l08a large strains
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MEM1M
MEMS Design
Material Behaviour at Large stress and strain
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Material failure
• Permanent deformation (yielding),plasticity
• Fracture: creation of new surfaces, andultimately separation of sample intotwo or more parts
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• Ductile material: a material that can undergo appreciable
amount of permanent deformation beforefracture
aluminum and some of its alloys copper
magnesium
lead
teflon
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• Brittle material: a material that can undergo only small
amount of yielding before fracture
silicon
quartz
concrete
stone
glass ceramic materials
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Brittle and ductile materials
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trength
• !eneral state of stress:• tandard tests:
tension test:
torsion test:
• "riteria for yield or fracture in unia#ial
tension or pure shear:
σ 1≥σ 2≥σ 3
σ 1=σ ≥σ
2=σ
3=0
σ 1=τ ≥σ
2=0≥σ
3=−τ
σ or τ
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• $ension test• Ductile material
shear fracture
plastic stretching, necking
then fracture
• Brittle material brittle fracture
essentially no prior yielding
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• $ension test
Silicon carbide tensile testspecimens, Sharpe J. MEMS,
2005
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• $ension test
Silicon carbide tensile testspecimens, Jackson, J. MEMS, 200
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• $ension test
Silicon carbide tensile testspecimens, Jackson, J. MEMS, 200
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Failure criteria
• %n practice we need
for ductile material:
criteria for onset of inelastic beha!iour,
yielding, under combined stress
for brittle material:
critera for fracture under combinedstress
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&ielding
• &ield criterium: Ma#imum shearing stress Moti'ated by e#perimental obser'ation of slip or
shear along crystalline plane
&ield onset when ma# shearing stress in materialeuals ma# shearing stress in simple tension test:
imple to use
grees well with e#periment
1
2 σ 1−σ 3=
1
2 σ yp
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&ielding
• &ield criterium: Ma#imum distortion energy Based on distortion energy, gi'en by
&ield onset when distortion energy in material eualsdistortion energy at yielding in simple tension test:
ometimes written as a condition on the 'on Mises
stress:
σ 1−σ
2 2 σ
2−σ
32 σ
3−σ
1 2=2σ
yp2
U 0d=U
0−
σ m2
2K , σ
m=
1
3σ
xσ
yσ
z
σ e= 12 [ σ 1−σ 2
2 σ 2−σ
32σ 3−σ
1 2 ]
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&ielding
• &ield criterium: Ma#imum octahedral shearing
stress Based on octahedral shearing stress, gi'en by
&ield onset when octahedral shearing stress in structureeuals that at yielding in simple tension test*
%s eui'alent to the ma#imum distortion energy
criterion
σ oct=
1
3 σ 1−σ
22 σ 2−σ 3 2 σ 3−σ
12
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• &ielding criteria, e#perimental comparison
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• &ielding criteria Ma#imum distortion energy+octahedral
shearing gi'e somewhat better agreementwith e#periments
Ma#imum shearing stress is simple and moreconser'ati'e, therefore also popular*
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Fracture
• Fracture criterium: Ma#imum principalstress Fracture when the magnitude of the largest
principal stress e#ceeds the ultimate strengthin a tension test
ltimate strength is the largest stress beforefracture in the tension test*
Does not account for difference incompression and tension
!ood e#perimental e'idence in cases with atleast one tensile principle stress
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Failure
• Fracture criterium: Mohr-s theory
Coulomb-Mohr
Coulomb
"racture #hen criticalshear stress on a plane isreached. ∣τ ∣= f σ