Elastic Deformation of Materials Flashcards
How can we get from displacement to strain?
Differentiate
How can we change strain for stress?
With Hooke’s law.
If displacement is constant everywhere is there any strain?
No, the body has been translated.
Define tensor shear strain.
Check
Write out the general strain tensor.
Check
Is the strain tensor symmetric?
Yes
Is stress a tensor property of a material?
Yes
Give an equation for stress using stress as a tensor.
F = ∑sigma A
By resolving the turning moment on a small element cube, show that sigmaij=sigmaji.
Check
Give an expression for hydrostatic stress in terms of normal stress.
Check
Give an expression for hydrostatic pressure of a body in terms of normal stress.
Check (should be minus the hydrostatic stress).
Note, not the same as external pressure
Relate stress and strain using the fact that the stress and strain tensors are symmetric.
Check, should get the compliance matrix.
When working with the compliance matrix, are the shear strains simple or pure?
Simple, so must convert to pure for Mohr’s circle.
State the Hooke’s law equations relating strain and stress by superposition of stress.
Check
Derive the Hooke’s law relation between stress and sum of strains.
Check
Give the Hooke’s law relation between stress and sum of strains.
Check
In the Hooke’s law relation between stress and sum of strains, does epsilon mm have any physical meaning?
Yes, it is the fractional change in volume.
Derive an expression for bulk modulus using Hooke’s law and hydrostatic pressure.
Check
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.
Show.
Derive the stress equilibrium equations.
Check
Why do we need strain compatibility equations?
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.
Derive a strain compatibility equation.
Check
What is the condition of displacement for the compatibility equations?
It must be single valued due to it being free of internal strain but an external force is applied.
Is a dislocation a compatible dislocation?
No as the displacement field is not single valued.
Derive the plane stress compatibility equation.
Check
State the Airy stress function for an edge dislocation.
Check
Define the Airy stress function in terms of plane stresses.
Check
Is an edge dislocation a situation of plane stress or plane strain or both?
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.
State the Mohr’s circle equations.
Check
For cylindrical polars, give the relation for normal strain between displacement and strain in the z direction.
Check
For cylindrical polars, give the relation for normal strain between displacement and strain in the r direction.
Check
For cylindrical polars, give the relation for circumferential strain and displacement in the r direction.
Check
For cylindrical polars, give the relation for circumferential strain between displacement and strain in the theta direction.
Check
Give expressions for shear strains in cylindrical polars.
Check
Give the solution for displacement in cylindrically symmetric systems.
Check
Solve the problem of a misfitting fibre in a rod generally finding the stresses in the fibre.
Check
Give the expressions for the displacements in a spherically symmetric system of radial dilation.
Check
Give the strains in a spherically symmetric system of radial dilation.
Check
State the solution for a spherically symmetric system of radial dilation.
Check
Solve the problem of a particle under pressure p generally finding the stresses and strain in the particle.
Check
Derive an expression for the interaction energy of an elastically deformed particle.
Check (use the fact it has deformed elastically, don’t need to integrate).
Calculate the strain energy density for the uniaxial deformation of a cube.
Check
Show that in 6x6 matrix form the compliance matrices Cij=Ckl, thus the matrix is symmetric and only has 21 independent components.
Check
Link stress and strain with the reduced compliance matrix.
Check
Link stress and strain with the reduced stiffness matrix.
Check
Since the reduced compliance matrix is symmetric, how can the Poisson’s ratios be related.
The component C12=C21
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?
Find solution.
Calculate the total energy for the uniaxial deformation of a cube.
Check
Calculate the elastic strain energy stored in a sphere under pressure p.
Check
Derive an expression for the elastic modulus when longitudinal loading of a fibre composite.
El = VfEf+VmEm
Derive an expression for the elastic modulus when transverse loading of a fibre composite.
1/Et = Vf/Ef +Vm/Em
When considering short fibre composites, what happens to the stress near the ends of the fibres?
It reduces.
How do we account for the reduce in stress near the end of the fibres in short fibre composites in the expression for modulus?
With a reinforcement factor Ω
What is the transfer length of a fibre in a composite?
The distance from the end of the fibre at which the stress increases to the long fibre value, epsilonEf.
Give an equation for the transfer length in a short fibre composite.
Check
Calculate the transfer length and longitudinal stiffness of an Al matrix containing 30% alumina fibres of 3µm diameter and 30µm length.
Check
Go through an example of stress and strain when an anisotropic material is loaded off axis.
Do it!! On a previous question sheet.
