Geometrically nonlinear Euler–Bernoulli and Timoshenko micropolar beam theories

Abstract

Two ways of incorporating moderate rotations of planes normal to the axis of a straight beam into the Euler–Bernoulli and the Timoshenko micropolar beam theories are presented. In the first case, the von Kármán nonlinear strains are used to incorporate the moderate rotations of normal planes into the beam theories. In the second case, appropriate approximations are made on the nonlinear Cosserat deformation gradient to reflect the condition of moderate rotations of the normal planes. The governing nonlinear differential equations and corresponding natural boundary conditions in both cases are derived using the principle of virtual displacements. A weak-form Galerkin displacement finite element formulation is presented for the developed nonlinear beam theories. The phenomenon of locking usually encountered in beam displacement finite elements is eliminated using higher-order finite elements with nodes located at spectral points. Finally, numerical examples are presented to illustrate the effect of coupling number and bending characteristic length scale on deflections and microrotations when a micropolar beam is modeled with the developed nonlinear beam theories.

Appendices

Appendix A Constitutive relations

Table 3 Constitutive relations for the von Kármán micropolar beam theories

Table 4 Constitutive relations for nonlinear micropolar beam theories

The constants in the constitutive relations of Tables 3 and 4 are defined as

$$\begin{aligned}&A_{11} = \int _{A}(\lambda + 2\mu + \kappa )\mathrm{d}A, \qquad D_{11} = \int _{A}(\lambda + 2\mu + \kappa )z^{2}\mathrm{d}A, \\&A_{44} = \int _{A}(2\mu + \kappa )\mathrm{d}A, \qquad A_{77} = \int _{A} \kappa \mathrm{d}A, \qquad E_{44} = \int _{A}, \gamma \mathrm{d}A, \end{aligned}$$

where A is the cross section area of the beam.

Appendix B Finite element stiffness matrices

1.1 Appendix B.1 Euler–Bernoulli beam theory

We use the following notation in representing the element stiffness and tangent stiffness matrices of Euler–Bernoulli beam theories.

$$\begin{aligned} ^{MN}S_{ij}^{xx}&= \frac{\mathrm{d} L_{i}^{(M)}}{\mathrm{d}x} \frac{\mathrm{d} L_{j}^{(N)}}{d x},\quad ^{MN}S_{ij}^{0x} = L_{i}^{(M)} \frac{\mathrm{d} L_{j}^{(N)}}{\mathrm{d}x}, \quad ^{MN}S_{ij}^{x0} = \frac{\mathrm{d} L_{i}^{(M)}}{\mathrm{d}x} L_{j}^{(N)}, \quad ^{MN}S_{ij}^{00} = L_{i}^{(M)} L_{j}^{(N)}, \\ ^{MN}S_{IJ}^{x^{2}x^{2}}&= \frac{\mathrm{d}^{2} H_{I}^{(M)}}{\mathrm{d}x^{2}} \frac{\mathrm{d}^{2} H_{J}^{(N)}}{\mathrm{d}x^{2}},\quad ^{MN}S_{IJ}^{xx} = \frac{\mathrm{d} H_{I}^{(M)}}{\mathrm{d}x} \frac{\mathrm{d} H_{J}^{(N)}}{\mathrm{d}x}, \quad ^{MN}S_{Ij}^{xx} = \frac{\mathrm{d} H_{I}^{(M)}}{\mathrm{d}x} \frac{\mathrm{d} L_{j}^{(N)}}{\mathrm{d}x}, \\ ^{MN}S_{iJ}^{xx}&= \frac{\mathrm{d} L_{i}^{(M)}}{\mathrm{d}x}\frac{\mathrm{d} H_{J}^{(N)}}{d x}, \quad ^{MN}S_{Ij}^{x0} = \frac{\mathrm{d} H_{I}^{(M)}}{\mathrm{d}x} L_{j}^{(N)}, \quad ^{MN}S_{iJ}^{0x} = L_{i}^{(M)}\frac{\mathrm{d} H_{J}^{(N)}}{\mathrm{d}x }, \end{aligned}$$

where $M,N = \{1,2,3\}$, $i,j = \{1,2,\dots n\}$ and $I,J = \{1,2,\dots 2n\}$, where n is the number of nodes in a typical finite element.

1.1.1 Appendix B.1.1. von Kaŕmán micropolar Euler–Bernoulli Beam

$$\begin{aligned} \begin{aligned} K_{ij}^{11}&= \int _{x_a}^{x_b}A_{11} \,^{11}S_{ij}^{xx}\mathrm{d}x, \qquad K_{iJ}^{12}=\int _{x_a}^{x_b} \frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\right) \,^{12}S_{iJ}^{xx}\mathrm{d}x, \qquad K_{Ij}^{21}=\int _{x_a}^{x_b}A_{11}\left( \frac{\mathrm{d} w_{0}^\mathrm{E}}{\mathrm{d}x}\right) \,^{21}S_{Ij}^{xx}\mathrm{d}x, \\ K_{IJ}^{22}&=\int _{x_a}^{x_b}\Biggl \{D_{11}\,^{22}S_{IJ}^{x^{2}x^{2}}+ \frac{A_{11}}{2}\left( \frac{\mathrm{d} w_{0}^\mathrm{E}}{\mathrm{d}x}\right) ^2\,^{22}S_{IJ}^{xx} + 2A_{77}\,^{22}S_{IJ}^{xx} \Biggr \}\mathrm{d}x, \\ K_{Ij}^{23}&= \int _{x_a}^{x_b}2A_{77}\,^{23}S_{Ij}^{x0}\mathrm{d}x, \qquad K_{iJ}^{32} = \int _{x_a}^{x_b}2A_{77}\,^{32}S_{iJ}^{0x} \mathrm{d}x, \qquad K_{ij}^{33}=\int _{x_a}^{x_b}\Biggl \{E_{44}\,^{33}S_{ij}^{xx}+ 2A_{77}\,^{33}S_{ij}^{00}\Biggr \}\mathrm{d}x. \end{aligned} \end{aligned}$$

The components of the element tangent stiffness matrix are the same as the components of the element stiffness matrix, except for the following terms:

$$\begin{aligned} \begin{aligned} T_{iJ}^{12}&= K_{iJ}^{12} + \int _{x_a}^{x_b} \frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\right) \,^{12}S_{iJ}^{xx}\mathrm{d}x, \qquad T_{IJ}^{22} = K_{IJ}^{22} +\int _{x_a}^{x_b} A_{11}\left( \frac{\mathrm{d} u_{0}^\mathrm{E}}{\mathrm{d}x}+\left( \frac{\mathrm{d} w_{0}^\mathrm{E}}{\mathrm{d}x}\right) ^2\right) \,^{22}S_{IJ}^{xx} \mathrm{d}x. \\ \end{aligned} \end{aligned}$$

1.1.2 Appendix B.1.2. Micropolar nonlinear Euler–Bernoulli beam

$$\begin{aligned}&\begin{aligned} K_{ij}^{11}&=\int _{x_a}^{x_b}A_{11}\,^{11}S_{ij}^{xx} \mathrm{d}x, \qquad K_{iJ}^{12}=-\int _{x_a}^{x_b} \frac{A_{11}}{2}\psi _{y}^\mathrm{E} \,^{12}S_{iJ}^{xx}\mathrm{d}x, \qquad K_{ij}^{13}=-\int _{x_a}^{x_b} \frac{A_{11}}{2} \frac{\mathrm{d} w_{0}^\mathrm{E}}{\mathrm{d}x} \,^{13}S_{ij}^{x0}\mathrm{d}x, \\ K_{Ij}^{21}&=-\int _{x_a}^{x_b}\frac{A_{11}}{2} \psi _{y}^\mathrm{E} \,^{21}S_{Ij}^{xx}\mathrm{d}x, \qquad K_{IJ}^{22}=\int _{x_a}^{x_b}\Biggl \{D_{11}\,^{22}S_{IJ}^{x^{2}x^{2}}+ \frac{A_{11}}{2}(\psi _{y}^\mathrm{E})^2\,^{22}S_{IJ}^{xx} + 2A_{77}\,^{22}S_{IJ}^{xx} \Biggr \}\mathrm{d}x, \\ K_{Ij}^{23}&= \int _{x_a}^{x_b}\Biggl \{\frac{A_{11}}{2} \left( \frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\psi _{y}^\mathrm{E}-\frac{\mathrm{d}u_{0}^\mathrm{E}}{\mathrm{d}x}\right) +2A_{77}\Biggr \}\,^{23}S_{Ij}^{x0}\mathrm{d}x, \qquad K_{ij}^{31} = -\int _{x_a}^{x_b}\frac{A_{11}}{2}\frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\,^{31}S_{ij}^{0x} \mathrm{d}x ,\\ K_{iJ}^{32}&= \int _{x_a}^{x_b}\Biggl \{\frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\psi _{y}^\mathrm{E}-\frac{\mathrm{d}u_{0}^\mathrm{E}}{\mathrm{d}x}\right) +2A_{77}\Biggr \}\,^{32}S_{iJ}^{0x} \mathrm{d}x, \\ K_{ij}^{33}&=\int _{x_a}^{x_b}\Biggl \{E_{44}\,^{33}S_{ij}^{xx}+\frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\right) ^{2}\,^{33}S_{ij}^{00}+ 2A_{77}\,^{33}S_{ij}^{00}\Biggr \}\mathrm{d}x, \end{aligned}\\&\begin{aligned} T_{iJ}^{12}&= K_{iJ}^{12} -\int _{x_a}^{x_b} \frac{A_{11}}{2}\psi _{y}^\mathrm{E} \,^{12}S_{iJ}^{xx}\mathrm{d}x, \qquad T_{ij}^{13} = K_{ij}^{13} -\int _{x_a}^{x_b} \frac{A_{11}}{2} \frac{\mathrm{d} w_{0}^\mathrm{E}}{\mathrm{d}x} \,^{13}S_{ij}^{x0}\mathrm{d}x, \\ T_{Ij}^{21}&= K_{Ij}^{21} -\int _{x_a}^{x_b}\frac{A_{11}}{2} \psi _{y}^\mathrm{E}\,^{21}S_{Ij}^{xx}\mathrm{d}x, \qquad T_{IJ}^{22} = K_{IJ}^{22} + \int _{x_a}^{x_b} \frac{A_{11}}{2}(\psi _{y}^\mathrm{E})^2\,^{22}S_{IJ}^{xx} \mathrm{d}x, \\ T_{Ij}^{23}&= K_{Ij}^{23} + \int _{x_a}^{x_b}\frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\psi _{y}^\mathrm{E}-\frac{\mathrm{d}u_{0}^\mathrm{E}}{\mathrm{d}x}\right) \,^{23}S_{Ij}^{x0}\mathrm{d}x + \int _{x_a}^{x_b}A_{11}\frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\psi _{y}^\mathrm{E}\,^{23}S_{Ij}^{x0}\mathrm{d}x ,\\ T_{ij}^{31}&= K_{ij}^{31} -\int _{x_a}^{x_b}\frac{A_{11}}{2}\frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\,^{31}S_{ij}^{0x} \mathrm{d}x, \\ T_{iJ}^{32}&= K_{iJ}^{32} + \int _{x_a}^{x_b}\frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\psi _{y}^\mathrm{E}-\frac{\mathrm{d}u_{0}^\mathrm{E}}{\mathrm{d}x}\right) \,^{32}S_{iJ}^{0x}\mathrm{d}x + \int _{x_a}^{x_b}A_{11}\frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\psi _{y}^\mathrm{E}\,^{32}S_{iJ}^{0x}\mathrm{d}x ,\\ T_{ij}^{33}&= K_{ij}^{33} + \int _{x_a}^{x_b}\frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{E}}{\mathrm{d}x}\right) ^{2}\,^{33}S_{ij}^{00}\mathrm{d}x. \end{aligned} \end{aligned}$$

1.2 Appendix B.2. The Timoshenko beam theory

We shall use the following notation in representing the element stiffness and tangent stiffness matrices of the Timoshenko beam theory:

$$\begin{aligned} ^{MN}S_{ij}^{xx} = \frac{\mathrm{d} L_{i}^{(M)}}{\mathrm{d}x} \frac{\mathrm{d} L_{j}^{(N)}}{d x},\quad ^{MN}S_{ij}^{0x} = L_{i}^{(M)} \frac{\mathrm{d} L_{j}^{(N)}}{\mathrm{d}x}, \quad ^{MN}S_{ij}^{x0} = \frac{\mathrm{d} L_{i}^{(M)}}{\mathrm{d}x} L_{j}^{(N)}, \quad ^{MN}S_{ij}^{00} = L_{i}^{(M)} L_{j}^{(N)}, \end{aligned}$$

where $M,N = \{1,2,3,4\}$, $i,j = \{1,2,\dots n\}$, where n is the number of nodes in a typical finite element.

1.2.1 Appendix B.2.1. The von Kaŕmán nonlinear Timoshenko beam

$$\begin{aligned} \begin{aligned} K_{ij}^{11}&=\int _{x_a}^{x_b}A_{11} \,^{11}S_{ij}^{xx} \mathrm{d}x, \qquad K_{ij}^{12}=\int _{x_a}^{x_b} \frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\right) \,^{12}S_{ij}^{xx}\mathrm{d}x, \qquad K_{ij}^{21}=\int _{x_a}^{x_b}A_{11}\left( \frac{\mathrm{d} w_{0}^\mathrm{T}}{\mathrm{d}x}\right) \,^{21}S_{ij}^{xx}\mathrm{d}x, \\ K_{ij}^{22}&=\int _{x_a}^{x_b}\Biggl \{\frac{A_{11}}{2}\left( \frac{\mathrm{d} w_{0}^\mathrm{T}}{\mathrm{d}x}\right) ^2 + \frac{\left( A_{44}+A_{77} \right) }{2} \Biggr \}\,^{22}S_{ij}^{xx} \mathrm{d}x, \qquad K_{ij}^{23}= \int _{x_a}^{x_b}\left( \frac{A_{44}-A_{77}}{2} \right) \,^{23}S_{ij}^{x0} \mathrm{d}x, \\ K_{ij}^{24}&= \int _{x_a}^{x_b} A_{77}\,^{24}S_{ij}^{x0} \mathrm{d}x, \qquad K_{ij}^{32}= \int _{x_a}^{x_b}\left( \frac{A_{44}-A_{77}}{2}\right) \,^{32}S_{ij}^{0x} \mathrm{d}x ,\\ K_{ij}^{33}&=\int _{x_a}^{x_b}\Biggl \{D_{11}\,^{33}S_{ij}^{xx} + \left( \frac{A_{44}+A_{77}}{2}\right) \,^{33}S_{ij}^{00} \Biggr \}\mathrm{d}x, \qquad K_{ij}^{34}=-\int _{x_a}^{x_b}A_{77}\,^{34}S_{ij}^{00} \mathrm{d}x ,\\ K_{ij}^{42}&= \int _{x_a}^{x_b} A_{77}\,^{42}S_{ij}^{0x} \mathrm{d}x, \qquad K_{ij}^{43}=-\int _{x_a}^{x_b}A_{77}\,^{43}S_{ij}^{00} \mathrm{d}x, \qquad K_{ij}^{44}=\int _{x_a}^{x_b}\Biggl \{E_{44}\,^{44}S_{ij}^{xx} + 2A_{77}\,^{44}S_{ij}^{00} \Biggr \}\mathrm{d}x. \end{aligned} \end{aligned}$$

Similarly, the components of the element tangent stiffness matrix are given by

$$\begin{aligned} T_{ij}^{MN} = K_{ij} ^{MN} \end{aligned}$$

except for the following terms:

$$\begin{aligned} T_{ij}^{12}&= 2K_{ij}^{12}, \qquad T_{ij}^{22} = K_{ij}^{22} + \int _{x_a}^{x_b}\left( \frac{A_{11}}{2}\left( \frac{\mathrm{d}u_{0}^\mathrm{T}}{\mathrm{d}x}\right) + A_{11}\left( \frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\right) ^{2}\right) \,^{22}S_{ij}^{xx} \mathrm{d}x. \end{aligned}$$

1.2.2 Appendix B.2.2. The micropolar nonlinear Timoshenko beam

$$\begin{aligned} \begin{aligned} K_{ij}^{11}&=\int _{x_a}^{x_b}A_{11}\,^{11}S_{ij}^{xx} \mathrm{d}x, \qquad K_{ij}^{12}=-\int _{x_a}^{x_b} \frac{A_{11}}{2}\psi _{y}^\mathrm{T}\,^{12}S_{ij}^{xx} \mathrm{d}x, \qquad K_{ij}^{14}=-\int _{x_a}^{x_b} \frac{A_{11}}{2}\frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\,^{14}S_{ij}^{x0} \mathrm{d}x, \\ K_{ij}^{21}&=-\int _{x_a}^{x_b}\frac{A_{11}}{2}\psi _{y}^\mathrm{T}\,^{21}S_{ij}^{xx} \mathrm{d}x, \qquad K_{ij}^{22}=\int _{x_a}^{x_b}\Biggl \{\frac{A_{11}}{2}(\psi _{y}^\mathrm{T})^{2} + \frac{\left( A_{44}+A_{77} \right) }{2} \Biggr \}\,^{22}S_{ij}^{xx} \mathrm{d}x ,\\ K_{ij}^{23}&= \int _{x_a}^{x_b}\left( \frac{A_{44}-A_{77}}{2} \right) \,^{23}S_{ij}^{x0} \mathrm{d}x, \qquad K_{ij}^{24}= \int _{x_a}^{x_b}\Biggl \{\frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\psi _{y}^\mathrm{T}-\frac{\mathrm{d}u_{0}^\mathrm{T}}{\mathrm{d}x}\right) + A_{77}\Biggr \} \,^{24}S_{ij}^{x0} \mathrm{d}x ,\\ K_{ij}^{32}&= \int _{x_a}^{x_b}\left( \frac{A_{44}-A_{77}}{2}\right) \,^{32}S_{ij}^{0x} \mathrm{d}x, \qquad K_{ij}^{33}=\int _{x_a}^{x_b}\Biggl \{D_{11}\,^{33}S_{ij}^{xx} + \left( \frac{A_{44}+A_{77}}{2}\right) \,^{33}S_{ij}^{00} \Biggr \}\mathrm{d}x ,\\ K_{ij}^{34}&=-\int _{x_a}^{x_b}A_{77}\,^{34}S_{ij}^{00} \mathrm{d}x, \qquad K_{ij}^{41}= -\int _{x_a}^{x_b} \frac{A_{11}}{2}\frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\,^{41}S_{ij}^{0x} \mathrm{d}x ,\\ K_{ij}^{42}&= \int _{x_a}^{x_b}\Biggl \{ A_{77} + \frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\psi _{y}^\mathrm{T}-\frac{\mathrm{d}u_{0}^\mathrm{T}}{\mathrm{d}x}\right) \Biggr \}\,^{42}S_{ij}^{0x} \mathrm{d}x, \qquad K_{ij}^{43} =-\int _{x_a}^{x_b}A_{77}\,^{44}S_{ij}^{00}\mathrm{d}x ,\\ K_{ij}^{44}&=\int _{x_a}^{x_b}\Biggl \{E_{44}\,^{44}S_{ij}^{xx}+ 2A_{77}\,^{44}S_{ij}^{00} + \frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\right) ^{2}\,^{44}S_{ij}^{00}\Biggr \}\mathrm{d}x. \end{aligned} \end{aligned}$$

Similarly, the components of the element tangent stiffness matrix are given by

$$\begin{aligned} T_{ij}^{MN} = K_{ij} ^{MN} \end{aligned}$$

except for the following terms:

$$\begin{aligned} T_{ij}^{12}&= K_{ij}^{12} -\int _{x_a}^{x_b} \frac{A_{11}}{2}\psi _{y}^\mathrm{T}\,^{12}S_{ij}^{xx}\mathrm{d}x, \qquad T_{ij}^{14} = K_{ij}^{14} -\int _{x_a}^{x_b} \frac{A_{11}}{2}\frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\,^{14}S_{ij}^{x0}\mathrm{d}x ,\\ T_{ij}^{21}&= K_{ij}^{21} -\int _{x_a}^{x_b} \frac{A_{11}}{2}\psi _{y}^\mathrm{T}\,^{21}S_{ij}^{xx}\mathrm{d}x, \qquad T_{ij}^{22} = K_{ij}^{22} + \int _{x_a}^{x_b} \frac{A_{11}}{2}(\psi _{y}^\mathrm{T})^{2}\,^{22}S_{ij}^{xx}\mathrm{d}x ,\\ T_{ij}^{24}&= K_{ij}^{24} -\int _{x_a}^{x_b} \frac{A_{11}}{2}\frac{\mathrm{d}u_{0}^\mathrm{T}}{\mathrm{d}x}\,^{24}S_{ij}^{x0}\mathrm{d}x + \int _{x_a}^{x_b} \frac{3A_{11}}{2}\frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\psi _{y}^\mathrm{T}\,^{24}S_{ij}^{x0}\mathrm{d}x ,\\ T_{ij}^{41}&= K_{ij}^{41} -\int _{x_a}^{x_b} \frac{A_{11}}{2}\frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\,^{41}S_{ij}^{0x}\mathrm{d}x ,\\ T_{ij}^{42}&= K_{ij}^{42} -\int _{x_a}^{x_b} \frac{A_{11}}{2}\frac{\mathrm{d}u_{0}^\mathrm{T}}{\mathrm{d}x}\,^{42}S_{ij}^{0x}\mathrm{d}x + \int _{x_a}^{x_b} \frac{3A_{11}}{2}\frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\psi _{y}^\mathrm{T}\,^{42}S_{ij}^{0x}\mathrm{d}x ,\\ T_{ij}^{44}&= K_{ij}^{44} + \int _{x_a}^{x_b} \frac{A_{11}}{2}\left( \frac{\mathrm{d}w_{0}^\mathrm{T}}{\mathrm{d}x}\right) ^{2}\,^{44}S_{ij}^{00}\mathrm{d}x. \end{aligned}$$