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

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## Abstract

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.

## Keywords

Cosserat theory Von Karman nonlinearity Large deformation Higher order shear deformation plate theory Skew 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- \(\varPsi_{{0x_{1} }}\)
Midplane rotation of the normal around the

*x*_{1}- \(\varPsi_{{0x_{2} }}\)
Midplane rotation of the normal around the

*x*_{2}*N*_{i}Membrane

*Q*_{i}Transverse shear force

*M*_{i}Bending moment per unit length

*P*_{i}Higher order bending moment

*R*_{i}Higher order shear force

*δU*Virtual strain energy

*δV*Virtual work

**s**Shape function matrix

- \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} }}\)
The general coordinate system

- 0
Superscript, before deformation

*t*Superscript, after deformation

- \({}^{0}a_{i}\)
Base vector of convective coordinate system

- \({}^{t}a_{i}\)
Base vector of convective coordinate system

- \({}^{0}{\mathbf{d}}\)
Director vector

- \({}^{t}{\mathbf{d}}\)
Director vector

- \(J = \frac{{{\text{d}}{}^{t}S}}{{{\text{d}}{}^{0}S}}\)
Jacobean transformation

*F*Deformation gradient tensor

- \(w_{\text{ext}}\)
Virtual work of the external forces

- \(\bar{W}\)
Nondimensional deflection

- \(\overline{{\sigma_{i} }}\)
Nondimensional stress

- \(\phi\)
Skew angle

## 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 three-dimensional 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 nanotube-reinforced 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 and Fox 1989; Simo et al. 1989, 1990a, b; Simo 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, 2011; Shahidi 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

*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. \(\varPsi_{{0x_{1} }}\) and \(\varPsi_{{0x_{2} }}\) are the midplane rotation of the normal around the

*x*

_{1}and

*x*

_{2}axes. Figure 1 shows the geometry of applied rectangular plate.

*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.

\(\delta V\) is the virtual work done by external forces and moments.

**S**is the shape function matrix and \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} }}\) is the general coordinate system. To determine parameter \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} }}\), the whole domain method was employed which is explained in “Numerical method”.

## Kinematic relations of Cosserat theory

*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.

In Eq. (17), subscript “0” denotes reference configuration.

In Eq. (19), subscript “*t*” denotes current configuration.

In Eq. (20), subscript “0” denotes reference configuration.

In Eq. (22), “*F*” is the deformation gradient tensor.

*W*

_{ext}is virtual work of the external forces. It can be written as follows:

**S**is the shape function matrix and \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} }}\) is the general coordinate system.

**E**

_{ αβ },

**K**

_{αβ}, and

**Y**would be determined through the Eqs. (32)–(34), respectively.

## Numerical method

- 1.Free$$S_{i} = \left[ {\begin{array}{*{20}l} 1 \hfill & {\xi_{i} } \hfill & {\xi_{i} (1 - \xi_{i}^{2} )} \hfill & {\xi_{i}^{2} (1 - \xi_{i}^{2} )} \hfill & \ldots \hfill \\ \end{array} } \right]\quad i = 1{\text{ or }}2.$$(40)
- 2.Simply support$$S_{i} = \left[ {\begin{array}{*{20}l} {1 - \xi_{i}^{2} } \hfill & {\xi_{i} (1 - \xi_{i}^{2} )} \hfill & {\xi_{i}^{2} (1 - \xi_{i}^{2} )} \hfill & {\xi_{i}^{3} (1 - \xi_{i}^{2} )} \hfill & \ldots \hfill \\ \end{array} } \right]\quad i = 1{\text{ or }}2.$$(41)
- 3.Clamped$$S_{i} = \left[ {\begin{array}{*{20}l} {(1 - \xi_{i}^{2} )^{{^{2} }} } \hfill & {\xi_{i} (1 - \xi_{i}^{2} )^{{^{2} }} } \hfill & {\xi_{i}^{2} (1 - \xi_{i}^{2} )^{{^{2} }} } \hfill & {\xi_{i}^{3} (1 - \xi_{i}^{2} )^{{^{2} }} } \hfill & \ldots \hfill \\ \end{array} } \right]\quad i = 1{\text{ or }}2.$$(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

*E*= 2 × 10

^{11}(Pa) and the poisson ratio

*υ*= 0.3. Transverse loading

*q*is considered as follows:

Comparison of the nondimensional deflection at the center of a rectangular plate under distributed load

\(\frac{a}{h}\) | \(\overline{W}\) | \(\frac{a}{h}\) | \(\overline{W}\) | \(\frac{a}{h}\) | \(\overline{W}\) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mechab et al. (2010) | Present model | Error (%) | Mechab et al. (2010) | Present model | Error (%) | Mechab et al. (2010) | Present model | Error (%) | |||

2 | 0.6993 | 0.7371 | 5.4078 | 8 | 0.3047 | 0.3256 | 6.8494 | 14.5 | 0.2890 | 0.2954 | 2.2366 |

2.5 | 0.5733 | 0.5943 | 3.6577 | 8.5 | 0.3084 | 0.3205 | 3.9481 | 15 | 0.2890 | 0.2944 | 1.8742 |

3 | 0.5213 | 0.5073 | 2.6791 | 9 | 0.2894 | 0.3155 | 9.0158 | 15.5 | 0.2890 | 0.2935 | 1.5713 |

3.5 | 0.4588 | 0.4623 | 0.7596 | 9.5 | 0.2878 | 0.3122 | 8.4606 | 16 | 0.2852 | 0.2927 | 2.6336 |

4 | 0.4068 | 0.4243 | 4.2961 | 10 | 0.2863 | 0.3089 | 7.8965 | 16.5 | 0.2702 | 0.2920 | 8.0825 |

4.5 | 0.3792 | 0.3967 | 4.6061 | 10.5 | 0.2847 | 0.3073 | 7.9414 | 17 | 0.2710 | 0.2911 | 7.4115 |

5 | 0.3552 | 0.3831 | 7.8684 | 11 | 0.2849 | 0.3075 | 7.9365 | 17.5 | 0.2711 | 0.2904 | 7.1533 |

5.5 | 0.3416 | 0.3660 | 7.1588 | 11.5 | 0.2794 | 0.3042 | 8.8869 | 18 | 0.2711 | 0.2898 | 6.8954 |

6 | 0.3385 | 0.3525 | 4.1308 | 12 | 0.2904 | 0.3026 | 4.1916 | 18.5 | 0.2711 | 0.2892 | 6.7014 |

6.5 | 0.3249 | 0.3494 | 7.5294 | 13 | 0.2889 | 0.2994 | 3.6275 | 19 | 0.2711 | 0.2887 | 6.5078 |

7 | 0.3161 | 0.3374 | 6.7402 | 13.5 | 0.2889 | 0.2977 | 3.0224 | 19.5 | 0.2711 | 0.2866 | 5.7344 |

7.5 | 0.3063 | 0.3306 | 7.9501 | 14 | 0.2890 | 0.2965 | 2.5994 | 20 | 0.2713 | 0.2795 | 3.0263 |

Comparison of the nondimensional stress at the center of a rectangular plate for two aspect ratios

| \(\overline{{x_{3} }} = \frac{{x_{3} }}{h}\) | \(\bar{\sigma }_{y}\) |
| \(\overline{{x_{3} }} = \frac{{x_{3} }}{h}\) | \(\bar{\sigma }_{y}\) | ||||
---|---|---|---|---|---|---|---|---|---|

Mechab et al. (2010) | Present model | Error (%) | Mechab et al. (2010) | Present model | Error (%) | ||||

2 | − 0.50 | − 0.6522 | − 0.6430 | 1.4061 | 3 | − 0.50 | − 0.3478 | − 0.3390 | 2.5386 |

− 0.40 | − 0.5870 | − 0.5790 | 1.3561 | − 0.40 | − 0.3261 | − 0.3180 | 2.4809 | ||

− 0.35 | − 0.5435 | − 0.5400 | 0.6403 | − 0.35 | − 0.2826 | − 0.2740 | 3.0466 | ||

− 0.25 | − 0.4783 | − 0.4690 | 1.9362 | − 0.25 | − 0.2609 | − 0.2520 | 3.4002 | ||

− 0.15 | − 0.4130 | − 0.4080 | 1.2202 | − 0.15 | − 0.2391 | − 0.2260 | 5.4907 | ||

0.00 | − 0.2609 | − 0.2570 | 1.4835 | 0.00 | − 0.1304 | − 0.1280 | 1.8706 | ||

0.10 | − 0.1739 | − 0.1690 | 2.8233 | 0.10 | − 0.0435 | − 0.0410 | 5.7038 | ||

0.25 | 0.2391 | 0.2270 | 5.0726 | 0.25 | 0.1739 | 0.1660 | 4.5483 | ||

0.30 | 0.4348 | 0.4290 | 1.3294 | 0.30 | 0.3043 | 0.2920 | 4.0547 | ||

0.35 | 0.6957 | 0.6880 | 1.0997 | 0.35 | 0.4783 | 0.4660 | 2.5635 | ||

0.40 | 1.0000 | 1.0800 | 8.0000 | 0.40 | 0.6087 | 0.5940 | 2.4134 | ||

0.45 | 1.5870 | 1.5740 | 0.8167 | 0.45 | 0.7826 | 0.7760 | 0.8433 | ||

0.50 | 1.9565 | 1.9490 | 0.3844 | 0.50 | 1.0870 | 1.0710 | 1.4683 |

### Validation the numerical results of Cosserat theory

Comparison of the nondimensional deflection at the center of a rectangular plate under distributed load

\(\frac{{q_{0} a^{4} }}{Dh} \times 10^{ - 6}\) | \(\overline{W}\) | \(\frac{{q_{0} a^{2} }}{Dh} \times 10^{ - 6}\) | \(\overline{W}\) | ||||
---|---|---|---|---|---|---|---|

Shahidi et al. (2007) | Present model | Error (%) | Shahidi et al. (2007) | Present model | Error (%) | ||

0 | 3.7975 | 3.7973 | 0.0045 | 10 | 28.0576 | 28.4550 | 1.4164 |

0.5 | 11.6429 | 11.8497 | 1.7762 | 12 | 28.8104 | 29.8691 | 3.6747 |

1 | 13.6655 | 14.0583 | 2.8744 | 14 | 29.8150 | 31.4102 | 5.3503 |

2 | 15.6881 | 16.1039 | 2.6504 | 16 | 30.8189 | 31.9304 | 3.6066 |

3 | 17.9613 | 18.1174 | 0.8691 | 20 | 32.3225 | 33.3278 | 3.1102 |

4 | 19.9832 | 20.6076 | 3.1246 | 25 | 35.3376 | 36.3831 | 2.9586 |

5 | 21.7513 | 22.1562 | 1.8615 | 30 | 37.3441 | 38.5839 | 3.3199 |

6 | 23.7720 | 24.2320 | 1.9350 | 35 | 39.8550 | 41.7791 | 4.8278 |

7 | 25.2862 | 25.5262 | 0.9491 | 40 | 42.1126 | 43.7732 | 3.9432 |

8 | 26.7999 | 26.8560 | 0.2093 | 50 | 47.3847 | 48.5269 | 2.4105 |

Cosserat theory gives acceptable results in very large deformation (Cosserat and Cosserat 1909). Table 3 shows the nondimensional deflection (\(\bar{W}\)) of the plate under uniform pressure. The pressure was increased to \(\bar{W} = 49.\) According to the revealed results in Table 3, there is a good agreement with the literature.

Comparison of the center deflection for a rectangular plate under concentrated load

\(\frac{{q_{0} a^{4} }}{Dh} \times 10^{ - 6}\) | \(\overline{W}\) | \(\frac{{q_{0} a^{2} }}{Dh} \times 10^{ - 6}\) | \(\overline{W}\) | ||||
---|---|---|---|---|---|---|---|

Shahidi et al. (2007) | Present model | Error (%) | Shahidi et al. (2007) | Present model | Error (%) | ||

1 | 17.0950 | 18.1126 | 5.9526 | 24 | 62.6816 | 63.5268 | 1.3484 |

1.5 | 22.7933 | 23.8332 | 4.5623 | 29 | 67.3743 | 67.607 | 0.3454 |

3.5 | 28.4916 | 29.3005 | 2.8391 | 35 | 71.3966 | 72.0926 | 0.9748 |

6 | 34.1899 | 35.6996 | 4.4156 | 43 | 76.4246 | 77.0712 | 0.8461 |

7.5 | 40.2235 | 41.1421 | 2.2837 | 50 | 80.1117 | 81.0354 | 1.1530 |

10 | 45.2514 | 46.4480 | 2.6443 | 60 | 85.4749 | 85.4461 | 0.0337 |

12 | 49.6089 | 49.9284 | 0.6440 | 70 | 89.162 | 89.8055 | 0.7217 |

15 | 53.6313 | 54.2571 | 1.1669 | 80 | 93.1844 | 93.9422 | 0.8132 |

19 | 57.9888 | 58.4119 | 0.7296 | 90 | 97.8771 | 98.1924 | 0.3221 |

### 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 \(\frac{a}{h} = 0.01\). The problem is solved for two boundary conditions: (1) all edges are roller and (2) fully clamped.

#### Skew plate under uniform pressure

## 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 \(\overline{W} = 5\) for simply supported boundary conditions and \(\overline{W} = 3\) 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.

## Notes

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