Mathematical Modeling:Mathematical Modeling:Stress Relaxation of Stress Relaxation of

Viscoelastic MaterialsViscoelastic Materials

Ryan Palmer

Faculty Advisor: Dr. Michael Shaw

The Main ObjectiveThe Main Objective Understand the viscoelastic material properties of

skin Apply knowledge towards healing chronic wounds

Pictures provided by Dr. Garner (USC)

ViscoelasticityViscoelasticity

Viscous: Fluid-like motion with high resistance to flowExample: Glass

Elastic: Spring-like motionExample: Rubber band

Gelatin Specimen Preparation at Gelatin Specimen Preparation at CLUCLU

R. Palmer, G. Toland

Experimental variables:– u*– du/dt– gelatin concentration Pictures and graphs provided by

Dr. Shaw (CLU)

Force-displacement-time Force-displacement-time schematic of experimentschematic of experiment

Force, F

Time, t(sec)

Displacement, u(mm)

t=0

t=180

u*=2.0 mmt=60

1. Apply controlled displacement until reach peak displacement, u*; monitor load

2. Hold peak displacement u*; monitor load relaxation

Force, F

Indentation displacement,

u

Indenter diameter = 12 mmGel diameter = 23 mmGel thickness ~ 4 mmPictures and graphs provided by Dr. Shaw (CLU)

Stress Relaxation

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Time (sec)

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N)

Ryan Palmer

Loading of specimensLoading of specimens

Stress Relaxation

93.7594.75

95.7596.7597.75

98.7599.75

100.75

55 75 95 115 135 155 175

Time (sec)

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Ryan Palmer

Stress-relaxation CurveStress-relaxation CurveStress Relaxation

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Viscoelastic modelsViscoelastic 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 ModelMaxwell Model

Maxwell Model:

Consists of a spring and dashpot in

series.

Voight ModelVoight Model

Voight Model:

Consists of a spring and dashpot in

parallel.

Kelvin ModelKelvin Model

Kelvin Model-

Consists of a spring in parallel with a Voight model.

Viscoelasticity and RC Viscoelasticity and RC CircuitsCircuits

Dashpot Resistor

Spring Capacitor

RC circuits are often used to simplify viscoelastic systems.Which help attain an equation to model its properties.

Stress Relaxation

94.00

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101.00

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Time (sec)

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Data

Predicted

Ryan Palmer

Stress-relaxation CurveStress-relaxation CurveStress Relaxation

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0 50 100 150

Time (sec)

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Stress Relaxation

98.00

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Data

Predicted

Ryan Palmer

Stress-relaxation CurveStress-relaxation CurveStress Relaxation

94.00

95.00

96.00

97.00

98.00

99.00

100.00

101.00

60 80 100 120 140 160 180

Time (sec)

Lo

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(N

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Data

Predicted

Y error = Σ (Ypred. – Ydata)2

X error = Σ (Xpred. – Xdata)2

X error

Y error