Dislocation-free Zone Research Articles

Transmission-scanning electron microscopy of interface fracture of ferrite deformation twins

AbstractDeformation twins in the ferrite matrix of an age-hardened duplex stainless steel have been observed using on-axis transmission Kikuchi diffraction (TKD) in a scanning electron microscope. This provided details of the lattice misorientation and dislocation arrangement, including the dislocation-free zone at the twin tip. These observations provide evidence for the link between microcracking of the irregular twin/parent interface and relaxation of the residual strains that arise from twin growth, offering new insights into fracture mechanics in these materials. Graphical abstract

A generalized antiplane singularity within a semi-infinite wedge of arbitrary angle

The state of stress arising within an elastic wedge generated by an antiplane singularity present within it is studied. An analytical solution is derived with a unified generic approach. The singularity may represent either an anti-plane concentrated force or a screw dislocation. To validate the general solution, two degenerate cases are presented. Further, the image force present on the screw dislocation due to the wedge free boundary is obtained. It is found that when the screw dislocations are placed in the vicinity of the wedge surface, the image force will drive dislocations to the free boundary where they will be annihilated. This implies that a dislocation-free zone may exist along the free surface of the wedge. To demonstrate the application of the fundamental solutions, a formulation for a slip band under anti-plane loading with Green’s function method is provided. The solutions developed in this study may be used as building blocks to model the damage of material near a V-notch under versatile anti-plane load conditions or torsional loading.

In-situ TEM observations of nanovoid initiation and crack evolution in FCC phase of AlCoCrFeNi2.1 eutectic high-entropy alloy

The changes in microstructures after the exhaustion of dislocation-gliding ability but before cracking in the face-centered cubic (FCC) phase of AlCoCrFeNi2.1 eutectic high-entropy alloy (EHEA) were investigated by using in-situ transmission electron microscopy (TEM) tensile tests. It was found that crystal thinning area occurred ahead of the crack tip through dislocation gliding during loading. After that, crystal rotation and/or twinning took place in the crystal thinning area, which were the two main procedures after the exhaustion of dislocation glide ability but before cracking for this alloy. Nanovoids initiate in the dislocation-free zone ahead of the crack tips through crystal rotation induced by local high-stress concentration. Once the high-stress concentration at crack tip overcomes the critical twinning stress, new nanovoids are initiated near the deformation twin through Shockley partial dislocation (1/6 〈112〉 type) glide emitted from the deformation twin. The connection of these nanovoids would lead to the formation of nanocracks. Eventually, the “Y” and “zigzag” crack propagation paths were formed in the FCC phase for AlCoCrFeNi2.1 EHEA.

A Parabolic Notch Interacting With a Generalized Antiplane Singularity

Abstract In this study, the interaction of a parabolic notch with a generalized antiplane singularity is studied, and its analytical solution is derived. The singularity may be either an antiplane concentrated force or a screw dislocation, and separate solutions for each of these are found. The driving force present on the screw dislocation due to the notch free boundary is obtained. It is found that a dislocation-free zone may exist beneath the notch root surface when the screw dislocation is placed on the notch geometric symmetry axis, as the driving force will pull dislocations to the free boundary where they will be annihilated. The solutions developed in this study may be used as building blocks to model the damage of material near a parabolic notch under antiplane load conditions and therefore serve as a step in quantifying crack nucleation conditions, which is the novelty of the current study.

The effect of specimen thickness on the dislocation-free zone model of fracture

We use the method of continuously distributed dislocations to study the dislocation-free zone (DFZ) model of fracture in an infinite plate of finite thickness under externally applied uniform anti-plane shear stress. We formulate the equilibrium condition in terms of a set of two coupled singular integral equations using the representation of the total interaction force between two screw dislocations in a thin plate obtained by Eshelby and Stroh (1951). This set of singular integral equations is then solved numerically using the Gauss-Chebyshev integration formula resulting in the dislocation distribution functions in the crack and in the two plastic zones, the total number of screw dislocations in the plastic zone, the DFZ condition for a crack in a thin plate and the local mode III stress intensity factor at the crack tip.

Interaction of a parabolic notch with a generalized singularity

In this paper, the interaction of a parabolic notch with a generalized singularity is studied and its analytical solution is derived. The generalized singularity may represent a concentrated force or an edge dislocation. As an example, the parabolic notch interacting with a dislocation is studied in details, and the driving force on the dislocation due to the notch configuration is obtained in terms of the vector J-integral. It is found that a dislocation-free zone may exist beneath the notch root surface. The solutions developed in this study give, on the one hand, basic information about the state of stress induced in the notch neighborhood, and, on the other hand, can be used as building blocks to model damage at the notch root under complex load conditions.

Nanotwin orientation on history-dependent stress decay in Cu nanopillar under constant strain

Cu with nanotwin (NT) possesses great electrical, mechanical, and thermal properties and has potential for electronic applications. Various studies have reported the effect of NT orientation on Cu mechanical properties. However, its effect on Cu stress-relaxation behavior has not been clarified, particularly in nano-scale. In this study, Cu nanopillars with various orientations were examined by a picoindenter under constant strain and observed by in situ TEM. The angles between the twin plane and the loading direction in the examined nanopillars were 0°, 60°, to 90°, and a benchmark pillar of single-crystal Cu without NT was examined. The stress drops were respectively 10%, 80%, 4%, and 50%. Owing to the interaction by NT, the dislocation behavior in nanopillars was different from that in bulk or in thin film samples. Especially, the rapid slip path of dislocations to go to the free surface of the nanopillar induced a dislocation-free zone in the 0° nanopillar, which led to work-softening. On the contrary, a high dislocation density was observed in the 90° nanopillar, which was generated by dislocation interaction and obstruction of dislocation slip by twin planes, and it led to work-hardening. The findings reveal the NT orientation in Cu nanopillars affected stress relaxation significantly.

Effect of Strain Amounts on Cold Compression Deformation Mechanism of Ti-55531 Alloy with Bimodal Microstructure

The plastic deformation mechanism of Ti-55531 alloy with bimodal microstructure was investigated by compression testing at room temperature. The bimodal microstructure was composed of equiaxed primary α phase (αp) and transformed β (βtrans) that consisted of acicular secondary α phase (αs) and residual β phase (βr). In the initial stage of deformation, the αp grains first underwent plastic deformation, the dislocations germinated and increased, forming the dislocation loop with the dislocation free zone in αp at the true stain of 0.083. With the true strain subsequently increasing to 0.105, the dislocation tangle and dislocation pile-up occurred in αp, and a lot of dislocations were also activated in most of αs. Moreover, the dislocation density was increasing gradually in βr with the adding of strain. Finally, the dislocation pile-up and dislocation tangle appeared in αs and βr at the true strain of 0.163. The whole deformation process was coordinated by αp, αs and βr. They accommodated mutually and completed deformation together.

Dislocation Emission and Crack Dislocation Interactions

An understanding of the crack initiation and crack growth in metals spanning the entire spectrum of conventional and advanced has long been a major scientific challenge. It is known that dislocations are involved both in the initiation and propagation of cracks in metals and alloys. In this review, we first describe the experimental observations of dislocation emission from cracks under stress. Then the role played by these dislocations in fatigue and fracture is considered at a fundamental level by considering the interactions of crack and dislocations emitted from the crack. We obtain precise expression for the equilibrium positions of dislocations in an array ahead of crack tip. We estimate important parameters, such as plastic zone size, dislocation free zone and dislocation stress intensity factor for the analysis of crack propagation. Finally, we describe very recent novel and significant results, such as residual stresses and relatively large lattice rotations across a number of grains in front of the crack that accompanies fatigue process.

Revealing the inhibition mechanism of grain size gradient on crack growth in gradient nano-grained materials

Gradient nano-grained (GNG) materials, in which grain size changes from nano-scale in the surface to micro-scale in the core, have shown superior fatigue property, good strength and ductility. However, crack growth behaviors of GNG materials are studied hardly. In this work, the effect of grain size gradient on crack growth is investigated to reveal the inhibition mechanism of crack growth in GNG materials. The developed model employs a distributed dislocation method to model the evolution of plastic zone. The influence of grain size gradient on the distribution of dislocations in grains and the local stress intensity factor is analyzed. The results show that the sizes of dislocation zone and dislocation-free zone in grains become larger as the gradient of grain size increases. The number of dislocations emitted from the crack tip increases with the grain size gradient increasing, so the shielding effect of emitted dislocations on the crack growth increases and crack blunting becomes more serious. This means that a sharper grain size gradient inhibits the crack growth more effectively. The results in this paper are important for understanding the fracture behaviors of materials with grain size gradient and are helpful for material design and structural optimization.

A unified theory for brittle and ductile shear mode fracture

ABSTRACT A unified theory captures both brittle and ductile fracture. The fracture toughness is proportional to the applied stress squared and the length of the crack. For purely brittle solids, this criterion is equivalent to Griffith's theory. In other cases, it provides a theoretical basis for the Irwin-Orowan formula. For purely ductile solids, the theory makes direct contact with the Bilby-Cottrell-Swinden model. The toughness is highest in ductile materials because the shielding dislocations in the plastic zone provide additional resistance to crack growth. This resistance is the force opposing dislocation motion, and the Peach-Koehler force overcomes it. A dislocation-free zone separates the plastic zone from and the tip of the crack. The dislocation-free zone is finite because molecular forces responsible for the cohesion of the surfaces near the crack tip are not negligible. At the point of crack growth, the length of the dislocation-free zone is constant and the shielding dislocations advance in concert. As in Griffith's theory, the crack is in unstable equilibrium. The theory shows that a dimensionless variable controls the elastoplastic behaviour. A relationship for the size of the dislocation-free zone is derived in terms of the macroscopic and microscopic parameters that govern the fracture.

Embrittlement of an elasto-plastic medium by an inclusion

A mathematical model for the embrittlement of a long elastic-plastic crack by a relatively small, misfitting inclusion is presented. The model makes direct contact with the Dugdale–Bilby–Cottrell–Swinden model as a limiting case. The particular case of an oxide inclusion with a triangular cross-section at the tip of an intergranular crack in the Ni-based superalloy RR1000 at 650,^{circ }hbox {C} is considered. The positive misfit of the intrusion provides an additional tensile load on the crack tip and on the plastic zone, raising the local stress intensity factor k_I and the crack tip opening displacement {varDelta } u above those when the inclusion is replaced by a dislocation-free zone of the same length. It is shown that for a given misfit strain and inclusion shape, the enhancement of k_I and {varDelta } u is controlled by a dimensionless parameter omega = (sigma /sigma _1)sqrt{c/(2l)} where sigma is the applied stress, sigma _1 is the yield stress, c is the crack length and l is the length of the inclusion. The anti-shielding effect of the intrusion is significant only when omega lesssim 6. As a result of the anti-shielding effect of the intrusion, the stress singularity at the crack tip always exceeds the compressive normal stress that exists within the thickest part of the intrusion when it is isolated. It is also shown that the gradient of the hydrostatic stress within the intrusion subjected to different applied stresses drives the oxygen diffusion and, hence, assists the oxidation at the grain boundary. The fracture toughness is considerably greater than that of a bulk sample of the oxide particle, which we attribute to the plastic zone.

The role of plastic strain gradients in the crack growth resistance of metals

Crack advance from short or long pre-cracks is predicted by the progressive failure of a cohesive zone in a strain gradient, elasto-plastic solid. The presence of strain gradients leads to the existence of an elastic zone at the tip of a stationary crack, for both the long crack and the short crack cases. This is in sharp contrast with previous asymptotic analyses of gradient solids, where elastic strains were neglected. The presence of an elastic singularity at the crack tip generates stresses which are sufficiently high to activate quasi-cleavage. For the long crack case, crack growth resistance curves are predicted for a wide range of ratios of cohesive zone strength to yield strength. Remarkably, this feature of an elastic singularity is preserved for short cracks, leading to a severe reduction in tensile ductility. In qualitative terms, these predictions resemble those of discrete dislocation calculations, including the concept of a dislocation-free zone at the crack tip.

Analysis of the effect of a micro-crack on plastic zone of the edge macro-crack tip by macroscopic and microscopic methods

The effect of the micro-crack on plastic zone ahead of the macro-crack tip (PZMA) is analyzed based on macroscopic and microscopic methods. From the macroscopic perspective, the plastic deformation is due to the local stress near crack tip is more than the yielding limit. So the complete stress field is calculated based on the distributed dislocation technique (DDT) and the plastic zone (PZ) is determined by von Mises yielding criterion (MYC). On the other hand, the crack tip plasticity is simulated by finite element method (FEM). From the microscopic perspective, the plastic deformation is caused by dislocations emitted from crack tip. So PZ is represented by dislocations, and it is determined based on DDT and dislocation-free zone (DFZ) model. The results show the micro-crack located in front of PZMA has the amplifying effect on PZMA. PZMA is separated by the micro-crack located in PZMA. The micro-crack behind PZMA has no effect on PZMA. PZMA has the shielding effect on the macro-crack propagation, and the shielding effect increases with PZMA increasing. The micro-crack has the amplifying effect on PZMA at about x0/a > 0.5, while it has the shielding effect at about x0/a < 0.5. These results are useful for predicting and interpreting the plastic behaviors at the crack tip.

Critical thickness phenomenon in single-crystalline wires under torsion

Analysis of idealised thin single-crystal wires under torsion based on the continuum theory of dislocations gives results in accordance with the critical thickness theory. The dislocation-free zone near the wire surface and the nearly-zero stress around the wire axis are predicted by both the continuum dislocation theory and critical thickness theory. It is demonstrated that the size effect at the onset of yielding, the distributions of stress and geometrically necessary dislocations in the thin wires in torsion, simply result from the critical thickness effect. A continuous increase of plastic strain from the neutral axis toward the wire surface is indicated. The plastic strain becomes (nearly) flat around the wire surface. Such a phenomenon is attributed to the fact that this is the region in which dislocations sources can operate, to provide the geometrically necessary dislocations required by the plastic strain gradient beneath. The results of continuum dislocation theory quantitatively elucidate the critical thickness phenomenon occurred in single-crystal wires under torsion. This links the continuum dislocation theory to the underlying physical picture of Matthews' critical thickness theory.

Progression of Strain Relaxation in Linearly-Graded GaAs1-yPy/GaAs (001) Epitaxial Layers Approximated by a Finite Number of Sublayers

We have investigated the residual in-plane strain and width of the surface misfit dislocation free zone in linearly-graded GaAs1-yPy metamorphic buffer layers as approximated by a finite number of sublayers. For this purpose we have developed an electric circuit model approach for the equilibrium analysis of these structures, in which each sublayer may be represented by an analogous configuration involving a current source, a resistor, a voltage source, and an ideal diode. The resulting node voltages in the analogous electric circuit correspond to the equilibrium strains in the original epitaxial structure. Utilizing this new approach, we show that the residual surface strain in linearly-graded epitaxial structures increases monotonically with grading coefficient as well as the number of sublayers, and is strongly dependent on the width of the misfit dislocation free zone, which diminishes with an increasing grading coefficient.

How to Measure a Dislocation’s Breakthrough Stress to Estimate the Grain Boundary Resistance against Slip Transfer Based on the DFZ-Model of Fracture

Stage-I-fatigue-cracks are used as highly localized dislocation sources with well-known Burger’s vectors to study the interaction between dislocations and grain boundaries. This interaction in the plastic zone is of particular interest to understand the fluctuating crack growth in the very short crack regime. In the case of a blocked slip band the dislocations pile up at the grain boundary causing a local stress concentration. The resulting local stress distribution is calculated based on measurements of the dislocation density distribution in the plastic zone. For this purpose the slip line profiles were measured by AFM, the dislocation density distribution was determined and the dislocation-free zone model of fracture (DFZ) was validated. With this it is possible to quantify the grain boundary resistance and to combine geometric and stress approach for grain boundary resistance against slip transfer.

High cycle fatigue behavior of Ti–5Al–5Mo–5V–3Cr–1Zr titanium alloy with bimodal microstructure

In this study, high–cycle fatigue (HCF) damage behavior of Ti–55531 alloy with bimodal microstructure (BM) was studied at room temperature. Fatigue crack initiation and propagation mechanisms of the alloy were thoroughly investigated by studying fracture morphology, crack front profiles, polished microstructure and dislocation structures beneath fatigue main–crack initiation sites of HCF samples. The results indicate that this alloy presents an excellent HCF strength (107cycles, R = −1) as equal to 656 MPa. Dislocation analyses exhibit that typical dislocation structures include straight prismatic slip lines, curved dislocation lines, dislocation tangles and {1¯011}α type twins in fatigued specimens. Primary αp particles and secondary αs lamellae accommodate more cyclic deformation than retained βr laths. Furthermore, the dislocation free zone can be observed in the αp/βtrans (β transformed microstructure) boundary. As a result, microcracks primarily nucleate at the αp/βtrans interface or at αp particles. Moreover, a few microcracks initiate at the αs/βr interface or at αs plates of βtrans microstructure. These fatigue crack initiation behaviors promote the fracture of Ti–55531 alloy.

Quantifying the grain boundary resistance against slip transfer by experimental combination of geometric and stress approach using stage-I-fatigue cracks

In recent studies, many groups have investigated the interaction of dislocations and grain boundaries by bi-crystals and micro-specimen experiments. Partially, these experiments were combined with supplementary simulations by discrete dislocation dynamics, but quantitative data for the grain boundary resistance against slip transfer is still missing. In this feasibility study with first results, we use stage-I-fatigue cracks as highly localised sources for dislocations with well-known Burgers vectors to study the interaction between dislocations in the plastic zone in front of the crack tip and selected grain boundaries. The stress concentration at the grain boundary is calculated with the dislocation-free zone model of fracture using the dislocation density distribution in the plastic zone from slip trace height profile measurements by atomic force microscopy. The grain boundary resistance values calculated from common geometric models are compared to the local stress distribution at the grain boundaries. Hence, it is possible to quantify the grain boundary resistance and to combine geometric and stress approach for grain boundary resistance against slip transfer to a self-contained concept. As a result, the prediction of the grain boundary resistance effect based on a critical stress concept is possible with knowledge of the geometric parameters of the grain boundary only, namely the orientations of both participating grains and the orientation of the grain boundary plane.

Distribution of dislocations in twisted bars

An asymptotically exact continuum dislocation theory of single crystal bars under torsion is proposed. The dislocation distribution minimizing energy of the bar with zero torque is shown to be uniform. If the applied torque is non-zero, the minimizer exhibits a dislocation-free zone at the outer ring of the bar's cross-section. The non-uniform distribution of dislocations in equilibrium as well as the twist angle per unit length are found in terms of the given torque. With the energy dissipation being taken into account, there exists an elastic core region, while dislocations are concentrated in a ring between two dislocation-free zones. This leads to the change of the stress distribution increasing the critical threshold of the torque.