A modified beam model based on Gurtin–Murdoch surface elasticity theory

Abstract

In this work, a modified surface-effect incorporated beam model based on Gurtin and Murdoch (GM) surface elasticity theory is established by satisfying the required balance equations on surfaces, which is often overlooked by researchers in this field. With the refinement, the proposed model is more rigorous in mathematics and mechanics compared with GM theory-based beam models in literature. To demonstrate the model, the problem for static bending of simply supported beam considering surface effects is solved by applying the general equations derived, and numerical results are obtained and discussed.

Appendix: The third-order beam relations without considering continuous surface shear stress

Start from Eq. (21) without considering surface shear stress continuum.

$$\begin{aligned} u_{x} & = u + \left[ {\psi - \frac{4}{{3h^{2} }}\left( {w_{{,x}} + \psi } \right)z^{2} } \right]z, \\ u_{z} & = w, \\ \end{aligned}$$

(53)

which are just the displacements assumed for the classical beam model [27]. The corresponding equations of motion incorporating surface effects are given by

$$\begin{aligned} & N_{{xx,x}}^{{**}} + r_{x}^{{\left( 0 \right)}} = \int_{{ - h/2}}^{{h/2}} \rho \ddot{u}_{x} {\text{d}}z + \rho _{0} \left( {\ddot{u}_{x}^{ + } + \ddot{u}_{x}^{ - } } \right), \\ & N_{{xz,x}}^{{**}} + p + r_{z}^{{\left( 0 \right)}} + \frac{4}{{3h^{2} }}\left( {P_{{xx,xx}}^{{**}} - 3R_{{xz,x}} + r_{{x,x}}^{{\left( 3 \right)}} } \right) \\ & \quad = \int_{{ - h/2}}^{{h/2}} \rho \left( {\ddot{u}_{z} + \frac{4}{{3h^{2} }}\ddot{u}_{{x,x}} z^{3} } \right){\text{d}}z + \rho _{0} \left[ {\left( {\ddot{u}_{z}^{ + } + \ddot{u}_{z}^{ - } } \right) + \frac{h}{6}\left( {\ddot{u}_{{x,x}}^{ + } - \ddot{u}_{{x,x}}^{ - } } \right)} \right], \\ & M_{{xx,x}}^{{**}} - N_{{xz}} + r_{x}^{{\left( 1 \right)}} - \frac{4}{{3h^{2} }}\left( {P_{{xx,x}}^{{**}} - 3R_{{xz}} + r_{x}^{{\left( 3 \right)}} } \right) \\ & \quad = \int_{{ - h/2}}^{{h/2}} \rho \ddot{u}_{x} \left( {z - \frac{4}{{3h^{2} }}z^{3} } \right){\text{d}}z + \frac{h}{3}\rho _{0} \left( {\ddot{u}_{x}^{ + } - \ddot{u}_{x}^{ - } } \right), \\ \end{aligned}$$

(54)

with the associated boundary conditions are prescribed as follows:

$$u\;{\text{or}}\;N_{{xx}}^{{**}} ,w\;{\text{or}}\;N_{{xz}}^{{**}} ,\psi \;{\text{or}}\;M_{{xx}}^{{**}} - \frac{4}{{3h^{2} }}P_{{xx}}^{{**}} ,w_{{,x}} \;{\text{or}}\;P_{{xx}}^{{**}} .$$

(55)

It is noted that although the extended resultant forces including the surface stress effects with two asterisk superscripts in Eqs. (54) and (55) have the same form as given in Eq. (7), the stress components are determined based on the displacements Eq. (53) but not Eq. (21).

For static cylindrical bending, and the body forces are not considered, the governing equations in Eq. (54) are reduced to

$$\begin{aligned} & N_{{xx,x}}^{{**}} = 0, \\ & N_{{xz,x}}^{{**}} + p + \frac{4}{{3h^{2} }}\left( {P_{{xx,xx}}^{{**}} - 3R_{{xz,x}} } \right) = 0, \\ & M_{{xx,x}}^{{**}} - N_{{xz}} - \frac{4}{{3h^{2} }}\left( {P_{{xx,x}}^{{**}} - 3R_{{xz}} } \right) = 0. \\ \end{aligned}$$

(56)

The resultant forces determined based on the displacements Eq. (53) are given by

$$\begin{array}{l}{N}_{xx}=\frac{2\mu h}{1-\nu }{u}_{,x}, {N}_{xz}=\frac{\mu h}{3}\left(2{w}_{,x}+2\psi \right), {M}_{xz}=0 , \\ {M}_{xx}=\frac{\mu {h}^{3}}{6\left(1-\nu \right)}\left(\left(\frac{{l}_{1\nu }}{h}-\frac{1}{5}\right){w}_{,xx}+\frac{4}{5}{\psi }_{,x}\right), {R}_{xx}=\frac{\mu {h}^{3}}{6\left(1-\nu \right)}{u}_{,x}, \\ {R}_{xz}=\frac{\mu {h}^{3}}{30}\left({w}_{,x}+\psi \right), {P}_{xx}=\frac{\mu {h}^{5}}{40\left(1-\nu \right)}\left[\left(\frac{{l}_{1\nu }}{h}-\frac{5}{21}\right){w}_{,xx}+\frac{16}{21}{\psi }_{,x}\right],\end{array}$$

(57)

and

$$\begin{array}{l}{N}_{xx}^{**}=2{\tau }_{0}+\frac{2\mu h}{1-\nu }\left(1+\frac{{l}_{0\nu }}{h}\right){u}_{,x}, {N}_{xz}^{**}=\frac{2\mu h}{3}\left[\left(1+\frac{3{l}_{1}}{h}\right){w}_{,x}+\psi \right], \\ {M}_{xx}^{**}=\frac{\mu {h}^{3}}{6\left(1-\nu \right)}\left[\left(\frac{{l}_{1\nu }-{l}_{0\nu }}{h}-\frac{1}{5}\right){w}_{,xx}+2\left(\frac{2}{5}+\frac{{l}_{0\nu }}{h}\right){\psi }_{,x}\right], \\ {R}_{xx}^{**}=\frac{{h}^{2}}{2}{\tau }_{0}+\frac{\mu {h}^{3}}{6\left(1-\nu \right)}\left(1+\frac{3{l}_{0\nu }}{h}\right){u}_{,x}, \\ {P}_{xx}^{**}=\frac{\mu {h}^{5}}{40\left(1-\nu \right)}\left[\left(\frac{{l}_{1\nu }}{h}-\frac{5}{3}\frac{{l}_{0\nu }}{h}-\frac{5}{21}\right){w}_{,xx}+\frac{10}{3}\left(\frac{8}{35}+\frac{{l}_{0\nu }}{h}\right) {\psi }_{,x}\right],\end{array}$$

(58)

where the parameters ${l}_{0}$, ${l}_{1}$, ${l}_{0\nu }$ and ${l}_{1\nu }$ are defined in Eq. (31).

By substituting Eqs. (57) and (58) in to Eq. (56), the governing equations can be expressed in terms of the displacements as

$$\begin{array}{l}\frac{2\mu h}{1-\nu }\left(1+\frac{{l}_{0\nu }}{h}\right){u}_{,xx}=0, \\ \frac{8}{15}\mu h\left[\left(1+\frac{15}{4}\frac{{l}_{1}}{h}\right){w}_{,xx}+{\psi }_{,x}\right] \\ -\frac{\mu {h}^{3}}{126\left(1-\nu \right)}\left[\left(1+7\frac{{l}_{0\nu }}{h}-\frac{21}{5}\frac{{l}_{1\nu }}{h}\right){w}_{,xxxx}-\frac{16}{5}\left(1+\frac{35}{8}\frac{{l}_{0\nu }}{h}\right){\psi }_{,xxx}\right]+p=0,\\ \mu h\left({w}_{,x}+\psi \right)+\frac{\mu {h}^{3}}{21\left(1-\nu \right)}\left[\left(1+7\frac{{l}_{0\nu }}{h}-\frac{21}{4}\frac{{l}_{1\nu }}{h}\right){w}_{,xxx}-\frac{17}{4}\left(1+\frac{35}{17}\frac{{l}_{0\nu }}{h}\right){\psi }_{,xx}\right]=0.\end{array}$$

(59)

For the static bending with the simply supported boundary conditions, the displacements Eq. (53) can be written as

$$u={U}^{**}\mathrm{cos}{q}_{n}x, w={W}^{**}\mathrm{sin}{q}_{n}x, \psi ={\Psi }^{**}\mathrm{cos}{q}_{n}x,$$

(60)

where ${q}_{n}=n\pi /l$ with n being a positive integral, and ${U}^{**}$, ${W}^{**}$ and ${\Psi }^{**}$ indicate maximal values of the displacement components.

By substituting Eq. (60) and the transverse loading $p={P}_{0}\mathrm{sin}{q}_{n}x$ to Eq. (59), the first Eq. in Eq. (59) gives ${U}^{**}=0$, and ${W}^{**}$ and ${\Psi }^{**}$ can be determined by the relations

$${W}^{**}=\frac{{a}_{22}^{**}}{{a}_{11}^{**}{a}_{22}^{**}-{a}_{12}^{**}{a}_{21}^{**}}{P}_{0}, {\Psi }^{**}=-\frac{{a}_{21}^{**}}{{a}_{11}^{**}{a}_{22}^{**}-{a}_{12}^{**}{a}_{21}^{**}}{P}_{0} ,$$

(61)

where

$$\begin{array}{l}{a}_{11}^{**}=\frac{8}{15}\mu h\left(1+\frac{15}{4}\frac{{l}_{1}}{h}\right){q}_{n}^{2}+\frac{\mu {h}^{3}}{126\left(1-\nu \right)}\left(1+7\frac{{l}_{0\nu }}{h}-\frac{21}{5}\frac{{l}_{1\nu }}{h}\right){q}_{n}^{4}, \\ {a}_{12}^{**}=\frac{8}{15}\mu h{q}_{n}-\frac{8\mu {h}^{3}}{315\left(1-\nu \right)}\left(1+\frac{35}{8}\frac{{l}_{0\nu }}{h}\right){q}_{n}^{3}, \\ {a}_{21}^{**}=\frac{8}{15}\mu h{q}_{n}-\frac{8\mu {h}^{3}}{315\left(1-\nu \right)}\left(1+7\frac{{l}_{0\nu }}{h}-\frac{21}{4}\frac{{l}_{1\nu }}{h}\right){q}_{n}^{3}, \\ {a}_{22}^{**}=\frac{8}{15}\mu h+\frac{34\mu {h}^{3}}{315\left(1-\nu \right)}\left(1+\frac{35}{17}\frac{{l}_{0\nu }}{h}\right){q}_{n}^{2},\end{array}$$

(62)

Define $s=l/h$ to be the length-to-thickness ratio of the beam, and considering $n=1$ for ${q}_{n}$, Eq. (62) can be rewritten as

$${a}_{11}^{**}=\frac{\mu }{h}{\bar{a}}_{11}^{**}, {a}_{12}^{**}=\mu {\bar{a}}_{12}^{**}, {a}_{21}^{**}=\mu {\bar{a}}_{21}^{**}, {a}_{22}^{**}=\mu h{\bar{a}}_{22}^{**},$$

(63)

where

$$\begin{array}{l}{\bar{a}}_{11}^{**}=\frac{8}{15}\left(1+\frac{15}{4}\frac{{l}_{1}}{h}\right){\left(\frac{\pi }{s}\right)}^{2}+\frac{1}{126\left(1-\nu \right)}\left(1+7\frac{{l}_{0\nu }}{h}-\frac{21}{5}\frac{{l}_{1\nu }}{h}\right){\left(\frac{\pi }{s}\right)}^{4}, \\ {\bar{a}}_{12}^{**}=\frac{8}{15}\left(\frac{\pi }{s}\right)-\frac{8}{315\left(1-\nu \right)}\left(1+\frac{35}{8}\frac{{l}_{0\nu }}{h}\right){\left(\frac{\pi }{s}\right)}^{3}, \\ {\bar{a}}_{21}^{**}=\frac{8}{15}\left(\frac{\pi }{s}\right)-\frac{8}{315\left(1-\nu \right)}\left(1+7\frac{{l}_{0\nu }}{h}-\frac{21}{4}\frac{{l}_{1\nu }}{h}\right){\left(\frac{\pi }{s}\right)}^{3}, \\ {\bar{a}}_{22}^{**}=\frac{8}{15}+\frac{34}{315\left(1-\nu \right)}\left(1+\frac{35}{17}\frac{{l}_{0\nu }}{h}\right){\left(\frac{\pi }{s}\right)}^{2},\end{array}$$

(64)

are non-dimensional values. The displacement amplitudes in Eq. (61) can thus be expressed in terms of the non-dimensional form ${\bar{W}}^{**}$ and ${\bar{\Psi }}^{**}$ as

$$W^{{**}} = \frac{{hP_{0} }}{\mu }\bar{W}^{{**}} ,\Psi ^{{**}} = \frac{{P_{0} }}{\mu }\bar{\Psi }^{{**}} ,$$

(65)

where

$$\bar{W}^{{**}} = \frac{{\bar{a}_{{22}}^{{**}} }}{{\bar{a}_{{11}}^{{**}} \bar{a}_{{22}}^{{**}} - \bar{a}_{{12}}^{{**}} \bar{a}_{{21}}^{{**}} }},\bar{\Psi }^{{**}} = - \frac{{\bar{a}_{{21}}^{{**}} }}{{\bar{a}_{{11}}^{{**}} \bar{a}_{{22}}^{{**}} - \bar{a}_{{12}}^{{**}} \bar{a}_{{21}}^{{**}} }}.$$

(66)