Length Scale Parameter Research Articles

Verification of strain energy splits of phase field fracture model using Westergaard’s problem under mixed-mode loading

One challenge in verifying strain energy splits in phase field fracture models is the lack of reference problems with exact solutions in mixed-mode loading that could serve as benchmark problems. In this work, different strain energy splits used in the variational phase field model have been applied to the classical Westergaard’s problem. In contrast to other benchmark problems used in the literature, the proposed problem not only includes the singular term as a boundary condition but also implicitly considers all terms. To the authors’ knowledge, this is the first time in the literature that this reference problem has been used to rigorously verify the accuracy of different energy splits of the phase field model. Several critical factors such as the pre-crack definition, mesh size, mixed mode ratio and the length scale parameter have been analysed. A good correlation between the numerical estimations and the analytical predictions was found for some pre-crack definition approaches under mode I. However, we demonstrated that the compatible regularisation length scale and mesh size are significantly lower under mode II loading (i.e., when the crack kinks) than under mode I.

A phase-field framework for modeling multiple cohesive fracture behaviors in laminated composite materials

This work proposes a phase-field (PF) framework to characterize the multiple fracture behaviors and quasi-brittle failure mechanisms in fiber reinforced laminated composites. The multi-phase field model considering the distinct failure mechanisms of composites is first derived based on thermodynamic principles. A hybrid PF formulation is proposed through the reasonable definition of the energy functional for each field, and a PF regularized cohesive zone model (CZM) combined with Hashin failure criteria is achieved for the typical failure modes in composites. The proposed PF-CZM is endowed with a linear softening behavior and is proven to be independent of the internal length scale parameter. The multiple fracture modeling of composite laminates is achieved by coupling the PF model for intralaminar failure with the cohesive interface element for interlaminar one. The model’s validation is demonstrated by several representative numerical examples, and results show that both the intralaminar fiber/matrix fractures and the induced interlaminar delamination, as well as their complex interactions, are well captured by the proposed model.

High-thermal free vibration analysis of functionally graded microplates using a new finite element formulation based on TSDT and MSCT

Recent advancements in additive manufacturing (AM) have revolutionized the design and production of complex engineering microstructures. Despite these advancements, their mathematical modeling and computational analysis remain significant challenges. This research aims to develop an effective computational method for analyzing the free vibration of functionally graded (FG) microplates under high temperatures while resting on a Pasternak foundation (PF). This formulation leverages a new third-order shear deformation theory (new TSDT) for improved accuracy without requiring shear correction factors. Additionally, the modified couple stress theory (MCST) is incorporated to account for size-dependent effects in microplates. The PF is characterized by two parameters including spring stiffness (kw) and shear layer stiffness (ks). To validate the proposed method, the results obtained are compared with those of the existing literature. Furthermore, numerical examples explore the influence of various factors on the high-temperature free vibration of FG microplates. These factors include the length scale parameter (l), geometric dimensions, material properties, and the presence of the elastic foundation. The findings significantly enhance our comprehension of the free vibration of FG microplates in high thermal environments. In addition, the findings significantly enhance our comprehension of the free vibration of FG microplates in high thermal environments. In addition, the results of this research will have great potential in military and defense applications such as components of submarines, fighter aircraft, and missiles.

Size-dependent vibration analysis of porous 3D-FG microshells in complex thermal environments using a neural network enhanced finite element model

This work aims to investigate the vibration behavior of porous three-directional functionally graded (3D-FG) microshells in thermal environments, where both mechanical and thermal material properties vary along three spatial directions following a power-law distribution, based on the modified couple stress theory (MCST). An isoparametric thermal element is employed for uncoupled heat conduction analysis, followed by dynamic analysis using a penalty solid element with three translation degrees of freedom (DOFs) and three rotation DOFs per node. However, due to the strong nonlinearity of the temperature field in 3D-FG microshells, obtaining exact temperature values at integration points in dynamic analysis requires a pretty dense mesh of the solid element. To overcome this issue, a back propagation (BP) neural network is utilized to predict the temperature field for various gradient indexes, enhancing the efficiency and precision of temperature distribution calculation. Numerical results demonstrate the effectiveness of the neural network enhanced finite element model. Furthermore, the effects of the material length scale parameter (MLSP), temperature difference, and gradient indexes on the natural frequencies are investigated. It is shown that higher structural stiffness weakens the impact of temperature difference on the frequency.

Rényi Holographic Dark Energy

AbstractIn this work, the holographic dark energy model is constructed by using the non‐extensive nature of the Schwarzschild black hole via the Rényi entropy. Due to the non‐extensivity, the black hole can be stable under the process of fixing the non‐extensive parameter. A change undergoing such a process would then motivate us to define the energy density of the Rényi holographic dark energy (RHDE). As a result, the RHDE with choosing the characteristic length scale as the Hubble radius provides the late‐time expansion without the issue of causality. Remarkably, the proposed dark energy model contains the non‐extensive length scale parameter additional to the standard model. The cosmic evolution can be characterized by comparing the size of the Universe to this length scale. Moreover, the preferable value of the non‐extensive length scale is determined by fitting the model to recent observations. The results of this work would shed light on the interplay between the thermodynamic description of the black hole with non‐extensivity and the classical gravity description of the evolution of the Universe.

Computer Simulation for Determination of Critical Buckling Temperatures of Sandwich Microplates with Cellular Core and CNT-RC Piezoelectric Patches via MSGT

This study introduces a computational simulation which leads to obtaining the critical buckling temperatures of a sandwich microplate which is modeled based on the modified strain gradient theory. This theory includes three material length scale parameters. The core of the microstructure is made from cellular materials which is treated with piezoelectric carbon nanotubes-reinforced composite patches. An advanced trigonometric hyperbolic shear deformation theory, eliminating the need for shear correction factors, is employed to analyze the microstructure. Thickness-dependent variations in structural characteristics are observed based on predefined functions. The equilibrium equations are derived using the virtual displacement principle. Fourier series functions provide an analytical way of solving these equations. The findings are verified by comparison with earlier research that was published and used fewer complex combinations. The study looks at how several important factors affect the structure’s reaction to thermal buckling.

Dynamic Response of Advanced Lightweight Porous Plates Integrated with Nanocomposite Face Sheets Resting on Elastic Substrate

This study centers on the analysis of a scale-dependent microplate configuration characterized by a porous core enveloped by nanocomposite patches embedded with graphene nanoplatelets. The microplate’s behavior is explored within a humid environment to comprehend the effects of moisture variations on its dynamic performance. Additionally, the microplate rests upon a Kerr foundation, a three-parameter elastic substrate. The material properties of all three layers are contingent upon thickness. To enhance precision, an innovative quasi-three-dimensional shear and normal trigonometric theory is employed, elucidating the kinematic interrelations of the microstructure. Notably, this novel theory accommodates the presence of transverse normal strain. For a comprehensive analysis of size influences, the modified couple stress theory is harnessed. This theory integrates a material length-scale parameter to anticipate outcomes at the micro-scale. By invoking Hamilton’s principle, differential motion equations are deduced and subsequently solved analytically. The investigation centers on probing the consequences of varied parameters on the natural frequencies. The findings underscore that the incorporation of GPLs amplifies the microplate’s stiffness, thus elevating its natural frequencies. In contrast, an escalation in the porosity index leads to a reduction in natural frequencies.

An adaptive phase field approach to 3D internal crack growth in rocks

Modeling the 3D internal crack under compression entails complex fracture mechanics (mode I-II-III fracture), resulting in substantial computational costs and challenges in characterizing fracture morphology characterization for Phase Field Method (PFM) simulations. In the present paper, we developed a 3D adaptive crack propagation model integrated into the PFM to study the internal cracks in rocks. The proposed locally refined solution was implemented within the isogeometric analysis (IGA) framework, utilizing PHT-splines and employing the phase field variable as an error indicator to accurately capture non-smooth crack surfaces. The simulation begins with a coarser mesh, and the spatial discretization is adaptively refined in real-time as the computation progresses and the phase variable threshold is reached. By comparing our results with previous numerical tests, we validated the effectiveness of our model in simulating crack growth. Notably, a smaller length scale parameter led to a higher peak force in notched square plates under tension, and our adaptive PFM successfully reduced errors caused by length scale parameters. We further validated our model by comparing it with uniaxial compression experiments on 3D-printed resin samples with internal penny-shaped cracks, achieving good agreement with the experimental data. Using the 3D adaptive PFM, we predicted changes in crack morphology and fracturing mechanisms by varying the inclination angles of internal penny-shaped cracks. Our numerical results showed that as the inclination angle increased, wing cracks initiated closer to the end tips of the original crack, and their length decreased due to the wings wrapping effect. The corresponding fracturing mechanisms transition from mode I-III fracture (0°) to mode I-II-III fracture (30°, 45°, 60°), and eventually to mode I fracture (90°) as inclination angles increase. Comparing 2D and 3D models, we found that the growth of wing cracks in 3D was significantly influenced by the intermediate principal stress. We conclude that the extended 3D adaptive PFM-based model accurately predicted complex mechanical behavior and crack growth patterns in rock.

Nonlocal strain gradient-based geometrically nonlinear vibration analysis of double curved shallow nanoshell containing functionally graded layers

It is a fact that while examining the mechanical behavior of nano/microscopic materials, size effects become apparent. This study aims at presenting a new approach to geometrically nonlinear vibration analysis of double curved shallow nanoshell on Kerr foundation containing functionally graded layers under the impacts of harmonic loading in the framework of nonlocal strain gradient theory and the classical shell theory. The three-parameter Kerr model is founded on the independence of the upper and lower elastic layers related to the shear layer. The motion equations of the double curved shallow nanoshell are deduced by introducing the Von Kármán strain-displacement relations. Governing equations are solved by displacement function method and Galerkin method to achieve the deflection differential equation, the fourth-order Runge-Kutta methods are implemented to solve the deflection differential equation of the dynamic system. In this study, the obtained outcomes are compared with other publications indicating that the theoretical results agree with the previously published ones. Afterward, the article investigates the effects of the material length scale parameter, the nonlocal parameter, the geometric parameters and the material properties on the geometrically nonlinear vibration analysis of the double curved shallow nanoshell containing functionally graded layers. This research paper can provide extensive references and valuable guidelines for the impact and application of double curved shallow nanoshell structures containing functionally graded layers.

Size-dependent nonlinear bending of microbeams based on a third-order shear deformation theory

In this paper, the size-dependent nonlinear bending of microbeams subjected to mechanical loading is studied using a finite element formulation. Based on the von Kármán nonlinear relationship and the third-order shear deformation theory, a size-dependent nonlinear beam element is derived by using the modified couple stress theory (MCST) to capture the microstructural size effect. The element with explicit expressions for the element vector of internal forces and tangent stiffness matrix is derived by employing the transverse shear rotation as a variable. Nonlinear bending of microbeams under different mechanical loading is predicted with the aid of Newton–Raphson iterative method. Numerical investigation shows that the derived element is efficient, and it is capable of giving accurate results by several elements. The obtained results reveal the importance of the micro-size effect on the nonlinear behavior of the microbeams, and the deflections are overestimated when the microstructural effect is ignored. The effects of the material length scale parameter, boundary conditions and loading type on the bending response of the microbeams are studied and highlighted.

Vibration of embedded restrained composite tube shafts with nonlocal and strain gradient effects

Torsional vibration response of a circular nanoshaft, which is restrained by the means of elastic springs at both ends, is a matter of great concern in the field of nano-/micromechanics. Hence, the complexities arising from the deformable boundary conditions present a formidable obstacle to the attainment of closed-form solutions. In this study, a general method is presented to calculate the torsional vibration frequencies of functionally graded porous tube nanoshafts under both deformable and rigid boundary conditions. Classical continuum theory, upgraded with nonlocal strain gradient elasticity theory, is employed to reformulate the partial differential equation of the nanoshaft. First, torsional vibration equation based on the nonlocal strain gradient theory is derived for functionally graded porous nanoshaft embedded in an elastic media via Hamilton’s principle. The ordinary differential equation is found by discretizing the partial differential equation with the separation of variables method. Then, Fourier sine series is used as the rotation function. The necessary Stokes' transformation is applied to establish the general eigenvalue problem including the different parameters. For the first time in the literature, a solution that can analyze the torsional vibration frequencies of functionally graded porous tube shafts embedded in an elastic media under general (elastic and rigid) boundary conditions on the basis of nonlocal strain gradient theory is presented in this study. The results obtained show that while the increase in the material length scale parameter, elastic media and spring stiffnesses increase the frequencies of nanoshafts, the increase in the nonlocal parameter and functionally grading index values decreases the frequencies of nanoshafts. The detailed effects of these parameters are discussed in the article.

Micropolar effects on the effective shear viscosity of nanofluids

The modified size-dependent Einstein's and Brinkman's solutions are established for the effective shear viscosity of rigid particle suspensions taking into account the micropolar effects in the base fluid. Solutions are obtained based on the homogenization approach and allow us to take into account the influence of the particle size. Two non-classical parameters arise in the considered micropolar solutions: the length scale parameter and the coupling (micropolarity) number of the base fluid. The solutions developed are validated using tests performed with polydimethylsiloxane based TiO2 nanofluids as well as other published data on the size-dependent shear viscosity of different nanofluids. Good agreement between the predictions and the experimental data is established across a wide range of volume fractions and size of nanoparticles. The possibility for unique identification (at given temperature) of the micropolar parameters of the base fluids is shown. Temperature-dependent values of non-classical rotational and spin viscosities of polydimethylsiloxane, ethylene glycol, and water are evaluated.

On nonlinear buckling of microshells

Investigation of the geometrical nonlinear action of doubly curved shell panels (DCSPs) in micro scale is the main target of this paper. The proposed microshell panels (MSPs) are assumed to be made of an auxetic honeycomb core (AHOC), leading to negative magnitudes of Poisson's ratio, covered by two nanocomposite enriched coating layers (NCECLs). To conduct the size-dependent nonlinear analysis and achieve the corresponding nonlinear equilibrium path (EQP) of the proposed MSPs, the nonlocal strain gradient theory (NLSGT) is utilized. The governing equations containing the equilibrium and compatibility nonlinear partial differential equations in terms of the deformation components are analytically solved based on the Galerkin technique for different types of simply-supported panels. The achieved results of the present solution exhibit the fact that nonlocal and material length scale parameters significantly affect the EQP of the proposed MSPs especially at their post-buckling stage during their snap-through instability. By solving several numerical examples, the effects of various parameters on the size-dependent EQP of the proposed MSPs are investigated. The results indicate that the influences of size-dependency are significantly affected by the curvature and also boundary conditions of the microshells.

Magneto-electro-mechanical responses of smart sensor/actuator for detection the information in football’s ball

A small-scale formulation is developed based on modified couple stress theory (MCST) for shear deformation analysis of piezomagnetic nano-panels. This theory employs the couple stress and symmetric curvature tensors to include the length scale parameter in the micro scale to capture size-dependency. The micro-panel is fabricated from an elastic core integrated with two piezoelectric/piezomagnetic face sheets subjected to initial electromagnetic loads. Using the principle of virtual work, the static governing equations are derived in terms of radial, axial, and circumferential displacements, axial and circumferential rotations, and electromagnetic potentials. An analytical method is employed to solve the governing equations. The structure is used for detection of the information in the football’s ball with changes of the position of the equipment in the sport. The results are estimated using the proposed model. Effect of important material and geometric parameters such as micro length scale parameter, initial electromagnetic potentials, and some geometric parameters such as length and span angle are also investigated on the bending features such as deformation and rotations components as well as maximum electromagnetic potentials.

A derivative-free phase-field theory for capturing local buckling induced damage in architected plates

Architected plates are commonly used to enhance the load-bearing capacity and stability of structures such as ship hulls and aircraft components. Local buckling-induced damage in these plates are critical events that occur when the plate is subjected to external transverse loading. Performing the analysis to accurately capture this damage and estimation of local buckling load requires explicit 3D- modeling of the architected plate, which is computationally expensive. We address this issue by proposing a derivative-free phase-field theory to capture local buckling-induced damage in architected plates. Here the architected plate is modeled using the shear deformable plate theory and the internal web-core structure is preserved in terms of length scale parameters. The numerical value of the length scale parameter depends on the shape of the webcore structure so that any complicated shape of the webcore is handled based on this parameter. This approach makes the analysis simple to predict the damage behavior with a lesser number of degrees of freedom. The extra energy required to attenuate the zero energy mode-induced oscillations in the solution is estimated from the in-plane buckling analysis of the architected plate. To showcase the efficiency, simulations are conducted based on the proposed approach with different loading cases. The local buckling load and the induced damage behavior of the webcore are compared with the 3D-finite element solution obtained from ABAQUS. The comparison shows that the damage variable is an equivalent estimator of out-of-plane stretch of the webcore in predicting the buckling behavior of the webcore with reduced computational cost and time.

Inextensional vibrations of thin spherical shells using strain gradient elasticity theory

We study inextensional vibrations of the spherical shell using strain gradient elasticity theory under Kirchhoff-Love hypotheses. For admissibility of the inextensional deformations the shell is punctured by making a tiny hole. The shell is assumed to be made of linearly elastic, homogeneous, and isotropic material. The closed form expression for the frequencies of inextensional modes of vibration of spherical caps of various polar angles and that of nearly complete spherical shell are derived using Rayleigh's method. Towards this, the displacement and velocity fields are obtained by setting the membrane strains to zero. Further, to facilitate the existence of inextensional vibrations, boundaries are assumed to be traction free. The frequencies are computed for the positive and negative signs in the strain gradient term of the constitutive law. The computed frequencies, employing negative sign are found to be higher than those computed using classical elasticity theory. The opposite effect is seen when the frequencies are computed with the positive sign. Further, the frequencies are found to saturate when the positive sign is used in the constitutive law. Parametric studies viz. variation in frequencies with the circumferential wavenumbers for different values of length scale parameter, variation in frequencies with circumferential wavenumbers for different values of polar angles and a given length scale parameter, variation of frequency with increasing polar angle for a given circumferential wavenumber, variation of saturation wavenumber with increasing polar angle, and increasing length scale parameter are carried out. The frequencies of nearly a spherical shell with increasing circumferential wavenumber are also computed.

Couple Stress-Based Flexoelectric Theory in Orthogonal Curvilinear Coordinates and its Application to an Infinitely Long Dielectric Cylinder

In this paper, the formulations of the couple stress based flexoelectric theory in orthogonal curvilinear coordinates are presented. By following the two rules of transforming tensor equations from Cartesian coordinates to curvilinear ones: replacing the partial differentiation symbol with the covariant differentiation symbol and ensuring that the identical indices are positioned diagonally, the governing equations, constitutive equations and boundary conditions in orthogonal curvilinear coordinates are obtained. Then the couple stress based flexoelectric theory is specified for cylindrical coordinates. The flexoelectric responses of an infinitely long dielectric cylinder under a uniform electric field are analyzed. Numerical results show that the radial displacement increases rapidly as the mechanical length scale parameter [Formula: see text] increases. When [Formula: see text] is larger than 5 [Formula: see text]m, the circumferential displacement decreases rapidly. [Formula: see text] has a large effect on the induced displacements but a small effect on the induced electric potential, while the electrical length scale parameter [Formula: see text] has a small effect on both the induced displacements and the induced electric potential.

An enhanced strain gradient model for temperature-dependent sound transmission loss in double-walled functionally graded microplates

With the growing use of functionally graded (FG) microplates in structural acoustic metamaterials for aerospace and automotive applications, accurate modeling of sound transmission loss is critical for effective vibration and noise control engineering. In this study, a theoretical model is formulated based on the first-order shear deformation theory (FSDT) to estimate sound transmission loss (STL) through air-filled rectangular double-walled FG microplates with simply supported boundary conditions. The microplates are subjected to a nonlinear thermal environment, and the modified strain gradient theory (MSGT) is employed, incorporating three material length scale parameters to account for the size effect. The material properties are temperature-dependent and vary across the thickness following a power-law distribution. Size-dependent coupled vibroacoustic equations are derived utilizing the sound velocity potential, normal velocity continuity conditions, and Hamilton's principle, and are subsequently solved using the Galerkin method. Comparative analysis is performed by contrasting the results obtained from the MSGT model with those from the modified couple stress theory (MCST) and classical continuum theory (CCT) models, allowing for the assessment of the accuracy and precision of the proposed solution. Additionally, the influence of various parameters, such as gradient index, length scale parameters, temperature variation, incident angles, and acoustic cavity depth, on the STL is investigated. Key findings demonstrate that in the stiffness-controlled region, length scale parameters significantly enhance STL, while power-law index and temperature variation reduce it.

A deviatoric couple stress Mindlin plate model and its degeneration

Based on the classical couple stress theory, the deviatoric couple stress theory (DCST), where both couple stress and curvature tensors are traceless, is developed. The deviatoric couple stress Mindlin plate is established based on a new hypothesis of the vanishing curvature of the transverse normal, and the governing equations and associated boundary conditions of the simplified size-dependent Mindlin plate are respectively derived by the principle of virtual work. In the absence of couple stresses, a simplified Mindlin plate model between the classical Kirchhoff plate model and classical Mindlin plate model is derived. To illustrate the simplified DCST-based Mindlin plate model, the static bending and free vibration of rectangular plates are studied. The dependence of the length scale parameter and curvature ratio on the deflection, rotation and natural frequency is analyzed. The numerical results show that the consideration of couple stresses and curvatures hardens the stiffness of the microstructure, resulting in a decrease in its deflection and rotation compared with that of the macroscopic structure under the same conditions, while the natural frequency increases, which is consistent with the size effect observed in experiments.

An investigation on the torsional vibration of a FG strain gradient nanotube

AbstractIn this work, an attention is paid to the prediction of torsional vibration frequencies of functionally graded porous nanotubes based on the Lam strain gradient elasticity theory. The nanotubes are formed of functionally graded porous nanomaterials that vary in the radial direction. This study also aims to obtain the analytical solution of the strain gradient model presented by Lam for torsional vibration response, in a simple manner, for different rigid or restrained boundary conditions. The torsion angle of a functionally graded nanotube is defined by an infinite Fourier series. Then, the Stokes’ transformation is applied to force the boundary conditions to the desired state. An eigenvalue problem is established with the help of the two systems of equations obtained. This eigenvalue problem, which includes deformable springs at both ends of the nanotube, appears as a general analytical solution that can find torsional vibration frequencies. It is shown that the vibrational responses can be significantly influenced by the through‐radius gradings of material, material length scale parameters and deformable springs of the functionally graded nanotubes and consequently can be predicted by giving proper values to torsional spring parameters.