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