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