A new displacement-based framework for non-local Timoshenko beams

Abstract

In this paper, a new theoretical framework is presented for modeling non-locality in shear deformable beams. The driving idea is to represent non-local effects as long-range volume forces and moments, exchanged by non-adjacent beam segments as a result of their relative motion described in terms of pure deformation modes of the beam. The use of these generalized measures of relative motion allows constructing an equivalent mechanical model of non-local effects. Specifically, long-range volume forces and moments are associated with three spring-like connections acting in parallel between couples of non-adjacent beam segments, and separately accounting for pure axial, pure bending and pure shear deformation modes. The variational consistency of the proposed non-local beam model is demonstrated by minimization of an appropriate total potential energy functional. Numerical results concerning the static behavior for different boundary and loading conditions are presented. It is shown that the proposed non-local beam model is able to capture experimental data on the static deflection of micro-beams, available in the literature.

Appendices

Appendix 1

In this Appendix, the derivation of identity (22) is presented. To this aim, first Eq. (22) is rewritten in the following form:

$$ \begin{aligned} \int_{0}^{L} {\int_{0}^{L} {\left[ {q_{x} \left( {x,\xi } \right)u\left( x \right) + q_{z} \left( {x,\xi } \right)v\left( x \right) + q_{\varphi \varphi} \left( {x,\xi } \right)\varphi \left( x \right) + q_{\varphi z} \left( {x,\xi } \right)\varphi \left( x \right)} \right]{\text{d}}x{\text{d}}\xi } } \hfill \\ = - \frac{1}{2}\int_{0}^{L} {\int_{0}^{L} {\left[ {q_{x} \left( {x,\xi } \right)\eta \left( {x,\xi } \right) + q_{\varphi \varphi } \left( {x,\xi } \right)\theta \left( {x,\xi } \right) + q_{\varphi z} \left( {x,\xi } \right)\psi \left( {x,\xi } \right)} \right]{\text{d}}x{\text{d}}\xi } } \hfill \\ \end{aligned} $$

(37)

where the definitions of the vectors ${\mathbf{u}}\left( x \right),{\tilde{\mathbf{q}}}\left( {x,\xi } \right),{\mathbf{e}}\left( {x,\xi } \right)$ and ${\mathbf{q}}\left( {x,\xi } \right)$ have been introduced [see Eqs. (20a, c), (23a, b)]. Then, Eq. (37) holds if the following identities are fulfilled:

$$ \int_{0}^{L} {\int_{0}^{L} {q_{x} \left( {x,\xi } \right)u\left( x \right){\text{d}}x{\text{d}}\xi } } = - \frac{1}{2}\int_{0}^{L} {\int_{0}^{L} {q_{x} \left( {x,\xi } \right)\eta \left( {x,\xi } \right){\text{d}}x{\text{d}}\xi } } ; $$

(38)

$$ \int_{0}^{L} {\int_{0}^{L} {q_{\varphi \varphi} \left( {x,\xi } \right)\varphi \left( x \right){\text{d}}x{\text{d}}\xi } } = - \frac{1}{2}\int_{0}^{L} {\int_{0}^{L} {q_{\varphi \varphi } \left( {x,\xi } \right)\theta \left( {x,\xi } \right){\text{d}}x{\text{d}}\xi } } ; $$

(39)

$$ \int_{0}^{L} {\int_{0}^{L} {\left[ {q_{z} \left( {x,\xi } \right)v\left( x \right) + q_{\varphi z} \left( {x,\xi } \right)\varphi \left( x \right)} \right]{\text{d}}x{\text{d}}\xi } } = - \frac{1}{2}\int_{0}^{L} {\int_{0}^{L} {q_{\varphi z} \left( {x,\xi } \right)\psi \left( {x,\xi } \right){\text{d}}x{\text{d}}\xi } } . $$

(40)

To prove Eq. (38), it is observed that due to the symmetry of g _x (x, ξ), one may write:

$$ \int_{0}^{L} {\int_{0}^{L} {q_{x} \left( {x,\xi } \right)u\left( x \right){\text{d}}x{\text{d}}\xi } } = - \int_{0}^{L} {\int_{0}^{L} {q_{x} \left( {x,\xi } \right)u\left( \xi \right){\text{d}}x{\text{d}}\xi } } . $$

(41)

Adding to both sides of Eq. (41) the term $\int_{0}^{L} {\int_{0}^{L} {q_{x} \left( {x,\xi } \right)u\left( x \right){\text{d}}x{\text{d}}\xi } }$, Eq. (38) can be obtained.

Equation (39) can be derived following a similar reasoning. Indeed, the symmetry of g _φ (x, ξ) allows us to write:

$$ \int_{0}^{L} {\int_{0}^{L} {q_{\varphi \varphi} \left( {x,\xi } \right)\varphi \left( x \right){\text{dxd}}\xi } } = - \int_{0}^{L} {\int_{0}^{L} {q_{\varphi \varphi} \left( {x,\xi } \right)\varphi \left( \xi \right){\text{d}}x{\text{d}}\xi } } . $$

(42)

Then, adding to both sides of the previous equation the term $\int_{0}^{L} {\int_{0}^{L} {q_{\varphi } \left( {x,\xi } \right)\varphi \left( x \right){\text{d}}x{\text{d}}\xi } }$, Eq. (39) is readily obtained.

Equation (40) can be split in the following two identities

$$ \int_{0}^{L} {\int_{0}^{L} {q_{z} \left( {x,\xi } \right)v\left( x \right){\text{d}}x{\text{d}}\xi } } = - \frac{1}{2}\int_{0}^{L} {\int_{0}^{L} {q_{\varphi z} \left( {x,\xi } \right)\left[ {2\left( {\frac{v\left( \xi \right) - v\left( x \right)}{\xi - x}} \right)} \right]{\text{d}}x{\text{d}}\xi } } ; $$

(43)

$$ \int_{0}^{L} {\int_{0}^{L} {q_{\varphi z} \left( {x,\xi } \right)\varphi \left( x \right){\text{d}}x{\text{d}}\xi } } = \frac{1}{2}\int_{0}^{L} {\int_{0}^{L} {q_{\varphi z} \left( {x,\xi } \right)\left[ {\varphi \left( x \right) +\, \varphi \left( \xi \right)} \right]{\text{d}}x{\text{d}}\xi } }. $$

(44)

To prove Eq. (43), let us first substitute in this equation the definitions (13b) and (14b) of q _z (x, ξ) and q _φz (x, ξ), respectively:

$$ \begin{aligned} \int_{0}^{L} {\int_{0}^{L} {2\frac{{\text{sgn}}\left( {\xi - x} \right)}{\left| {\xi - x}\right|} g_{z} \left( {x,\xi } \right)\psi \left( {x,\xi } \right)v\left( x \right){\text{d}}x{\text{d}}\xi } } \hfill \\ &=& - \frac{1}{2}\int_{0}^{L} {\int_{0}^{L} {g_{z} \left( {x,\xi } \right)\psi \left( {x,\xi } \right)\left[ {2\left( {\frac{v\left( \xi \right) - v\left( x \right)}{\xi - x}} \right)} \right]{\text{d}}x{\text{d}}\xi } }. \hfill \\ \end{aligned} $$

(45)

Due to the symmetry of the attenuation function g _z (x, ξ), the following relationship holds:

$$ \begin{aligned} \int_{0}^{L} {\int_{0}^{L} {2\frac{{\text{sgn}}\left( {\xi - x} \right)}{\left| {\xi - x}\right|} g_{z} \left( {x,\xi } \right)\left[ {2\left( {\frac{v\left( \xi \right) - v\left( x \right)}{\xi - x}} \right)} \right]v\left( x \right){\text{d}}x{\text{d}}\xi } } \hfill \\ \quad= - \int_{0}^{L} {\int_{0}^{L} { 2\frac{{\text{sgn}}\left( {\xi - x} \right)}{\left| {\xi - x}\right|} g_{z} \left( {x,\xi } \right)\left[ {2\left( {\frac{v\left( \xi \right) - v\left( x \right)}{\xi - x}} \right)} \right]v\left( \xi \right) {\text{d}}x{\text{d}}\xi } }. \hfill \\ \end{aligned} $$

(46)

Then, adding to both sides of Eq. (46) the integral on the l.h.s. of this equation, the following identity is obtained:

$$ \begin{aligned} \int_{0}^{L} {\int_{0}^{L} { 2\frac{{\text{sgn}}\left( {\xi - x} \right)}{\left| {\xi - x}\right|} g_{z} \left( {x,\xi } \right)\left[ {2\left( {\frac{v\left( \xi \right) - v\left( x \right)}{\xi - x}} \right)} \right]v\left( x \right){\text{d}}x{\text{d}}\xi } } \hfill \\ \quad = - \frac{1}{2}\int_{0}^{L} {\int_{0}^{L} { g_{z} \left( {x,\xi } \right)\left[ {2\left( {\frac{v\left( \xi \right) - v\left( x \right)}{\xi - x}} \right)} \right]2 \left( {\frac{v\left( \xi \right) - v\left( x \right)}{\xi - x}} \right) {\text{d}}x{\text{d}}\xi } }. \hfill \\ \end{aligned} $$

(47)

where ${\text{sgn}}(\xi - x)/|\xi - x| = 1/(\xi - x)$ has been set on the r.h.s. Furthermore, the symmetry of the function g _z (x, ξ) allows us to write:

$$ \begin{aligned} \int_{0}^{L} {\int_{0}^{L} { 2\frac{{\text{sgn}}\left( {\xi - x} \right)}{\left| {\xi - x}\right|} g_{z} \left( {x,\xi } \right)\left[ {\varphi \left( x \right) + \varphi \left( \xi \right)} \right] v\left( x \right) {\text{d}}x{\text{d}}\xi } } \hfill \\ = - \int_{0}^{L} {\int_{0}^{L} { 2\frac{{\text{sgn}}\left( {\xi - x} \right)}{\left| {\xi - x}\right|} g_{z} \left( {x,\xi } \right)\left[ {\varphi \left( x \right) + \varphi \left( \xi \right)} \right]v\left( \xi \right) {\text{d}}x{\text{d}}\xi } }. \hfill \\ \end{aligned} $$

(48)

Then, adding to both sides of Eq. (48) the integral on the l.h.s., yields:

$$ \begin{aligned} \int_{0}^{L} {\int_{0}^{L} { 2\frac{{\text{sgn}}\left( {\xi - x} \right)}{\left| {\xi - x}\right|} g_{z} \left( {x,\xi } \right)\left[ {\varphi \left( x \right) + \varphi \left( \xi \right)} \right]v\left( x \right){\text{d}}x{\text{d}}\xi } } \hfill \\ = - \frac{1}{2}\int_{0}^{L} {\int_{0}^{L}{ g_{z} \left( {x,\xi } \right)\left[ {\varphi \left( x \right) + \varphi \left( \xi \right)} \right] 2\left( {\frac{v\left( \xi \right) - v\left( x \right)}{\xi - x}} \right) {\text{d}}x{\text{d}}\xi } } . \hfill \\ \end{aligned} $$

(49)

where, as in Eq. (47), ${\text{sgn}}(\xi - x)/|\xi - x| = 1/(\xi - x)$ has been set on the r.h.s. Subtracting both sides of Eqs. (47) and (49) and recalling the definition (10) of ψ(x, ξ), Eq. (43) is obtained.

Finally, to prove Eq. (44), it is observed that due to the symmetry of g _z (x, ξ), the following identity holds:

$$ \begin{aligned} \int_{0}^{L} {\int_{0}^{L} { g_{z} \left( {x,\xi } \right)\psi \left( {x,\xi } \right)\varphi \left( x \right){\text{d}}x{\text{d}}\xi } } \hfill \\ = \int_{0}^{L} {\int_{0}^{L} { g_{z} \left( {x,\xi } \right)\psi \left( {x,\xi } \right)\varphi \left( \xi \right){\text{d}}x{\text{d}}\xi } } . \hfill \\ \end{aligned} $$

(50)

Then, adding the l.h.s of Eq. (50) to both sides of the same equation and taking into account the definition (14b) of q _φz (x, ξ), Eq. (44) is obtained.

Appendix 2

The purpose of this Appendix is to briefly illustrate the behavior of the proposed non-local beam model when β ₁ < 1 is set in Eq. (4) for the local terms in Eqs. (32b, c). This is of interest to show, as discussed in the Conclusions, that the proposed non-local beam model is potentially capable of predicting non-local solutions that may be either stiffer or softer with respect to the classical local TM beam response.

A simply-supported beam is considered. Parameters and loading conditions are taken as in Sect. 5.1.1, while β ₁ in Eq. (4) is given the following values: β ₁ = 0.4; 0.6; 0.8. The dimensionless deflection v(x)/L versus the non-dimensional location x/L is reported in Fig. 13, for different values of the internal length l. For comparison, the classical local TM beam response, corresponding to β ₁ = 1 and no long-range resultants in Eqs. (32b, c), is also reported.

Fig. 13

Simply-supported beam: non-local and local dimensionless deflection for different β ₁ in Eq. (4) and l (internal length)

For a given value β ₁, it is seen that the non-local response may be either stiffer or softer than the classical local TM beam response, depending on the internal length l. This behavior can be explained considering that, for β ₁ < 1, the solution provided by the local terms only in Eqs. (32b, c), i.e. without long-range resultants, is obviously softer than the classical local TM beam response [corresponding to β ₁ = 1 and no long-range resultants in Eqs. (32b, c)]. The long-range resultants provide additional stiffness with respect to that of the local terms in Eqs. (32b, c), but such additional stiffness may not be enough to make the non-local response stiffer than the classical local TM beam response. In particular, Fig. 13 shows that the non-local response becomes progressively stiffer with increasing l, consistently with the fact a larger internal length l corresponds indeed to a larger amount of mutually interacting non-adjacent beam segments, with a consequent stiffening (see also comments on Figs. 4 and 5). The comments above explain also the fact that, in Fig. 13, a softer non-local response is obtained as parameter β ₁ decreases, for a given internal length l.

The rotation response is in accordance with the deflection response in Fig. 13 and pertinent results are not reported for conciseness. Likewise, results for a cantilever beam subjected to a tip load agree with those for the simply-supported beam and are omitted.