Metal Forming Initial

8/12/2019 Metal Forming Initial

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Elastic stress-strain relationship

How can we relate stress tensor and strain tensor?This can be done by introducing material properties in the elastic region

Elastic stress can be related to elastic strain by Youngs modulus (Hookes law)

x = E x; where E is the Youngs modulus

3-D elastic stress-strain relations:

A tensile force along X-direction can produce elongation in that direction andproduces contraction along Y and Z directions. The strains in three differentdirections can be related by Possions ratio, .

y = z = - x = - x /E,

where = 0.33 for most of the bulk materials

=> X

This means by applying x to the solid, strains y, z, and x can be createdand are related as above. This holds good for y and z also.


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The following table details this effect.

Finally by superposition of strain components from the table, we get elastic stress-strain relations in 3-D

We have assumed that, (1)material is isotropic, and(2) normal stresses doesnot produce shear strainon x, y, z planes and shear

stresses does not producenormal strains on x, y, zplanes

Shear stresses will create shear strains i.e.,

xy = G xy; yz = G yz; xz = G xzwhere G is modulus of rigidity

Constitutive equations in elasticdeformation region

Hookeslaw in 3-D


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Bulk modulus, K = Hydrostatic pressure / volume strain = m /

Four elastic constants viz., E, , G, K can be related as,

Refresh these relations fromsolid mechanics

Also we have assumed isotropicnature in the elastic deformation.Anisotropy of elastic deformationis possible. Since elastic part issmall during metal deformation, we

will not discuss much onanisotropy of elastic deformation


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(2) Plastic deformation

Important points to remember in plastic deformation

- Hookes law is not valid in plastic deformation

- Irreversible process material will not come back

to its original dimension- Plastic strain depends on the loading path by whichfinal state is achieved (elastic deformation dependson initial and final states)

- There is no easily measured constant relating stressand strain in plastic deformation region, unlikeelastic deformation region

- Important phenomenon STRAIN HARDENING has to be addressed in this regime

- Plastic anisotropy, Bauschinger effect areimportant

- Criteria for yielding has to be developed

eO A

BC

D

E

Elastic deformation ( el)

Plastic deformation ( el+ pl)


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Flow curve and flow equation

Description of stress-strain curve

0

A

a

A

b

Unloading at A, total strain willdecrease from 1 to 2. Thisstrain decrease is calledrecoverable elastic strain . Someplastic strain will disappear withtime ( 2 3). This is known asanelastic behavior

Unloading at A, stress decreaseswith strain parallel to elastic

regime & upon reloading curvebend over and reaches A, afterwhich it takes the original shapeof stress-strain curve

b < a

Bauschinger effect

0

123

A

Recoverableelastic strain

Anelasticbehavior

Strain hardeningregime


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The true stress-strain curve is called as Flow curve and the eqn. that describes

the curve is called Flow equation. Generally the flow equations are empirical,fit equations.

For eg., Hollomons equation, = K n where is the true stress forparticular true strain , K is the strength coefficient, n is the strainhardening exponent is a flow equation. It should be noted that this eqn. isvalid from onset of plastic deformation to maximum load at which neckingstarts.

Rigid ideal plastic material Ideal plastic material withelastic deformation regime

Linear strain hardeningmaterial

= 0 + K n (Ludwik equation; 0 yield stress)= K ( 0+ n) (Swift law; 0 Pre-strain value)= B-(B-A)exp(-n 0 ) (Voce law; A, B, n 0 material parameters)


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Relating engineering and true quantities

Dimensions of the specimen change with deformation, hence true quantitiesare better indicator of forming than engineering quantities. Engg. quantitiesdepend on the original dimensions of the sample unlike true quantities.

True strain, = ln (L/L0)

Engg. strain, e = L/L0 = (L-L 0)/L0 = (L/L 0) 1 => 1+e = L/L 0

Hence, = ln (1+e) (This equation is valid up to maximum load and invalid after that point. This is because of localization of neck after which the gage strain can not be

referred for measurement); = ln (L/L 0 ) is always useful.

=> Remember volume of solid remains constant during plastic deformation.

Hence x + y + z = 1 + 2 + 3 = 0 during plastic deformation.

=> This is not true in elastic deformation regime i.e.,

= e x + e y + e z = (1-2 )/E [ x + y + z ]; will be zero only if =

Refresh derivations for this from solid mechanics


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True stress, = Load / Instantaneous cross-section area = P/A

Engineering stress, S = Load / Initial cross-section area = P/A 0

= P/A = (P/A 0 ) (A 0 /A) = S (1+e); where A0 /A = L/L 0 = volume constancyprinciple

= ln (1+e); = S (1+e)

stress

strain

Yield strength

a

au

u

f

f

True stress-strain curve

Engg. stress-straincurve

Comparison betweentrue stress-strain and

engg. stress-straincurve


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Definition of plastic deformation properties

D Ultimate tensile strength, maximum load/A.C.S

E Fracture

eu

uniform elongation, elongation before necking begins

ef total elongation, elongation till fractureeO A

BC

D

E

Elasticdeformation

Plastic deformation

Uniformplasticdeformation Non-uniform

plasticdeformation


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Strain hardening and tensile instability

After yield point, further plastic deformation requires anincreasing load but at decreasing rate. Work hardeningstrengthens the material, but at the same time area ofcross-section is decreasing. The combined effect of thesetwo phenomenon results in typical load-progressioncurve.

From yield point to ultimate load, work hardening isdominant. At ultimate load, condition of tensile

instability occurs. Till the ultimate load, deformation ofsample gage length is uniform.

Necking a local constriction begins along the gagesection. From this point, incompatibility between strainhardening and area decrease arises. As a result, the loadrequired for further progression decreases. This meansthat load carrying capacity of the sample decreases.

After this, practically all plastic deformation isconcentrated in the small necked region. Finally failure

occurs in the necked region.

Observation during tensile test

Step 1: Initial shape and size of thespecimen with no load.

Step 2: Specimen undergoing uniform

elongation.Step 3: Point of maximum load and ultimatetensile strength.

Step 4: The onset of necking (plastic/tensileinstability).

Step 5: Specimen fractures.

Step 6: Final specimen length.

Image from public domain


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Hollomons equation, = K n where is the true stress for particular truestrain , K is the strength coefficient, n is the strain hardeningexponent. It should be noted that this eqn. is valid from onset of plasticdeformation to maximum load at which necking starts

ln ( ) = ln K + n (ln ) => Y = C + mX Plot graph between ln ( ) and ln ( )

Fitting stress-strain curve

If the true stress-strain plot is non-linear, it shows that material does not truly obeythe Hollomon s eqn. and n is not a constant. In this case, generally, n will bedefined w.r.t. strain.

Strain hardening exponent, n = d (ln )/d (ln )

a

b

Slope, n = a/b

K is stress at = 1

ln ( )

ln ( )

Linearrelation

Start ofnecking

Range: 0 < n < 1

n = 0.1 to 0.5 for mostof the metals


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Physical restrictions of the hardening law:

Eqn. is valid between strain of 0.04 and strain at which necking begins Predicting yield strength using this eqn. should be avoided. Offset method has tobe followed => For eg., Yield Strength = K (0.002) n is incorrect.

Exclusion of elastic and transition regions leads to little error

Considere criterion for tensile instability:

At maximum load, dP = 0 (Not really true)(Not really true)

We know that P = A; dP = dA + A d = 0d / = - dA/A = d (by definition) => = d /d

Assuming = K n , d /d = K n n-1 = = K n

n/ = 1 and so n =Since this is related to ultimate load, n = u

The general form of Considere criterion is, d (ln )/d (ln ) =

at the onset of instability

n = u d (ln )/d (ln ) =

This criterion can not be used to findn value of any material; Standardpractice has to be followed.


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Note that the rate of strain hardening d /d is not identical with strainhardening exponent n.

n = d (ln )/d (ln ) = / (d /d )

d /d = n ( / )

At instability, u = n => d /d =

The point of necking at maximum load can be obtained from the true stress-strain curve by finding the point on the curve having a subtangent of unity

or the point at where the rate of strain hardening equals the stress .


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Behavior after necking

Till necking we will have only uni-axial stress state during tensile testing; i.e., 1 = ; 2 = 0; 3 = 0. Hence effective stress ( ) is equivalent to true stress ( ).

During necking, this is not true as we will have tri-axial stress state in neck.

Reason: Constraint given by un-deformed region to the necked region; This causes

radial and circumferential stresses (longitudinal stress exist already) in the sample.

Here Bridgman correction (1944) is important


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Effect of inhomogeneity on uniform strain

a

b

b

F

F

Presence of inhomogeneities in tensile material

Consider a tensile test sample having homogeneous properties butdiffering dimensions in two regions, a and b.

Inhomogeneity factor, f = A a0 /Ab0 ; Aa0 a Aa = b Ab

Using power hardening law and true strain relationship

K an Aa0 e

- a= K b

n Ab0 e

- b=> f a

ne

- a= b

ne

- b

For given values of f and n, above eqn. give a as a function of b up toa value of a = n where necking would occur.

0.99

a

b

0.25

f = 1

f = 0.995

0.98

n = 0.25b

*

f (1 to 0.9)

0.25 for f = 10.208 for f = 0.996


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Effect of strain rate

The rate at which strain is applied to the sample is strain rate, = d /dt, (1/s)

Increasing strain rate increases flow stress and strain rate dependence ofstrength increases with increase in temperature (see figure).

Conventional strain rate, e = de/dt

= d (L-L o)/Lo = (1/L o) (dL/dt) =

dt e = v/L o

The conventional strain rate is proportional tocross-head velocity as Lo is constant.

Cross-head speed =v = dL/dt

6xxx Al

The true strain rate is given by, = d /dt = d(ln (L/Lo))/dt = (1/L) (dL/dt) = v/L

The above eqn. indicates that for a constant cross-head speed ( v), the true strain rate willdecrease when specimen elongates. In order to maintain constant true strain rate, the

deformation velocity should increase in proportion to the increase in sample length, v = LOexp ( t)


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As shown in the earlier figure, relationship between flow stress and strain

rate at constant strain and temperature is, = C m Where m is strain rate sensitivity .

m can be obtained from log log plot. However a better method is

presented in the figure.

For metals m is low (


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At room temperature deformation, tensile sample will not neck as long

as d /d > . But in the case of deformation at high temperatures, effectof strain rate is considerable. Here necking is prevented by strain ratehardening as in superplastic materials. Consider a superplastic materialrod of C.S.A. A and subjected to axial load P .

Combining above two eqns.,

This eqn. states that so long as m


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This eqn. states that so long as m


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Temperature effect on tensilebehavior of mildsteel

Temperature effect on yield strength

Ni FCC

Ta, W, Mo, Fe - BCC

The temperature dependence of flow stress at constant strain and strainrate is,

Influence of temperature


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Yield surface Assume that loading, unloading of square sheetof material is done in two directions, in anyproportion

1

2

X 1

X 2

In this chose specific stress path => 2 = 2

Load to 2,

1(1, 2, ) shown as symbols e, p

Unload this each time and look for permanentdeformation (plastic) to occur

Onset of plastic deformation between 1 (5) & 1 (6)

Continue this for varied 2 values - 2, 2 .

2

1

2

1 2 3 4 5 6 7

Elastic

deformation

Plastic

deformatione p

1

1

12

3

45

6 7

e p

2

= constant

Plasticity

e

O A

BC

D

E

Elasticdeformation

Plasticdeformation

YS

Yield strength inuni-axial loading


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2-D mapping of discrete stress states that first cause plastic deformation

Stress states within the red contour representELASTIC DEFORMATION REGIME

Red line represents onset of PLASTICDEFORMATION

The loci of all stress combinations that firstcause plastic deformation is called Yield locior surface .

2

1e

p

eee p

p

p

Onset of plasticdeformation

Elasticdeformation

1

2

1

2

Radial or proportional paths Other stress paths

2

1Possible shape of yield locusfrom discrete measurements


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Yield function

Yield function defines yield surface

Assume that yield surface is closed, smooth surface

At any instant of time, yield surface is defined as

f ( ij ) = f ( 11 , 22 , 33 , 23 , 13 , 12 ) = k => This is 6-D surface with eachdimension represent one of the stress components

Assume isotropic material same properties in all directions; In this case,

we can write in terms of principal stresses only 1 , 2 , 3 and surface isreduced to 3-D . We need cubic eqn. to relate these stresses to principalstresses, 3 I 1

2 I 2 I 3 = 0.

The stress invariants are related to principal stresses, I 1 = 1 + 2 + 3 ;

I 2 = - ( 1 2 + 2 3 + 1 3 ); I 3 = 1 2 3

With isotropic assumption , we can write k = f (I 1 , I 2 , I 3 ) or k = f ( 1 , 2 , 3 )

First assumption k = f (I 1 , I 2 , I 3 )


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Neglecting Bauschinger effect, we can neglect I 3 and hence,

Isotropic, pressure independent, no Bauschinger effect : c = f (I 2 )

Third assumptionThis is the final form ofyield function

Now, I 2 = - ( 1 2 + 2 3 + 1 3)

Put 1 , 2 in terms of principal stresses, we get

I 2 = 1/6 [( 1- 2 ) 2 + ( 1- 3 )

2 + ( 2- 3 ) 2 ]

=> C = ( 1- 2 ) 2 + ( 1- 3 ) 2 + ( 2- 3 ) 2

Yield function in principal

coordinate system

C = ( 11- 22 ) 2 + ( 11- 33 ) 2 + ( 22- 33 ) 2 + 6 12 2 + 6 13 2 + 6 23 2

Yield function in general

coordinate system

C = ( 1-

2 ) 2 + (

1-

3 ) 2 + (

2-

3 ) 2

In plane stress, C = ( 1- 2 ) 2 + 1 2 + 2 2

c = f (I 2 )

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Metal Forming Initial