Lecture 4: Plasticity
4.1: Stiffness-limited design
Material indices for design
Identify the material index to be minimized or maximized
- For stiffness-limited design
Minimizing weight: A light, stiff tie-rod loaded in tension
The objective function (to describe the quantity to be minimized or maximized)
The goal is to minimize the value of the objective function within the given constraints
Object: minimize mass
Constraint: Section area A must be big enough to provide a specific stiffness of S*
Eliminate the free variable A:
Where S* and L are specified
So: the lightest tie that will provide a stiffness S* is that one made of a material with the smallest
value of / E or the highest value of E /
Material index: Mt = E / So, all materials with the same E / will function equally well
Minimizing weight: A light, stiff panel loaded in bending
Objective function:
Stiffness constraint:
Second moment of area:
Stiffness S*, length L, and width b are specified; thickness h is free
Eliminating the free variable h:
So the lightest panel is that one made of a material with the smallest value of / E1/3 or the highest
value of E1/3 /
Material index Mp = E1/3 / So, all materials with the same E1/3 / will function equally well
Minimizing weight: A light, stiff beam loaded centrally
Objective function:
Stiffness constraint:
Second moment of area:
Eliminating the free variable A:
Material index: Mb = E1/2 / So, all the materials with the same E1/2 / ratio will perform equally
Shape factor
- By reshaping the cross-section of a beam, it is possible to increase I, thus increasing
stiffness – without increasing the total area
- The ratio of I for the shaped section to that for a solid square section with the same area
is defined as the shape factor
Minimizing material cost instead of weight
Objective function:
With A and L specified, the goal of a material selection would be to minimize C m or maximize 1/Cm
Lecture 4.2: Beyond Elasticity, Plasticity and Strength
Strength, toughness are microstructure-sensitive properties
Metals
Yield stress: 0.2% proof stress, parallel line to elastic region with 0.2% offset of zero strain.
Tensile strength: maximum stress
Area between yield stress and tensile stress is strain/work-hardening region, in this region the
material undergoes uniform plastic deformation.
After tensile stress it undergoes necking until fracture, during necking the material shows non-
uniform plastic deformation.
Failure/fracture strain is determined by removing the elastic deformation
Plastic strain ductility
Plastic work: the area under the plastic region of the stress/strain curve, W p , dissipated work
Nominal stress and strain vs. true stress and strain
Nominal: using area the start of the test
True: correcting to the deformations happening during the test relevant after the yield stress
How to test material yield behaviour on a small scale and not destroying the product?
Indentation:
Polymers
Yield stress is defined at 1% strain
Stress strain behaviour = f(T vs Tg)
At T<<Tg brittle
At T = Tg cold drawing in thermoplastics
At T>>Tg viscous flow in thermoplastics
Failure due to propagation of dominant flaw
Strength in polymers in plastic regime
Drawing aligns polymer chains in the direction in which the material is stretched – this can increase
strength and stiffness by a factor 8
Polymers with a high Tg can not be drawn at RT – they craze forming small crack-shaped regions
within the polymer. Crazes scatter light; can develop in to cracks and failure
When crazing limits ductility in tension, large plastic strains may still be possible in compression by
shear banding
Deformation and temperature in polymers
Above the glass temperature, molecular segments become mobile, so when we apply a (gentle)
stress for a longer time, the deformation will continue with time and the material will deform
plastically (even at elastic stress level!!). This is called creep.
Ceramics and glasses
Brittle at room temperature
They have yield stress but so high that failure occurs before it
can be measured
Compressive crushing strength = elastic stress
Ideal strength?
No material gets close to this ideal strength but polymers get closer
Reason: microstructure and imperfections
