Two Simple Models of Thermal Stress Voller-Guzina-Stelson University of Minnesota 2. Crack Patterns in Thermal Processing Residual stress in solidification

With all the bells and whistles The Approach A basic approach Use A Full FEM Solution With all the bells and whistles CAN lead to A Kitchen Sink Model “Phenomenological Noise” An Alternative/Supplemental Simple Semi-Analytical Lower Dimensional Models CAN lead to Leading to Back-of-the-envelope-calculations and Insight

When a Polymer is solidified in a A Residual Stress Model On final Solidification A Residual Stress will be Observed When a Polymer is solidified in a Rectangular mold Q z = 0 surface z = zg z = b mid-plane Solid Liquid Mold tension + - compression z = 0 surface z = b mid-plane need a non-linear and/or rate dependent behavior

Why ? 1. Flow Shear solid liquid In solid: straight coils will try compression tension - + Shear will straighten polymer coils solid liquid In solid: straight coils will try and re-coil leading to tension in high shear regions and compression in low shear In liquid –high shear flow will straighten out polymer coils Opposite Of Observation

Glass-Transition increases Why ? 2. Rate Effect Glass-Transition increases with cooling rate compression tension - + v Q fast slow T Tg Cooled-- FAST SLOW Consider isolated lamella Opposite Of Observation At room temp. Isolated lamella At surface will Shrink MORE If Lamella Are Stuck Together Tension Compression

Why ? 3. Flow Strain Layer at solid-liquid Front under goes tension compression tension - + Layer at solid-liquid Front under goes A FLOW STRAIN To join existing solid Q SOLID Initial “flow” strain in surface lamella is smaller than initial Flow strain in center lamella Consider isolated lamella BUT Once Solid undergo same thermal deformation If Lamella Are Stuck Together As Required Compression Tension

A Simple Model Of Residual Stress Based On Flow-Strain Concept z (After Osswald and Menges) At ant time t uniform strain in the solid is Q SOLID elastic thermal flow Flow strain at a given position zs is “frozen in place” at point of solidification This flow strain will be the average of the thermal and Flow strain in the existing solid If We know Temperature history in Space and Time we can calculate flow strain—and determine stress at room temp. compression tension - +

zs Use A HEAT BALANCE INTEGRAL WITH VAM tension + - compression Actual Temperature Profile Approximate Temperature Profile compression tension - + Use A HEAT BALANCE INTEGRAL WITH VAM zs VAM  Real Temp Linear Approx. Fit with numerical model

On an injected molded starch based polymer blends Comparison with LHBI-VAM Model and Experimental (surface removal) measurements On an injected molded starch based polymer blends Journal Of Thermal Stress 25 (2002)

Spacing in Jointed Rock A Crack Spacing Model Consider a Film placed on a substrate and subjected to a thermal strain Often Observe Characteristic Crack Spacing Ceramic Film- 2.56% substrate strain Cooled Asphalt Spacing in Jointed Rock 150 meter 150 micron Bai, Pollard &Gao, Nature, 403, 753-756

Elastic Rod with an Elastoplastic Restraint imposed at the film/substrate interface Interface shear stress at failure Spring coefficient (Winkler Foundation)

Elastoplastic Restraint elastic plastic substrate stiffness x =l/2 xt x =l

S For a given temperature drop there is a Characteristic size Smaller and max stress can not reach S Larger and maximum stress Will exceed Strength S Also-- with increasing temp. drop would expect increase in crack density (cracks per unit length) UPTO where failure along the ENTIRE interface is plastic Increase Strain Can-Not Increase Stress Fixed slope

Ceramic Film—Use Model to Predict Properties tf 0.233 GPa 521 GPa sres 10.4 GPa Esteel 190 GPa nsteel 0.3 S 0.853 GPa

Simple Models Can:-- Identify Contributing Phenomena compression tension - + Predict Material Properties Shear Strength Residual Stress