HANDOUT 2 - Component design and composites

Question 1

What is the equation for Poisson’s ratio, v?

Answer 1

v = - Lateral strain/Tensile strain

(note minus sign, so v is positive)

Note: Laterial strain is not due to volume conservation, but reflects the way atomic bonds deform under load.

Question 2

What does Poisson’s ratio matter in design?

Answer 2

• not important in most deisgn with uniaxial loads(e.g truss)
• important when stress state is 2D or 3D (constrained expansion)
• important in vibration of plates ( affects frequencies of vibration modes )
• important in large strain bending, giving anticlatic curvature.

Question 3

Consider a unit cube of material, under a general set of normal stresses.

Find the resulting strains.

Answer 3

Strains due to stress σ_{1 :}

ε₁ = σ₁/E

ε₂ = (-vσ₁)/E

ε₃ = (-vσ₁)/E

Repeat for each stress in turn, and sum the strains:

ε₁ = (σ₁ -vσ₂ - vσ₃)/E

ε₂ = (-vσ₁ σ₂ - vσ₃​)/E

ε₃ = (-vσ₁ -vσ₂ + σ₃​)/E

Question 4

Give the equation for dilation, ∆

(also known as volumetric strain)

Answer 4

∆ = △V/Vo

Question 5

Consider the unit cube, for a general strain state ( ε₁ , ε₂ , ε₃ ):

Initial volume : V_o= 1

Find the final volume.

Answer 5

Final cube dimensions : (1+ε₁ , 1+ε₂ , 1+ε₃ )

v = (1+ε₁) x (1+ε₂) x (1+ε₃)

= 1 + ( ε1 + ε₂ + ε₃) + (higher order terms)

so ∆ = ε₁ + ε₂ + ε₃

For small strains (ε <<1)

Question 6

A state of hydrostatic stress is when all three normal stresses are equal, under uniform external pressure p: σ₁ = σ₂ = σ₃ = -p

This loading occurs in ceramic manufacturing.

Calculate the strain of each axis of the cube and hence find the dilation and also the bulk modulus, K.

Answer 6

ε₁ = ε₂ = ε₃ = (-p + vp + vp)/E = -p(1-2v)/E

Hence dilation is: ∆ = -3p(1-2v)/E

K = E/(3(1-2v))

Question 7

What is the equation for bulk modulus, K?

(units: GPa)

Answer 7

K = Hydrostatic stress/Volumetric strain

Question 8

Why is rubber incompressible? ( Don’t explain in terms of microstructure, explain in terms of posion ratio)

Answer 8

Rubber has a poison ratio of v = 0.5 which leads to a bulk modulus of infinity.

Question 9

Define shear stress

Answer 9

Force per unit area carried parallel to a plan within the material.

Question 10

What is the equation for the shear modulus G?

Answer 10

G = Shear stress/Shear strain = τ/γ (units: GPa)

Shear strain: γ = w/l_o

Question 11

What is the eqatuon that relates shear modulus G and Young’s modulus E?

Answer 11

G = E/(2(1+v))

Question 12

define isotropic

Answer 12

same properties in all directions

Question 13

Consider a cube of material fitted into a square-section slot in a rigid plate, and loaded with a compressive stress σ₁.

Find the “effective modulus”. (σ₁/ε₁)

Answer 13

ε₃ = 0 = -vσ₁/E + σ₃/E ( note σ₂ = 0 )

Hence: σ₃ = vσ₁ (both compressed)

Strain in the 1-direction (due to both stresses) is given by:

ε₁ = σ₁/E - vσ₃/E = σ₁/E - v²σ₁/E = σ₁(1-v²)/E

Hence the “effective modulus”, σ₁/ε₁ = E/(1-v²)

Question 14

What is the equation for thermal strain, ε_thermal?

Answer 14

ε_thermal​ = α x ∆T

When total strain = 0, εelastic = -ε_thermal

Question 15

If a metal railway track experiences a thermal expansion during a hot day what can be done to its design to stop it buckling?

Answer 15

• Leaving expansion gaps
• Install under tension

Question 16

Find the strain the top layer experiences, when a temperature drop of ∆T causes the two layers to thermally expand by different amounts.

Top layer has a thermal expansion coefficient of α₁ and bottom layer has a thermal expansion coefficient of α₂.

Answer 16

final length = that of top layer = l_o(1-α₂∆T)

Superpose tensile stress in surface layer, to increase its length from its contracted length, to match that of substrate:

• change in length: ∆l = l_oα₁∆T - l_oα₂∆T
• strain in surface layer: ∆l/(l_o(1-α₁∆T))

= ((α₁-α₂)∆T)/(1-α₁∆T)

Since (α₁∆T) <<1, strain = (α₁ - α₂)∆T

Question 17

Give the different ways to measure the Youngs moldus, E, of a material and give the advantages/disadvantages for these methods.

Answer 17

Tensile Testing:

• elastic extensions are small hence difficult to measure precisely.
• measurement from machine must allow for flexure of machine.

Bending stiffness of a beam:

Beam of uniform cross-section loaded in 3-point bending.

• Equation can be found in structures data book.
• Bending gives more more deflection for given load than tension.
• E sensitive to L and D: requires accurate measurement of dimensions.

Natural frequency of vibration:

• Beam supported at nodal points and set vibrating.
• Measuring frequency more accurate than deflection
• Calculated E still sensitive to beam/plate dimensions.

Speed of Sound in the material:

Measure E by measuring v_t:

• strike a bar of material on one end
• time the longitudinal wave reflected from far end of the bar.
• accuracy depends on precise time measurement, which is relatively easy with piezoelectric transducers.

Question 18

What are the properties of amorphous metals?

Answer 18

• mechanically hard; magnetically - may be hard or soft
• very low damping (little energy lost in elastic collisions).

Question 19

What are the two mechanisms for “alloying” polymers?

Answer 19

Copolymers: more than one monomer polymerised together - only a few combinations will do this.

Polymer blends: molecular-scale mixtures of two polymer chains, without cross-linking.

Question 20

What is the glass transition temperature?

Answer 20

In polymers, the weaker secondary bonds are overcome by thermal energy at a lower temperature: the glass transition temperature, T_g.

Question 21

Explain how amorphous thermoplastics behave above T_g.

Answer 21

Melt to a viscous liquid (entangled molecules slide over one another).

Question 22

Explain how semi-crystalline thermoplastics behave above T_g.

Answer 22

Amorphous region melt, crystalline regions survive to higher melting point, T_m, above which a viscous liquid forms.

Question 23

Explain how elastomers and thermosets behave above T_g.

Answer 23

Secondary bonds melt at T_g but cross-links do not - on heating the polymer does not melt, but decomposes or burns.

Question 24

Give the consequences for processing and environmental impact of:

Thermoplastics

Elastomers/Thermosets

Answer 24

Thermoplastics:

Easy to re-mould, weld, and recycle

Viscosity falls with T: mould 150*C above T_g

Elastomers/Thermosets:

Mould once only

cannot recycle (limited re-use)

Question 25

Draw the Young’s modulus vs Temperature graph for semi-crystalline thermoplastics, one with a high degree of crystallinity and one with a low degree of crystallinity.

Answer 25

• At high crystalline fractions, glass transition has no effect on E.
• At T_m crystalline regions melt: viscous flow.
• crystalline regions stiffer than amorphous: higher E below T_g.

Question 26

Draw the Young’s modulus vs Temperature graph for thermosets( high degree of cross-linking ) and for elastomers ( low degree of cross-linking ).

Answer 26

• Highly cross-linked, no effect of glassy transition.
• Stiffer than thermoplastics in the glassy region; E falls slowly on heating.

Question 27

What is the equation used to find out the E of a foam material?

Answer 27

E_foam/E_solid = (p_foam/p_solid)²

Question 28

Give the three main composite geometries (with diagrams).

Answer 28

Particulate:

metal -ceramic: Al-SiC

polymer - ceramic: “filled polymers”

Fibres:

carbon/glass/kevlar fibre - polymer

Laminates:

plywood

Question 29

Give the composite processing for the three types of composites.

Answer 29

Particulate composites:

• Add micron-scale particles to melt before casting or moulding.

Fibre composites:

• Short chopped fibres: mix with resin, shape in a mould.

Laminates:

• wood etc: stack and glue thin layers
• long-fibre composites: stack multiple layers of prepreg with different fibre orientations.

Question 30

define anisotropic

Answer 30

Stiffness differs parallel and perpendicular to the layers

Question 31

What is the equation for longitudinal modulus ( Young’s modulus parallel to layers of a composite ) ?

Answer 31

E_c=σ_c/ε_c = V_fE_f + (1-V_f)xE_m

(denoted E_II in Databook)

Question 32

What is the equation for the transverse modulus of a laminate composite?

Answer 32

E_c= (V_f/E_f + (1-V_f)/E_m)^-1

Question 33

Are particulate composites upper bound or lower bound on the graph shown?

Are fibre composites upper bound or lower bound on the graph shown?

Answer 33

Particulate composites:

• isotropic ( same in all directions )
• close to lower bound (equal stress)

Fibre composties:

• anisotropic (stiffer parallel to fibres)
• parallel to fibres: upper bound exact(equal strain)
• perpendicular to fibres: close to lower bound (equal stress)