Planar Double Slip Research Articles

Formation of GNDs in copper single crystal under pure bending considering planar double slip

In the present work a finite strain non-local crystal plasticity model is implemented via UMAT in ABAQUSTM. The main objective of this work is to study the distribution of geometrically necessary dislocations in pure bending. Ashby (1969) with his hypothesis showed that the gradient of strains caused by inhomogeneous deformation plays a major role in the mechanical behavior of crystals. In this work we study the evolution of geometrically necessary dislocations (GNDs) in case of pure bending of Cu single crystals. In the present work we develop a crystal plasticity model considering the evolution of GNDs. For verification of current implementation, we have used a planar double slip model of FCC crystals derived by Asaro (1979). We show that in the absence of screw dislocations the total GNDs are in form of only edge type dislocations. The distribution of statistically stored dislocations (SSDs), GNDs and their components are in good agreement with Dai (1997).

The concept of control points in hybrid discontinuous Galerkin methods—Application to geometrically nonlinear crystal plasticity

SummaryA new concept for hybrid discontinuous Galerkin (DG) methods is presented: control points. These are defined on the interelement boundaries. The concept makes it possible to formulate element shape functions without nodes. Moreover, the theory is not restricted to certain element shapes. Furthermore, one can formulate the discrete model such that the displacement is either continuous or discontinuous at the control points. Classical continuous isoparametric elements are included as special case. As an additional new feature, a regularization technique for very high strain rate sensitivity exponents up to 1000 in finite single crystal viscoplasticity is presented and implemented into the new hybrid DG framework. In addition, the numerical linearization used in an earlier work is carried out analytically in this work. To the knowledge of the authors, this work presents the first hybrid DG implementation of geometrically nonlinear plasticity, here in the context of single crystal plasticity. The regularization method in combination with the DG formulations facilitates a very simple implementation leading to a numerically efficient, robust, and locking‐free model. Two examples are investigated: the deformation of a planar double slip single crystal exhibiting localization in the form of shear bands and an oligocrystal under uniaxial load.

Cyclic crystal plasticity analyses of stationary, microstructurally small surface cracks in ductile single phase polycrystals

ABSTRACT This paper explores the effects of microstructural heterogeneity on the cyclic crack tip opening and sliding displacements for stationary, microstructurally small transgranular surface cracks in a single phase metallic polycrystal using planar double slip crystal plasticity computations. Crack tip displacements are examined under plane strain conditions for stationary cracks of different lengths relative to grain size as a function of the applied nominal strain amplitude for tension‐compression and cyclic shear. Nominal strain amplitudes range from well below to slightly above the nominal cyclic yield strength for each type of loading condition. Results indicate the complex nature of the crack tip sliding and opening displacements as functions of nominal strain amplitude and orientation of the nearest neighbour grains, the influence of the free surface in promoting the cyclic opening displacement even for cracks in the first surface grain, the rather restricted limits of applicability of linear elastic fracture mechanics, and very interesting crack tip plasticity effects which include crack tip displacement ratcheting or progressive accumulation, even for completely reversed, proportional applied loading. Results are compared for cases with and without crack face friction.

Crack tip displacements of microstructurally small surface cracks in single phase ductile polycrystals

A planar double slip crystal plasticity model is applied to the evaluation of crack tip opening (CTOD) and sliding (CTSD) displacements for microstructurally small stationary cracks under monotonic loading for a material with nominal stress–strain behavior that is representative of a relatively high strength helicopter rotor hub material. Two-dimensional plane strain finite element calculations are presented for CTSD and CTOD of microstructurally small transgranular surface cracks in a polycrystal subjected to monotonic loading. The effects of crack length relative to grain size, orientation distribution of nearest neighbor grains, stress state and stress level are considered for nominal stress levels below the macroscopic yield strength. The CTOD and CTSD are computed for stationary crystallographic surface cracks with various realizations of crystallographic orientations of surrounding grains. It is found that (i) the opening displacement is dominant for remote tension even for crystallographic cracks oriented along the maximum shear plane in the first surface grain, (ii) there is a strong dependence of the CTOD on the proximity to grain boundaries, but lesser dependence of the CTSD, and (iii) that the elastic solutions for CTOD and CTSD are valid below about 30% of the 0.2% offset-defined yield strength.

The double slip plane model for the study of short cracks

Apart from crack closure which has been employed extensively to explain apparent differences in the behaviour of short and long cracks, dislocation crack tip shielding is investigated as an additional contributing factor. The double slip plane (DSP) crack model is employed and a numerical approach is followed for the calculation of the dislocation density distributions on the crack and the slip planes and the plastic zone size. By accounting for dislocation crack tip shielding, the critical stress σcr and applied stress intensity factor KA,cr which define the onset of crack advance, exhibit a crack length dependence which is a manifestation of the anomalous short crack behaviour. It is shown, that below a critical crack length KA,cr is not constant, but rather decreases with decreasing crack length. Furthermore, a critical stress smaller than that predicted by linear elastic fracture mechanics is found. A method to calculate crack propagation rates based on dislocation density distributions is also presented. It is shown that under cyclic loading, a short crack propagates below the threshold stress intensity of a long crack and exhibits faster crack growth rates for the same nominal crack driving force. For the mild crack growth regime, Paris power-law exponents of 2 and 2.7 are predicted, depending on the value of an adjustable parameter in the configuration of the DSP model.

Shear band post-bifurcation in oriented copper single crystals

By in situ SEM observation performed in oriented copper single crystals strained in tension, the formation and propagation processes of macroscopic shear bands (MSB) as well as the consequences of their interactions are analysed. We found that an MSB consisted of a collection of parallel shear bands on slip lines scale. The observed lengthening and thickening of MSB were related to saturation of the individual shear bands and to nucleation of new ones in the adjacent matrix. In order to understand this phenomenon, a post-bifurcation model is applied to an idealized planar double slip single crystal. We show that shear banding is effectively characterized by the three observed stages: bifurcation, large shear strain localization and saturation. The bifurcation is accompanied by unloading in the matrix. The saturation is due to the fact that the geometrical softening leading to the bifurcation decreases with localized shear.

Shear band post-bifurcation analysis in a planar double slip single crystal model

Abstract A shear band post-bifurcation analysis for a double slip ductile single crystal is presented. The crystal geometry is idealized in terms of a planar double slip model. The crystal is assumed to be rigid-plastic and rate-independent. The crystallographic slip is governed by Schmid's law. The strain hardening is expressed by a latent hardening matrix. By analysing kinematics and statics of shear banding, conditions for shear band bifurcation and post-bifurcation are obtained. The obtained bifurcation condition is a necessary and sufficient condition. The post-bifurcation condition allows us to determine the magnitudes of the crystallographic slips inside the band as well as those outside the band (matrix). The results show that shear band can form in a hardening single crystal without any defect, that its formation and its development are characterized by a rigid matrix during shear banding, and that they depend strongly on crystal slip geometry and on strain hardening properties, ruled by the “geometrical” softening due to the lattice rotation accompanied with shear banding. By analysis of shear banding kinematics we find that, when the slip geometry is such that the two slip directions make an angle greater than 90°, the localized shearing has a limit value; however, in the inverse case, the localized shearing has kinematically no limit. The static conditions allow us to determine in the latter case the saturation of the shear band. It is shown that shear banding can be described by a scalar parameter C. C = 0 corresponds to the bifurcation. If C after the bifurcation, the plastic flow will entirely be localized inside the band, while the matrix will be unloaded. If C becomes positive during shear banding, the matrix will restart to be deformed. Consequently, the following development processes of a shear band are predicted: bifurcation, large shear strain localization and saturation. These predictions are in agreement with experimental observations of shear band propagation mechanism on a sample scale.

A constitutive algorithm for rate-dependent crystal plasticity

An algorithm is proposed for the finite element implementation of a rate-dependent crystal plasticity model. A ‘planar double slip’ two-dimensional crystal model is used. The constitutive equations are derived in detail, and it is shown that they reduce to a set of three ordinary differential equations: two for the resolved shear stresses on the slip planes, and one for the angle that defines the slip systems’ orientation. The proposed algorithm, which is suitable for implementation in an explicit finite element code such as PRONTO2D, is based on a weighted residual method for the approximate solution of these equations. It turns out that the performance of the algorithm does not restrict the selection of the global time increment. The efficiency and performance of the algorithm is demonstrated through a comparison with the so-called rate-tangent method. Also, numerical examples are given to demonstrate the model's capability to capture shear localization phenomena at high strain rates, using the PRONTO2D explicit finite element code.

Crack tip stress intensity factor of the double slip plane crack model: Short cracks and short short-cracks

The crack tip stress intensity factorK1 for a short crack is determined using the double slip plane (DSP) crack model. It is shown thatK1 for a stationary crack is larger than the nominal stress intensity factorK. This result differs from the case of the stationary DSP long crack for whichK1 =K. The physical cause whyK1 >K is the fall off with distance of the dislocation shielding/antishielding factor I. at a rate faster than an inverse square root dependence when the distance from a dislocation to a crack tip is of the order of or larger than the crack half widtha. The value ofL for a dislocation situated at an arbitrary position about a crack is derived in this paper. (The Rice-Thomson expressions forL are valid only if a dislocation is very close to a crack tip.) The short short-crack is also analysed using the DSP crack model. (A short short-crack crack is defined to be a short crack whose plastic zone behind the crack tip extends to the center of the crack.) The value of K1 for the short short-crack is a constant and is larger than K. Finally, it is shown that if the crack length is smaller than a critical value that is inversely proportional to the yield stress and is proportional to the critical stress intensity factor Kcb of a Griffith crack that K1 must be smaller than K,t, regardless of how closely the applied stress approaches the yield stress. These results imply that fatigue crack growth of short cracks in the DSP crack model occurs at a faster rate than for long cracks when the conventional cyclic stress intensity factor is above the threshold value and that short cracks can grow under cyclic stress intensity factors smaller than the threshold value.

Crack growth for the double slip plane and the modified dsp crack model-II. fatigue crack growth with and without blunting

The double slip plane (DSP) crack model was used to obtain crack growth equations for the mode II and III crack under a monotonically increasing stress in the Part I paper. In this paper the growth law under cyclic stress (the Paris fatigue crack growth equation) is found. The success of the analysis depends upon the fact that the DSP crack closely approximates a Bilby-Cottrell-Swinden (BCS) crack when the slip zone is large compared with the slip plane to crack plane spacing. Consequently the dislocation distribution on the slip planes approximates the BCS crack one ahead of the crack tip. The results obtained are: if no work hardening of the slip planes takes place and the crack does not blunt a 2nd power Paris equation is found. If blunting occurs the exponent of the Paris equation is reduced to a 4 3 rd power. Crack advance starts well below the peak stress intensity factor K p is reached. For both the blunting and the nonblunting cases work hardening reduces the value of the Paris exponent. A modified DSP plane is also investigated. In this model the dislocations on the slip behind the crack tip move into the crack and are annihilated. When no blunting occurs a 4th power Paris equation is obtained. When blunting occurs both 4th power and 2nd power Paris equations can be derived. If no work hardening occurs the fatigue crack does not advance. Crack advance starts well below a stress intensity factor of K p when the power exponent is 2 or smaller. The advance starts almost at the level K p for a 4th power equation.

Crack growth for the double slip plane crack model-I. The R-curve

The double slip plane (DSP) crack model of Weertman, Lin and Thomson has been used to obtain crack growth equations for the mode II or III crack under a monotonically increasing stress (the R-curve) in this paper. [In a companion paper the growth under cyclic stress (the Paris fatigue crack growth equation) is determined.] The success of the analysis depends upon the fact that the DSP crack closely approximates a Bilby-Cottrell-Swinden (BCS) crack when the slip zone is large compared with the slip plane to crack plane spacing. Consequently the dislocation distribution on the slip planes approximates the BCS crack one ahead of the crack tip. Behind the crack tip a result fortuitous for the analysis is found that the gradient of the dislocation density on a slip plane is proportional to the shear stress on the slip plane. The results obtained are: if the friction stress on a slip plane is constant the crack never propagates catastrophically. Instead crack extension occurs which is proportional to K 2 where rK is the stress intensity factor. Were the surface energy of a solid to be suddenly reduced, as it might be by the sudden introduction of an active environment, the distance the tip of a stressed stationary crack jumps is proportional to K. The distance jumped, for a large drop in surface energy, is smaller than the crack advance that occurs if the active environment were always present and the crack is monotonically loaded to the same value of K. If work hardening of the friction stress takes place stable crack growth takes place up to a critical K c value. The R-curve equation is given by an integral which is simple in form but requires a numerical integration or series expansion. The critical K c value is proportional to the critical K cb stress intensity factor of a Griffith crack raised to the power ( m + 1 )/2 m where m is the power exponent of the simulated plastic stress-strain curve. We believe that this paper demonstrates the double slip plane crack model is the first crack model since the Griffith crack model and its variants that can give an explicit fracture equation starting from first principles.

Explicit calculation of the crack tip stress intensity factor for the double slip plane model crack

A more rigorous analysis is presented for the stationary double slip plane (DSP) crack loaded in mode III with constant friction stress on the slip planes. This analysis gives the dislocation distributions on the slip and crack planes. More important, the analysis leads to an explicit calculation of the crack tip stress intensity factor K t without resort to use of the J-integral or the Rice-Thompson expression for dislocation crack tip shielding/antishielding factor. It is shown that K t ≈ K for the stationary crack. With the results of this paper there are now essentially three independent methods of obtaining K t for a DSP crack.

Material rate dependence and localized deformation in crystalline solids

Nonuniform deformations of rate dependent single crystals subject to tensile loading are analyzed numerically. The crystal geometry is idealized in terms of a planar double slip model. In addition to allowing the effects of material rate sensitivity to be explored, the present rate dependent formulation permits the analysis of a range of material strain hardening properties and crystal geometries that could not be analyzed within a rate independent framework. Two crystal geometries are modeled. One is a planar model of an f.c.c. crystal undergoing symmetric primary-conjugate slip. For this geometry, a direct comparison with a previous rate independent calculation shows that material rate sensitivity delays shear band development significantly. Our present rate dependent formulation also enables a more complete exploration of the effects of high (i.e. greater than Taylor) latent hardening ratios on “patchy” slip development. In particular we show that strong latent hardening and patchy slip can give rise to kinematical constraints that prevent shear bands from propagating completely across the gage section. The second geometry models a b.c.c. crystal oriented so that there is approximately a double mode of slip with the slip systems inclined by more than 45° to the tensile axis. This calculation displays the formation of a localized band of conjugate slip. The lattice rotations accompanying this mode eventually lead to a decrease in the resolved shear stress on the more active system in the band so that the bands do not accumulate large strains and catastrophic shear bands do not form. The implications of material rate sensitivity for uniqueness are also discussed with reference to implications for the prediction of mechanical properties of polycrystals.

Double slip plane crack model

In this paper a simple crack model is proposed and the behavior of this crack under stress is explored. The crack consists of an ordinary Griffith crack, but on either side of this crack, at a distance w, exist two slip planes that are parallel to the crack plane. It is assumed that slip can only take place on these two planes. Elsewhere the material is elastic. When w is set equal to zero the crack becomes a Bilby-Cottrell-Swinden crack. This crack model simulates in a simple way the elastic crack tip enclave model of a crack. Because w has a finite value the material around the crack tip is elastic. The crack is considered to be stressed in either mode II (plane strain shear) or mode III (anti-plane strain shear). It is found that for a virgin, stationary crack the stress intensity factor at the crack tip is equal to the conventional stress intensity factor when the stress is raised under a monotonically increasing load. However, when the crack tip advances under such a load the crack tip stress intensity factor is smaller than the conventional stress intensity factor. The fracture stress is proportional to the surface energy of the solid raised to a power. In general, this power is not equal to one half. For cyclic loading by using qualitative arguments it is shown that the crack can grow an incremental distance each cycle, and the growth law is a fourth power Paris equation.

Application of the double slip plane crack model to temperature and strain rate sensitive BCC metals

The double slip plane crack model proposed by Weertman, Lin and Thomson (1982) has been applied to model the effect of temperature and strain rate on the stress intensity factor at a crack tip in temperature and strain rate sensitive materials. Increase in temperature or decrease in strain rate (as well as a decrease in slip plane spacing) are shown to increase the shielding of the crack tip by dislocation distributions on the slip planes. Furthermore, the effect of temperature on the fracture toughness, K llc , at various strain rates was shown to exhibit the same sigmoidal shaped curve seen for K lc data in typical alloy steels.

An analysis of nonuniform and localized deformation in ductile single crystals

The nonuniform and localized deformations of ductile single crystals subject to tensile loading are analyzed numerically. The crystal is modelled by a rate independent, elastic-plastic relation based on Schmid's law which precisely accounts for lattice rotations. Both self hardening and latent hardening of the slip systems are included in the model. The crystal geometry is idealized in terms of a planar double slip model. Initial imperfections are specified in the form of slight thickness inhomogeneities and the calculations follow the crystal deformation through diffuse necking and the formation of shear bands. The pattern of shear bands depends on the initial imperfection, but, independent of the particular small imperfection, the material planes of the bands are inclined at a characteristic angle to the slip planes. Also, the lattice misorientation across the shear band, which is such as to cause geometrical softening of the bands, is not sensitive to the imperfection form. For high strength, low hardening crystals a comparison with existing experimental data shows remarkably good qualitative and quantitative agreement between the calculations and observations. We also model a relatively soft high hardening crystal which undergoes more diffuse necking than the strong high hardening crystal. Diffuse necking leads to lattice rotations which produce geometrical softening and hence promote shear band formation. Furthermore, we carry out a calculation for a high strength low hardening crystal with the latent hardening rate prescribed somewhat larger than for isotropic hardening. In this case a ‘patchy’ pattern of slip emerges. However, the course of shear band development is unaffected.