A Ritz type solution with exponential trial functions for laminated composite beams based on the modified couple stress theory

doi:10.1016/j.compstruct.2018.02.025

Composite Structures

Volume 191, 1 May 2018, Pages 154-167

https://doi.org/10.1016/j.compstruct.2018.02.025 Get rights and content

Abstract

This paper proposes novel Ritz functions for the size-dependent analysis of micro laminated composite beams with arbitrary lay-ups. Displacement field is based on a higher-order deformation beam theory and size effect is captured by the modified couple stress theory. Lagrange’s equations are used to obtain the governing equations of motion. The present beam model, which can recover the classical one by neglecting the material length scale parameter, is used to predict the size-dependent responses of micro composite beams. The results indicate that the present study is efficient for bending, vibration and buckling problems of micro composite beams. Some new results are given to serve as benchmarks for future studies.

Introduction

Composite materials are commonly used in many engineering fields because they have many advantages such as high stiffness-to-weight and strength-to-weight ratios as well as the ability to change fiber orientations or density to meet design requirements. In addition to their extensive use in practice, composite materials also attract many academic researchers [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. In these studies, classical continuum theories are used to analyse macro composite beams. It should be noted that the classical continuum theories are unable to capture size effects. The experiment works made by Stolken and Evans [15] and Fleck et al. [16] indicated that the behaviour of microstructures is size-dependent.

A review of non-classical continuum mechanics models for size-dependent analysis of small-scale structures can be found in [17]. These models for size-dependent analysis can be divided into three groups: nonlocal elasticity theory, micro continuum theory and strain gradient family. Nonlocal elasticity theory was proposed by Eringen [18], [19], Eringen and Edelen [20], and its recent applications can be found in [21], [22], [23], [24]. In this theory, the stress at a reference point is considered as a function of strain field at all points of the body, and thus the size effect is captured by means of constitutive equations using a nonlocal parameter. Micro continuum theory in which each particle can rotate and deform independently regardless of the motion of the centroid of the particle was developed by Eringen [25], [26], [27]. The strain gradient family is composed of the strain gradient theory [16], [28], the modified strain gradient theory [29], the couple stress theories [30], [31], [32] and the modified couple stress theory (MCST) [33]. In the strain gradient family, both strains and gradient of strains are considered in the strain energy. The size effect is accounted for using material length scale parameters (MLSP). The MCST introduced an equilibrium condition of moments of couples to enforce the couple stress tensor to be symmetric. Consequently, MCST needs only one MLSP instead of two as the couple stress theories, or three as the modified strain gradient theory. This feature makes the MCST easier to use and more preferable to capture the size effect because the determination of MLSP is a challenging task.

Chen et al. [34], [35] developed Timoshenko and Reddy beam models to analyse the static behaviours of cross-ply simply supported microbeams. Chen and Si [36] suggested an anisotropic constitutive relation for the MCST and used global-local theory to analyse Reddy beams using Navier solution. By using a meshless method, Roque et al. [37] analysed the static bending response of micro laminated Timoshenko beams. A size-dependent zigzag model was also proposed by Yang et al. [38] for the bending analysis of cross-ply microbeams. Abadi and Daneshmehr [39] analysed the buckling of micro composite beams using Euler-Bernoulli and Timoshenko models. Mohammadabadi et al. [40] also predicted the thermal effect on size-dependent buckling behaviour of micro composite beams. The generalized differential quadrature method was used to solve with different boundary conditions (BCs). Chen and Li [41] predicted dynamic behaviours of micro laminated Timoshenko beams. Mohammad-Abadi and Daneshmehr [42] used the MCST to analyse free vibration of cross-ply microbeams by using Euler-Bernoulli, Timoshenko and Reddy beam models. Ghadiri et al. [43] analysed the thermal effect on dynamics of thin and thick microbeams with different BCs. Most of the above-mentioned studies mainly focused on cross-ply microbeams. Therefore, the study of micro general laminated composite beams (MGLCB) with arbitrary lay-ups is necessary.

Despite in fact that numerical approaches are used increasingly [6], [21], [22], [37], [44], [45], Ritz method is still efficient to analyse structural behaviours of beams [3], [4], [12], [13], [46], [47], [48], [49]. In Ritz method, the accuracy and efficiency of solution strictly depend on the choice of trial functions. An inappropriate choice of the trial functions may cause slow convergence rates and numerical instabilities [13]. The trial functions should satisfy the specified essential BCs [50]. If this requirement is not satisfied, the Lagrangian multipliers and penalty method can be used to handle arbitrary BCs [4], [14], [51]. However, this approach leads to an increase in the dimension of the stiffness and mass matrices and causing computational costs. Therefore, the objective of this study is to propose trial functions for Ritz type solutions that give fast convergence rate, numerical stability and satisfy the specified BCs.

In this study, new exponential trial functions are proposed for the size-dependent analysis of MGLCB based on the MCST using a refined shear deformation theory. Lagrange’s equations are used to obtain the governing equations of motion. The accuracy of the present model is demonstrated by verification studies. Numerical results are presented to investigate the effects of MLSP, span-to-thickness ratio and fiber angle on the deflections, stresses, natural frequencies and critical buckling loads of micro composite beams with arbitrary lay-ups.

Section snippets

Theoretical formulation

A MGLCB with rectangular cross-section shown in Fig. 1 is considered. $L$ , $b$ and $h$ denote are the length, width and thickness of the beam, respectively. It is composed of $n$ plies of orthotropic materials in different fiber angles with respect to the x-axis.

Convergence and accuracy studies

Convergence and verification studies are conducted to demonstrate the accuracy of the present study. Laminates, which are made of the same orthotropic materials, have equal thicknesses with material properties in Table 2. The beam is under a uniformly distributed load $q = q_{0}$ or a sinusoidal load $q = q_{0} \sin (\frac{π x}{L})$ . Unless otherwise stated, the following non-dimensional terms are used: $\bar{w} = \frac{100 w_{0} E_{2} {bh}^{3}}{q_{0} L^{4}}, {\bar{σ}}_{x} = \frac{{bh}^{2}}{q_{0} L^{2}} σ_{x} (\frac{L}{2}, z), {\bar{σ}}_{xz} = \frac{bh}{q_{0} L} σ_{xz} (0, z), {\bar{N}}_{cr} = N_{cr} \frac{L^{2}}{E_{2} {bh}^{3}}$ $\bar{ω} = \frac{ω L^{2}}{h} \sqrt{\frac{ρ}{E_{2}}} for Material (MAT) I and \bar{ω} = \frac{ω L^{2}}{h} \sqrt{\frac{ρ}{E_{1}}} for MAT III$

Conclusions

The size effect, which is included by the modified couple stress theory, on bending, vibration and buckling behaviours of micro composite beams with arbitrary lay-ups is investigated in this study. The governing equations of motion are derived from Lagrange’s equations. New trial functions are developed to solve problems. The frequencies, critical buckling loads, displacements and stresses of micro composite beams with various BCs are obtained. The results indicate that the present study is

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 107.02-2015.07.

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A Ritz type solution with exponential trial functions for laminated composite beams based on the modified couple stress theory

Abstract

Introduction

Section snippets

Theoretical formulation

Convergence and accuracy studies

Conclusions

Acknowledgements

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