Elastic-Plastic Fracture Mechanics - Elastic-Plastic Fracture Mechanics Introduction...

download Elastic-Plastic Fracture Mechanics - Elastic-Plastic Fracture Mechanics Introduction ¢â‚¬¢ When does one

of 32

  • date post

    12-Mar-2020
  • Category

    Documents

  • view

    5
  • download

    0

Embed Size (px)

Transcript of Elastic-Plastic Fracture Mechanics - Elastic-Plastic Fracture Mechanics Introduction...

  • Elastic-Plastic Fracture Mechanics

    Introduction

    • When does one need to use LEFM and EPFM?

    • What is the concept of small-scale and large-scale yielding?

    Contents of this Chapter

    • The basics of the two criteria used in EPFM: COD (CTOD), and J-Integral (with H-R-R)

    • Concept of K- and J-dominated regions, plastic zones

    • Measurement methods of COD and J-integral

    • Effect of Geometry

    Background Knowledge

    • Theory of Plasticity (Yield criteria, Hardening rules)

    • Concept of K, G and K-dominated regions

    • Plastic zone size due to Irwin and Dugdal

  • LEFM and EPFM

    LEFM

    • In LEFM, the crack tip stress and displacement field can be uniquely characterized by K, the

    stress intensity factor. It is neither the magnitude of stress or strain, but a unique parameter that

    describes the effect of loading at the crack tip region and the resistance of the material. K filed is

    valid for a small region around the crack tip. It depends on both the values of stress and crack size.

    We noted that when a far field stress acts on an edge crack of width “a” then

    for mode I, plane strain case

    σ

    σ

    τ π

    θ

    θ θ

    θ θ

    θ θ

    x x

    y y

    x y

    IK

    r

    R S |

    T|

    U V |

    W| =

    +

    L

    N

    M M M M M MM

    O

    Q

    P P P P P PP

    2 2

    1 2

    3

    2

    1 2

    3

    2

    2

    3

    2

    c o s

    s i n ( ) s i n ( )

    s i n ( ) s i n ( )

    s i n ( ) s i n ( )

    σ σ ν σ σzz zz xx yy= = +0 for plane stress; for plane strain( )

    u

    u

    K r k

    k

    x

    y

    I RST UVW

    = − +

    + −

    L

    N

    M M MM

    O

    Q

    P P PP

    2

    2 1 2

    2

    2 1 2

    2

    2

    2µ π

    θ θ

    θ θ2

    cos ( sin ( ))

    sin ( cos ( ))

  • LEFM concepts are valid if the plastic zone is much smaller than the singularity zones.

    Irwin estimates

    Dugdale strip yield model:

    r K

    p

    I

    ys

    = 1

    2

    2

    π σ ( )

    r K

    p I

    ys

    = 1

    8

    2 ( ) σ

    ASTM: a,B, W-a 2.5 , i.e. of specimen dimension.≥ ( ) KI

    ys σ

    2 r p ≤

    1

    50

    LEFM cont.

    Singularity dominated region

    σ

    σ

    τ π

    x x

    y y

    x y

    IK

    r

    R S |

    T |

    U V |

    W |

    =

    L

    N M M M

    O

    Q P P P2

    1

    1

    0

    For =0 θ

    For = 2

    all ijθ θ

    σ, = 0

  • EPFM

    • In EPFM, the crack tip undergoes significant plasticity as seen in the following diagram.

    sharp tip

    I d e a l e la s ti c b r i t t l e b e h a v i o r

    c l e a v a g e f r a c tu r e

    P : A p p l ie d l o a d

    P : Y i e ld l o a dy D is p la c e m e n t , u

    L o ad

    ra ti

    o ,

    P /P

    y

    1 .0

    F r ac tu re

    B lu n t t i p

    L im ite d p la st ic it y a t c r ac k

    t ip , s t ill c l e a v ag e f ra c tu r e

    D isp la cem en t, u

    L o ad

    ra ti

    o ,

    P /P

    y

    1 .0

    F r ac tu re

  • Blunt tip

    Void formation & coalescence

    failure due to fibrous tearing Disp lacement, u

    L o ad

    ra ti

    o ,

    P /P

    y

    1 .0 Fracture

    l a r g e s c a l e

    b l u n t i n g

    L a r g e s c a l e p l a s t i c i t y

    f i b r o u s r a p t u r e / d u c t i l e

    f a i l u r e D is p la c e m e n t , u

    L o ad

    ra ti

    o , P /P

    y

    1 .0 F rac tu re

  • EPFM cont.

    • EPFM applies to elastoc-rate-independent materials, generally in the large-scale plastic

    deformation.

    • Two parameters are generally used:

    (a) Crack opening displacement (COD) or crack tip opening displacement (CTOD).

    (b) J-integral.

    • Both these parameters give geometry independent measure of fracture toughness.

    δSharp crack

    Blunting crack

    y

    x

    Γ ds

  • EPFM cont.

    • Wells discovered that Kic measurements in structural steels required very large thicknesses for

    LEFM condition.

    --- Crack face moved away prior to fracture.

    --- Plastic deformation blunted the sharp crack.

    δ Sharp crack

    Blunting crack

    • Irwin showed that crack tip plasticity makes the crack behave as if it were longer, say from size a to a + rp -----plane stress

    From Table 2.2,

    Set ,

    r K

    p I

    y s

    = 1

    2

    2

    π σ ( )

    u K r

    ky I= + −

    2 2 2 1 2

    2

    2

    µ π θ θ

    s in ( ) [ c o s ( ) ]

    θ π= u k K r

    y I

    y =

    + 1

    2 2µ π a ry+

    θ π=

    δ π σ

    = =2 4 2

    2 u

    K

    E y

    I

    y s

    Note:

    since

    k E= − +

    = + 3

    1 2 1

    ν ν

    µ ν and ( )

    δ π σ

    = =CTOD 4 G

    ys

    G K

    E

    I= 2

  • CTOD and strain-energy release rate

    • Equation relates CTOD ( ) to G for small-scale yielding. Wells proved that

    Can valid even for large scale yielding, and is later shown to be related to J.

    • can also be analyzed using Dugdales strip yield model. If “ ” is the opening at the end of the strip.

    δ π σ

    = =CTOD 4 G

    ys

    δ δ

    δ δ

    δ σ

    ys

    Consider an infinite plate with a image crack subject to a

    Expanding in an infinite series,

    σ σ∞ =

    δ σ

    π π σ

    σ = = ⋅2

    8 u

    a

    E y

    y s

    y s

    li n s e c ( 2

    )

    δ σ

    π π σ

    σ π σ

    σ = ⋅ + ⋅ +

    8 1

    12

    2 4ys

    ys ys

    a

    E [ 1

    2 (

    2 (

    2 ) ) . . . ]

    If , and can be given as:

    In general,

    δ σ

    π σ σ

    = + ⋅ K

    E

    I

    y s y s

    2

    2[ 1 1

    6 (

    2 ) ]

    σ σ

    σ σ δ σ σys

    ys

    ysE

    G ≈

  • Alternative definition of CTOD

    δSharp crack

    Blunting crack

    δBlunting crack

    Displacement at the original crack tip Displacement at 900 line intersection, suggested by Rice

    CTOD measurement using three-point bend specimen

    W

    P

    a

    r p(W-a)

    z

    Vp

    δ p

    '

    '

    'δ p l p p

    p

    r W a V

    r W a a z =

    − + +

    ( )

    ( )

    displacement

    expandingδ

  • Elastic-plastic analysis of three-point bend specimen

    δ δ δ σ

    = + = + −

    − + +el p l I

    ys

    p p

    p

    K

    m E

    r W a V

    r W a a z

    2 ( )

    ( )

    Where is rotational factor, which equates 0.44 for SENT specimen.δ pl

    • Specified by ASTM E1290-89 --- can be done by both compact tension, and SENT specimen

    • Cross section can be rectangular or W=2B; square W=B

    KI is given by

    δ ν

    σe l I

    y s

    K

    E =

    −2 21 2

    ( )

    K P

    B W f

    a

    W I = ⋅ ( )

    δ p l p p

    p

    r W a V

    r W a a z =

    − + +

    ( )

    ( )

    lo a d

    Mouthopening

    υ p υ e

    V,P

  • CTOD analysis using ASTM standards

    Figure (a). Fracture mechanism is purely cleavage, and critical CTOD

  • More on CTOD

    The derivative is based on Dugdale’s strip yield model. For

    Strain hardening materials, based on HRR singular field.

    By setting =0 and n the strain hardening index based on

    *Definition of COD is arbitrary since

    A function as the tip is approached *Based on another definition, COD is the distance between upper

    and lower crack faces between two 45o lines from the tip. With this

    Definition

    2

    or ICOD y y

    K J

    E δ

    σ σ = =

    ( ) 11

    1 ,

    n

    n

    n i y i

    y y n

    J u r u n

    I αε θ

    ασ ε

    + +

      =  

     

    1

    3

    2

    n

    y ije

    y y y

    ε σσα ε σ σ

    −  

    =     ( ) ( ), 0 , 0y yu x u xδ + −= −

    ( ) 1

    1nx +−

    COD n

    y

    J dδ

    σ =

  • Where ranging from 0.3 to 0.8 as n is varied from

    3 to 13 (Shih, 1981)

    *Condition of quasi-static fracture can be stated as the

    Reaches a critical value .