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 Time, t (sec) Displacement, u (mm) t=0 t=180 u*=2.0 mm t=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 mm Gel diameter = 23 mm Gel thickness ~ 4 mm Pictures and graphs provided by Dr. Shaw (CLU)

Ryan Palmer Loading of specimens

Ryan Palmer Stress-relaxation Curve

Viscoelastic models Viscoelastic materials have been modeled by a mixture of Maxwell, Voight and Kelvin models 3. These models consist of spring and dashpot setups. Dashpot: A pneumatic or hydraulic cushion for a falling weight, to prevent shock 5.

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 Model- Consists 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.

Ryan Palmer Stress-relaxation Curve

Ryan Palmer Stress-relaxation Curve Y error = Σ (Y pred. – Y data ) 2 X error = Σ (X pred. – X data ) 2 X error Y error