A high-order theory of a thermoelastic beams and its application to the MEMS/NEMS analysis and simulations

Curved and Layered Structures

2017

New models for plane curved rods based on linear couple stress theory of elasticity have been developed. 2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and rotation along with body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko's and EulerBernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scales when taking into account couple stress and rotation effects.

“…By substituting kinematic relations (74) to the generalized Hooke's law (75) we obtain constitutive relations in the form…”

Section: Euler-bernoulli Couple Stress Theory Of the Curved Rodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

Curved and Layered Structures

2017

“…Thermoelastic contact problems of plates and shells when mechanical and thermal conditions are changed during deformation have been considered in [42,43]. Then the proposed approach and methodology were further developed and extended to thermoelasticity of the laminated composite materials with the possibility of delamination along with mechanical and thermal contact in the temperature field in [44], the pencil-thin nuclear fuel rods modeling in [45], modeling of MEMS and NEMS in [49,50] and the functionally graded shells in [46,51]. Analysis and comparison with classical theory of elastic and thermoelastic plates and shells has been done in [47,48].…”

Section: Introductionmentioning

confidence: 99%

“…In our previous publications [42][43][44][45][46][47][48][49][50][51] the approach based on the use of Legendre's polynomials series expansion has been applied to the development of high order models of shells, plates and rods. Thermoelastic contact problems of plates and shells when mechanical and thermal conditions are changed during deformation have been considered in [42,43].…”

Section: Introductionmentioning

confidence: 99%

Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

Curved and Layered Structures

2017

New models for micropolar plane curved rods have been developed. 2-D theory is developed from general 2-D equations of linear micropolar elasticity using a special curvilinear system of coordinates related to the middle line of the rod and special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First stress and strain tensors, vectors of displacements and rotation and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then in the same way as in the theory of elasticity, system of differential equations in term of displacements and boundary conditions for Fourier coefficients have been obtained. The Timoshenko's and EulerBernoulli theories are based on the classical hypothesis and 2-D equations of linear micropolar elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scale when taking into account micropolar couple stress and rotation effects.

Higher order theory of micropolar plates and shells

2018

Z Angew Math Mech

New higher order models of micropolar plates and shells have been developed here. The 3‐D dynamic equations of the micropolar elasticity have been presented in an orthogonal system of coordinates using a generalized variational principle. For the creation of 2‐D models of plates and shells the curvilinear system of coordinates related to the middle surface of the shell have been used along with a special hypothesis based on assumptions that take into account the fact that the rod is thin. Higher order theory is based on a generalized variational principle and the expansion of the 3‐D equations of the micropolar theory of elasticity into Fourier series in terms of Legendre polynomials. The stress and strain tensors, as well as vectors of displacements and rotation have been expanded into Fourier series in terms of Legendre polynomials with respect to thickness. Thereby, all equations of the micropolar theory of elasticity (including generalized Hooke's law) have been transformed to the corresponding equations for the Legendre polynomials coefficients. Then, in the same way as in the classical theory of elasticity, a system of differential equations in terms of displacements and rotation with initial and boundary conditions for the Legendre polynomials coefficients have been obtained. All equations for higher order theory of micropolar plates in Cartesian and polar coordinates as well as for cylindrical and spherical shells in coordinates related to the shells geometry have been developed and presented here in detail. The obtained equations can be used for calculating the stress‐strain and for modelling thin walled structures in macro, micro and nano scale when taking into account micropolar couple stress and rotation effects.