Non-conforming Trefftz finite element implementation of orthotropic Kirchhoff plate model based on consistent couple stress theory

Abstract

This work proposes a new 4-node non-conforming plate element for static and dynamic analyses of small-scale orthotropic thin plates within the consistent couple stress theory. By applying the Kirchhoff thin plate bending constraint to the three-dimensional consistent couple stress elasticity, the non-classical orthotropic thin plate model is established and the Trefftz functions are derived. Then, the new element is formulated in a straightforward manner based on the generalized conforming theory, by taking the obtained Trefftz functions as the basic function for construction and employing a novel set of point conforming conditions to enforce the compatibility requirement in weak sense. Several benchmark examples are examined and the results show that the element has good numerical accuracy and mesh-distortion tolerance in prediction of the size-dependent bending behavior of the small-scale orthotropic thin plates.

Appendices

Appendix 1

The matrix ${{\varvec{\uplambda}}}$ in Eq. (54) is composed of two parts, as follows:

$${{\varvec{\uplambda}}} = \left[ {\begin{array}{*{20}c} {{{\varvec{\uplambda}}}^{w} } \\ {{{\varvec{\uplambda}}}^{\psi } } \\ \end{array} } \right]$$

(73)

The first submatrix ${{\varvec{\uplambda}}}^{w}$ is expressed as

$${{\varvec{\uplambda}}}^{w} = \left[ {\begin{array}{*{20}c} {\hat{w}^{1} \left( {x_{1} ,y_{1} } \right)} & {\hat{w}^{2} \left( {x_{1} ,y_{1} } \right)} & \cdots & {\hat{w}^{14} \left( {x_{1} ,y_{1} } \right)} \\ {\hat{w}^{1} \left( {x_{2} ,y_{2} } \right)} & {\hat{w}^{2} \left( {x_{2} ,y_{2} } \right)} & \cdots & {\hat{w}^{14} \left( {x_{2} ,y_{2} } \right)} \\ {\hat{w}^{1} \left( {x_{3} ,y_{3} } \right)} & {\hat{w}^{2} \left( {x_{3} ,y_{3} } \right)} & \cdots & {\hat{w}^{14} \left( {x_{3} ,y_{3} } \right)} \\ {\hat{w}^{1} \left( {x_{4} ,y_{4} } \right)} & {\hat{w}^{2} \left( {x_{4} ,y_{4} } \right)} & \cdots & {\hat{w}^{14} \left( {x_{4} ,y_{4} } \right)} \\ {\hat{w}^{1} \left( {x_{5} ,y_{5} } \right) + \hat{w}^{1} \left( {x_{7} ,y_{7} } \right)} & {\hat{w}^{2} \left( {x_{5} ,y_{5} } \right) + \hat{w}^{2} \left( {x_{7} ,y_{7} } \right)} & \cdots & {\hat{w}^{14} \left( {x_{5} ,y_{5} } \right) + \hat{w}^{14} \left( {x_{7} ,y_{7} } \right)} \\ {\hat{w}^{1} \left( {x_{6} ,y_{6} } \right) + \hat{w}^{1} \left( {x_{8} ,y_{8} } \right)} & {\hat{w}^{2} \left( {x_{6} ,y_{6} } \right) + \hat{w}^{2} \left( {x_{8} ,y_{8} } \right)} & \cdots & {\hat{w}^{14} \left( {x_{6} ,y_{6} } \right) + \hat{w}^{14} \left( {x_{8} ,y_{8} } \right)} \\ \end{array} } \right]$$

(74)

while the second part ${{\varvec{\uplambda}}}^{\psi }$ is given by

$${{\varvec{\uplambda}}}^{\psi } = {\mathbf{T}}_{n} {{\varvec{\Phi}}}$$

(75)

in which

$${{\varvec{\Phi}}} = \left[ {\begin{array}{*{20}c} {\hat{\psi }_{x}^{1} \left( {x_{A1} ,y_{A1} } \right)} & {\hat{\psi }_{x}^{2} \left( {x_{A1} ,y_{A1} } \right)} & {...} & {\hat{\psi }_{x}^{14} \left( {x_{A1} ,y_{A1} } \right)} \\ {\hat{\psi }_{y}^{1} \left( {x_{A1} ,y_{A1} } \right)} & {\hat{\psi }_{y}^{2} \left( {x_{A1} ,y_{A1} } \right)} & {...} & {\hat{\psi }_{y}^{14} \left( {x_{A1} ,y_{A1} } \right)} \\ \vdots & \vdots & \ddots & {...} \\ {\hat{\psi }_{x}^{1} \left( {x_{B4} ,y_{B4} } \right)} & {\hat{\psi }_{x}^{2} \left( {x_{B4} ,y_{B4} } \right)} & {...} & {\hat{\psi }_{x}^{14} \left( {x_{B4} ,y_{B4} } \right)} \\ {\hat{\psi }_{y}^{1} \left( {x_{B4} ,y_{B4} } \right)} & {\hat{\psi }_{y}^{2} \left( {x_{B4} ,y_{B4} } \right)} & {...} & {\hat{\psi }_{y}^{14} \left( {x_{B4} ,y_{B4} } \right)} \\ \end{array} } \right]$$

(76)

and

$${\mathbf{T}}_{n} = \left[ {\begin{array}{*{20}c} {{\mathbf{T}}_{1} } & {} & {} & {} \\ {} & {{\mathbf{T}}_{2} } & {} & {} \\ {} & {} & {{\mathbf{T}}_{3} } & {} \\ {} & {} & {} & {{\mathbf{T}}_{4} } \\ \end{array} } \right],\;\;\; {\mathbf{T}}_{i} = \left[ {\begin{array}{*{20}c} { - \frac{{y_{ij} }}{{L_{ij} }}} & {\frac{{x_{ij} }}{{L_{ij} }}} & {} & {} \\ {} & {} & { - \frac{{y_{ij} }}{{L_{ij} }}} & {\frac{{x_{ij} }}{{L_{ij} }}} \\ \end{array} } \right],\;\;\;\left( {ij = 12,\;23,\;34,\;41} \right)$$

(77)

where $x_{ij} = x_{i} - x_{j}$ and $y_{ij} = y_{i} - y_{j}$.

The matrix ${{\varvec{\Lambda}}}$ in Eq. (54) takes the form

$${{\varvec{\Lambda}}} = \left[ {\begin{array}{*{20}c} {{{\varvec{\Lambda}}}^{w} } \\ {{{\varvec{\Lambda}}}^{\psi } } \\ \end{array} } \right]$$

(78)

in which

$${{\varvec{\Lambda}}}^{w} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & 1 & 0 & 0 & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & 1 & 0 & 0 & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & 1 & 0 & 0 \\ \frac{1}{2} & { - \frac{{x_{12} }}{8}} & { - \frac{{y_{12} }}{8}} & \frac{1}{2} & {\frac{{x_{12} }}{8}} & {\frac{{y_{12} }}{8}} & \frac{1}{2} & { - \frac{{x_{34} }}{8}} & { - \frac{{y_{34} }}{8}} & \frac{1}{2} & {\frac{{x_{34} }}{8}} & {\frac{{y_{34} }}{8}} \\ \frac{1}{2} & {\frac{{x_{41} }}{8}} & {\frac{{y_{41} }}{8}} & \frac{1}{2} & { - \frac{{x_{23} }}{8}} & { - \frac{{y_{23} }}{8}} & \frac{1}{2} & {\frac{{x_{23} }}{8}} & {\frac{{y_{23} }}{8}} & \frac{1}{2} & { - \frac{{x_{41} }}{8}} & { - \frac{{y_{41} }}{8}} \\ \end{array} } \right]$$

(79)

and ${{\varvec{\Lambda}}}^{\psi }$ is given by

$${{\varvec{\Lambda}}}^{\psi } = \left[ {\begin{array}{*{20}c} {{{\varvec{\Lambda}}}_{11}^{\psi } } & {{{\varvec{\Lambda}}}_{12}^{\psi } } & {} & {} \\ {} & {{{\varvec{\Lambda}}}_{22}^{\psi } } & {{{\varvec{\Lambda}}}_{23}^{\psi } } & {} \\ {} & {} & {{{\varvec{\Lambda}}}_{33}^{\psi } } & {{{\varvec{\Lambda}}}_{34}^{\psi } } \\ {{{\varvec{\Lambda}}}_{41}^{\psi } } & {} & {} & {{{\varvec{\Lambda}}}_{44}^{\psi } } \\ \end{array} } \right]$$

(80)

with

$${{\varvec{\Lambda}}}_{ii}^{\psi } = \left[ {\begin{array}{*{20}c} 0 & { - \frac{{1 + {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}}}{{2L_{ij} }}y_{ij} } & {\frac{{1 + {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}}}{{2L_{ij} }}x_{ij} } \\ 0 & { - \frac{{1 - {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}}}{{2L_{ij} }}y_{ij} } & {\frac{{1 - {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}}}{{2L_{ij} }}x_{ij} } \\ \end{array} } \right],\;\;\; {{\varvec{\Lambda}}}_{ij}^{\psi } = \left[ {\begin{array}{*{20}c} 0 & { - \frac{{1 - {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}}}{{2L_{ij} }}y_{ij} } & {\frac{{1 - {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}}}{{2L_{ij} }}x_{ij} } \\ 0 & { - \frac{{1 + {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}}}{{2L_{ij} }}y_{ij} } & {\frac{{1 + {1 \mathord{\left/ {\vphantom {1 {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}}}{{2L_{ij} }}x_{ij} } \\ \end{array} } \right]$$

(81)

where $ij = 12,\;23,\;34,\;41$.

Appendix 2

As previously discussed, the analytical solutions of some of the numerical tests are relatively easy to obtain and these are summarized in this section.

With respect to the static bending behavior of the square thin plate under uniformly distributed transverse load q, the deflection of the simply supported orthotropic case can be obtained using the Navier method:

$$w = \frac{{16qL^{4} }}{{\pi^{6} }}\sum\limits_{n = 1}^{\infty } {\sum\limits_{m = 1}^{\infty } {\frac{1}{mn}\frac{{\sin \frac{m\pi }{L}x\sin \frac{n\pi }{L}y}}{{\left( {D_{x} + D_{l} } \right)m^{4} + 2\left( {H + D_{l} } \right)m^{2} n^{2} + \left( {D_{y} + D_{l} } \right)n^{4} }}} }$$

(82)

in which L is the edge length of the square plate. Then, by substituting Eq. (82) into the definitions given in Sect. 2, the resultants can be also obtained. From Eq. (82), the deflection of the simply supported isotropic case is delivered:

$$w = \frac{{16qL^{4} }}{{\left( {D + D_{l} } \right)\pi^{6} }}\sum\limits_{m = 1}^{\infty } {\sum\limits_{n = 1}^{\infty } {\frac{1}{mn}\frac{{\sin \frac{m\pi }{L}x\sin \frac{n\pi }{L}y}}{{\left( {m^{4} + 2m^{2} n^{2} + n^{4} } \right)}}} }$$

(83)

Note that it is difficult to derive the solution of the clamped orthotropic square plate analytically, only that of the clamped isotropic case is provided by using the superposition method [53]:

$$w = \frac{q}{{8\left( {D + D_{l} } \right)}}\left[ {\left( {\frac{{L^{2} }}{4} - x^{2} } \right)\left( {\frac{{L^{2} }}{4} - y^{2} } \right) - \frac{{L^{3} }}{2}\left( {\sum\limits_{n = 1,3...}^{\infty } {\frac{{A_{n} Y_{n} }}{{n^{2} }}\frac{{\cos \frac{n\pi x}{L}}}{{\cosh^{2} \frac{n\pi }{2}}} + \sum\limits_{m = 1,3...}^{\infty } {\frac{{A_{m} X_{m} }}{{m^{2} }}\frac{{\cos \frac{m\pi y}{L}}}{{\cosh^{2} \frac{m\pi }{2}}}} } } \right)} \right]$$

(84)

in which

$$X_{m} = L\sinh \frac{m\pi }{2}\cosh \frac{m\pi x}{L} - 2x\cosh \frac{m\pi }{2}\sinh \frac{m\pi x}{L}$$

(85)

$$Y_{n} = L\sinh \frac{n\pi }{2}\cosh \frac{n\pi y}{L} - 2y\cosh \frac{n\pi }{2}\sinh \frac{n\pi y}{L}$$

(86)

and the coefficients $A_{n}$ and $A_{m}$ are determined in accordance with the boundary conditions and summarized in Table

Table 13 The coefficients used in Eq. (84)

13.

Besides, the reference frequency of the free vibration of the simply supported isotropic square plate is given by

$$\theta_{mn} = \pi^{2} \left(\frac{{m^{2} }}{{L^{2} }} + \frac{{n^{2} }}{{L^{2} }}\right)\sqrt {\frac{{D[1 + 24(1 - \nu )\frac{{l^{2} }}{{h^{2} }}]}}{{\rho h + \frac{\rho }{12}h^{3} \pi^{2} (\frac{{m^{2} }}{{L^{2} }} + \frac{{n^{2} }}{{L^{2} }})}}} ,\;\;\;m\;{\text{and}}\;n = 1,2,3 \ldots$$

(87)

With respect to the static bending behavior of the circular thin plate under uniformly distributed transverse load q, the deflection of the clamped orthotropic case can be obtained:

$$w = \frac{{q\left( {R^{2} - r^{2} } \right)^{2} }}{{8\left[ {3\left( {D_{x} + D_{l} } \right) + 2\left( {H + D_{l} } \right) + 3\left( {D_{y} + D_{l} } \right)} \right]}}$$

(88)

in which R is the radius of the circular plate and $r^{2} = x^{2} + y^{2}$. From Eq. (88), the deflection of the clamped isotropic case is further deduced:

$$w = \frac{q}{{64\left( {D + D_{l} } \right)}}\left( {R^{2} - r^{2} } \right)^{2}$$

(89)

In addition, the deflection of the simply supported isotropic circular plate is given by

$$w = \frac{{qR^{4} }}{{64\left( {D + D_{l} } \right)\left[ {\left( {1 + \nu } \right)D + 2D_{l} } \right]}}\left( {{\text{A}}\frac{{r^{4} }}{{R^{4} }} + {\text{B}}\frac{{r^{2} }}{{R^{2} }} + {\text{C}}} \right)$$

(90)

with

$${\text{A}} = \left( {1 + \nu } \right)D + 2D_{l} {,}\;\;\;{\text{B}} = - 2\left( {3 + \nu } \right)D - 8D_{l} {,}\;\;\;{\text{C}} = \left( {5 + \nu } \right)D + 6D_{l}$$

(91)