Damage Fracture using Continuum Damage Mechanics Prakash M. Dixit - - PowerPoint PPT Presentation

Damage Fracture using Continuum Damage Mechanics

Prakash M. Dixit Department of Mechanical Engineering Indian Institute of Technology Kanpur Acknowledgements

• Dr. Sankar Dhar, ME Dept., Jadavpur University
• Dr. Sachin S. Gautam, ME Dept., IIT Guwahati
• Mr. Manoj Kumar, ME Dept., IIT Kanpur

1

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Plan of Presentation

• Introduction to Ductile Fracture
• Continuum Damage Mechanics (CDM)
• Damage Growth Law
• Crack Initiation Criterion (Critical Damage)
• Ductile Fracture in Taylor Rod and Tube Impact Problems
• Damage Growth Law (Revisited)

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Micro-Structural Observation: Ductile fracture in metals involves the following three stages.

Void coalescence, i.e., due to necking

• f

inter-void matrix, leading to micro-crack initiation. Growth of voids induced by plastic straining where the triaxiality may also increase. Nucleation of microvoids : due to fracture of secondary particles or their debonding from ductile matrix.

Figure 1 : Three stages of ductile fracture

Introduction to Ductile Fracture

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Introduction to Ductile Fracture... Two commonly used continuum models for the prediction of ductile fracture based on micro-structural observations of void nucleation, growth and coalescence:

• 1. Porous Plasticity Model

(Gurson, 1977; Tvergaard and Needleman 1984)

• 2. Continuum Damage Mechanics (CDM) Model

(Chaboche, 1981; Lemaitre, 1985; Rousselier, 1987)

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

( )

2 2 2 1 3

3 2 cosh 1 2

eq m y y

q q f q f σ σ σ σ     Φ = + − + =            

Material with voids idealized as a porous material.

Gurson (1977) proposed the following expression for plastic potential:

Porous Plasticity Model

1 2 3

= void volume fraction (porosity), = mean stress of the porous aggrgrate, = equivalent stress of the porous aggrgrate, = yield stress of Gurson t he as mat sume rix mate d ri Tve al, =1.

m eq y

q f q q σ σ σ = =

( )

2 1 2 3 1

rgaard 1981 proposed , better agreement of the model with numerical predictio 1.5, = 1 for ns q q q q = =

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

• Evolution Law for Void Volume Fraction (Porosity):

, rate of change of porosity due to void nucleation (1 ) , rate of change of porosity due to void gr t ,

• w h

p nucleation eq p g nucleation growth rowth kk

f A f f f f f ε ε = = = + −       

Constant, Mean plastic strain rate, Equivalent plastic strain rate

p kk p eq

A ε ε = = =  

Porous Plasticity Model …

• The critical value of porosity is obtained by either a void

coalescence condition or experimentally.

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

• In CDM,

the effect of void growth on material behaviour is incorporated by introducing a continuum variable, called damage, in the formulation

• Damage: Assumed isotropic.

It quantifies the void density at a point.

It is defined as

lim ; 0 1 Av D D A A ∆   = ≤ ≤   ∆ → ∆  

ΔAv= Area of void (and crack) traces contained in ΔA Continuum Damage Mechanics (CDM)

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Continuum Damage Mechanics (CDM) … In a damaged material, the specific Helmholtz free energy (ψ) or thermodynamic potential is given by:

1 (1 ) 2 Density, Elastic part of strain tensor Fourth order elasticity tensor Absolute temperature, Specific entropy

e e ijkl ij kl e ij ijkl

C D Ts C T = s ψ ε ε ρ ρ ε = − − = = = =

Elastic constitutive relation is obtained as

(1 ), Cauchy stress tensor

e ij ijkl kl ij e ij

C D ψ σ ρ ε σ ε ∂ = = − = ∂

Conservative part of the thermodynamic force Y corresponding to damage is given by:

1 2

e e ijkl ij kl

Y C D ψ ρ ε ε ∂ = = − ∂

These are called State Laws

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Continuum Damage Mechanics (CDM) … ( )

2 2

2 1-

eq m eq

Y f E D σ σ σ   = −       ( ) ( )

2

2 1 3 1- 2 3

m m eq eq

f σ σ ν ν σ σ         = + +                  

• Using Isotropic stress-strain relations, the conservative part of

the thermodynamic force Y corresponding to damage can be expressed in terms of stress:

1/2

E, Elastic constants, (1/ 3) ; Mean stress, (3 / 2) ; Equivalent stress, Deviatoric part of the stress tensor, The ratio is called

m ii eq ij ij ij m eq

ν σ σ σ σ σ σ σ σ = = ′ ′   =   ′ = Triaxiality

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Continuum Damage Mechanics (CDM) … Dissipation Potential (per unit volume): = Entropy production rate per unit volume (i for irreversible)

( )

For process, depends on the following state variables and their rates: , , , , ,

p p p p ij eq ij eq

isothermal D D ε ε ε ε Φ Φ = Φ   

Plastic part of strain tensor; Plastic part of strain rate tensor; 2 ; ; (Equivalent plastic strain and strain rate) 3

p p ij ij t p p p p p eq eq eq ij ij

dt ε ε ε ε ε ε ε = = = =

∫

    

: ; ;

ij p p ij eq

R Y D σ ε ε ∂Φ ∂Φ ∂Φ = − = − = ∂ ∂ ∂ Complementary Laws   

• and -

: Dissipative parts of the thermodynamic forces corresponding to and respectively. is the negative of the corresponding conservative

p eq

R Y D Y Y ε −

i

TS Φ = ≥ 

i

S 

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Continuum Damage Mechanics (CDM) … Dissipation Potential and Complementary Laws: Legendre-Fenchel transformation:

( ) ( )

, , , , , , , , Dual of

p p p p ij eq ij eq ij

D D R Y ε ε ε ε σ

∗

Φ ⇒ Φ − − Φ   

The complementary laws can be expressed as the evolution laws

• f the rates of state variables:

p ij ij

ε σ

∗

∂Φ = ∂ 

( )

p eq

R ε

∗

∂Φ = ∂ − 

( ) D Y

∗

∂Φ = ∂ − 

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

• The dissipation potential Φ* is maximized subject to the constraint

that the yield function/plastic potential should be zero on the yield surface:

Here, δ is the variation and is the Lagrange multiplier

Continuum Damage Mechanics (CDM) ... Dissipation Potential and Evolution Laws:

• Since Φ and Φ* are governed by an equality,

evolution laws not useful for computations

( ) F δ λ

∗

Φ − = 

λ 

( , , )

ij

F F R Y σ = − −

( )

; ; ( )

p p ij eq ij

F F F D R Y ε λ ε λ λ σ ∂ ∂ ∂ = = = ∂ ∂ − ∂ −      

• The above laws called the plastic flow rules.

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

• Assume: Plastic potential can be decomposed as (Lemaitre, 1985)
• Where F1 and FD are the plastic potentials associated with

yielding/hardening and damage respectively. The variables (-R) and (-Y) in F1 replaced by and D respectively. Plastic Potential due to Yielding/Hardening:

• von Mises yield function for damaged material:

( )

( )

1

, ,

p ij eq D

F F D F Y σ ε = + − Continuum Damage Mechanics (CDM) ... Plastic Potential

1

, 1 The variable yield stress usually modelled by a power law ( ) ( ) ( ) ( ) Initial yield stress; Hardening parameters = Hardening function

eq Y p p n Y eq Y eq Y

F D H K K, n = H σ σ σ ε σ ε σ = − − ≡ = + =

p eq

ε

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Continuum Damage Mechanics (CDM) ...

Incremental Elasto-Plastic Constitutive Relation

• First Plastic Flow Rule in Incremental Form:

1

3 ; 1 2

eq ij p ij ij ij ij ij eq

F d d d a a D σ σ λ ε λ σ σ σ ′ ∂ ∂ = = = = ∂ − ∂

• The scalar dλ found from the consistency condition:

1 1 1 p ij eq p ij eq

F F dF d d σ ε σ ε ∂ ∂ ≡ + = ∂ ∂

• Incremental Elasto-Plastic Constitutive Relation:

( )

p EP ij ijkl ij ij ijkl ij

d C d d C d σ ε ε ε = − =

Substitution of the plastic flow rule and expression for dλ in the above equation leads to

,

ijmn mn pq pqkl EP EP ij ijkl ij ijkl ijkl rs rsuv uv

C a a C d C d C C a C a H σ ε = = − ′ +

( )

( )

( )

'

:

n p p eq Y eq

H H K ε σ ε = = + hardenin Derivative of the function g

In the above relation, dσij has to be objective. One common choice is the product of the Jaumann stress rate and incremental time.

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Damage Growth Law

• For isothermal case, Lemaitre (1985) proposed the following expression for FD

(Plastic potential associated with damage, called Damage Potential):

( )

2

1 ( ) , S is a material constant (1 ) 2

D

Y F D S − = −

• Then, from incremental form of the third plastic flow rule

1

D

F d Y dD d ( Y ) D S λ λ ∂ − = = ∂ − −

It can be shown that

1

p eq

d d D λ ε = −

Elimination of 1 gives d D λ − / ( )

( )

p eq

Y dD d S ε − =

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Damage Growth Law ...

Determination of the material constant S in Lemaitre’s Damage Growth Law:

• Assuming that the triaxiality remains constant even after necking and the material

behaves like a non-hardening material near fracture, Lemaitre (1985) expressed the constant S in terms of the four measurable quantities in Tension Test: εo = Equivalent plastic strain at the damage initiation/threshold, Do = Damage at the threshold, εc = Equivalent plastic strain at fracture, Dc = Damage at fracture called the critical damage

=

p c m eq c eq

D D dD f dε σ ε ε σ   −     −  

Then,

Lemaitre’s damage growth law is linear in equivalent plastic strain. However, in most metals, the variation of damage with equivalent plastic strain is observed to be non-linear.

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Damage Growth Law ...

LeRoy et al (1981) measured damage in AISI 1015, 1045 and 1090 steels in tension test. The results are:

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Dhar’s (1995) Damage Growth Law for AISI1090 Steel Damage Growth Law ...

• Based on experimental results of LeRoy et al (1981), Dhar proposed the

following damage growth law:

1 2 1 2

( )( ) ( , , : )

p p eq eq

dD cd a a D Y d c a a ε ε = + + − material constants

To determine the material constants, Dhar (1995) adopted the following approach:

• The above equation is arranged as a relation between
• While doing this, the triaxiality term in (-Y) is expressed as a function of equivalent

plastic strain by substituting Bridgeman’s (1952) expression

• From the experimental results of Leroy et al (1981), the slopes are calculated at

various levels of equivalent plastic strain

• Finally, the material constants are obtained by the least square curve fitting method.

The values are:

/ and

p p eq eq

dD dε ε

( ) ( )

1 1 02 04 1 2

1.898x10 , 9.8x10 MPa , 1.84 MPa c a a

− − − −

= = =

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Crack Initiation Criterion (Critical Damage)...

• Void coalescence is caused by the necking of the inter-void matrix
• Thomason (1990) estimated the critical stress (i.e., maximum

principal stress of the macroscopically homogeneous stress state) needed for the onset of necking

• Employed unit cell method where the void size and inter-void

spacing were obtained from the statistical average. Further, ellipsoidal void was replaced by equivalent square-prismatic void.

• A kinematically admissible velocity field was proposed and upper

bound theorem for perfectly plastic material was used to estimate the critical stress.

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

• Analysis of Rice and Tracy (1969) was used to express the void

spacing and inter-void spacing in terms of the strain field (for the case of small strain/rotation).

• Thus, the void coalescence condition was obtained as a condition

involving the maximum principal stress (σ1), equivalent plastic strain ( ) and the yield stress (σy)

• Dhar (1995) extended the method to make it valid for large strain/

rotation and hardening material (variable yield stress):

Crack Initiation Criterion (Critical Damage)...

eq p

ε

( ) ( )

1 eq 1/2 eq

1.2 0.1 exp 1 exp / 2 σ ε σ ε     − + −       − −    

p Y p

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Crack Initiation Criterion (Critical Damage)...

• Dhar (1995) developed 2-D/Axisymmetric, Large Deformation,

Damage-Coupled, Elasto-Plastic, Finite Element Code using his Damage Growth Law (for AISI1090 Steel) Employed the code to analyse the following 5 problems

• Tension Test for Cylindrical Specimen (imperfection provided)
• Tension Test for Plane Strain Specimen (imperfection provided)
• Tension Test for Cylindrical Pre-Necked Specimen
• Tension Tests for Plane Strain Side-Grooved Specimen
• Tension Test of Cracked Plate

He estimated the damage at fracture (i.e., at micro-crack initiation) in all these 5 problems using the void coalescence condition.

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Crack Initiation Criterion (Critical Damage)...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

• Crack initiation depends on two continuum parameters: equivalent

plastic strain and triaxiality

• Thus, the void coalescence condition (for predicting the micro-crack

initiation) is represented as a graph of triaxiality verses the equivalent plastic strain (called the failure curve) Crack Initiation Criterion (Critical Damage)

Property Value Young’s modulus (E) 210 GPa Poisson’s ratio (ν ) 0.3 Initial yield stress ((σy)o ) 464 MPa Hardening coefficient (K) 1115 MPa Hardening exponent (n) 0.19

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Crack Initiation Criterion (Critical Damage)...

• He found the damage at the fracture initiation in all the 5 problems to

be the same. This value of the damage is called the critical damage, denoted by Dcr .

• Dhar (1995) estimated the damage at the fracture (i.e., at micro-crack

initiation) in all these 5 problems using the void coalescence condition.

• The value of critical damage can also be estimated from the experi-

mental curve of damage verses equivalent plastic strain. The value of damage at which the graph becomes almost vertical (i.e., the slope becomes approximately equal to 10) can be taken as the critical value.

• Thus, the crack initiation criterion can be expressed in terms in

terms of the critical damage. If at a point, the damage reaches the critical value, crack initiation takes place at that point.

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Ductile Fracture in Taylor Rod and Tube Impact Problems

Gautam (2009) studied ductile fracture in impact problems (in AISI1045 steel).

• He considered the effects of both high temperature and high strain rate on

material behaviour.

• He used the Johnson-Cook model for the dependence of the yield stress
• n high temperature and high strain rate:

( ) ( ) 1 1 ln

m p ref eq p Y Y eq p m ref ref

T T K B T T ε σ σ ε ε         −     = + − +             −              

(σy)0 =Initial yield stress; K, n = Hardening parameters; Tm = Melting temperature; Tref = Reference temperature; m,B = Material constants; = Reference equivalent plastic strain rate

p ref

ε 

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Ductile Fracture in Taylor Rod and Tube Impact ...

• Additive decomposition was assumed for the elastic, thermal and

plastic parts of the incremental strain tensor.

• Now, the dissipation potential also depends on T. Following the same

procedure, the elasto-plastic constitutive equation becomes

( )

EP T ij ijkl ij ij ij

d C d d R σ ε ε = − +

;

ijmn mn pq pqkl EP ijkl ijkl p Y rs rsuv uv eq p eq

C a a C C C a C a d σ ε ε = − ∂ + ∂

p Y Y ijkl kl eq p eq ij p Y rs rsuv uv eq p eq

C a d dT T R a C a d σ σ ε ε σ ε ε   ∂ ∂ +     ∂ ∂   = ∂ + ∂  

3 ; 2

ij ij eq

a σ σ ′ =

(1 )

T ij ij

d dT ε α δ = +

(α = Coefficient of thermal expansion, Cijkl= Fourth order elasticity tensor)

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Ductile Fracture in Taylor Rod and Tube Impact ...

• No data was available about how the damage growth depends on

the temperature and strain rate. Therefore, the same damage growth law, as developed by Dhar (1995), was used:

1 2 1 2

( )( ) ( , , : )

p p eq eq

dD cd a a D Y d c a a ε ε = + + − material constants

• In each increment, first, the incremental deformation equations are solved

using the temperature data of the previous increment. Then, the unsteady heat conduction equation is solved to determine the temperature rise distribution. The heat generated due to plastic deformation and friction is incorporated. The whole procedure is repeated in every increment.

• Gautam (2009) developed 3-D, Large Deformation, Damage-Coupled, Dynamic,

Elasto-Plastic, Finite Element-Finite Difference Code having Contact Module and Coupled with Unsteady Heat Conduction Equation based on the Above Formulation

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Ductile Fracture in Taylor Rod and Tube Impact ...

Damage Calculation:

• Since the triaxiality becomes negative in some parts of the domain,

the cut-off value of (-1/3) on negative triaxiality (as suggested by Bao and Wierzbicki (2005)) is used. Thus, the damage increment is computed only when the triaxiality is greater than (-1/3)

• This is to take care of the observation that the micro-voids/cracks

may not initiate in the presence of negative triaxiality.

• However, the crack closure effect of negative triaxiality is not

incorporated.

• Critical damage criterion is used for micro-crack initiation.

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Ductile Fracture in Taylor Rod and Tube Impact ...

• Thus, the damage increment is set zero if the damage exceeds the

critical value. An element is deemed to have failed if the average damage inside the element reaches the critical value

• The spread of fracture is simulated by tracing the path of the

failed elements. The stiffness of the failed elements is set to zero.

• Since the geometry and loading are axisymmetric, the damage

growth also would be axisymmetric. However, the experimentally

• bserved fracture patterns are not axisymmetric.
• To simulate them, a material imperfection (in the form of some

small initial values of and equivalent plastic strain) is provided at some random locations. Damage Calculation (contd)...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Ductile Fracture in Taylor Rod and Tube Impact ...

Material properties of AISI1045 Steel, Other Parameters and Geometric Properties

Young’s modulus (E) 210 GPa Other Parameters Poisson’s ratio (ν) 0.3 Reference temperature (Tref) 25

0C

Initial yield stress ((σy)o ) 302 MPa Fraction of plastic work converted to heat 0.9 Hardening coefficient (K) 796 MPa Fraction of friction heat entering rod/tube 0.1 Hardening exponent (n) 0.59 Geometric properties of Taylor rod Material constant (m) 1 Length of Taylor rod 25.4 mm Material constant (B) 1 Diameter of Taylor rod 7.11 mm Reference equivalent Plastic strain rate 1 Geometric properties of tube Melting temperature (Tm) 1460

0C

Length of tube 62.75 mm Thermal conductivity (k) 52 W/m

0C

Outer diameter of tube 12.55 mm Specific heat (c) 432.6 J/Kg

0C Thickness of the tube

0.78 mm Coefficient of thermal expansion (α) 1.1x10

• 5 /

0C

Coefficient of friction 0.05

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Ductile Fracture in Taylor Rod and Tube Impact ...

Taylor Rod Problem:

• Gautam (2009) simulated two different fracture patterns:

(i) Tensile Splitting (ii) Confined Fracture and Fragmentation leading to Separation of Conical Region and Outer Pieces

• In both the cases, the imperfection is provided along some radial lines selected

randomly.

• At the impact velocity of 300 m/s, the damage initiates at three locations:

(i) along the axis at the impact surface (centre of the impact surface) (ii) along the axis at some distance above the impact surface (say at point P) (iii) at the outer edge of the impact surface at four points (outer edge)

• The triaxiality at the middle portion of the impact surface is negative. Therefore,

the fracture initiates at the outer edge and propagates inward along the radial

• direction. This is called Tensile splitting.

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Tensile Splitting in Taylor Rod Problem

• Tensile splitting in Taylor rod at the impact velocity of 300 m/s
• This fracture pattern is found to be consistent with the experimental
• bservations of Woodward et al (1992)

Ductile Fracture in Taylor Rod and Tube Impact ...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Taylor Rod Problem (Contd)…

Ductile Fracture in Taylor Rod and Tube Impact ...

• At the impact velocity of 400 m/s, the damage growth is fast at point P as

well as at the outer edge. Therefore, the fracture initiates at both these locations. However, it does not initiate at the centre of the impact surface

• Since P is an interior point, the fracture at P is called confined fracture.

Thus, both the confined fracture and tensile splitting occurs simultaneously

• At the impact velocity of 500 m/s, the fracture zones initiated at point P and at the
• uter edge meet and cause fragmentation into central conical region and four outer

pieces from the impacted end.

• Both the predicted fracture patterns are found to be consistent with the experimental
• bservations of Woodward et al (1992)

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Confined Fracture and Fragmentation in Taylor rod Problem

• Impact velocity of 400-500 m/s
• Confined Fracture: Fracture at point P. Fragmentation: Separation
• f a central conical region and outer pieces from the impacted end.
• This fracture pattern is found to be consistent with the experimental
• bservations of Woodward et al (1992)

Ductile Fracture in Taylor Rod and Tube Impact ...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Ductile Fracture in Taylor Rod and Tube Impact ...

• The imperfection is provided along some radial lines selected randomly
• At the impact velocity of 250 m/s, mushrooming of the impacted end takes

place followed by the buckling of the tube along its length. But, no fracture initiation is observed.

• At the impact velocity of 350 m/s also, mushrooming of the impacted

end takes place but no buckling is observed. Further, a crack initiates at the mid-point of the tube wall in contact with the impacted end and then progresses first radially outward and then inward becoming a through-the-thickness crack.

• The predicted fracture pattern is found to be consistent with the

experimental observations of Wang and Lu (2002)

Tube Impact Problem

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Fracture in Tube Impact Problem:

• Fracture at the impacted end at the impact velocity of 350 m/s
• This fracture pattern is found to be consistent with the experimental
• bservations of Wang and Lu (2002)

Ductile Fracture in Taylor Rod and Tube Impact ...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Kumar (2014) proposed a Damage Growth Law for IS 2062: 2006 GR E410W A steel based on experimental measurement of damage in tension test.

The test specimens of standard shape and size are prepared from the plate provided by IGCAR and tested in displacement control mode using 200 KN capacity UTM (BISS Inc.) to predefined plastic strain levels (5%, 10%, … up to fracture).

• The samples, in the form of thin slices, are prepared by sectioning

the specimens at the measurement zone (that includes the necked cross-section) using Electric Discharge Machining (EDM) process.

• Then, one side of each slice is polished to reveal the true micro-
• structure. Next, the damage is measured at several Representative

Surface Elements (RSE) on the polished surface.

Damage Growth Law (Revisited)...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Measurement of Damage...

• Using the Scanning Electron Microscopy (SEM) imaging technique

and a magnification factor of 1000 (as suggested by Lemaitre and Dufailly (1987)), high resolution image of each RSE (Representative Surface Element) is obtained at 30 KV voltage.

• The voids in each RSE are identified using an image processing

program from the MATLAB. The damage for each RSE is obtained as the ratio of Av/A where Av is the area of the RSE and A is the area of void traces contained in.

• Finally, the damage at the polished cross-section is obtained as the average
• f the damage values over all the RSE’s of the surface.

Damage Growth Law (Revisited)...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Damage Growth Law (Revisited)...

Damage is measured at these points

Current Diameter (d) d/2

Location of ‘Representative Surface Elements (RSE)’ and

Magnified SEM Image of a ‘Representative Surface Element (RSE)’ Locations of six ‘Representative Surface Elements (RSE)’

Voids

X1000

Magnified SEM Image of a ‘Representative Surface Element (RSE)’

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Measurement of Average Equivalent Plastic Strain

• In tension test, the shear strains are zero (upto necking), and the normal strains

are uniform over any cross-section. Therefore, the equivalent plastic strain becomes equal to the plastic part of the logarithmic axial strain. Since the elastic deformation is negligible, it becomes equal to the total logarithmic axial strain.

• After the necking, the shear strains may develop in the necked zone and the

normal/shear strains may not be uniform over the cross-section.

• However, it seems reasonable to assume that the shear strains would be small,

except at the fracture.

• Then, the average equivalent plastic strain at the minimum cross-section can

be approximated as where do and d are respectively the initial and the current diameters of the minimum (necking) cross-section.

2ln ε   =    

p eq

d d

Damage Growth Law (Revisited)...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Damage Growth Law (Revisited)... Experimental Variation of Average Damage with Average Equivalent Plastic Strain

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Measurement of Triaxiality Triaxiality at various levels of plastic strain level is estimated from Bridgeman’s (1952) axisymmetric analysis of necking problem: a = Radius of the minimum (necked) cross-section, R = Radius of curvature of the minimum cross-section Both were measured experimentally at every level of plastic strain

2 2

1 2 ln 3 2

m eq av

a aR r aR σ σ       + −   = +                

r z Necked zone

a

Radius of curvature R at minimum cross- section Image Size :2.38mm X 1.79mm

Image of necked zone in longitudinal plane

Minimum cross-section

Damage Growth Law (Revisited)…

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Damage Growth Law (Revisited)... Experimental Variation of Average Triaxiality with Average Equivalent Plastic Strain

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Measurement of Equivalent Stress

• In tension test, the axial stress (called the true stress) at the necked

cross-section is measured as the axial force divided by the current cross-sectional area.

• However, after necking, it is not equal to the equivalent stress.

The equivalent stress is calculated from the measured axial stress using a correction factor based on Bridgeman’s (1952) axisymmetric analysis

• f necking problem

Damage Growth Law (Revisited)...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Based on experimental measurements, Kumar (2014) proposed the following expression for the Damage Potential

( )

{ }

exp (1 )

D

a F b Y b D = − −

Then, the damage growth law becomes

( )

{ }

exp

p eq

dD a b Y dε = −

• The material constants are determined from the experimental values of

damage, triaxiality and equivalent stress (at various plastic strain levels)

• For this purpose, the above equation is rearranged as

( )

{ }

( ) ( )

exp ln ln

p p eq eq

dD dD a b Y a b Y d d ε ε   = − ⇒ = + −      

Damage Growth Law (Revisited)...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

• From the measured values of damage and plastic strain, the slope ( ) is

calculated using a forward difference formula, and then its natural logarithm is taken. This is done at every plastic strain level

• At every plastic strain level, the values of (-Y) are calculated from the measured

values of equivalent stress, damage and triaxiality using the expression:

• Then, the above straight line is fitted through the calculated values of
• Then, the values of the material constants become: a0 = 0.0045, b0 = 2.62 MPa-1

/

p eq

dD dε

( ) ( ) ( )

2 2 2

2 1 3 1- 2 3 2 1-

eq m eq

Y E D σ σ ν ν σ       = − + +             ln( / ) and ( ) ε −

p eq

dD d Y

Damage Growth Law (Revisited)...

( ) ( )

ln ln

p eq

dD a b Y dε   = + −      

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

Damage Growth Law at High Temperature

Kumar (2014) conducted similar experiments for the following temperatures: 350, 425, 500, and 575 Kelvin by attaching a Thermal Chamber to the 200 KN capacity UTM (BISS Inc.) (on the same material: IS 2062: 2006 GR E410W A steel). Based on this, he proposed 2 versions of the damage growth law at high temperature:

Version 1:

( )

{ }

exp

p eq

dD a b Y dε = −

( )

2

a T 0 0045 0 046 T = +

*

( ) . . ,

( )

b T 2 62 2 219 T = +

*

( ) . .

ref m ref

T T T T T −   =     −  

*

Version 2:

( )

{ }

*

exp 1 ( ) ε     = − +    

p m eq

dD a b Y d c T

a0 = 0.0045, b0 = 2.62 MPa-1 (Same as at Room Temperature), c0 = 6.606, m = 1

Damage Growth Law at High Strain Rate

Kumar (2014) conducted similar experiments for the following strain Rates: 600, 1200, and 2400 per sec by using a Tensile Split Hopkinson Pressure Bar (TSHPB)

Damage Growth Law (Revisited)...

Prakash M Dixit, Department of Mechanical Engineering, IIT Kanpur, INDIA

THANK YOU!