Comparison of nonlinear Von Karman and Cosserat theories in very large deformation of skew plates

Prediction of plate behavior in large deformation is one of the important problems in plate theories. Cosserat theory is an advanced theory for simulation of plates in very large deformation, but it is complex from mathematical viewpoint. Another theory that has been used extensively for large deformation problems is nonlinear Von Karman theory which is easy for formulation and computation. In this paper, these theories were compared for rectangular and skew plates in simply supported and clamped boundary conditions to propose the acceptable range of using nonlinear Von Karman in very large deformation as a simpler theory. Higher order shear deformation plate theory was used with Von Karman nonlinearity. Whole domain method was employed for numerical solution. Each theory was validated with the literature for verification of the numerical method. Deflection and stress distribution were compared from small to very large deformations. The obtained results show that two theories were matched up to the maximum nondimensional deflection of 5 and 3 for simply supported and clamped boundary conditions, respectively. Moreover, by increasing the skew angle, the consistency of two theories would decrease even in deflections smaller than the thickness of the plate.


List of symbols h
Thickness of the plate u 01 Midplane displacements in x 1 direction u 02 Midplane displacements in x 2 direction u 03 Displacement of plate in x 3 direction 0x 1 Midplane rotation of the normal around the x 1 0x 2

Introduction
Nonlinear deformation of plates and shells has been studied as an interesting and challenging subject in recent years. Skew plates are widely used in industrial modern structures such as decks of bridges, ship hulls, and buildings as reinforced slabs, stiffened, and decks (Shahidi et al. 2007). Deflection and stress distribution is one of the fundamental problems for designing of plates with large deformation. Von Karman is one of the most applicable nonlinear theories for large deformation of plates and shells which is suitable for moderately large deformation (Ugural 1981). This nonlinear theory does not have any mathematical complexities. Moreover, higher order shear deformation theory, considered Von Karman term, has been employed for large deformation of plates by many researchers. Derived equations have been solved by various numerical and analytical methods. Kumar et al. (2011) and Shi (2007) worked on bending analysis of plates using higher order shear deformation plate theory (HSDT) and nonlinear Von Karman theory analytically. Tahouneh and Naei (2016a, b) investigated the effect of bidirectional continuously graded nanocomposite materials on free vibration of thick shell panels rested on elastic foundations. They utilized 2-D differential quadrature method as an efficient numerical tool to discretize governing equations and implement boundary conditions. Their new results for natural frequencies of the shell included the effects of elastic coefficients of foundation, boundary conditions, and material and geometrical parameters. Moreover, the obtained results indicated that a graded nanocomposite volume fraction in two directions had a higher capability to reduce the natural frequency than the conventional 1-D functionally graded nanocomposite materials. They also analyzed free vibration and vibrational displacements of thick laminated curved panels with finite length resting on two-parameter elastic foundations based on the threedimensional elasticity theory. The effects of two-parameter elastic foundation modulus, and geometrical and material parameters together with the boundary conditions on the frequency parameters of the laminated functionally graded panels were investigated. They found that the outer functionally graded material layers have significant effect on the vibration behavior of cylindrical panels. Indeed, their study served as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analysis of laminated curved panels (Tahouneh and Naei 2016a, b). Tahouneh (2016) evaluated 3-D elasticity solution for free vibration analysis of continuously graded carbon nanotubereinforced rectangular plates resting on two-parameter elastic foundations. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) were assumed to be graded in the thickness direction. Moreover, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach was employed to estimate the effective constitutive law of the elastic isotropic medium with oriented, straight carbon nanotubes. A semi-analytical approach composed of differential quadrature method (DQM) and series solution was adopted to solve the equations of motion. Their novelty was to exploit Eshelby-Mori-Tanaka approach to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various coefficients of foundation, and different combinations of free, simply supported, and clamped boundary conditions on the vibrational characteristics of CGCNTR rectangular plates. For numerical solution of this problem, Nguyen-Xuan et al. (2013), Shariyat (2010a) and Neff (2004) can be mentioned. Analytical solution of buckling and free vibration of plates using Von Karman theory studied by Fazzolari et al. (2013), Sahmani and Ansari (2013a) and Swaminathan and Naveenkumar (2014). Phung- Van et al. (2013), Sahmani and Ansari (2013b), Sahmani et al. (2014) and Shariyat (2010b) has used numerical method for solution of this problem. Amiri Rad and Panahandeh-Shahraki (2014), Duc and Cong (2013), Duc and Tung (2011), Lee and Kim (2014), Ovesy et al. (2012) and Yang et al. (2006) have applied HSDT and nonlinear Von Karman theory for post buckling analysis of plates. Cosserat brothers suggested Cosserat theory for the first time in 1909 (Cosserat and Cosserat 1909). Naghdi shell model according to Cosserat theory is another advanced theory for very large deformation of plates and shells (Simo 1993;Simo et al. , 1990aSimo and Kennedy 1992). Some researchers have been solved large deflection of isotropic plates and shells with this theory by numerical methods (Ghassemi et al. 2010(Ghassemi et al. , 2011Shahidi et al. 2007). Despite of accurate results, this theory has mathematical complexities in formulation and computation. For the first time, comparison of Cosserat and Von Karman theories has been studied in the present work. For this aim, large deflection and stress distribution of skew plates were obtained by two methods: first, HSDT by considering Von Karman nonlinearity and second, Cosserat theory. To consider the numerical solution, the whole domain method was employed. In the proposed method, the plate was considered as one element and the shape functions were developed up to very high order. The main purpose of this paper was to make a comparison between two theories from small to very large deflection of the skew plates under transverse loading. This comparison can give acceptable range of using Von Karman theory in large deflections. By determining this range, some large deflection analysis can be done using Von Karman as a simpler theory. The outline of this paper is as follows. In "Kinematic equations of HSDT", the theory of higher order shear deformation plate by considering Von Karman nonlinearity is explained. The next section is about the kinematic relations of Cosserat theory followed by which numerical method is presented. The subsequent section gives the numerical results. Finally, conclusions are given.

Kinematic equations of HSDT
According to the HSDT, displacement field is considered as follows (Reddy 1984): 1 3 where h is the thickness of the plate, and u 01 and u 02 are midplane displacements in x 1 and x 2 directions, respectively.
u 03 is displacement of plate in x 3 direction. 0x 1 and 0x 2 are the midplane rotation of the normal around the x 1 and x 2 axes. Figure 1 shows the geometry of applied rectangular plate.
Strain field can be achieved by displacement in Eq.
(2). It is notable that Von Karman theory is added due to large deformation: where Midplane curvatures in Eq. (2) are as follows: For an elastic isotropic material, constitutive equations are as follows: (1) (4) x 1 x 2 .
The stress resultants and couple stresses are given by Eq. (7): where N i and Q i are the membrane and transverse shear forces and M i is the bending moment per unit length. P i and R i are the higher order bending moment and shear force, respectively. Therefore, the constitutive relations of the plate can be rewritten by substituting Eq. (5) into Eq. (7) as follows: where the matrix coefficients are as follows:

3
The principle of virtual work is as follows: where U is the virtual strain energy and this term can be written as follows: V is the virtual work done by external forces and moments.
According to Eq. (7), virtual strain energy can be rewritten as follows: For numerical solution of Eq. (10), let displacement vector as follows: This displacement vector can be discretized as follows: where S is the shape function matrix and ⌢ is the general coordinate system. To determine parameter ⌢ , the whole domain method was employed which is explained in "Numerical method". Figure 2 shows an element of a plate before and after deformation. Superscripts "0" and "t" are implied as the status of "before deformation" and "after deformation" respectively. 0 a i and t a i are the base vectors of convective coordinate system. In addition, in this figure, 0 d and t d are the director vectors. These vectors are considered normal to the midplane status.

Kinematic relations of Cosserat theory
The base vectors in Current configuration can be written as follows ): Using Eq. (15), the first and second fundamental tensors of the surface can be written as follows: By this definition membrane and bending strains are expressed as follows: In Eq. (17), subscript "0" denotes reference configuration. According to the above equation, consider strain vector, t 0 , as: Considering Naghdi shell model, invariant form of Cauchy stress resultants is obtained as follows: In Eq. (21), J = d t S d 0 S is the Jacobean transformation. This term is the ratio between the element surface after and before deformation: In Eq. (22), "F" is the deformation gradient tensor. According to Eq. (20), the resultant stress vector can be defined as follows: Based on the Hooke's law for an isotropic linear elastic material, resultant stress and strain vectors are related as follows (Simo and Kennedy 1992): where: Principle of virtual work can be used for equilibrium condition in current and reference configuration. These equations are written as follows: W ext is virtual work of the external forces. It can be written as follows: For numerical solution of Eq. (27) or (28), displacement vector should be discretized by considering In Eq. (19), subscript "t" denotes current configuration.
In the initial configuration, second Piola stress resultants can be written as follows: In Eq. (20), subscript "0" denotes reference configuration. The Cauchy and Piola stresses are related as follows:

3
Displacement vector can be discretized as follows: where S is the shape function matrix and ⌢ is the general coordinate system. So: By substitution of Eqs. (32)-(36) in Eq. (28), the following relation can be achieved: where E αβ , K αβ , and Y would be determined through the Eqs. (32)-(34), respectively.

Numerical method
Whole domain method was applied for numerical solution of virtual work equations for both theories. In this method, whole domain of the plate will be considered as one element. First, the domain of the plate is mapped to a standard square (Fig. 3).
Hierarchical finite-element shape function was employed for interpolation of displacement field. Equations (40)-(42) show shape functions of free, simply supported, and clamped boundary conditions in one direction:

Clamped
For two-dimensional cases, shape functions matrix is as follows: Therefore, displacement in each direction can be discretized as follows: (42)

Numerical results
To validate the present numerical method, the results of the two theories are compared separately with the literatures. Then, the difference between two theories is compared to rectangular and skew plates from small to very large deformation.

Validation the numerical result of HSDT by consideration Von Karman nonlinearity
In this section, HSDT which is solved by whole domain method is compared with exact solution presented by  Mechab et al. (2010). To this aim, consider a rectangular plate with roller boundary conditions as follows (Fig. 4): The Young's modulus of the plate is E = 2 × 10 11 (Pa) and the poisson ratio υ = 0.3. Transverse loading q is considered as follows: The dimensionless formulation of deflection and stresses are as follows: Table 1 reveals the nondimensional deflection at the center of a rectangular plate for various thicknesses. As   Table 2 shows nondimensional stresses in x 2 direction at x 1 = a 2 , x 2 = b 2 . Two aspect ratios of a b = 2, 3 are tested. As the results shown in Table 2, the stresses are in close agreement with literature.

Validation the numerical results of Cosserat theory
In this section, results of Cosserat numerical solution are compared for two examples. At the first example, consider a simply supported (SSSS) rectangular plate under uniform distributed load (Fig. 5).
In Table 3, W is as follows: Cosserat theory gives acceptable results in very large deformation (Cosserat and Cosserat 1909). Table 3 shows  According to the revealed results in Table 3, there is a good agreement with the literature. As the second example, consider a rectangular plate under concentrated load (Fig. 6). Table 4 shows nondimensional deflection at the center of the plate. According to Table 4, by presented numerical solution, deflection of the plate is very close to literature up to very large deformation.

Comparison of Von Karman and Cosserat theories from small to very large deformation
Nonlinear Von Karman theory results are acceptable for moderately large deformation (Ugural 1981), and noticeably, this theory is not very complex from mathematical point of view. Cosserat theory is a nonlinear advanced theory for simulation of plate behavior in very large deformation, but it is mathematically complicated. In this section, Von Karman and Cosserat theories are compared from small to very large deformation for rectangular and skew plates. This comparison can give acceptable range of Von Karman nonlinear theory for large deformation problems.

Rectangular plate under uniform pressure
Consider a rectangular plate with a h = 0.01 . The problem is solved for two boundary conditions: (1) all edges are roller and (2) fully clamped. Figure 7 shows nondimensional deflection ( W ) at the center of the plate using two theories. In addition, a better view of Fig. 7 is drawn. It can be seen from Fig. 7b that two theories are very close to each other until W = 3 for fully clamped boundary condition and W = 5 for roller boundary condition. As the compared results of membrane force shown in Fig. 8, for clamped case, membrane forces until N a 2 D×10 4 = 1.3 are close to each other and for simply supported case up to N a 2 D×10 4 = 1.7 , the results of the two theories are matched. It can be seen from Fig. 8 that membrane forces of the two theories are matched up to a range of large deformation.

Skew plate under uniform pressure
Consider a skew plate according to Fig. 9. In this section, two theories are compared for five skew plates by = 15 • , 30 • , 45 • , 60 • , 75 • . The plate boundary condition is considered simply supported. Figure 10 shows nondimensional deflection at the center of the plate versus nondimensional uniform pressure. Numerical solution was applied from small to very large deformation. The results showed that two theories are matched as follows: It is obvious that by increasing the skew angle, consistency of two theories decreases. In addition, Fig. 11 indicates nondimensional membrane force in x 1 direction versus nondimensional deflection at the center of the plate. This figure was drawn up to large deformation. According to Fig. 11, membrane force of two theories is matched as follows:

Conclusions
In this study, two theories of HSDT (considering Von Karman) and Cosserat were compared up to large deformation ranges of the skew plates in simply supported and clamped boundary conditions for all edges. Whole domain method was applied for numerical solution. It is notable that according to the obtained results, Von Karman theory was simpler than Cosserat theory in formulation and computation. This comparison suggested acceptable range of Von Karman theory in very large deformation. Moreover, for the rectangular plates, the results obtained by the two theories matched when the maximum nondimensional deflection was approximately W = 5 for simply supported boundary conditions and W = 3 in clamped boundary conditions. Hence, Von Karman theory was valid for deflections greater than thickness up to a specific range for presented boundary condition. In addition, for the skew plates, consistency of two theories depended on skew angle. By increasing the skew angle, consistency of two theories decreases. The results illustrated that for plates with great skew angle, Von Karman theory deviated from Cosserat theory even in deflections smaller than thickness.
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