断裂与损伤力学导论.pdf

2022-11-23 1下载 2.96MB

断裂与损伤力学导论


Introduction to Fracture
and Damage Mechanics
Wolfgang Brocks
Five Lectures
at
Politecnico di Milano
Milano, March 2012
Contents
I. Linear elastic fracture mechanics (LEFM)
Stress field at a crack tip
Stress intensity approach (IRWIN)
Energy approach (GRIFFITH)
J-integral
Fracture criteria – fracture toughness
Terminology
II. Plasticity
Fundamentals of incremental plasticity
Finite plasticity (deformation theory)
Plasticity, damage, fracture
Porous metal plasticity (GTN Model)
III. Small Scale Yielding
Plastic zone at the crack tip
Effective crack length (Irwin)
Effective SIF
Dugdale model
Crack tip opening displacement (CTOD)
Standards: ASTM E 399, ASTM E 561
IV. Elastic-plastic fracture mechanics (EPFM)
Deformation theory of plasticity
J as energy release rate
HRR field, CTOD
Deformation vs incremental theory of plasticity
R curves
Energy dissipation rate
V. Damage Mechanics
Deformation, damage and fracture
Crack tip and process zone
Continuum damage mechanics (CDM)
Micromechanisms of ductile fracture
Micromechanical models
Porous metal plasticity (GTN Model)
Seite 1
Concepts of Fracture Mechanics
Part I: Linear Elastic Fracture Mechanics
W. Brocks
Christian Albrecht University
Material Mechanics
Milano_2012 2
Overview
I. Linear elastic fracture mechanics
(LEFM)
II. Small Scale Yielding
III. Elastic-plastic fracture mechanics (EPFM)
Seite 2
Milano_2012 3
LEFM
Stress Field at a crack tip
Stress intensity approach (Irwin)
Energy approach (Griffith)
J-integral
Fracture criteria – fracture toughness
Terminology
Milano_2012 4
Stress Field at a Crack Tip
boundary conditions
( 0, 0) ( , ) 0
( 0, 0) ( , ) 0
yy
xy r
x y r
x y r
θθ
θ
σ σ θ π
σ σ θ π
≤ = = = =
≤ = = = =
Hooke’s law of linear elasticity
2
1 2ij ij kk ijG νσ ε ε δ
ν
⎛ ⎞= +⎜ ⎟−⎝ ⎠ Inglis [1913], Westergaard
[1939], Sneddon [1946], ...
Williams’ series [1957]
0ΔΔΦ =Airy’s stress function
( )
( )
2 cos cos 2
sin sin 2
r A B
C D
λΦ λθ λ θ
λθ λ θ
+ ⎡= + + +⎣
⎤+ + + ⎦
21 15 3 5 3 31
04 2 4 2 4 2 4 2
9 3 5 9 15 5
1 14 2 4 2 4 2 4 2
cos cos sin cos 4 cos
cos cos sin cos ( )
rr
A C A
r r
A r C r r
θ θ θ θ
θ θ θ θ
σ θ− −= − + − + + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
+ + + + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ O
Seite 3
Milano_2012 5
Fracture Modes
Milano_2012 6
LEFM: Stress Intensity Approach
Irwin [1957]: mode I
1 1
II( , ) ( )
2j j ji ii
Kr f
r
Tσ θ θ δ
π
δ= +
T = non-singular T-stress
Rice [1974]: effect on plastic zone
General asymptotic solution
I II III
I II III
1( , ) ( ) ( ) ( )
2ij ij ij ijr K f K f K f
r
σ θ θ θ θ
π
⎡ ⎤= + +⎣ ⎦stresses
displacements I II III
I II III
1( , ) ( ) ( ) ( )
2 2i i i i
ru r K g K g K g
G
θ θ θ θ
π
⎡ ⎤= + +⎣ ⎦
I (geometry)K a Yσ π∞=
KI = stress intensity factor
Seite 4
Milano_2012 7
Angular Functions in LEFM
Milano_2012 8
A.A. Griffith [1920]
LEFM: Energy Approach
Elastic strain energy of a panel of thickness
B under biaxial tension in a circular
domain of radius r
( )( ) ( )
2 2
2 2e
0 1 1 2 1
16
BrU
G
π σ κ λ λ∞ ⎡ ⎤= − + + −⎣ ⎦
elliptical hole, axes a,b
( ) ( ) ( )
( )(

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