Prof. Ramesh Singh

Mechanics Review-III

• Elasticity• Elements of Plasticity• Material Models• Yielding criteria, Tresca and Von Mises• Levy-Mises equations• Strengthening mechanisms

Prof. Ramesh Singh

Elastic Stress-Strain

• Linear stress-strain

• The extension in one direction is accompanied by contraction in other two directions, for isotropic material

xx Ees =

xzy neee -==

Prof. Ramesh Singh

Hooke’s Law• In x direction, strain produced by stresses

( )zyx

x

zyxx

zz

yy

xx

EE

EEE

E

E

E

ssnse

nsnsse

nss

nss

ss

+-=

--=

-®

-®

®

ing,Superimpos

stresses various toduedirection in x Strains

Prof. Ramesh Singh

Hooke’s Law

( )

( )

( )

xzxz

yzyz

xyxy

yxz

z

zxy

y

zyx

x

G

G

GEE

EE

EE

gt

gt

gt

ssnse

ssnse

ssnse

=

=

=

+-=

+-=

+-=

Prof. Ramesh Singh

Poisson’s Ratio• Adding the strains

• For fully plastic,

( )

condition elasticfor strains gEngineerin ,

21

zyx

zyyzyx

eeeVV

E

++=D

++-

=++ sssneee

0=++ zyx eeen = 0.5, Typically, -1<n<0.5, Most metals, 0.3

Prof. Ramesh Singh

Stretching these two-dimensional hexagonal structures horizontally reveals thephysical origin of Poisson's ratio. a, The cells of regular honeycomb or hexagonal crystals elongate and narrow when stretched, causing lateral contraction and so a positive Poisson's ratio. b, In artificial honeycomb with inverted cells, the structural elements unfold, causing lateral expansion and a negative Poisson's ratio.

Prof. Ramesh Singh

Constitutive Behavior Equations

Linear elastic (simplest) – Young’s modulus, E = s/e

e

s E

1

Thomas Young1773-1829

Robert Hooke1635-1703

Prof. Ramesh Singh

#2 - Match

a) Elastic – plastic material

b) Perfectly plastic material

c) Elastic – linear strain hardening material

1)

2)

3)

s

s

s

e

e

e

Prof. Ramesh Singh

Actual Material Behavior

fracturest

e

K

1

Ylinear

plasticUTS

fracture

elastic

s

e

Y

st = Ken

K = strength coefficient n = strain hardening coefficient

Prof. Ramesh Singh

Strain Rate Effect (1)

C = strength coefficientm = strain rate sensitivity coefficient

1mlog s

log

mCes !=

e!

C

1

Prof. Ramesh Singh

Yield Criteria

• How do you know if a material will fail?Compare loading to various yield criteria– Tresca– von Mises

• Key concepts – plane stress– plane strain

Richard von Mises1883-1953

Henri Tresca1814-1885

Prof. Ramesh Singh

#3 - Matcha) Tresca yield

criterionb) Von Mises yield

criterionc) Maximum

distortion energy criterion

d) Maximum shear stress criterion

1) tmax > ½ syield

2) (s1-s2)2 + (s2-s3)2

+ (s3-s1)2 = 2Y2

3) s3-s1 = 2tyield

Stress-Strain Relationship• Strength of a material or failure of the material is determined from

uniaxial stress-strain tests

• The typical stress-strain curves for ductile and brittle materials are shown below.

Prof. Ramesh Singh

Theories of FailureStrength of a material or failure of the material is deducedgenerally from uni-axial tests from which stress straincharacteristics of the material are obtained.

The typical stress-strain curves for ductile and brittle materialsare shown below.

Material Strength parameters are ORyS uS

Theories of FailureIn the case of multidimensional stress at a point we have a morecomplicated situation present. Since it is impractical to test everymaterial and every combination of stresses !1, !2, and !3, afailure theory is needed for making predictions on the basis of amaterial’s performance on the tensile test., of how strong it willbe under any other conditions of static loading.

The “theory” behind the various failure theories is that whateveris responsible for failure in the standard tensile test will also beresponsible for failure under all other conditions of staticloading.

Need for a failure theory • In the case of multiaxial stress at a point we have a more

complicated situation present. Since it is impractical to test every material and every combination of stresses s1, s2, ands3, a failure theory is needed for making predictions on the basis of a material’s performance on the tensile test.

Prof. Ramesh Singh

Prof. Ramesh Singh

Importance of Yield CriterionAssume three loading conditions in plane stress:

sss s

p

p

ss

p

p

s1 – s3 = Y

s = Ys + p = Y s = ?

At the onset of yielding,

Prof. Ramesh Singh

Tresca Yield Criteriontmax > k or tflow (shear yield stress) a material property

• Simple tensions1 = Y = 2k=2tflow

• Under plane stresss1 – s3 = Y

smax - smin = Y = 2k=2tflow k2σστ minmax =

-=

Derivation of Distortion Energy

• Done in class

Prof. Ramesh Singh

Prof. Ramesh Singh

Von Mises Yield CriterionBased on Distortion Energy

(s1-s2)2 + (s2-s3)2 + (s3-s1)2 = 2Y2

Y = uni-axial yield stress

( ) ( ) ( ) ( ) 2222222 26 Yxzyzxyxzzyyx =+++-+-+- tttssssss

Prof. Ramesh Singh

VonMises/Tresca Criterion

• The locus of yielding for VonMises

Y

Y

Y

=-

==

>>

=-+

31

31

2

31

231

23

21

line, deg 45 a be it willfourth and secondquadrant thirdandfirst in , of valuemaximum stress planefor

00,0

quadrantfirst in Criterion Tresca Using

ss

sss

ss

ssss

Prof. Ramesh Singh

Locus of Yield

s3

s1

Y

Y-Y

-Y

Prof. Ramesh Singh

Levy Mises Flow Eqn.

ed

s

( )

( )

( )

( )úûù

êëé +-=

úûù

êëé +-=

úûù

êëé +-=

+-=

2133

3122

3211

212121

Equation Elastic toAnalogous

ssssee

ssssee

ssssee

ssnse

dd

dd

dd

EE zyx

x

s

For Brittle Materials

• Brittle Materials – Hardened steels exhibit symmetry tension

and compression so preferred failure is maximum normal stresses or principal stresses

– Some materials which exhibit tension compression asymmetry such cast iron need a different failure theory such Mohr Coulomb

Prof. Ramesh Singh

Failure criteria a case study in composites

23

Matrix failure(Mohr-coulombfailure criteria)

Interface failure(Traction separation law)

Fiber failure(Maximum principlestress failure criteria)

Delamination between plies(Traction separation law)

Failure stress for glass-fibers (Weibull Plot)

24

In(I

n(1/

(1-P

(σ))

))

In (σi)

y = 6.2176x - 46.41R² = 0.9007

-5

-4

-3

-2

-1

0

1

2

7 7.2 7.4 7.6 7.8

In(I

n{1/

(1−P

(σ)

)})

In(σi )

Test dataLinear (Test data)

Failu

re p

roba

bilit

y

Stress (MPa)

0

0.2

0.4

0.6

0.8

1

1.2

0 500 1000 1500 2000 2500Fa

ilure

pro

babi

lity

Stress (MPa)

Test data

• Mean value of the tensile strength is 1623.3 MPa• Maximum principle stress criteria is used for modeling fiber failureFo

rce

(N)

Displacement (mm)

00.10.20.30.40.50.60.70.8

0 0.05 0.1 0.15 0.2 0.25

Forc

e (N

)

Displacement (mm)

Glass fiber

Matrix failure and debonding

25

Mohr–Coulomb model for matrix failure• τ = c – σ tan φC -cohesion (yield strength of matrix under pure shear loading) = 30 MPaφ -friction angle (100-300) = 100

Friction angle ‘θ’

Cohesion ‘C’σ

τ

FailedStable

q Yielding takes place when the shear stress, τ, acting on a specific plane reaches a critical value, which is a function of the normal stress, σn ,acting on that plane.

Maximum nominal stress criterion

For cohesive layer, Knn= Kss = Ktt = 35 Gpatnn = tss = ttt = 30 Mpa

þýü

îíìúûù

êëé

þýü

îíì

=

εt

εs

εn

Ktt

Kss

Knn

tt

ts

tn

Elastic definition for traction separation law