𝑃𝑥

Friction of Extensible Strips: an Extended Shear Lag Model with

Experimental EvaluationAhmad R. Mojdehi1,3, Douglas P. Holmes2, Christopher B. Williams3, Timothy E. Long3, David A. Dillard1,3

1Department of Biomedical Engineering and Mechanics, Virginia Tech, Blacksburg, VA2Department of Mechanical Engineering, Boston University, Boston, MA

3Macromolecules and Interfaces Institutes (MII), Virginia Tech, Blacksburg, VA

Motivation Experiments Results

• Friction plays an important role in our daily life• Tires, shoes, wearable sensors, etc.• Material stiffness varies widely depending on

their application, material properties andgeometry

• National Operating Committee on Standards forAthletic Equipment (NOCSAE) DOC (ND) 019 –10m15a

• Standard Test Method and PerformanceSpecification for Newly Manufactured FootballPlayers Hand Coverings

• Requirement: Static Coefficient of Friction ≤ 4.5• But the effect of stiffness on COF is not

considered

Background

Shear Lag Model

Conclusion

• A scaling law hasbeen developed by Dr.Crosby’s group torelate the shearadhesion strength tothe stiffness

Bartlett, Michael D., et al. Advanced Materials 24.8 (2012): 1078-1083.

𝐹𝑐~ 𝐺𝑐𝐴

𝐶

• Shear Lag model was developed by Hart-Smith for analysis ofAdhesive-Bonded Single-Lap Joints experiencing adhesive’splasticity

Hart-Smith, L., Adhesive-bonded single-lap joints. 1973: NASA Technical Report

Sled

Strip

Substrate

Strip glued to the sledLoad Cell

Actuator

𝑃𝑥

𝑓𝑒

𝑓𝑏

𝑓 +𝑑𝑓

𝑑𝑥∆𝑥 𝑓

𝑓𝑒∆𝑥

𝑓𝑏∆𝑥

𝜕2𝛿

𝜕𝑥2−𝑓𝑒 − 𝑓𝑏𝐸𝐴

= 0

• The governing differential equations are obtained from a differentialelement at contact interfaces and equilibrium condition, for eachregion

• Boundary conditions are satisfied by continuity of the displacementand force at the boundaries of each region

• The displacement field and therefore the friction force can beobtained by solving the corresponding differential equations

• A custom built friction setup is used to measure the friction vsdisplacement of the strips

• A translational actuator pulls a steel sled (mass 190g, length100mm, and width 10mm), the nose of which secures the endof the extensible strip

• Different strip materials and substrates were used, e.g.elastomer-coated fabric, braided elastic strip, glass, and tape

• The intrinsic friction force was obtained by rigidly adhering thestrips to the sled

𝑓𝑒 =

𝑘1𝛿 0 ≤ 𝛿 ≤ ∆𝑒𝑛 𝑓𝑒𝑠 − 𝑘2(𝛿 − ∆𝑒𝑠 ∆𝑒𝑛≤ 𝛿 ≤ ∆𝑒𝑠

𝑓𝑒𝑠 𝛿 ≥ ∆𝑒𝑠

𝑓𝑏 = 𝑘3(𝛿 𝐿 − 𝛿 0 ≤ 𝛿 ≤ ∆𝑏𝑛

𝑓𝑏𝑠 𝛿 ≥ ∆𝑏𝑠

P

Slip

TransitionNo-Slip

Direction of slip zone propagation between elastomer and substrate

Direction of

sled motion

esen et

P

Direction of slip zone propagation between sled and backing

Direction of sled

motion

bnbs

Friction zones between elastomer and substrate Friction zones between backing and sled

Intrinsic friction responses of two contact surfaces

τ

x

PP

E

lt

η

tx

G τ

ϒe ϒp

τpG

ϒ

T2+ΔT2T2

M2V2

T3+ΔT3T3

M3V3

Analogy between Elastic-Plastic stress and Static-Kinetic friction:

𝑃

Friction vs displacement

f

𝛿

Static

Tran

siti

on

Kinetic

Shear stress vs strain

τ

ϒ

Elastic Plastic

PP

E

lt

η

tx

G

http://www.discounttiredirect.com

http://www.ife.ee.ethz.ch/

Displacement of different points anddevelopment of slippage zone across the lengthof a strip. The least-square fits (dash lines)confirm the formation of transition and slip zones

Normalized friction force per unit lengthacross the length of the strip for twocases obtained from shear lag model

Increasin

gaxialstiffn

ess

Comparison of friction force versus displacementresponse for strips with three different backingstiffnesses (shear lag model and experiment).Clouds’ widths correspond to two standarddeviations

Comparison of friction force versusdisplacement response for strips withdifferent lengths and backing stiffnesses.Clouds’ width correspond to two standarddeviations

Fabric

Elastomer

Steel

Glass

Fric

tio

n

Displacement

Fric

tio

n

Displacement

Fabric

Fabric

Steel

Glass

Fric

tio

n

Displacement

Fric

tio

n

Displacement Fabric

Fabric

Tape

Tape

Fric

tio

n

Displacement

Fric

tio

n

Displacement

Friction force vs displacement response ofan elastic strip with two fabric sides incontact with steel on the top and glass onthe bottom

Friction force vs displacement response ofan elastic strip with two fabric sides incontact with tape on both sides(symmetric)

• Effective stiffness has a profound effect on both static and kineticfriction of extensible strips

• There are three distinct regions along the length of the strip, namelyno-slip, transition, and slip zones

• The static friction decreases with effective stiffness whereas thekinetic friction increases by decreasing the effective stiffness

• An extended shear lag model is developed to predict the frictionalresponse of extensible strips

• The analysis resembles that obtained when shear lag theory isapplied to lap shear joints experiencing adhesive layer plasticity

AcknowledgementThe authors would like to thank the Macromolecules and InterfacesInstitute (MII) at Virginia Tech for travel support of ARM, and theBiomedical Engineering and Mechanics (BEAM) Department for use ofequipment