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Mesh Convergence Requirements
for Composite Damage Models
Carlos G. DávilaStructural Mechanics and Concepts Branch
NASA Langley Research Center
Hampton, VA
Mechanical Engineering Dept.,
TU Eindhoven, NL
17 March 2016
https://ntrs.nasa.gov/search.jsp?R=20160007808 2018-05-21T18:10:04+00:00Z
2/34
Outline
• Strength or Toughness?
• Cohesive Laws
• Process Zones and Load Redistribution
• Continuum Damage Models
• Transverse Matrix Cracks
• Summary
3/34
Scales of Damage in Composites
“Physics” of failure:• At each scale, damage is described by different physical observations
Matrixcrack
Delamination
Matrixcrack
Fiberkink
-45
0
Structure Microstructure
4/34
Effect of Material on Structural Response
• Quasi-isotropic laminates, three-dimensional (3-D) scaling
• [45m/90m/-45m/0m]ns, m, n = 1, 2, 4, 8
• d = 1/8 in., ¼ in., ½ in.
w/d = 5 d
Gripping region Gauge section
Brittle Pull-out Delamination
Thin plies Thick plies
*Green and Wisnom, 2007
Identical material, different modes of failure!
5/34
Scaling: the Effect of Structure Size on Strength
Fracture Mechanics (Griffith, 1921)
(Galileo, 1638)
(Mariotte, 1686)Statistical Theory of Size Effect
“weakness is due to the
presence of flaws”
(da Vinci, 1505)
(Weibull, 1939)
Strength of Materials
a
GE cu
6/34
Damage Length Scales in Structural Materials
Strength versus toughness:
• Strength and toughness is the result of the interplay between a number of individual
mechanisms originating at different length scales.
• Some damage mechanisms inhibit crack propagation.
Idealization of fracture processes
R. Ritchie, 2011
7/34
Size Effect - the Issue of Scale
Scaling from test coupon to structure
Structural Size, in.
Yield or Strength Criteria
Linear Elastic Fracture Mechanics
log n
log D
(Z. Bažant)
Scaling Laws
Normal testing
a
GE cu
Num
ber
of Tests
8/34
Cohesive Laws
Bilinear Traction-Displacement Law
cc Gd
0 )(
Two material properties:
• Gc Fracture toughness
• c Strength
2c
cc
GEl
Characteristic Length:
t0 Yield or Strength Criteria
log n
log D
a
GE cu
9/34
Mesh Convergence Studies
Analytical solution
lelem=2 lc
lelem=lc
lelem=lc /3
lelem=lc /5
Converged
meshes
Non-converged
meshes
Applied displacement D, mm
Ap
plie
d F
orc
e, N
Converged
meshes
2c
cc
GEl
10/34
Crack Length and Process Zone
Quasi-
brittle
Force, F
Applied displacement, D
LEFM error
2c
cp
GEl
a0
F, D
Gc= constant
Initiation
Steady-state
LEFM analytical solution
11/34
Crack Length and Process Zone
0
0
0
5
5100
100
al
lal
la
p
pp
p
Brittle:
Quasi-brittle:
Ductile:Long crack Brittle
Quasi-
brittle
ShortcrackDuctile
LEFM error
Force, F
Applied displacement, D
LEFM error
LEFM error
2c
cp
GEl
a0
F, D
12/34
Fracture-Dominated Failure
E
aG
2
2a
G
a0 Da
max
R = GIc
unstable
E
2
:Slope
Crack propagates unstably once driving force G(, a0) reaches GIc
G(, a0)
13/34
Fracture-Dominated Failure: R-Curve
2a E
aG
2
G
a0
GR(Da)
Da
init
max
GInit
unstable
Gc
2(a+Da)
G(, a0)
Crack propagates stably when driving force G(, a0) > GInit
Unstable propagation initiates at cInit GGG
2
c
cstable
EGa
D
14/34
Length of the Process Zone
mm4.36.02
c
cc
GEl
mm7.4pzl
A
B
C
D
E
F
Symmetry
Sym
me
try
h/ao = 1
Maximum Load
ao
h
CT Sun,
Purdue U2ao
Short Tensile TestLexan Polycarbonate
2h
Cohesive
elements
15/34
• The use of cohesive laws to predict the
fracture in complex stress fields is explored
• The bulk material is modeled as either
elastic or elastic-plastic
0
1000
2000
3000
4000
5000
6000
0 1 2 3 4
Fmax, N
h/a
Test (CT Sun)
LEFM
Cohesive
h/a = 0.25 (long process zone)
Observations:
• LEFM overpredicts tests for h/a<1
Lexan Plexiglass tensile specimens (CT Sun)
h/a=0.25h/a=0. 5h/a=1h/a=2h/a=4
2h
2a
h/a=1 (short process zone)
mm.7.4czl
Widthczl
Cohesive Laws - Prediction of Scale Effects
16/34
fi
i
Kd
K
,)1(
,
L+ +
K
F,
Gc
c
(1-d)K
Quasi-Brittle Response and Fracture
K
EL
EAAF
D
K
EEGL
G
EAAF
c
c
c
c
D
2
2
2
Before damage After damage
For “long” beams, the response is unstable, dynamic, and independent of Gc
0D
F2
2
c
cEGL
For stable fracture under D control:
D
F
StableUnstable
0D
F
17/34
Outline
• Strength or Toughness?
• Cohesive Laws
• Process Zones and Load Redistribution
• Continuum Damage Models (CDM)
• Transverse Matrix Cracks
• Summary
18/34
Failure Criteria and Material Degradation
Failure criterion
E
1
Residual=E/100
Strain
Stress
e/e0
Progressive Failure Analysis
1
Elastic
property
Benefits• Simplicity (no programming needed)
• Convergence of equilibrium iterations
Drawbacks• Mesh dependence
• Dependence on load increment
• Ad-hoc property degradation
• Large strains can cause reloading
• Errors due to improper load redistributions
19/34
Failure Criteria and Material Degradation
Failure criterion
E
1
Residual=E/100
Strain
Stress
e/e0
Progressive Failure Analysis
1
Elastic
property
Failure criterion
E
1
Strain
Stress
e/e0
Progressive Damage Analysis – Regularized Softening Laws
1
Elastic
property
Increasing lelem
Increasing lelem
20/34
Failure Criteria and Material Degradation
Critical Element size (snap-back):
2
2
c
celem
EGl
Strain
Stress
e/e01
Increasing lelem
unsta
ble
Bar
Element
lelem
21/34
Advanced Failure Criteria for Laminated Composites
2a0
22, YT
t12, S
L
is is
a=53º
Matrix Tension & Shear
Fiber Compression
Matrix Compression and Shear
LaRC04 Criteria
• In-situ matrix strength
prediction
• Advanced fiber kinking
criterion
• Prediction of angle of fracture
LaRC02 - Dávila, Rose et al., 2003
LaRC04 - Pinho, et al., 2005
Failure Criteria: equations based on stresses and strengths that
represent the initiation of damage mechanisms
22/34
Size Effect and Material Softening Laws
Two material properties:
• c Strength
• Gc Fracture toughness
Damage Evolution Laws:
Each damage mode has its
own softening response
Fracture Tests
Strength
Tests
c
Gc / l
E
Material length scale
26.0
c
cc
GEl
23/34
Continuum Damage Model (CDM) (Maimí/Camanho 2007)
Damage Modes:
Tension Compression
Damage Evolution:
Thermodynamically-consistent material
degradation takes into account energy
release rate and element size for each mode.
LaRC04 Criteria
• In-situ matrix strength prediction
• Advanced fiber kinking criterion
• Prediction of angle of fracture (compression)
• Criteria used as activation functions within
framework of damage mechanics
Critical (maximum) finite
element size:
Bazant CBT:2*
2*
2
2
iii
ii
XlGE
XlA
2
* 2
i
ii
X
GEl
ii
i
i fAf
d 1exp1
1
fi: LaRC04 failure criteria as activation functionssyy MMMFFi ;;;;
F
yM
F
yM
e
Xi
24/34Log (diameter, mm)
Log (
Str
ength
, M
Pa)
Scale Effect in Open Hole Tension (OHT)
Prediction of size effects in notched composites
• Stress-based criteria predict no size effect.
• CDM damage model predicts scale effects w/out calibration.
(P. Camanho, 2007)
Hexcel IM7/8552 [90/0/45/-45]3s laminate
0.2 0.4 0.6 0.8 1.0 1.2
2.9
2.8
2.7
2.6
2.5
2.4
25/34
Log (diameter, mm)
Log (
Str
ength
, M
Pa
)Process Zone and Scale Effect in OHT
“Both energy and stress criteria are necessary conditions
for fracture but neither one nor the other are sufficient.”(Leguillon, E J of Mech and Solids, 2001)
Scale effect is due to
relative size of process zone
Cohesive law Stress distribution
2c
cp
GEl
lp lp
0.2 0.4 0.6 0.8 1.0 1.2
2.9
2.8
2.7
2.6
2.5
2.4
26/34
Outline
• Strength or Toughness?
• Cohesive Laws
• Process Zones and Load Redistribution
• Continuum Damage Models
• Transverse Matrix Cracks
• Summary
27/34
Transverse Matrix Cracks
Transverse matrix cracks
propagate in thickness
direction and in the fiber
direction.
Cracks shield the adjacent
material and prevent new
cracks.
Shielded area
Propagation
28/34
Matrix Cracking ̶ In Situ Effect
29/34
𝑓(x)
Material Inhomogeneity
F Leone, 2015
Initial crack density in a uniformly stressed laminate is
strictly a function of material inhomogeneity
x
• Strength scaled by 𝑓, Fracture toughness scaled by 𝑓2
• Constant 𝑓 along each crack path
10 elts.
Inhomogeneity applied to 3 levels of mesh refinement
Cra
ck d
en
sity
Stress
Stochastic
Deterministic
2 elts.
1 elt.
init sat
1.41
1
0.575
0°
90°
0°
0°
90°
0°
0°
90°
0°
30/34
Effect of Transverse Mesh Density on Crack Spacing
F Leone, 2015
0°
90°
0°
Analyses with 3 levels of mesh refinement
31/34
Transverse Matrix Cracks w/ One Element Per Ply
Multi-element model:
correct crack evolution
Conventional single-element:
no opening w/out delam.Modified single-element:
correct Energy Release Rate
t
Ply thickness, mm
Typical
90,
MP
aK
t
EK
2
24
Van der Meer, F.P. & Dávila, C., JCM, 2013
Uncracked
Cracked
32/34
Crack Initiation, Densification, and Saturation
Van der Meer, F.P. & Dávila, C., JCM, 2013
= 182 MPa = 273 MPa
= 372 MPa = 679 MPa
Cohesive zone
Traction-free cohesive zone
Delamination
33/34
Comparison of X-FEM Single-Ply Model and Full 3-D
Van der Meer, F.P. & Dávila, C., JCM, 2013
= 372 MPa
Cra
ck d
en
sity,
1/m
m
Applied stress, MPa
Full 3-D model
Single elem./ply
Experiment
34/34
Summary
• Strength and fracture issues are:
• Interrelated
• Subject to size effects
• Cohesive laws:
• Account for strength and fracture toughness
• More than 3 elements are required in the
process zone
• Continuum damage models
• Snap-back imposes maximum element size
• Transverse matrix cracks:
• More than one element per ply is required
• Other mesh issues?
Mesh Requirements
for Damage Models