Fracture mechanics – fundamentals

Stress at a notch Stress at a crack

Stress intensity factors Fracture mechanics based design

Failure modes Failure can occur in a number of modes: -  plastic deformation -  (brittle) fracture - instability (static & dynamic

Failure mode will depend on load conditions, material, temperature etc

The course focuses on fracture

Example – fracture of a finger

Fracture – details A crack normally propagates in mode I (perpendicular to the load)

Possible reasons for branching:

- altered load direction

- anisotropy (path of least resistance)

- influence of free edge

Stress at a notch!

!  The elastic stress at the notch edge is defined by the stress concentration factor: !max = Kt!!nom

!  At an elliptic hole in a large plate, the maximum stress is !max = (1+2a/b)!!nom

!  When b"0, (the case of a crack) we get !max = ! (result of elastic mtrl)"and Kt = ! (useless measure)#

!  We need something else to charactarize the state of stress!

!

"nom

!

"nom!

Kt = 3

!

" = 3#" nom

!

" = $"nom

!

Kt "# då b" 0

!

$nom

!

$nom

!

2a

!

Kt = 1+2 ab

% & '

( ) *

!

$nom

!

$nom

!

2b

!

2a

Crack tip blunting!

!  Metals: plastic deformation crack tip opening displacement (CTOD) can be used as a (ambiguous) measure of crack loading

!  Polymers: crazing

!  Ceramics (e.g. concrete): damage zones

Loading of a crack !  Three loading modes:

"  mode I – opening "  mode II – shearing "  mode III – tearing

!  In fatigue mode I loading dominates: "  often (quasi-)uniaxial loading "  shear driven growth more

difficult the longer the crack "  the crack grows in mode I

!  Exceptions: "  small crack growth "  contact fatigue "  anisotropic or inhomogenous

material

Stresses ahead of a crack tip !  In mode I, stresses ahead of

a crack can be expressed as !xx = KI/($2%r)·cos(#/2)[1–"

sin(#/2)sin(3#/2)]+…#!yy = KI/($2%r)·cos(#/2)[1–"

sin(#/2)sin(3#/2)]+…#!xy = KI/($2%r)·"

cos(#/2)sin(#/2)cos(3#/2)]+…#!  KI depends on loading and

geometry !  The expression is only valid

close to the crack tip !  Similar expressions are valid

for loading in modes II and III

Stress intensity factor !  The elastic stress field at a

crack tip is characterized by KI#

!  The stress intensity factor K represents “how fast the stress goes to infinity” at the crack tip

!  For modes II and III we get similar results (KII and KIII)

!  For the stress directly ahead of the crack tip, we get !xx = !yy = KI/$(2$x)#

!  The dimension of the stress intensity factor is [N"m/m2] or [MPa"m]

!  Do not confuse the stress intensity factor K with the stress concentration factor kt [–].

!  Some presumptions for K to be valid are: "  linear elastic material "  isotropic material "  ideal crack geometry "  loading in one mode

How to estimate K !  K can be derived from the

stress distribution for instance in the following manner: "  analytical – match K to the

stress distribution "  FE-simulations – employ

asymptotic formulas (typically the crack face displacement, %=$, gives the best results)

"  handbooks of solved cases, K described as K = !$($a) · f where " is nominal stress and f a load–geometry-factor

Superposition of K !  Superposition can be used

to derive K under combined or two (or more) loads

!  Demands: "  same geometry "  load in the same mode

Fracture mechanics based design !  If K is assumed to reflect the

stress at the crack tip, we can assume fracture to occur at a certain magnitude of K KI = KIC

!  KIC is called the fracture toughness

!  KIC is valid at plane strain (small plastic zone size)

!  Criterion for plane strain: a, t, (W-a) & 2.5(KIC/!y)2 "must be fulfilled for LEFM to be valid (note that dimensions must agree!)

Fracture mechanics design !  Determine allowable crack

size: "  Specify location of the crack "  Determine stress intensity

factor K a.f.o. {a, !,f(a)}#"  from handbooks "  from FE-simulations

"  Find ac that yields KI = KIC "  Assure that LEFM is valid

a, t, (b-a), h & 2.5(KIC/ ! y)2#

"  If not, apply corrections "  If true, apply safety factors

!  Determine allowable stress: "  Specify crack location "  Specify allowable crack size "  Determine stress intensity

factor K a.f.o. {a, !,f(a)}#"  from handbooks "  from FE-simulations

"  Find ! that yields KI = KIC "  Assure that LEFM is valid

a, t, (W-a), h & 2.5(KIC/ !y)2#

"  If not, apply corrections "  If true, apply safety factors

Plastic zone size !  The elastic stress is

theoretically infinite at the crack tip, but in reality plasticity will occur

!  The plastic zone must not be so large that it obscures the ”K-field”

Plastic zone size – continued !  The plastic zone must not

extend to a free edge !  Ensure that the distance

towards the free edge is sufficient

!  Also ensure a small plastic size as compared to the crack length a, (b–a), h ! 4"(K/"0)2/!

!  Additionally you can assure plane strain conditions (ASTM standard conditions) t, a, (b–a), h ! 2.5"(K/"0)2

!  This is the most usual criterion for static LEFM

!  Conservative or non-conservative to not follow these?

Plane strain vs plane stress !  If plane strain conditions are not

ensured the fracture toughness will depend on the component geometry

Estimation of K !  For cases when K is

unknown, it can be estimated from known cases

Advanced estimation of K !  Example:

"  a hole in the middle of a crack will have a small influence if the crack is long

"  for a large hole and small crack, the crack will just experience the nearby stress i.e. a two-sided crack will be similar to a one-sided crack

"  There is no influence of a stress parallel to a long crack

"  At the hole this stress will give an effect. Why?

Leak before break

Fracture toughness !  Fracture toughness is mainly

a material parameter, but dependent on factors as: "  geometry (plane stress,

plane strain) "  ductility "  temperature "  crack size

!  Optional methods if LEFM is not valid: "  Irwin correction (crack length

correction, see book) "  J-integral (energy measure)

Fracture toughness, continued

Fracture toughness, continued

Some practical comments !  LEFM in static and dynamic

loading !  Final fracture criterion (and

limits of validity) in static and dynamic loading

!  Plane stress – plane strain (also in the same component)

!  Environmental influence !  Elastic vs elastoplastic

modelling !  Overloads (final fracture vs

crack growth)

!  FE simulations "  Derivation of K

"  J-integral "  asymptotic formulas

"  Crack modelling "  mesh "  crack face contacts

!  Handbook solutions "  Choice of standard case

"  approximations "  superpositions

"  Derivation of “nominal stresses”

!  Testing and scatter