Dislocation Mean Free Path Research Articles

Efficient modeling of microstructure evolution in magnesium by energy minimization

The description of the complex interplay between deformation-induced twinning and dislocation slip, typical for metals showing an hcp structure such as magnesium, is of utmost importance for understanding their deformation behavior. In the present paper, an incremental energy principle is presented for that purpose. Within this principle, dislocation slip is modeled by crystal plasticity theory, while the phase decomposition associated with twinning is considered by a mixture theory. This mixture theory naturally avoids the decomposition of the twinning process into so-called pseudo- dislocations followed by a reorientation of the total crystal. By way of contrast, the proposed model captures the transformation of the crystal lattice due to twinning in a continuous fashion by simultaneously taking dislocation slip within both, possibly co-existent, phases into account. The shear strain induced by twinning as well as the deformation history are consistently included within the twinned domain by an enhanced multiplicative decomposition of the deformation gradient. Kinematic compatibility between the different phases is enforced by a Hadamard-type compatibility condition, while compatibility with respect to the boundary conditions requires the introduction of a boundary layer. The evolution of all state variables such as the twinning volume and the plastic strains associated with dislocation slip follow jointly and conveniently from minimizing the stress power of the total crystal. This canonical variational principle is closely related to the postulate of maximum dissipation and guarantees thermodynamical consistency of the resulting model. Particularly, the second law of thermodynamics is fulfilled. In contrast to previous models suitable for the analysis of the deformation systems in magnesium, the Helmholtz energy of the twinning interfaces and that of the aforementioned boundary layer are considered. Analogously, the energy due to twinning nucleation and that related to twinning growth are accounted for by suitable dissipation functionals. By doing so, the number of twinning laminates becomes an additional unknown within the minimization principle and thus, the thickness of the lamellas can be computed. Interestingly, by interpreting this thickness as the mean free path of dislocations, a size effect of Hall–Petch-type can naturally be included within the novel model. The predictive capabilities of the resulting approach are finally demonstrated by analyzing the channel die test. For that purpose, a certain rank-two laminate structure is considered. However, it bears emphasis that the proposed framework is very general and consequently, it can also be applied to other materials.

Size effects in yield and plasticity under uniaxial and non-uniform loading: experiment and theory

The size of microstructural features has long been known to determine the strength of materials, as in the Hall–Petch effect of grain size. More recently, the importance of the size of the structure itself (thin foil, wire or pillar) or of the loaded volume (indentation) has been recognised. Many phenomenological theories have been proposed to account for the size effect. Here, experimental data of very high accuracy are reported on copper wire in torsion which distinguish the size effects of grain size and structure size on the elastic limit and on the flow stress at low plastic strain. The data are compared with less accurate data from similar wires in tension. The size effects are shown to arise by first-principles analysis of dislocation behaviour, exploiting the slip-distance theory of Conrad together with forest (Taylor) work-hardening theory, but with modifications to both theories to account for finite structure size. The resulting theory is compatible with the concepts of dislocation starvation and of strain-gradient plasticity. The size effect in the elastic limit is due to the constraint the size puts on dislocation curvature, while the size effect on the flow stress is due to the constraint size puts on dislocation mean free path and to the fate of a dislocation after it has moved.

Intrinsic Effects due to the Reduction of Thickness on the Mechanical Behavior of Nickel Polycrystals

Uniaxial tensile tests were performed on high-purity nickel polycrystals with thicknesses (t) ranging from 12.5 to 3200 μm. The grain size (d) of each sample was set to a value close to 100 μm in order to avoid grain size effects. Experimental results highlight the intrinsic effect of the thickness on the mechanical properties of nickel. For plastic strains higher than 0.01, a strong softening of the flow stress is clearly observed when the t/d ratio decreases from four to one. This critical range is independent of the plastic strain. Transmission electron microscopy (TEM) investigations of dislocation structures were carried out in core and surface regions of samples plastically strained in the second hardening stage. It is shown that the mean dislocation cell diameter obtains values two or three times higher near free surfaces than in core areas for specimens with t/d ratios lying at a range of 1 to 4. This observation can be interpreted in terms of an increase in the mean free path of mobile dislocations in the second hardening stage. The transition between polycrystal and single-crystal behavior for nickel with constant grain size and various thicknesses is, therefore, an intrinsic effect related to progressively stronger surface softening effects.

Deformation twinning as a strengthening mechanism in microtruss cellular materials

The use of deformation twinning as a strengthening mechanism in microtruss cellular materials is described for the first time. Plastic strain was imparted to the high-Mn steel microtruss struts during stretch–bend fabrication. The extensive twinning reduced the mean free path of subsequent dislocations, increasing the flow stress and work-hardening rate within the microtruss struts. Over the range of architectures studied, the presence or absence of these deformation twins determined the overall compressive strength of the cellular material.

Dislocation subgrain structures and modeling the plastic hardening of metallic single crystals

A single crystal plasticity theory for insertion into finite element simulation is formulated using sequential laminates to model subgrain dislocation structures. It is known that local models do not adequately account for latent hardening, as latent hardening is not only a material property, but a nonlocal property (e.g. grain size and shape). The addition of the nonlocal energy from the formation of subgrain structure dislocation walls and the boundary layer misfits provide both latent and self-hardening of a crystal slip. Latent hardening occurs as the formation of new dislocation walls limits motion of new mobile dislocations, thus hardening future slip systems. Self-hardening is accomplished by an evolution of the subgrain structure length scale. The substructure length scale is computed by minimizing the nonlocal energy. The minimization of the nonlocal energy is a competition between the dislocation wall energy and the boundary layer energies. The nonlocal terms are also directly minimized within the subgrain model as they affect deformation response. The geometrical relationship between the dislocation walls and slip planes affecting the dislocation mean free path is taken into account, giving a first-order approximation to shape effects. A coplanar slip model is developed due to requirements while modeling the subgrain structure. This subgrain structure plasticity model is noteworthy as all material parameters are experimentally determined rather than fit. The model also has an inherit path dependence due to the formation of the subgrain structures. Validation is accomplished by comparison with single crystal tension test results.

Flow stress analysis of TWIP steel via the XRD measurement of dislocation density

In this study, the rate of dislocation accumulation in the tensile strained twinning induced plasticity (TWIP) steel was calculated via the X-ray diffraction (XRD) measurements and compared with other fcc metals and alloys. The results indicated that the XRD technique is an alternative method to estimate the dislocation density. Moreover, flow stress analysis of Fe–31Mn–3Al–3Si TWIP steel with the grain size of about 18 μm indicated that, beside a direct effect of the dislocation interactions on the flow stress, another strengthening mechanism is also required to describe the flow behavior. For this reason, the strengthening contribution due to the formation of mechanical twins was considered as a reduction of dislocation mean free path. Interestingly, the estimated flow stress equation consisting of the strengthening effects of both dislocation interactions and dynamic microstructure refinement due to mechanical twinning (i.e., the dynamic Hall–Petch effect) are in good agreement with the experimental data and equation proposed by Ludwigson for low SFE materials.

The deformation stage II of face-centered cubic crystals: Fifty years of investigations

Abstract This article critically reviews progress in the understanding of strain hardening during the deformation stage II of pure face-centered cubic crystals since its discovery in the mid-1950s. A wealth of models attempted explaining why stage II exhibits a linear slope on the base of specific dislocation configurations and interactions associated with short- or long-range internal stresses. Slip trace observations and more recent investigations on dislocation avalanches led to the identification of intermittent elementary slip events that are now investigated in terms of dislocation mechanisms. Numerical estimates showed that the overcoming of junctions and locks accounts for most of the flow stress and the hardening rate, as assumed by the present forest models. A multiscale analysis of stage II is outlined with emphasis on the modeling of dislocation mean free paths by dislocation dynamics simulations.

Microstructure evolution during cold rolling in a nanocrystalline Ni–Fe alloy determined by synchrotron X-ray diffraction

Stress softening after cold rolling is observed in an electrodeposited nanocrystalline Ni–Fe alloy. The grain-size distribution becomes much broader after the cold rolling. Microstructure changes, though moderate, such as simultaneously decreased dislocation and twin densities with grain growth during cold rolling, are systematically proved by synchrotron high-energy X-ray diffraction, transmission electron microscopy and differential scanning calorimetry (DSC). The amorphous fractions in the form of grain boundaries are evidenced by the diffuse-background scatterings and large DSC values. Partial dislocation separation calculation, a dislocation mean free path and annihilation model, and texture development together reveal that the current nanocrystalline Ni–Fe alloy exhibits the combined behavior of perfect dislocation slip and grain-boundary mediated deformation.

Electric stimulation of magnetoplasticity and magnetic hardening in crystals

The effect of electric field E on the magnetoplastic effect (MPE) has been investigated in NaCl crystals with different impurities, which provide either the plasticization of the samples in the magnetic field (positive MPE) or their magnetic hardening (negative MPE). The mobility of individual dislocations under the joint action of the magnetic and electric fields and the mechanical load on the crystals has been studied. The sharp electric stimulation of the MPE of both signs has been revealed, i.e., an increase or a decrease in the mean free path of dislocations that is roughly proportional to exp(±E/E0) at E ≳ E0 ∼ 1–10 kV/m. In particular, in the negative-MPE NaCl(Pb) crystals, the accompanying electric field enhances the magnetic suppression of plasticity. The results are attributed to the electrically induced transformation of the additional part of the pinning impurity ions Me++ to the magnetically active state of Me+ on the dislocations. The subsequent magnetic transformation of the structure of these pinning centers should lead to a sharper variation of the dislocation pinning force (either an increase or a decrease, depending on the MPE sign).

Modeling dislocation storage rates and mean free paths in face-centered cubic crystals

The first part of a dislocation-based constitutive formulation for strain hardening in face-centered cubic crystals is presented. This multiscale approach is based on the storage–recovery framework expanded at the scale of slip systems. A parameter-free formulation is established for the critical stress and the storage rate, taking advantage of recent results yielded by dislocation dynamics simulations. The storage rate of dislocations in the presence of forest obstacles is modeled for the first time at the level of dislocation intersections and reactions. The mean free path per slip system is found to be inversely proportional to the critical stress. It also depends on the number of active slip systems, which leads to an orientation dependence of stage II strain hardening in agreement with experimental data. The total storage rate is obtained by including three additional contributions, notably that of the self-interaction, which leads to a model for stage I hardening.

Dislocation Mean Free Paths and Strain Hardening of Crystals

Predicting the strain hardening properties of crystals constitutes a long-standing challenge for dislocation theory. The main difficulty resides in the integration of dislocation processes through a wide range of time and length scales, up to macroscopic dimensions. In the present multiscale approach, dislocation dynamics simulations are used to establish a dislocation-based continuum model incorporating discrete and intermittent aspects of plastic flow. This is performed through the modeling of a key quantity, the mean free path of dislocations. The model is then integrated at the scale of bulk crystals, which allows for the detailed reproduction of the complex deformation curves of face-centered cubic crystals. Because of its predictive ability, the proposed framework has a large potential for further applications.

Plastic deformation of Al and AA5754 between 4.2 K and 295 K

Plastic deformation and work-hardening behaviour of high-purity Al and AA5754 have been studied by a combination of measurements of mechanical response, electrical resistivity and TEM. The results show that materials deformed at very low temperatures exhibit an unprecedented level of strength and unusual work-hardening behaviour. Strain rate sensitivity measurements suggest that deformation of high-purity Al is governed by dislocation–dislocation interactions in a broad range of temperatures, whereas Al alloys exhibit a larger thermal component of flow stress due to the presence of solute atoms in the matrix. The electrical resistivity data provide information about the evolution of both the dislocation density and the dislocation mean free path and establish the contributions of dislocation storage and dynamic recovery during plastic flow of materials at 4.2 K. The results suggest that fracture is initiated by the collapse of the dislocation network at places where dislocations develop a critical spacing for spontaneous annihilation. This spacing is estimated as ∼8 nm at 4.2 K and ∼12 nm at 78 K.

A Dislocation-Crystal Plasticity Simulation on FCC Single Crystal Considering Geometrically Necessary Dislocation Density and Incompatibility

In the previous paper, the geometrically necessary (GN) incompatibility is newly defined and a new annihilation term of dislocation pairs due to the dynamic recovery is introduced into an expression of dislocation density. Furthermore, a multiscale model of crystal plasticity is proposed by considering the GN dislocation density and incompatibility. However, details of dislocation-crystal plasticity calculation are not given. In this paper, we explain a method of dislocation-crystal plasticity analysis. A finite element simulation is carried out for an f.c.c. single crystal under plane strain tension. It is numerically predicted that micro shear bands are formed at large strain, and sub-GNBs: small angle tilt boundaries are induced along these bands. Furthermore, the annihilation of dislocation pairs and the increase of dislocation mean free path characterizing stage III of work-hardening are computationally predicted.

Recent progress on the study of the microstructure and mechanical properties of ECAE copper

Results on the microstructure and the tensile properties of equal channel angular extruded (ECAE) copper processed for one to 16 passes are presented and compared with the available literature data. With increasing number of passes (N), the microstructure changes from a strongly elongated shear band structure after N = 1 and 2, towards a more equiaxed subgrain and grain structure. This is accompanied by a decrease in the cell wall or subgrain-boundary widths and an increase in recovered or even recrystallised grain structures with low dislocation densities. Electron backscatter diffraction measurements have indicated that for lower N, the location of Σ3 boundaries is restricted to shear bands, while at greater N, Σ3 boundaries were found to be more widely distributed. Texture measurements indicate close similarity with simple shear texture components and a spread of the orientation components with greater N. Upon comparing the tensile behaviour of as-ECAE Cu with the surveyed literature, broad agreement on the strength of the material is achieved. However, a strong variation in the percentage elongation to failure is also noted. Strain hardening and deformation kinetic analysis via strain rate jump tests indicate an evolution from stage III to V hardening during post-ECAE compression and a saturation in the strain rate sensitivity after N = 4 resulting in maximum values of ∼0.02. Our results suggest that rather than a change in deformation mechanism, the increase in ductility with increasing N is associated with an increase in the mean free path of dislocations—with the grain boundaries remaining actively involved as the transmitter of plastic strain and their interaction with dislocations being the rate controlling deformation mechanism.

Recent progress on the study of the microstructure and mechanical properties of ECAE copper

Results on the microstructure and the tensile properties of equal channel angular extruded (ECAE) copper processed for one to 16 passes are presented and compared with the available literature data. With increasing number of passes (N), the microstructure changes from a strongly elongated shear band structure after N = 1 and 2, towards a more equiaxed subgrain and grain structure. This is accompanied by a decrease in the cell wall or subgrain-boundary widths and an increase in recovered or even recrystallised grain structures with low dislocation densities. Electron backscatter diffraction measurements have indicated that for lower N, the location of Σ3 boundaries is restricted to shear bands, while at greater N, Σ3 boundaries were found to be more widely distributed. Texture measurements indicate close similarity with simple shear texture components and a spread of the orientation components with greater N. Upon comparing the tensile behaviour of as-ECAE Cu with the surveyed literature, broad agreement on the strength of the material is achieved. However, a strong variation in the percentage elongation to failure is also noted. Strain hardening and deformation kinetic analysis via strain rate jump tests indicate an evolution from stage III to V hardening during post-ECAE compression and a saturation in the strain rate sensitivity after N=4 resulting in maximum values of ∼0.02. Our results suggest that rather than a change in deformation mechanism, the increase in ductility with increasing N is associated with an increase in the mean free path of dislocations—with the grain boundaries remaining actively involved as the transmitter of plastic strain and their interaction with dislocations being the rate controlling deformation mechanism.

The mean free path of dislocations in nanoparticle and nanorod reinforced metal composites and implication for strengthening mechanisms

The mean free path length of dislocations plays an important role in the plastic behavior of metals, which may be significantly enhanced by the addition of nanoparticles. The effects of particle distribution pattern, particle size, shape and volume fraction on the dislocation mean free path length and average obstacle distance are analyzed in two- and three-dimensional models. As the particle volume fraction increases, the dislocation mean free path length and average distance between dislocations obstacles can be significantly reduced, implying enhancement of strength. The random particle distribution exhibits the best combination of dislocation mean free path length and average obstacle distance. In addition, both dislocation mean free path and average obstacle distance can be significantly reduced by changing the particle shape from nanosphere to nanorod. The present analysis may provide useful information for designing the particle enforced composite materials.

On the strain hardening behaviour of magnesium at room temperature

The strain hardening behaviour of randomly oriented polycrystals deformed in tension and compression is analysed using the Kocks–Mecking phenomenological approach. At low stresses, an extended regime of linear hardening consistent with an athermal forest hardening mechanism, akin to that observed in face-centered cubic metals, is observed. The increase in yield strength with decreasing grain size can also be accounted for by relating the mean free path of dislocations to the grain size. Profuse twinning in compression seems to have little or no effects on the overall strain hardening behaviour.

A Dislocation-Crystal Plasticity Simulation on Large Deformation Considering Geometrically Necessary Dislocation Density and Incompatibility (2nd Report, Application to FCC Single Crystal)

In the previous paper, the GN incompatibility is newly defined and a new annihilation term of a dislocation pair due to the dynamic recovery is introduced into an expression of dislocation density. Furthermore, a multiscale model of crystal plasticity is proposed by considering GN dislocation density and incompatibility. However, details of dislocation-crystal plasticity simulation are not given. In this paper, we explain a method of dislocation-crystal plasticity analysis. A finite element simulation is carried out for an f.c.c. single crystal under plane strain tension. It is numerically predicted that micro shear bands are formed at large strain, and sub-GNBs : small angle tilt boundaries are induced along these bands. Furthermore, the annihilation of dislocation pair and the increase of dislocation mean free path characterizing stage III of work-hardening are computationally predicted.

2605 結晶粒方位差に基づく転位パターニングに関する転位-結晶塑性シミュレーション(S18-1 組織変化と強度,S18 材料の強度と組織)

A multiscale model on dislocation patterning of cell structure and subgrain for polycrystal is newly developed on the basis of reaction-diffusion theory. We introduce a threshold value of slip for stage transition corresponding to the initial crystal orientation into a dislocation mean free path model so as to express the dependence of three-stage hardening on the initial crystal orientation of each grain. A FD simulation for dislocation patterning and a FE one for crystal deformation are simultaneously carried out for a FCC polycrystal under severe strain condition. Reflecting stress value on stress-effect coefficients, it is numerically predicted that the evolution of dislocation pattern in a polycrystal is different in response to the stress condition of each grain. Moreover, the formation of GN boundaries with large misorientation angle is observed along subgrain walls due to large deformation.

Plasticity of Copper with Small Grain Size

Based on the Taylor theory, a critical length scale is defined as the minimum dislocations cell size obtained at the maximum work-hardening for metals and alloys. When grain size is smaller than this length scale, corresponding also to a critical dislocation mean free path, new behaviours occur; such as ductility and strength, near perfect elasto-plasticity, high strain-rate sensitivity. Bulk samples are fabricated from Cu nanopowders (particle size 50 nm) by powder metallurgy techniques. The final grain size is comprised between the critical mean free path, evaluated at 130 nm and the size where transition to the so-called nano regime occurs (when unit dislocation no longer exists below 30 nm for Cu). Tensile tests are carried and microstructural analysis are performed before and after deformation.