Strain Gradient Plasticity Research Articles

Fractional strain gradient plasticity and ductile fracture of metals

We present an optimal scaling analysis based on (possibly fractional) strain-gradient plasticity. The analysis yields optimal scaling laws, in the sense of upper and lower bounds of a power-law type with matching exponents, connecting macroscopic fracture properties, such as the critical elongation at failure and the specific fracture energy, to microscopic mechanisms such as cleavage and microplasticity. We show that an optimal upper bound can be derived from an exceedingly simple test deformation that opens up a sheet of parallelepipedic voids. We also show that an optimal lower bound can be obtained by relaxing compatibility between transverse fibers, which effectively renders the analysis one-dimensional. The analysis predicts a ‘gating effect’ of the surface energy. Specifically, a critical surface energy arises from the analysis that marks a sharp transition between brittle and ductile behavior. When the surface energy of the material exceeds the threshold value, the macroscopic specific fracture energy is predicted to rise sharply as a power of the surface energy with an exponent defined precisely by the theory.

Toward the development of plasticity theories for application to small-scale metal structures.

Experiments are performed on micron-scale single-crystal prototypical structural elements experiencing combined torsion and bending to gather data on their load-carrying capacity in the range of size and strain relevant to micron-scale structures for which little data are available. The observed strengthening dependence on size for the structural elements is in general accord with trends inferred from prior tests such as indentation and pure torsion. In addition, the experiments systematically reveal the strengthening size-dependence of structural elements whose surface has been passivated by a very thin Cr coating, an effect shown to have substantial strengthening potential. A state-of-the-art strain gradient plasticity theory is used to analyze the structural elements over the entire range of size and loading. While the computed trends replicate the experimental trends with reasonable fidelity, the predictive exercise, which is representative of those that will be required in micron-scale structural analysis, brings to light constitutive and computational issues which will have to be addressed before micron-scale plasticity theory can serve as effectively at the micron scale as conventional plasticity does at larger scales.

Indentation Size Effect in Electrodeposited Nickel with Different Grain Size and Crystal Orientation

Indentation size effect at shallow indentation depths still remains a challenge as it cannot be correctly described by the Nix–Gao model based on the concept of strain gradient plasticity and geometrically necessary dislocations. The reasons for this discrepancy may be various, and multiple microstructural factors may play a role at the nanoscale. In the present paper, the breakdown of the Nix–Gao model was explored in electrodeposited nickel with different grain size/shape and crystallographic orientation. Crystallographic orientation has no significant effect on the indentation process at shallow depths if plastic deformation has already developed. On the other hand, decreasing the grain size leads to constrained plastic deformation in the grains below the indenter and to an effective plastic zone expansion. Further grain refinement down to the nanograin material leads to a change in the plastic deformation mechanisms to grain boundary-mediated deformation and a more pronounced breakdown of the Nix–Gao model.

Coupling effect of strain gradient strengthening and thermal softening on the microscale plastic behavior of metallic materials

Size effects and thermal effects together determine the microscale plastic response of metallic structures in high-integrated microelectromechanical systems (MEMS) at different temperatures. To investigate the microscale plastic behavior of metallic materials at different temperatures, a thermo-mechanical coupled microscale plasticity model is developed. In the mechanical part of this model, the framework of conventional plasticity theory is maintained, and the plastic strain gradient is introduced as an internal variable to increase the tangential hardening modulus without the introduction of higher-order stress and higher-order boundary conditions. The influence of temperature on the mechanical parameters (e.g. the yield strength) is calibrated by experiment results. In the thermal part of this model, the heat generation in the plastic deformation stage is calculated by the difference between the plastic work and the hardening stored energy influenced by the plastic strain gradient. Due to the strong nonlinearity of the coupled equations, a finite element solution algorithm that combines the radial return method and the staggered solution scheme is proposed. The effectiveness of this model and its solution algorithm is verified by comparisons with the experiment results and a numerical benchmark example. Finally, taking the tensile behavior of a plate with a hole in its center as an example, the coupling effect of strain gradient strengthening and thermal softening on the microscale plastic behavior of metallic materials is investigated. The results show that the microscale plastic behavior of metallic materials at high temperatures depends on the competition between thermal softening and strain gradient strengthening. Our study provides a theoretical basis and a reliable simulation method for the design of MEMS at different temperatures.

Strain gradient plasticity with nonlinear evolutionary energetic higher order stresses

This paper explores the effect of evolutionary-type nonlinear energetic higher order stress inside the strain gradient plasticity framework that allows nonlinearity in dissipation. Based on the fundamental principles of thermodynamics, a multi-term Chaboche-type model has been formulated to describe the evolution of energetic moment stresses that depend on strain gradient and effective plastic strain. The derived evolutionary relation shows a coupling between energetic and dissipative length scales. The energetic part of the higher order stress, related to the GND formation, saturates as effective plastic strain increases during plastic flow. Higher order stress update equation and material tangent are derived using implicit time discretization. The proposed nonlinear energetic moment stress model confirms its applicability compared to experimentally observed responses. Our study demonstrates the importance of incorporating a nonlinear kinematic hardening rule to accurately represent the Bauschinger effect in samples with a thickness of less than 1μm. The cyclic bending experiment with a single slip sample shows that the nonlinear kinematic hardening rule can effectively capture nonlinearity in the experimental observation. Compressed shear layer simulation reveals that consideration of thickness-dependent relaxation coefficient accurately models the experimental observations. Thus, it makes further relevant consideration of a more advanced kinematic hardening rule through the proposed approach.

Operando neutron diffraction reveals mechanisms for controlled strain evolution in 3D printing

Residual stresses affect the performance and reliability of most manufactured goods and are prevalent in casting, welding, and additive manufacturing (AM, 3D printing). Residual stresses are associated with plastic strain gradients accrued due to transient thermal stress. Complex thermal conditions in AM produce similarly complex residual stress patterns. However, measuring real-time effects of processing on stress evolution is not possible with conventional techniques. Here we use operando neutron diffraction to characterize transient phase transformations and lattice strain evolution during AM of a low-temperature transformation steel. Combining diffraction, infrared and simulation data reveals that elastic and plastic strain distributions are controlled by motion of the face-centered cubic and body-centered cubic phase boundary. Our results provide a new pathway to design residual stress states and property distributions within additively manufactured components. These findings will enable control of residual stress distributions for advantages such as improved fatigue life or resistance to stress-corrosion cracking.

Non-Quadratic Strain Gradient Plasticity Theory and Size Effects in Constrained Shear

Abstract A previously proposed strain gradient plasticity theory is extended to incorporate a non-quadratic power law function of the plastic strain gradient in the free energy expression with an exponent of N + 1. The values of N are taken to vary from N = 1 to N = 0. A simple shear problem of a metal layer between rigid boundaries is analyzed. Two stages of plastic deformation are considered. In stage I, the plastic strain is taken to be zero at the boundaries. Stage I ends when a specified magnitude of the plastic strain gradient is attained at the boundaries. In stage II, the magnitude of the plastic strain gradient at the boundaries is fixed at the specified value. With N = 0, a critical plastic strain gradient cannot be specified at the boundaries because the plastic strain gradient is infinite at the boundaries. The theory with N = 0 predicts a constant plateau stress immediately after initial yield, and the dependence of the plateau stress on the layer thickness can fit experimentally observed plateau stress values. However, with N = 0, a stress gap occurs between the initial yield stress and the plateau stress. The theory with 0 < N ≤ 1 and with stage II also can reproduce the experimentally observed dependence of the plateau stress on the layer thickness for any value of N in that range, with an appropriate value of critical plastic strain gradient at the boundaries. The solution for 0 < N ≤ 1 includes that for N = 0 as a limiting case.

Spherical cavity expansion method dependent on strain, strain rate, and temperature

This paper presents a spherical cavity expansion method based on the self-similar field considering dynamic hardening, which includes plastic strain, plastic strain rate, and temperature for an infinite compressible medium under J2 plasticity. This study proposes an approach based on the gradients of plastic strain and temperature under self-similarity transformation to deal with dynamic hardening models such as the Johnson-Cook (JC) and the modified Johnson-Cook (mJC) models. A theoretical model shows that the effective and radial stresses are reduced near the cavity surface and increased away from the cavity under the mutual influence of hardening and thermal softening. The JC hardening model is used to analyze the behavior of the cavity expansion for AISI 4340 steel. The proposed theoretical model was validated by comparing the result of the explicit finite element method (FEM). This study discusses the behavior of the expanding cavity problem for thermal softening and strain rate sensitivities. In addition, rigid penetration problems are simulated to analyze the application of the presented method. FEM and the proposed cavity expansion method were compared with the penetration depth of AA6061-T651 target using the mJC hardening model for a hemispherical-nosed penetrator. A comparison is made for the performance of the penetration depth with an ogive-nose projectile for AA7075-T651 and AA6061-T651 materials using the mJC hardening model. This study concludes that the proposed method can be applied without modifying the hardening models to spherical cavity expansion problems that involve dynamic hardening models dependent on strain, strain rate, and temperature under adiabatic heat.

A theory of coupled strain-gradient plasticity with species transport at small deformations

This work presents a mechanical isotropic rate-independent theory for plastically deformed materials coupled with species transport. The mass and virtual power balances are natural ingredients used in this work to obtain appropriate local balance laws for species transport, macroscopic, and microscopic forces. The second law of thermodynamics is another key tool used to obtain thermodynamically consistent constitutive relations for the species flux and microscopic stresses. The free energy is approximated as a quadratic form and used to obtain the energetic microscopic stresses as linear combinations of their respective energy conjugates and the species densities. Rate-independent Mises flow rule is deduced in terms of accumulated plastic strain and species density. Furthermore, variational formulation for the coupled theory is obtained as a variational inequality.

Interface strain gradient enabled high strength and hardening in laminated nanotwinned Cu

Interfaces play a crucial role in mechanical behaviors of laminated materials. In this study, a series of hard/soft nanotwinned Cu laminates with varying interface spacing from 200 to 33 μm are prepared by means of direct current electrodeposition. Simultaneous improvement of strength and work hardening with decreasing interface spacing is found in tensile tests. Extra strengthening and geometrically necessary dislocations (GNDs), but without any strain concentration, are found in the vicinity of interfaces. An obvious strain discrepancy appears across the interface and decreases with decreasing interface spacing. Most importantly, a highest plastic strain gradient appearing in the vicinity of interfaces, regardless of interface spacing, indicates a significant deformation compatibility across the interfaces. The spatially-distributed strengthening mediated by interface contributes for increasing strength and decreasing strain discrepancy by decreasing the interface spacing.

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.

The elastic threshold for strain-gradient plasticity, and comparison of theoretical results with experiments

This work continues an earlier study, aimed at determining the threshold of elastic behaviour for specimens at the microscale. Given the size-dependent nature of the response the theory makes use of a model of rate-independent strain-gradient plasticity to establish conditions for lower and upper bounds to the threshold. The model is based on a family of yield or dissipation functions. In the earlier work bounds were established for one member of the family, in which the dissipation depends on the root mean square of the magnitude of plastic strain and its gradient. In this work bounds are derived based on a second alternative, in which the dependence is linear. These two alternatives are most relevant in determining length scales through fits to experimental results. The second aim of this work is to carry out comparisons between the bounds obtained theoretically, with experimental data on initial yielding for torsion tests. The correlations are in some cases reasonable, while in others there are discrepancies. Generally they illustrate the complexities – for example in accurate experimental determination of the elastic threshold, and of the length scale – in predicting aspects of behaviour at the microscale.

Strain gradient plasticity based on saturating variables

Standard scalar-based strain gradient plasticity models generally incorporate the gradient of the cumulative plastic strain into the constitutive modeling. The following limitations of this approach have been pointed out in the literature: Unlimited enhanced hardening at large strains, unlimited cyclic hardening in bending, strain localization band broadening at large applied strains for softening materials with saturation, and possible vanishing of the yield stress for too high values of the Laplace term in the enhanced hardening function. The present work proposes an alternative scalar-based micromorphic approach that solves most of the listed difficulties. The theory incorporates the effect of the gradient of saturating variables instead of the gradient of ever-increasing cumulative plastic strain. A phenomenological theory is presented that includes an exponentially saturating internal variable and its gradient. The approach is also applied to dislocation density based plasticity where the dislocation density is known to saturate at large strains. The benefits of the new theory are demonstrated by finite element simulations of various boundary value problems: Monotonic and cyclic tension and simple glide of an infinite strip, strain localization in a plate, bending and torsion of bars. Both hardening and softening plasticity laws are addressed illustrating the size-dependent hardening in the former case and the evolution of the finite width of localization bands in the latter. Weak points of the approach, namely the possibility of negative dissipation and of negative yield stress, are also identified.

On elastic gaps in strain gradient plasticity: 3D discrete dislocation dynamics investigation

Although presenting attractive features in dealing with small-scale size effects, strain gradient plasticity (SGP) theories can lead to uncommon phenomena for some boundary value problems. Almost all non-incremental (Gurtin-type) SGP theories including thermodynamically-consistent higher-order dissipation predict elastic gaps under certain non-proportional loading conditions. An elastic gap is defined as a finite change in the equivalent yield stress after an infinitesimal change in the strain conditions, at the occurrence of the non-proportional loading source. The existence of such gaps in reality is largely questioned and represents a major source of uncertainty preventing the development of robust SGP theories for real small-scale applications. Using 3D discrete dislocation dynamics (3D-DDD), the present paper aims at investigating size effects within micron-scale single crystal structures under various non-proportional loading conditions, including tension–compression–passivation, bending–passivation and tension–bending. An in-depth investigation of the occurrence of elastic gaps under these conditions, which are known to entail such gaps when using classical non-incremental SGP theories, is conducted. The obtained 3D-DDD results reproduce well known experimentally confirmed size effects like Hall–Petch effect, Asaro’s type III kinematic hardening and reversible plasticity. However, no evidence of the phenomenon of elastic gaps is found, which constitutes a first indication that this phenomenon may not exist in reality. The simulations are performed on face-centered cubic (FCC) Nickel single grains with cuboid shapes ranging from 2μm to 15μm and different orientations.

Surface layer characterization of 2205 duplex stainless steel subjected to abrasive water jet descaling

Abrasive water jet (AWJ) is an environmentally friendly and acid-free descaling technology, that has the added benefit of strengthening the material surface. However, the effects of AWJ descaling on the surface layer of heterostructured (HS) materials and the application of deformable abrasives involving high cycle times for AWJ descaling have not been extensively reported. In this study, martensitic stainless steel particles with high cycle times were subjected to AWJ descaling to investigate the effect of AWJ on the surface layer characterization of 2205 duplex stainless steel (DSS). The properties of the 2205 DSS surface layer were characterized by the surface morphology, surface roughness, microstructure, and microhardness. When exposed to an AWJ intensity of 0.245 mmA, the surface of 2205 DSS exhibited the formation of extrusion craters and flat debris. Additionally, the surface roughness Ra was measured at 3.778 µm. The gradient hardening layer was formed on the surface of the AWJ specimen, and the depth of the hardening layer was 550 µm. The maximum microhardness was 395 HV, which is approximately 30% higher than that of the original specimen. Within the AWJ specimens, dislocation networks were observed in the ferrite, while the austenite exhibited dislocation cells, deformation twins, and strain-induced martensite. Furthermore, the experimental results indicate that plastic deformation was unevenly distributed between the two phases, with the majority of deformation occurring within the austenite phase. Owing to the requirement of deformation compatibility, several of geometrically necessary dislocations (GNDs) accumulated near the austenitic phase boundary to regulate the plastic strain gradient, and the hetero-deformation induced (HDI) stress generated by these dislocations strengthens the austenite. The AWJ refines the surface material structure; increases the density of dislocations, phase boundaries, and stacking faults; forms strain-induced martensite; and introduces a residual compressive stress field, which improves the material surface hardness.

Contact of Rough Surfaces: An Incremental Model Accounting for Strain Gradient Plasticity

In the contact of rough surfaces, most contact patches are at the scale of micrometers, and thus, their contact deformation can be dominated by the size-dependent plasticity. In this paper, we propose a new strategy to analyze the role of strain gradient plasticity in the contact response between a realistic rough surface and a rigid plane, which modifies the incremental contact model based on the mechanism-based gradient plasticity (MSGP) theory. For several different rough surfaces with their topography measured experimentally, the relations between applied load and real contact area are derived in a simple but effective way. It is found that strain gradient plasticity significantly increases the level of mean contact pressure. The hardening effect caused by strain gradient plasticity weakens somewhat as the contact area increases. Compared with previous methods, the present model might be more efficient and of wider application.

Strain gradient plasticity phenomenon in surface treated plain carbon steel

Surface engineering is essential for materials being used for functional and structural applications. There have been various approaches and mechanisms used for improving, mechanical, tribological and surface properties of materials over the years. One phenomenon that require much attention is indentation size effect (ISE). Mechanistic and phenomenological theories have been developed to explain the observed size dependence of mechanical properties in materials. However, the dislocations microstructure evolution, deformation mechanisms and the contributions of length scale parameters are not fully understood. This paper explores the effects of surface carbo-nitriding on material length scale and dislocation microstructure mechanisms of 1045 steel. The material was carbo-nitrided at 500 °C and 900 °C, to induce nitrogen and carbon diffusion respectively, prior to, microstructural and mechanical (micro/nano-indentation) characterization. The statistically stored and geometrically necessary dislocation densities increased with increasing carbo-nitriding temperature. A bi-linear relationship was also established based on the indentation depth and hardness measurements. Three regimes of dislocation behaviour (dislocations source–limited/starvation, discrete dislocations, and continuum dislocations) are observed. The observed trends are explained using dislocation densities that were estimated using dislocation theories. The implications of the results are also discussed for the carbonitrided steels.

One-Dimensional Failure Modes for Bodies with Non-convex Plastic Energies

In this paper, a complete picture of the different plastic failure modes that can be predicted by the strain gradient plasticity model proposed in Del Piero et al. (J. Mech. Mater. Struct. 8:109–151, 2013) is drawn. The evolution problem of the elasto-plastic strain is formulated in Del Piero et al. (J. Mech. Mater. Struct. 8:109–151, 2013) as an incremental minimization problem acting on an energy functional which includes a local plastic term and a non-local gradient contribution. Here, an approximate analytical solution of the evolution problem is determined in the one-dimensional case of a tensile bar. Different solutions are found describing specific plastic strain processes, and correlations between the different evolution modes and the convexity/concavity properties of the plastic energy density are established.The variety of solutions demonstrates the large versatility of the model in describing many failure mechanisms, ranging from brittle to ductile. Indeed, for a convex plastic energy, the plastic strain diffuses in the body, while, for a concave plastic energy, it localizes in regions whose amplitude depends on the internal length parameter included into the non-local energy term, and, depending on the convexity properties of the first derivative of the plastic energy, the localization band expands or contracts. Complex failure processes combining different modes can be reproduced by assuming plastic energy functionals with specific convex and concave branches.The quasi-brittle failure of geomaterials in simple tension tests was reproduced by assuming a convex-concave plastic energy, and the accuracy of the analytical predictions was checked by comparing them with the numerical results of finite element simulations.

Dispersed strain bands promote the ductility of gradient nano-grained material: A strain gradient constitutive modeling considering damage effect

Gradient nano-grained (GNG) metals have achieved superior strength-ductility synergy than their homogeneous counterparts. The high strength is usually attributed to the grain size effect and hetero-deformation-induced strengthening. However, incommensurate work on ductility leads to an incomplete understanding of the strength-ductility combination. In this work, a dislocation density-based strain gradient plasticity model coupling with a damage model is developed to describe the strain hardening and softening behavior of GNG material. A grain size-dependent back stress model derived from the generation of dislocation pileups is invoked to describe the widely-concerned back stress hardening. Finite element implementation of the model quantitatively predicts the tensile response of GNG nickel with various degrees of grain size gradient. The results reveal that dispersed strain bands propagate stably in the nano-grained surface layer of GNG material, which is totally different from those occurring in a freestanding nano-grained material. The stabilization of dispersed strain bands enables the nano-grained layer to deform uniformly, thus premature failure of the whole GNG material is suppressed and improved ductility is achieved. Furthermore, increasing the grain size gradient renders the strain bands more stable, leading to enhanced ductility. The method developed in this work is helpful for understanding the strength-ductility synergy of GNG materials and for optimizing the microstructure gradient in GNG materials.

A self - Consistent higher-order strain gradient plasticity analysis of micropolar dual-phase crystalline FGM made of Titanium-boride/ Titanium

A self-consistent higher order couple stress model is based on Fleck - Hutchinson Strain Gradient Plasticity to consider flexural behavior of brittle - ductile microscale structures. The dual phasic functionally graded material made of Titanium-boride/Titanium structure is modeled by defining a material flow stress which reflects a parameter of combined effective length scale, dependent on Geometrically Necessary Dislocations. The dual phasic crystalline micro-beam is built with the properties having continuously and non-linearly variation along the thickness. The perfectly plastic behavior of micropolar functionally graded material is described by incorporating a ductile –like mixture law, so-called TTO model, possessing a simple possibility of transition through defining a graded phase acting as an equilibrium mixed phase. Further, the effects of the elastic foundation and temperature rise on the plastic deformations are investigated. While an energy method is used to derive governing equations, the obtained results are verified individually in three steps.