Dislocation Solution Research Articles

Effect of Dislocation Magnification and Solute Atoms Concentration Variation on the Electroplastic Effect of 2024 Aluminum Alloy

Effect of Dislocation Magnification and Solute Atoms Concentration Variation on the Electroplastic Effect of 2024 Aluminum Alloy

Axisymmetric planar cracks in finite hollow cylinders of transversely isotropic material: part I—dislocation solution for infinite cylinders

This paper presents the solution for the crack tip stress intensity factors of a system of coaxial axisymmetric planar cracks in a transversely isotropic infinite hollow cylinder. The cylinder is under uniform axial tensile loading at infinity. First, the state of stress in an infinite hollow cylinder with transversely isotropic material containing axisymmetric prismatic and radial dislocations is studied. To this end, the solutions are represented in terms of biharmonic stress functions. Next, using the distributed dislocation technique, the mixed mode crack problem is reduced to Cauchy-type singular integral equations for dislocation densities on the surfaces of the cracks. The dislocation densities are employed to determine stress intensity factors for axisymmetric interacting cracks. For validation, the problems including one single crack in an infinite hollow cylinder with a circumferential edge crack or an annular crack are reexamined.

Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders

• New analytical dislocation solution in a transversely isotropic cylinder is developed. • Material anisotropy has an efficient effect on stress intensity factors. • Interaction of cracks implies that need for mixed-mode analysis is essential. • Loading type and crack length have also a crucial effect on stress intensity factors. The present study examined mixed mode cracking in a transversely isotropic infinite cylinder. The solutions to axisymmetric Volterra climb and glide dislocations in an infinite circular cylinder of the transversely isotropic material are first obtained. The solutions are represented in terms of the biharmonic stress function. Next, the problem of a transversely isotropic infinite cylinder with a set of concentric axisymmetric penny-shaped, annular, and circumferential cracks is formulated using the distributed dislocation technique. Two types of loadings are considered: (i) the lateral cylinder is loaded by two self-equilibrating distributed shear stresses; (ii) the curved surface of the cylinder is under the action of a distributed normal stress. The resulting integral equations are solved by using a numerical scheme to compute the dislocation density on the borders of the cracks. The dislocation densities are employed to determine stress intensity factors for axisymmetric interacting cracks. Finally, a good amount of examples are solved to depict the effect of crack type and location on the stress intensity factors at crack tips and interaction between cracks. Numerical solutions for practical materials are presented and the effect of transverse isotropy on stress intensity factors is discussed.

Response of an orthotropic half-plane subjected to transient anti-plane loading with multiple edge cracks

This study deals with the problem of multiple edge cracks in an elastic orthotropic half-plane under transient loading. The dislocation solution is utilized to derive integral equations for multiple interacting edge cracks in an orthotropic half-plane. These equations are solved numerically thereby obtaining the dislocation density function on the crack faces and stress intensity factors at crack tips. Numerical results are obtained to illustrate the variation of the dynamic stress intensity factors as a function of crack length and material properties.

Analysis of cracked bars with rectangular cross-section and isotropic coating layer under torsion

The solution to problem of a Volterra-type screw dislocation in a rectangular cross section bar with an isotropic coating is first achieved by means of a finite Fourier cosine transform. The bar is under axial torque which is governed by Saint-Venant torsion theory. The series solution is then derived for warping function and stress fields in the rectangular cross section with an isotropic coating. The dislocation solution is employed to derive a set of Cauchy singular integral equations for the analysis of smooth cracks. The solution of these equations is used to determine the torsional rigidity of bar and the stress intensity factors for the crack tips. Finally, several examples are presented to show the accuracy and efficiency of the dislocation technique in Saint-Venant torsion problems.

Effects of martensitic transformability and dynamic strain age hardenability on plasticity in metastable austenitic steels containing carbon

We investigated the effects of solute carbon concentration on the mechanical properties of Fe–19Cr–8Ni–0.05C and Fe–19Cr–8Ni–0.14C metastable austenitic steels. These steels showed an FCC(γ) → HCP(e) → BCC(α′) martensitic transformation, resulting in transformation-induced plasticity (TRIP). The presence of excess solute carbon reduced the transformability because of an increase in the austenite stability. However, the work hardening capability was enhanced by a combined effect of the TRIP and dynamic strain aging (DSA). DSA requires a high diffusivity of carbon. Thus, the FCC (low diffusivity) to BCC (high diffusivity) transformation favors DSA. The hardening capability of BCC-martensite per volume is enhanced by the dislocation pinning and solution hardening effect of the carbon atmosphere, despite a decrease in the transformation rate per strain by carbon addition. Moreover, carbon addition stabilizes the deformation-induced HCP-martensite against the BCC-martensite, improving the hardening capability of the HCP-martensite through suppression of the window effect, which affects the plastic accommodation mechanism. According to our study, the steel with a low carbon content demonstrated extraordinary work hardening rates owing to a high transformation rate per strain. In contrast, the steel with a high carbon content showed sustained and high work hardening rates because of DSA. Both the steels showed approximately the same tensile strength, but completely different work hardening behavior.

High-strength state of ultrafine-grained martensitic steel produced by high pressure torsion

The paper presents the study results on the effect of severe plastic deformation (SPD) via high pressure torsion (HPT) on the structure and properties of martensitic steel. The contribution of different strengthening mechanisms in the strength of steel has been analyzed. It is shown that independently of the deformation temperature the main contribution in hardening belongs to grain boundaries (about 50 %), whereas the dislocation and solid solution components achieve 15 and 25 %, respectively.

Dislocation configuration and solute redistribution of low angle kink boundaries in an extruded Mg–Zn–Y–Zr alloy

The microstructural and chemical features of deformation-induced interfaces are one of key issues in engineering materials because they determine plastic deformation behavior and thus affect mechanical properties of the materials. Using atomic-resolution high-angle annular dark-field scanning transmission electron microscopy, we characterized deformation-induced low angle kink boundaries (LAKBs) in long period stacking ordered (LPSO) structures in an extruded Mg–2.3Zn–6.6Y–0.56Zr (wt%) alloy. We clarified that the LAKB in LPSO phase consists of an array of <a> dislocations, while the LAKB in Mg interlayers sandwiched between LPSO phases is composed of an array of dissociated <c+a> and/or <a> dislocations. Correspondingly, the former and the latter LAKBs are depleted and segregated with Zn/Y/Zr elements, respectively. I2 stacking fault (SF) is meanwhile generated in Mg layers, and its energy is evaluated approximately 0.1–1.6mJm−2. Deformation-induced LAKBs, the resultant redistribution of solute elements, and precipitated I2 SFs, are proposed to be responsible for the high strength of extruded Mg alloys containing LPSO structures.

Anti-plane analysis of an infinite plane with multiple cracks based on strain gradient theory

The anti-plane stress analysis in an infinite elastic isotropic plane is carried out by using the distributed dislocation techniques in the framework of strain gradient theory. The technique involves the solution of a screw Volterra dislocation in the region. Employing a distributed dislocation solution indicated that the stress components reveal the familiar hypersingularity at the site of the dislocation. The solution for stress fields in integral form is obtained for a plane containing a Volterra-type screw dislocation via Fourier transform of the biharmonic equation. This method is extended to allow the solution of integral equations in the regions weakened by multiple smoothed curved cracks under anti-plane deformation. The integral equations are solved numerically for the density of dislocation on the crack surfaces. The numerical method in Chebyshev series forms is used to solve the hypersingular integral equations. The effects of interaction among cracks with different arrangements and those of the size effect on the stress intensity factors are discussed. To confirm the validity of the formulations, numerical values of the results are compared. The predicted results via the aforementioned approach are in excellent agreement with those in the literature. Some new examples are solved to demonstrate the applicability of the procedure.

Analysis of Micro-Cracks Using Modified Couple-Stress Elasticity under Anti-Plane Deformation

Based on the modified couple stress theory the solution to a screw dislocation is obtained, in an isotropic elastic plane, via Fourier integral transform method. The asymptotic analysis of displacement field at the tip of a stationary crack reveals that stress field is not singular. A crack under anti-plane deformation is modeled by the distribution of screw dislocations. The ensuing integral equations are solved numerically to determine the density of dislocation on a crack surface. The dislocation solution is used to study the interaction between two parallel non-collinear micro-cracks.

Torsion analysis of finite solid circular cylinders with multiple concentric planar cracks

First, using the separation of variables technique a series solution for torsion of an intact finite cylinder under self‐equilibrating lateral shear tractions is given. Next, an integral form solution for torsion of an intact infinite cylinder under the aforementioned lateral loading is obtained. After that the solution of an axisymmetric rotational Somigliana ring dislocation in the infinite circular cylinder is obtained. The distributed dislocation technique is used to construct integral equations for stress analysis of the infinite cylinder with a set of coaxial axisymmetric cracks. These defects are penny‐shaped, annular and circumferential edge cracks. The infinite cylinder is under the action of the loading similar to the intact cylinder. The integral equations are solved numerically to obtain the dislocation density on the surfaces of the cracks. The stress intensity factors for the cracks are determined by employing the dislocation densities. The problem of a cracked finite cylinder is treated by cutting method. That is, a similar cracked infinite cylinder is sliced by extending two additional annular axisymmetric cracks. To validate the cutting method, the solution of the sliced intact infinite cylinder is compared to that of an intact finite cylinder which both of them are under the same self‐equilibrating lateral shear tractions. The solution is applied to several examples to study the effect of crack type/location on the resulting stress intensity factors at tips of the cracks.

Analytic solutions of multiple moving cracks in an orthotropic layer bonded to an orthotropic FGM coating

In this paper, the dynamic behavior of an orthotropic substrate weakened by moving cracks and reinforced by a non-homogenous coating is studied. First, the solution to the screw dislocation in an orthotropic strip with imperfect orthotropic functionally graded coating is obtained. Then, for the anti-plane analysis of cracks, the screw dislocations are distributed along the crack lines and the dislocation solution is used to derive integral equations for dislocation density functions on the surface of cracks. The effects of non-homogeneity parameters, geometrical parameters and bonding coefficient on the stress intensity factors are investigated.

Effect of Twin Boundary–Dislocation–Solute Interaction on Detwinning in a Mg–3Al–1Zn Alloy

In the present study, the influence of solute atoms together with dislocations at { 10 1 ¯ 2 } twin boundary (TB) on mechanical behavior of a detwinning predominant deformation in a Mg alloy AZ31 plate was systematically studied. The results show that a large number of { 10 1 ¯ 2 } twins disappear during recompression along the normal direction. Both the TB–dislocation interaction and TB–solute–dislocation interaction can greatly enhance the yield stress of the recompression along the normal direction (ND). However, the solute segregation at { 10 1 ¯ 2 } TBs with an intensive interaction with dislocations cannot further enhance the yield stress of ND recompression. The samples with TB–dislocation interaction show a similar working hardening performance with that subjected to a TB–solute–dislocation interaction. Both the TB–dislocation interaction and TB–solute–dislocation interaction greatly reduce the value of work hardening peaks during a detwinning predominant deformation.

Saint-Venant torsion of orthotropic bars with a circular cross-section containing multiple cracks

In this paper, the problem of the circular orthotropic bars with multiple cracks is investigated based on the Saint-Venant torsion theory. The solution to the problem of an orthotropic bar weakened by a Volterra-type screw dislocation is first obtained by means of the finite Fourier sine transform. In this research, the bar is assumed to be subjected to an axial net torsion when finding the dislocation solution. The closed form solution is then derived for displacement and stress fields in the bar. At the next step, the dislocation solution is employed to derive a set of Cauchy singular integral equations for analysis of the circular bar with curved cracks. The solution to these equations is used to determine the torsional rigidity of the bar and the stress intensity factors of the tips of the cracks. The paper is furnished with several examples of a single crack and multiple cracks.

Strengthening mechanisms of (Al3Zrmp + ZrB2np)/AA5052 hybrid composites

To establish the correlation between grain size, dislocations, dispersed particles (size and vol.%) along with their solid solution strengthening effects in alloy and combined effect of all on the strengthening of advanced composite materials (Al3Zrmp + ZrB2np)/AA5052 hybrid composites have been selected for the investigation. (Al3Zrmp + ZrB2np)/AA5052 hybrid composites have been synthesized by the direct melt reaction of AA5052 alloy and inorganic salts (K2ZrF6 and KBF4). These composites have been characterised by X-ray diffractometer, optical microscopy, scanning-electron microscopy with energy-dispersive spectroscopy, transmission-electron microscopy, tensile and hardness test. Results indicate the successful formation of second phase reinforcement particles namely Al3Zr and ZrB2 in the AA5052 alloy matrix. Al3Zr particles exhibit rectangular and polyhedron morphology within an average of micron size while ZrB2 show hexagonal and rectangular within an average of nano size. Grain refinement of Al-rich phase observed in the composites, increases with increasing vol.% of reinforcement particles. TEM observation shows the presence of dislocations in the composite matrix which help to improve the strength parameters. Tensile results show the improvement in strength parameters which improve with the increasing amount of particles whereas percentage elongation also improves up to certain vol.% of particles and beyond that decrease. However, bulk hardness shows an increasing trend continuously with vol.% of particles. The strengthening mechanisms, namely dislocation, Orowan, grain-refined and solid solution are quantified for the hybrid (Al3Zrmp + ZrB2np)/AA5052 composites and the total of above are in agreement with experimental results. Solid-solution and Orowan are the predominant strengthening mechanisms in the composites.

Surgical Management of Old Lower Cervical Dislocations With Locked Facet

Prospective study. To evaluate the role of the canal and vertebrae sagittal diameter (C/V) ratio in the treatment of old dislocations of the lower cervical spine. Few studies have reported the management of old dislocations of the lower cervical spine. Conservative treatments including the use of a Halo vest, neck brace, and prolonged traction have been problematic. Operative treatment consisted of a primary or staged reduction and fusion using an anterior, posterior, or combined approach. Fourteen consecutive patients with old dislocations of the lower cervical spine were included in this series. The preoperative C/V value was calculated based on the measurement on the neutral sagittal computed tomography at the most narrow place of the dislocated segments. Closed reduction was attempted in 9 patients with moderate stenosis (C/V>0.5). Five patients with severe stenosis (C/V≤0.5) were treated with a primary combined anterior and posterior operation. Patient's radiographic information, pain, and neurological function were assessed and recorded before and after surgery. Closed reduction followed by anterior cervical discectomy and fusion was performed in 3 of 9 patients with moderate stenosis. Eleven patients underwent circumferential release, posterior reduction, and fixation followed by anterior fusion. No severe complications were found. The average operative time was 138±43 minutes. The average blood loss was 239±140 mL. The postoperative C/V value was significantly increased. The neurological status improved at least one grade in all 13 neurologically impaired cases except for 2 who had complete spinal cord injuries. Bony fusion was obtained in all patients at 1-year follow-up. The C/V value plays an important role in determining surgical solutions for old lower cervical dislocations with locked facets. Favorable clinical outcomes can be achieved using closed reduction and surgical procedures with anterior or anterior plus posterior approaches.

In-plane stress analysis of dissimilar materials with multiple interface cracks

In the present paper, the behavior of an interface crack in two bonded dissimilar materials subjected to in-plane loading is studied. First, dislocation solution is presented to construct integral equations for dissimilar medium weakened by several interface cracks. These equations are of Cauchy singular type at the location of dislocation, which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation solution is then applied to calculate the mixed mode stress intensity factors of multiple interface cracks. The effects of nonhomogeneity parameter on the stress intensity factors are investigated.

Investigation of the thermo-mechanical behavior of neutron-irradiated Fe-Cr alloys by self-consistent plasticity theory

The thermo-mechanical behavior of non-irradiated (at 223 K, 302 K and 573 K) and neutron irradiated (at 573 K) Fe-2.5Cr, Fe-5Cr and Fe-9Cr alloys is studied by a self-consistent plasticity theory, which consists of constitutive equations describing the contribution of radiation defects at grain level, and the elastic-viscoplastic self-consistent method to obtain polycrystalline behaviors. Attention is paid to two types of radiation-induced defects: interstitial dislocation loops and solute rich clusters, which are believed to be the main sources of hardening in Fe-Cr alloys at medium irradiation doses. Both the hardening mechanism and microstructural evolution are investigated by using available experimental data on microstructures, and implementing hardening rules derived from atomistic data. Good agreement with experimental data is achieved for both the yield stress and strain hardening of non-irradiated and irradiated Fe-Cr alloys by treating dislocation loops as strong thermally activated obstacles and solute rich clusters as weak shearable ones.

Multiple crack problems in nonhomogeneous orthotropic planes under mixed mode loading conditions

Analytical method based on continuous dislocation technique is developed to examine the mixed mode behavior of a functionally graded orthotropic plane. First, the elastic stress fields in a functionally graded orthotropic infinite plane containing single Volterra types climb and glide edge dislocations are obtained. Using the dislocation solutions, the problem of interacting cracks is reduced to a system of singular integral equations for dislocation density functions on the surfaces of smooth cracks which are solved numerically for the dislocation density. Then, the dislocation density function is applied to calculate the stress intensity factors for several different cases of crack configurations and arrangements.

Analysis of an orthotropic strip containing multiple defects bonded between two piezoelectric layers

In this paper, the problem of multiple defects in an orthotropic layer bonded between two piezoelectric layers is considered. The analysis is based on the stress fields caused by Volterra-type screw dislocation in the orthotropic strip. The solution for the dislocation is obtained by means of the complex Fourier transform. The dislocation solution is then employed as strain nuclei to derive singular integral equations for a medium weakened by multiple defects. These equations, as a class of Cauchy singular equations, are solved numerically for dislocation density functions. A number of examples is given for various crack orientations and material properties. At the end, it is shown that the effect of the properties and defect geometries on the stress intensity factors and hoop stress for cavities can be highly significant.