Thermal Behavior of Materials
ME 2105
Dr. R. Lindeke
Some Definitions
• Heat Capacity: the amount of heat (energy)
required to raise a fundamental quantity of a
material 1 K˚
• The quantity is usually set at 1 gm-atom (elements) or 1
gm-mole (compounds)
• Given by the formula: C = q/(mT) in units of J/gmatom* K˚ or J/gm-mole* K˚
• Specific Heat: a measure of the amount of
heat energy to raise a specific mass of a
material 1 K˚
Heat Capacity
• Heat capacity is reported in 1 of two ways:
• Cv – the heat capacity when a constant volume of
material is considered
• Cp – the heat capacity when a constant pressure is
maintained while higher than Cv these values are nearly
equal for most materials
• Cp is most common in engineering applications (heat
stored or needed at 1 atm of pressure)
• At temperature above the Debye Temperature Cv  3R 
Cp
Definitions
• Thermal Expansion is the “growth” of materials due to
increasing vibration leading to larger inter-atomic distances
and increasing vacancy counts for materials as temperature
increases
Thermal Expansion
• Linear thermal expansion is given by this
model:
dL

LdT
• As an Example:
• A gold ring (diameter = 12.5 mm) is worn by a person,
they are asked to wash the dishes at their apartment –
water temperature is 50˚C – how big is the ring while it
is submerged?
Thermal Expansion is
“Temperature Dependent”
Solving:
L   L0 T
L0  d 0 *   12.5*3.14159  39.2699
 50   27  50  14.1
 50 

 527   27 16.5  14.1
data from table 7.2
 50  14.21x10 (mm/mm C )
6
L   avg L0 T  14.21*106 *39.27 *12.5  0.007 mm
Df
39.27  .007 



 12.502mm
Definition:
• Thermal
dQ
Conductivity: the
dt
k 
transfer of heat
dT
A
energy through a
dx
material (analogous at steady state conditions:
to diffusion of mass)
Q
• Modeled by:
k 
t



x
A T

Note, k is a function of temperature (like  was)
• Modeling Fourier’s Law of Thermal Conduction (heat
flow thru a bounded area)
Thermal Conductivity
• Involves two primary (atomic level) mechanisms:
• Atomic vibrations – in ceramics and polymers this dominates
• Conduction by free electrons – in metals this dominates
• Focusing on Metals:
• thermal conductivity decreases as temperature increases
since atomic vibrations disrupt the primary free electron
conduction mechanism
• Adding alloying “impurities” also disrupts free electron
conduction so alloys are less conductive than pure metals
Thermal Conductivity
• Focusing on Ceramics and Polymers:
• Atomic/lattice vibrations are “wave-like” in nature and
impeded by structural disorder
• Thermal conductivity will, thus, drop with increasing
temperature
• In some ceramics, which are “transparent” to IR radiation,
TC will eventually rise at elevated temperatures since radiant
heat transfer will begin to dominate “mechanical”
conduction
• Porosity level has a dramatic effect on TC (pores are filled
with low TC gases which limits overall TC for a structure
(think fiberglass insulation and ‘strya-foam’ cups)
And Continuing:
Thermal conductivity of several ceramics over a range
of temperatures.
(From W. D. Kingery, H. K.
Bowen, and D. R. Uhlmann,
Introduction to Ceramics,
2nd ed., John Wiley & Sons,
Inc., New York, 1976.)
Definition:
• Thermal Shock: it is simply defined as the fracture of a
material (often a brittle ceramic) as the result of a
(sudden) temperature change and is dependent on the
interplay of the two material behaviors: thermal
expansion and thermal conductivity
• Thermal Shock can be explained in one of two ways:
• Failure stress can be built up by constrained thermal expansion
• Rapid temperature changes lead to internal temperature gradients
and internal residual stresses based in finite thermal conductivity
reasoning
By Constrained Thermal Expansion:
Thermal shock resulting from constraint of uniform thermal expansion. This
process is equivalent to: a. free expansion followed by; b. mechanical compression
back to the original length.
Let’s Consider an Example:
• A 400 mm long ‘rod’ of Stabilized ZrO2 ( =
4.7x10-6 mm/mm˚C) is subject to a thermal
cycle in a ‘ceramic’ engine – it’s the crank
shaft! – from RT (25˚C) to 800˚C. Determine
the induced stress and determine if it is likely
to fail?
• E for Stabilized ZrO2 is 150 GPa
• MOR for Stabilized ZrO2 is 83 MPa
 TI  E TI
where:  TI 
lT exp
l0
  TI  T  4.7 *106 *  800  25   0.00364 mm
mm
 TI  150GPa *0.00364 mm mm  0.546GPa  546MPa
Since the Inducted Compressive Stress exceeds the MOR one might expect the
‘rod’ to fail or rupture – unless it is allowed to expand into a designed in
‘pocket’ built into the engine block to accept the shaft’s expansion
By Thermal Conductivity (induced)Temperature
Gradients:
Thermal shock resulting from temperature gradients created by a finite thermal conductivity.
Rapid cooling produces surface tensile stresses and Griffith Crack Generation.
(From W. D. Kingery, H. K.
Bowen, and D. R. Uhlmann,
Introduction to Ceramics, 2nd
ed., John Wiley & Sons, Inc.,
New York, 1976.)
Thermal quenches that produce failure by thermal shock are illustrated. The temperature drop
necessary to produce fracture (T0 − T) is plotted against a heat-transfer parameter (rmh). More
important than the values of rmh are the regions corresponding to given types of quench (e.g.,
water quench corresponds to an rmh around 0.2 to 0.3).
Thermal Shock Resistance
• Occurs due to: uneven heating/cooling.
• Ex: Assume top thin layer is rapidly cooled
rapid quench

tries to contract during cooling
T2
resists contraction
T1
Temperature difference that
can be produced by cooling:
(T1  T2 ) 
quench rate
k
from T1 to T2:
Tension develops at surface
  E(T1  T2 )
Critical temperature difference
for fracture (set  = f)
(T1  T2 )f racture 
f
E
set equal
• Result:
f k
(quench rate) f or f racture 
E
• Large thermal shock resistance when
is large.  f k
E
22
Thermal Shock Resistance Parameter
fk
TSR 
E l
Where:
• f is fracture strength of a material
• l is coeff. Of linear thermal expansion
• k is thermal conductivity of material
• E is modulus of elasticity