## Abstract

Although the cantilever beam has been widely used as a sensor to measure various physical quantities, important issues such as how residual stress affects its bending stiffness and what are the underlying physical origins have not been fully understood. We perform both theoretical analyses and finite-element simulations to demonstrate for the first time that without changing the material tangent stiffness, residual stress within the beam can directly influence the bending stiffness of the beam. This direct influence arises from two origins: geometry nonlinearity and Poisson’s ratio effect. For a cantilever beam with adsorbed macromolecules on its surfaces, we find that longer macromolecular chains have lower normal stiffness and larger intermolecular forces, which makes the effect of the residual stress more pronounced. The excellent agreement between our theoretical predictions and finite-element calculation results validate our analysis. The present work provides an important framework for improving the sensitivity of a cantilever beam as a sensor.

## 1. Introduction

By detecting the resonant frequency change, micro- or nano-cantilever is one of the few effective sensors over the small scale to measure physical quantities, such as the mass and the amount of adsorbed biomolecules (Fritz *et al.* 2000; Raiteri *et al.* 2001; Lavrik *et al.* 2004; Saya *et al.* 2004; Eom *et al.* 2007; Zhang & Shan 2008; Wang & Feng 2009). Experimental, theoretical and computational studies on nanomechanical resonators as well as their applications in chemical/biological sensing and detecting were summarized and discussed by Eom *et al.* (2011) in their recent review paper. It was believed in previous studies that the mass change and the material stiffness change due to the adsorption affect the resonant frequency of cantilever (Chun *et al.* 2007). However, the role of residual surface stress on the frequency is still controversial. Gurtin *et al.* (1976) divided the surface stress into the strain-dependent part and the strain-independent residual part, and the latter is the surface stress in the prestressed configuration before bending or vibration. According to their linear elastic analysis based on a classical continuum beam model, they claimed that the strain-independent residual surface stress (SIRSS) does not influence the resonant frequency. A similar conclusion was also reached by Lu *et al.* (2005) with more elaborate investigation on various cases.

Inconsistency between experimental results and predictions from the classical linear elastic beam theory on the role of residual stress has attracted a great deal of effort to explore its possible mechanisms (Raiteri *et al.* 2001; Lavrik *et al.* 2004). The atomistic feature and realistic interatomic interaction in the surface layer were adopted in some studies (Huang *et al.* 2006; Park & Klein 2008; Zhang *et al.* 2008*a*,*b*; Yi & Duan 2009), in which the effects of the surface elasticity and the SIRSS on the resonant frequency were coupled owing to the nonlinear surface stress–strain relation. These studies concluded that the SIRSS can only affect the frequency indirectly, i.e. by changing the surface material tangent stiffness. He & Lilley (2008) investigated the surface stress effect with different boundary conditions. Wang & Feng (2007) noted that the residual surface tension can generate a distributed loading on the curved surface, and thus studied its effect on natural frequency of the beam. It was pointed out by some previous studies (Huang & Wang 2006; Huang & Sun 2007; Wang *et al.* 2010) that for a curved surface, the surface tension affects the elastic behaviour if the reference and the current configurations are discriminated, and based on this, the effective Young’s modulus of Al nanowires was derived. The finite deformation effect of the surface stress in an energy term was also investigated by Ru (2010). Song *et al.* (2011) developed a new formulation of the Euler–Bernoulli beam model for nanowires with geometrical nonlinearity considered, and demonstrated that surface and surface-induced initial stress have significant effects on size- and boundary-condition-dependent mechanical properties of nanowires. Lachut & Sader (2007, 2009) conducted a full three-dimensional finite element analysis, and found that the effect of the surface stress is related to Poisson’s ratio *ν*, and *ν*=0 leads to a negligible effect on the stiffness.

From these previous studies, we found that two important questions have not been answered convincingly: can the residual stress directly affect the bending stiffness without changing the material tangent stiffness? What are the real underlying mechanisms responsible for the residual stress effect? In order to address these questions, a systematic study is carried out. Several examples are introduced and analysed one by one in the following sections to demonstrate that the residual stress, or SIRSS, can directly affect bending stiffness without changing the material tangent stiffness. Analytical formulae for the bending stiffness of cantilevers with residual stress included are derived for each case. Section 2 presents two simple idealized examples. Two more realistic cantilevers are investigated in §§3 and 4, respectively. The conclusions are summarized in §5.

## 2. Two simple examples for understanding the effect of residual stress

Under the framework of linear elastic mechanics, the self-equilibrium residual stress does not affect any stiffness. However, it is widely noted that the residual tensile stress in the string of an instrument can change the tone, i.e. the lateral stiffness and frequency easily. As demonstrated in the following examples, this inconsistency is because that the linear elastic mechanics is incapable of accounting for geometrical nonlinearity and differentiating the configurations before and after deformation.

### (a) Example 1: prestressed T-shape truss

Figure 1 shows a T-shape truss with residual tensile force *f*_{0} in two horizontal bars and no internal residual force in the vertical bar. Obviously, the structure is in equilibrium state. To determine the vertical stiffness at point *A* of the structure, it is assumed that point *A* undergoes an infinitesimal displacement *δ* vertically, then the corresponding energy change in bar *AC* is
2.1and the energy change in bars *BA* and *AD* is
2.2where *k*_{1} and *k*_{2} are the stiffness of bars *BA* and *AD*, and *AC* respectively; *l*_{0} is the bar length before deformation; *l* is the bar length of bar *BA* and *AD* after deformation. Differentiating the total energy of the structure twice with respect to *δ*, the vertical stiffness of point *A* can be determined as
2.3Here, we would like to emphasize that this structural stiffness depends not only on the material stiffness, but also on the residual force *f*_{0}. It is also interesting to note that although the relative extension of bar *BA* (or *AD*) is , one order smaller than that of bar *AC*, i.e. *δ*/*l*_{0}, it can still affect the lateral stiffness due to the finite residual internal force. Therefore, one should not ignore the small quantities from geometrical nonlinearity in the analysis if the residual stress is present.

### (b) Example 2: truss cantilever beam

Figure 2*a* shows a relatively more realistic example, a cantilever composed of triangle-lattice truss before bending. The length of bars is *l*_{0}. The surface bars have a stiffness of *k*_{s} and residual internal force *f*_{s0}. To simplify the analysis, the inner bars are assumed to have an infinite stiffness, i.e. . Assume that the whole structure is under pure bending; one representative unit in the deformed configuration is schematically shown in figure 2*b*. *θ*_{d} is the bending angle of the unit, and the lengths of upper and lower surface bars can be obtained as
2.4and
2.5Owing to the infinite rigidity of the inner bars, the strain energy can only be stored in the upper and lower surface bars, and the corresponding energy change due to the deformation is
2.6and
2.7Then the bending stiffness can be determined analogously as
2.8where *θ*=*θ*_{d}/*l*_{0} is the bending angle per unit length.

Equation (2.8) clearly indicates that the tensile surface stress can reduce the bending stiffness. We may understand this conclusion with a more detailed analysis. From equations (2.4)–(2.7), the forces in the upper and lower surface bars can be determined as
2.9and
2.10It is easy to know that in computing the total moment with respect to point *A* (the centre point in figure 2*b*), if the arm lengths of *f*_{u} and *f*_{l} are different, the contributions of *f*_{s0} from upper and lower bars to the moment will not be cancelled. This is just the case when a beam is under bending, in which the upper surface bar moves closer to point *A* while the lower one moves slightly further away. It is noted that this truss example is also a typical tensegrity structure, and the prestress will change the stiffness according to the corresponding theory (Juan & Tur 2008).

## 3. The effect of residual stress on the bending stiffness of continuum cantilever beam

Next, we will move on to a more general problem. Figure 3*a* shows a continuum cantilever beam of thickness *h* with its Young’s modulus *E*, Poisson’s ratio *ν* and residual stress *σ*_{0} varying along thickness. For the non-bent self-equilibrium configuration, the non-vanishing components of *σ*_{0} satisfy the following force and moment balance equations:
3.1
3.2
3.3
and
3.4

In this section, finite deformation analysis is adopted to accurately account for the geometrical nonlinearity. The non-bent self-equilibrium configuration is chosen as the reference configuration, and the corresponding engineering stress and strain (or Biot stress and strain) *σ* and *ε* are used. *z* is a coordinate defined in this reference configuration. When this beam is under pure bending of an infinitesimal bending angle per unit length *θ*, the Cauchy stress for this bent configuration is introduced to construct the force balance equation. The free-body diagram for an upper slab of the beam under pure bending is shown in figure 3*b*. The width of the slab at the reference configuration is taken as 1, and becomes 1+*ε*_{y}(*z*) for the bent configuration. The curvature radius of the bottom surface of the slab
3.5The axial force of the slab
3.6From the free-body diagram of the bent configuration, we can get the following vertical balance condition:
3.7Noting the relation between Cauchy stress and engineering stress (Popov & Balan 1998)
3.8together with equations (3.5) and (3.6), equation (3.7) becomes
3.9It can be found later in this paper that the second- and higher-order terms of *θ* do not contribute to the bending stiffness when differentiation with respect to *θ* is performed.

Obviously, the shear stress and shear strain components are zeroes in pure bending cases, and the constitutive equations are
3.10
3.11
and
3.12We may further transform these three equations into
3.13
3.14
and
3.15Under pure bending, the normal plane to the middle plane still remains plane due to the symmetry, and the following kinetic relation can be obtained based on figure 4:
3.16where *u*_{z}(0)=0 is assumed without losing generality. Differentiating both sides of the equation (3.16) yields
3.17Considering that *ε*_{z} is of the same order of *θ*, the solution to equation (3.17) can be expressed as
3.18where is the strain of the middle plane.

Only the plane strain case is studied in this paper (the analysis in other cases are similar), i.e. 3.19is independent of coordinates.

Up to now, according to equations (3.9)–(3.19), all stress and strain components can be expressed in terms of and the middle plane strain . They can be determined from the force balance equations of bending configuration
3.20and
3.21We then can obtain
3.22and
3.23where
3.24and
3.25and
3.26
3.27
3.28
and
3.29The bending moment can be computed as
3.30where
3.31is the displacement along the *z*-direction.

It should be emphasized that different from classical linear elastic analysis on a beam, equation (3.30) is more accurate by accounting for the change of moment arm length during the deformation. Two effects are responsible for this change: one is Poisson’s effect, and the other is from the surface-stress-induced normal pressure that appears in a configuration with a non-flat surface, as demonstrated in equation (3.9) and figure 3*b*.

Finally, the bending stiffness can be determined as
3.32This is a general formula for the beam with non-uniform material constants and residual stress, which apparently depends on the residual stress *σ*_{x0}. In the following, two special cases are studied and discussed.

### (a) Symmetric case: a cantilever beam with symmetric upper and lower surface layers

As shown in figure 5, the material constant of the beam is uniform, and the residual stress has symmetrical distributions as
3.33where *σ*_{s0} is the residual stress and *h*_{s} is the thickness of the surface layers. According to equations (3.24)–(3.29),
3.34From equation (3.32), the corresponding bending stiffness becomes
3.35

### (b) Asymmetric case: a cantilever beam with only an upper surface layer

If only the upper surface layer is resulted with residual stress, as shown in figure 6, then the residual stress distribution can be assumed to be 3.36where 3.37and 3.38Substituting equation (3.36) into equation (3.32) yields the corresponding bending stiffness 3.39

### (c) Discussions and validations by finite-element simulations

Both equations (3.35) and (3.39) confirm the direct influence of residual stress on the bending stiffness. It should be pointed out that all derivations in this paper are performed by a mathematical software MAPLE. Moreover, for the symmetric case, we also carry out a finite-element simulation to validate the theoretical predictions.

Considering the details in the earlier-mentioned analytical analysis, numerical finite deformation analysis has been performed with the commercial FEM software ABAQUS. In all the simulations, the thicknesses of upper and lower surface layers are both fixed at *h*_{s}/*h*=0.25. Poisson’s ratio is taken to be *ν*=0.3. The residual stress is assigned as *σ*_{s0}/*E*=0.1, which is large enough to show the effects on bending stiffness. When a bending angle per unit length *θ*=10^{−5} is imposed to the beam, stresses *σ*_{x} and *σ*_{z} and displacement *u*_{z} are then studied in sequence in comparison with their counterparts in the analytical analysis. Their variations along thickness are plotted side by side in figure 7*a*–*c*. For each variable evaluated, the two solutions from the two approaches match so well that the differences between them cannot be seen at all in the three graphs. Since stress *σ*_{x} and displacement *u*_{z} are the two key factors in computing the moment (see equation (3.26)), the excellent agreements imply that the analytical analysis is well formulated and the key nonlinear geometry deformation features pertinent to bending stiffness are captured.

In most cases, since *σ*_{s0}≪*E* and *h*_{s}≪*h*, the second and third terms in equations (3.35) and (3.39) are much smaller than the classical first term, and the residual stress has only a slight effect on the bending stiffness. However, for the cantilever beam with adsorbed macromolecules, this effect may become prominent, as discussed in §4.

## 4. The effect of residual stress on the bending stiffness of cantilever beam with macromolecules adsorbed on surfaces

Figure 8*a* schematically shows a non-bent cantilever beam with macromolecules adsorbed on surfaces with spacings *l*_{0} and *b* along the *x*- and *y*-directions, respectively. The macromolecules are represented by vertical springs with spring constant *k*_{n} and free length *a*_{0}. The interactions between neighbouring macromolecules are simplified into horizontal springs with spring constant *k*_{s} and residual force *f*_{s0}. The non-bent beam is assumed to have uniform material constants and residual stress *σ*_{x0}. It should be pointed out that the following study is not limited to the linear elastic interaction among macromolecules, even though the linear spring model is adopted. For a nonlinear intra/inter-molecule interaction (normally the case in reality), the linear springs can be taken to characterize the linear relation between the small increments of force and deformation among the macromolecules around the state of interest. This is essentially a typical approach using a linear system to represent and study a nonlinear system near specific stationary states.

The self-equilibrium condition of whole beam yields the relation between *f*_{s0} and *σ*_{x0} as
4.1where *h* is the height of the cantilever beam.

We then study the force and strain of the pure bending configuration as shown in figure 8*b*. The upper and lower spring forces are related to the spring lengths by
4.2and
4.3where *l*_{u} and *l*_{l} are the upper and lower horizontal spring lengths, respectively.

For the whole structure, the force balances along *x*- and *y*-directions require
4.4and
4.5The force balances at points *A* and *B* in figure 8*b* yield
4.6and
4.7where *a*_{u} and *a*_{l} are the upper and lower vertical spring lengths, *θ*_{d} is the bending angle of the unit.

The pressure on the upper surface of the beam due to the adsorbed macromolecules can be estimated as
4.8Similar to equation (3.9), the normal stress along the *z* direction is
4.9The following kinetic relations provide all the necessary connections among various geometry parameters and deformation variables,
4.10
4.11
4.12
4.13
4.14
4.15
4.16
and
4.17where *R*_{m}, *R*_{u} and *R*_{l} are curvature radii of the middle plane, the upper and the lower surfaces of the beam shown in figure 8*b*, respectively.

Since the upper and lower surface spring forces also contribute to the moment, the moment becomes 4.18Substituting equations (4.1)–(4.17) into equation (4.18), the bending stiffness can be determined as 4.19

If the adsorption occurs only on the upper surface, as shown in figure 9, the residual stress in the beam is not a constant owing to the asymmetry, and can be assumed as 4.20Using the similar approach and the following force balance equation 4.21the bending stiffness for one-side adsorption is 4.22

From equations (4.19) and (4.22), we can see that, in the case of macromolecules adsorption, the normal stiffness *k*_{n} is usually very small; therefore the effect of residual force *f*_{s0} on the bending stiffness becomes significant. In particular, we will discuss the influence from the last term of equation (4.19). The relative deviation due to this term is given by the ratio between the last term and the first traditional term 24*l*_{0}*f*^{2}_{s0}(1−*ν*^{2})/(*k*_{n}*bEh*^{3}). It is reasonable to assume that *k*_{n} is inversely proportional to the macromolecular chain length 2*a*_{0} because a longer macromolecular has lower stiffness, while the intermolecular force *f*_{s0} is proportional to 2*a*_{0}. Totally, the relative deviation will increase with power 3 of the macromolecular chain length, and decrease with power 3 of the cantilever thickness *h*. Therefore, the higher sensitivity can be achieved by using the thinner cantilever to detect macromolecules with longer chains. In addition, it is also worth discussing the influence of the horizontal spring constant *k*_{s}, i.e. the second term in equation (4.19). Different from *k*_{n}, *k*_{s} represents the interaction between neighbouring macromolecules, and it should be in proportion to the contour length (i.e. the size of the molecule). The contribution of the *k*_{s} term to the overall bending stiffness can be estimated by the ratio of the second term over the first term of equation (4.19). As the spacing in the *x*-direction *l*_{0} is roughly equal to that in the *y*-direction (i.e. *b*) and the free-length *a*_{0} is much smaller than the beam height *h*, the ratio can be reduced to
*Eh*, the transversal stiffness of the solid beam, can be expected to be several orders larger than *k*_{s} in magnitude, which suggests through the reduced ratio formulas that the effect of the *k*_{s}-related term can be neglected.

We further estimate the magnitude of this relative deviation. First, assume that adsorbed macromolecules have the same spacing along *x*- and *y*-directions, i.e. *l*_{0}=*b*. Other parameters are taken from the following typical values, *E*=100 GPa, *ν*=0.3, *h*=100 nm. If the contour length of the macromolecules is taken as 10 nm, the normal stiffness *k*_{n} can be estimated as 1.3×10^{−5} N m^{−1} based on equation (2.2) in Bao’s review article (Bao 2002), and the intermolecular force *f*_{s0} can be estimated from Mao *et al.*’s study (2010) as of the order of several nanonewtons. Then the relative deviation due to the last term of equation (4.19) might be larger than 10 per cent (when *f*_{s0}>2.5 nN). It is interesting to note that the normal stiffness *k*_{n} is inversely proportional to the contour length of the macromolecules, while the intermolecular force *f*_{s0} is proportional to this length. Totally the deviation of the last term of equation (4.19), 2*l*_{0}*f*^{2}_{s0}/*k*_{n}*b*, is proportional to the cube of the molecular contour length. Therefore, the longer the macromolecule, the more prominent the effect of residual stress on the bending stiffness, which may suggest that nano-cantilevers are more suitable for detecting larger molecule adsorption. Since a larger quantity of adsorbed molecules generally indicates stronger interaction between neighbouring molecules, i.e. larger residual force *f*_{s0}, the formula 2*l*_{0}*f*^{2}_{s0}/*k*_{n}*b* implies that a larger quantity of molecules adsorbed would also make the deviation larger. Typically, there are two modes to sense adsorbed molecules with a cantilever-based sensor, i.e. by measuring its resonant frequency change or static deflection (Eom *et al.* 2011). The bending stiffness of a beam is directly related to both its resonant frequency and static deflection, and therefore the analysis here would benefit all sensing techniques based on the two typical modes. Although the studies here just focus on rectangular cross section beams, similar behaviour and conclusions can be expected for axisymmetric cross section beams, for example circular beams.

## 5. Conclusions

In this paper, the direct effect of residual stress on the bending stiffness of various cantilevers has been investigated analytically and numerically, and the following conclusions have been reached.

— The bending stiffness of a cantilever beam with residual stress is actually a structural stiffness, which not only relies on the material stiffness, but also includes the direct contribution from residual stress.

— To correctly reflect the influence of the residual stress, the geometrical nonlinearity of the finite deformation must be taken into account, and the bending moment should be computed based on the configuration after deformation.

— For a continuum cantilever under bending, the normal pressure induced by residual surface stress and Poisson’s effect lead to different moment arm lengths of the upper and lower surface layers, through which the residual stress contributes to the bending stiffness.

— For a cantilever beam with adsorbed macromolecules, it is found that longer macromolecular chains, with lower normal stiffness and larger intermolecular forces, would make the effect of the residual stress more significant.

It is anticipated that the direct influence of residual stress on bending stiffness revealed by the present work, together with the change of the mass and the material stiffness of the surface layer, may help improve understandings of the micro- or nano-cantilever behaviours and potentially enhance their performance. However, with respect to macromolecular adsorption, our study here is quite preliminary and many details at atomic level, such as thermal effect, van der Waals interaction, chemical forces, etc. are simplified or ignored. More realistic models and atomistic simulations are needed for further studies on macromolecular sensing with nano-cantilever beams in the future.

## Acknowledgements

B.L. acknowledges the support from National Natural Science Foundation of China (grant nos. 11090334, 10732050, 90816006 and 10820101048), and National Basic Research Programme of China (973 Programme) grant nos. 2007CB936803 and 2010CB832701. We also gratefully acknowledge the support from Tsinghua University Initiative Scientific Research Program (no. 2011Z02173).

- Received November 5, 2011.
- Accepted February 29, 2012.

- This journal is © 2012 The Royal Society