Mindlin's Strain Gradient Research Articles

On mechanics of piezocomposite shell structures

This study presents an original and novel investigation into the mechanics of piezo-flexo-magneto-elastic nanocomposite doubly-curved shells (PFMDCSs) and the ability to detect the lower and higher levels of electro-magnetic fields. In this context, by utilizing the first-order shear deformation shell model, stresses and strains are acquired. By imposing Hamilton's principle and the von Kármán approach, the governing equations have been obtained. The intelligent shell model consists of size-dependent influences, viz., strain gradients. This will take place via Mindlin's strain gradient elasticity theory and the subsequent re-establishing of the mathematical framework by incorporating this concept. The strain gradient results in a flexoelectric/flexomagnetic effect. The converse effect of the magnetic field on the basis of a close circuit has been assumed. The developed bending equations have been transferred into the algebraic ones by substituting an analytical technique based on homogeneous immovable simple support for the four edges. The problem has been solved according to the Newton-Raphson iteration scheme, and transverse deflections have been computed. For researching the rightness and precision of the shell models together with the solution process, a comparison is prepared by the finite element method (FEM) results for simplified shells, and a good correlation has been observed. At last, by examining several factors governing the problem, the conditions under which the magnetic effects can be noticeable and dominant in doubly-curved shells have been sought. This study could serve as a benchmark reference for piezoceramic-DCSs, as the presented governing equations are original. The most interesting outcome of this research is that the electro-magnetic response of intelligent structures can be entirely geometry-dependent.

Hölder continuity of exponential pullback attractors for Form II Mindlin's strain gradient viscoelastic plate

Hölder continuity of exponential pullback attractors for Form II Mindlin's strain gradient viscoelastic plate

Pullback exponential attractors in nonlocal Mindlin's strain gradient porous elasticity

ABSTRACT In this paper, we study the long-time dynamics of a pullback exponential attractors for non-autonomous one dimensional system in nonlocal Mindlin's strain gradient porous elastic theory recently developed by Aouadi [Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin's strain gradient porous elasticity. Z Angew Math Phys. 2022;73:111]. In this theory, the second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution. Then we establish a Lipschitz stability result. We also prove the existence of pullback exponential attractors which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. Finally, we prove the upper-semicontinuity of minimal pullback attractors with respect to the perturbations parameter in natural energy space.

On well‐posedness of the first boundary‐value problem within linear isotropic Toupin–Mindlin strain gradient elasticity and constraints for elastic moduli

AbstractWithin the linear Toupin–Mindlin strain gradient elasticity we discuss the well‐posedness of the first boundary‐value problem, that is, a boundary‐value problem with Dirichlet‐type boundary conditions on the whole boundary. For an isotropic material we formulate the necessary and sufficient conditions which guarantee existence and uniqueness of a weak solution. These conditions include strong ellipticity written in terms of higher‐order elastic moduli and two inequalities for the Lamé moduli. The conditions are less restrictive than those followed from the positive definiteness of the deformation energy.

Asymptotic behavior in nonlocal Mindlin's strain gradient thermoelasticity with voids and microtemperatures

A nonlocal theory for thermoelastic materials with voids and microtemperatures based on Mindlin's strain gradient theory was derived in this paper. The obtained system of equations is a coupling of a two second-order in time equations with higher gradients terms due to the strain gradient length scale parameter l and the elastic nonlocal parameter ϖ coupled with two parabolic equations. This poses some new mathematical difficulties due to the lack of regularity. Using the semigroup theory, we show the well-posedness of the one dimensional problem. By an approach based on the Gearhart-Herbst-Prüss-Huang theorem, we prove that the associated semigroup is exponentially stable; but not analytic.

Determination of length scale parameters of strain gradient continuum theory for crystalline solids using a computational quantum mechanical model based on density functional theory

Although the classical continuum theory is advantageous in finding solutions to a wide range of engineering problems, it cannot describe some phenomena such as dispersion of acoustic waves, effects of surfaces and interfaces on the mechanical behavior of small-scale structures, and microstructure contribution in special materials. Owing to this fact, several enhanced continuum theories have evolved in the literature. However, the difficulty in determination of the length scale parameters that appear in the governing equations of such theories hampers their widespread use in practice. To date, except for a very limited number of materials, there is no known experimental procedure for the identification of these parameters. In this research, the internal length scales for an augmented continuum theory, i.e., Mindlin's strain gradient theory, have been theoretically determined for some crystalline materials with cubic structure that are of engineering interest, using ab initio DFT. According to the values obtained for these parameters, it can be perceived that the strain gradient theory is a valuable tool for capturing the size effects at even the smallest scales comparable to the dimensions of a unit cell of a crystal lattice.

Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory

Nonlinear plate bending within Mindlin's strain gradient elasticity theory (SGT) is investigated by employing somewhat non-standard finite element methods. The main goal is to compare the bending results provided by the geometrically nonlinear three-dimensional (3D) theory and the geometrically nonlinear Reissner–Mindlin plate theory, i.e., the first-order shear deformation plate theory (FSDT), within the SGT. For the 3D theory, the nonlinear Green–Lagrange strain relations are adopted, while the von Kármán nonlinear strains are employed for the FSDT. The matrix-vector forms of the energy functionals are derived for both models. In order to perform the corresponding finite element discretizations, a quasi-C1-continuous 4-node tetrahedral solid element and a quasi-C1-continuous 6-node triangular plate element are employed for the 3D model and plate model, respectively. The first-order derivatives of the primal problem quantities are employed as additional nodal values to respond to the continuity requirements of class C1. A variety of computational results highlighting the differences between the 3D and FSDT models are given for two different plate geometries: a rectangular plate with a circular hole and an elliptical plate.

Uniformly moving screw dislocation in strain gradient elasticity

In the present paper, Mindlin's strain gradient theory of elasticity (form II) is utilized to determine the deformation and strain fields due to a straight screw dislocation that moves uniformly at a subsonic velocity in an infinite isotropic body. The theory employed herein, in addition to the effect of strain gradient, is capable of accounting for that of acceleration gradient through the incorporation of the micro-inertia term into the analysis of the problem. Integral representations are obtained for the displacement and strain fields, and it is shown that the utilization of the strain gradient theory for the continuum description of a moving screw dislocation leads to discarding the discontinuity of the induced displacement field, consequently resulting in the removal of the classical singularities of the associated strain and stress fields.

Spherical inclusion with time-harmonic eigenfields in strain gradient elasticity considering the effect of micro inertia

In the present paper, the complete form of Toupin–Mindlin strain gradient theory of elasticity is utilized to determine the elastic field of a spherical inclusion undergoing a time-harmonic eigenfield in an infinite isotropic body. The theory adopted herein involves three characteristic lengths, two ones of which are present therein due to the incorporation of strain gradients in the strain energy density function of the material under consideration, and the presence of the third one arises from taking into account the role of velocity gradients in its kinetic energy density function, reflecting the effect of micro inertia. After the derivation of an expression for the associated Green’s function of the problem, closed-form solutions are obtained for the interior and exterior elastic fields of a spherical inclusion subjected to a time-harmonic eigenfield whose spatial distribution is expressible as a function of the radial distance of points from the center of the inclusion.

Isogeometric finite element analysis of mode I cracks within strain gradient elasticity

A variational formulation within an H2 Sobolev space setting is formulated for fourth-order plane strain/stress boundary value problems following a widely-used one parameter variant of Mindlin's strain gradient elasticity theory. A corresponding planar mode I crack problem is solved by isogeometric Cp-1-continuous discretizations for NURBS basis functions of order p \ge 2. Stress eld singularities of the classical elasticity are shown to be removed by the strain gradient formulation.

Concrete wave dispersion interpretation through Mindlin's strain gradient elastic theory

Classical elastic wave features like pulse velocity and attenuation have been used for decades for concrete condition characterization. Relatively recently the effect of frequency has been studied showing no doubt over the dispersive behavior of the material. Despite the experimental evidence, there is no unified theory to model the material and explain this phase velocity change at frequencies below 200 kHz. Herein, the Mindlin's strain gradient elastic theory including the additional micro-stiffness and micro-inertia parameters is considered as an alternative of multiple scattering theory. Experimental results are produced from material with dictated microstructure using a specific diameter of glass beads in cement paste. Results show that Mindlin's theory provides conclusions on the microstructure of the material and is suitable for describing the observed dispersion in different length scales (from millimeters in the case of mortar to several centimeters in the case of concrete).

Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin's strain gradient theory

Presented herein is a comprehensive study on the size-dependent coupled longitudinal-transverse-rotational free vibration behavior of post-buckled functionally graded (FG) micro- and nano-beams based on the most general Mindlin's strain gradient theory. The current model enables us to incorporate size effects via introducing material length scale parameters and is developed in the framework of the first-order shear deformable beam model and the von Karman geometric nonlinearity. The FG micro- and nano-beams, whose volume fraction is expressed by using a power law function, are assumed to be made of a mixture of metals and ceramics. By using Hamilton's principle, the nonlinear governing equations and associated boundary conditions are derived for FG micro- and nano-beams in the postbuckling domain. Afterwards, the governing equations and boundary conditions are discretized using the generalized differential quadrature (GDQ) method in conjunction with a direct approach without linearization, before solving numerically by Newton's method. The effects of length scale parameter, length-to-thickness ratio, material gradient index and boundary conditions on the postbuckling path and frequency of FG micro- and nano-beams are carefully investigated. Finally, numerical results obtained from both the modified strain gradient theory (MSGT) and modified couple stress theory (MCST) are compared.

Size-dependent thermo-mechanical vibration and instability of conveying fluid functionally graded nanoshells based on Mindlin's strain gradient theory

The free vibration and instability characteristics of nanoshells made of functionally graded materials (FGMs) with internal fluid flow in thermal environment are studied in this paper based upon the first-order shear deformation shell theory. In order to capture the size effects, Mindlin's strain gradient theory (SGT) is utilized. The mechanical and thermal properties of FG nanoshell are determined by the power-law relation of volume fractions. The Knudsen number is considered to analyze the slip boundary conditions between the flow and wall of nanoshell, and the average velocity correction parameter is used to obtain the modified flow velocity of nano-flow. The governing partial differential equations of motion and associated boundary conditions are derived by Hamilton's principle. An analytical solution method is also employed to solve the governing equations under the simply-supported end conditions. Then, some numerical examples are presented to investigate the effects of fluid velocity, longitudinal and circumferential mode numbers, length scale parameters, material properties, temperature difference and compressive axial loads on the natural frequencies, critical flow velocities and instability of system.

Nonlinear Bending Analysis of First-Order Shear Deformable Microscale Plates Using a Strain Gradient Quadrilateral Element

Based on Mindlin's strain gradient elasticity and first-order shear deformation plate theory, a size-dependent quadrilateral plate element is developed in this paper to study the nonlinear static bending of microplates. In comparison with the classical first-order shear deformable quadrilateral plate element, the proposed element needs 15 additional nodal degrees-of-freedom (DOF) including derivatives of lateral deflection and rotations with respect to coordinates, which means a total of 20DOFs per node. Also, the developed strain gradient-based finite-element formulation is general so that it can be reduced to that on the basis of modified couple stress theory (MCST) and modified strain gradient theory (MSGT). In the numerical results, the nonlinear bending response of microplates for different boundary conditions, length-scale factors, and geometrical parameters is studied. It is revealed that by the developed nonclassical finite-element approach, the nonlinear behavior of microplates with the consideration of strain gradient effects can be accurately studied.

Vibration and buckling of first-order shear deformable circular cylindrical micro-/nano-shells based on Mindlin's strain gradient elasticity theory

Abstract In this research, a size-dependent first-order shear deformable model is developed based on the Mindlin's strain gradient elasticity theory to analyze the free vibration and axial buckling of circular cylindrical micro-/nano-shells. The size-dependent governing equations and corresponding boundary conditions are established through Hamilton's principle. For some specific values of the gradient-based material parameters, the most general form of shell formulation can be reduced to those based on simple forms of the strain gradient elasticity theory such as the modified strain gradient theory (MSGT), modified couple stress theory (MCST) and strain gradient theory (SGT). To illustrate the characteristics of a micro-/nano-shell obtained by the size-dependent shell formulation, the axial buckling and free vibration problems of a simply-supported (SS) microshell are analyzed by employing a Navier-type solution. Selected numerical results are presented to get an insight into the effects of dimensionless material length scale parameters, length-to-radius ratio and circumferential mode number on the non-dimensionless natural frequencies and buckling loads. For comparison purpose, the non-dimensional natural frequencies and buckling loads predicted by MSGT, MCST, SGT and classical theory (CT) are also presented. It is shown that the effect of small scale is more prominent for lower values of dimensionless length scale parameter.

A novel size-dependent microbeam element based on Mindlin’s strain gradient theory

Based on Mindlin's strain gradient elasticity and Euler---Bernoulli beam theory, a non-classical beam element capable of considering micro-structure effects is developed. To accomplish this aim, the higher-order tensors of energy pairs in the energy functional are vectorized and written in the quadratic representation, from which the stiffness and mass matrices of the element are obtained. In comparison with the classical Euler---Bernoulli beam element, the new element needs one additional nodal degree of freedom (DOF) which results in a total of three DOFs per node. The formulation of the paper is general so that it can be reduced to that based on the modified couple stress theory, the modified strain gradient theory, and the classical elasticity theory. To show the reliability of the proposed element, the bending and free vibration problems of microbeams under different kinds of end conditions are addressed. It is revealed that the present finite element results are in excellent agreement with the ones achieved through analytical solutions.

Formulation of Toupin–Mindlin strain gradient theory in prolate and oblate spheroidal coordinates

Abstract The Toupin–Mindlin strain gradient theory is reformulated in orthogonal curvilinear coordinates, and is then applied to prolate and oblate spheroidal coordinates for the first time. The basic equations, boundary conditions, the gradient of the displacement, strain and strain gradient tensors of this theory are derived in terms of physical components in these two coordinate systems, which have a potential significance for the investigation of micro-inclusion and micro-void problems. As an example, using these formulae, we formulate and discuss the boundary-value problem of a spheroidal cavity embedded in a strain gradient elastic medium subjected to uniaxial tension. In addition, the previous results given by Zhao and Pedroso (Int. J. Solids. Struct. (2008) 45, 3507–3520) in cylindrical and spherical coordinates are amended.

Computational homogenization of cellular materials

In this work we propose to study the behavior of cellular materials using a second-order multi-scale computational homogenization approach. During the macroscopic loading, micro-buckling of thin components, such as cell walls or cell struts, can occur. Even if the behavior of the materials of which the micro-structure is made remains elliptic, the homogenized behavior can lose its ellipticity. In that case, a localization band is formed and propagates at the macro-scale. When the localization occurs, the assumption of local action in the standard approach, for which the stress state on a material point depends only on the strain state at that point, is no-longer suitable, which motivates the use of the second-order multi-scale computational homogenization scheme. At the macro-scale of this scheme, the discontinuous Galerkin method is chosen to solve the Mindlin strain gradient continuum. At the microscopic scale, the classical finite element resolutions of representative volume elements are considered. Since the meshes generated from cellular materials exhibit voids on the boundaries and are not conforming in general, the periodic boundary conditions are reformulated and are enforced by a polynomial interpolation method. With the presence of instability phenomena at both scales, the arc-length path following technique is adopted to solve both macroscopic and microscopic problems.

On a strain gradient elastic Timoshenko beam model

AbstractConsidering the influence of the microstructure, the Timoshenko beam model is revisited, invoking Mindlin's strain gradient strain energy density function. The equations of motion are derived and the bending equilibrium equations are discussed. The adopted strain energy density function includes new terms. Those terms introduce the strong effect of the beam cross‐section area. The influence of those terms is more evident in thin beams where the cross‐section area is far bigger than its moment of inertia. Applications have been worked out exhibiting the difference of the present theory not only from the classical Timoshenko beam, but also from the existing variations including couple stresses. The solution of the static problem, for a simply supported beam loaded by a force at the middle of the beam, is defined and the first (least) eigen‐frequency is found. The present model is proved to be stiffer.

BEM Solutions for 2D and 3D Dynamic Problems in Mindlin's Strain Gradient Theory of Elasticity

BEM Solutions for 2D and 3D Dynamic Problems in Mindlin's Strain Gradient Theory of Elasticity