STRESS CONCENTRATION...concentration 1 STRESS CONCENTRATION - Cracks always initiate at points of stress concentration. - A crack, once initiated, becomes an intense stress concentrator

concentration 1

STRESS CONCENTRATION

- Cracks always initiate at points of stress concentration. - A crack, once initiated, becomes an intense stress concentrator itself. - Two fundamental cases of plane elasticity: INFINITE PLATE CONTAINING A CIRCULAR HOLE (Kirsh, G, (1898), V.D.I., 42, 797-807)

→ Stress Concentration

INFINITE PLATE CONTAINING AN ELLIPTICAL HOLE (Kolosoff, G.V., On an application of complex function theory to a plane problem of the mathematical theory of elasticity, Yuriev, 1909; Inglis, C.E., (1913), Stresses in a plate due to the presence of cracks and sharp corners,Transactions of the Royal Institute of Naval Architectes, 60, 219-241)

→ Stress Concentration

concentration 2

- In the limit of (minor axis) / (major axis) → 0 : INFINITE PLATE CONTAINING A CRACK (Wieghardt, K., (1907), Z. Mathematik und Physik, 55, 60-103; translated by Rossmanith (1995), On splitting and cracking of elastic bodies, Fatigue Fract. Engrg. Mater. Struct., 18, 1371-1405) Muskhelishvili, N.I., Some basic problems of the mathematical theory of elasticity, in Russian 1933 (in English 1953, Noorhoff-Groningen). Westergaard, Bearing Pressures and cracks, (1937), J. Applied Mechanics, 6, A49-53. Williams, M.L. (1952), Stress singularities resulting from various boundary conditions in angular corners of plates in extension, J. Applied Mech., 19, 526-528. Williams, M.L. (1957), On the stress distribution at the base of a stationary crack, J. Applied Mech., 24, 109-114.)

→ Stress Intensification

concentration 3

INFINITE PLATE CONTAINING A CIRCULAR HOLE (Kirsh, G, (1898), V.D.I., 42, 797-807) - Consider infinite plate containing a circular hole of radius R and subject to a remote tensile stress σx = σ.

- Consider portion of plate within concentric circle of radius R' >> R so that stress field is not perturbed by hole (Saint- Venant’s Principle) - Stress field at r = R' (Mohr's circle):

( )

θσσ

θσσ

θ 22

212

sin

cos

−=

+=

r

r

concentration 4

- Decompose problem into:

Problem (1)

02 == θσσσ rr ,/ Problem (2)

( ) θσσθσσ θ 22

22

sin cos −== rr ,

- Solution problem (1): thick cylindrical tube under tension (Lamè):

22

2

222

22

22

2

222

22

RR'R'

rRR'R'R

RR'R'

rRR'R'R

−+

−+=

−+

−−=

21

2

21

2

σσσ

σσσ

θ

r

- For R' → ∞

++=

−+=

2

2

2

2

rR

rR

12

12

σσ

σσ

θ

r

concentration 5

- Solution problem (2): Assume Airy stress function:

θcos2)(rf=Φ Impose compatibility, ∇4Φ = 0,

0112

2

2

22

2=Φ

∂

∂+

∂∂

+∂

∂

θrrrr

Substitute stress function and get the ordinary differential equation to determine f(r):

0412

22

2=

−+ frrrr d

dddcos2θ

General solution:

DrCBrArf +++= 2

42

Boundary conditions:

0)(0)(

22

)(

22

)(

==

−=

+=

RR

sinR'

cosR'

θ

θ

σσ

θσσ

θσσ

r

r

r

r

concentration 6

Recall:

θθσ

σ

θσ

θ

θ

∂∂Φ∂

−∂Φ∂

+=

∂

Φ∂+=

∂

Φ∂+

∂Φ∂

+=

rr1

r1r

r1

rr1

2

2

22

2

2

2

r

r

Substitute into general solution to get constants:

For R' → ∞ : 24 RD ,RC ,B ,A24

04

σσσ=−==−=

- Solution problems (1) + (2):

θσσ

θσσσ

θσσσ

θ

θ

22312

2312

12

24312

12

sinrR

rR

cosrR

rR

cosrR

rR

rR

2

2

4

4

4

4

2

2

2

2

4

4

2

2

+−−=

+−

+=

−++

−=

r

r

concentration 7

For r = R:

0221

0

=

−==

θσ

θσσσ

θ

r

r)( cos

Maximum stress: σσθ 3= for θ = π/2, 3π/2 Minimum stress: σσθ −= for θ = 0, π Note: - stress concentration factor = 3, independent of R - compression for −π/6 ≤ θ ≤ π/6 and −5π/6 ≤ θ ≤ 7π/6 - circumferential stress for θ = π/2:

++= 4

4

2

2

rR

rR 32

2σσθ

⇒ hole has a localized character

for r = 4R: ).( 0401+= σσθ

⇒ solution applicable to finite plates with width > 4R. - stress field satisfies plane-strain and generalized plane

stress.

massabo

Rectangle

massabo

Rectangle

concentration 8

- INFINITE PLATE WITH CIRCULAR HOLE SUBJECT TO BIAXIAL STRESS

Apply superposition principle to get stresses at r = R: 1) Biaxial tension: Uniform stress: σσθ 2= 2) Tension/compression (pure shear): Maximum stress: σσσσθ 43 =+= for θ = π/2, 3π/2 Minimum stress: σσσσθ 43 −=−−= for θ = 0, π

concentration 9

INFINITE PLATE CONTAINING AN ELLIPTICAL HOLE (Kolosoff, G.V., On an application of complex function theory to a plane problem of the mathematical theory of elasticity, Yuriev, 1909; Inglis, C.E., (1913), Stresses in a plate due to the presence of cracks and sharp corners,Transactions of the Royal Institute of Naval Architectes, 60, 219-241) Hp: elliptical hole with: a = major axis, b = minor axis.

- Solution: Kolosof (complex function theory); Inglis (Conformal Mapping, elliptical coordinates) (see Carpinteri, Meccanica dei materiali e della frattura, 1992, Pitagora, for details). - For β = π/2 (tensile stress perpendicular to major axis): a = major axis, b = minor axis

σθ

θσθ

+

+=

2mcos2-m12cos22m-m-1

2

2

m=(a-b)/(a+b).

θ

σ

concentration 10

Maximum circumferential stress:

σσσσ θθ

+=

+

+==ba

m-1m1max 21210)(

- Other cases:

Tensile stress perpendicular to

major axis

+=

ba

max 21σσ

Tensile stress perpendicular to

minor axis

+=

ab

max 21σσ

Uniform stress σσ

ba

max 2=

Pure shear

+=

ba12σσmax

Pure shear parallel to major and minor axes

ab

b)(a 2max

+= σσ

- For a = b, solution for a circular hole (Kirsh); - For b/a → 0 and tensile stress perpendicular to major

axis: stress intensification;

- For b/a → 0 and tensile stress perpendicular to minor axis: uniform stress equal to applied load,

no stress concentration

massabo

Rectangle

massabo

Rectangle

intensification 1

STRESS INTENSIFICATION

Plate with a crack: stress intensification at the crack tip - What is the critical load in the cracked plate? - What is the "fine structure" of the stress field at the crack tip? - What is the power of the singularity? (e.g., 2D problem of a point load acting on a semi-infinite plane (Kelvin): stresses = O(r −1), displacements = O(log r))

intensification 2

Traditional design approach: (2 parameters: σapplied, σu)

σapplied < σu σu = yield or tensile strength ⇒ (a): σapplied < σu / 3 → safe ⇒ (b): σapplied < σu / (1+a/b) → safe ⇒ (c): σapplied ≠ 0 → unsafe for any σy

??? Fracture Mechanics approaches in design: (3 parameters: σapplied, GIC (or KIC), a) - Energy Approach:

the crack propagates when the energy available for crack growth

overcomes the material resistance (fracture toughness) - Stress Intensity Factor Approach:

the crack propagates when a local measure of the singular stress field

reaches a critical value (fracture toughness)

intensification 3

- Energy Approach: (Griffith, A.A., The phenomena of rupture and flow in solids, Philosofical Transactions, Series A, vol. 221, 1921, 163-198; Griffith, A.A., The theory of rupture, First Int. Congress of Applied Mechanics, Delft, 1924, 55-63;) - Stress Intensity Factor Approach: (Wieghardt, K., (1907), Z. Mathematik und Physik, 55, 60-103; translated by Rossmanith (1995), On splitting and cracking of elastic bodies, Fatigue Fract. Engrg. Mater. Struct., 18, 1371-1405) Muskhelishvili, N.I., Some basic problems of the mathematical theory of elasticity, in Russian 1933 (in English 1953, Noorhoff-Groningen). Westergaard, Bearing Pressures and cracks, (1937), J. Applied Mechanics, 6, A49-53. Williams, M.L. (1952), Stress singularities resulting from various boundary conditions in angular corners of plates in extension, J. Applied Mech., 19, 526-528. Williams, M.L. (1957), On the stress distribution at the base of a stationary crack, J. Applied Mech., 24, 109-114.)

intensification 4

CRACK TIP SINGULARITY IN AN INFINITE PLATE CONTAINING A CRACK

WESTERGAARD METHOD

(Westergaard, Bearing Pressures and cracks, (1937), J. Applied Mechanics, 6, A49-53.) (see Carpinteri, Meccanica dei materiali e della frattura, for details)

Hp: homogeneous, isotropic, linearly elastic body

plane stress crack length = 2a biaxial load:

k )(z

)(z

x

y

σσ

σσ

=∞=

=∞=

with k = real constant, k- ∞≤≤∞ Problem: define stress field at crack tip

intensification 11

Stress field ahead of the crack tip in an infinite cracked plate (symmetry about x axis)

( )

( )

23

22

23

22

23

22 2

θθθ

θθθ

θθθ

πσ

πσ

πσ

sincossinr 2

K

sinsin1cosr 2

K

Bsinsin-1cosr 2

K

I

I

I

=

+=

+=

xy

y

x

Note: - General expression for the stress components:

)(θπ

σ ijij II f

r 2K

=

- inverse square root singularity in all stress components: - power of the singularity and fIij(θ) depending on crack face boundary conditions and unaffected by remote boundary conditions - stress field univocally defined by KI (now still unknown): → KI is a measure of the singular stress field → single parameter description of crack tip conditions - KI units: [F] [L] -3/2 - Singularity dominated zone

massabo

Rectangle

massabo

Rectangle

intensification 13

Recall πIKag(a) = then:

aKI πσ= KI = Stress Intensity Factor (SIF) in an infinite cracked plate subject to biaxial loading (symmetry about x axis): Note: - σx(∞) does not affect KI. - KI and stress field at the crack tip depend on crack length a - σx = σ(k-1) for θ = π, along crack faces: σx = -σ uniaxial load (k = 0) σx = 0 biaxial load, (k = 1 or σx(∞)=σy(∞)=σ)

intensification 14

EFFECT OF FINITE SIZE: MODE I PROBLEMS IN FINITE BODIES

- Stresses at the crack tip:

)(θπ

σ ijij II f

r 2K

=

- Stress Intensity Factor in finite body:

F aKI πσ= F dimensionless function, typically a polynomial expression, depending on geometry and loading conditions.

massabo

Rectangle

intensification 15

Examples: - Cracked strip in tension:

1/2

1/2

I

2hasec

haFF

2hasec aK

=

=

=

π

ππσ

guess based on numerical solution

haF

- Three point bending beam:

9/27/25/2

3/21/2

3/2I

ha 38.7-

ha 37.6-

ha 21.8

ha 4.6-

ha2.9

haF

haF

thPlK

+

+

=

=

intensification 16

CRACK OPENING DISPLACEMENT

- From plane stress elasticity:

)( xyy

y E1

yu

νσσε −=∂

∂=

then:

terms motionbody rigid dy u yy += ∫ε - substituting the stresses in the singularity dominated region:

dy )ByImZ-(ReZE

-dy )ByImZ(ReZE1dy εu '

II'IIyy ∫∫∫ +−+==

ν

- It is easy to check that the following uy satisfies previous equation:

ByE

1yReZE

1ZImE2u IIy

νν +−

+−=

intensification 17

- using the complex potential:

CBξ2ξ2KZ 1/2I

I ++=π

- and polar coordinates:

C)Br(r2

2KZ 1/2II ++++= θθθθ

πsincos)

2sin

2(cos ii

-so that:

1/2I1/2

yy r EK24uuCOD

=−=−==π

πθπθ )()(

Note: - COD depends on KI/E and increases on increasing KI/E - COD is parabolic along the crack, COD ≈ r1/2

- COD has a vertical tangent in r = 0

massabo

Rectangle

intensification 18

ELEMENTARY MODES OF LOADING APPLIED TO A CRACK

Mode I: Opening symmetric loading about crack plane Mode II: In-plane shear or in-plane sliding anti-symmetric (or skew-symmetric) loading about x axis Mode III: Tearing or out of plane shear anti-symmetric (or antiplane) loading about x-z plane

intensification 19

INFINITE CRACKED PLATE IN MODE II

(Westergaard, Bearing Pressures and cracks, (1937), J. Applied Mechanics, 6, A49-53.)

- Airy stress function: IIII ZyRe−=Φ

- ………

- Stresses at crack tip:

( )

( )23

22

23

22

23

22

1 θθθ

θθθ

θθθ

πσ

πσ

πσ

sinsin-cosr 2

K

cossincosr 2

K

coscos2sinr 2

K

II

II

II

=

=

+−=

xy

y

x

- )(θπ

σ ijij IIII f

r 2K

=

- Stress Intensity Factor:

aKII πτ=

intensification 20

STRESS FIELD AHEAD OF THE CRACK TIP

Mode I: ( )

( )

stress plane in strain, plane in

sincossinr 2

K

sinsin1cosr 2

K

Bsinsin-1cosr 2

K

I

I

I

0)(

0

2

23

22

23

22

23

22

=+=

==

=

+=

+=

zyxz

yzxz

xy

y

x

σσσνσ

σσπ

σ

πσ

πσ

θθθ

θθθ

θθθ

Mode II:

( )

( )

stress plane in strain, plane in

sinsin-cosr 2

K

cossincosr 2

K

coscos2sinr 2

K

II

II

II

0)(

0

1 23

22

23

22

23

22

=+=

==

=

=

+−=

zyxz

yzxz

xy

y

x

σσσνσ

σσπ

σ

πσ

πσ

θθθ

θθθ

θθθ

Mode III:

0

2

2

====

=

−=

xxyyx

yz

xz

σσσσπ

σ

πσ

θ

θ

cosr 2

K

sinr 2

K

III

III

intensification 21

CRACK TIP DISPLACEMENT FIELDS Mode I:

=u2

2sin + 1)- 2

cos 2r

GK = u

2 2cos - 1) sin

2r

GK =u

z

2 Ix

2 Iy

0

(2

(22

+

θκθπ

θκθπ

Mode II:

0 =u2

cos 2+ 1) sin 2r

2GK u

2 sin 2- 1)

2 cos

2r

2GK =u

z

2IIx

2IIy

++=

−−

θκθπ

θκθπ

(2

(

Mode III:

0 =u0 u

sin2r GK =u

y

x

III z

=

− 2

θπ

where: κ = 3−4ν (plane strain) κ = (3−ν)/(1+ν) (plane stress)

intensification 23

SIF EVALUATION

- Close form solutions:

complex function theory (conformal mapping, boundary collocation method, Laurent series expansion,… integral transforms (Fourier, Mellin, Hanckel transforms) eigenfunction expansion

limited to very simple cases - Computational solutions (FEM, BEM, FDM, …) - Experimental solutions (photoelasticity, moire interferometry,…) - Fracture handbooks Tada, Paris and Irwin, (1985), The stress analysis of cracks handbook, Paris Prod., Inc., St. Luis (II edition). Murakami (1987), Stress Intensity Factor handbook, Pergamon Press, NY. Rooke and Cartwright, (1976), Compendium of Stress Intensity Factors, Her Majesty’s Stationary Office, London.

intensification 24

Tada, Paris and Irwin, (1985), The stress analysis of cracks handbook

intensification 28

STRESS INTENSITY FACTOR SUPERPOSITION PRINCIPLE

- For linear elastic materials individual components of stress, strain and displacement are additive. - Similarly, stress intensity factors are additive:

...

...

...

+++=

+++=

+++=

(C)(B)(A)III

(C)(B)(A)II

(C)(B)(A)I

IIIIIIIII

IIIIII

III

KKKK

KKKK

KKKK

where:

(i)(i)(i)IIIIII

K ,K K , (i = A, B, C, ….) are mode I, mode II and mode III stress intensity factors for the ith applied load. Example:

(SENT)

I IKK =

(SENB)

I IKK =

(SENB)(SENT)

I IIKKK +=

intensification 31

SUPERPOSITION PRINCIPLE

Any loading configuration can be represented by appropriate tractions applied directly to the crack faces Proof:

intensification 32

SUPERPOSITION PRINCIPLE

- Infinite plate subject to uniaxial load with a crack of length 2a oriented at an angle β with the x axis

- For β ≠ 0 the crack experiences Mode I and Mode II - Introduce a new coordinate system x' - y' with x' coincident with the crack orientation - Define loads in the new system using Mohr's Circle

cos sin

cos

sin

yx

2y

2x

ββσσ

βσσ

βσσ

=

=

=

''

'

'

- Stress intensity factors

acos K 2I πβσ=

acossin KII πββσ=

mixed mode 1

MIXED MODE FRACTURE

Assumptions: homogeneous, isotropic, linear elastic material plane problem (Mode I + Mode II)

Griffith’s Energy Criterion applies only to collinear crack growth A Criterion for Mixed Mode fracture must define: a) the direction of crack growth b) the critical values of the fracture parameters MAXIMUM CIRCUMFERENTIAL STRESS CRITERION (Erdogan and Sih, (1963), On crack extension in plates under plane loading

and transverse shear, J. Basic Engineering, 85, 519-527)

mixed mode 2

Example of Application - Cracked plate subject to biaxial load σ1 - σ2

- Assume: m = σ1 / σ2 β = crack direction about σ2

- Crack tip stress intensity factors: a K

a K

II

I

πτ

πσ

β

β

=

=

mixed mode 3

Direction of crack growth The crack propagates in a direction normal to the maximum circumferential stress σθ

Special cases: 1) m = 1 , σ1 = σ2 ⇒ Mohr’s circle degenerates

into a point symmetry about crack direction

01

=

=

β

β

τ

σσ ⇒ KII = 0, θ = 0

⇒ collinear crack growth 2) m = 0, σ1 = 0 (uniaxial tension)

if β = 0 ⇒ θ = 0 collinear crack growth

if β ≠ 0 ⇒ °==→ + 70.6)0,(m

0lim

βθβ

mixed mode 4

Criterion for crack growth The Mixed-Mode crack will propagate along θ when σθ = σθcr(Mode I)

Fracture locus

- the fracture locus is symmetric about the K*I axis - the fracture locus is not defined for KI < 0

energy approach 2

GRIFFITH ENERGY CRITERION

(Griffith, A.A., The phenomena of rupture and flow in solids, Philosofical Transactions, Series A, vol. 221, 1921, 163-198; Griffith, A.A., The theory of rupture, First Int. Congress of Applied Mechanics, Delft, 1924, 55-63;) - First law of Thermodynamics: when a system goes from a nonequilibrium state to equilibrium there will be a net decrease in energy → A crack can form in a body only if such a process causes the total energy to decrease or remain constant (Griffith) - Consider an infinite plate of unit width subject to biaxial load in plane stress conditions. Imagine to create a crack of length 2a while keeping the remote displacements constant

The elastic strain energy released is proportional to the strain energy contained in a circle of radius a (after Inglis):

EaW

22

eσπ=

energy approach 3

The strain energy, W, of the cracked plate is then given by:

Ea-WW-WW

22

0e0σπ==

where W0 is the strain energy of the uncracked plate.

- The energy required to create the crack surface is:

ss aW γ4= - An incremental crack extension da is possible if:

0da

dWdadW s ≤+

s

24

Ea γσπ ≥2

aEs

πγσ 2

≥

(substitute E with E’ for plane strain)

massabo

Rectangle

energy approach 4

Fracture Energy:

GIC = 2 γs

= energy required to create a unitary crack area. (crack area = 2a; surface area = 4a) - measure of the fracture toughness of the material - to be defined through standard tests

⇓ Griffith equation: Stress for equilibrium crack growth:

aEIC

max πσ

G=

growthdynamic aEIC

πσ G

>

growth no aEIC

πσ G

<

energy approach 5

Stress for equilibrium crack growth:

aEIC

max πσ G

=

growthdynamic aEIC

πσ G

>

growth no aEIC

πσ G

<

- crack growth is unstable: the load decreases on increasing the crack length - two asymptotes: • for a →∞, σmax → 0

• for a →0 σmax → ∞ ???? analogy with buckling collapse vs. strength collapse: for a<a0 the strength collapse precedes the fracture collapse - The crack length a0 corresponding to σmax = σy ( = σp) is an equivalent measure of the microcracks, flaws and defects of the virgin material:

2y

IC0

Eaσπ

G1=

energy approach 6

MODIFIED GRIFFITH EQUATION

(Irwin, (1948), Fracture Dynamics, Fracturing of Metals, ASM Cleveland, 146-166; Orowan,(1948), Fracture and strength of solids, Reports on Progress in Physics, XII, p. 195.) - In actual structural materials: a) the energy needed to cause fracture is much higher than the surface energy (orders of magnitude higher) b) inelastic deformations arise around the crack front → linear elastic medium with infinite stresses at the crack tip unrealistic - Griffith equation can be modified to include the plastic work γP dissipated at the crack front

a)E( ps

πγγ

σ+

≥2

- Orowan estimated γP ≈ 103 γs in typical metals. - The modified Griffith equation is correct only if: the size of the plastic zone around the crack tip is very small compared to the crack size (small-scale yielding conditions) → the details of the crack tip stress do not affect the stress field in the elastic bulk of the medium → purely elastic solutions can be used to calculate the rate of energy available for fracture (!!!!!!!!)

energy approach 7

ENERGY BALANCE IN BRITTLE FRACTURE

- Griffith criterion refers to a special case, i.e. infinite cracked plate, biaxial loading, fixed-grip conditions - In general, the First Law of Thermodynamics yields:

0dA

dWdAdW

dAdE sT ≤+=

A = crack area ET = total energy Ws = energy required to create new crack surfaces W = total potential energy W = U - L U = potential strain energy L = potential of external forces - From the definition of fracture energy:

ICss 2

dAdW

G== γ

- Energy criterion for brittle crack growth (Mode I):

ICdAdW

G≥−

energy approach 8

THE STRAIN ENERGY RELEASE RATE (Irwin, (1957), Analysis of stresses and strains near the end of a crack traversing a plate. ASME Journal of Applied Mechanics, 24, 361-364) - Irwin introduced the Strain Energy Release Rate

dAdW

−=G

as the energy available for an increment of crack extension, given by the total potential energy released due to the formation of a unit crack area - G is also called Crack Driving Force or Crack Extension Force - Energy criterion for brittle crack growth (Mode I):

ICI GG =

energy approach 9

ENERGY APPROACH VS. STRESS APPROACH (CRITICAL CONDITIONS)

- The strain energy release rate is the fracture parameter describing the global behavior of the body. - The stress intensity factor if the fracture parameter describing the local behavior (stresses, strains and displacements near the crack tip) of the body - Stress criterion for brittle crack growth (Mode I):

ICI KK ≥

KIC = critical stress intensity factor KIC is a measure of the material fracture toughness to be defined through standard tests - ICG and KIC are related through a fundamental relationship

EK2

ICIC =G

derived from:

conditiony instabilitenergy aEIC

πσ G

≥

conditiony instabilit stress a

KICπ

σ ≥

⇓ EK ICIC G=

energy approach 17

STRAIN ENERGY RELEASE RATE IN A DCB SPECIMEN

- Assume built-in end conditions for the two arms. From beam theory:

3

33

EBh4Pa

3EIPa/2 ==Δ

- The elastic compliance is:

3EI2a

PC

3=

Δ=

- The strain energy release rate in the DCB specimen is:

3

2

2

2

Eha

BP 12

3EI6a

2BP

dadC

2BP 222

===G

- Same conclusion from:

3

2

2

232

Eha

BP 12

EIa

BP

3EIaP

dad

B1P

21

dad

B1

BdadW 22

==⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎠⎞

⎜⎝⎛ Δ−−=−=G

energy approach 18

- The critical load for crack propagation (Griffith criterion) is

12BEh

a1P

23IC

crG

=

⇒ unstable crack growth Note: Strain energy release rate and critical load have been defined through a beam theory approximation.

If the geometry satisfies the assumptions of beam theory the solution is correct. However, the local fields at the crack tip are not correctly given by beam theory

energy approach 19

ENERGY RELEASE RATE IN A BODY SUBJECT TO DIFFERENT LOADING SYSTEMS

Hp: self-similar crack growth - From the stress intensity factors of a body subject to different loading systems (A), (B), (C),…..

...

...

...

+++=

+++=

+++=

(C)(B)(A)III

(C)(B)(A)II

(C)(B)(A)I

IIIIIIIII

IIIIII

III

KKKK

KKKK

KKKK

and the relationship between G and K

2GK

EK

EK 2

III2II

2I ++=G

⇒ the strain energy release rate is

2G....)K(K

E...)K(K

E....)KK(K 2(B)

III(A)III

2(B)II

(A)II

2(C)I

(B)I

(A)I ++

+++

++++

=G

energy approach 20

- The strain energy release rate contributions for each mode of fracture are addictive

2GK

EK

EK 2

III2II

2I ++=++= IIIII GGGG

- The strain energy release rate contributions for each loading system are not addictive, e.g. for mode I

...+++≠+++ (C)I

(B)I

(A)I

...(C)(B)(A)I GGGG

cohesive crack model 1

NONLINEAR FRACTURE MECHANICS THE COHESIVE CRACK MODEL

(Dugdale, D.S.: Yielding of steel sheets containing slits, J. Mechanics Physics Solids 8 (1960), 100-104 Barenblatt, G.I.: The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks, J. Applied Mathematics and Mechanics 23 (1959), 622-636. Barenblatt, G.I.: The mathematical theory of equilibrium cracks in brittle fracture, in H.L. Dryden and T. von Karman (eds.), Advanced in Applied Mechanics, Academic Press, New York, 1962, pp. 55-129.)

ductile materials ⇒ Dugdale’s model purely brittle materials ⇒ Barenblatt’s model quasi brittle materials ⇒ Hillerborg’s Fictitious (e.g. concrete) Crack Model brittle matrix composites ⇒ Bridged crack model quasi brittle matrix composites ⇒ Cohesive crack model


Dugdale’s Model for ductile fracture

Fracture of elasto-plastic materials Assumptions: - Mode I crack in an infinite sheet under uniform tensile stress

- the material is ductile and the plastic deformations localized in a thin zone coplanar with the crack

- the plastic zone is modeled through a fictitious crack (of unknown length) and a uniform distribution of cohesive tractions σP = yield stress

Length of the cohesive zone The length of the cohesive zone aP is calculated by imposing the condition of smooth closure of the crack faces:

0KKK PIII =+= σσ

where: )( PI aa K += πσσ


The SIF due to the plastic stresses is calculated using the SIFs due to a pair of concentrated forces F acting at x:

1/2

I

1/2

I

xaxa

aF(B)K

xaxa

aF(A)K

⎟⎠⎞

⎜⎝⎛

+−

=

⎟⎠⎞

⎜⎝⎛

−+

=

π

π

so that:

∫ ⎥⎦

⎤⎢⎣

⎡++−+

+−+++

+=

P

P

a

a P

P

P

P

P

PI dx

x)a(ax)a(a

x)a(ax)a(a

)aa- K

(πσ

σ

)()(2

P

PPI aa

aarccosaa- (a)K P ++

=π

σσ

From the condition for smooth closure:

0aa

aarccosaaaaP

PPP =

++

−+)(

)(2)(π

σπσ

PP 2cos

aaa

σπσ

=+

In the limit σ = 0 ⇒ aP = 0

σ → σP ⇒ aP → ∞


Performing a Taylor series expansion (if σ << σP): 2

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

+ PP 2211

aaa

σπσ

2P

22

P

P

8aaa

σσπ

=+

and, as a function of KIσ,

2P

2I

PK

8 a

σπ σ=

Length of the cohesive zone in critical conditions Following Irwin's approach:

2P

2IC

PCK

8 a

σπ

=

where: EK ICIC G=

and: ==∫ aP0

P σdσa

δδδ

ICG

with δa the crack tip opening displacement given by:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=

P

Pa 2

seclnE

a8 σ

πσπσδ or E

K P

2I

a σδ = (if σ << σP)

(use Castigliano's method)


Barenblatt’s Model for purely brittle fracture “In our work the question concerning equilibrium cracks forming during brittle fracture of a material is presented as a problem in the classical theory of elasticity, based on certain very general hypotheses concerning the structure of a crack and the forces of interaction between its opposite sides, and also on the hypothesis of finite stresses at the ends of the crack, or, which amounts to the same thing, the smoothness of the joining of opposite sides of the crack at its ends” (Barenblatt, 1959)

Consider a Mode I crack in a homogeneous, isotropic, linear-elastic infinite medium under uniform loading

Represent the atomic bonds holding together the two halves of the body separated by the crack as cohesive forces acting along the edge regions of the crack and attracting one side of the crack to the other b = interatomic distance (order = 10-7 mm)

maximum intensity = ideal strength b

EE sc

γπ

σ ≈≈


Determination of the cohesive forces The accurate determination of the cohesive forces acting along the edge regions is difficult - Assumptions: 1) The dimension of the edge region is small in comparison with the size of the whole crack 2) The displacements in the edge region, for a given material under given conditions, is always the same for any acting load ⇒ during crack propagation the edge region simply translate forward 1) + 2) = small scale yielding assumption 3) The opposite sides of the crack are smoothly joined at the ends or, which amounts to the same thing, the stress at the end of a crack is finite

⇒ K = 0 (zero stress intensity factor)

These assumptions lead to the definition of the critical state of mobile equilibrium which depends on the modulus of cohesion depending on the integral of the cohesive tractions along the edge region


Equivalence of Griffith and Barenblatt approaches Willis (by means of the complex variable method) and Rice (by means of the J integral) proved that: Barenblatt’s theory based on atomic forces is equivalent to Griffith’s energy approach provided the integral of the cohesive forces is equal to the fracture energy

IC

*y

0dy f G=∫

Barenblatt also showed that the cohesive forces essentially have effect only on the displacement field close to the edge of the crack and not on those in the main part of the crack


Rice’s proof of the equivalence between Barenblatt’s model and Griffith’s energy approach

- Assume the crack to be in a state of mobile equilibrium - Consider cohesive forces σ(δ) acting along the crack - Let δ∗ to be the separation distance beyond which the cohesive tractions vanish

- Consider the path Γ for which: dy = 0, ds = dx on Γ− and ds = -dx on Γ+ , T1 = 0, T2 = σ(δ) on Γ+ , T2 = -σ(δ) su Γ− -Evaluate the J integral along Γ:

⇒∂∂

=∂∂

=

=∂∂

∂∂

=∂∂

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

⋅−=

∫∫

∫∫∫∫∫++

+

+

+

ΓΓΓ +−

dxx

)(-)dxu-(ux

)(-

(-dx)xu)(-(dx)

xu)(--ds

xu-ds

xu-ds

xuTdyUJ

da

a

da

a

-22

a2

da22

22

2i

id

δδσδσ

δσδσada

TT

∫=*

)(δ

δδσ0

dJ

⇒ The value of J which will cause crack extension is given by the integral of the cohesive forces:

⇒ IC

*

0d )( G=∫ δδσ

δ

Documents

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