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Temperature and Its Effects Part 1-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, Expansion, Contraction, Change, properties, Coefficient, Thermal, Strain, Mechanical, Elasticity, Compliance, Unconstrained, Deformation

Typology: Exercises

2011/2012

Uploaded on 07/26/2012

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Thus far we have discussed mechanical loading and the
stresses and strains caused by that. We noted, however, that
the environment can have an effect on the behavior of
materials and structures. Let’s first consider:
Temperature and Its Effects
2 basic effects:
expansion / contraction
change of material properties
Look, first, at the former:
Concept of Thermal Stresses and Strains
Materials and structures expand and contract as the temperature
changes. Thus:
εT = α ( T)
thermal temperature change
strain coefficient of thermal expansion (C.T.E.) units: 1
degrees
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materials and structures. Let’s first consider:the environment can have an effect on the behavior ofstresses and strains caused by that. We noted, however, thatThus far we have discussed mechanical loading and the

Temperature and Its Effects

2 basic effects:

expansion / contraction

change of material properties

Look, first, at the former:

Concept of Thermal Stresses and Strains

changes. Thus: Materials and structures expand and contract as the temperature

ε T

= (^) α (^) ( ∆ T)

thermal

temperature change

strain

coefficient of thermal expansion (C.T.E.)

units:

degrees

then “thermal stresses” can arise.If these thermal expansions / contractions are resisted by some means,

However

, “thermal stresses” is a

misnomer, they are really…

“stresses due to thermal effects”

-- stresses are

always

“mechanical”

(we’ll see this via an example)

--> Consider a 3-D generic material.

Then we can write:

T

ε ij

=

α ij ∆ T

i, j = 1, 2, 3

(as before)

α ij (^) = 2nd order tensor

The

(^) total strain

of a material is the sum of the

mechanical strain

(^) and the

thermal strain

mechanical

thermal

M

T

ε ij =

ε ij

ε ij

total

Unit 9 - p. 3

σ kl

=

E

ijkl

ij −

E

ijkl

ij

T

thermal terms: This is the same equation as we had before except we have the

E

ijkl

α ij ∆ (^) T

--> so how does a “thermal stress” arise?

Consider this example

mechanical strain is zero and thus the “thermal stress” is zero.That is, the thermal strain is equal to the total strain. Thus, theSince it is unconstrained it expands freely and no stresses occur. If you have a steel bar lying on a table and heat it, it will expand.

Figure 9.

Free thermal expansion of a steel bar

Fall, 2002

--> However, if the bar is constrained, say at both ends:

Figure 9.

Representation of constrained steel bar

the same as in the previous case Then, as it is heated, the rod cannot lengthen. The thermal strain is

but

(^) now the total strain is zero (i.e.,

Starting with (in one direction):no physical deformation).

ε

ε M

(^) ε T

with:

ε (^) = 0

Thus, the mechanical strain is the negative of the thermal strain.

Fall, 2002

Values of C.T.E.

’s

Note

α ij (^) = (^) α ji

Anisotropic Materials 6 possibilities:

α 11 , α 22 , α 33 , α 12 , (^) α 13 , (^) α 23

T can cause shear strains

not true in “engineering” materials

Orthotropic Materials 3 possibilities:

α 11 , (^) α 22 , (^) α 33

T only causes

extensional

(^) strains

Notes

are interested only inGenerally we deal with planar structures and

α 11

and

α 22

principal material axes, we can “have” anIf we deal with the material in other than the

α 12

Transformation obeys same law as strain (it’s a tensor).

2-D form:

α˜ αβ^

l

l βγ (^) α σγ

ασ

˜

˜

α 11 , α 22

(in - plane values)

α 12

0 (in material axes)

3-D form:

α ij =

l

l j l˜ α kl

i k˜

So, in describing deformation in some axis system at an angle

(^) θ (^) to

the principal material axes…..

Figure 9.

Representation of 2-D axis transformation

y^ ~ 1 y^2

θ

y 2

θ

+ CCW

y 1

Notes

Graphite/epoxy has a

negative

C.T.E. in the fiber direction so it

contracts when heated

Implication

: by laying up plies with various orientation can

desired directionachieve a structure with a C.T.E. equal to zero in a

C.T.E. of a structure depends on C.T.E.

(^) and

elastic constants of

the parts

E = 0 (perfectly compliant) E = E

1

C.T.E. = 5

2 examples

total

α (^) of

C.T.E. = 2

structure

E = E

1

C.T.E. = 5

total

α (^) of

C.T.E. = 2

structure

E =

(perfectly rigid)

Fall, 2002

α (^) =

α (T)

C.T.E. is a function of temperature (see MIL HDBK 5 for metals). Can be large difference.

Implication

: a zero C.T.E. structure may not truly be

attainable since it may be C.T.E. at T

1 but not at T

2 (^)!

(^) Sources of temperature differential (heating)

shadow, tropics, etc.)ambient environment (engine, polar environment, earth

aerodynamic heating

radiation (black-body)

(^) Constant

T

(with respect to spatial locations)

In many cases, we are interested in a case where

T (from some reference

temperature) is constant through-the-thickness, etc.

thin structures

structures in ambient environment for long periods of time

Relatively easy problem to solve. Use:

equations of elasticity

equilibrium

stress-strain

Example 3

  • bimetallic strip

Each metal has different

α ’s.

What will happen?

Bending!

--> Concept of

(^) self-equilibrating stresses

Must always be in equilibrium. General equation is:

σ dA

F

where: F = externally applied force

If F = 0, can we still have stresses?

Yes

, but they must be “self-equilibrating” (satisfy equilibrium in and

of themselves):

σ dA

spatially-varyingof structures with varying properties or This is the case of free expansion (thermal)

T

(^) (we’ll address this in a bit)

If α 1 (^) > (^) α 2 and

T > 0

T

T

ε ij

ε ij

(^1)

2

(we’ll see more about this bending later)

Bimetallic strip used as temperature sensors!

T varies spatially

(and possibly with time as well, we analyze at

any constant in time)

Must determine

T by looking at heat flux into structure. Three basic

methods:

induction

convection

radiation

Source of heat here is from air boundary layer:

q = h

(T

AW

(^) - T

s ) surface temperature of body

heat flux

heat transfer

[watts/M

2 ]

(convective constant)coefficient

(h is determined from boundary layer theory)

--> 2 considerationsatmosphere). Always important but especially in space (or at high temperature in Radiation

Emissivity

--> surface emits heat

q = -

ε σ

T

s 4

surface temperature

fluxheat

Stefan-Boltzmann constant

(a material property) emissivity

Absorptivity

heat

q =

α (^) I s (^) λ

angle factor

flux

intensity of source

(a material property)absorptivity

Figure 9.

Representation of heat flux impinging on structure

structure angle of

like the sun

I s (^) (intensity of source)

Figure 9.6 consider:

Representation of structure exposed to two environments

Environment 1

Environment 2

look at a strip of width dz:

Figure 9.

Representation of heat flow through infinitesimal strip of

material

T

q z

T

T

q z + ∂ q z

dz

∂ z