Modeling of Failure and Response to Laminated Composites ... · FIBER BREAKAGE A hypothesis was...

N94-22612

Modeling of Failure and Response to LaminatedComposites Subjected to In-Plane Loads

Iqbal Shahid and Fu-Kuo ChangStanford University

Stanford, CA

PtMIC_DINE; PAGE BLANK NOT FIt.MEID

83

https://ntrs.nasa.gov/search.jsp?R=19940018139 2020-08-03T03:24:37+00:00Z

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CURRENT STATUS

T\

An analytical model has been developed for predicting the response of laminated composites withor without a cutout and subjected to in-plane tensile and shear loads. Material damage resulting from theloads in terms of matrix cracking, fiber-matrix shearing, and fiber breakage was considered in the model.Delamination, an out-of-plane failure mode, was excluded from the model.

TENSION SHEAR

GIVEN : GEOMETRY, LAYUP, LOADS

PREDICT: ° DAMAGE IN COMPOSITES

MATRIX CRACKING

FIBER-MATRIX SHEARING

FIBER BREAKAGE

° RESIDUAL STIFFNESS

° RESPONSE AS A FUNCTION OF LOADS

o FAILURE

" 'PAGE_---INT_NT_O_A_¥BLANK85

WHAT DO WE NEED?

In order to accurately predict the response of the laminates, the model must be capable of predictingthe state of damage as a function of the applied load, relating the damage state to the loss of material

properties, and calculating stresses and swains everywhere inside the materials. Accordingly, theproposed analytical model consists of three parts: constitutive modeling, failure analysis and stressanalysis.

LOAD

DAMAGE STATE

° FAILURE MODE

° EXTENT OF DAMAGE

CONSTITUTIVE EQUATIONS

° DAMAGE STATE

STRESS ANALYSIS

RESPONSE

FAILURE MODES

The three basic in-plane failure modes of a single unidirectional ply considered in the model arematrix cracking, fiber-matrix shearing, and fiber breakage.

MATRIX FIBER-MATRIX FIBERCRACKING SHEAR-OUT BREAKAGE

87

CONSTITUTIVE MODELING

The constitutive equations of a unidirectional ply in an undamaged state can be characterized bystandard mechanical testing. However, once damage occurs in a ply within a multidirectional laminate, thematerial properties of the ply need to be determined in order to construct the constitutive equations for thedamaged laminate. Therefore, the proposed model was based on continuum mechanics whereby thedamaged ply in a laminate was treated as a continuous body with degraded material properties.

PLY STIFFNESS (UNDAMAGED STIFFNESS, DAMAGE STATE)

UNDAMAGED PLY:MECHANICAL TESTING

Ex, Ey, Es, Vxy _ [Q]

DAMAGED PLY: ( IN LAMINATE )

O MATRIX CRACKING

O FIBER-MATRIX SHEAR-OUT

O FIBER BREAKAGE

D D D D

Ex, Ey, Es, Vxy [QD] = .9

88

MATRIX CRACKING

In order to determine the effect of matrix cracking on the reduction of the stiffness of aunidirectional ply in a laminate, crack density was selected as the damage parameter for characterizing thedamage state of matrix cracking.

FOR EACH PLY

X2

--_X 1

LAMINATION EFFECT

MATRIX CRACK DENSITY

EFFECTIVE PLY STIFFNESS

[QD] =I

-Qll (t_) Q12 (_):0!

Q21 (¢) Q22 (¢)! 0

0 0m

!

IQ66( _ )

89

MATRIX CRACKING - APPROACH

A constitutive model was developed for characterizing the material properties of a ply in a

symmetric laminate as a function of its own crack density. For a given crack density in a ply whose fiberdirection may not be parallel to the global x-axis, the model f'wst rotates the laminate such that the fiberdirection of the cracked ply is aligned with the x-axis. It is then assumed that all the matrix cracks in theply are uniformly distributed. As a result, a unit-cell of the laminate can be selected as a representativevolume of the cracked laminate. The representative volume may be comprised of up to three sublaminates

labeled as 1, 2 and 3 in the figure.

• ORTHOTROPIC SUBLAMINATES ASSUMPTION

• 2-D ELASTICITY ANALYSIS

TENSION

z' _ ') O t'3 0 O t'3 0/ ////////

Q 11 (_) Q 12 ((_)

Q21 (t_) Q22 ((_)

Z ! ..m

SHEAR

®®®® ®y,

3 _2tx

2Ix

St',)OOOOOO

/ ////////

f Q66 ((_)

• REPEAT PROCEDURE FOR ALL PLIES OF THE LAMINATE

90

MATR_ CRACKING

In the constitutive model it was further assumed that the sublaminates 2 and 3 could be treated as

homogeneous and orthotropic materials. Accordingly, the three-dimensional volume could be reduced to atwo-dimensional element. By applying a far-field tensile or shear load, the material properties of the

cracked ply (sublaminate 1) as a function of the crack density could be calculated from a two-dimensionalelasticity theory. The aforementioned procedure was then applied to each of the plies in a laminate for any

given crack density.

APPROACH

/,

MATRIX

CRACK

y'=xt_X l _ X'=Xl

Zzy!

r////

REPRESENTATIVE ELEMENT

91

FIBER-MATRIX SHEAR-OUT

Once the applied load continued to increase, the plies in the laminate may have failed due to eitherfiber-matrix shearing or fiber breakage, leading to catastrophic failure of the laminate. Fiber-matrix shear-out failure could be attributed to interracial debonding and slipping or nonlinear elasticity of the material.

The aforementioned elasticity theory for matrix cracks could not be applied to characterize the reduction ofmaterial properties resulting from the shear-out failure. To account for interracial debonding and slipping,continuum damage mechanics was adopted based on the concept proposed by Krajcinovic and Fonseka.Nonlinear material response was considered in the model through the shear stress-shear strain relationship.

HIGH SHEAR DEFORMATION

• FIBER-MATRIX INTERFACE DEBONDING, SLIP ETC.

• NONLINEAR SHEAR DEFORMATION

CONTINUUM DAMAGE MECHANICS: (Krajcinovic and Fonseka, 1981)

-( )Q66 = Q66(_) ds ds = e

0 = SATURATION CRACK DENSITY

11 = SHAPE PARAMETER

• PLY SHEAR STRESS-SHEAR STRAIN:

3

Hahn.. _/12 - GI2 + (_ ((_12)

Q_6 Q_6

0"12

F

92

FIBER BREAKAGE

Based on Rosen's cumulative weakening failure theory, failure of a unidirectional ply undertension occurs only when there are enough fiber breaks that occur within a critical area characterized by thefiber interaction distance 8, which is the maximum distance within which one fiber break would affect the

stresses of the neighboring fibers. Accordingly, not only stresses but also the area within which fiberbreaks occur are essential for characterizing fiber failure of a unidirectional composite.

UNIDIRECTIONAL COMPOSITE:

CUMULATIVE WEAKENING FAILURE ( Ro_en, !,964 )

FAILURE OF UNIDIRECTIONAL PLY OCCURS AT THEWEAKEST CROSS SECTION

,t

4l

B , m B

m

5m

5

_ FIBER INTERACTION ZONE

93

FIBER BREAKAGE

A hypothesis was postulated that stiffness reduction of a unidirectional composite due to fiberbreakage is related to the extent of the area in which fiber breakage occurs.

NOTCHED COMPOSITE:

HYPOTHESIS;

STIFFNESS REDUCTION IS FUNCTION OF

FIBER BREAKAGE AREA (A)

df= e( -_2)_

A = FIBER BREAKAGE AREA

= SHAPE PARAMETER

[QD] =m

Qll Q12 0

Q21 Q22 0

0 0 Q66

- -df

0

0n mm

0 0 -

df 0

0 df

F==

94

CONSTITUTIVE MODEL

The effective material properties of a single ply within a symmelric laminate can be related toundamaged material properties and damage state with three different failure modes.

• WITHOUT SHEAR NON-LINEARITY

{o} = [QD ] {e }

[QD]

MATRIX CRACKING

-QI1 (_) Q12 (_) o

Q21(t_) Q22(t_) 0

0 0 Q66( d_ )Ilm

FIBER-MATRIXSHEAR-OUT

lm

1 0 0

0 1 0

0m

0

FIBER BREAKAGE

l ili

df 0 0

0 df 0

0 0 df

• WITH SHEAR NON-LINEARITY

{dO} = [QD it{dE}

95

DAMAGE GROWTH CRITERIA

Modified Hashin Failure Criteria were adopted for predicting the mode and state of damage of a plyin a laminate. The stresses used in the criteria are the effective stresses obtained from the effective

properties. The effective strengths of the ply are no longer treated as constants, but may vary as a functionof crack density (damage state).

PREDICT:MODE OF FAILURE AND DAMAGE STATE

2 ;MATRIX CRACKING ( Yt (_)) S (_))

>1mm

FIBER-MATRIX (G1,;+( (_12 ; >1SHEAR-OUT X t S (_))

FIBER BREAKAGE ( (_11 ; >1Xt =

Yt(¢)=? S(_))=? ]

96

EFFECTIVE STRENGTHS

The effective transverse tensile and shear strengths at crack density _ are defined as the minimum

stresses that are required to generate crack density _bin the ply. A model was proposed based on the

elasticity theory and fracture mechanics to characterize the effective strengths as a function of crack

density.

S t (_)) = MINIMUM TRANSVERSE STRESS REQUIRED TO GENERATECRACK DENSITY

S (_)) = MINIMUM SHEAR STRESS REQUIRED TO GENERATECRACK DENSITY _)

Z !

TENSION SHEAR

y'

3

O000OO0 kk

/////////

• FRACTUREMECHANICS

AU@)=AGIc

• 2-D ELASTICITY

|

Z _,

®®®

0000000/////////

63 6)Y'

3 ",

• FRACTURE MECHANICS

AU@) =AGIIc

• 2-D ELASTICITY

s(_)

97

FLOWCHART

A f'mite element analysis has been developed based on the proposed model. The flowchart of theanalysis is presented.

I

i AT1CONSTITUTIVE RELATION

UPDATE STRESSES

TFAILURE

MODE

STATE

I GEOMETRY 1

MATERIAL AND LAYUP

LOADING

I STRESS ANALYSIS 1 -_

. GROWT!CRITERIA_

MATRIX CRACKING

SHEAR-OUT

FIBER BREAKAGE

I DELAMINATION ]

( ULTIMATE FAILURE _

t INCREASE

LOAD I CYCLE II

STATIC) I(FATIGUE) 1_

NO TFAILURE

98

AS4/3501 [0/902]s

Comparisonbetween the model prediction and the test data. A [0/902]s composite subjected to a10° off axis uniaxial tensile load.

AS4/3501-6

[0/90 2 ] s

10*Off-Axis Tensile Loading

(Daniel and tsai, 1991)

50

40

_' 30

20

00.000

L DATA

• i 1 t I i

0.002 0.004 0.006

STRAIN (in/in)

0.008

_40

30

r,.)0

0

L DATAMODEL

,_zx

10 20 30 40

APPLIED STRESS (ksi)

50

i 99

AS4/3502 [60/90/-60/60/90/-60/90]s

Comparison between the model prediction and the test data. A [60/90/-60/60/90/-60/90]_composite subjected to a uniaxial tensile load.

AS4/3502

[60/90/.60/90/60/90/-60/90] s

(Kistner et al., 1985)

4O

20

00.00

O DATA

MODEL

°i | I

0.01 0.02

STRAIN (in/in)

0.03

_, lOO

_ 8O

_ 40

_ 20

0.00

A

0.01 0.02- 0.03

STRAIN (in/in)

100

AS4/3502 [0/90/0/90/0/90/0/90]s

Comparison between the model prediction and the test data_ A [0/90/0/9010/90/0/90]s compositesubjected to a uniaxial tensile load.

r_Z

.<

,-_ 200elml

r._ 150

_ 100

_ 5o

O DATA--_ MODEL

• I _ I i , I

0.005 0.010 0.015

8O

60

40

20

00.000

A DATA /

(Avg. of outer 90 s)//

A

A A

- i-_ I I

0.005 0.010 0.015

80

6O

40

20

' 00.020 0.000

STRAIN (in/in)

AS4/3502

[0/90/0/90/0/90/0/9"0] s

|

0.020

A DATA (center 90 )

- _ AA

0.005 0.010 0.015

STRAIN (in/in)

|

0.020

101

AS4/3501

Comparison between the model prediction and the test data. Cross-ply composites subjected to auniaxial tensile load.

5O

_ 40

20

< I0

00

=__J

om

Z

<

50

40

30

20

10

00

AS/3501-6 [0 2/90] s

A DATA

/y

40 80 120 160

1[02/902]s [

40 80 120


160

50

40

30

20

10

00

WANG et al., 1984

_ TENSION

[02/903 ]S]

A

40 80 120


160

102

AS4/3502 [45/90/-45/90/45/90/.45/90]s

Comparison between the model prediction and the test data. A [45/90/-45/90/45/90/-45/90]5composite subjected to a uniaxial tensile load.

AS4/3502

[45/90/-45/90/45/90/.45/90] s

(Kistner et al., 1985)

r/3

I-f/3

,d

<

40

30

20

10

00.000

O

DATA

MODEL

0.005 O.OLO o.015 0.020

STRAIN (in/in)

100

am

8o

[- 60

Z40

20

00.000

AA

AA

A_DAT A 1

0.005 0.010 0.015 0.02t

STRAIN (in/in)

103

RAIL SHEAR SPECIMEN

A typical finite element mesh used in the calculation for rail shear specimens.

1

t!

1114

IN-PLANE SHEAR STRENGTH

Comparison between the prediction of rail shear strength and the measurement.

16000

12000

z

8000

r_

Z<

4000

|

Z

00

T300/976 o

o

DATA

MODEL

2 4 6 8 10

NUMBER OF CLUSTERED PLIES, n

105

RAIL SHEAR TEST SIMULATION

The predicted matrix crack density distribution in a [04/902]s shear specimen near 90% of the f'malfailure load.

r

MATRIX CRACK DENSITY DISTRIBUTION

0 90

rl

|m i

Ill I

T300/976[04/902]s I

---] O<=CD <5

D 5 (=CD < 10

D 10 _-CD ( 20

D 20 <=CD, 25

CD>25

106

SHEAR LOAD

The predicted load-deflection response of cross-ply rail shear specimens.

m

<O

<

5000

4000

3000

2000

1000

0.06

T300/976

[0/90]

....... [03/903 ]s

.......... [06/906 Is

APPLIED EDGE DISPLACEMENT (inch)

107

PROGRESSIVE FAILURE PREDICTION (VERIFICATION - NOTCHED LAMINATE)

Numerical simulation of damaged extension of notched laminated composites as a function of

applied load under uniaxial tension.

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MATERIAL: AS4/350 I-6

LAYUP: [45/90/-45/0]s

D= 0.072(i n)

W/D= 4.0

LOAD = 5328(Ibs)

I _ DAMAGE MODE I

Fiber Breakage

Fiber-Matrix Shear-Out

_'{Z Matrix Cracking

108

PROGRESSIVE FAILURE PREDICTION

Numerical simulation of damaged extension of notched laminated composites as a function ofapplied load under uniaxial tension.

MATERIAL: AS4/3501-6

LAYUP: [45/90/-45/0]s

D= 0.872(in)

W/D= 4.0

LOAD = 6720(Ibs)

DAMAGE MODE

I Fiber Breakage

I Fiber-Matrix Shear-Out

_ Matrix Cracking

109




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MATERIAL: AS41350 I-6

LAYUP: [45/90/-45/01s

D= 0.872(i n)

W/D= 4.0

LOAD = 7447(Ibs)

DAMAGE MODE

I Fiber Breakage


_ Matr|x Cracking

110



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MATERIAL: AS4/3501-6

LAYUP: [45/90/-45/0]s

D= 0.872(in)

W/D= 4.0

LOAD = 3847(Ibs)

DAMAGE MODE

I Fiber Breakage


Matrix Cracking

111

RESIDUAL STRENGTH

The residual strength distribution of notched [45/90/-45/0]s composites as a function of laminatewidth. Comparison between the prediction and the test data.

<

Z

[-r_

<

r,13

80

60

40

20AS4/3501-6

[+45/90/-45/0] s

W/D = 4

0 i I i I i l

0 1 2 3

i i

WIDTH OF PANEL, W (inch)

D DATA

--- MODEL

I |

4

112

RESIDUAL STRENGTH

width.data.

The residual strength distribution of notched [Crown- 1] composites as a function of laminateComparison between the predictions based on the model and the existing methods and the test

80

z 40

<20

g_

0 m I t I _ I _ I _ I ,

0 2 4 6 8 10 12


113

RESIDUAL STRENGTH

Theresidualstrengthdistributionof notched [0/90/0/90]s composites as a function of laminatewidth. Comparison between the prediction and the test data.

e_

80

[..,

Z

m[-ct_

<

m

4O

20

[] DATA

-- MODEL

[]

,i

AS4/3501-6

[0/90/0/90] s

W/D = 4

0

0 2 4 6 8 10 12


114

RESIDUAL STRENGTH

The residual strength distribution of notched [Crown-1] tow-composites as a function of laminate

width. Comparison between the predictions based on the present model and the Mar-Lin model and thetest data.

80

z 40

[-

r,.) 20[-

Z

w

%

", [] DATA: ..... Mar Lin\

•. m MODEL

"",,,,,.

_umlmm_lmmalj

AS4/3501-6 (Tow)

CROWN-1

W/D = 4

0 t I t I

0 2 4

n I _ I

6 8


!

10 12

115



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MATERIAL: AS4/350 I-6 (TOW)

LAYUP: CROWN- I

D= 0.872(in)

W/D= 4.0

LOAD = 3612(Ibs)

DAMAGE MODE

I Fiber Breakage


_ Matri x Cracki ng

116



MATERIAL: AS4/3501-6 (TOW)

LAYUP: CROWN- |

D- 0.872(in)

W/D= 4.0

LOAD = 7771(Ibs)

DAMAGE MODE

I Fiber Breakage


_. Matrix Cracking

117




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LAYUP: CROWN- I

D= 0.872(in)

W/D= 4.0

LOAD = 10202(lbs)

DAMAGE MODE

I Fiber Breakage


_:_:iiMatrix Cracking

118



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LAYUP: CROWN- l

D= 0.872(in)

W/D= 4.0

LOAD = 8612(Ibs)

DAMAGE MODE

I Fiber Breakage


_. Matrix Cracking

119

FUTURE WORK

•

I. IMPLEMENTATION

IMPLEMENTATION OF THE CURRENT MODEL TO EXISTING

FEM CODES

II. DAMAGE MODELLING

1. CRACK GROWTH MODEL

2. DELAMINATION INITIATION AND GROWTH MODEL

3. FATIGUE MODEL

II1. COMPUTATIONAL MECHANICS

1. MESH SENSITIVITY

2. DAMAGE SIMULATION

3. GLOBAL-LOCAL FEM

4. PARALLEL PROCESSING

120

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Modeling of Failure and Response to Laminated Composites ... · FIBER BREAKAGE A hypothesis was...

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