Two Simple Models of Thermal Stress Voller-Guzina-Stelson University of Minnesota

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

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

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

4. tension + - compression Why ? 1. Flow Shear 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

5. tension + - compression Why ? 2. Rate Effect Glass-Transition increases with cooling rate 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 Compression Tension

6. 3. Flow Strain tension + - compression Why ? 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

7. tension + - compression A Simple Model Of Residual Stress Based On Flow-Strain Concept (After Osswald and Menges) z At ant time t uniform strain in the solid is Q elastic thermal flow SOLID 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.

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

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

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

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

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

13. 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

14. Ceramic Film—Use Model to Predict Properties

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