Mindlin's Strain Gradient Theory Research Articles

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.

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.

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.

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.

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

Strain-Gradient Effects on Force Transfer Between Embedded Microflakes

In this paper the plane strain problem concerned with shearing stresses along the fiber-matrix interface due to a fracture of the adjacent fiber is solved anew based on Mindlin's strain-gradient theory of linear elasticity. Classical method of selecting proper stress functions and determining superposition constants to satisfy all homogeneous boundary conditions is used to obtain the auxiliary solution for the problem. Applying a Fourier integral transformation to the auxiliary solution both homogeneous and non-homogeneous boundary conditions are simultaneously satisfied. The exact solu tion is thus obtained in integral form. Numerical results of the solution for some selected elastic constants have been worked out and compared with results previously obtained based on classical theory and couple-stress theory.