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Temperature and Its Effects Part 2-Material and Structures-Lecture Handout, Exercises of Structures and Materials

This lecture note is part of Material and Structures course. It was provided by Prof. Aparijita Singh at Andhra University. It includes: Temperature, Effects, Balance, Energy, Heat, Flux, Density, Specific, heat, Capacity, Fourier, Equation, Conductivity, Diffusivity, Distribution, Internal, Stress

Typology: Exercises

2011/2012

Uploaded on 07/26/2012

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Do a balance of energy:
T
qz
T
qz
T + qz
dz = ρC T dz
z t
heat flux heat flux change in stored
-=
on side 1 on side 2 heat in strip
where:
ρ = density
C = specific heat capacity
t = time
this becomes:
T
q
z
= ρC T
z t
Recalling that:
T
q
z
= k
T
T
z
z
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12

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Do a balance of energy:

T

q z T

q^ z T

q z  

dz

ρ C

T

dz

 ∂ z  ∂ t

heat flux

heat flux

change in stored

on side 1

on side 2

heat in strip

where:

ρ

= density

t = time C = specific heat capacity

this becomes:

T

− (^) ∂ q z = ρ C ∂ T

z

t

Recalling that:

T

q z

(^) k T

T

z

z

we have:

k z T

∂ T  ^

ρ C

T

z

z (^) 

t

If k

z T and

ρ c are constant with respect to z (for one material they are),

then we get:

k T ∂ 2 T ∂ T z

=

ρ C ∂ z 2 ∂ t

Fourier

s equation

We call:

k z T

thermal diffusivity

ρ C

More generally, for 3-D variation:

k T

∂ T 

+ ∂  T ∂ T  ∂  T ∂ T

 ^

C

ρ

T

∂ x  x ∂ x

y

(^) k y

y ^ 

z

k z

z (^) 

t

thermal conductivities in x, y, and z directions

could get other cases where T peaks in the center, etc.

Result

Interna

l stresses

(generally)

arise if T varies spatially

. (unless it is

a linear variation which is unlikely given the governing equations).

Why

consider an isotropic plate with

T varying only in the y-direction

∆∆∆

Figure 9.

Representation of isotropic plate with symmetric y-variation

of

T about x-axis

(for the time being, limit

T to be symmetric with

respect to any of the axes)

locally

two elements side by side In order to attain this deformation, stresses must arise. Consider

Undeformed

Deformed

T

greater

piecestresses present to compress the top piece and elongate the bottom These two must deform the same longitudinally, so there must be

Thus:

ε x =

ε x (x)

ε y =

ε y (y)

This

physical

argument shows we have thermal strains, mechanical

strains and stresses.

self-equilibrating

a

− a

σ y dx

b

− b

σ x dy

Causes

Degradation of Material Properties

(due to

thermal effects)

Here there are two major categories

Static

Properties

temperature (generally, TModulus, yield stress, ultimate stress, etc. change with

property

transition at “glass transition temperature”Fracture behavior (fracture toughness) goes through a

ductile

brittle

T

g

(see Rivello)

Figure 9.

Representation of variation of ultimate stress with

temperature

Figure 9.

Representation of change in stress-strain behavior with

temperature

ductile as T increases)

(generally, behavior is more

--> Thus, must use properties at appropriate temperature in analysis

MIL-HDBK-5 has much data

“Other” Environmental Effects

may be important in both areasTemperature tends to be the dominating concern, but others

atomic oxygen degrades properties

UV degrades properties

etc.

strains:Same effects may cause environmental strains like thermal

Example - moisture

“swelling coefficient” =Materials can absorb moisture. Characterized by a

β ij

Same “operator” as

α ij (C. T. E.) except it operates on moisture

concentration, c:

s

ε ij

=

β ij c

“swelling”

moisture concentration

strain

swelling

coefficient

and then we have:

ε ij

=

ε ij M

ε ij T

ε ij S

total

effect is: This can be generalized such that the strain due to an environmental

environmental

strain

E

ε ij

=

χ ij X

environmental

environmental

operator

scalar

This deals with the field of A strain of this “type” has become important in recent work.environmental strains and the total strain is the sum of the mechanical strain(s) and the

(consider the case with only mechanical and piezoelectric strain)And we add this strain to the others to get the total strain

ε ij

=

ε ij M

ε ij p

Again, only the mechanical strain is related

directly

to the stress:

ε ij

=

S ijmn

σ mn

d ijk

E k

inverting gives:

σ ij

=

E

ijmn

ε mn

E ijmn

d mnk

E

k

(watch the switching of indices!)

thus we have “piezoelectric-induced” stresses of:

E

ijmn

d mnk

E

k

Again, equilibrium if the piezoelectric expansion is physically resisted.

( ∫ (^) σ

=

F)

must

be satisfied.

But

, unlike environmental cases, the electric field is not just an external

equation:parameter from some uncoupled equation of state but there is a coupled

D

i

=

e ik

E k

d inm

σ mn

dielectric constant from previous equationnote switch in indices since this is transpose of

where:

e ik = dielectric constant

D

i = electrical charge

is assumed. That is, E --> “Normally”, when piezoelectric materials are utilized, “E-field control”

k is the independent variable and the electrical

when charge constraints are imposed the simultaneous set of equations:charge is allowed to “float” and take on whatever value results. But,

σ mn

E

mnij

ε ij

E mnij

d ijk

E k

D

i

e ik

E k

d inm

σ mn

(mechanical, etc.) must be solved. This is coupled with any other sources of strain