Material Length Scale Research Articles

Vibration and buckling optimization of functionally graded porous microplates using BCMO-ANN algorithm

A BCMO-ANN algorithm for vibration and buckling optimization of functionally graded porous (FGP) microplates is proposed in this paper. The theory is based on a unified framework of higher-order shear deformation theory and modified couple stress theory. A combination of artificial neural network (ANN) and balancing composite motion optimization (BCMO) is developed to solve the optimization problems and predict stochastic vibration and buckling behaviors of functionally graded porous microplates with uncertainties of material properties. The characteristic equations are derived from Hamilton’s principle and approximation of field variables under Ritz-type exponential series. Numerical results are obtained to investigate the effects of the material distribution, material length scale, porosity density and boundary conditions on natural frequencies and critical buckling loads of functionally graded porous microplates. The novel results derived from this paper can be used as future references.

Size-dependent free vibration of functionally graded beams with a tip nanoparticle and a nonlinear spring constraint via strain gradient theory

In this article, the free vibration of a functionally graded Euler–Bernoulli beam with a tip nanoparticle and a nonlinear/linear spring constraint is presented for the first time using strain gradient theory. It is assumed that there is a nonlinear/linear spring and a nanoparticle at one end of the beam while the other is clamped. The influence of different parameters, such as the material length scale parameter, the mass of the nanoparticle, and the nonlinear/linear spring constant on the natural frequencies, is discussed in detail. The presented benchmark tables can be used as an excellent database to verify approximate or analytical solutions in future works. The presented results can be useful for those designing micro/nano electromechanical devices.

Decoupling toughness and strength through architected plasticity

We investigate the elastic–plastic fracture of architected materials through experiments and theory, with a focus on understanding the combined effects of material length scale and geometry, using a pillar array as a model structure. We show that load sharing across the pillars, and hence toughness, can be controlled by changing the spatial distribution and height of the pillars. A simple relation is presented to relate the extent of the plastic fracture process zone to the pillar array structure and the resulting toughness. This relation allows for quantitative prediction of failure loads of specimens with plasticity localized to a structured region and reveals that strength and toughness can be decoupled through architecture. Our findings establish a foundation for design of architected materials with enhanced fracture toughness.

Length scale parameters to estimate fatigue lifetime of 3D-printed titanium alloy Ti6Al4V containing notches in the as-manufactured condition

The accuracy of the Theory of Critical Distances, Gradient Elasticity and the Averaged Strain Energy Density criterion in estimating fatigue lifetime of notched additively manufactured Ti6Al4V is assessed against numerous experimental results generated under load ratios equal to −1 and 0.1. The 3D-printed Ti-alloy under investigation was tested by keeping the notches in the as-manufactured condition. The common feature of the considered design approaches is that they all make use of a material length scale. The validation exercise based on the generated experimental results demonstrates that the length scale concept can be extended successfully also to the fatigue assessment of notched 3D-printed metallic components.

A better understanding of the mechanics of borehole breakout utilizing a finite strain gradient-enhanced micropolar continuum model

Borehole breakout denotes the failure in rock mass subjected to drilling, caused by stress concentrations exceeding the material strength. Depending on the material properties, the preexisting in situ stress state, and the borehole dimensions, different types of borehole breakout, such as spiral-shaped breakout or v-shaped breakout, are distinguished in the literature. In the present work, we address the influence of the material properties on the predicted borehole breakout mode in a comprehensive finite element study. To this end, we employ a gradient-enhanced micropolar damage-plasticity model based on the Mohr–Coulomb strength criterion, formulated in the finite strain regime, which is calibrated based on experimental results from plane strain compression tests on Red Wildmoor sandstone. In the numerical study, the influence of the in situ stress state, the material friction angle, plastic dilation, post peak residual strength, and the inherent material length scale parameters are investigated. Thereby, we demonstrate that depending on the material parameters, substantially different failure modes, characterized by strongly localized shear bands or diffuse failure zones, are predicted. It is shown that in particular the brittleness of the material in the post peak regime has a major influence on the predicted breakout type. Moreover, a statistical validation of the results is obtained by considering different random field distributions of the initial material strength.

Cation Short-Range Ordering of MgAl2O4 and NiAl2O4 Spinel Oxides at High Temperatures via In Situ Neutron Total Scattering.

Complex oxides that adopt the isometric spinel structure (AB2O4) are important for numerous technological applications and are relevant for certain geological processes, which involve exposure to extreme environments such as high pressures and temperatures. Recent studies have shown that the changes to the spinel structure caused by these environments are complex and depend on the material length scale under consideration. In this study, we have expanded this approach to the behavior of spinels under high temperatures. In situ neutron total scattering experiments, coupled with pair distribution function analysis, performed on two spinel compositions with various levels of pre-existing disorder (MgAl2O4 and NiAl2O4) revealed that both compositions trend to a state of maximum disorder where the A and B cations are randomly distributed among the two available sites. Temperature-induced cation inversion, conventionally understood as an exchange of cations on the A and B sites, is locally expressed as an atomic rearrangement to a tetragonal symmetry, a correlation that is retained up to the maximum temperature studied (1000 °C). A complex thermal expansion behavior is revealed wherein the oxide materials expand heterogeneously at the level of coordination polyhedra with an apparent dependence on bond strength.

Vibration analysis of size dependent micro FML cylindrical shell reinforced by CNTs based on modified couple stress theory

In this manuscript, the sequel of agglomeration on the vibration of fiber metal laminated (FML) cylindrical shell in the micro phase using developed couple stress theory (MCST). Hamilton’s principle has been carried out for deriving the non-classical equations of motion of size-dependent thin micro cylindrical shell on the basis of Love’s first approximation theory. Mori-Tanaka and extended rule of mixture are utilized to estimate the mechanical attributes of carbon nanotubes (CNTs) and equivalent fiber, respectively. These four phases CNTs/fiber/polymer/metal laminated (CNTFPML) micro cylindrical shell is analyzed applying beam modal function model for several boundary limitations. Then, an investigation is performed to study the impacts of differing input parameters namely material length scale parameter, agglomeration, the distributions of agglomerated CNTs, the mass fraction of equivalent fiber and the volume fraction of CNTs on the frequency response of micro agglomerated CNTFPML cylindrical shell. The main output illustrated that the growth of frequencies is directly dependent to the increase of material length scale parameter for this agglomerated CNTFPML cylindrical shell so that through increasing the values of agglomeration parameters η and μ and material length scale parameter l altogether, the frequencies of this cylindrical shell grow.

A Hermite-Family C1 Finite Layer Method for the Three-Dimensional Free Vibration Analysis of Exponentially Graded Piezoelectric Microplates Based on the Consistent Couple Stress Theory

Based on the consistent couple stress theory (CCST), we develop a Hermite-family [Formula: see text] finite layer method (FLM) for the three-dimensional (3D) free vibration analysis of a simply-supported, exponentially graded (EG) piezoelectric microplate under open- and closed-circuit surface conditions. In the formulation of the FLM, the microplate is artificially divided into a number of finite microlayers, and Fourier functions and Hermite polynomials are used to interpolate the in-plane and out-of-plane variations of a number of primary variables, respectively, including elastic displacement components and the electric potential variable for each individual layer. The Hermite-family [Formula: see text] FLM for analyzing EG piezoelectric microplates is reduced to the Hermite-family [Formula: see text] FLM for analyzing EG piezoelectric macroscale plates and functionally graded (FG) elastic microplates by assigning a value of zero to the material length scale parameter and by ignoring the piezoelectric and flexoelectric effects in the formulation, respectively. The accuracy and convergence rate of the FLM are assessed by comparing their solutions with the benchmark solutions of both the EG piezoelectric macroplates and the power-law-type FG elastic microplates that are available in the relevant literature. We examine and discuss some key effects on the free vibration characteristics of an EG piezoelectric microplate, including the impact of the material length scale parameter, the material-property gradient index, the length-to-thickness ratio, the piezoelectric effect, and the flexoelectric effect.

Nonlinear vibration control of a piezoelectric graphene sheet reinforced microplate with thermal uncertainty: Feedback linearization and sliding mode approaches

In this study, the nonlinear vibration control of a simply supported sandwich microplate with piezoelectric facesheets reinforced with graphene sheets is investigated regarding uncertainty about the temperature of the operating environment. The goal is that despite the variation of the temperature of the operating environment, the steady state deflection of the mid-point of the microplate subjected to an external transverse load remains identical to the corresponding value at room temperature. The temperature dependency of the material properties of the piezoelectric facesheets, the matrix phase, and the graphene sheets are regarded. When the material temperature dependency is accompanied by the rising of the operating ambient temperature, we need a controller with the ability to control a system with uncertainties. The motion equations of the piezoelectric microplate are derived in reference to the Kirchhoff’s plate hypothesis alongside the modified couple stress theory. The free vibrations outcomes are achieved by implementing the Navier’s method. For nonlinear vibration control of piezoelectric graphene sheets microplate, sliding mode and feedback linearization controllers are exploited. Some numerical examples are released in order to present the impressions of material length scale constant, graphene sheets distribution pattern, and temperature on the dynamic response, and the controlled response of piezoelectric graphene sheets microplate. It is found that the sliding mode controller operates more appropriately relative to the feedback linearization controller with a smaller energy cost. Moreover, the sliding surfaces for various graphene sheet models coincide with each other after an opening divergence.

Size dependent large displacements of microbeams and microframes

The size dependent large displacement behavior of planar microbeams and microframes is studied in this paper using a corotational beam element. To account for the size effect, the modified couple stress theory (MCST) is employed in conjunction with Euler-Bernoulli beam theory in deriving the internal force vector and the tangent stiffness matrix of the beam element. The Newton-Raphson based iterative procedure is used in combination with the arc-length method to solve the nonlinear equilibrium equation and to trace the equilibrium paths. Various microbeams and microframes are analyzed to show the influence of the size effect on the large deflection behavior of the microstructure. The obtained result reveals that the size effect plays an important role on the large deflection response, and the displacements of the structure are over estimated by ignoring the size effect. A parametric study is carried out to highlight the influence of the material length scale parameter on the large displacement behavior of the microbeams and microframes.

A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory

In this work, a finite element formulation for a size dependent frame system is presented. Size dependency is discussed via the modified couple stress theory. The nodal displacement and rotation analyses of a frame system with total of three elements, including two columns and one beam element connecting these two columns, are considered. The classical stiffness and size dependent stiffness matrices of frame system are derived. Then, solution procedure for this problem is explained. Lastly, a numerical application is realized and effect of material length scale parameter on nodal displacements and rotations is discussed. To present the numerical application, it is assumed that the elements of the nanoframe are composed of silicon carbide nanotubes.

On the dynamics and stability of size-dependent symmetric FGM plates with electro-elastic coupling using meshless local Petrov-Galerkin method

In present study, the equations of motion of sandwich symmetric functionally graded size-dependent plates integrated with piezoelectric layers are derived using Hamilton’s variational principle (with modified Lagrangian) based on a higher-order nonlocal strain gradient theory in conjunction with Reddy’s third-order shear deformation plate kinematics. For the first time, a set of sixth-order partial differential equations are derived and solved by meshless local Petrov-Galerkin method to investigate the bifurcation buckling and free vibration of plates under electromechanical in-plane forces with considered nanostructure and various boundary conditions. The shifted and scaled moving least-squares approach is utilized for approximation of unknown variables, and the Gauss weight function is used as test function for deriving local weak form of governing equations. The Gauss-Legendre quadrature is employed for numerical evaluation of weak forms. A power-law distribution and a half cosine variation are applied to model the variation of materials properties and electric potentials, respectively. The obtained results are verified with well-known solutions in the literature. The effect of nonlocal parameter, material length scale parameter, power-law index, predefined electric field, axial compressive and tensile forces, volume ratio of FG core and piezoelectric layers, and stiffness of Pasternak foundation on the critical buckling loads and natural frequencies are presented and discussed comprehensively. The results obtained can be used in the analysis, design and control of composite nanostructures that are widely used in MEMS and NEMS devices.

An energy-based criterion for mixed-mode I/II/III fracture considering effective critical distances

Traditionally, fracture criteria have been established based on a crack-tip material length scale which famously is called the critical distance. This parameter is considered to be independent of the mode mixity and measured under pure mode I loading conditions as a material parameter. However, recent studies show this parameter may considerably change between the fracture modes depending on the mode mixity. The mode-dependent critical distance, called the effective critical distance, is found to be a reasonable approach when implemented into a strain-based framework. In this paper, the concept of effective critical distance (ECD) is combined with the maximum tangential strain energy density criterion (MTSED), and a novel mixed-mode I/II/III brittle fracture criterion, called ECD-MTSED, is suggested. The new criterion is used to predict the experimental data reported in the literature for the mixed-mode I/II/III fracture toughness and fracture angles along with four other criteria – namely, maximum tangential stress (MTS), maximum principal stress (MPS), strain energy density (SED), and maximum principal strain using an effective critical distance (ECD-SN). The results not only showed the superiority of the suggested criterion but also accentuated the necessity of employing effective critical distance in the three-dimensional brittle fracture theories.

Size-dependent dynamic stability of nanocomposite enriched micro-shell panels in thermal environment using the modified couple stress theory

This work aims to evaluate dynamic stability of micro-shell panels enriched by nanocomposites based on an analytical approach. In order to capture the effects of size on the mechanical behaviour of the micro-shell panels, the modified couple stress theory is employed which uses the material length scale parameter. The fundamental equations incorporating kinematic, constitutive, equilibrium and compatibility equations, are developed based on the first-order shear deformation theory (FSDT) which supports analysis of thin and moderately thick micro-shells. The extended fundamental differential equations are solved by the Galerkin method which results a closed-form solution for evaluating the dynamic stability of the proposed micro-shell panels. Then, some comparative studies are conducted to validate accuracy of the proposed method. Finally, by performing an extended parametric study, effects of the material length scale parameter on the dynamic stability of the proposed micro-shell panels are investigated.

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

This paper presents an investigation of the dynamic behavior of bi-directionally functionally graded (BDFG) micro/nanobeams excited by a moving harmonic load. The formulation is established in the context of the surface elasticity theory and the modified couple stress theory to incorporate the effects of surface energy and microstructure, respectively. Based on the generalized elasticity theory and the parabolic shear deformation beam theory, the nonclassical governing equations of the problem are obtained using Lagrange’s equation accounting for the physical neutral plane concept. The material properties of the beam smoothly change along both the axial and thickness directions according to power-law distribution, accounting for the gradation of the material length scale parameter and the surface parameters, i.e., residual surface stress, two surface elastic constants, and surface mass density. Using trigonometric Ritz method (TRM), the trial functions denoting transverse, axial deflections, and rotation of the cross sections of the beam are expressed in sinusoidal form. Then, with the aid of Lagrange’s equation, the system of equations of motion are derived. Finally, Newmark method is employed to find the dynamic responses of BDFG subjected to a moving harmonic load. To validate the present formulation and solution method, some comparisons of the obtained fundamental frequency and dynamic response with those available in the literature are performed. A parametric study is performed to extensively explore the impact of the key parameters such as the gradient indices in both directions, moving speed, and excitation frequency of the acting load on the dynamic response of BDFG nanobeams. The obtained results can serve as a guideline for assessing the multi-functional and optimal design of micro/nanobeams acted upon by a moving load.

Transverse free vibration of nanobeams with intermediate support using nonlocal strain gradient theory

In the present study, the transverse free vibration of intermediately supported nanobeams is investigated in the framework of size-dependent nonlocal strain gradient theory. Unlike the classical elasticity theory, nonlocal strain gradient theory considers the micro/nano scale effects in mechanical analysis of size-dependent structures. The potential and kinetic energies have been derived for the nanobeam model and the Ritz method has been used to determine the natural frequencies of intermediately supported nanobeams. A cantilever nanobeam model with intermediate support is considered in the analysis. By changing the position of the intermediate support, dimensionless vibration frequencies of the nanobeam model have been obtained. Obtained results from the present nonlocal strain gradient theory showed that hardening or softening material responses have been observed according to the classical elasticity theory depending on the magnitude of the material length scale parameter and nonlocal scale parameter. This mechanical behavior can provide an advantage to designers while modeling the micro/nanostructures. Present results can be used for the design of nano-sensors, nanotube-based resonators, and oscillators.

Nonlinear static and dynamic isogeometric analysis of functionally graded microplates with graphene-based nanofillers reinforcement

The main contribution of this study is to propose an efficient and straightforward numerical model for exploring size-dependent geometrically nonlinear static and dynamic characteristics of functionally graded microplates reinforced by graphene nanofillers. To accomplish this purpose, modified couple stress theory including only one material length scale parameter is utilized to capture the size-dependent effect while the displacement field of small-scale structures is calculated by using four-variable refined plate theory and von-Kármán assumptions within the framework of isogeometric analysis. In this study, four dispersion patterns of graphene-based nanofillers along the thickness direction of microplate models are investigated under various static or dynamic loads. Solutions of nonlinear governing equation of motion can be obtained based on the Newton–Raphson iterative procedure and Newmark's time integration scheme. The Rayleigh damping is included, for the first time, to explore the influence of damping on the oscillations of functionally graded microplates reinforced by graphene nanofillers under three dynamic loads including step, triangular and half-sine pulses. Herein, several numerical examples are conducted to investigate the influences of a wide range of significant parameters on the geometrically nonlinear behaviors of functionally graded microplates with graphene nanofillers reinforcement. The significant outcomes of the present study can be referred to as benchmark results as well as provide valuable guides for designing small-scale structures with excellent features in the future.

Parametrically-upscaled continuum damage mechanics (PUCDM) model for plain weave woven composites: Part II model validation and parametric studies

In this second of the two-part paper, the two-level parametrically-upscaled continuum damage mechanics models (PUCDM-1 and PUCDM-2) developed in the first part Xiao et al. (2022) are applied for multiscale analysis of plain weave composite structures. The material damage response in the intra-yarn unidirectional microstructures and the woven yarn–matrix mesostructures are respectively represented by level-1 and level-2 PUCDM models. The models bridge material length scales through explicit functions of the lower-scale morphological descriptors (representative aggregated micro/mesostructural parameters or RAMPs) and material properties (Xmat) with high computational efficiency. The paper focuses on model validation with experiments and parametric studies. The model is validated with observations from tensile tests on an hourglass-shaped specimen with good agreement at multiple scales. The validated PUCDM model is then employed for the analysis of multiscale deformation and damage mechanisms in single-edge notched bending (SENB) tests on woven composite beams. Parametric studies are performed to investigate the effect of lower-scale morphology and material properties on the structural damage response of the woven composite beams.

Stochastic vibration and buckling analysis of functionally graded microplates with a unified higher-order shear deformation theory

Based on equations of the elasticity, this paper proposes a unified higher-order shear deformation theory for stochastic vibration and buckling analysis of the functionally graded (FG) microplates. The governing equations of motions are derived from Hamilton’s principle. The solutions are approximated by bi-directional series in which hybrid shape functions are proposed, then the stiffness and mass matrix are explicitly derived. In order to investigate the stochastic responses of the FG microplates, the polynomial chaos expansion (PCE) is used. The multiple uncertain material properties are randomly changed via the lognormal distributions. Numerical results are presented for different configurations of the FG microplates such as the power-law index, material length scale parameter, length-to-thickness ratio and boundary conditions on their critical buckling loads and natural frequencies. The results from PCE are evaluated by comparing with those from Monte Carlo simulation to show the efficiency and accuracy of the present approach. Some new results for stochastic analysis of the FG microplates are presented and can be used for future references.

Detection of thermal size-dependent dynamic instability of rotating CNTRC microbeams with damping under parametric resonance excitation

Small-scale dynamic instability analysis of a rotating nanocomposite damped microbeam reinforced with carbon nanotube exposed to a temperature growth is addressed in this research. The modified couple stress theory, along with the Timoshenko beam hypothesis is explored to develop the motion equations. The practical thermo-mechanical characteristics are determined by resorting to the modified rule for mixtures. The Chebyshev–Ritz approach is exerted on the nonlinear motion equations to achieve the discretized form of the motion equations. The influence of structural damping is integrated into the governing equations. Considering a varying rotating speed, a possible instability resulting from principal parametric resonance stimulation is examined using the Bolotin's route and the Floquet hypothesis. The influences of the material length scale constant, distribution type of carbon nanotubes, temperature, and the damping constant on the dynamic instability boundaries are investigated. The outcomes enlighten that a rotating nanocomposite microbeam with O distribution pattern for carbon nanotubes is more sensitive to the changing temperature.