Couple Stress Components Research Articles

Couple-stress elasticity of intrinsic and extrinsic dislocations in three-dimensional multilayered materials

This work investigates the influence of couple stresses on intrinsic and extrinsic dislocations within three-dimensional heterogeneous multilayered structures. The materials consist of orthotropic and dissimilar layers with different elastic properties, incorporating the microstructural lengths from a special version of the anisotropic couple-stress elasticity. Double Fourier series expansions are used to describe extended displacements and tractions, considering rotation and couple-stress components and accounting for arbitrary displacement and rotation discontinuities associated with dislocation and disclination loops as well as periodic dislocation and disclination networks at semi-coherent interfaces. The dual-variable and position technique is used to recursively determine the field responses at any point in the multilayered laminate. The application examples focus on dislocation problems with couple stresses, revealing the effects of the core-spreading dislocation technique, free surfaces, anisotropic elasticity, and couple stresses on the field solutions. The analysis also uncovers pressure and von Mises stress amplitudes, non-zero rotation, and non-symmetric stress profiles due to couple stresses in copper/niobium bimaterials. Changes in the sign and magnitude of the Peach-Koehler forces are observed when couple stresses are introduced, which deviate from the classical theory of dislocations. The investigation finds a significant 29% reduction in energy per unit area at a semi-coherent interface between copper and niobium influenced by couple stresses. These results deepen our understanding of how couple stresses affect dislocation behavior in multilayered materials, and provide prospects for future discrete dislocation and disclination dynamics simulations with engineering applications in various nano- and micro-structured materials.

Photo-thermo-elastic interactions in micropolar generalized thermoelasticity theory in the framework of Green-Naghdi theory

The purpose of the current study is to establish a novel mathematical model in the micropolar theory of generalized thermoelasticity under the framework of photothermal theory. For the solution of the required problem, the integral transforms along with the potential displacement approach has been utilized. For graphical representation of different physical quantities such as displacement components, force and couple stress components, carrier density as well as the temperature distribution, Matlab software has been used.

Analysis of a Reiner–Rivlin liquid sphere enveloped by a permeable layer

The present article investigates the axisymmetric flow of a steady incompressible Reiner–Rivlin liquid sphere enveloped by a spherical porous layer using the cell model technique. The Brinkman-extended Darcy model is deployed for the porous medium hydrodynamics, and isotropic permeability is considered. The stream function solutions of the governing equations are obtained, which involves the Gegenbauer functions and the modified Bessel functions. An asymptotic series expansion in terms of the Reiner–Rivlin liquid parameter S has been employed to determine the expression of the flow field for the Reiner–Rivlin liquid. Boundary conditions on the cell surface corresponding to the Happel, Kuwabara, Kvashnin, and Cunningham models are considered. Analytical expressions are derived for dimensionless pressure, tangential stress, and the couple stress components using the method of separation of variables and Gegenbauer functions/polynomial. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the porous zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the Reiner–Rivlin fluid on the porous sphere are derived with corresponding expressions for the drag coefficient. Mathematical expression of the drag coefficient, pressure distribution, velocity profile, and separation parameter is established. On the basis of viscosity ratio, permeability parameter, dimensionless parameter, and the volume fraction, variations of the drag coefficient, velocity profiles, fluid pressure, and separation parameter (SEP) are investigated with their plots. The effects of the streamline pattern make the flow more significant for the Mehta–Morse when compared to the other models. Additionally, the mathematical expression of the separation parameter (SEP) is also calculated, which shows that no flow separation occurs for the considered flow configuration and is also validated by its pictorial depiction. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behavior often arises in addition to porous media effects.

Mathematical modelling of couple stress fluid flow around a semi‐permeable sphere enclosing a solid core

AbstractThe focus of this article is to study theoretically the steady‐axisymmetric creeping flow dynamics of a couple stress fluid external to a semi‐permeable sphere containing a solid core. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behaviour often arises in addition to porous media effects. The non‐Newtonian Stokes’ couple stress fluid model features couple stresses and body couples that are absent in the classical Navier–Stokes viscous model. It provides a robust framework for simulating emulsions, complex suspensions and other liquids which possess microstructure. The physical regime is delineated into two zones – the interior of the semi‐permeable zone (region II) which engulfs the solid core and the external couple stress fluid zone (region I). The model is formulated using a spherical polar coordinate system in terms of the stream function ψ. The Brinkman‐extended Darcy model is deployed for the porous medium hydrodynamics and isotropic permeability is considered. Analytical expressions are derived for dimensionless pressure, tangential stress and the couple stress components using the method of separation of variables and Gegenbauer functions of the first kind. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the semi‐permeable zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the couple stress fluid on the semi‐permeable sphere and volumetric flow rate are also derived with corresponding expressions for the drag coefficient and non‐dimensional volumetric flow rate. The influence of permeability (k), separation parameter (l), couple stress viscosity coefficient (η), couple stress inverse length dependent parameter (λ = √(μ/η)) and couple stress viscosity ratio (τ = η/η/) on all key variables is studied graphically. Additionally, streamline contours are computed for a range of parameters including inverse permeability parameter (α = 1/k). The computations show that increasing couple stress inverse length dependent parameter (λ) greatly reduces the dimensionless volumetric flow rate in particular at high values of separation parameter (l). Flow rate is, however, markedly enhanced with permeability (k) and also couple stress viscosity parameter (η). In the presence of the solid core, a much greater drag coefficient is observed at all values of couple stress inverse length dependent parameter (λ) relative to the case without a solid core. A significant distortion in streamlines is computed with increasing separation parameter (l) with a dual vortex structure emanating. With greater couple stress viscosity parameter (τ), no tangible modification is observed in the streamline contours. The present work generalizes the earlier study of Krishnan and Shukla to consider a semi‐permeable (porous medium) outer sphere and furthermore presents detailed streamline visualizations of the creeping flow for a range of emerging parameters of relevance to non‐Newtonian chemical engineering processes including emulsion droplet dynamics in porous materials.

Thermally induced nonlinear stability and imperfection sensitivity of temperature- and size-dependent FG porous micro-tubes

An attempt is made in the current research to analyse the nonlinear thermal stability and imperfection sensitivity of functionally graded (FG) porous micro-tubes. Temperature-dependent properties of the geometrically imperfect micro-tube are graded across the radius of cross-section. It is assumed that the micro-tube with different end conditions is in contact with a two-parameter elastic foundation. The nonlinear component of the elastic foundation can be of the hardening or softening type. The equilibrium equations are obtained within the framework of von Karman nonlinear assumptions and high-order shear deformation tube theory. The governing equations are reformulated for the case of imperfect micro-tubes based on the modified couple stress theory. The system of nonlinear differential equations is solved using the two-step perturbation technique and Galerkin procedure. The analytical solutions are obtained for three different types of immovable boundary conditions which are clamped-rolling, simply-supported and clamped–clamped. The closed-form expressions are given to obtain the large deflection in the micro-tube as a function of the elevated temperature. Novel parametric studies are given to explore the thermal stability and imperfection sensitivity analysis of the perfect and imperfect micro-tubes, respectively. The effects of boundary conditions, couple stress components, porosity coefficient, elastic foundation, FG pattern, temperature dependence and geometrical parameters are studied.

A size-dependent model for shear deformable laminated micro-nano plates based on couple stress theory

In this paper, a modified first order shear deformation theory is proposed for laminated micro-nano plates. Couple stress components are involved in the differential equations of equilibrium for shear deformable micro-nano plates, which are established by considering the deformation mechanism of an element cut from the micro-nano plate. Based on the mechanics of composite materials and Koiter’s theory, the constitutive equations for laminated micro-nano plates are established to characterize the relationships between Cauchy stress tensor and the strain tensor, the couple-stress tensor and the curvature tensor. Governing equations with size-dependent effect are derived and are expressed in terms of the displacements and the rotation functions. The proposed theory is applied to investigate the eigenvalue buckling and free vibration of laminated micro-nano plates simply supported at the four sides. Matlab program based on the analytical solutions is developed and is testified by comparing the solutions for isotropic micro-nano plates from the proposed theory against those from an available model and from ABAQUS analysis. Various examples are studied to investigate the influence of scale ratio, aspect ratio, length-to-thickness ratio, material properties and stacking sequence on the buckling and free vibration response of the laminated micro-nano plates. The size-dependent effect is captured for all examples, particularly for micro-nano isotropic laminated plate with geometry size similar to its material length scale parameter, where the buckling load may be nearly up to 10 times larger than that given by the FSDT theory. A good agreement among solutions from the Matlab program, the existing literature and ABAQUS analysis demonstrates the efficiency and accuracy of the proposed theory.

Singularity-free defect mechanics for polar media

We present singularity-free solution for cracks within polar media in which material points possess both position and orientation. The plane strain problem is addressed in this study for which the generalized continua including micropolar, nonlocal micropolar, and gradient micropolar elasticity theories are employed. For the first time, the variationally consistent boundary conditions are derived for gradient micropolar elasticity. Moreover, having reviewed the solution to line defects including glide edge dislocation and wedge disclination from the literature, the fields of a climb edge dislocation within micropolar, nonlocal micropolar and gradient micropolar elasticity are derived. This completes the collection of line defects needed for an inplane strain analysis. Afterward, as the main contribution, using both types of line defects (i.e., dislocation being displacement discontinuity and disclination being rotational discontinuity), the well-established dislocation-based fracture mechanics is systematically generalized to the dislocation- and disclination-based fracture mechanics of polar media for which we have three translations together with three rotations as degrees of freedom. Due to the application of the line defects, incompatible elasticity is employed throughout the paper. Cracks under three possible loadings including stress and couple stress components are analyzed, and the corresponding line defect densities and stress and couple stress fields are reported. The singular fields are obtained once using the micropolar elasticity, while nonlocal micropolar, and gradient micropolar elasticity theories give rise to singularity-free fracture mechanics.

Reflection and transmission of elastic waves in the multilayered orthotropic couple-stressed plates sandwiched between two elastic half-spaces

• A solving method that is suitable for the wave reflection and transmission of anisotropic microstructures is presented. • Anisotropic multilayered couple-stressed plates between two elastic half-spaces are studied. • The incident P wave is influenced by one length scale parameter but the SH wave is influenced by two length scale parameters. • The length scale parameter lx has much more significant influence on SH waves than the length scale parameter ly . In this paper, a high efficient computational approach, the extended Legendre orthogonal polynomial method (LOPM) is provided to investigate the reflection and transmission of elastic waves in orthotropic couple-stress layered plates sandwiched between two elastic half-spaces. In this approach, the anisotropic couple-stress theory is introduced into the LOPM to calculate the reflection and transmission coefficients of the orthotropic interlayers. The stress components, couple-stress components, rotation vectors and governing equations are derived in terms of the Legendre orthogonal polynomial. The present method does not require calculations of the displacement solutions of each partial wave in anisotropic multilayered microstructures, but expands the displacement vector of each layer into a Legendre orthogonal polynomial series with the expansion coefficients to be calculated. The incident P wave and SH wave are calculated, respectively. The effects of length scale parameters in three different directions are studied. It is found that the reflection and transmission coefficients of the incident P wave are only related to the length scale parameter in the z direction. For incident SH wave, the influence of the length scale parameter along the thickness direction is much more significant than that of the length scale parameter in the y direction.

Reflection of Longitudinal Micro-Rotational Wave at Viscoelastically Supported Boundary of Micropolar Half-Space

AbstractThe reflection of longitudinal micro-rotational wave at the viscoelastically supported boundary of micropolar half-space is studied in this paper. The viscoelastic boundary is described by spring-dashpot model with parallel or serial connection. Both the spring and the dashpot contribute to the displacements and micro-rotation and the boundary conditions include the force stress and couple stress components. From the boundary conditions, the amplitude ratios and phase shifts of reflection waves with respect to the incident wave are obtained. Further, the energy flux ratios of the reflection waves to the incident wave are estimated. In order to validate the numerical results, the energy flux conservation with consideration of the energy dissipation of dashpot is used. Based on the numerical results, the influences of elastic parameters and viscous parameters are studied, respectively. It is found that the elastic parameters and the viscous parameters have evident influences on the amplitude ratio, the phase shift and the energy partition. The causes resulting in these deviations are related with the instantaneous elasticity of spring and the time-delay effects of dashpot.

Size-dependent dynamic pull-in analysis of geometric non-linear micro-plates based on the modified couple stress theory

This paper focuses on the size-dependent dynamic pull-in instability in rectangular micro-plates actuated by step-input DC voltage. The present model accounts for the effects of in-plane displacements and their non-classical higher-order boundary conditions, von Kármán geometric non-linearity, non-classical couple stress components and the inherent non-linearity of distributed electrostatic pressure on the micro-plate motion. The governing equations of motion, which are clearly derived using Hamilton's principle, are solved through a novel computationally very efficient Galerkin-based reduced order model (ROM) in which all higher-order non-classical boundary conditions are completely satisfied. The present findings are compared and successfully validated by available results in the literature as well as those obtained by three-dimensional finite element simulations carried out using COMSOL Multyphysics. A detailed parametric study is also conducted to illustrate the effects of in-plane displacements, plate aspect ratio, couple stress components and geometric non-linearity on the dynamic instability threshold of the system.

The Influence of Couple Stress Components and Electrostatic Actuation on Free Vibration Characteristics of Thin Micro-Plates

The present paper deals specifically with the frequently asked question of are there any differences between classical non-actuated mode-shapes of plate-type MEMS and the electrically deformed ones obtained by the non-classical modified couple stress theory. To this end, a Kirchhoff’s thin plate model together with the modified couple stress theory is considered to extract the eigenvalue problem which governs on free vibrations of the system. Using the free vibration equation of the system, it is proved that neither the electrostatic actuation nor the couple stress components influence on linear mode-shapes of the system. To emphasis on the validity of the present proof, the extended Kantorovich method is employed to provide analytical solutions for an electrically actuated micro-plate with clamped boundary conditions. The solutions, whose accuracy is verified by available results in the literature, also indicate that the linear mode-shapes of an electrically actuated micro-plate are totally independent from the influences of size and electrical loading.

Analytical determination of size-dependent natural frequencies of fully clamped rectangular microplates based on the modified couple stress theory

This paper presents an analytical and size-dependent model for vibrational analysis of fully clamped rectangular microplates. Modified couple stress theory (MCST) and the Kirchhoff plate model are considered, and Hamilton’s principle is employed to derive the sizedependent equation of motion that accounts for the effect of residual stresses. The natural frequencies of the microplate are extracted analytically by extended Kantorovich method. The present findings are validated with the available results in the literature, and an excellent agreement is observed between them. In addition, a parametric study is conducted to demonstrate the significant effects of couple stress components on the natural frequencies of fully clamped microplates. The ratio of MCST natural frequencies to those obtained with classical theory depends only on the Poisson’s ratio of the plate and is independent of the aspect ratio of the plate for cases with no residual stresses.

Size-dependent free vibration analysis of electrostatically pre-deformed rectangular micro-plates based on the modified couple stress theory

The modified couple stress theory (MCST) is a non-classical continuum theory which is capable of capturing size-dependent behavior of small structures occurring in micron and sub-micron scales. The objective of the present paper is to investigate the size-dependent free vibration characteristics of rectangular micro-plates pre-deformed by an electric field based on the MCST. To this end, a size-dependent Kirchhoff׳s plate model is considered and the equation of motion which accounts for the effect of residual and couple stress components as well as the inherent non-linearity of distributed electrostatic excitation is derived using Hamilton׳s principle. The eigenvalue equation corresponding to the free vibration of electrostatically pre-deformed rectangular micro-plates is also extracted from the equation of motion. This equation is solved numerically using the finite element method (FEM). The results are compared and validated by available analytical and semi-analytical findings for flat micro-plates as well as empirical pull-in observations in the literature and a very good agreement between them is observed. A parametric study is also conducted to investigate the effects of couple stress components as well as electrostatic attraction on both natural frequencies and mode-shapes of fully clamped and simply supported micro-plates. It is found that the size effect on natural frequencies is quite negligible for flat and electrostatically pre-deformed micro-plates when the ratio of plate thickness to the material length scale parameter is larger than 10 and 20, respectively. Furthermore, it is found that couple stress components and electrostatic attraction do not have a sizeable effect on micro-plate mode-shapes.

Response Due to Concentrated Force in Micropolar Elastic Solid with Voids

Abstract The eigen value approach, following Laplace and Fourier transforms has been employed to find the general solution of the field equation in a micropolar elastic solid with voids for the plane strain problem. An application of an infinite space with impulsive force has been taken to illustrate the utility of the approach. The integral transformations have been inverted by using a numerical inversion technique to get result in physical domain. The result in the form of normal displacement, volume fraction, normal force stress, tangential force stress and tangential couple stress components has been obtained numerically and illustrated graphically to depict the effect of micropolarity and voids.

Response of harmonic sources in couple stress thermoelastic medium with state space approach

Purpose – The purpose of this paper is to investigate the two-dimensional problem in couple stress thermoelastic medium for a half space is established and state space approach has been applied to solve the problem. Design/methodology/approach – Normal mode analysis is used to obtain the exact expressions for normal stress, tangential stress and couple stress. Numerical calculation is prepared for these quantities and depicted graphically for a special model. Findings – The expressions for normal stress, tangential stress and couple stress are obtained numerically and depicted graphically to see the couple stress effect. Originality/value – It is found that couple stress effect decrease the value of normal stress components for circular frequency equal to 0.5 for small values of the wave number and then increases whereas the values of normal stress components decrease first and then increase monotonically for circular frequency equal to 0.1 when the force is applied in normal direction and the values of tangential stress components and couple stress components decrease for all values of wave number. But the values for normal stress components, tangential stress components and couple stress components increase when the force is applied in tangential direction.

Size-dependent dynamic pull-in analysis of beam-type MEMS under mechanical shock based on the modified couple stress theory

Size-dependent stability analysis of a fully clamped micro-electro-mechanical beam under the effect of shock acceleration pulse is the objective of present paper. The size-dependent Euler–Bernoulli beam model based on the modified couple stress theory (MCST) with von Kármán-type geometric non-linearity is utilized in theoretical formulations. The non-linear governing differential equation of motion is derived using Hamilton’s principle and solved using a simple and computationally efficient single degree-of-freedom (SDOF) approach. The model’s predictions based on the classical theory (CT) are compared with those obtained using the finite element method (FEM) and six modes Galerkin approximations in previous studies and an excellent agreement between them is achieved. It is shown that the present SDOF predictions agree better with the FE results than those obtained using six modes approximations for high shock accelerations. Furthermore, the present model can remove the limitation of previous models in capturing dynamic pull-in instability under enormous shock accelerations. A parametric study is also conducted to show the significant effects of couple stress components on micro-beam motion. It is found that the size effect on both dynamic pull-in voltage and maximum amplitude of micro-beam oscillations is usually negligible, when the ratio of beam thickness to the material length scale parameter is larger than 15.

Eigen Value Approach in Micropolar Elastic Medium with Voids

The eigen value approach, following the Laplace and Hankel transformation has been employed to find a general solution of the field equations in a micropolar elastic medium with voids for an axisymmetric problem. An infinite space with the mechanical source has been applied to illustrate the utility of the approach. The integral transformations has been inverted by using a numerical inversion technique to get the result in physical domain. The results in the form of normal displacement, volume fraction, normal force stress, tangential force stress and tangential couple stress components have been obtained numerically and illustrated graphically.

Numerical modeling of bending of micropolar plates

In this paper we present the Finite Element modeling of the bending of micropolar elastic plates. Based on our recently published enhanced mathematical model for Cosserat plate bending, we present the micropolar plate field equations as an elliptic system of nine differential equations in terms of the kinematic variables. The system includes an optimal value of the splitting parameter, which is the minimizer of the micropolar plate stress energy. We present the efficient algorithm for the estimation of the optimal value of this parameter and discuss the approximations of stress and couple stress components. The numerical algorithm also includes the method for finding the unique solution of the micropolar plate field equations corresponding to the optimal value of the splitting parameter. We provide the results of the numerical modeling for the plates made of polyurethane foam used in structural insulated panels. The comparison of the numerical values of the vertical deflection for the square plate made of dense polyurethane foam with the analytical solution of the three-dimensional micropolar elasticity confirms the high order of approximation of the three-dimensional (exact) solution. The size effect of micropolar plate theory predicts that plates made of smaller thickness will be more rigid than would be expected on the basis of the Reissner plate theory. We present the numerical results for plates of different shapes, including shapes with rectangular holes, under different loads.

Generation of waves in an infinite micropolar elastic solid body under initial stress

The aim of the present paper is to investigate generation of waves in an infinite micropolar elastic medium under the influence both of initial stressp and body forces X. The equation of motion has been solved applying the Fourier-Hankel transform. The final results, the displacement, the stress, the rotation, and the couple stress components have been obtained in analytical form as integrals involving Bessel function of first kind and of zero order.

Circular cylinder oscillating about a mean position in incompressible micropolar fluid

The motion of a micropolar incompressible fluid arising from the rectilinear oscillation of a circular cylinder is examined. The velocity, micro-rotation, stress and couple stress components are expressed in terms of the modified Bessel functions with complex arguments. The force on the cylinder is obtained. It is found that it does not experience any couple.