l08a Large Strains

8/12/2019 l08a Large Strains

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MEM1M

MEMS Design

Material Behaviour at Large stress and strain



Material failure

• Permanent deformation (yielding),plasticity

• Fracture: creation of new surfaces, andultimately separation of sample intotwo or more parts



• Ductile material: a material that can undergo appreciable

amount of permanent deformation beforefracture

aluminum and some of its alloys copper

magnesium

lead

teflon



• Brittle material: a material that can undergo only small

amount of yielding before fracture

silicon

quartz

concrete

stone

glass ceramic materials



Brittle and ductile materials





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



• $ension test• Ductile material

shear fracture

plastic stretching, necking

then fracture

• Brittle material brittle fracture

essentially no prior yielding



• $ension test

Silicon carbide tensile testspecimens, Sharpe J. MEMS,

2005



• $ension test

Silicon carbide tensile testspecimens, Jackson, J. MEMS, 200



• $ension test

Silicon carbide tensile testspecimens, Jackson, J. MEMS, 200



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



&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



&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 ]



&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



• &ielding criteria, e#perimental comparison



• &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*



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



Failure

• Fracture criterium: Mohr-s theory

Coulomb-Mohr

Coulomb

"racture #hen criticalshear stress on a plane isreached. ∣τ ∣= f σ

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l08a Large Strains