In this article, we have analyzed the interaction of arbitrarily oriented cracks under anti-plane deformation in a bi-directional functionally graded material (bi-FGM) using strain gradient elasticity (SGE) theory. The first crack is aligned along the x 1 -axis, and the second crack is aligned along the x 2 -axis, each obtained by rotating the global x y -coordinate system to their respective local systems. The material gradation within the bi-FGM follows an exponential distribution in the x y -plane. Using SGE theory, which incorporates two characteristic lengths of the material, ℓ and ℓ ′ , to account for the effects of volumetric and surface strain gradients, we adopt a robust methodological framework. This involves the application of Fourier transforms and a novel approach of hypersingular integro-differential equations. By solving the resulting system of equations with Chebyshev polynomial expansion techniques and appropriate collocation points, analytical expressions are obtained for stress intensity factor (SIF), strain distributions, stresses, and crack surface displacement (CSD) profiles for both cracks. An illustrative numerical case study is presented to show the influence of the orientation angle on the various fracture parameters. In addition, we explore the impact of intercrack distances, providing a thorough understanding of the interaction between crack geometry and bi-directional material gradation. Moreover, the variation in the CSD profile under linear and quadratic loading conditions is examined to highlight the influence of loading patterns on crack behavior. These findings offer a deeper understanding of the fracture mechanics in bi-FGMs under the influence of strain gradient elasticity.

In this research, we conduct a thorough analysis of an arbitrarily oriented mode-III crack in a bidirectional functionally graded material (FGM) using strain gradient elasticity (SGE) theory. The focus is on understanding the growth and behavior of the crack when it is positioned at an angle counterclockwise to the x-axis. The material gradation in the bidirectional FGM is assumed to follow an exponential distribution within the xy-plane. By transforming the global coordinate system into a local system, the x_1 -axis is aligned with the crack’s direction, forming a specific angle with the x-axis. The SGE theory uses two material characteristic lengths, ℓ and ℓ ^' , to account for volumetric and surface strain gradient factors, respectively. To solve the crack boundary value problem, we utilize a methodology that combines Fourier transforms with an innovative hyper-singular integrodifferential equation approach. This methodological framework allows us to derive a comprehensive system of equations, which are then solved using Chebyshev polynomial expansion techniques and the selection of suitable collocation points. Our study includes a detailed examination of the crack surface displacement under various material parameter configurations. We also analyze the stress intensity factors and the energy release rate at the crack tips, providing critical insights into the mechanical behavior of cracks in bidirectional FGMs under the influence of strain gradient elasticity.

In contrast to classical mechanics, which primarily relies on continuum assumptions and neglects microstructural effects, the strain gradient elasticity (SGE) theory represents a paradigm shift in understanding the mechanical behavior of materials at small length scales. In this article, the influence of the bi-directional material gradation on a mode-III crack in functionally graded material via SGE theory is studied. The SGE theory uses two material characteristic lengths, l and l', to account for volumetric and surface strain-gradient factors, respectively. Our investigation is centered on a material gradation model assumed to vary exponentially, with the shear modulus represented as G(x, y) = G(0)e(beta x+gamma y), where beta and gamma. are material gradation constants. To address the crack boundary value problem under consideration, we employ a methodology combining Fourier transforms and an innovative hyper-singular integro-differential equation approach. Using this approach, we systematically formulate a system of equations, which can be solved by selecting suitable collocation points. The closed-form analytical expressions are derived for the standard fracture parameters such as crack surface displacement (CSD), stress intensity factor (SIF), and energy release rate (ERR). Numerical studies are illustrated for the derived standard fractures, and the influence of these parameters beta, gamma, l, l', and applied shear load is graphically presented. Through comprehensive analysis, our aim is to provide insights into the complex interplay between material parameters, loading conditions, and crack behavior in functionally graded materials.

In this study, we extend the investigation of mechanical behavior in functionally graded materials (FGMs) to include bi‐directional exponential variations in material properties, addressing the case of two unequal collinear mode‐III cracks weakened FGM plate. Employing the strain gradient elasticity theory, which incorporates two characteristic lengths, and , to capture volumetric and surface strain‐gradient effects, respectively, we analyze the shear modulus as a function of both spatial coordinates, represented as . This bi‐directional gradation introduces new complexities into the system, necessitating advanced numerical techniques. We construct a system of hyper‐singular integral‐differential equations to derive numerical solutions for various fracture parameters. Our analysis covers different variations of bi‐directional material gradation parameters, and , and strain gradient parameters, and to explore their influence on stress intensity factor, strain distributions, stresses, and crack surface displacement. Additionally, we examine the effect of varying inter‐crack distances, providing a comprehensive understanding of the interplay between crack geometry and bi‐directional material gradation. The findings demonstrate significant variations in fracture behavior and highlight the importance of considering bidirectional material gradation in FGMs.

In this paper, we have studied the behaviour of two symmetric mode-III collinear cracks in a functionally graded material (FGM). The fundamental goal of this paper is to provide insight on the interaction of two cracks in FGMs with the strain gradient effect. To assess the influence of gradient elasticity, we have considered two key parameters ℓ and ℓ ^' , which describe the size scale effect caused by the underlying microstructure and are related to volumetric and surface strain energy, respectively. The crack boundary value problem have been solved by the approach involving Fourier transforms and the innovative hyper-singular integro-differential equation method, where the integral equation contains the two terms in integrals for the both cracks. A system of equations has been constructed by employing the Chebyshev polynomial expansion and then by choosing the suitable collocation points the system of equation have been solved. Our investigation involves the determination of stress intensity factors at both crack tips. These factors are vital for understanding the material’s fracture behavior and structural integrity. Furthermore, we explore the variations in the displacement profile when the distance between the cracks is reduced to close proximity. This particular scenario is of significant interest as it provides insights into how the interaction between the cracks impacts the overall structural response.

Within the framework of linear elasticity, the M-integral is connected to energy changes due to a self-similar expansion of a configuration. In this contribution, we will calculate the M-integral by evaluating a path-independent contour integral and by calculating the virial of material tractions active at boundaries and interfaces for the simple case of a hollow circular cylinder having an inclusion with and without misfit, loaded by external forces under plane stress/plane strain conditions. The connection between the similarity transformation of the total elastic potential and the M-integral is explored.

In this article, the strain gradient elasticity (SGE) theory is used to investigate the mechanical behaviour of a functionally graded material (FGM) weakened by two unequal collinear mode-III cracks. The SGE theory uses two material characteristic lengths, l and l ', to account for volumetric and surface strain-gradient factors, respectively. The directions of both cracks coincide with the material gradation, which is oriented parallel to the x-axis. The numerical outcomes of the problem are obtained by constructing the system of equations using the hyper-singular integral-differential equation technique. The influence of the material gradation parameter on various fracture parameters, including the stress intensity factor (SIF), strain and crack surface displacement (CSD), is numerically shown. In addition, different loads are considered while analysing CSD profiles in connection to the gradation parameter beta. Furthermore, the effect of inter-crack distance on CSD profiles is comprehensively explored, showing information between crack geometry and material gradation.

The last couple of years has motivated and triggered more in-depth research in viruses. With the recent advancement in nanotechnology and nanomechanic, the mechanical perspective of virus is an important perspective to explore. Multidisciplinary research is needed in order to gain a full understanding of how viruses work mechanically which encompasses the understanding of defect mechanics, nanomechanics, nonlinear finite element analysis, molecular dynamic and contact mechanic, just to name a few. We discuss methods for describing the physical processes underlying viral structure and self-assembly. We consider an elasticity theory and its properties in the curved space, particularly for application to obtain mechanical properties of viral capsids. We furthermore review some applications to biomechanics and nanomechanical detection of viruses. This review is aimed to spark and gain further interest in the mechanical perspective of viruses which can ultimately lead to the development of new technologies to prevent future pandemics.

In the present work, the extended finite element method (XFEM) is successfully implemented for the thermo-elastic analysis of edge dislocations. Volterra type edge dislocation is modeled using Heaviside and core enrichment functions. The singularity at the dislocation core is captured through infinite domain solution at the core. The Peach-Koehler force is numerically evaluated using the domain form of the J-integral from the XFEM solution of thermo-elastic fields. Two problems i.e. an edge dislocation in the semi-infinite domain and an edge dislocation near the bi-material interface, are solved for the thermo-elastic case. The problems of dislocation dipole are evaluated for the calculation of Peach-Koehler force. The displacement and traction boundary conditions are applied in different problems along with the thermal boundary conditions. Three different cases i.e. constant heat flux parallel to the glide plane, constant heat flux perpendicular to the glide plane, and constant temperature are considered for the analysis. The numerical simulations are performed at different temperatures to examine its influence on the Peach-Koehler force.

The effects of lattice-mismatched strain on the optical properties of type-I core/shell quantum dot nanostructures are investigated theoretically using the simple continuum elasticity and the effective mass quantum mechanics models. The misfit strains are calculated based on the spherically anisotropic linear elasticity solution for a misfitted inclusion embedded in a finite matrix. The effective mass quantum mechanics model for the case of a CdSe/CdS core/shell quantum dot (QD) shows that the subband energy is blue shifted by the compressive strain in the QD core resulting from the lattice-mismatched heterojunction. Using a very simple spherically anisotropic strain model, we calculated the subband, interband transition, and binding energies for a zinc-blende (ZB) CdSe/CdS QD. These results compare favorably with those obtained from the finite element analysis (FEA) considering the rectilinear cubic crystal anisotropy.

Analytical expressions are derived for the stresses and electric fields induced in a piezoelectric multilayer deposited on a substrate with lattice misfit and thermal expansion coefficient mismatch. The piezoelectric multilayer can be subjected to an externally applied force, moment and electric potential. The derived formulations can model any number of layers using recursive relations that minimize the computation time. A proper rotation matrix is utilized to generalize the derived expressions to accommodate various orientations allowing each layer to have hexagonal crystal symmetry. The influence of lattice misfit and thermal expansion coefficient mismatch in the presence of externally applied force, moment and electric potential on the state of the electroelastic fields in each layer is evaluated for various applications. Comparison with finite element analysis results shows excellent agreement. The analytical expressions developed here can be useful in designing electromechanical sensors, actuators and optoelectronic devices made from piezoelectric multilayers.

The analytical model to predict the density of misfit dislocations at different interfaces in a piezoelectric multilayer with two thin-film layers deposited on a thick substrate has been formulated considering the two-stage relaxation with the aim to generalize further for the multilayer system in future. Different lattice parameters of the film layers and the substrate give rise to high misfit strain. Misfit dislocation formation at the interface releases excessive misfit strain. The internal energy of the piezoelectric thin film at the critical thickness and after relaxation through ‘n’ number of misfit dislocations together with the energy of the misfit edge dislocation have been utilized to develop the formulation. This formulation has been used to estimate dislocation density at the interface of the gallium nitride film layer with various cases of misfit strains. It has rightly predicted higher dislocation density for higher misfit strain cases as well as for greater film thickness. It will be used to predict the dislocation densities at the gallium nitride–indium gallium nitride interface and the sapphire–gallium nitride interface of the LED device in future. The theoretical model developed in this work can be beneficial in estimating the optoelectronic performance of the LED devices correctly in the presence of defects. The model can also be helpful in developing strategies to reduce the dislocation density by accurately predicting them for the given misfit strain in the film.

Configurational forces acting on two-dimensional (2D) elastic line singularities are evaluated by path-independent J-, M-, and L-integrals in the framework of plane strain linear elasticity. The elastic line singularities considered in this study are the edge dislocation, the line force, the nuclei of strain, and the concentrated couple moment that are subjected to far-field loads. The interaction forces between two similar parallel elastic singularities are also calculated. Self-similar expansion force, M, evaluated for the line force shows that it is exactly the negative of the strain energy prelogarithmic factor as in the case for the well-known edge dislocation result. It is also shown that the M-integral result for the nuclei of strain and the L-integral result for the line force yield interesting nonzero expressions under certain circumstances.

Exact closed-form expressions have been derived for the stresses and the electric fields induced in piezoelectric multilayers deposited on a substrate with lattice misfit and thermal expansion coefficient mismatch. The derived formulations can model any number of layers using recursive relations that minimize the computation time. A proper rotation matrix has been utilized to generalize the expressions so that they can be used for various growth orientations with each layer having hexagonal crystal symmetry. As an example, the influence of lattice misfit and thermal expansion coefficient mismatch on the state of electroelastic fields in different layers of GaN multi quantum wells has been examined. A comparison with the finite element analysis results showed very close agreement. The analytical expressions developed herein will be useful in designing optoelectronic devices as well as in predicting defect density in multi quantum wells.

Linear piezoelectric formulations were employed to analytically evaluate the electroelastic fields generated by a single threading edge/screw dislocation in piezoelectric gallium nitride (GaN) thin film. All possible growth orientations, namely c-plane (polar), a- and m-planes (non-polar), and (112¯2)-plane (semi-polar) of GaN layers were considered for the analysis. Single piezoelectric threading edge and screw dislocations were also modeled by a commercial finite element analysis code, ABAQUS 6.14. The edge dislocation was modeled by invoking one-dimensional thermal strain analogy to effectively introduce an extra plane of atoms. The screw dislocation was modeled by creating a slip in opposite surfaces by applying opposite displacements of magnitude equal to half the Burgers vector parallel to the direction of the dislocation line. We also evaluated the Peach-Kohler force acting on a piezoelectric dislocation as well as the material force calculated by the J-integral. The effects of piezoelectricity and the line charge on electroelastic fields induced by a single dislocation were investigated for various growth orientations. We also evaluated the role of spontaneous polarization along c-direction in the presence of electroelastic fields generated by the dislocation. These results can be utilized in studying the influence of piezoelectric strains and electrical fields on the optoelectronic performance of GaN-based devices in the presence of dislocations. They can also be helpful in understanding the mechanics of misfit dislocations that form between the substrate and the film.

Piezoelectric behavior of quantum dot core/shell structures in zinc-blende and wurtzite crystals are investigated in the framework of linear piezoelectricity. Strains arising from lattice mismatch in the core/shell structure are modeled as eigenstrains resulting from size-mismatched inclusion embedded in a finite spherical piezoelectric medium. Assuming that the core/shell piezoelectric structure exhibits spherically-hexagonal anisotropy, an exact solution is obtained through the eigenvalue decomposition method and the analytical expressions of the electroelastic fields are found using appropriate boundary and continuity conditions. It is found that the electroelastic fields can become singular or vanish at the center of the core for certain cases of spherical anisotropy. The analytical solutions are verified with the finite element analysis (FEA) and found to be in excellent agreement. FEA for the case of rectilinear anisotropy was also performed which showed considerable difference from the spherically anisotropic analytical solution. The closed-form solution obtained in this work can be used for any two-layer piezoelectric structures having spherical geometry subjected to various electromechanical loads.

Since blood viscosity is a basic parameter for understanding hemodynamics in human physiology, great amount of research has been done in order to accurately predict this highly non-Newtonian flow property. However, previous works lacked in consideration of hemodynamic changes induced by heterogeneous vessel networks. In this paper, the effect of bifurcation on hemodynamics in a microvasculature is quantitatively predicted. The flow resistance in a single bifurcation microvessel was calculated by combining a new simple mathematical model with 3-dimensional flow simulation for varying bifurcation angles under physiological flow conditions. Interestingly, the results indicate that flow resistance induced by vessel bifurcation holds a constant value of approximately 0.44 over the whole single bifurcation model below diameter of 60μm regardless of geometric parameters including bifurcation angle. Flow solutions computed from this new model showed substantial decrement in flow velocity relative to other mathematical models, which do not include vessel bifurcation effects, while pressure remained the same. Furthermore, when applying the bifurcation angle effect to the entire microvascular network, the simulation results gave better agreements with recent in vivo experimental measurements. This finding suggests a new paradigm in microvascular blood flow properties, that vessel bifurcation itself, regardless of its angle, holds considerable influence on blood viscosity, and this phenomenon will help to develop new predictive tools in microvascular research.