Mathematical Modeling of Creep

1
Mathematical Modeling of Creep

• Dr. Robert Jones
• November 15, 2003

2
Objectives

• Increase awareness of the centrality of
  mathematical modeling of physical data to the
  practice of engineering
• Provide examples of applications of linear
  fitting and natural logarithms in the management
  of creep deformation
• Provide a classroom experiment which can yield
  raw data for building a mathematical model of
  creep.

3
Course Background

• Introduction to Materials Engineering
• Prerequisites Chemistry I, Algebra
• Course taken by Freshmen and Sophomores

4
Weaknesses in Student Preparation for Engineering

• Conceptual linkage between mathematical
  expressions and physical processes.
• Manipulation of data to derive constants
  exponents for modeling real processes
• Management of data in graphical formats
• Use of function tables (erf, Bessel, etc)

5
Creep

• Time dependent deformation of a
  material/component under load.
• Generally occurs at temperatures above 40 of the
  melt point of a material (on an absolute scale)

6
Definitions

• Strain intensity of deformation
• strain change in length/original length
• Stress intensity of force
• Stress force/ area (MPa, psi)
• Strain rate rate of change in strain with time,
  units of time-1

7
Time Dependent Deformation
Strain
Stress
TIME
8
Creep Temperature

• Will Lead Products Creep at Room Temperature?
• TM 327 ?C 600 ?K
• TROOM 23 ?C 296 ?K
• 100 x 296 / 600 49.3 ? YES!!

9
Where Does it Occur?

• Combustion chambers (gas oil fired furnaces)
• Pressurized piping (steam power plants, nuclear
  plants, heat exchangers)
• Blades and components in gas turbines (Jet
  engines, power plant turbines)
• Polymeric systems

10
Turbine Blades(Titanium)
http//www.moelleraircraft.com/airfoil2.htm
11
Power Turbine Blades
Source Dresser-Rand http//www.dresser-rand.com/e
-tech
12
Polymer Creep

• Most Commonly encountered in seals and gaskets
  that no longer seal
• All polymers will creep to some extent when under
  constant load

13
Polymeric SystemsValve Cover Gasket
http//www.trustmymechanic.com
14
Creep Modeling Approaches

• Total time to failure (rupture)
• Strain rate (which is used to determine when part
  has deformed too much for use)

15
Semi-Log DataCreep Rupture
David Woodford http//www.creep-rupture.com/paper1
.htm
16
Strain Rate Modeling

• Slope of secondary region is steady-state creep
  rate

17
Real DataHandout

• Graphical data (non-digital) is and will remain
  common in engineering for many years.
• Students should be familiar with principles of
  manual graphical data extraction.

18
Creep With Time in a Polymer
19
Creep Rate Stress

• d? K2 ?n exp (-QC / RT)
• dt
• For T constant
• d? K1 ?n
• dt

20
Linearize Creep Rate

• d? K1 ?n
• dt

21
Real Data

• If the proposed model is accurate, a plot of
  ln(strain rate) vs. ln(stress) should yield a
  straight line

22
Log-Log Data
Chris Wilson and Brett Marmo http//www.virtualex
plorer.com.au/VEjournal
23
Real Example

• Solve for n graphically on plot
• Solve for n using simultaneous equations

24
Other References

• NIST Database of Solder Properties
• http//www.boulder.nist.gov/div853/lead20free/pro
  ps01.html
• More Constitutive Models
• www.ntu.edu.sg/home/mwzhou/ Papers5CZhou2020022
  0-20J20E20Packaging20-20Creep20Model.pdf

25
Data Collection in Class

• Electrical Solder will Creep at room temperature
  and failure will occur in a reasonable time.
• Creep curves can be constructed from tests under
  two different loads (P 3P)
• Obtain steady-state rate from slope
• Determine constants from two stress level
• Predict strain rate at a third stress level (2P)
• Run test at that level and compare strain rate to
  prediction