MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING

CAMBRIDGE, MASSACHUSETTS 02139 3.22 MECHANICAL PROPERTIES OF MATERIALS

PROBLEM SET 4 SOLUTIONS

*1. * *Consider a 500 nm thick aluminum film on a 500 µm thick silicon wafer, 200 mm in diameter. After *
*deposition at 60°C, the wafer is cooled to room temperature. What is the radius of curvature of the *
*film? (E**Si **= 150 GPa, ν*_{Si }*= 0.17, α*_{Si }*= 3×10*^{-6 }^{°}*C*^{-1}*, E**Al **= 69 GPa, ν*_{Al }*= 0.33, α*_{Al }*= 23×10*^{-6 }^{°}*C*^{-1}*) *
The use of the Stoney formula is appropriate for this system because the aluminum film is very thin
compared to the relatively thick substrate. The Stoney formula is given by

2

6

*s s*
*m f*

*E h*
ρ *h*

= σ ,

where ρ is the radius of curvature, *E*_{s}*is the biaxial modulus of the substrate, h**s** and h**f** are the substrate *
and film thickness, and σ is the stress at the interface. _{m}

The biaxial modulus of the substrate is calculated by 150 GPa

180.7 GPa

1 1-0.17

*s* *s*
*s*

*E* *E*

= ν = =

− .

The stress at the interface in the aluminum film is given by 1

*f*

*m* *m* *f* *m*

*f*

*E* *E*

σ ε ε

= = ν

− .

To find the strain at the interface, we must calculate the mismatch strain due to the thermal expansion coefficient mismatch.

6 -1 6 -1

( ) (3 10 C 23 10 C ) (20 C 60 C) 0.0008

*m* *s* *f* *T*

ε = α α− ⋅ ∆ = × ^{−} ^{D} − × ^{−} ^{D} ⋅ ^{D} − ^{D} =
Solving for the mismatch stress,

69 GPa

0.0008 82.39 MPa

1 1 0.33

*f*

*m* *m* *f* *m*

*f*

*E* *E*

σ ε ε

= = ν = =

− −

The radius of curvature of the film-on-substrate system can be calculated from the Stoney formula.

2 6 2

9

(180.7 GPa)(500 10 m) 182.8 m 6 6(0.08239GPa)(500 10 m)

*s s*
*m f*

*E h*
ρ *h*

σ

−

−

= = × =

×

The positive curvature (concave) indicates that the film is in tension after cooling to room temperature.

Because the thermal expansion coefficient for aluminum is larger than silicon, the aluminum attempts to contract more than the silicon during the cooling. However, at the interface, the aluminum cannot contract fully due to the constraining silicon, and as a result the aluminum is in tension.

*2. A bilyer is composed of a 1 mm thick coating (E*=350 GPa,α = ×9 10^{−}^{6} ^{D}C^{−}^{1}*) on a 5 mm substrate *
*(* *) and has a width of 5 mm. The bilayer has zero curvature at the initial *
*temperature of 20°C. The bilayer is then heated to 100°C. Calculate the resulting curvature and the *
*maximum tensile and compressive stresses. Where do the maximum internal stresses occur? *

70 GPa, 23 106 C
*E*= α = × ^{−} ^{D} ^{−}^{1}

To find the curvature of the bilayer, we can use the equation for the most general bilayer derived in class

2 1 0

1 1 2 1

1 1 2 2

( )( )

2( ) 1 1

2

*T T*
*E I* *E I*
*h*

*h* *E a b* *E a*

κ= α α− −

+ + +

* ^{b}*

where the coating properties are denoted by the subscript 1 and the substrate properties by 2. To calculate the curvature, we need the moments of inertia for each layer.

3 3

13 2 1

1

(0.005 m)(0.001 m)

4.1667 10 m

12 12

*I* =*ba* = = × ^{−}

3 3

11 4 2

2

(0.005 m)(0.005 m)

5.20833 10 m

12 12

*I* =*ba* = = × ^{−}

Entering the parameters of the bilayer system into the given equation, we find that the curvature of the
bilayer is κ =0.25 m^{−}^{1} .

The maximum stresses for a bilayer were derived in class and exist at the interface of the two layers.

1 1 2 2 1 1

max,1

1

1 1 2 2 2 2

max,2

2

2( )

107 MPa 2

2( )

56.4 MPa 2

*E I* *E I* *a E*
*ha b*

*E I* *E I* *a E*
*ha b*

σ κ

σ κ

+

= + =

+

= − + = −

*3. The structure and mechanics of cork: The structure and properties of cork are approximately *
*axisymmetric. In the schematics on the next page, we see a macroscopic view of cork as it exists on a *
*tree, a microscopic look at the arrangement of cork cells, and finally a view of an individual cork cell. *

*Ignoring the natural variations in cell structure, we can model this cellular material as a linear-elastic *
*solid by *

*ij*

*S*

*ijkl*

*kl*

## ε = σ

.*Axisymmetry reduces the number of independent compliances to five (note that the axis of symmetry is *
*the x**1** axis). *

11 1111 11 1122 22 1122 33

22 1122 11 2222 22 2233 33

33 1122 11 2233 22 2222 33

23 2222 2233 23

31 1212 31

12 1212 12

( )

2 2

*S* *S* *S*

*S* *S* *S*

*S* *S* *S*

*S* *S*

*S*
*S*

ε σ σ σ

ε σ σ σ

ε σ σ σ

ε σ

ε σ

ε σ

= + +

= + +

= + +

= −

=

=

*Measurement of four of the five compliances is straightforward. Simple tensile or compressive tests are *
*done on the cork along the axis of symmetry (x**1** axis) and orthogonal to this direction (x**2** and x**3** axes) to *
*measure the Young’s modulus and Poisson’s ratio in these directions. *

1 1111

2 3 2222

12 1 13 1 21 2 31 2 1122

23 2 32 2 2233

1

1 1

*E* *S*

*E* *E* *S*

*E* *E* *E* *E* *S*

*E* *E* *S*

ν ν ν ν

ν ν

=

= =

= = = = −

= = −

*The modulus G**23** in the x**2**, x**3**-plane is obtained from these measurements by *

23 23 2 2222 2233

1*G* =2(1+ν ) *E* =2(*S* −*S* )*. *

*To determine the fifth compliance value, S**1212**, which is related to the shear modulus, G**12**, we rotate the *
*cork through 45° about the x**3**-axis and cut a cube with one face normal to the x**3**-axis, and the other two *
*at 45° to the x**2**-axis and x**1**-axis. A simple compression test in the new x**1** direction then gives a new *
*Young’s modulus, *

*E′*

*.*

*a. What is the relationship between the shear modulus, G**12**, and S**1212**? *

*The shear modulus G*12 relates the engineering shear strain and shear stress. Using matrix
notation, we have

6 66 6

12

*S* 1

*G* ^{6}

ε = σ = σ .

To convert between matrix and tensor notation, we have the relation that

66 4 1212

*S* = *S* .
Thus we find that

12 1212

14
*G* = *S* .

*b. By rotating the S*_{ijkl}* through 45° about the x**3**-axis, obtain an expression for * * in terms of the *
*compliances in the original orientation. *

*S′*1111

*The direction cosines for this change in orientation by rotation about the x*3-axis is

1 1 0

2 2

cos 45 sin 45 0

1 1

sin 45 cos 45 0 0

2 2

0 0 1 0 0

*ij*

−

1

−

= =

D D

D D

A .

The tensor equation for the transformation of a fourth order tensor is *S** _{ijkl}*′ =A A A A

_{mi nj ok}

_{pl}*S*

*. To find the*

_{mnop}*S′*

_{1111}component, we use the transformation rule.

1111 *m*1 *n*1 *o*1 *p*1 *mnop*

*S*′ =A A A A *S*

Although the expanded form of the tensor transformation has 81 terms per equation, we note that A and A are the only nonzero direction cosine terms that appear in the tensor

transformation. The resulting expanded form is

11 21

1111 11 11 11 11 1111

11 11 11 21 1112 11 11 21 11 1121 11 21 11 11 1211 21 11 11 11 2111 11 11 21 21 1122 11 21 21 11 1221 11 21 11 21 1212

21 11 11 21 2112 21 11 21 11 2121 21 21 11

*S* *S*

*S* *S* *S* *S*

*S* *S* *S*

*S* *S*

′ = +

+ + +

+ + +

+ +

A A A A "

A A A A A A A A A A A A A A A A "

A A A A A A A A A A A A "

A A A A A A A A A A A A_{11 2211}

21 21 21 11 2221 21 21 11 21 2212 21 11 21 21 2122 11 21 21 21 1222 21 21 21 21 2222

*S*

*S* *S* *S* *S*

*S*

+

+ + +

"

A A A A A A A A A A A A A A A A "

A A A A

+

+

From the symmetry of the strain tensor and stress tensor that are related by the compliance tensor,
we can simplify the expression. (Note that *S*_{1112} and *S*_{2212} are zero)

4 3

1111 ( 11) 1111 4( 11) ( 21) 1112

*S*′ = A *S* + A A *S* +2(A_{11}) (^{2} A_{21})^{2}*S*_{1122}+4(A_{11}) (^{2} A_{21})^{2}*S*_{1212}+4(A_{21}) (^{3} A_{11})*S*_{2221} + A( _{21}*) S*^{4} _{2222}

In tensor notation, the expression for the compliance in the new direction is

1111 1111 1122 1212 2222

1 1 1

4 2 4

*S*′ = *S* + *S* +*S* + *S*

*c. Use your answers from parts (a) and (b) to find an expression for the shear modulus G**12**, in *
*terms of experimental parameters E E*_{1}, _{2},ν_{12}, and*E′ . *

From part (a) we have

1212 12

1 4*S*

*G* = .

*From the answer to part (b) we can solve for S*1212 and substitute this expression into the
equation above.

1111 1111 1122 2222

12

1 1 1 1

4 *S* 4*S* 2*S* 4*S*
*G*

′

= − − −

Replacing the compliance values with the corresponding experimental values leads to the result
(Note that the value of the compliance in the new direction is related to the new Young’s
modulus by *S*_{1111}′ =1*E*′.)

12

12 1 1 2

12

12 1 2

1 1 1 1 1 1 1

4 4 2 4

1 2

1 4 1

*G* *E* *E* *E* *E*

*G* *E* *E* *E*

ν ν

−

= ′− − −

= − − −

′

.

*d. For the isotropic case, what does the equation derived in part (c) become? *

For an isotropic material, which is more symmetric than the transversely isotropic symmetry assumed above, the moduli and Poisson’s ratios do not vary with direction. Hence, we can rewrite the answer in part (c) as

1 4 1 2 1

*G* *E* *E* *E*

ν

= − − − . Simple algebra leads to the result

1 4 1 2 1

1 2 2

2(1 )

*G* *E*

*G* *E*

*G* *E*
ν
ν

ν

− + −

=

= +

⇒ =

+

Thus, the expression in part (c) correctly reduces to the relationship between the elastic modulus, shear modulus and Poisson’s ratio for the isotropic case.

*4. An isotropic sample of material subjected to a compressive stress σ = − is confined so that it cannot *_{z}*p*
*deform in either the x- or y-directions. *

*a. Do the stresses occur in the material in the x- and y-directions? If so, how are they related to *
*σ ? **z*

If the sample were not confined in the x- and y-directions, we would expect the material to expand in the transverse direction upon compressive loading. The fact that the material is constrained in the transverse direction indicates that a stress must evolve to maintain zero strain in the transverse direction. Looking at Hooke’s law for an isotropic material in three dimensions, we find

( )

*x* *x* *y*

*E* *E*

σ ν

ε = − σ +σ* _{z}* = . 0

Due to the symmetry of the problem, we know σ* _{x}* =σ

*. Thus, we can find a relationship between the transverse stress and the applied compressive stress*

_{y}*x* *y* 1 *z*

σ σ ν σ

= = ν

− .

*b. Determine the stiffness E*′ =σ ε_{z}_{z}* in the direction of the applied stress in terms of the isotropic *
*elastic constants and E* *ν for the material. Is E′ equal to the elastic modulus from uniaxial *
*loading? Why or why not? *

Again using Hooke’s law to find the strain in the z-direction, 2 2

( ) 1

1

*z* *z*

*z* *x* *y*

*E* *E* *E*

σ ν σ ν

ε σ σ

ν

= − + = − − .

Using this expression for the strain in the z-direction, we can calculate the stiffness in the z- direction.

2 2

2 2

1 1

1 1

*z* *z*

*z* *z*

*E* *E*

*E*

σ σ

ε σ ν ν

ν ν

′ = = =

− −

− −

The stiffness calculated here is not equal to the elastic modulus in uniaxial loading. In uniaxial loading, the transverse stresses are zero and do not contribute to the uniaxial deformation. However, in this case the transverse stresses do affect the strain in the direction of loading.

*c. What happens if the Poisson’s ratio for the material approaches 0.5? *

As Poisson’s ratio approaches 0.5, the strain in the z-direction goes to zero for any applied
compressive stress. This can be understood without derivation. We know that materials with a
Poisson’s ratio of 0.5 are incompressible (ε* _{x}*+ε

*+ε*

_{y}*= ). Because the transverse directions are 0 constrained (zero strain), the strain in the loading direction must be zero to satisfy the*

_{z}incompressibility condition, regardless of the applied compressive stress.

*5. Consider a flat plate of isotropic material that lies in the x-y plane and which is subjected to applied *
*loading in this plane only. Such a plate is under plane stress, so that *σ* _{z}* =τ

*=τ*

_{yz}*=0 MPa*

_{xz}*.*

*a. Does the thickness of the plate usually change when the plate is loaded? *

For all non-zero Poisson’s ratio materials, there will be some change in the thickness of the plate due to the Poisson effect. By inspecting Hooke’s law for a three dimensional isotropic material

( )

*z*

*z* *E* *E* *x* *y*

σ ν

ε = − σ +σ

we notice that even in a state of plane stress, the stresses in the orthogonal directions contribute to the strain in the out-of-plane direction.

*b. Under what conditions does the thickness not change? That is, when is this state of plane stress *
*also a state of plane strain? *

Looking at Hooke’s law again, it is clear that if the stresses in the orthogonal directions to the out-of- plane direction are equal and opposite, the out-of-plane strain is necessarily zero, indicating no thickness change.