Micropolar Plates Research Articles

Exploring acoustic wave transmission in micropolar plate in air

This study explores the behavior of sound wave propagation across a micropolar plate at diverse incident angles, specifically crafted from expanded polystyrene. Through a blend of theoretical formulations and practical experimentation, we scrutinize the alterations in transmission coefficients within the frequency spectrum, shedding light on the dynamic shifts in various vibrational modes. Utilizing a comprehensive setup comprising a loudspeaker, microphone, and National Instruments acquisition system, we meticulously discerned the initial trio of longitudinal vibration modes across a broad frequency span extending up to 70 kHz. Notably, despite the plate's inherent dispersion and attenuation traits, this experimental setup proved effective. A distinctive facet of our approach lies in the deliberate incorporation of multiple incident angles, which yielded crucial insights. Notable is the identification, within the experiments, of a critical angle, marking a transitional zone. Below this threshold, distinct Lamb modes dominate, while beyond it, the emergence of micropolar modes becomes apparent. These empirically validated findings, in alignment with theoretical projections, substantially enrich our comprehension of these materials' engineering applications.

Observation of micropolar modes in the transmission of acoustic waves at oblique incidence by polystyrene plates.

This study investigates micropolar plates through the analysis of transmitted acoustic waves at various incident angles. Using a combined theoretical and experimental approach, we explore the non-classical elasto-dynamic behavior of these materials. Theoretical transmission coefficients are tailored to highlight generalized Lamb modes and a new type of vibration mode called micropolar modes, considering the constant thickness of the plate and the effects of material micropolarity on linear elasticity. Overlaying diverse transmission coefficients for different angles provides insight into micropolar behavior, aiding in the understanding of mode evolution in the frequency domain and the estimation of the critical angle. We employ two types of low-density closed-cell foams made from polystyrene (expanded in one case, extruded in the other), covering a frequency range from 4 to 60 kHz using a tweeter loudspeaker and a microphone integrated with a National Instruments acquisition system. Identification of the first vibration mode of the plates is facilitated by the frequency range rich in low frequencies. Additionally, all responses exhibit supersonic velocities with substantial attenuations. The originality of this paper is twofold: it provides theoretical predictions of micropolar modes, along with their experimental observations above a critical angle of incidence. Consequently, this research enriches our understanding of micropolar materials and their potential contribution to effective noise reduction strategies in acoustic engineering.

A theory for three-dimensional response of micropolar plates

A theory for three-dimensional response of micropolar plates

Influence of diverse boundary conditions on SH and P-SV wave dynamics in micropolar plates

Studying wave propagation characteristics in plates is vital for comprehending the material dynamics to enhance the strength of structures and for optimizing the effectiveness of non-destructive testing devices. Micropolar elasticity displays microcontinuum behavior by integrating rotational and coupling effects through displacements and micro-rotations. The present study thoroughly investigates horizontally polarized shear waves (SH) within a thin micropolar plate by employing diverse boundary conditions namely stress-free, clamped, and mixed conditions. Shear horizontal waves propagate within a plate by shearing the material along the surface, excluding any motion in perpendicular direction. Moreover, for a more comprehensive grasp of wave behavior in a micropolar plate, the study investigates P-SV type waves under plane strain conditions. Analytical techniques are utilized to derive the dispersion relations for SH and P-SV type waves in a micropolar plate. Graphical representations are provided to showcase the impacts of various micropolar parameters such as coupling number, characteristic length, and plate thickness on the phase velocities of SH and P-SV type waves. The study also includes a dispersion analysis of the multimode aspect of SH and P-SV waves. This thorough investigation can significantly augment the comprehension of wave propagation phenomena within plate-like structures.

Boundary Value Problems in a Theory of Bending of Thin Micropolar Plates with Surface Elasticity

Boundary Value Problems in a Theory of Bending of Thin Micropolar Plates with Surface Elasticity

Hexahedral finite elements with enhanced fixed‐pole interpolation for linear static and vibration analysis of 3D micropolar continuum

AbstractThe spotlight of this research is on the application of the fixed‐pole interpolation, sometimes used in the analysis of three‐dimensional (3D) geometrically non‐linear beams, but for which no attempts have been made to apply it to linear analysis so far. Particular attention is given to the correlation between the linearised forms of the fixed‐pole and helicoidal interpolation with the linked interpolation. Knowledge of this interdependence is crucial for identifying paths for possible enhancement and extension from the Timoshenko beam of arbitrary order to the new hexahedral finite element of arbitrary order for linear analysis of 3D micropolar continuum. After ensuring the convergence of the newly developed micropolar element through a set of patch tests, three numerical examples of a 3D micropolar continuum in static equilibrium and free vibration of 3D micropolar plates with different geometric properties and boundary conditions have been analysed. Based on these results, the newly proposed finite elements have been critically assessed against the conventional elements.

Analytical solution for the bending problem of micropolar plates based on the symplectic approach

Abstract An analytical solution for the bending problem of micropolar plates is derived based on the symplectic approach. By applying Legendre's transformation, we obtain the Hamiltonian canonical equation for the bending problem of a micropolar plate. According to the method of separation of variables, the homogenous Hamiltonian canonical equation can be transformed into an eigenvalue problem of the Hamiltonian operator matrix. We derive the eigensolutions of the eigenvalue problem for the simply-supported, free and clamped boundary conditions at the two opposite sides. Based on the adjoint symplectic orthogonal relation of the eigensolutions, the solution of the bending problem of micropolar plate is expressed as a series expansion of eigensolutions. Numerical results confirm the validity of the present approach for the bending problem of micropolar plates under various boundary conditions and demonstrate the capability of the proposed approach to capture the size-dependent behavior of micropolar plates.

Mixed variational principle for micropolar elasticity and an accurate two-dimensional plate model

A mixed variational principle for the micropolar elasticity theory is established in which the displacements, microrotations, force stresses and couple stresses are considered as independent fields. The presented mixed variational principle not only collects all the intrinsic features of the model in one statement, but, it helps to overcome some inconsistencies between introduced simple kinematic assumptions and the boundary conditions. The mixed variational statement is used to present improved 2D model for micropolar plates. A simple displacement vector accounting for shear deformation and normal stain effects is introduced, as well as, microrotations are taken varying linearly through the plate thickness. Furthermore, force stresses and couple stresses are assumed in complete consistent with the loading conditions on the external faces of the plates. Bending and free vibration applications are presented for micropolar homogenous and functionally graded (FG) plates with different edges conditions. The validity of the presented model is examined by comparing some present numerical results with other published results due to 2D and 3D micropolar plate models. Numerical and graphical analyses are presented to demonstrate the importance of the normal strain and microrotation effects on the bending and free vibration responses of the micropolar plates.

Erratum to: Natural Vibrations of Micropolar Elastic Flexible Plates and Shallow Shells

Erratum to: Natural Vibrations of Micropolar Elastic Flexible Plates and Shallow Shells

Natural Vibrations of Micropolar Elastic Flexible Plates and Shallow Shells

A mathematical model representing the dynamics of geometrically nonlinear (flexible) micropolar elastic thin plates in Cartesian and curvilinear coordinates is constructed (the approach is generalized to the case of micropolar flexible shallow shells as well). The model is developed under the assumption that the elastic deflection of a plate is comparable with the plate thickness but is small compared to the characteristic plate size in plan. Based on the given model of micropolar elastic flexible plates, the problem on free vibrations is solved for rectangular and circular plates and shallow shells. Effective manifestations of characteristic features of a micropolar material are considered in comparison with the corresponding classical material.

A new deformation measure for micropolar plates subjected to in-plane loads

A new deformation measure for micropolar plates subjected to in-plane loads

Carrera unified formulation (CUF) for the micropolar plates and shells. III. Classical models

It is shown that the classical theories of micropolar plates and shells can be obtained using the Carrera Unified Formulation (CUF) approach as a special case of approximation. The theory of micropolar plates and shells based on the hypotheses of Timoshenko–Mindlin and Kirchhoff–Love is considered in detail. The stress and strain tensors, as well as the displacement and rotation vectors are presented as linear expansion along the shell thickness coordinates. Then, all the equations of the micropolar theory of elasticity (including the generalized Hooke's law) were transformed into the corresponding equations for the expansion coefficients in the coordinates of the shell thickness. All equations of the theory of micropolar plates and shells based on the hypotheses of Timoshenko–Mindlin and Kirchhoff–Love are presented here. Micropolar plates in Cartesian and polar coordinates, as well as micropolar shells of cylindrical, conical, spherical and shallow geometry are considered in detail. The equations obtained can be used to calculate the stress-strain state and simulate thin-walled structures at the macro, micro and nanoscale, taking into account the micropolar stresses and the effects of rotation.

Isogeometric dynamic analysis of shells based on the nonlinear micropolar theory

Studying the large-amplitude forced vibration characteristics of micropolar shell-type structures is the main objective of this paper. In order to obtain the full geometrically nonlinear model of micropolar theory, the micromorphic seven-parameter shell kinematic is first developed. Via a nonlinear micro-motion map, the extracted formulation is fitted for the micropolar continuum. As a result, possessing three stress–strain fields with eighteen independent elastic parameters of a linear material, the proposed micropolar shell theory is able to capture the inhomogeneity and size-effects. On the basis of Hamilton’s principle, the variational form of governing equations of motion is determined. The finite element isogeometric solution approach is then utilized to produce the geometry and approximate the displacement field in mid-surface area of shell. This would be helpful for resolving the locking and instability issues of the traditional low-order standard finite element shell models. By discretizing the period of harmonic vibration, a numerical method is also implemented in time domain. To demonstrate the microstructural effects, dynamic behavior of micropolar plate- and shell-type structures are investigated in several parametric studies. Since the classical strain and stress tensors are retained in present micropolar elasticity, both the large macro- and micro-deformations are captured for the first time.

Carrera unified formulation for the micropolar plates

Starting from the variational principle of virtual power for the three-dimensional equations of the micropolar theory of elasticity and using generalized series in terms of the plate thickness coordinates a new higher order models of orthotropic micropolar plates have been developed here for the first time. Following carrera unified formulation, the stress and strain tensors, as well as the vectors of displacements and rotation, have been expanded into series in terms of the plate thickness coordinates. Then, all the equations of the micropolar theory of elasticity (including generalized Hooke’s law) have been transformed to the corresponding equations for the coefficients of the series expansion on the plate thickness coordinates. A system of differential equations in terms of the displacements and rotation vectors and natural boundary conditions for the coefficients of the series expansion of the plate thickness coordinates have been obtained in the same way as in the classical theory of elasticity. All equations for the higher order theory of micropolar plates have been developed and presented here. The case of complete linear expansion has been considered in detail and compared with the theories based on shear deformation and Kirchhoff hypothesis. The obtained equations can be used for calculating the stress-strain and for modeling thin walled structures in macro, micro, and nanoscale when taking into account micropolar couple stress and rotation effects.

Geometrically nonlinear models of static deformation of micropolar elastic thin plates and shallow shells

AbstractThe present paper considers micropolar plates and shallow shells, the elastic deflections of which are comparable with their thickness and are small in comparison with characteristic cross‐section size. At the same time, both rotation angles of the normal to the median surface before deformation and their free rotations are small. Also, in the tensors of deformation and flexures‐torsions, the nonlinear members in the gradients of the displacement are considered. Hypothesis method is developed, on the basis of which general applied models of static deformation of the micropolar elastic flexible plates and shallow shells are constructed. On the basis of these models, specific problems for the micropolar elastic flexible rectangular plates and shallow shells rectangular in cross‐section are solved in case, when bounds are hinge‐supported. On the basis of numerical analysis, certain effective features of the micropolar material are established in comparison with the corresponding classical material.

Design space for bifurcation buckling of laser-welded web-core sandwich plates as predicted by classical and micropolar plate theories

The strength of laser-welded web-core sandwich plates is often limited by buckling. In design of complex thin-walled structures the combination of possible structural and material combinations is basically infinite. The feasibility of these combinations can be assessed by using analytical, numerical and experimental methods. At the early design stages such as concept design stage, the role of analytical methods is significant due to their capability for parametric description and extremely low computational efforts once the solutions have been established for prevailing differential equations. Over the recent years significant advances have been made on analytical strength prediction of web-core sandwich panels. Therefore, aim of the present paper is to show impact of this development to the design space of web-core sandwich panels in buckling. The paper reviews first, briefly the differential equations of a 2-D micropolar plate theory for web-core sandwich panels and the Navier buckling solution for biaxial compression recently derived by Karttunen et al. (Int J Solids Struct 170(1):82–94, 2019) by exploiting energy methods. By comparing the micropolar and widely-used classical first-order shear deformation plate theory (FSDT) solutions, it is shown that the different equivalent single layer (ESL) formulations and plate aspect ratios have a significant impact on the practical outcomes of the feasible design space and this way motivating further developments for micropolar formulations from practical structural engineering viewpoint.

Stress distribution around an elliptic hole in a plate with ‘implicit’ and ‘explicit’ non-local models

Understanding the effects of defects is crucial due to their deliberate or unintentional presence in many materials. Classical theory of elasticity may not be the best candidate to describe behaviour of structures with defects of comparable size of its underlying material organization, as it lacks in internal scale parameters. In this respect, present study focused on comparison of two well-established non-local theories; ‘implicit/weak’, as micropolar (Cosserat), and ‘explicit/strong’, as Eringen’s model, with that of classical model to highlight their differences in a common case study: infinite plates weakened with an elliptic hole of different aspect ratios, under remote uniaxial tension. Fraction coefficient, providing identical stress concentration factor with micropolar plates, is searched for two-phase local/nonlocal Eringen’s model. Results are obtained by adopting finite element method with quadrilateral elements. To account for the discontinuities within domain, Eringen’s model is modified by using geodetical distance instead of Euclidean one, and a computationally very efficient procedure is developed to exploit the symmetric character of the problem without losing long-range interactions. The results suggest that non-local effects, reducing the maximum stress, become more pronounced with increasing geometric discontinuity quantified by the aspect ratio of ellipse which also influences equivalency between characteristic lengths of non-local models.

Closed-form solution for the micropolar plates: Carrera unified formulation (CUF) approach

Navier’s closed-form solution for the higher order theory of micropolar plates based on the CUF approach has been developed here. Obtained using a principle of virtual displacements the 2D system of the differential equations for the higher order theory of micropolar elastic plates is solved here for the case of simply supported plates using Navier’s method of the variables separation. For the higher order theory of micropolar plates developed here, which is based on CUF, some numerical examples have been done and the influence the rotation field on the stress–strain fields has been analyzed. Methods of determination of the classical and micropolar elastic moduli of different materials have been analyzed, and available experimental data have been presented in Nowacki’s notations. The obtained equations can be used for calculating the stress–strain and for modeling thin walled structures in macro-, micro- and nanoscale when taking into account micropolar couple stress and rotation effects.

Carrera unified formulation (CUF) for the micropolar plates and shells. I. Higher order theory

Starting from the variational principle of virtual power for the 3-D equations of the micropolar theory of elasticity in orthogonal curvilinear coordinates and using generalized series in terms of the plate thickness coordinates a new higher order model of orthotropic micropolar plates and shells have been developed here. Following Carrera Unified Formulation (CUF), the stress and strain tensors, as well as the vectors of displacements and rotation, have been expanded into series in terms of the shell thickness coordinates. Then, all the equations of the micropolar theory of elasticity (including generalized Hooke’s law) have been transformed to the corresponding equations for the coefficients of the series expansion on the plate thickness coordinates. A system of differential equations in terms of the displacements and rotation vectors and natural boundary conditions for the coefficients of the series expansion of the shell thickness coordinates have been obtained in the same way as in the classical theory of elasticity. All equations for the higher order theory of micropolar plates and shells have been developed and presented here. The obtained equations can be used for calculating the stress-strain and for modeling thin walled structures in macro, micro, and nanoscale when taking into account micropolar couple stress and rotation effects.

Carrera unified formulation (CUF) for the micropolar plates and shells. II. Complete linear expansion case

New higher-order models of orthotropic micropolar plates and shells have been developed using Carrera Unified Formulation (CUF). Here, a complete linear expansion case (CLEC) has been considered in detail. The stress and strain tensors, as well as the vectors of displacements and rotation, have been presented as linear expansion in terms of the shell thickness coordinates. Then, all the equations of the micropolar theory of elasticity (including generalized Hooke’s law) have been transformed to the corresponding equations for the coefficients of the expansion on the shell thickness coordinates. A system of differential equations in terms of the displacements and rotation vectors and natural boundary conditions for the coefficients of the expansion of the shell thickness coordinates has been obtained. All equations for the case of CLEC theory of micropolar plates and shells have been developed and presented here. The obtained equations can be used for calculating the stress-strain and for modeling thin walled structures in macro, micro, and nanoscale when taking into account micropolar couple stress and rotation effects.