On thermomechanics of multilayered beams

doi:10.1016/j.ijengsci.2020.103364

International Journal of Engineering Science

Volume 155, October 2020, 103364

https://doi.org/10.1016/j.ijengsci.2020.103364 Get rights and content

Highlights

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A non-local stress-driven nanobeam model for composite multilayered beams is proposed.
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Thermoelastic material behavior is assumed.
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Governing equations are solved by Laplace transforms.
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Coupling effects enable a lot of possibilities in design of nanomechanical engineering structures.

Abstract

In this paper, the mechanical behavior of multilayered small-scale beams in nonisothermal environment is investigated. Scale phenomena are modeled by means of the mathematically well-posed and experimentally consistent stress-driven integral formulation of elasticity. The present research extends the treatment in Barretta, Čanađija, Luciano, and de Sciarra (2018b) confined to elastically homogeneous nano-scopic structures. It is shown that the non-locality leads to a complex coupling between axial and transverse elastic displacements. Such a size-dependent phenomenon makes the solution of the relevant nonlocal thermoelastostatic problem, governed by a system of two ordinary differential equations with ten standard boundary conditions and non-classical constitutive boundary conditions, significantly more involved with respect to treatments in literature. Thus, a novel solution methodology, based on Laplace transforms, is proposed and illustrated by examining simple structural schemes of current applicative interest in Nanomechanics and Nanotechnology.

Section snippets

Introduction, motivation and outline

Analysis, modelling and assessment of elastic responses of small-scale structures has been a subject of great interest in the community of Engineering Science (Farajpour, Ghayesh, & Farokhi, 2018) due to the need to significantly design and optimize nanocomposites (Eyvazian, Shahsavari, Karami, 2020, Karami, Shahsavari, Li, Karami, Janghorban, 2019, Malikan, Krasheninnikov, Eremeyev, 2020, Omari, Almagableh, Sevostianov, Yaseen, et al., 2020, Qi, Zhou, Li, 2016) and smaller and smaller

Geometry, materials and coordinate system

This research considers multilayered beams, that is beams consisting of n layers. The model will accommodate beams in which each layer is made of a different material and has a rectangular cross-section of different dimensions. The beam has length L and it is assumed that no spatial changes take place regarding cross-section and material along the beam axis. Height and width of each layer will be denoted by h_i and b_i, $i \in {1, 2, \dots, n},$ respectively. Only cross-sections symmetric with respect to one

Equilibrium and constitutive boundary conditions

With the foundations of the problem provided in the previous section, the link between displacements and stress resultants can be set-up. The formulation relies on linking displacements u₀(x) and w(x) to stress resultants N(x) and M(x), i.e. the axial force and the bending moment, respectively. With (10), (11) being the starting point, the normal strain in replaced with displacements by enforcing Bernoulli-Euler kinematics, Eq. (3): $\frac{σ}{E_{i}} = - L_{λ, i}^{2} (u_{0}^{(3)} - w^{(4)} (z - ζ_{0}) - α_{i} Δ θ) + u_{0}^{(1)} - w^{(2)} (z - ζ_{0}) - α_{i} Δ θ$ and

Variational formulation

The complete differential formulation augmented with boundary conditions is now obtained by enforcing the strict varational approach. The governing potential is defined by: $Π (u_{0}, w) = U_{i} - U_{e},$ where the internal potential U_i for a beam $B$ is: $U_{i} = \int_{B} \int_{0}^{ε} σ d \bar{ε} d V .$ The external potential U_e is: $U_{e} = \int_{L} q_{x} u_{0} d x + \int_{L} q_{z} w d x + N_{0} u_{0} (0) + N_{L} u_{0} (L) + T_{0} w (0) + T_{L} w (L) - M_{0} w^{(1)} (0) - M_{L} w^{(1)} (L),$ where $N_{0}, N_{L}$ and $T_{0}, T_{L}$ are external axial and transverse forces at x ∈ {0, L}, respectively. $M_{0}, M_{L}$ are external moments at same positions. q_z(x) and q_x(x

Solution of the coupled problem

Solution of the coupled non-homogenous ordinary differential problem (42), (43), (44) is not straightforward. Nowadays, automated differential equations solvers like Mathematica, Matlab or alike are typically used for such purpose. In the present case, a direct approach by means of such software does not appear to have much success. Thus, two alternative solution strategies are considered below.

Nonisothermal cantilever nanobeam in a homogeneous thermal field

The introductory example considers a simple cantilever two-layer beam, free from external mechanical loading but situated in a homogeneous temperature field $Δ θ = 0.1$ . For present purposes, measurment units are deemed not to be important, so geometrical and material properties are assumed to be dimensionless. In that way, the unit length of the non-local beams is assumed $L = 1$ . Also: $b_{1} = b_{2} = 1,$ $h_{1} = h_{2} = 1,$ $E_{2} = 2 E_{1} = 2,$ $α_{2} = 2 α_{1} = 0.2$ .

The beam’s cross-section is not symmetrical with respect to material

Closing remarks

Thermomechanical analyses of multilayered micro- and nano-beams have been performed in this paper by the stress-driven nonlocal integral approach of elasticity. Main contributions and findings are summarized as follows.

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Nonisothermal extension of existing isothermal nonlocal beam stress-driven integral formulations available in literature has been provided. The conceived methodology is suitable to model and assess the nonlocal behaviour of multilayered beams assembled of arbitrary number of

Declaration of Competing Interest

The authors declare that they do not have any financial or nonfinancial conflict of interests.

Acknowledgments

This work has been fully supported by Croatian Science Foundation under the project IP-2019-04-4703.

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On thermomechanics of multilayered beams

Highlights

Abstract

Section snippets

Introduction, motivation and outline

Geometry, materials and coordinate system

Equilibrium and constitutive boundary conditions

Variational formulation

Solution of the coupled problem

Nonisothermal cantilever nanobeam in a homogeneous thermal field

Closing remarks

Declaration of Competing Interest

Acknowledgments

Composites Part B: Engineering

Composites Part B: Engineering

Composite Structures

Composites Part B: Engineering

European Journal of Mechanics - A/Solids

Composites Part B: Engineering

International Journal of Engineering Science

Composite Structures

International Journal of Engineering Science

International Journal of Engineering Science

Composite Structures

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Engineering Science

Composite Structures

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Mechanical Sciences

Composites Part B: Engineering

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Engineering Science

Composite Structures

International Journal of Engineering Science

Composites Part B: Engineering

International Journal of Mechanical Sciences

Composites Part B: Engineering

Composites Part B: Engineering

International Journal of Engineering Science