polymers are of utmost importance when these materials are deployed in microstructural applications, e.g. MEMS and NEMS. The classical continuum plasticity theories become ill-posed in the case of highly localized deformations and they result in mesh dependent Finite Element (FE) solutions. Then the continuum theories may not directly be applicable to study stress–strain responses in nano- and micro-scale structures. The non-local modeling approach is one solution to this deficiency in which the non-local effects, associated with the presence of highly localized plastic deformation fields, are integrated into the constitutive relations. The microstructural information together with the intrinsic material length scale are incorporated in these theories to take into account the non-local effects and to ensure the well-posedness of the governing equations. While non-local theories have been extensively investigated for Metals (class M materials), this work aims to develop a rate dependent plasticity theory with strain gradient effects for semicrystalline Polymers (class P materials).

Microstructure to Macro-Scale Using Gradient Plasticity with Temperature and Rate Dependent Length Scale

Gradient plasticity theory formulates a constitutive framework on the continuum level that bridges the gap between the micromechanical plasticity and classical continuum plasticity by incorporating the material length scale. A micromechanical-based model of variable material intrinsic length scale is developed in the present work which allows for variations in temperature and strain rate and its dependence on the grain size and accumulated plastic strain. The material constants of the proposed model are calibrated using the size e_ect encounter in nanohardness experiments. In this regard, two di_erent physically based models for Temperature and Rate Indentation Size E_ects (TRISE) are also developed in this work for single and polycrystalline metals by considering di_erent expressions of the geometrical necessary dislocation (GND) density. The results of indentation experiments performed on various single- and polycrystalline materials are then used here to implement the aforementioned framework in order to predict simultaneously the TRISE and variable length scale at di_erent temperatures, strain rates and various grain sizes.

Determination of nanoindentation size effects and variable material intrinsic length scale for body-centered cubic metals

It is well-known by now that the micro and nanoindentation hardness of metallic materials displays a strong size effect. The objective of this work is to formulate a micromechanical-based model for Temperature and Rate Indentation Size Effects (TRISE) for body centered cubic (BCC) metals encountered in nanoindentation experiments. In this regard, two physically based models are proposed here in order to capture the TRISE in single and polycrystalline materials by considering different expressions of the geometrical necessary dislocations (GNDs) density. The gradient plasticity theory formulates a constitutive framework on the continuum level that bridges the gap between the micromechanical plasticity and the classical continuum plasticity by incorporating the material length scale. A micromechanical-based model of variable material intrinsic length scale is also developed in the present work. The proposed length scale allows for variations in temperature and strain rate and its dependence on the grain size and accumulated plastic strain. The results of indentation experiments performed on niobium, tungsten, and single- and polycrystalline commercially pure iron (very similar to iron alloys) are used here to implement the aforementioned framework in order to predict simultaneously the TRISE and variable length scale at different temperatures, strain rates and various distances from the grain boundary. Numerical analysis is performed using the ABAQUS/VUMAT software with a physically based viscoplastic constitutive model.

Dislocation Density-Based Finite Element Method Modeling of Ultrasonic Consolidation

A dislocation density based constitutive model has been developed and implemented into a crystal plasticity quasi‐static finite element framework. This approach captures the statistical evolution of dislocation structures and grain fragmentation at the bonding interface when sufficient boundary conditions pertaining to the Ultrasonic Consolidation process are prescribed. Hardening is incorporated using statistically stored and geometrically necessary dislocation densities (SSDs and GNDs), which are dislocation analogs of isotropic and kinematic hardening respectively. The GND considers strain‐gradient and thus renders the model size‐dependent. The model is calibrated using experimental data from published refereed literature and then validated for the Aluminum 3003 alloy. Introduction As a direct result of ongoing research efforts in ultrasonic consolidation (UC) worldwide [1], it has become apparent that a new approach to modeling of UC bonding is needed. A model which provides a better understanding of the effects of process parameter changes on grain refinement, plastic deformation and bonding during UC will better enable researchers to predict which materials will bond, how the mechanical properties of UC-produced parts can be improved, and how to better design the next generation of UC equipment. The continuum properties of parts made using UC are strongly dependent upon the micromechanics of the bonded interface [1]. Interfacial-scale microstructures can be studied fundamentally using electron microscopy [1] and can be used to correlate atomic and mesoscopic mechanisms of deformation to their continuum counterparts. A dislocation density-based crystal plasticity Finite Element Model (FEM) can capture the statistical distribution of dislocations, partials and various deformation mechanisms at the bonding interface as inputs to predict macroscopic deformation profiles as a function of energy input characteristics. These input characteristics are a function of the process parameters used in a UC machine, namely vibration amplitude, normal force, ultrasonic frequency, welding speed, sonotrode geometry and temperature. Problem Formulation It has been shown that material sheets subjected to UC undergo inhomogeneous plastic deformation through their thickness [1]. Classical continuum plasticity theories do not fully explain this phenomenon [2]. Therefore, a study of strain localization and grain refinement at the material interfaces during UC bonding is required. The following steps lead to the calculation of these localized strains and their effects. Large Deformation Quasi-Static Crystal Plasticity Description The deformation map in space and time is described by the total deformation gradient tensor F (Figure 1). Applying the Kroner-Lee assumption, F is decomposed into elastic Fe and plastic gradient Fp tensors using multiplicative operator theory

Simulations of micro and nanoindentations

The paper reviews recent research and developments on simulated indentation tests at micron and nanometer levels. For indentation at the maximum depth of several microns or hundreds of nanometer, classical continuum plasticity framework incorporating Taylor dislocation model via strain gradient plasticity embedded in the constitutive equation may be adopted to take care of the size effect. As higher-order stress components and higher-order continuity requirements can be made redundant, only C 0 finite elements incorporating strain gradient plasticity have to be formulated. This results in the significant ease and convenience in finite element implementation requiring minimal additional computational effort and resources. Alternatively, when the indentation depth is lower at nanometer level, either a large scale molecular dynamics model or a hybrid finite element and molecular dynamics simulation has to be adopted. The article includes certain results from the former approach on nanoindentation based on combination of both Morse potential and embedded-atom model potential.

Mechanism-based strain gradient plasticity in C0 axisymmetric element

Non-uniform plastic deformation of materials exhibits a strong size dependence when the material and deformation length scales are of the same order at micro- and nano-metre levels. Recent progresses in testing equipment and computational facilities enhancing further the study on material characterization at these levels confirmed the size effect phenomenon. It has been shown that at this length scale, the material constitutive condition involves not only the state of strain but also the strain gradient plasticity. In this study, C 0 axisymmetric element incorporating the mechanism-based strain gradient plasticity is developed. Classical continuum plasticity approach taking into consideration Taylor dislocation model is adopted. As the length scale and strain gradient affect only the constitutive relation, it is unnecessary to introduce either additional model variables or higher order stress components. This results in the ease and convenience in the implementation. Additional computational efforts and resources required of the proposed approach as compared with conventional finite element analyses are minimal. Numerical results on indentation tests at micron and submicron levels confirm the necessity of including the mechanism-based strain gradient plasticity with appropriate inherent material length scale. It is also interesting to note that the material is hardened under Berkovich compared to conical indenters when plastic strain gradient is considered but softened otherwise.

Determination of the Material Intrinsic Length Scale of Gradient Plasticity Theory

The enhanced gradient plasticity theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical plasticity and the classical continuum plasticity. The later cannot predict the size effects since it does not posses an intrinsic length scale. To assess the size effects, it is indispensable to incorporate an intrinsic material length parameter l into the constitutive equations. However, the full utility of gradient-type theories hinges on one's ability to determine the constitutive length-scale parameter l that scales the gradient effects. Thus, the definition and magnitude of the intrinsic length scale are keys to the development of the new theory of plasticity that incorporates size effects. The classical continuum plasticity is also unable to predict properly the evolution of the material flow stress since the local deformation gradients at a given material point are not accounted for. The gradient-based flow stress is commonly assumed to rely on a mixed type of dislocations: those that are initially randomly or statistically distributed, which are referred to as statistically stored dislocations (SSDs), and those formed to account for the additional strengthening mechanism associated with the deformation gradients, which are referred to as geometrically necessary dislocations (GNDs). In this work two micromechanical models to assess the coupling between SSDs and GNDs are discussed. One in which the SSDs and GNDs are simply summed (model-I) and one in which, implicitly, their accompanying strength are added (model-II). These two dislocation interaction models, which are based on Taylor's hardening law, are then used to identify the deformation-gradient-related intrinsic length-scale parameter l in terms of measurable microstructural physical parameters. The paper also presents a method for identifying the material intrinsic length parameter l from micro hardness results obtained by conical or pyramidal indenters.

Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments

The enhanced gradient plasticity theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical plasticity and the classical continuum plasticity. They are successful in explaining the size effects encountered in many micro- and nano-advanced technologies due to the incorporation of an intrinsic material length parameter into the constitutive modeling. However, the full utility of the gradient-type theories hinges on one's ability to determine the intrinsic material length that scales with strain gradients, and this study aims at addressing and remedying this situation. Based on the Taylor's hardening law, a micromechanical model that assesses a nonlinear coupling between the statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs) is used here in order to derive an analytical form for the deformation-gradient-related intrinsic length-scale parameter in terms of measurable microstructural physical parameters. This work also presents a method for identifying the length-scale parameter from micro- and nano-indentation experiments using both spherical and pyramidal indenters. The deviation of the Nix and Gao [Mech. Phys. Solids 46 (1998) 411] and Swadener et al. [J. Mech. Phys. Solids 50 (2002) 681; Scr. Mater. 47 (2002) 343] indentation size effect (ISE) models’ predictions from hardness results at small depths for the case of conical indenters and at small diameters for the case of spherical indenters, respectively, is largely corrected by incorporating an interaction coefficient that compensates for the proper coupling between the SSDs and GNDs during indentation. Experimental results are also presented which show that the ISE for pyramidal and spherical indenters can be correlated successfully by using the proposed model.

Interface crack problems with strain gradient effects

In this paper, the strain gradient theory proposed by Chen and Wang (2001a, 2002b) is used to analyze an interface crack tip field at micron scales. Numerical results show that at a distance much larger than the dislocation spacing the classical continuum plasticity is applicable; but the stress level with the strain gradient effect is significantly higher than that in classical plasticity immediately ahead of the crack tip. The singularity of stresses in the strain gradient theory is higher than that in HRR field and it slightly exceeds or equals to the square root singularity and has no relation with the material hardening exponents. Several kinds of interface crack fields are calculated and compared. The interface crack tip field between an elastic-plastic material and a rigid substrate is different from that between two elastic-plastic solids. This study provides explanations for the crack growth in materials by decohesion at the atomic scale.

On the relation between continuum plasticity and dislocation theories

A theory of elastic-plastic flow based upon dislocation mechanics is presented. The logical structure of the theory parallels the Green-Naghdi plasticity theory, thereby permitting direct interpretation of the classical continuum plasticity assumptions in terms of dislocation mechanics. Thermodynamical restrictions derivable from the Clausius-Duhem inequality are discussed in terms of dislocation motion.