Transcript Mathematical Modeling: Stress Relaxation of Viscoelastic Materials Ryan Palmer Faculty Advisor: Dr. Michael Shaw.

Mathematical Modeling:
Stress Relaxation of
Viscoelastic Materials
Ryan Palmer
Faculty Advisor: Dr. Michael Shaw
The Main Objective

Understand the viscoelastic material properties of
skin
 Apply knowledge towards healing chronic wounds
Pictures provided by Dr. Garner (USC)
Viscoelasticity

Viscous: Fluid-like motion with high resistance to flow
 Example: Glass

Elastic: Spring-like motion
 Example: Rubber band
Gelatin Specimen Preparation at
CLU
R. Palmer, G. Toland

Experimental variables:
– u*
– du/dt
– gelatin concentration
Pictures and graphs provided by
Dr. Shaw (CLU)
Force-displacement-time
schematic of experiment
Force, F
Indentation
displacement,
u
Force, F
1. Apply controlled displacement
until reach peak displacement, u*;
monitor load
t=0
t=60
u*=2.0 mm
Time, t
(sec)
t=180
2. Hold peak displacement u*;
monitor load relaxation
Pictures and graphs provided by Dr. Shaw (CLU)
Displacement, u
(mm)
Indenter diameter = 12 mm
Gel diameter = 23 mm
Gel thickness ~ 4 mm
Loading of specimens
Stress Relaxation
120
Load (N)
100
80
60
40
20
0
0
50
100
150
Time (sec)
Ryan Palmer
Stress-relaxation Curve
Stress Relaxation
120
80
60
40
20
0
0
50
100
150
Time (sec)
Stress Relaxation
Load (N)
Load (N)
100
100.75
99.75
98.75
97.75
96.75
95.75
94.75
93.75
55
75
95
115
135
155
175
Time (sec)
Ryan Palmer
Viscoelastic models

Viscoelastic materials have been modeled
by a mixture of Maxwell, Voight and Kelvin
models3.
 These models consist of spring and dashpot
setups.

Dashpot:
A pneumatic or hydraulic cushion for a falling
weight, to prevent shock5.
Maxwell Model
Maxwell Model:
Consists of a spring
and dashpot in
series.
Voight Model
Voight Model:
Consists of a spring
and dashpot in
parallel.
Kelvin Model
Kelvin ModelConsists of a spring
in parallel with a
Voight model.
Viscoelasticity and RC Circuits
Dashpot
Resistor
Spring
Capacitor
RC circuits are often used to simplify viscoelastic systems.
Which help attain an equation to model its properties.
Stress-relaxation Curve
Stress Relaxation
120
80
60
40
20
0
0
50
100
150
Time (sec)
Data
Stress Relaxation
Predicted
101.00
100.00
99.00
Load (N)
Load (N)
100
98.00
97.00
96.00
95.00
94.00
60
80
100
120
Time (sec)
140
160
180
Ryan Palmer
Stress-relaxation Curve
Data
Stress Relaxation
Predicted
101.00
100.00
Y error = Σ (Ypred. – Ydata)2
X error = Σ (Xpred. – Xdata)2
98.00
97.00
96.00
95.00
94.00
60
80
100
120
140
160
180
Time (sec)
Data
Stress Relaxation
Predicted
99.00
X error
98.80
Load (N)
Load (N)
99.00
98.60
Y error
98.40
98.20
98.00
60
65
70
Time (sec)
75
80
Ryan Palmer