## Abstract

We show that the mixed-mode fracture/adhesion energy of an interface with periodically varying cohesive interactions generally depends on the size of the cohesive zone near the tip of a crack along the interface: it is equal to the average cohesive energy of the interface, if the cohesive zone size is much larger than the period of cohesive interaction but becomes the peak value of the local cohesive energy when the opposite is true. It is also interesting that the cohesive zone size can be strongly influenced by the geometry and velocity of the crack. As an example of geometry-constrained cohesive zone, we consider peeling of a thin film on substrate and show that the cohesive zone size under 90° peeling scales with the bending stiffness of the film, while that under 0° peeling scales with the tension stiffness of the film. As an example of a velocity-constrained cohesive zone, we consider crack propagation along an interfacial layer of weak molecular bonds joining two elastic media and show that the cohesive zone size can be altered by an order of magnitude over feasible regimes of crack velocity. These results suggest possible strategies to control fracture/adhesion strength of interfaces in both engineering and biological systems.

## 1. Introduction

The apparent fracture/adhesion energy of an interface with periodic cohesive interactions is of interest to the understanding of adhesion and friction properties of micro- or nano-patterned surfaces (Sun *et al*. 2006; Jiang *et al*. 2007). Our recent study (Chen *et al*. 2008*a*), hereafter referred to as Part I, has revealed that the apparent fracture/adhesion energy of such interfaces under mode I loading depends on the ratio between the period of cohesive interaction and the size of the cohesive zone near the tip of a crack along the interface: it is equal to the average cohesive energy of the interface, if the former is much smaller than the latter but becomes the peak value of the local cohesive energy when the opposite is true. It has also been shown in Part I that the cohesive zone size of a thin film under 90° peeling from a substrate can be highly constrained by the film thickness, thereby providing a feasible explanation of recent molecular dynamics simulations by Shi *et al*. (2005), which showed that the apparent adhesion energy of a single-stranded DNA (ss-DNA) on a graphite sheet is equal to the peak, rather than the average value of the interaction energy between the ss-DNA and the substrate.

In the current work, we generalize the study in Part I to interfaces under general mixed-mode loading. Our analysis will show that most of the conclusions from Part I remain valid under mixed-mode loading. We will further show that the cohesive zone size of a thin film under 90° peeling from a substrate scales with the bending stiffness of the film, while that under 0° peeling scales with the tension stiffness of the film. For fracture/adhesion in soft materials, the crack velocity can also influence the cohesive zone size. We will show that the cohesive zone size associated with fracture along an interfacial layer of weak molecular bonds joining two elastic media can be altered by an order of magnitude over feasible regimes of crack velocity. These findings are of fundamental importance in understanding adhesion and fracture phenomena at micro- and nano-scales, and suggest possible strategies to control fracture/adhesion strength of interfaces in both engineering and biological systems.

## 2. Mixed-mode fracture/adhesion energy of a semi-infinite crack along an interface joining two infinite elastic media with a locally varying cohesive interactions along the fracture path

Figure 1 shows a semi-infinite crack along an interface, which joins two semi-infinite elastic media. The crack is under far-field mixed-mode loading. The two solids are assumed to have the same elastic properties with Young's modulus *E* and Poisson's ratio *v*. The stress field far from the crack tip is characterized by a far-field stress intensity factor with the angle of mode-mixity defined as , where is the mode I component and the mode II component of *K*_{app}. Locally, the cohesive strength is assumed to vary periodically along the prospective fracture path. Following the analysis approach adopted in Part I, the apparent fracture/adhesion energy of the interface will be studied via a cohesive zone model (Dugdale 1960; Barenblatt 1962). For convenience, we adopt a moving coordinate *x* that follows the tip of the cohesive zone as it develops along the interface and a fixed coordinate *X* which coincides with *x* before the loading is applied.

To illustrate the basic concept without loss of generality, we consider the following simple cohesive law (Dugdale 1960):(2.1)where *σ*_{c} is the traction; *δ*_{c} the local separation of two points along the interface that initially coincide with each other; *σ*_{0} the strength and *δ*_{0} the extent of cohesive interaction. Here, the cohesive law is assumed to be isotropic so that it does not depend on the orientation of cohesive bonds with respect to the interface. The cohesive energy associated with the interaction law in equation (2.1) is *γ*=*σ*_{0}*δ*_{0}. We let the cohesive strength, *σ*_{0}, vary as a sinusoidal function of *X* while keeping the interaction range *δ*_{0} constant, i.e.(2.2)where is the average cohesive strength along the interface; *c*_{σ} is the amplitude; *l*_{p} the wavelength; and *α* the phase angle of the cohesive strength variation. For this interaction law, the average cohesive energy along the interface is . Upon loading, a cohesive zone develops at the crack tip to a length of *l*_{c} while the surface separation at the left end of the cohesive zone is equal to *δ*_{0}.

Under the far-field mixed-mode loading, the cohesive bonds are stretched in a direction inclined to the interface. As a first order approximation, we assume all cohesive bonds to be aligned with the same angle that is related to the mode mixity of far-filed loading by 90°−*θ*.

The surface separation at the beginning of the cohesive zone is(2.3)where the first term on the right-hand side is the contribution owing to the far-field applied loading and the second term from the traction within the cohesive zone. Following Dugdale (1960), the cohesive zone size is determined by eliminating the stress singularity at the right end of the cohesive zone, i.e.(2.4)where the equation (2.1) involving cos *θ* corresponds to the mode I stress intensity and the equation (2.2) involving sin *θ* corresponds to the mode II stress intensity. Crack propagation becomes imminent as the surface separation at the beginning of the cohesive zone reaches the maximal interaction range, i.e. . From equations (2.3) and (2.4), the cohesive zone size *l*_{c} can be determined as a function of the phase angle *α* according to(2.5)Here, we have introduced two normalized length scales(2.6)where(2.7)*E*′ being the effective modulus equal to *E* under plane stress and under plane strain. Note that corresponds to the ‘reference cohesive zone size’ of a semi-infinite crack with uniform cohesive interaction along the interface.

For a given *α*, we can determine from equation (2.5) which, together with equations (2.1)–(2.3), leads to the far-field stress intensity factor as a function of *α*, i.e. . The apparent fracture toughness/adhesion energy of the interface is defined as the maximum of the far-field energy release rate(2.8)over *α*, i.e. .

Comparing the above analysis to that of Part I, we see that the governing equations for the apparent fracture/adhesion energy of a mixed-mode crack are identical to those for a mode I crack. In fact, all conclusions for a mode I crack can be directly extended to the mixed-mode case. For example, similar to Part I, we can conclude that the far-field energy release rate should be equal to the local cohesive energy at the crack tip when the period of cohesive interaction is much larger than the cohesive zone size, but it should be equal to the average cohesive energy of the interface when the period of cohesive interaction is much smaller than the cohesive zone size. In addition, for general , the apparent fracture energy *G*_{c} under mixed-mode loading should follow the same empirical expression(2.9)where is given as a polynomial function of in Part I. These conclusions are valid as long as we adopt an isotropic cohesive law, i.e. when the cohesive energy is a material constant independent of the mode-mixity.

## 3. Constrained cohesive zones in thin films on substrates

In Part I, we have investigated the cohesive zone size of a thin film under 90° peeling from a substrate, which was shown to be highly constrained by the film thickness. Here, we will show that an alternative interpretation is that the cohesive zone size under 90° peeling scales with the bending stiffness of the film, and we will further show that the cohesive zone size under 0° peeling scales with the tension stiffness of the film.

Let us first consider the cohesive zone of a thin film under 0° peeling on a rigid surface, as shown in figure 2. According to Kendall's (1975) model, the critical peel-off force for 0° peeling is(3.1)where *H* is the film thickness and *γ* is the cohesive energy along the interface.

Assuming that the deformation under 0° peeling is dominantly mode II and adopting Dugdale's cohesive law of equation (2.1), a balance between the shear forces within the cohesive zone and the critical peel-off force yields(3.2)as the cohesive zone size.

Comparing equation (3.2) and the corresponding result for 90° peeling given in eqn (3.3) of Part I, we find that a general expression for the cohesive zone size is(3.3)where *γ* and denote the cohesive energy and cohesive strength of the interface,(3.4)is the characteristic stiffness of the dominant deformation mode within the cohesive zone, and(3.5)is a power exponent that can be found from a dimensional analysis.

Equations (3.3)–(3.5) show that the cohesive zone size of a thin film on substrate scales with the bending stiffness of the film with a power index of 1/4 under 90° peeling and scales with the tension stiffness of the film with a power index of 1/2 under 0° peeling. For very thin film, the bending stiffness of the film can be much smaller than the tension stiffness, hence, the cohesive zone size under 90° peeling can be much smaller than that under 0° peeling. This difference can have a significant effect on the peeling strength. Interestingly, equation (3.3) also gives the correct result for a semi-infinite crack with *K*=*E* and *P*=1, i.e. .

For an atomic-scale structure such as a ss-DNA molecule adhering on a graphite substrate (Shi *et al*. 2005), the thickness of the structure may not be well defined. On the other hand, the concepts of bending and tension stiffnesses can often be determined through simulations or experiments without the assumption of a continuum structure. Equation (3.3) allows us to estimate the cohesive zone size without having to define the thickness of the structure. Clearly, the cohesive zone size of a thin film on substrate should depend on the peeling angle.

## 4. Finite element simulations

To demonstrate the concepts discussed in §§2 and 3, we have performed finite element simulations of a thin film adhered on a rigid surface under 0° peeling.

### (a) An elastic thin film under 0° peeling from a rigid surface

We consider a thin film under 0° peeling from a rigid substrate. This problem is motivated by the adhesion of a thin spatula pad on the gecko's toe on a vertical surface (Tian *et al*. 2006). Chen *et al*. (2008*b*) have discussed the role of spatulae in robust attachment and easy detachment of the gecko. Owing to surface roughness effects (Persson & Gorb 2003), discrete contact sites could be a general feature of adhesion of a spatula pad on substrate. Here, we simulate the 0° peeling of a spatula pad adhering on a rigid surface with single or multiple contacting sites. The simulation results will be compared with the theoretical predictions in §2.

In our simulations, the parameters of the pad are taken from those of the spatula of *Gekko gecko* provided by Tian *et al*. (2006). A spatula pad is approximately 0.3 μm in length, 0.2 μm in width and 5 nm in thickness. Its elastic modulus is approximately 2 GPa. Figure 2 shows an elastic thin film of unit width representing a spatula pad adhering on a rigid surface.

Either Abaqus/Standard or Abaqus/Explicit can be employed to simulate the 0° peeling process. The cohesive energy within the contact region of the interface is taken to be 0.01 J m^{−2}. Plain-strain solid elements and two-dimensional cohesive elements are used to model the film and interfacial adhesive interaction, respectively. The constitutive response of the cohesive elements is defined in terms of the traction–separation law and damage of cohesive elements is initiated when the maximum nominal stress reaches 20 MPa, taken to be the strength of dry van der Walls interaction. A more general and realistic traction–separation law than that given in equation (2.1) is used in our numerical simulations, which has a linear part at the beginning, followed by a gradual decline of traction (damage). Damage evolution is based on an isotropic dependence of adhesion energy on mode mixity. When Abaqus/Standard is used, a displacement boundary condition is enforced at the end of the peeling arm to model the peeling process and the RIKS method in Abaqus/Standard is adopted for stable simulation. When Abaqus/Explicit is used, a velocity boundary condition, 1 m s^{−1}, is enforced at the end of the peeling arm. This velocity is much lower than the wave speeds of the thin film and loading can be regarded as quasi-static.

We first employ Abaqus/Standard to determine the cohesive zone size of the spatula pad under 0° peeling. In our simulation, the pad is attached to a rigid surface at one end and the peel-off force is found to be . This value agrees well with that predicted from Kendall's model using equation (3.1). Contour of stiffness degradation of the cohesive elements is shown in figure 3. The degradation spans approximately 20 nm, while the thickness of spatula is 5 nm, in agreement with that predicted from equation (2.7) or equation (3.2).

Next, we consider multiple contacting sites between the thin pad and the underlying rigid surface. We assume that the contacting sites are periodically distributed along the interface and have the same size, as schematically shown in figure 4. Different periods, 5, 30 and 50 nm, were investigated. Our simulations show that the pad starts to be detached away from the rigid surface as the peeling force exceeds a critical value, followed by a ‘stick-slip’ phase. The time history of the peeling force is presented in figure 4, where it can be seen that the peel-off force varies periodically with time. When the period of contact is 50 nm, the maximum peel-off force agrees with the prediction of equation (3.1) based on the peak value 0.01 N m^{−1} of the cohesive energy. When the period of contact is 5 nm, the peel-off force is close to that predicted from equation (3.1) based on the average value 0.005 N m^{−1} of the cohesive energy along the interface. These results are consistent with the theoretical predictions from §2. That is, if the cohesive zone size is smaller than the period of cohesive interaction, the apparent adhesion energy could be equal to the peak value of cohesive energy along the interface. Oppositely, if the cohesive zone size is larger than the period of cohesive interaction, the apparent adhesion energy could be the average cohesive energy along the interface.

### (b) Constraint effect of film thickness on cohesive zone size

The thin-film structure to be simulated is shown in the inset of figure 4, where the bonded part of the film is periodically attached to the substrate with a period of *l*_{p}. Within one period, the size of the attachment region is equal to that of the detachment region. We alter the film thickness *H* while keeping *l*_{p} and fixed in our simulation.

In Part I, we have simulated the same thin-film structure under 90° peeling. The material parameters for the film were taken to be *E*=1.0 MPa and *v*=0.3; the cohesive energy within the attachment region was assumed to be 0.01 J m^{−2}. Abaqus/Standard was employed and failure of cohesive element occurred via progressive degradation of material stiffness, wherever the nominal/equivalent stress reached 2 KPa. The period of adhesion patches was fixed at *l*_{p}=125 μm. According to equation (2.7), . The film thicknesses were selected to be *H*=100 and 50 μm. According to equation (3.3), the cohesive zone size was estimated to be approximately 120 μm when *H*=100 μm, while it decreases to 71 μm when *H*=50 μm. We would thus expect that the 90° peeling strength of the *H*=50 μm film should be higher than that of the *H*=100 μm film, since the cohesive zone size is larger for the latter case. This was confirmed in the simulations discussed in Part I, where the peeling strength of the *H*=50 μm film was found to be 50 per cent higher than that of the *H*=100 μm film.

Here, we simulate the peeling strength of the same thin film under 0° peeling. The material parameters for the film are chosen as *E*=3.0 GPa and *v*=0.3. The period of adhesion patches is fixed at *l*_{p}=20 nm. The cohesive energy within the attached regions is 0.01 J m^{−1}. The thickness of the film is taken to be *H*=1.5 nm or *H*=0.5 nm. According to equation (3.2), the cohesive zone size is estimated to be approximately 15 nm when *H*=1.5 nm, while it decreases to 9 nm when *H*=0.5 nm. In this case, we expect that the 0° peeling strength of the *H*=0.5 nm film should be higher than that of the *H*=1.5 nm film, since the cohesive zone size of the *H*=0.5 nm film is smaller than that of the *H*=1.5 nm film. In our simulation, it is observed that the degradation of cohesive elements spans a size of 17 nm when *H*=1.5 nm, while it only spans a size of 10 nm when *H*=0.5 nm. Our simulation also shows that the apparent adhesion energy *γ* is 0.006 N m^{−1} when *H*=1.5 nm and it becomes 0.009 N m^{−1} when *H*=0.5 nm, which is consistent with our theoretical considerations.

## 5. Velocity-constrained cohesive zone

Fracture/adhesion of soft materials/structures often involves stochastic rupturing and rebinding of molecular bonds along an interface. In contrast to dynamic crack propagations in brittle solids, where only crack velocities near the elastic wave speeds can exert significant influences on the cohesive zone size (Freund 1990), here we demonstrate that, in adhesive binding via molecular bonds, even relatively low crack speeds can have strong influences on the cohesive zone size.

The breaking/rupture of molecular bonds in biological systems is statistical in nature and has been widely modelled based on Bell's (1978) theory. As a first order approximation, Buehler & Ackbarow (2007) assumed that the bond break-off speed is a linear product of bond break-off rate and bond breaking distance, *d*, between the equilibrated state and the transitional state of a bond. In this way, Bell's (1978) theory can be extended to describe the dependence of bond breaking speed, , on bond breaking force, *F*, as(5.1)where *v*_{0} is the natural bond-breaking speed (when no load is applied), *θ* is the angle between the direction of the reaction pathway of bond breaking and the direction of the applied load, and *kT* is the product of Boltzmann's constant and temperature. Equation (5.1) is consistent with the experimental finding on the unfolding force of folded domains in Titin reported in Rief *et al*. (1997).

We consider the same generic crack propagation problem shown in figure 1, except that the interface is now assumed to consist of a layer of weak molecular bonds. Based on equation (5.1), a simple expression between the bond strength and bond breaking speed is(5.2)where *σ*_{0} is a reference constant.

For steady-state crack propagation at speed *V*, we have the relation(5.3)where *δ*_{0} is the bond extension or *d*, and is the cohesive zone size.

Using equation (2.7) and equations (5.2) and (5.3), is determined from(5.4)

To demonstrate the effect of crack speed on the cohesive zone size, we consider protein-like materials with Young's modulus *E*′=1 GPa and density *ρ*=10^{3} Kg m^{−3}, and hydrogen bonds with *δ*_{0}=1.85 Å and *v*_{0}=10^{−8} m s^{−1} (Buehler & Ackbarow 2007), and *σ*_{0}=10 MPa. As shown in figure 5, as the crack speed increases from 2×10^{−6} m s^{−1} to 20 m s^{−1}, the cohesive zone size decreases by an order of magnitude. Note that these crack speeds are low compared to the elastic wave speeds, which are on the order of 1000 m s^{−1}.

The reason for the change in cohesive zone size with crack speed shown in figure 5 is owing to the logarithmic increase in bond strength with speed, which is a salient feature of soft molecular bonds. In our analysis, we have chosen hydrogen bonds, which have a relatively short bond extension, to demonstrate that cohesive zone can be highly constrained even at relatively low crack speeds. It would be interesting to include bond rebinding and consider behaviours of ligand–receptor bonds in cell adhesion that typically have bond extension on the order of 10 nm. These effects are beyond the scope of this paper.

## 6. Conclusion

In this paper, we have extended our previous study (Chen *et al*. 2008*a*) on the mode I fracture/adhesion energy of interfaces with periodically varying cohesive interaction to general mixed-mode loading. We have shown that the mixed-mode fracture toughness/adhesion energy generally depends on the size of the cohesive zone near the tip of a crack along the interface: it is equal to the average cohesive energy of the interface if the cohesive zone size is much larger than the period of cohesive interaction but becomes the peak value of the local cohesive energy when the opposite is true. We have further shown that the cohesive zone size of a thin film under 90° peeling from a substrate scales with the bending stiffness of the film while that under 0° peeling scales with the tension stiffness of the film. These results that have been confirmed by finite-element numerical simulations, also suggest a general expression for the cohesive zone size of a very thin structure. For fracture/adhesion along a layer of molecular bonds in soft materials/structures, we have shown that crack speeds far below the elastic wave speeds could have a strong effect on the cohesive zone size. This effect can be attributed to the statistical behaviour of molecular bonds which leads to a dependence of the cohesive strength on the break-off rate. Our studies provide insights into the fracture/adhesion properties of small-scale structures in which the interplay between discrete/heterogeneous adhesion patches and cohesive zone is of dominant importance and suggest possible strategies to control fracture/adhesion strength of interfaces in both engineering and biological systems.

## Acknowledgments

The work of B.C. is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The work of H.G. is partly supported by the A^{*}Star Visiting Investigator Program ‘Size Effects in Small Scale Materials’ hosted at the Institute of High Performance Computing in Singapore.

## Footnotes

- Received September 11, 2008.
- Accepted November 7, 2008.

- © 2008 The Royal Society