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GP3 > Elastic Deformation of Materials > Flashcards

Elastic Deformation of Materials Flashcards

1
Q

How can we get from displacement to strain?

A

Differentiate

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2
Q

How can we change strain for stress?

A

With Hooke’s law.

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3
Q

If displacement is constant everywhere is there any strain?

A

No, the body has been translated.

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4
Q

Define tensor shear strain.

A

Check

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5
Q

Write out the general strain tensor.

A

Check

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6
Q

Is the strain tensor symmetric?

A

Yes

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7
Q

Is stress a tensor property of a material?

A

Yes

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8
Q

Give an equation for stress using stress as a tensor.

A

F = ∑sigma A

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9
Q

By resolving the turning moment on a small element cube, show that sigmaij=sigmaji.

A

Check

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10
Q

Give an expression for hydrostatic stress in terms of normal stress.

A

Check

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11
Q

Give an expression for hydrostatic pressure of a body in terms of normal stress.

A

Check (should be minus the hydrostatic stress).

Note, not the same as external pressure

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12
Q

Relate stress and strain using the fact that the stress and strain tensors are symmetric.

A

Check, should get the compliance matrix.

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13
Q

When working with the compliance matrix, are the shear strains simple or pure?

A

Simple, so must convert to pure for Mohr’s circle.

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14
Q

State the Hooke’s law equations relating strain and stress by superposition of stress.

A

Check

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15
Q

Derive the Hooke’s law relation between stress and sum of strains.

A

Check

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16
Q

Give the Hooke’s law relation between stress and sum of strains.

A

Check

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17
Q

In the Hooke’s law relation between stress and sum of strains, does epsilon mm have any physical meaning?

A

Yes, it is the fractional change in volume.

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18
Q

Derive an expression for bulk modulus using Hooke’s law and hydrostatic pressure.

A

Check

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19
Q

Show how for Hooke’s law relating shear stress and strain, Mohr’s circle can be used to turn it into a normal stress and strain problem.

A

Show.

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20
Q

Derive the stress equilibrium equations.

A

Check

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21
Q

Why do we need strain compatibility equations?

A

There are 6 components of strain at every point.
There are 3 components of displacement at every point.
The strain components then cannot be independent.

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22
Q

Derive a strain compatibility equation.

A

Check

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23
Q

What is the condition of displacement for the compatibility equations?

A

It must be single valued due to it being free of internal strain but an external force is applied.

24
Q

Is a dislocation a compatible dislocation?

A

No as the displacement field is not single valued.

25
Q

Derive the plane stress compatibility equation.

A

Check

26
Q

State the Airy stress function for an edge dislocation.

A

Check

27
Q

Define the Airy stress function in terms of plane stresses.

A

Check

28
Q

Is an edge dislocation a situation of plane stress or plane strain or both?

A

Plane strain only as by using Hooke’s law for stress in terms of strain, the sum of normal strain will always be non-zero so a stress will arise in every direction.

29
Q

State the Mohr’s circle equations.

A

Check

30
Q

For cylindrical polars, give the relation for normal strain between displacement and strain in the z direction.

A

Check

31
Q

For cylindrical polars, give the relation for normal strain between displacement and strain in the r direction.

A

Check

32
Q

For cylindrical polars, give the relation for circumferential strain and displacement in the r direction.

A

Check

33
Q

For cylindrical polars, give the relation for circumferential strain between displacement and strain in the theta direction.

A

Check

34
Q

Give expressions for shear strains in cylindrical polars.

A

Check

35
Q

Give the solution for displacement in cylindrically symmetric systems.

A

Check

36
Q

Solve the problem of a misfitting fibre in a rod generally finding the stresses in the fibre.

A

Check

37
Q

Give the expressions for the displacements in a spherically symmetric system of radial dilation.

A

Check

38
Q

Give the strains in a spherically symmetric system of radial dilation.

A

Check

39
Q

State the solution for a spherically symmetric system of radial dilation.

A

Check

40
Q

Solve the problem of a particle under pressure p generally finding the stresses and strain in the particle.

A

Check

41
Q

Derive an expression for the interaction energy of an elastically deformed particle.

A

Check (use the fact it has deformed elastically, don’t need to integrate).

42
Q

Calculate the strain energy density for the uniaxial deformation of a cube.

A

Check

43
Q

Show that in 6x6 matrix form the compliance matrices Cij=Ckl, thus the matrix is symmetric and only has 21 independent components.

A

Check

44
Q

Link stress and strain with the reduced compliance matrix.

A

Check

45
Q

Link stress and strain with the reduced stiffness matrix.

A

Check

46
Q

Since the reduced compliance matrix is symmetric, how can the Poisson’s ratios be related.

A

The component C12=C21

47
Q

Consider a thin wall pressure vessel radius r, wall thickness t and filled with gas pressure p. A small hole is made in the wall of the pressure vessel. What is the stress in the vicinity of the hole?

A

Find solution.

48
Q

Calculate the total energy for the uniaxial deformation of a cube.

A

Check

49
Q

Calculate the elastic strain energy stored in a sphere under pressure p.

A

Check

50
Q

Derive an expression for the elastic modulus when longitudinal loading of a fibre composite.

A

El = VfEf+VmEm

51
Q

Derive an expression for the elastic modulus when transverse loading of a fibre composite.

A

1/Et = Vf/Ef +Vm/Em

52
Q

When considering short fibre composites, what happens to the stress near the ends of the fibres?

A

It reduces.

53
Q

How do we account for the reduce in stress near the end of the fibres in short fibre composites in the expression for modulus?

A

With a reinforcement factor Ω

54
Q

What is the transfer length of a fibre in a composite?

A

The distance from the end of the fibre at which the stress increases to the long fibre value, epsilonEf.

55
Q

Give an equation for the transfer length in a short fibre composite.

A

Check

56
Q

Calculate the transfer length and longitudinal stiffness of an Al matrix containing 30% alumina fibres of 3µm diameter and 30µm length.

A

Check

57
Q

Go through an example of stress and strain when an anisotropic material is loaded off axis.

A

Do it!! On a previous question sheet.

GP3 (5 decks)

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