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Materials selection for mechanical design 1
• Breaking down a design problem • Identifying function, objectives, constraints • Optimizing performance • Ashby plots
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• Mechanical response: how does a material respond to loads? • Elastic deformation
• strain: unitless change in dimension • stress: force per area • elastic deformation is reversible upon release of applied forces
• Plastic deformation • permanent deformation is what’s left over • response depends on material history (microstructure)
• Fracture • preexisting cracks/flaws in material, propagate under applied stress • catastrophic failure of material
• Creep • time-dependent permanent deformation to time-independent load • thermally activated response
• Fatigue • fracture response to time-dependent cyclic loading • cycling promotes crack nucleation and growth; catastrophic failure
Mechanical response of materials 2
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• Uniaxial tension test: • Stress: what we do to the material • Strain: how the material responds
• Quantify: • Initial gauge length L0 and area A0; instantaneous length L and area A
Stress-strain diagram 3
Engineering stress
�E =P
A0
�T =PA
True stress
Engineering strain
✏E =�L0=
L � L0
L0
True strain
✏T = lnLL0= ln
A0
A= ln(1 + ✏E)
schematic steel stress-strain curve (not to scale)
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• Engineering vs. true quantities • Engineering: loads and deformation from initial geometry • True: what the material experiences (stress), accumulated/additive
dimension change (strain: 10% true + 10% true = 20% true) • Yield strength: highest stress that the material can withstand without
undergoing significant plastic (irreversible) deformation • May be defined by a yield point (rapid drop in stress at yield) • May be defined as 0.2% offset (stress to get 0.2% plastic strain)
• Ultimate strength: is the maximum value of stress (engineering stress) that the material can withstand
• Fracture stress: the value of stress at fracture • Stiffness: ratio of stress to strain, primarily of interest in the elastic
region. (elastic moduli) • Ductility: Materials that undergo large strain before fracture are classified
as ductile materials. Necks before failure • Percent elongation: 100(Lf−L0)/L0 • Percent reduction in area: 100(A0−Af)/A0
4Stress-strain diagram: features
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! Rupture occurs along a cone-shaped surface that forms an angle of approximately 45° with the original surface of the specimen (“cup-cone” shape)
! Shear is primarily responsible for failure in ductile materials
! Axial loading: maximum shear stress occurs at 45o
Necking Rupture
5Stress-strain diagram: ductile materials
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Ashby plots
Materials selection for mechanical
design:
choose best material among
competing properties
Our goal: understanding mechanisms
responsible for behavior
M. Ashby “Materials Selection in Mechanical Design”
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PM(M)PG(G)
Materials selectionDesign concerns
function - what a component does constraints - what must/must not occur objective - what is maximized/minimized
M. Ashby, “Materials Selection in Mechanical Design”
function constraint objective
Example: tie-rod stretches to carry load, must not yield, and be lightweight
Rank different designs performance as a function P(F, G, M): functional needs (F) geometry (G) material properties (M)
We assume a separable form: P(F, G, M) = PF(F) PG(G) PM(M) so that material choice can be optimized independent of design specifics, with flexibility
The goal: optimize performance
pmax =2↵⇡r·0BBBB@
K2Ic
�YS
1CCCCA
Ex: maximum pressure in cylindrical vessel to leak, but not fracture:
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Materials propertiesMaterial properties
mechanical: modulus, yield stress, fracture toughness, ... transport/thermal: heat capacity, thermal expansion, resistivity economic: density, cost/mass
price includes: cost to extract, cost/energy to process, cost/energy to form, cost of disposal, regulation cost
M. Ashby, “Materials Selection in Mechanical Design”
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Relative material cost/mass
Based on Callister, 6th ed.
Reference material: rolled A36 carbon steel
Relative costs fluctuate less than actual costs over time.
Cost includes both extraction and processing, but not production.
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Materials properties and selectionMaterial properties
mechanical: modulus, yield stress, fracture toughness, ... transport/thermal: heat capacity, thermal expansion, resistivity economic: density, cost/mass
price includes: cost to extract, cost/energy to process, cost/energy to form, cost of disposal, regulation cost
Materials selection involves 1. determining combination of properties to maximize (function,
constraint, and objective) 2. selecting material/material class to fill that need
M. Ashby, “Materials Selection in Mechanical Design”
We do selection via an “Ashby plot”: log-log plot of two material properties
PM(M) =M↵11 ·M
↵22 · · ·
log PM(M) = ↵1 log M1 + ↵2 log M2 + · · ·
Why log-log?
Constant (equal) performance is a straight line on an Ashby plot
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Light, strong tie-rod 12
A tie-rod carries load along its length. We want it to carry the load without yielding, and if it’s in a vehicle, we want a low mass. First: what is the function? A. low mass B. low area C. carry load D. long length E. not yield
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Light, strong tie-rod 13
A tie-rod carries load along its length. We want it to carry the load without yielding, and if it’s in a vehicle, we want a low mass. Next: what is the objective? A. low mass B. low area C. carry load D. long length E. not yield
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Light, strong tie-rod 14
A tie-rod carries load along its length. We want it to carry the load without yielding, and if it’s in a vehicle, we want a low mass. Finally: what is the constraint? A. low mass B. low area C. carry load D. long length E. not yield
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Light, strong tie-rod 15
A tie-rod carries load along its length: • Functional needs: carry load F • Geometry: length L, area A • Constants? F, L • Variables? area A, material • Constraint? stress below yield stress • Performance? mass m
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Light, strong tie-rod 16
A tie-rod carries load along its length: • Functional needs: carry load F • Geometry: length L, area A • Constants? F, L • Variables? area A, material • Constraint? stress below yield stress • Performance? mass m
m = SLFρ
σYS
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Yield strength vs. density
M. Ashby, “Materials Selection in Mechanical Design”
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Yield strength vs. density
M. Ashby, “Materials Selection in Mechanical Design”
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lines of constant performance?
A B
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Yield strength vs. density
M. Ashby, “Materials Selection in Mechanical Design”
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Yield strength vs. density
M. Ashby, “Materials Selection in Mechanical Design”
increasing YS/density
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Light, stiff tie-rod 21
A tie-rod carries load along its length. We want it to carry the load without extending more than length δ, and if it’s in a vehicle, we want a low mass. • Functional needs: carry load F • Geometry: length L, area A • Constants? F, L • Variables? area A, material • Constraint? extension below δ • Performance? mass m
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Light, stiff tie-rod 22
A tie-rod carries load along its length. We want it to carry the load without extending more than length δ, and if it’s in a vehicle, we want a low mass. • Functional needs: carry load F • Geometry: length L, area A • Constants? F, L • Variables? area A, material • Constraint? extension below δ • Performance? mass m
m =SFL2
δρE
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Young’s modulus vs. density
M. Ashby, “Materials Selection in Mechanical Design”
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Young’s modulus vs. density
increasing E/density
M. Ashby, “Materials Selection in Mechanical Design”
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Light, strong cantilever 25
A cantilever is fixed at one end, and carries load perpendicular to its length. We want it to carry the load without yielding, we want a low mass. • Functional needs: carry load W • Geometry: length L, diameter d • Constants? W, L • Variables? diameter d, material • Constraint? stress below yield • Performance? mass m
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Light, strong cantilever 26
A cantilever is fixed at one end, and carries load perpendicular to its length. We want it to carry the load without yielding, we want a low mass. • Functional needs: carry load W • Geometry: length L, diameter d • Constants? W, L • Variables? diameter d, material • Constraint? stress below yield • Performance? mass m
m ∝ρ
σ2/3YS
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Yield strength vs. density
M. Ashby, “Materials Selection in Mechanical Design”
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Yield strength vs. density
increasing YS2/3/density
M. Ashby, “Materials Selection in Mechanical Design”
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What about cost?Material properties
mechanical: modulus, yield stress, fracture toughness, ... transport/thermal: heat capacity, thermal expansion, resistivity economic: density, cost/mass
price includes: cost to extract, cost/energy to process, cost/energy to form, cost of disposal, regulation cost
M. Ashby, “Materials Selection in Mechanical Design”
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Young’s modulus vs. cost
increasing E/cost
M. Ashby, “Materials Selection in Mechanical Design”
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Young’s modulus vs. cost
increasing E1/2/cost
M. Ashby, “Materials Selection in Mechanical Design”
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• External load does work on a body: change in internal energy • Work = integral of force × distance
• Force = (stress) × (area) • Distance = (strain) × (length)
• stress × strain → energy/volume • If the deformation is recoverable
then so is the energy.
Strain energy density 32
W =Z
F · dr
=
Z(�A0)(L0 d✏)
= V0
Z� d✏
Elastic energy storage density: modulus of resilience
Total strain energy density from fracture: modulus of toughness
ur =12�pl✏pl
=12
�2pl
E
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Young’s modulus vs. yield strength
M. Ashby, “Materials Selection in Mechanical Design”
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Young’s modulus vs. yield strength
increasing YS2/E
M. Ashby, “Materials Selection in Mechanical Design”
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Pressure tanks 35
A pressure tank holds a fluid at pressure, and carries load perpendicular to its thickness. We want it to carry the load to yield before fracture, and so that the critical flaw size is larger than the thickness (leak before fracture). • Functional needs: hold pressure p • Geometry: wall thickness t, diameter d • Constants? d • Variables? thickness t, material • Constraint? stress below yield, flaw size above t • Performance? pressure p
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Fracture toughness vs. yield strength
increasing KIc/YS
increasing YS
M. Ashby, “Materials Selection in Mechanical Design”
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Fracture toughness vs. yield strength
M. Ashby, “Materials Selection in Mechanical Design”
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• Beginning of design and materials selection • Breaking down a design problem • Identifying function, objectives, constraints • Optimizing performance • Ashby plots
Summary 38