Document Type: Research Paper
Authors
University of Kashan
Abstract
Keywords

Mechanics of Advanced Composite Structures 3 (2016) 5362 

Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://macs.journals.semnan.ac.ir 
Sizedependent Bending of Geometrically Nonlinear of MicroLaminated Composite Beam based on Modified Couple Stress Theory
A.R. Ghasemi^{*}, M. Mohandes
Department of Solid Mechanics, University of Kashan, Kashan, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 20160317 Revision Received 20160421 Accepted 20160427 
In this study, the effect of finite strain on bending of the geometrically nonlinear of micro laminated composite EulerBernoulli beam based on Modified Couple Stress Theory (MCST) is studied in thermal environment. The GreenLagrange strain tensor according to finite strain assumption and the principle of minimum potential energy is applied to obtain governing equation of motion and boundary conditions. The equation of motion with boundary conditions is solved using a generalized differential quadrature method and then, the deflection of the beam in classical elasticity and MCST states is drawn and compared with each other. Considering the bending of the beam, which has been made of carbon/epoxy and glass/epoxy materials specified, it can be seen there is a significant difference between the finite strain and vonKarman assumptions particularly for . Also, the results show that the thermal loadings have a remarkable effect on the glass/epoxy beam based on the finite strain particularly for simply supported boundary condition. 



Keywords: Sizedependent Finite strain Modifided couple stress theory Laminated composite



© 2016 Published by Semnan University Press. All rights reserved. 
Using composite materials in the most advanced engineering fields such as aerospace, mechanical and civil has increased in the past decades. The reason of this increase is the outstanding engineering properties of composite materials, such as high value of strengthtoweight and stiffnesstoweight ratios. One of the most important subjects for the composite structures is the bending analysis of composite beams. Many researchers have investigated the bending of composite beams using different theories such as EulerBernoulli, Reddy and Timoshenko [16].
One of the most important subjects in industrial sections is the bending of various structures and it is completely obvious that the bending of nonlinear structures is more accurate and capable than the linear one. The nonlinear structures are divided into two main groups: (1) structures with large deformation and small strains, and (2) structures which are affected by finite strain [710]. In theories with finite strain assumption, not only deformations are large, but also the strains are not limited to infinitesimal strain. The finite strain assumption is the most accurate state to study the bending of nonlinear systems.
It is necessary to say that with decreasing the size scale, the stiffness and strength of materials can increase, which is called size effects. The classical continuum mechanics theory is not capable of accounting the size effects in micro and nano scale structures. So, in order to overcome this problem, some higher order continuum theories that contain additional material constants have been presented to determine the size effect. . Some of the theories are micropolar theory [11], nonlocal elasticity theory [12], surface elasticity theory [13], strain gradient theory [14], couple stress theory [15] and Modified Couple Stress Theory (MCST) [16]. The couple stress and strain gradient theories are applied for the microscale structures [17 23]. There is a difference between these two theories that rotation and strain are applied as a variable to describe curvature in the couple stress and strain gradient theories, respectively. Therefore, it is known that the couple stress theory is a special format of the strain gradient theory.
The static bending and free vibration of a Timoshenko micro beam subjected to simply supported condition based on the MCST were observed by Ma et al. [24]. In 2010, Asghari et al. [25] investigated the static bending and free vibration of a nonlinear Timoshenko micro beam model subjected to simply supported condition on the basis of MCST. To solve the static bending and free vibration equations of the beam the numerical and analytical methods were used, respectively. In 2011, Chen et al. [26] reported a new model for the crossply laminated composite beam with firstorder shear deformation based on the MCST. In 2013, Roque et al. [27] used the MCST and meshless method to study the bending of laminated composite Timoshenko micro beam. They demonstrate that the obtained numerical results have a good agreement with the analytical ones. In 2013, Simsek and Reddy [28] proposed a new higher order theory for the static bending and free vibration of Functionally Graded (FG) micro beam on the basis of the MCST. Moreover, they indicate the Poisson effect decreases and increases the static deflection and vibration frequencies, respectively.
In this research, the effect of the finite strain on the bending of the micro laminated composite EulerBernoulli beam is investigated under thermal loading. The small scale structures such as micro structures, need a high accuracy. So, in this study, the finite strain assumption is employed in order to study micro structures with the highest degree of accuracy. The equation of motion is derived using the principle of minimum potential energy and the Generalized Differential Quadrature Method (GDQM) is used to solve the governing equation of motion with boundary conditions. The results indicate that there is a difference between the finite strain and vonKarman assumptions so that the slope of the deflection curves based on the finite strain is less than the vonKarman hypothesis. Also, the results demonstrate that material properties have a remarkable effect on the behavior of the finite strain micro beam.
According to the MCST, the strain energy can be expressed as what follows [29,30]:
(1) 
Where
(2) 

(3) 

(4) 

(5) 
In which and are components of the symmetric stress and strain tensors, respectively. Also, and are components of the deviatoric part of the symmetric couple stress tensor and symmetric curvature tensor, respectively. In addition, and denote the two Lame constants, is the material length scale parameter and is the Kronecker delta. Moreover, components of the displacement and rotation vectors are represented by and , respectively, which the rotation vector is related to the displacement vector as follows [31, 32]:
(6) 
Using the Cartesian coordinat system (x, y, z) as shown in Fig. 1, where the xaxix is coincident with the centroidal axis of the undeformed beam, , and are length, width and thickness of the beam, respectively.
The axial and transverse displacement fields in an EulerBernoulli beam can be described by the following equations [3335]:
(7) 

Where , and are the displacements of a point along the , and coordinates, respectively.
(a)

(b) 
(c) 
Figure 1. The schematic of (a) the simply supported beam, (b) the clamped beam and (c) the cross section of the beam 
The nonlinear structures are modeled by either vonKarman assumption or finite strain hypothesis. Although the vonKarman assumption is applied for the structures with large deformation and small strains, it is not used for large strains. So, if high accuracy is needed, the vonKarman assumption cannot present such accurate responses. To reach the most accurate responses, the bending of nonlinear structures is studied by the finite strain assumption. In theories with finite strain assumption, not only the deformations are large, but also the strains are not limited to infinitesimal strain. Therefore, based on the finite strain assumption, none of the transformation terms are eliminated in the GreenLagrange strain tensor. Therefore, by substituting Eq. (7) into the Eq. (3), the nonlinear straindisplacement relations of the beam can be written as the following equations for the axial strain:
(8) 
The components of rotation vector can be obtained from Eqs. (6) and (7) as follow:
(9) 
Substituting Eq. (9) into Eq. (5) gives:
(10) 
Also, using Eq. (10) in Eq. (4) gives:
(11) 
The equation of motion with boundary conditions can be derived using the principle of minimum potential energy which can be considered as the following equation [36, 37]:
(12) 
Where is the total potential energy, and and are the strain energy and virtual work done by external loads, respectively.
According to Eq. (1), the variation form of the strain energy can be written as what follows:
(13) 
The constitutive relations for the orthotropic composite beam under thermal loading are as what follows [35]:
(14) 
Where denotes the stressreduced stiffness of the orthotropic beam. Also, and are the coefficients of thermal expansion and temperature changes, respectively.
The couple moment and the stress resultants including inplane force , bending moment and high order resultant of normal stress are expressed as follows [38, 39]:
(15) 
The stiffness components including , and , which are extensional stiffness, bending stiffness and additional stiffness coefficient matrices, respectively, can be represented by the following equation [40]:
(16) 
Substituting Eq. (8) into Eq. (14) and using Eqs.(15) and (16) leads to the following equations:
(17) 

(18) 

(19) 

(20) 
Where
(21) 

(22) 
Where the width is .
The variation form of the work done by the externally transverse loading is defined by what follows [34]:
(23) 
Where is the external load. Substituting Eqs. (13) and (23) into Eq. (12) and using Eq. (15), the equation of motion and boundary conditions for the micro laminated composite EulerBernoulli beam based on finite strain is obtained. After that, substituting Eqs. (17) and (19) into the equation of motion and boundary conditions gives:
(24) 
The boundary conditions at and are what follow:
(25a) 

or


(25b) 

or

In this section, the static bending of a nonlinear sizedependent laminated EulerBernoulli beam is studied based on the finite strain. Here, we have . According to these assumptions and introducing the following dimensionless parameters
(26) 
We have the following expressions:
(27) 
The boundary conditions at and are what follow:
or 
(28a) 
or 
(28b) 
The dimensionless form of the equation of motion (27) and boundary conditions (28a) and (28b) on the basis of the finite strain can be derived as:
(29) 
The boundary conditions at and are what follow:
(30a) 

or


(30b) 

or

Where the dimensionless parameters are the following expressions:
(31) 

In this study, the GDQM has been applied to solve the nonlinear equations of the finite strain vibration of composite beam. In the GDQM, the differential function and its derivatives at all grid point in the whole domain of spatial coordinate are demonstrated as a weighted linear sum of the all functional values. In other words, the governing differential equations using weighting coefficients change to the firstorder algebraic equations [41]. In the present study, the used GDQM was derived by Du et al. [42]. In this method, the firstorder derivative of function can be approximated as a linear sum of the weighting coefficients and function values for all grid points in the domain.
(32) 
Where is the number of grid point in the domain, is the function in the point of and is the weighting coefficient of the firstorder derivate. The weighting coefficient for the firstorder derivate is expressed as the following:

(33) 
The r^{th}order approximation of function in the GDQM for domain is given as the following [42]:
(34) 
(35) 
In this research, the ChebyshevGuassLobatto sample points [43] have been used to calculate the weighting coefficients.
(36) 
In this section in order to validate the accuracy of the presented model, the bending of the crossply laminated linear beam solved by the GDQM is compared to the bending of the crossply laminated linear EulerBernoulli beam, which has been studied by Chen et al. [26]. The material properties are , , and . Also, other parameters are , and . The shown comparison in Fig 2 illustrates that the presented numerical model has a good agreement with the analytical model.
In order to study the static bending of a crossply laminated micro beam, the sizes of the beam model are considered with a thickness of and a width of . Furthermore, the material properties of carbon/epoxy are , , and and the material properties of glass epoxy are , , and .To study the bending of the beam, the nonlinear equation (29) with boundary conditions is solved using the GDQM, which is developed by Du et al. [42].The normalized static deflection of the crossply micro beam without thermal loading influence based on the finite strain is shown in Fig. 3 for , and various . The boundary conditions are adopted simply supported. The Figure demonstrates that the deflection of the beam in MCST is smaller than that in the classical elasticity method.
To compare the finite strain and vonKarman assumptions, the deflection of the crossply micro beam without thermal loading effect is considered in Figs. 4 and 5 for and . The Figures indicate that there is a remarkable difference between the finite strain and vonKarman particularly for . Also, the effect of thermal loading on the basis of the finite strain assumption subjected to different boundary conditions and is investigated for carbon/epoxy and glass/epoxy in Figs. 6 and 7, respectively.
Figure 2. The deflection of the linear beam with for and 
Figure 3. The deflection of the beam with , and for and 
Figure 4.The deflection comparison between the finite strain and vonKarman for , and

Figure 5.The deflection comparison between the finite strain and vonKarman for , and 
Figure 6. The deflection of the beam based on the finite strain for carbon/epoxy materials with different boundary conditions and thermal loadings 
Figure 7. The deflection of the beam based on the finite strain for glass/epoxy materials with different boundary conditions and thermal loadings 
It is noted that , , and . As depicted, the effect of thermal loading on the carbon/epoxy is very small. Unlike the carbon/epoxy, the influence of the thermal loading on the glass/epoxy is remarkable. These Figures have shown that with increasing the thermal effect, the deflection of the beam is increased, too. In addition, Fig 7 indicates that the difference between different thermal loadings for the SimplySupported (SS) beam is more than that difference for the ClampedClamped (CC) beam.
In this study, the influence of finite strain on the bending of the micro laminated composite EulerBernoulli beam in thermal environment based on the modified couple stress theory was investigated. The bending of the micro beam undergoing finite strain assumption for the most accurate state was studied. The governing equation of motion and boundary conditions were obtained using the Hamilton’s principle, and the GDQM was utilized for obtaining numerical results. The numerical results show that the slope of the deflection curves based on the finite strain is less than the vonKarman hypothesis. Also, the bending results demonstrate that there is a difference between the finite strain and vonKarman assumptions, particularly the difference is considerable for . In addition, the numerical results indicate that the material properties have a remarkable effect on the behavior of the finite strain micro beam. With increasing the thermal loading, although the deflection of the carbon/epoxy laminated composite materials beam for different boundary conditions does not change dramatically, the deflection of the glass/epoxy laminated composite materials beam undergoes a significant change especially for simply supported boundary condition.
Acknowledgements
The authors are grateful to the University of Kashan for supporting this study by Grant No. 574605/06.
References
[1] Chandra R, Stemple AD, Chopra I. Thinwalled Composite Beams under Bending, Torsional, and Extensional Loads. J Aircraft 1990; 27: 619626.
[2] Maiti DK, Sinha PK. Bending and Free Vibration Analysis of Shear Deformable Laminated Composite Beams by Finite Element Method. Compos Struct 1994; 29; 421431.
[3] Khdeir AA, Reddy JN. An Exact Solution for the Bending of Thin and Thick Crossply Laminated Beams. Compos Struct 1997; 37: 197203.
[4] Xiong J, Ma L, Stocchi A, Yang J, Wu L, Pan S. Bending response of carbon fiber composite sandwich beams with three dimensional honeycomb cores. Compos Struct 2014; 108: 234242.
[5] Sayyad AS, Ghugal YM, Naik NS. Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory. Curved Layer Struct 2015; 2: 279289.
[6] Shen HS, Chen X, Huang XL. Nonlinear bending and thermal postbuckling of functionally graded fiber reinforced composite laminated beams with piezoelectric fiber reinforced composite actuators. Compos Part B 2016; 90: 326335.
[7] Ghasemi AR, TaheriBehrooz F, Farahani SMN, Mohandes M. Nonlinear Free Vibration of an EulerBernoulli Composite Beam Undergoing Finite Strain Subjected to Different Boundary Conditions. J Vib Cont 2014; DOI: 10.1177/1077546314528965.
[8] Mohandes M, Ghasemi AR. Finite Strain Analysis of Nonlinear Vibrations of Symmetric Laminated Composite Timoshenko Beams Using Generalized Differential Quadrature Method. J Vib Cont 2016; 22: 940954.
[9] Mohandes M, Ghasemi AR. Modified Couple Stress Theory and Finite Strain Assumption for Nonlinear Free Vibration and Bending of micro/nanolaminated Composite EulerBernoulli Beam Under Thermal Loading. J Mech Engin Sci Part C 2016; DOI: 10.1177/0954406216656884.
[10] Ghasemi AR, Mohandes M. Nonlinear Free Vibration of Laminated Composite EulerBernoulli Beams Based on Finite Strain Using GDQM. Mech Adv Mat & Struct. 2016; DOI: 10.1080/15376494.2016.1196794.
[11] Eringen AC. Theory of micropolar plates. Zeitschrift fur angewandte Mathematik und Physik 1967; 18: 1230.
[12] Eringen AC. Nonlocal polar elastic continua. Int J Eng Sci 1972; 10: 116.
[13] Gurtin ME, Weissmuller J and Larche F. The general theory of curved deformable interfaces in solids at equilibrium. Phil Mag A 1998; 78: 10931109.
[14] Aifantis EC. Strain gradient interpretation of size effects. Int J Fracture 1999; 95: 299314.
[15] Yang F, Chong ACM, Lam DCC, Tong P. Couple stress based strain gradient theory for elasticity. Int J Solid Struct 2002; 39: 27312743.
[16] Park SK, Gao XL. EulerBernoulli beam model based on a modified couple stress theory. J Micromech Microengin 2006; 16: 2355.
[17] Salamattalab M, Nateghi A, Torabi J. Static and Dynamic Analysis of Thirdorder Shear Deformation FG Micro Beam Based on Modified Couple Stress Theory. Int J Mech Sci 2012; 57: 6373.
[18] Wang YG, Lin WH, Liu N. Large Amplitude Free Vibration of Sizedependent Circular Microplates Based on the Modified Couple Stress Theory. Int J Mech Sci2013; 71: 5157.
[19] Jung WY, Park WT, Han SC. Bending and Vibration Analysis of SFGM Microplates Embedded in Pasternak Elastic Medium Using the Modified Couple Stress Theory. Int J Mech Sci 2014; 87: 150162.
[20] Kahrobaiyan MH, Asghari M, Ahmadian MT. A Timoshenko Beam Element Based on the Modified Couple Stress Theory. Int J Mech Sci 2014; 79: 7583.
[21] Shaat M, Mahmoud FF, Gao XL, Faheem AF. Sizedependent Bending Analysis of Kirchhoff Nanoplates Based on a Modified CoupleStress Theory Including Surface Effects. Int J Mech Sci 2014; 79: 3137.
[22] Farokhi H, Ghayesh MH. Nonlinear Dynamical Behaviour of Geometrically Imperfect Microplates Based on Modified Couple Stress Theory. Int J Mech Sci 2015; 90: 133144.
[23] Mohammadimehr M, Mohandes M. The Effect of Modified Couple Stress Theory on Buckling and Vibration Analysis of Functionally Graded Doublelayer Boron Nitride Piezoelectric Plate Based on CPT. J Solid Mech, 2015; 7: 281298.
[24] Ma HM, Gao XL, Reddy JN. A Microstructuredependent Timoshenko Beam Model Based on a Modified Couple Stress Theory. J Mech Phys Solids 2008; 56: 33793391.
[25] Asghari M, Kahrobaiyan MH, Ahmadian MT. A Nonlinear Timoshenko Beam Formulation Based on the Modified Couple Stress Theory. Int J Eng Sci 2010; 48: 17491761.
[26] Chen W, Li L, Xu M, A Modified Couple Stress Model for Bending Analysis of Composite Laminated Beams with First Order Shear Deformation. Compos Struct 2011; 93: 27232732.
[27] Roque CMC, Fidalgo DS, Ferreira AJM, Reddy JN. A Study of a Microstructuredependent Composite Laminated Timoshenko Beam Using a Modified Couple Stress Theory and a Meshless Method. Compos Struct 2013; 96: 532537.
[28] Simsek M, Reddy JN. Bending and Vibration of Functionally Graded Micro Beams Using a New Higher Order Beam Theory and the Modified Couple Stress Theory. Int J Eng Sci 2013; 64: 3753.
[29] Ghayesh MH, Farokhi H, Amabili M. Nonlinear Dynamics of a Micro Scale Beam Based on the Modified Couple Stress Theory. Compos Part B Eng 2013; 50: 318324.
[30] Ilkhani MR, HosseiniHashemi SH. Size dependent vibrobuckling of rotating beam based on modified couple stress theory. Compos Struct 2016; 143: 7583.
[31] Mohammadimehr M, Mohandes M, Moradi M. Size Dependent Effect on the Buckling and Vibration Analysis of Doublebonded Nanocomposite Piezoelectric Plate Reinforced by Boron Nitride Nanotube Based on Modified Couple Stress Theory. J Vib Cont 2016; 22: 17901807.
[32] AkbarzadehKhorshidi M, Shariati M, Emam SA. Post Buckling of Functionally Graded Nanobeams based on Modified Couple Stress Theory under General Beam Theory. Int J Mech Sci 2016; 110: 160169.
[33] Akgoz B, Civalek O. Free Vibration Analysis of Axially Functionally Graded Tapered EulerBernoulli Micro Beams Based on the Modified Couple Stress Theory. Compos Struct 2013; 98: 314322.
[34] Reddy JN. Mechanics of Laminated Composite Plates and Shells. CRC Press; 2003.
[35] Sourki R, Hoseini SAH. Free vibration analysis of sizedependent cracked microbeam based on the modified couple stress theory. Appl Phys A 2016; 122: 413.
[36] Reddy JN. Energy Principles and Variational Methods in Applied Mechanics John Wiley; 2002.
[37] Simsek M, Yurtcu HH. Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos Struct 2013; 97: 378386.
[38] Pradhan SC, Murmu T. Small Scale Effect on the Buckling of Singlelayered Graphene Sheets under Biaxial Compression via Nonlocal Continuum Mechanics. Comput Mater Sci 2009; 47: 268274.
[39] Wang YG, Lin WH, Liu N. Nonlinear Free Vibration of a Microscale Beam Based on Modified Couple Stress Theory. Phys E 2013; 47: 8085.
[40] Zhang LW, Lei ZX, Liew KM, Yu JL. Static and Dynamic of Carbon Nanotube Reinforced Functionally Graded Cylindrical Panels. Compos Struct 2014; 111: 205212.
[41] Bert CW, Malik M. Differential quadrature method in computational mechanics: a review. Appl Mech Reviews 1996; 49: 127.
[42] Du H, Lim MK, Lin RM. Application of Generalized Differential Quadrature to Vibration Analysis. J Sound Vib 1995; 181: 279293.
[43] Wu YL, Shu C. Development of RBFDQ method for derivative approximation and its application to simulate natural convection in concentric annuli. Comput Mech 2002; 29: 477485.