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Research Papers

# Frictional Energy Dissipation in Wavy SurfacesOPEN ACCESS

[+] Author and Article Information
Lejie Liu

Department of Mechanical Engineering,
1513 University Avenue,

Melih Eriten

Department of Mechanical Engineering,
1513 University Avenue,
e-mail: eriten@wisc.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 18, 2016; final manuscript received August 13, 2016; published online September 9, 2016. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 83(12), 121001 (Sep 09, 2016) (7 pages) Paper No: JAM-16-1305; doi: 10.1115/1.4034461 History: Received June 18, 2016; Revised August 13, 2016

## Abstract

Accurate estimation and tuning of frictional damping are critical for proper design, safety, and reliability of assembled structures. In this study, we investigate how surface geometry and boundary conditions affect frictional energy dissipation under microslip contact situations. In particular, we investigate the frictional losses of a two-dimensional (2D) deformable wavy surface in contact with rigid plate under specific normal and tangential loading. We also propose a dissipation tuning mechanism by tension-induced wrinkling of a composite surface. This surface is made of stiff strips printed on a compliant substrate. We show that the contact geometry of wrinkling surfaces can be altered significantly by tensile loading and design of the composite surface. Using this, we present frictional dissipation maps as functions of applied tension and one of the geometric parameters in the composite design; spacing between stiff strips. Those maps illustrate the dissipation tuning capability of wrinkled surfaces, and thus present a unique mean of damping control.

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## Introduction

The effect of surface geometry on frictional dissipation was first embedded in the response of spherical contacts to cyclic loading by Mindlin et al. [16]. The dissipation was found to scale inversely with contact radius under constant normal load and tangential fluctuation amplitude. More recently, frictional dissipation was investigated extensively in arbitrary 2D contacts [17,18], and rough surfaces [15,19,20]. In those works, surface geometry and compliance were found as major factors affecting dissipation. Utilizing those observations, we aim to employ surface wrinkling in this paper to tune frictional energy dissipation in microslip contacts. Tension-induced wrinkling of elastic thin films was studied by Cerda and Mahadevan [21], and strain incompatibility due to Poisson effect was found as the major wrinkling mechanism. Large mismatch in stiffness of constituents was also a key factor in creating wrinkles on composite surface layers such as soft biological tissues and soft-hard materials interfaces [2224]. Inspired by the mismatch factor, Chen and Elbanna recently showed that a composite film with periodic arrangement of stiff strips transfer-printed on a compliant substrate wrinkles under tension, and the tension and periodic arrangement control the amplitude of wrinkles (corrugation) [25]. We will investigate a similar configuration as in Ref. [25], and tune frictional dissipation on wrinkled (corrugated) surfaces.

The remainder of the paper is organized as follows. The finite-element model (FEM) of a 2D wavy (wrinkled) surface in contact with a rigid plate will be developed and validated in Sec. 2. The effects of corrugation and wavelength of wrinkled surfaces on frictional dissipation under different boundary conditions are presented in Sec. 3. Section 3 also features a theoretical and numerical wrinkling analysis leading to tunable dissipation via loading and geometric parameters. Conclusions are listed in Sec. 4.

## Finite-Element Model and Validation

###### Sinusoidal Surface in Contact With Rigid Plate.

We will estimate the energy dissipation due to microslip in a 2D wavy surface in contact with a rigid flat plate by a finite-element model (Fig. 1). During microslip, part of the contact slips while remaining part sticks. Loading conditions and friction coefficient determine the extent of stick and slip.

The loading in normal direction is set as a constant, i.e., constant displacement for displacement control, and constant force for load control, and tangential loading fluctuates periodically, i.e., Display Formula

(1)$P(t)=P0$
Display Formula
(2)

where , and T are the constant normal load, and amplitude and period of tangential load fluctuations. Assuming that the local Coulomb friction law with the friction coefficient of f holds over the contact, total frictional dissipation in a loading cycle can be estimated by Display Formula

(3)

where $p(x)$ and $u(x)$ are the contact pressure and relative tangential (slip) displacement at a contact point with a distance of $x$ from the origin of the contact. For a 2D sinusoidal surface in contact with a rigid flat plate, the contact half-length are given in Ref. [26] as Display Formula

(4)$a(P0)=λπ sin−1(P0γΔ)1/2$

where $γ=(1−ν2)/(πE)$, and $λ$, $Δ$, $E$, and $υ$ are the spatial wavelength, peak-to-peak height (Fig. 1), Young’s modulus, and the Poisson’s ratio of the wavy surface, respectively. A 2D finite-element model (FEM) is created in abaqus to obtain frictional dissipation. Six-node modified quadratic plane strain triangle elements (CPE6MH) are used and the mesh is finest nearby the contact and gradually coarsens away from the contact region.

Surface to surface contact formulation with Lagrange multipliers for tangential behavior is used to model the contact case. Several models with different geometry are built for parametric investigation of corrugation effects. The modeling parameters used are tabulated in Table 1. Note that the frictional dissipation is mostly converted into thermal energy. In case of low thermal conductivity, that thermal energy can heat up the contact, and thus, alter the contact geometry and material properties in the vicinity of contact. In this study, we neglect the thermal effects by assuming that the thermal conductivity and heat flow rate at the contact are sufficiently high to inhibit any significant rise in temperature within a loading cycle.

Another assumption we make in our model is that the contact loads are small and localized to the peaks of the corrugated profile, and hence do not alter the corrugation. Larger contact loads, however, could result in stress fields diffusing into the soft substrate and modify the corrugation. Hence, our study and results are applicable only to the small contact loads and deformations.

We validate our FEM by known analytical solutions of the 2D contact problem. As described in Eq. (3), shear traction will contribute to the energy dissipation, and the contact pressure for a sinusoidal surface under normal loading is given in Ref. [26] as Display Formula

(5)$p(x,P0)=2P0λcos(πxλ) sin2(πaλ) sin2(πaλ)−sin2(πxλ), |x|

The FEM results of normal contact for different geometry and loading parameters are compared with the analytical expressions in Eqs. (4) and (5). The average errors in contact half-length and pressures are within 5% and 16%, respectively, for all the FE models with the parameters listed in Table 1.

In Ref. [27], the extent of the stick zone is given as Display Formula

(6)$QfP=1− sin2(πb/λ) sin2(πa/λ)$

where $b$ is the half-length of the stick zone. When the contact region is small relative to the whole wavelength, the contact parameters approach to the ones given for a cylinder in contact with rigid plate. So, the shear traction under partial slip takes the following form [26]: Display Formula

(7)$q(x)={fp0{(a2−x2)1/2−(b2−x2)1/2}/a,0≤x

where $p0$ is the maximum normal contact pressure. Figure 2 shows the comparison between the tangential tractions predicted by the analytical expressions in Eq. (7) and the FEM for a specific case (λ = 1.8 mm, $Δ$ = 0.06 mm, constant loading condition) under three different loading conditions: (a) $Q/fP=0.2$, (b) $Q/fP=0.3$, and (c) $Q/fP=0.4$. The tractions correlate well within the stick region, and the maximum mismatch happens at the boundary of the stick–slip zone. The maximum mismatch does not exceed 20%.

###### Frictional Energy Dissipation.

“Frictional dissipation” parameter in abaqus is used to trace the mechanical losses due to friction in the FEM. We compared this parameter against the integral definition of energy dissipation in Eq. (3) for the same case as described in Fig. 2 of $Q/fP=0.3$ for a quarter loading period. Contact pressure and slip displacements used in the integral formulation are retrieved from the FEM, and numerical integration over the contact patch is performed. As shown in Fig. 3, the maximum error between the integral formulation and readily available “frictional dissipation” parameter is about 14%. The mismatch is mostly due to the mismatch in shear tractions at the edge of the stick–slip boundary, and error in contact half-length predictions. Similar error prevails for all of the loading conditions, and thus, does not introduce any superficiality to the results. Therefore, we will report the frictional dissipation obtained directly from the FEM.

## Results

###### Energy Dissipation Under Constant Normal Displacement.

First, we consider the case with the constant normal displacement at the center of the rigid plate. Then, we apply a varying tangential load as described in Eq. (2) while keeping the normal displacement constant. Figure 4 shows the influence of corrugation amplitude, $Δ$ on the frictional energy dissipation. Here, we choose the wavelength as 1.8 mm, and $Δ$ varies an order of magnitude (from 0.06 mm to 0.6 mm). Energy dissipation shown here is normalized to the smallest value obtained at Q/fP = 0.3. As evident from the figure, less corrugated surfaces dissipate significantly higher energy. For a given normal displacement, flatter surfaces distribute the normal load over larger contact areas and the contact pressure across those areas are less concentrated. Both of those observations lead to increase in contact slip and the extent of slip regions, and thus, facilitate higher frictional dissipation. Figure 4 also shows the trends for different tangential loads. As expected, higher tangential loads lead to higher dissipation thanks to increase in shear tractions, slip, and the extent of slip zones. Flatness of the contact patches is not only controlled by the corrugation amplitude but also by the wavelength. Next, we show the effects of both parameters on the frictional dissipation as a contour plot.

To obtain the contour plot shown in Fig. 5, we vary the wavelengths from 0.9 to 3.6 mm, and the corrugation from 0.06 to 0.6 mm, while keeping the tangential load as Q/fP = 0.3. We follow the same normalization in Fig. 4 and find out that the energy dissipation varies approximately an order of magnitude. The contours show that larger wavelengths and smaller corrugations yield higher energy dissipation. This is mainly because flatter surface profiles lead to more frictional slip. In contrast, when the wavelength is small, the contact tip is sharper, and that leads to more concentrated contact pressures and less slip. This trend is expected to prevail for other loading cases with varying contrast in contours. As evident in Fig. 4, the maximum energy dissipation for $Q/fP$  = 0.4 is nearly twice as the maximum for $Q/fP$  = 0.2. Those differences will impact the contrast in the contour plots.

Similar trends are expected to hold for 3D contacts. For instance, in a sphere-on-flat contact, the energy dissipation can be estimated by [16] Display Formula

(8)$ΔW=9f2P0210a(2−ν1G1+2−ν2G2){1−(1−Q1fP0)53−5Q16fP0[1−(1−Q1fP0)23]}$

where $G1$, $G2$ and $ν1$, $ν2$ are the shear moduli and Poisson’s ratios for sphere and flat surfaces, respectively. For a given $Q/fP$ and constant $f$, the energy dissipation given in Eq. (8) scales with $(P02/a)$. Substituting Hertzian contact relations for a fixed normal penetration, $a∼R1/2$, and $P0∼R1/2$; the energy dissipation can be related to the sphere radius as $ΔW∼(P02/a)∼R1/2$. Therefore, flatter surfaces are expected to dissipate more energy than corrugated surfaces under constant normal penetration in both 2D and 3D contact configurations.

###### Energy Dissipation Under Constant Normal Load.

Next, we study the case with constant normal load applied at the center of the rigid plate. Similar to the case with the displacement boundary condition, we generate the normalized energy dissipation versus corrugation and the contour plots as shown in Figs. 6 and 7. Energy dissipation is normalized to the smallest dissipation value obtained for Q/fP = 0.3. In Fig. 6, the wavelength is fixed at 1.8 mm, and corrugation amplitude is varied from 0.06 to 0.6 mm. For the contour plot shown in Fig. 7, both the wavelength and corrugation are varied as in Fig. 5. In contrast with the displacement controlled case, both Figs. 6 and 7 show that more corrugated surfaces dissipate more energy. For a given normal load, more corrugated surfaces lead to smaller contact radii and slip zones, but higher contact pressures and shear tractions over the slip zones. Those competing factors nearly cancel out each other, and thus, variation in corrugation results in only slight change in energy dissipation. This competition and weak-dependence on corrugation can be seen in 3D contacts too. As evident from the analytical expression given in Eq. (8), the energy dissipation in spherical contacts is inversely proportional to the contact area under a constant normal load. Therefore, more corrugation leading to smaller contact areas is expected to result in higher frictional dissipation. Mathematically, we can express the contact area in terms of applied load and sphere radius, and substitute in Eq. (8), and show that $ΔW∼R−1/3$ for the constant loading condition. In other words, flatter surfaces will dissipate less energy. However, the energy dissipation depends weakly on the sphere radius when compared to the displacement-controlled case. This is also reflected in the contrast of the contours presented in Figs. 5 and 7. Displacement-controlled case leads to an order of magnitude variation in dissipation in contrast to twofold change in the load-controlled case. We also observe in Fig. 6 that the energy dissipation increases significantly with tangential load. As described before, this is due to substantial increase in shear tractions and the size of the slip zone.

###### Tuning Energy Dissipation by Surface Wrinkling.

The results presented above suggest that the frictional dissipation can be tuned by surface wrinkling. Surface wrinkling can be induced by various methods discussed in Sec. 1. Here, we study a wrinkling model consisting of stiff strips of length L and height t periodically embedded in a long compliant substrate with height h [28]. As discussed in Ref. [28], the high contrast in elastic properties of the substrate and embedded strips result in significant shifts of the neutral axis off the center of the substrate. Therefore, any distant concentric loading as shown in Fig. 8 leads to spatially varying moments and bending, and thus, wrinkling. The spacing between periodic strips, s and one strip length determine the half wavelength of the periodic-wrinkled profile due to the distant loading imposed by the uniaxial tension p shown in Fig. 8. In Secs. 3.3.1 and 3.3.2, we will first solve this problem numerically and analytically, and relate the wrinkled profile to the distant loading and spacing of the strips. Then, we will utilize the results of Secs. 3.1 and 3.2 to investigate the effect of surface wrinkling on frictional dissipation.

###### Surface Wrinkling Analyses.

A finite-element model of the geometry shown in Fig. 8 is created in Mathematica (second-order triangular elements). Two-dimensional plane strain formulation is used, and both the substrate and strips are assumed as linear isotropic elastic materials. Note that in Ref. [28], the substrate was assumed as hyperelastic, and thus, finding analytical solution to the problem was difficult especially at large deformations. Despite the large mismatch between the elastic properties of the substrate and strips, they are assumed perfectly bonded. The geometry and material properties used are summarized in Table 2.

In the finite-element simulations, the spacing between the strips, s and the distant uniaxial tensile strain, $εa=p/Esubs$ will be varied, and the distance between the peak and the valley of the upper-edge profile will be reported as the corrugation amplitude, i.e., assuming a sinusoidal profile. The wavelength of that profile will be $λ=2(s+L)$. A representative contour plot of the axial strain is shown in Fig. 9 within the composite strip–substrate domain of one wavelength (shown at the top of Fig. 9). This contour is obtained from the FE model with $s$  = 0.5 mm and $εa$  = 0.08. Axial strains vanish near the stiff strips, and reaches maximum on the edge of the substrate away from the strips. Also, the strain values exceed the applied tension at those edges because of bending-induced tension. Together with this bending-induced excessive tension, alternating periodic arrangements of the stiff strips on different edges of the substrates (top/bottom) result in periodic corrugation. Figure 10(a) shows the wrinkled profile at the top edge of the substrate (y = 0.2 mm in the undeformed configuration) as obtained from the same FE simulations in comparison with the sinusoidal curve that has the same wavelength and corrugation amplitude. Wrinkled profile follows the analytical curve reasonably well throughout one wavelength, and thus, can be assumed sinusoidal. Next, we analyze the spatial variation of axial strains along the neutral axis shown in Fig. 10(b). The axial strain is drawn along the neutral axis, $N0$ shown at the top diagram of Fig. 10(b), of the middle part of the composite. Figure 10(b) reveals the distribution of axial strains along one period of the composite. One can obtain the strains in the whole composite domain by mirroring the strains periodically.

Combined section analysis gives the distance of neutral axis from the free surface as Display Formula

(9)$Y0=nt(h+t2)+h22nt+h$

where $n=Estrip/Esubs$. The axial strain on the neutral axis is zero for almost 60% of the length of stiff strips, exhibit discontinuity at the edges of the strips, and transition to values close to the applied axial strain ($εa$  = 0.08 for the case shown) around the space between two consecutive strips. In line with this observation, one can model the problem as a simple beam loaded by pure moment, M up to the 60% of the strip lengths. Considering only quarter of the wavelength $((L+s)/2)$, the corrugation at the 60% of half of the strip length can be found in Ref. [29] as Display Formula

(10)$yx=0.6L2=M(0.6L/2)22EsubsIcomb$

where the moment projected on the neutral axis $M=Esubsεawh(Y0−(h/2))$, $w$ is the width of the composite structure, the combined moment of inertia is the sum of that of the strip and substrate, i.e., $Icomb=Istrip+Isubs$, and the moment of inertias of the strip and substrate follow from the parallel-axis theorem as Display Formula

(11)$Istrip=112nwt3+nwt(h+t2−Y0)2$
Display Formula
(12)$Isubs=112wh3+wh(Y0−h2)2$

Assuming the transition region between two consecutive strips experiences no net moment, the additional corrugation can be found by the product of the slope at the 60% of half of strip length and the sum of 40% of half of the strip length and half of the spacing, and so the corrugation amplitude at the quarter wavelength can be estimated by Display Formula

(13)

Figure 11 shows the corrugation amplitudes obtained from the FEM and analytical expression of Eq. (13) for different strip spacing and applied strain. The corrugation amplitude increases linearly with the applied strain. Also spacing between the strips influences the corrugation amplitude to a great extent. Influence of applied strain and spacing is well captured by Eq. (13). Yang et al. studied the wrinkling of a PDMS substrate with stiff silicon ribbons by FEA [28]. For small strains (2.5% elongation in their study), they reported corrugations of 13.5, 11, and 8.5 μm for s/L ratios of 150/500, 250/400, and 350/300 μm, respectively. Using the same material properties and geometry of Ref. [28] in Eq. (13), we predict the corrugation amplitudes of 12.9, 10.9, and 8.7 μm for the same s/L ratios. Therefore, we will use Eq. (13) to predict the corrugation amplitudes and tune frictional damping accordingly.

###### Tuning Energy Dissipation by the Strip Spacing and Applied Tension.

After establishing the relationship between the wrinkled profile (wavelength and corrugation), and the applied strain and the strip spacing, we transform the contour plots presented in Figs. 5 and 7 to the applied strain–strip spacing domain (Fig. 12: (a) for displacement and (b) for force boundary conditions). The transformation from the wavelength-corrugation to the applied strain–strip spacing domain is achieved by the corrugation expression given in Eq. (13), and the approximate geometric relation of $λ=2(s+L)$. Applied strains in those contour plots are limited to 0.5 to ensure relevance in practice.

The displacement boundary condition case presented in Fig. 12(a) shows that increasing the strip spacing and decreasing the applied strain will lead to flatter surface profiles, i.e., less corrugation, and thus, higher frictional dissipations. At a fixed strip spacing of 1.5 mm, the frictional dissipation can be reduced approximately an order of magnitude by increasing the applied strain from 0 to 0.5. Note that Eq. (13) includes the combined effect of the applied strain and strip spacing, and hence, one-to-one transformation between the wavelength-corrugation, and the applied strain–strip spacing domains is not possible. This results in differences between Figs. 12(a), 12(b), 5, and 7.

For the normal load boundary condition case shown in Fig. 12(b), larger strip spacing and larger applied strain leads to higher energy dissipation. In this case, the energy dissipation does not vary as much as in the displacement boundary condition case for the reasons explained above (see Fig. 7).

## Conclusions

The effect of geometry of a 2D sinusoidal surface, specifically the corrugation amplitude and wavelength on the contact frictional energy dissipation is studied by numerical and analytical techniques. The FEM results are first validated with known analytical expressions, and then, the frictional dissipation is linked to the corrugation and wavelength of the surface profile. Finally, tension-induced surface wrinkling of a composite surface consisting of a compliant substrate and stiff strips attached periodically is studied. The parameters influencing the wrinkled profile (applied tension and strip spacing) are then used to tune the frictional dissipation. Key findings of the paper can be listed as follows:

• For a 2D sinusoidal surface in contact with rigid plate, large wavelength, and small corrugation will lead to high frictional dissipation under constant normal displacement boundary condition. This is due to the flatter surfaces are expected to dissipate more energy than the corrugated surfaces under constant normal penetration.

• This behavior is reversed and only slight change in energy dissipation is observed under constant normal force boundary condition because the competing factors that cause the change in frictional dissipation are nearly canceled out each other.

• Wrinkling of a composite surface in the form of stiff strips embedded periodically in a compliant substrate can be controlled by tension. The spatial distribution of the stiff strips and applied tension can be used to tune the corrugation of the wrinkled profile and thus frictional dissipation.

## Acknowledgements

This work was supported by the National Science Foundation under Grant No. NSF-CMMI-1462870.

## References

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Hartog, J. P. D. , 1930, “ LXXIII. Forced Vibrations With Combined Viscous and Coulomb Damping,” Philos. Mag. Ser. 7, 9(59), pp. 801–817.
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Dieterich, J. H. , 1978, “ Time-Dependent Friction and the Mechanics of Stick-Slip,” Pure Appl. Geophys., 116(4–5), pp. 790–806.
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## References

Lazan, B. J. , 1968, Damping of Materials and Members in Structural Mechanics, 1st ed., Pergamon Press, Oxford, New York.
Hartog, J. P. D. , 1930, “ LXXIII. Forced Vibrations With Combined Viscous and Coulomb Damping,” Philos. Mag. Ser. 7, 9(59), pp. 801–817.
Cattaneo, C. , 1938, “ Sul Contatto di Due Corpo Elastici,” Atti Accad. Naz. Lincei Sci. Fis. Mater. Nat., 27(6), pp. 342–348.
Mindlin, R. , 1949, “ Compliance of Elastic Bodies in Contact,” ASME J. Appl. Mech., 16(3), pp. 259–268.
Dieterich, J. H. , 1978, “ Time-Dependent Friction and the Mechanics of Stick-Slip,” Pure Appl. Geophys., 116(4–5), pp. 790–806.
Dieterich, J. H. , 1979, “ Modeling of Rock Friction: 1. Experimental Results and Constitutive Equations,” J. Geophys. Res. Solid Earth, 84(B5), pp. 2161–2168.
Ruina, A. , 1983, “ Slip Instability and State Variable Friction Laws,” J. Geophys. Res. Solid Earth, 88(B12), pp. 10359–10370.
Williams, E. J. , and Earles, S. W. E. , 1974, “ Optimization of the Response of Frictionally Damped Beam Type Structures With Reference to Gas Turbine Compressor Blading,” ASME J. Manuf. Sci. Eng., 96(2), pp. 471–476.
Beards, C. F. , and Woowat, A. , 1985, “ The Control of Frame Vibration by Friction Damping in Joints,” ASME J. Vib. Acoust., 107(1), pp. 26–32.
Gaul, L. , and Nitsche, R. , 2000, “ Friction Control for Vibration Suppression,” Mech. Syst. Signal Process., 14(2), pp. 139–150.
Beards, C. F. , 1983, “ The Damping of Structural Vibration by Controlled Interfacial Slip in Joints,” ASME J. Vib. Acoust., 105(3), pp. 369–373.
Griffin, J. H. , and Menq, C.-H. , 1991, “ Friction Damping of Circular Motion and Its Implications to Vibration Control,” ASME J. Vib. Acoust., 113(2), pp. 225–229.
Menq, C.-H. , Chidamparam, P. , and Griffin, J. H. , 1991, “ Friction Damping of Two-Dimensional Motion and Its Application in Vibration Control,” J. Sound Vib., 144(3), pp. 427–447.
Jang, Y. H. , and Barber, J. R. , 2011, “ Effect of Phase on the Frictional Dissipation in Systems Subjected to Harmonically Varying Loads,” Eur. J. Mech. A: Solids, 30(3), pp. 269–274.
Putignano, C. , Ciavarella, M. , and Barber, J. R. , 2011, “ Frictional Energy Dissipation in Contact of Nominally Flat Rough Surfaces Under Harmonically Varying Loads,” J. Mech. Phys. Solids, 59(12), pp. 2442–2454.
Mindlin, R. D. , Mason, W. P. , Osmer, J. F. , and Deresiewicz, H. , 1952, “ Effects of an Oscillating Tangential Force on the Contact Surfaces of Elastic Spheres,” 1st U.S. National Congress of Applied Mechanics, Chicago, IL, June 11–16, p. 203.
Davies, M. , Barber, J. R. , and Hills, D. A. , 2012, “ Energy Dissipation in a Frictional Incomplete Contact With Varying Normal Load,” Int. J. Mech. Sci., 55(1), pp. 13–21.
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## Figures

Fig. 1

Abaqus model of 2D wrinkling surface in contact with a rigid plate

Fig. 2

Shear traction of analytical results and FEM results for (a) Q/fP = 0.2, (b) Q/fP = 0.3, and (c) Q/fP = 0.4

Fig. 3

Comparison of normalized energy dissipation for a specific case between the integral formulation from Eq. (3) and “Frictional dissipation” parameter in abaqus

Fig. 4

Normalized energy dissipation versus the corrugation of the sinusoidal surface under constant normal displacement

Fig. 5

Contour plot of energy dissipation for normal displacement boundary conditions based on varying wavelength and corrugation

Fig. 6

Normalized energy dissipation versus the corrugation of the sinusoidal surface under constant normal load

Fig. 7

Contour plot of energy dissipation for normal load boundary conditions based on varying wavelength and corrugation

Fig. 8

Surface wrinkling model including transfer-printing stiff strips with length L and height t periodically on a long substrate with height h under uniform tensile stress p

Fig. 12

Contour plot of energy dissipation for (a) normal displacement boundary condition, and (b) normal load boundary condition based on varying strip spacing and applied strain

Fig. 11

Corrugation extracted by analytical expression (lines) from Eq. (13) and FEM (circles) for the spacing value, s = 0.05, 0.5, and 1.4 mm

Fig. 10

(a) Comparison of the sinusoidal profile of a FE model that s = 0.5 mm and εa  = 0.08 (red dotted line) and an analytical sinusoidal curve (blue line) which has the same wavelength and corrugation amplitude (in the plane parallel to the uniform tensile stress). (b) Axial strains along with the green dashed line N0 and its corresponding wrinkling model for the same FE model.

Fig. 9

Contour plot of axial strain within the composite strip–substrate domain of one wavelength shown at the top

## Tables

Table 1 Geometry, materials and loading values used in FE models
Table 2 Geometry and material parameters of substrate and strips

## Errata

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