Mechanism-based Strain Gradient Plasticity Research Articles

Effect of stress/strain partition on the mechanical behavior of heterostructured laminates: A strain gradient plasticity modeling

Based on the structure of natural nacres, laminated metal materials have been successfully developed to possess desirable mechanical properties for diverse engineering applications. These heterostructured (HS) laminates consist of layers with different mechanical performances, resulting in asynchronous deformation and intricate stress/strain partitioning behavior. These distinctive phenomena are crucial in achieving the strength-ductility synergy of HS laminates. Nonetheless, further investigations are imperative to gain a comprehensive understanding of local stress and strain evolutions during deformation, and to thoroughly explore the influence of stress/strain partition on the mechanical behavior of the HS laminates. This study utilized the conventional mechanism-based strain gradient (CMSG) plasticity theory to conduct finite element simulations. The tensile deformation of HS laminate comprising nanostructured (NS) bronze and coarse-grained (CG) copper layers was simulated as a benchmark. The simulation results revealed the following: (1) Stress partition and stress transfer between the layers occur during deformation and are influenced by the layer thickness; (2) Both strain partition and strain banding exist in the HS laminates with relatively thinner layers, and the formation of dispersed strain bands is related to the plastic strain gradient; (3) Heterogeneous deformation promotes strain delocalization and strain hardening, consequently leading to improved uniform elongation. This study provides valuable insights into the significant effect of stress/strain partition on the strength-ductility synergy of HS laminates from the perspective of constitutive modeling.

The coupling effects of strain gradient and damage on Mode I crack tip stress fields

The cleavage fracture of metallic materials is determined by the crack tip stress field depending on plastic strain gradient and microstructural damage. The synergistic effects of strain gradient and damage on the stress fields near a Mode I crack tip in metallic materials are investigated using a modified conventional theory of mechanism-based strain gradient plasticity (CMSG). Besides preserving the essence of classical CMSG, a stress-dependent damage variable is introduced to characterize the effect of microstructural damage on the material's intrinsic length, elastic modulus and plastic yielding criterion. Based on the present theoretical model, it is found that the strain gradient and microstructural damage effects both are prominent in a small region near the Mode I crack tip, at which the damage evolution is determined by the strain gradient. Although the crack tip stress field with strain gradient and damage effects is lower than the existing ones influenced only by strain gradient, it is still significantly higher compared with the yielding strength of the material so that the cleavage fracture at the crack tip can be well explained. The damage-induced stress level reduction indicates that the stress concentration at the crack tip can be alleviated to inhibit the crack propagation. As a result, when the damage effect is considered, not only a larger external load is required to induce crack propagation but also the plastic region in front of the crack tip is enlarged, consequently leading to increases of fracture strength and energy dissipation in the material. The present research provides a more precise prediction of the crack tip stress distribution in metallic materials, which is helpful for better understanding the microscopic mechanism of the cleavage fracture phenomenon.

Micromechanical modeling of damage evolution and fracture behavior in particle reinforced metal matrix composites based on the conventional theory of mechanism-based strain gradient plasticity

The particle/matrix interface of metal matrix composites (MMCs) can give rise to extra strength and then affect the local deformation behavior. This strengthening effect originates from the plastic strain gradients due to the incompatibility of plastic deformation between the particle and matrix. However, only limited researches utilized the strain gradient plasticity to study the damage evolution and fracture behavior, in which only one or two of the damage mechanisms (i.e., matrix damage, interface debonding, and particle fracture) was considered. In this work, all of these damage mechanisms were coupled into the finite element model under conventional theory of mechanism-based strain-gradient (CMSG) plasticity. Besides, a new numerical algorithm of geometrically necessary dislocations (GNDs) was proposed for a multi-scale model, and then this model was used to analyze the damage evolution and failure behavior of SiCp/AZ91 composite. The results show that the strengthening effects of plastic strain gradients can describe interface debonding, break the monotonicity of the effective plastic strain with the effective stress at the local area close to the particle/matrix interface, and give a more reasonable distribution of stress and plastic strain compared to the classical J2 flow theory. If the probability distribution of the interfacial strength is considered, the CMSG model has the potential capacity to capture the crack initiation in the matrix. When using the continuum damage mechanics approach to describe the fracture process based on the multi-scale model, the weakening exponent value should be considered.

An Improved Mechanism-Based Strain Gradient Plasticity Model and Its Application to Size Effect Under Complex Loading

The ingredient devices tend to be designed and fabricated with microscale, high accuracy, and complex geometry and are subjected to complex loading. Many experiments have proved that the size effect plays a significant role in designing and manufacturing an engineering component. This size effect is largely attributed to the grain size and the strain gradient. While the grain size effect was omitted in conventional strain gradient theories, an extended model that considers both effects of grain size and strain gradient has been proposed in the current work (denoted as GMSG). The proposed GMSG model has been implemented into the finite element method (denoted as GMSG-FEM) to investigate the size effect of copper wires under complex working conditions. The simulation results show that the hollow structure can improve the bearing capacity of the micro-wires under torsion. Among micro-wires with different cross-sections, the bearing capacity of the micro-wire with a circular section is the largest, followed by those with square and triangle sections. Complex loading also has an important influence on stress distribution. Based on the current study, it can be envisaged that the GMSG-FEM could be a useful tool in engineering applications where the size effect has to be considered.

Contact analysis between rough surfaces considering the size-affected deformation behaviour of multi-scale asperities

Contact parameters such as the real contact area of rough surfaces depend on the interaction between asperities and the characteristic scale in which they remain. The high-stress deformation mechanism of the local contact spots exhibits size-dependent behaviour when the size of the asperities reaches a certain scale because of the scale effect. This causes deviations in the contact analysis, which can affect several tribological properties of the structural interface such as interface stiffness, fretting wear, and contact resistance. In this study, the deformation process affected by the scale effect is analysed using a finite element method that employs a conventional mechanism-based strain gradient plasticity (CMSGP) constitutive theory, whereas a full-scale elastoplastic contact model is developed for asperities at different size levels. Next, the contact analysis of rough surfaces is performed using the multi-scale contact theory proposed by Jackson and Streator (JS) considering this individual asperity model. The proposed model is validated using comparison with the refined finite element simulation analysis of rough surface contact that employs the CMSGP theory; further, the effect of morphology features on contact behaviour is discussed. The research results are conducive to the tribological design of small-size engineering interfaces.

Cohesive zone modelling of hydrogen assisted fatigue crack growth: The role of trapping

We investigate the influence of microstructural traps in hydrogen-assisted fatigue crack growth. To this end, a new formulation combining multi-trap stress-assisted diffusion, mechanism-based strain gradient plasticity and a hydrogen- and fatigue-dependent cohesive zone model is presented and numerically implemented. The results show that the ratio of loading frequency to effective diffusivity governs fatigue crack growth behaviour. Increasing the density of beneficial traps, not involved in the fracture process, results in lower fatigue crack growth rates. The combinations of loading frequency and carbide trap densities that minimise embrittlement susceptibility are identified, providing the foundation for a rational design of hydrogen-resistant alloys.

Inversion of dislocation densities under mixed mode fracture

Mixed-mode crack growth is here investigated through experiments and computations for 34X and P2M steels, 7050 aluminium, and Ti-6Al-4V alloys in a compact tension shear (CTS) specimen. In this study, we use the mechanism-based strain gradient (MSG) plasticity theory to evaluate both crack tip dislocation density behaviour and the coupled effect of the material plastic properties and the intrinsic material length on local stress distributions. The constitutive relations are based on Taylor’s dislocation model, which allows to gain insights into the role of the increased dislocation density associated with large gradients in plastic strain near cracks. The material model is implemented in a commercial finite element (FE) software package using a user subroutine, and the nonlinear stress intensity factors (SIFs) are evaluated as a function of the intrinsic material length. As a result of the FE calculations of dislocation density distributions, the effects of both the fracture mode and the stress–strain state are determined. Strain gradient effects associated with dislocation hardening mechanisms elevate crack tip stresses relative to conventional plasticity. Dislocation densities, stress fields and nonlinear SIF solutions are determined for experimental curvilinear crack paths by taking into account the transition from the initial Mode II crack to the mixed-mode fracture.

Experimental and numerical investigations of micro-meso damage evolution for a WC/Co-type tool material

Commercial Co/WC/diamond composites with 90vol.% Co also belong to hard metals and, as a kind of tool materials, are very useful. Their deformation behavior can be both ductile and quasi-brittle, determined by the diamond portion and local morphology. Another characteristic is that submicron-sized WC particles, possessing non-negligible strengthening influence due to the size effect, cannot be fully present in a representative microstructure. This work emphasizes the local damage evolutions’ dependence on microstructural features. Rice&Tracey damage and cohesive zone model describe the ductile and quasi-brittle damage behavior. The mechanism-based strain gradient plasticity takes the size effect of submicron-sized WC particles into consideration. Both real and artificial microstructures are used. Besides homogeneous boundary conditions (BCs), the periodic BCs are also applied in a 2D damage simulation. This work proves that FE models with two phases, the homogenized Co-WC matrix and diamond particles, can correctly predict damage evolution. FE results show that the WC phase has a higher mean stress value than the diamond phase, which is proved by the nano-indentation test. From FE simulation results, local hot spots appear in the matrix closed to sharp diamond corners/edges and crossing regions of shear bands. The experimental and numerical results are compared on micro and macro scales. For the local strain distribution and the damage development, numerical predictions match the reality well, even in morphological details. Furthermore, since the published data about WC-Co type tool materials with Co>50vol.% are rare, the obtained knowledge in this work also contributes to the data collection.

Crack root radius effect on stress fields under nonlinear

The influence of crack root radius and plastic material properties on the crack tip fields was evaluated by strain gradient plasticity theory, and classical plasticity theory. A conventional mechanism-based strain gradient (CMSG) plasticity theory was used in the study. The stress fields in single edge tension specimen are investigated numerically for a wide range of the crack tip radius variations. Five values of the crack tip curvature radius ρ = 0 (mathematical notch); ρ = 25 nm, ρ = 30 nm, ρ = 60 nm and ρ = 100 nm were investigated. The strain hardening exponent varied from N = 0.075 to N = 0.4. A stress analysis by CMSG plasticity was performed for two values of the Taylor’s length parameter l = 1μm and l = 10μm. The boundaries of the local areas, where the influence of the crack tip radius is realized were determined for both HRR and CMSG plasticity. The sizes of the dominance area of the strain gradient plasticity were established and their approximation equations were presented. The influence of the plastic properties and the Taylor’s length scale parameter on the stress fields in the vicinity of rhe crack tip was estimated for the strain gradient plasticity.

2021 Timoshenko Medal Acceptance Speech

2021 Timoshenko Medal Acceptance Speech

Mode I and mode II stress intensity factors and dislocation density behaviour in strain gradient plasticity

In this study, we use the mechanism-based strain gradient plasticity theory to evaluate both crack tip dislocation density behaviour and the coupled effect of the material plastic properties and the intrinsic material length on non-linear amplitude factors. The two planar classical stress–strain states are examined, namely, plane strain and plane stress, both under pure mode I and pure mode II loading conditions. The constitutive relations are based on Taylor’s dislocation model, which enables gaining insights into the role of the increased dislocation density associated with large gradients in plastic strain near cracks. The material model is implemented in a commercial finite element (FE) software package using a user subroutine, and the nonlinear stress intensity factors (SIF) are evaluated as a function of the intrinsic material length, characterising the scale at which gradient effects become significant. As a result of the FE calculations of dislocation density distributions, the effects of both the fracture mode and the stress–strain state are determined. In pure mode I, the geometrically necessary dislocation (GND) density is located symmetrically with respect to the blunted crack tip. On the contrary, under pure mode II, the GND density becomes concentrated in the blunted and sharp parts of the crack tip. In this case, fracture initiation is shown to be likely to occur near the blunted region of the crack tip, where both the stress triaxiality and the GND density are at their maximum. The relation between the equilibrium state of dislocation densities and the intrinsic material length as well as the plastic SIF as a function of the work hardening exponent is discussed.

Effective acquisition of elastoplastic and creep parameters of lead-free solder alloy from high-temperature micro-indentation eliminating the size effect

Effective acquisition of the related mechanical properties of lead-free solder alloys through micro-indentation tests is of great significance in the research on electronic interconnection design and reliability for the multi-functionalization and high integration of electronic products. The temperature-dependent elastoplastic and creep parameters of Sn–3.0Ag–0.5Cu (SAC) solder are systematically investigated by micro-indentation with plural indenters and the “rapid loading-holding-rapid unloading” test mode. Based on the depth dependence of indentation hardness from mechanism-based strain gradient plasticity and the stress independence of creep compliance for linear viscous assumption, the complete indentation load-depth curves eliminating the indentation size effect of SAC solder under different temperature and two pyramidal indenters with different equivalent semi-cone angles are reconstructed. Combining with the temperature-dependent elastic modulus measurements from cylindrical indentation, corrected load-depth curves for pyramidal indentation, and the concept of representative stress and representative plastic strain, the elastoplastic parameters eliminating the indentation size effect under different temperature are uniquely determined through numerical simulation and inversion analysis, as well as the corresponding creep parameters. The temperature dependence of elastoplastic and creep parameters of SAC solder is analyzed, and related deformation mechanism is discussed. Results of forward and inverse analyses indicate the effectiveness of proposed characterization method.

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.

Experimental and Numerical Investigations of Micro-Meso Damage Evolution for a WC/Co-Type Tool Material

Commercial Co/WC/diamond composites with 90vol.% Co also belong to hard metals and, as a kind of tool materials, are very useful. Their deformation behavior can be both ductile and quasi-brittle, determined by the diamond portion and local morphology. Another characteristic is that submicron-sized WC particles, possessing non-negligible strengthening influence due to the size effect, cannot be fully present in a representative microstructure. This work emphasizes the local damage evolutions’ dependence on microstructural features. Rice&Tracey damage and cohesive zone model describe the ductile and quasi-brittle damage behavior. The mechanism-based strain gradient plasticity takes the size effect of submicron-sized WC particles into consideration. Both real and artificial microstructures are used. Besides homogeneous boundary conditions (BCs), the periodic BCs are also applied in a 2D damage simulation. This work proves that FE models with two phases, the homogenized Co-WC matrix and diamond particles, can correctly predict damage evolution. FE results show that theWC phase has a higher mean stress value than the diamond phase, which is proved by the nano-indentation test. From FE simulation results, local hot spots appear in the matrix closed to sharp diamond corners/edges and crossing regions of shear bands. The experimental and numerical results are compared on micro and macro scales. For the local strain distribution and the damage development, numerical predictions match the reality well, even in morphological details. Furthermore, since the published data about WC-Co type tool materials with Co>50vol.% are rare, the obtained knowledge in this work also contributes to the data collection.

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.

Size effect on the damage evolution of a modified GTN model under high/low stress triaxiality in meso-scaled plastic deformation

Micro-forming is a micro-manufacturing method with broad application prospects, in which the size effect on damage development under different stress states is an allimportant problem, but has not been systematically explained. This study established a coupled model by the combination of mechanism-based strain gradient plasticity (MSG) and a shear modified GTN damage model. The proposed model is implemented in UMAT subroutine of ABAQUS and validated by experimental observation and numerical simulation of notched specimens with different shapes. Fracture photographs suggested that the failure mechanism of materials under high/low stress triaxiality is nucleation, growth and coalescence of voids and shear-induced slip respectively. The fracture displacement and loading force obtained by the numerical simulation considering the coupling model are basically consistent with the tensile test results. Furthermore, the size effect on two damage parameters is qualitatively described, that the MSG theory can promote the evolution of shear damage and inhibit the development of microvoids. In addition, the influence mechanism of the strain gradient is discussed in detail, and the relationship among the equivalent stress, stress triaxiality and damage parameters is revealed. The present work thus facilitates the in-depth understanding of the size effect on damage evolution and fracture formation in meso-scale of metallic materials.

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.

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 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.