Polymers- Plastic Deformation Flashcards
Nominal stress
Force over original area
Nominal stress vs strain graph
Linear for strains below 0.001. Reaches a rounded peak (yield point) and then goes back down a bit (strain 0.001-0.01). Then horizontal (dσ/dε is 0) for strains up to 0.1. Finally curves up to failure
Temperature dependence of stress strain behaviour
Lower temperature has steeper linear region and yield point at same strain. So lower temperature has higher overall curve
Constant volume condition
AiLi=AL
A is CSA, L is length
Means Ai=Aλ where λ is extension ratio L/Li
True stress
σt=F/A=λF/Ai=λ σn
Relations at the yield point
dσn/dεn=dσn/dλ=(1/λ)dσt/dλ - σt/λ^2
But at yield dσn/dεn=0
So
dσt/dλ=σt/λ
Eyring’s model and assumptions
There is an initial state (above some ground level) and then a jump of ΔH as a shear stress is applied and then comes back down to a final state (below the ground level). Like bell curve.
Assumes impose strain rate is proportional to the net jump rate in the direction of the shear stress and the dominant shear stress in a tensile test is the maximum shear stress and at the yield point σs=σy/2
Jump rates in Eyring’s model
Before shear stress applied jump rate=αexp(-ΔH/RT)
After applied forward jump rate=αexp(-(ΔH-σsV)/RT)
Net jump rate = αexp(-ΔH/RT)exp(σsV/RT)
Imposed strain rate in Eyring’s model
At yield point ε•=dε/dt is:
εy•=ε0•exp(-ΔH/RT)exp(σyV*/2RT)
Formula for yield stress in Eyring’s model
σy/w=(2/V*)(ΔH/T + 2.303Rlog(εy•/ε0•)
So yield strength over T is linear increase with log(strain rate) (line lower for greater temperatures
Tresca yield criterion
Yield occurs when a critical value of maximum shear stress (σs) is reached.
(σ1-σ3)2=σs
Where σ1>σ2>σ3
Von Mises yield criterion
Yield occurs when the shear-strain energy in the material reaches a critical value.
(σ1-σ2)^2+(σ2-σ3)^2+(σ3-σ1)^2=2σy^2
Graph of σ1/σy vs σ2/σy for Tresca and Von Mises
Tresca has squares in first and third quadrants with diagonal lines between the corners through the other two quadrants. Von Mises forms an oval that goes through all the outer corners of Tresca. Inside the shapes on the graph is elastic deformation and outside is plastic
Where does a craze form?
On a free surface of the object with its length perpendicular to the tensile stress applied
Mechanism of crazing
A local-yielding and cold-drawing process which takes place in a constrained zone of a material. Local stresses at the tip of a surface flaw or a growing craze are higher than the overall tensile stress applied to the specimen. Yielding takes place locally at these stress concentrations. Stress required is below that to cause general yielding in the rest of the specimen.
Formula for maximum tensile strain
ε1=1/E(σ1-ν σ2-ν σ3)
ν is poisson’s ration
When does crazing occur?
At the point when ε1=εc
Accounting for hydrostatic tension εc=C+D/p
σ1 - ν σ2 - ν σ3=X+Y/(σ1+σ2+σ3)
X and H are time and temperature dependent constants
Graph of σ1 vs σ2 for crazing
Same sir if shape as Von Mises but top right is cut away as in p13 week 10 and crazing happens beyond this new outline. Shear yielding occurs elsewhere on the unmodified outline
Why is crazing more likely for tensile stresses?
Because crazing increases the volume of the specimen and compression would act against this effect
