Damage Instability and

Transition from Quasi-Static

to Dynamic Fracture

1

Carlos G. Dávila

Structural Mechanics & Concepts Branch

NASA Langley Research Center

Hampton, VA

USA ICCST/10

Instituto Superior Técnico

Lisbon, Portugal, 2-4 September 2015

https://ntrs.nasa.gov/search.jsp?R=20160006280 2020-07-18T00:29:43+00:00Z

Loading Phases:

• 0) to A) – Quasi-static (QS) loading

• A) to B) – Dynamic response

F, d

A)

B)

Force

Displacement

No QS

solution exists

A)

B)0)

Snapback behavior:• More strain energy available than

necessary for fracture

Quasi-Static Loading and Rupture

2

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

3

Failure Criteria and Material Degradation

Failure criterion

E

1

Residual=E/100

Strain

Stress

e/e01

Elastic

property

Failure criterion

E

1

Strain

Stress

e/e01

Elastic

property

Increasing lelem

Increasing lelem

Progressive Failure Analysis

Progressive Damage Analysis – Regularized Softening Laws

4

fi

i

Kd

K

,)1(

,

L+ +

K

F,

Gc

c

(1-d)K

Strength-Dominated Failure

K

EL

EAAF

K

EEGL

G

EAAF

c

c

c

c

2

2

2

Before damage After damage

For “long” beams, the response is unstable, dynamic, and independent of Gc

0

F2

2

c

cEGL

For stable fracture under control:

F

StableUnstable

0

F

5

Fracture-Dominated Failure

E

aG

2

2a

G

a0 a

max

R = GIc

unstable

E

2

:Slope

Crack propagates unstably once driving force G(, a0) reaches GIc

G(, a0)

6

Fracture-Dominated Failure

2a E

aG

2

G

a0 astable

GR(a)

a

init

max

GInit

unstable

Gc

2(a+a)

G(, a0)

Crack propagates stably when driving force G(, a0) > GInit

Unstable propagation initiates at cInit GGG

7

Mechanics of Crack Arrest

G

a0

a

max

R = GIc

unstable arrest

aarrest

Crack arrest due to decreasing G

8

Mechanics of Crack Arrest

G

a0

a

max

unstable

arrest?

R rate sensitive

Large strain rates often result in lower fracture toughness

and delayed arrest

9

Griffith growth criterion

Griffith Criterion and Stability

Stability of equilibrium propagation

Wimmer & Pettermann

J of Comp. Mater, 2009

10

Stability of Propagation with Multiple Crack Tips

P, v

Wimmer & Pettermann

J of Comp. Mater, 2009

a1

a2

a2, mm

a1,

mm

Displacement, mm

Fo

rce, N

0/90/…/90/0

15 plies total

P, vCurved laminate with through-the-width delamination

2.25 mm

11

Scaling: The Effect of Structure Size on Strength

Scaling from test coupon to structure

Structural size, in.

Yield or Strength Criterialog n

log D

(Z. Bažant)

Scaling Laws

Normal testing

12

Cohesive Laws

Bilinear Traction-Displacement Law

cc Gd

0 )(

Two material properties:

• Gc Fracture toughness

• c Strength

2c

cp

GEl

Characteristic Length:

Final debond length

t0

unloading reloading

Yield or Strength Criteria

log n

log D

Gc

Kp

c

0 c 13

Crack Length and Process Zone

0

0

0

5

5100

100

al

lal

la

p

pp

p

Brittle:

Quasi-brittle:

Ductile:

Quasi-

brittle

Force, F

Applied displacement,

LEFM error

2c

cp

GEl

a0

F,

Gc= constant

14

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,

LEFM error

LEFM error

2c

cp

GEl

a0

F,

15

2c

cp

GEl

a0

F,

Strength and Process Zone

As the strength c decreases,

1. the length lp of the process

zone increases

2. the error of the Linear

LEFM solution increases

a0

a0+lp

Force, F

Applied displacement,

LEFM error

Gc = constant

Decreasing

c

16

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

2c

cc

GEl

e17

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 continuum damage mechanics

(CDM)

e

Critical (maximum) finite

element size:

Bazant Crack Band Theory:

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

Progressive Damage Analysis (Maimí/Camanho 2007)

18

log (diameter, mm)

log (

Str

ength

, M

Pa

)

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 CFRP laminate

Experimental (mean)

Analysis

Predicting Scale Effects with Continuum Damage Models

19

log (diameter, mm)

log

(S

trength

, M

Pa

)

Scale effect is due to

relative size of process zone

Cohesive law Stress distribution

(P. Camanho, 2007)

Process Zone and Scale Effect in Open Hole Tension

20

Length of the Process Zone (Elastic Bulk Material)

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

21

• 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

22

Study of size effect: measuring the R-curve

Double-notched compression specimens

G

a0 astable

GR(a)

a

Increasingw

a

2w

a0 a0

5.00 w

a

eff

u

E

a

w

aG

2

By FEM analysis From test

2 3 4 5 6 7 8 9

, MPa-2 *10-5

a0, mm

2

u3.2

2.8

2.4

2.0

1.6

1.2

0.8

0.4

0.0

Catalanotti, et al., Comp A, 2014

E

aG

2(Similar to )

23

Characterization of Through-Crack Cohesive Law

Cle

vis

Anti-

buckling

guide

Specimen

𝑥

𝑦

𝑧

Experimental setup

0°𝑃

𝑎0 ∆𝑎

𝑊

1.18𝑊

𝑥

𝑦

Thin

multidirectional

laminate

Compact Tension (CT) Specimen Characterization Procedure:

1. Measure R-curve from CT

test

2. Assuming a trilinear

cohesive law, fit analytical

R-curve to the measured

R-curve

3. Obtain the cohesive law

by differentiating the

analytical R-curve

𝐺𝑅 =𝑃2

2𝑡

𝜕𝐶

𝜕𝑎

𝜎 𝛿 =𝜕𝐽fit𝜕𝛿

𝜂 =

𝑖

𝑛𝑠

𝐽fit𝑖 − 𝐺𝑅

𝑖

𝜎𝑐

𝛿

𝐺𝑐

𝐾

𝜎

Trilinear cohesive law

Bergan, 2014

24

Size-Dependence of R-Curve

(a) ‘S’

(b) ‘L’

25 mm

Plotting the R-curve as a function of

the notch displacement removes the

size-dependency

Bergan, 2014

Displacements

measured through

digital image

correlation (DIC)

Small

Large

aeff, mm aeff, mm

GR,N

/mm

GR,N

/mm

GR,N

/mm

𝛿, mm 25

R-Curve Effect in Fiber Fracture

𝐽𝑅 =

0

𝛿𝑐

𝜎 𝛿 𝑑𝛿

Curve fit assuming bilinear

𝑃 [kN]

𝛿, mm

Analysis

Test

0.0 1.0 2.0 3.0 4.0

6.

4.

2.

0.

P,

𝑎0 ∆𝑎

Cohesive elements

w/ characterized

cohesive law

Bergan, 2014

0

200

400

600

800

0.0 0.5 1.0 1.5 2.0

[MPa]

[mm]

Cohesive response

for fiber failure

𝛿, mm

,M

Pa

P,kN

𝛿, mm

GR,N

/mm

26

Mode II-Dominated Adhesive Fracture

Tip of adhesive

Teflon

Adhesive thickness: 0.13 mm

27

ENF J-Integral from DIC

MMB Test - Analysis Results

Nominally identical bonded MMB specimens sometimes fail in

quasi-static mode and others dynamically. Why?

Displacement, mm

Applie

d load,

NMixed mode bending (MMB) test fixture

29

Double Delamination in MMB Tests

Composite

delamination

Adhesive

failure

Failure

Surfaces

• Unexpected failure mechanism

• Two delamination fronts run in

parallel: one in the adhesive,

the other in the composite

• When the fiber bridge breaks, the crack grows unstably in the

composite causing the drop in the load-displacement curve30

Modeling the Double Delamination

Fiber bridge

Cohesive layer

Adhesive Layer

• A model was developed to evaluate the observed double

delamination phenomenon

• The model contains two additional cohesive layers within the

composite arms

• This failure mechanism is often observed in bonded joints

Displacement, mm

Applie

d load,

N

MMB test specimen

Model of MMB specimen with double delamination

Model with double delamination

Model with single delamination

Experimental result

31

Representative Volume Element and Micromechanics

32

Why Micromechanics?

Assumption:

“Micromechanics has more built-in physics because it is closer to

the scale at which fracture occurs”

Why NOT Micromechanics? (Representative Volume Element [RVE])

• Problem of localization

• Randomness of unit cell configurations

• Lengthscales missing

• Characterization of material properties, especially the interface

• Computational expense

RVE: 1) Problem of Localization

33

Scale of RVE

cannot be

eliminated

Linear

RVE, Schapery Theory,

homogenization

Localization;

regularized CDM,

nonlocal methods

Softening

Softening

Linear

Hardening

Hardening

LocalizedSmeared

e

RVE: 2) Randomness of Unit Cell Configurations

34

Melro et al. IJSS, 2013.

Bloodworth, V., PhD Dissertation,

Imperial College, UK, 2008.

Fracture is a combination of interacting discrete and diffuse damage mechanisms

RVE: 3) Issue of Length Scales

35

RVE may not account for:• Ply thickness

• Longitudinal crack length

• Crack spacing

Crack spacing = RVEShielding

Matrix Cracking ̶ In Situ Effect

36

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

K

t

EK

2

24

t

Van der Meer, F.P. & Dávila, C., JCM, 2013 Ply thickness, mm

Typical

90,

MP

a

37

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 38

𝑓(x)

Material Inhomogeneity

39F 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.

Effect of Transverse Mesh Density on Crack Spacing

40

F Leone, 2015

Commercial finite element vendors and

developers are providing more and more

tools for progressive damage analysis.

… more analysis tools=

more rope!

But, if the load incrementation

procedures do not converge…

What Happened to Quadratic Convergence!!??

41

• Viscoelastic Stabilization

• Delayed damage evolution

• Implicit dynamics or Explicit solution

• Arc-length techniques

• Dissipation-based arc-length

Constant energy

dissipation in each

load increment

Gutiérrez, Comm Numer Meth Eng (2004)

Verhoosel et al. Int J Numer Meth Eng (2009)

g

Techniques for Achieving Solution Convergence

42

Van der Meer, Eng Fract Mech, 2010

QS Solution of Unstable OHT Fracture

43

Open Questions

• Is the QS solution physical?

• Are the dynamic effects necessary?

• Which solution provides more

insight into failure modes?

F

d

F

d

A)

Implicit Explicit

44

Concluding Remarks

• A typical structural tests usually consist of three stages:

1. QS elastic response without damage

2. QS response with damage accumulation

3. Dynamic collapse/rupture

• Most structural failures exhibit size effects that depend on load

redistribution that occurs during the QS phases

• Correct softening laws based on strength and toughness considerations

are required

• Dynamic collapse/rupture is a result of the interaction between

damage propagation and structural response

• A stable equilibrium state often does not exist after failure under either

load or displacement control

• Onset of instability (failure) occurs when more elastic strain energy can

be released by the structure than is necessary for damage propagation

• Simulation of unstable rupture is often needed to ascertain mode of

failure and to compare to test results

45