Strain Gradient Plasticity Theory Research Articles

Three-Dimensional Elastoplastic Contact Analysis of Rough Surface Considering a Micro-Scale Effect

Abstract The micro-surface asperity scale of grinding metal parts is within several microns. When two grinding surfaces are in contact, the unevenness of the plastic deformation of the asperities at the micro-scale leads to greater plastic hardening strength of the material. The results of the nano-indentation experiment conducted in this paper confirmed this phenomenon. Based on conventional mechanism-based strain gradient (CMSG) plasticity theory, the micro-scale plastic constitutive equation of materials is given and then is verified by the nano-indentation experiment. Finite element software abaqus and the user-defined element (UEL) subroutine are used to build three-dimensional rough surface elastoplastic contact models. By calculating the grinding rough surface contact in the macro-scale constitutive model based on J2 theory and in the CMSG plasticity constitutive model, the influence law of plastic micro-scale effect on contact performance is obtained.

The role of rough surface in the size-dependent behavior upon nano-indentation

The hardness measured by nano-indentation on metallic material with rough surface shows a significant deviation from the Nix-Gao relation, which established an important relation between the hardness and indentation depth at small scale (Nix and Gao, 1998). Such effect has been investigated by nano-indentation experiments, and accounted for in a few proposed nano-indentation models in terms of surface roughness. However, the predicted roughness effects based on these models are quite different and even contradictory in some aspects. It is believed that the surface profile and the indentation position nano-indentation tests on copper specimens with fabricated nanoscale rough surfaces were carefully performed. Therefore, current work investigated this effect of nanoscale rough surface by both nano-indentation tests and finite element simulations. The conventional theory of mechanism-based strain gradient plasticity was implemented into ABAQUS for modeling the size-dependent indentations on both nano-grooved and sine-waved surfaces. The effect of the rough surface was analyzed in terms of varying the amplitude, the wavelength and the aspect ratio between them. An updated Nix-Gao relation that accounts for the surface roughness was proposed. The new relation captured well the available nano-indention behavior of Sb–Pd alloy and Nickel.

Closed-form functions of cross-scale indentation scaling relationships based on a strain gradient plasticity theory

ABSTRACT Indentation scaling relationships intend to guide indentation experiments and help understand the indentation hardness. In the present study, dimensionless functions were obtained by fitting a series of numerical results of the cross-scale indentation tests based on a strain gradient plasticity theory. The closed-form indentation scaling relationships were found between dimensionless material properties and indentation responses, which were non-dimensionalised by a representative stress. The closed-form expressions were of high accuracy with both the numerical results and the experimental data. Forward and reverse analysis were then established and discussed, providing the indentation scaling relationships a quantitative result. The explicit definition of the representative stresses made it possible to obtain the specific relationship between indentation responses and each material parameter. Together with some pre-determined material parameters, the material intrinsic length was able to be derived from the indentation tests. The present work significantly enhanced the value of the cross-scale indentation scaling relationships, providing great convenience in guiding instrumented tests.

Microstructural design strategy to maintain the brittle fracture toughness of hard-soft dual phase steel after cyclic plastic strain

Superstructures are seriously damaged because of plastic deformation when natural disasters, such as large earthquakes or unexpectedly large waves, occur. Therefore, there is a need to develop materials that are resistant to damage and fracture while maintaining a certain level of strength. In this paper, the difference in material toughness changes when cyclic prestrain occurs among different ferrite/pearlite two-phase microstructures were investigated. Synthetic analysis using strain gradient plasticity (SGP) theory, one of the crystal plasticity theories, was carried out, and conventional macroscopic material damage consideration rules were employed. Then, it was suggested that the amount of material damage changes among the different microstructures. In addition, a parametric study using SGP theory is performed to suggest directions for the microstructural design of steel with a lower risk of brittle fracture due to plastic deformation. As a result, damage-resistant materials with sufficient strength can be proposed.

Scalar-based strain gradient plasticity theory to model size-dependent kinematic hardening effects

A common belief in phenomenological strain gradient plasticity modeling is that including the gradient of scalar variables in the constitutive setting leads to size-dependent isotropic hardening, whereas the gradient of second-order tensors induces size-dependent kinematic hardening. The present paper shows that it is also possible to produce size-dependent kinematic hardening using scalar-based gradient theory. For this purpose, a new model involving the gradient of the equivalent plastic strain is developed and compared with two reference scalar-based and tensor-based theories. Theoretical investigations using simple monotonic loading conditions are first presented to assess the ability of the proposed model to solve some issues related to existing scalar-based gradient theories. Simulations under cyclic loading conditions are then provided to investigate the nature of the resulting hardening. These simulations show that the proposed model is capable of producing size-dependent kinematic hardening effects at more affordable costs, compared to existing tensor-based strain gradient plasticity theories.

Effect of crack curvature radius on the stress fields under nonlinear deformation

The influence of crack curvature radius and plastic properties of material on the crack tip fields is evaluated by conventional mechanism-based strain gradient (CMSG) plasticity theory, and classical plasticity theory. Stress fields in single edge tension specimen are investigated numerically for a wide range of the crack curvature radius variations. The strain hardening exponent varied from N=0.075 to N=0.4. The local areas of the influence of the crack tip curvature radius are established for the strain gradient plasticity and classical plasticity. The couple effects of material length scale parameter and plastic properties of material on crack tip fields are established for strain gradient plasticity. The boundaries of CMSG plasticity dominated zone are determined.

Investigation of the Hall-Petch Relationship Using Strain Gradient Plasticity Model for Finite Deformation Framework

The Hall-Petch relationship in metals is investigated using the strain gradient plasticity theory within the finite deformation framework. For this purpose, the thermodynamically consistent constitutive formulation for the coupled thermomechanical gradient-enhanced plasticity model is developed. The corresponding finite element method is performed to investigate the characteristics of the Hall-Petch relationship in metals. The proposed model is established based on an extra Helmholtz-type partial differential equation, and the nonlocal quantity is calculated in a coupled method based on the equilibrium conditions. An excellent agreement between the simulation results and the test data is resulted in the Hall-Petch graph. Furthermore, it is observed that the Hall-Petch constants do not remain unchanged but vary with the strain level.

Characterizing geometrically necessary dislocations using an elastic–plastic decomposition of Laplace stretch

In this paper, the geometric dislocation density tensor and Burgers vector are studied using an elastic–plastic decomposition of Laplace stretch $$\varvec{\mathcal {U}}$$ . The Laplace stretch arises from a $$\mathbf {QR}$$ decomposition of the deformation gradient and is very useful, as one can directly and unambiguously measure its components by performing experiments. The geometric dislocation density tensor $${\tilde{\mathbf {G}}}$$ is obtained using the classical argument of failure of a Burgers circuit in a suitable configuration $$\tilde{\kappa }_p$$ where the deformation of a body is solely due to the movement of dislocations. The geometric features of space $$\tilde{\kappa }_p$$ are explored. It is shown that the derived geometric dislocation tensor is related to the torsion of $$\tilde{\kappa }_p$$ , which serves as a measure of incompatibility in this space. Additionally, $${\tilde{\mathbf {G}}}$$ vanishes only when the space $$\tilde{\kappa }_p$$ is compatible. A balance law for geometric dislocations is derived taking into account the effect of the dislocation flux and source dislocations. The physical meaning of the plastic Laplace stretch, and consequently, of the derived geometric dislocation tensor proves to be particularly useful in the classification of dislocations. Finally, the significance of the dislocation density tensor is discussed. The derived geometric dislocation density tensor could be specifically useful in developing a strain-gradient and size-dependent theory of plasticity.

Crack tip fields and fracture resistance parameters based on strain gradient plasticity

The crack tip mechanics of strain gradient plasticity solids is investigated analytically and numerically. A first-order mechanism-based strain gradient (MSG) plasticity theory based on Taylor’s dislocation model is adopted and implemented in the commercial finite element package ANSYS by means of a user subroutine. Two boundary value problems are considered, a single edge tension specimen and a biaxially loaded plate. First, crack tip fields are characterized. Strain gradient effects associated with dislocation hardening mechanisms elevate crack tip stresses relative to conventional plasticity. A parametric study is conducted and differences with conventional plasticity predictions are quantified. Moreover, the asymptotic nature of the crack tip solution is investigated. The numerical results reveal that the singularity order predicted by the first-order MSG theory is equal or higher to that of linear elastic solids. Also, the crack tip field appears not to have a separable solution. Moreover, contrarily to what has been shown in the higher order version of MSG plasticity, the singularity order exhibits sensitivity to the plastic material properties. Secondly, analytical and numerical approaches are employed to formulate novel amplitude factors for strain gradient plasticity. A generalized J-integral is derived and used to characterize a nonlinear amplitude factor. A closed-form equation for the analytical stress intensity factor is obtained. Amplitude factors are also derived by decomposing the numerical solution for the crack tip stress field. Nonlinear amplitude factor solutions are determined across a wide range of values for the material length scale l and the strain hardening exponent N. The domains of strain gradient relevance are identified, setting the basis for the application of first-order MSG plasticity for fracture and damage assessment.

Accurate Estimation of Brittle Fracture Toughness Deterioration in Steel Structures Subjected to Large Complicated Prestrains

Studies have suggested that brittle fractures occur in steel because microcracks in the brittle layer at grain boundaries propagate as a result of the increase in piled-up dislocations. Therefore, prestraining can approach the limits of a material, which could lead to a decrease in fracture toughness. However, strains are tensors comprising multiple components, so the effect of prestrain on fracture toughness is not simple. Additionally, the mechanism of change in critical stress due to prestrain has not been thoroughly investigated. For the lifetime evaluation of steel structures with a complicated load history, it is important to generalize the effect of complicated prestrain on the decrease in fracture toughness. In this paper, a single prestrain was applied in a direction different from the crack opening direction. A general three-point bending test was employed for fracture evaluation. Numerical analyses using the strain gradient plasticity (SGP) theory, which is a method based on the finite element method (FEM) are carried out; conventional macroscopic material damage rules are considered as well. Using these FEM analyses, the critical stress is calculated. Finally, the change in critical stress can be expressed by the yield point increase and dislocation density and formulated based on the identified micromechanisms.

Strain-gradient effect on the crack tip dislocations density

In this study, the influence of a material’s plastic properties on the crack tip fields and dislocation density behavior is analytically and numerically analyzed using the conventional mechanism-based strain-gradient plasticity (CMSGP) theory established using the Taylor model. The material constitutive equation is implemented in a commercial finite element code by a user subroutine, and the crack tip fields are evaluated with novel parameters in the form of the intrinsic material length, characterizing the scale over which gradient effects become significant. As a consequence of the strain-gradient contribution, FE results show a significant increase in the magnitude of the stress fields of CMSGP when the material length parameter is considered. It is found that the density of geometrically necessary dislocations (GND) is large around the crack tip, but it rapidly decreases away from the crack tip. On the contrary, the density of statistically stored dislocations (SSD) is not as large as geometrically necessary dislocations around the crack tip, but it decreases much slower than GND away from the crack tip. A couple effect of material work hardening and the crack tip distance is identified.

A Review on Strain Gradient Plasticity Approaches in Simulation of Manufacturing Processes

Predicting the performances of a manufactured part is extremely important, especially for industries in which there is almost no room for uncertainties, such as aeronautical or automotive. Simulations performed by means of numerical methods such as Finite Element Methods represent a powerful instrument in achieving high level of predictability. However, some particular combinations of manufactured materials and manufacturing processes might lead to unfavorable conditions in which the classical mathematical models used to predict the behavior of the continuum are not anymore able to deliver predictions that are in good agreement with experimental evidence. Since the first evidences of the shortcomings of the classical model were highlighted, many non-classical continuum mechanics theories have been developed, and most of them introduce dependencies at different levels with the Plastic Strain Gradient. This manuscript aims at gathering the milestone contributions among the Strain Gradient Plasticity Theories developed so far, with the object of exploring the way they interface with the requirements posed by the challenges in simulating manufacturing operations. Finally, the most relevant examples of the applications of Strain Gradient Plasticity Theories for manufacturing simulations have been reported from literature.

Effect of KDP-Crystal Material Properties on Surface Morphology in Ultra-Precision Fly Cutting.

To study the effect of material properties on the surface morphology of potassium dihydrogen phosphate (KDP) crystals, an ultra-precision fly cutting machine tool with a single-point diamond tool was used to perform a cutting experiment on (100) crystal plane of the KDP crystal. The elastic modulus, shear modulus, hardness, and dislocation of KDP crystals are taken into the cutting force model by introducing the strain gradient plasticity theory. Since the size effect and dynamic response will affect the surface roughness during ultra-precision machining, the surface roughness of workpieces in ultra-precision fly cutting is hard to predict. Based on the previously established strain gradient plasticity theoretical model, cutting force model, and the dynamic characteristics of the ultra-precision fly cutting system, a surface morphology prediction model under the influence of KDP crystal material properties was established. Finally, the accuracy of the surface morphology prediction model was verified by ultra-precision fly cutting experiments, and identified the frequency range of the characteristic signal caused by the anisotropy of the KDP crystal from the frequency, thereby verifying the KDP crystal material properties has a significant effect on the surface of the machined workpiece roughness.

On dissipative gradient effect in higher-order strain gradient plasticity: the modelling of surface passivation

The phenomenological flow theory of higher-order strain gradient plasticity proposed by Fleck and Hutchinson (J. Mech. Phys. Solids, 2001) and then improved by Fleck and Willis (J. Mech. Phys. Solids, 2009) is used to investigate the surface-passivation problem and micro-scale plasticity. An extremum principle is stated for the theory involving one material length scale. To solve the initial boundary value problem, a numerical scheme based on the framework of variational constitutive updates is developed for the strain gradient plasticity theory. The main idea is that, in each incremental time step, the value of the effective plastic strain is obtained through the variation of a functional in regard to effective plastic strain, provided the displacement or deformation gradient. Numerical results for elasto-plastic foils under tension and bending, thin wires under torsion, are given by using the minimum principle and the numerical scheme. Implications for the role of dissipative gradient effect are explored for three non-proportional loading conditions: (1) stretch-passivation problem, (2) bending-passivation problem, and (3) torsion-passivation problem. The results indicate that, within the Fleck–Hutchinson–Willis theory, the dissipative length scale controls the strengthening size effect, i.e. the increase of initial yielding strength, while the surface passivation gives rise to an increase of strain hardening rate.

A modified conventional theory of mechanism-based strain gradient plasticity considering both size and damage effects

The in-service deformation damage cannot be avoided, even for micro-scale metallic materials. To characterize the coupling effect of size and damage in micro-metallic materials, a modified incremental constitutive model is proposed on the basis of the conventional theory of mechanism-based strain gradient plasticity (CMSG), in which the assumptions of both small strain and isotropic damage are adopted. Other than retaining the essential features of the CMSG theory, the in-service deformation damage effects on the elastic moduli, plastic yielding criterion and intrinsic length scale, are well included. Typical micro-scale experimental results, including those from the torsion of thin wires, bending of ultra-thin beams, and tension and compression of micro-particle-reinforced metal matrix composites (MPMMCs), are theoretically analyzed using the new modified theory. Good agreements between the theoretical predictions and the experimental measurements are achieved, especially for the mechanical behaviors of MPMMCs subjected to relatively large deformation. Furthermore, it is interesting to find, upon increasing the uniaxial tension, the dominant damage mode in MPMMCs will transit from interface debonding to matrix damage, while matrix damage is always dominated in MPMMCs under uniaxial compression. In this study, we provide a convenient and precise way to characterize the in-service mechanical behaviors of micro-scale material systems, especially the commonly used composites with a characteristic length scale of micrometers.

Simulation of micro-milling Inconel 718 considering scale effect

To study the scale effect during micro-milling process, we establish a modified Johnson-Cook (JC) constitutive model to describe the strengthening behavior of materials on microscale based on the theory of strain gradient plasticity. The VUMAT constitutive subroutine of Inconel 718 is programmed. The radius of the tool edge and its influence on the rake angle are considered. Based on the modified JC constitutive equation, the simulation of micro-milling Inconel 718 process is achieved. The micro-milling Inconel 718 experiments are conducted. The accuracy of the simulation of micro-milling Inconel 718 process considering scale effect is verified by comparing cutting force from experiments with simulation outputs based on the traditional as well as modified JC constitutive equation. It is proved that the simulation outputs of the micro-milling process based on the modified JC constitutive equation are more consistent with the experimental results.

Characterization of strain amplitude-dependent behavior of hardness and indentation size effect of SS400 structural steel

In this paper, the continuous stiffness measurement (CSM) indentation is employed to investigate fatigue mechanical properties of structural steel under cyclic loading. For this purpose, several representative analytical approaches were introduced to estimate the basic mechanical properties including Young’s modulus and indentation hardness from the characteristics of the loading/unloading curves. Several experiments including CSM nanoindentation, low-cycle fatigue experiment for four strain amplitude levels, optical microscope (OM), and transmission electron microscopy (TEM) examinations were conducted to observe the variation characteristics of mechanical properties at the microscale and their micro-mechanisms. The microstructural evolution of the specimens deformed by the low-cycle fatigue was observed using the OM and TEM examinations. The standard nanoindentation experiments were then performed at different strain rate levels to characterize the influences of strain rate indentation on hardness of the material. The micro-mechanisms established based on the microstructural evolution and strain gradient plasticity theory were introduced to be responsible for the variation of indentation hardness under cyclic loading. Finally, the indentation size effect (ISE) phenomenon in SS400 structural steel was investigated and explained through the strain gradient plasticity theory regarding geometrically necessary dislocations underneath the indenter tip. The experimental results can be used for practical designs as well as for understanding the fatigue behavior of SS400 structural steel.
 Keywords:
 cyclic loading; fatigue; nanoindentation; indentation size effect; strain rate sensitivity; structural steel.

Strain gradient finite element model for finite deformation theory: size effects and shear bands

In this work, a thermodynamically consistent constitutive formulation for the coupled thermomechanical strain gradient plasticity theory is developed in the context of the finite deformation framework. A corresponding finite element solution is presented to investigate the microstructural features of metallic volumes. The developed model is established based on an extra Helmholtz-type partial differential equation, and the nonlocal quantity is calculated in a coupled method based on the equilibrium conditions. This approach is well known for its computational strength, however, it is also commonly accepted that it cannot capture the size effect phenomenon observed in the micro-/nanoscale experiments during hardening. In order to resolve this issue, a modified strain gradient approach which can capture the size effects under the finite deformation is constructed in this work. The shear problem is then solved to carry out the feasibility study of the developed model on the size effect phenomenon. Lastly, a plane strain problem under uniaxial tensile loading with shear bands is examined to perform the mesh sensitivity tests of the model during softening.

Comparative Modeling of Power Hardening Micro-scale Metallic Plates Based on Lower and Higher-Order Strain Gradient Plasticity Theories

Present study investigates a comparative study of lower and higher-order strain gradient plasticity (SGP) theories involving the size-dependent micromechanically flexural behaviors of crystalline thin plates. The investigation includes the Mechanism-Based and the Chen–Wang SGP models established on the Taylor dislocation hardening by evoking the statistically stored dislocations and geometrically necessary dislocations. In addition, these models are conjugated with a multiple plastic work-hardening law proposed for the microstructural applications of the SGP. An analytical approach based on energy minimizing method is used for obtaining deflection values in terms of the length scale, exponent of the work-hardening and the tangential module. The obtained results indicate a meaningful dependence of the deflections to the length scale, plastic work hardening and other parameters as well.

Establishment of damage estimation rules for brittle fracture after cyclic plastic prestrain in steel

The microprocess of brittle fracture in steel is suggested to occur because microcracks appearing in the brittle layer at the grain boundary propagate driven by increased piled-up dislocation. Thus, loading plastic deformation, equivalent to the increase in dislocation, is directly connected to the increase in brittle fracture risk. Because of these mechanisms, prestraining means approaching the limits of the material, which could lead to deterioration of fracture toughness. For a lifetime evaluation of steel structure, it is important to generalize the effect of various cyclic prestrains.In this paper, material toughness changes when tensile and compression prestrain occur were investigated. A general three-point bending test was employed to evaluate fracture. Synthetic analysis using Conventional Mechanism-based Strain Gradient Plasticity (CMSGP) theory, in which dislocation density is used to determine the amount of material damage, was carried out as well as employing conventional macroscopic material damage consideration rules. The consistency of the simulation and the experimental results were supported by the Electron Back Scattered Diffraction Pattern (EBSD) observation. Additionally, critical stress was calculated, and the change in critical stress for prestrain conditions was investigated. Then, the mechanism of critical stress changes could be expressed by the yield point increase and dislocation density.