Lecture #12: Elasticity and failure of fiber-reinforced ... · D. Mohr 2/15/2016 Lecture #12...

2/15/2016 1 1Lecture #12 – Fall 2015 1D. Mohr

151-0735: Dynamic behavior of materials and structures

by Dirk Mohr

ETH Zurich, Department of Mechanical and Process Engineering,

Chair of Computational Modeling of Materials in Manufacturing

Lecture #12:

• Elasticity and failure of fiber-reinforced composites

© 2015



Intelligent Lightweight Engineering



Intelligent Lightweight Engineering



Elasticity of Fiber-reinforced laminae and laminates (cont.)



DefinitionsDenote the three orthotropy direction by i, j, k. Then consider a uniaxial tensionexperiment along the orthotropy direction xi.

ii1

jj1

iiii

Plus thickness strain measurement

ii

iiiE

ii

jj

ij

kkt

t 1

0

We can define (& determine):

• Young’s modulus

• Poisson’s ratios

ii

kkik



Plane stress law for an orthotropic material (2D)

For notational convenience, the elastic stress-strain relationship for anorthotropic material is also written as

1221

2

21

2

21

2

1

2

212

2

222

1/1

E

EE

E

EE

EQ

with

12

22

11

66

2212

1211

12

22

11

00

0

0

Q

QQ

QQ

1221

1

21

2

21

1

1

2

212

2111

1/1

E

EE

E

EE

EEQ

1221

121

1

2

212

212112

1

E

EE

EEQ

1266 2GQ



Isotropy check

2221

E

Q

2111

E

Q

2121

EQ

2661

)1(

E

Q

1122 QQ

11

12

Q

Q

121111

11

1266 1 QQQ

Q

QQ

66

2212

1211

00

0

0

Q

QQ

QQ

Q1122 QQ

121166 QQQ defines an isotropic material if



Constitutive equation for lamina As the same transformations are valid for the strain vector, the relationshipamong the stress and strain components in the (ex, ey)-frame is then given by

xy

yy

xx

xy

yy

xx

xy

yy

xx

Q

QQ

QQQ

Q

QQ

QQ

66

2622

161211

66

2212

1211

1

ˆ

ˆˆ

ˆˆˆ

00

0

0

TT

22

4

6612

22

11

4

11 )(2ˆ QsQQscQcQ

Upon evaluation, we find the components

12

44

662211

22

12 )()2(ˆ QcsQQQscQ

)(2)(2ˆ662212

3

661211

3

16 QQQcsQQQscQ

22

4

6612

22

11

4

22 )(2ˆ QcQQscQsQ

)(2)(2ˆ662212

3

661211

3

26 QQQscQQQcsQ

66

44

66122211

22

66 )()2(2ˆ QscQQQQscQ

Note that the mathematical definition of the shear strain is applied. Many textbooks and FE programs adopt theengineering definition. In that case, the third column of the stiffness matrix Q needs to be multiplied with 0.5.

sym

]cos[c

]sin[s



Membrane response of laminates

Let denote the stiffness matrix of the i-th lamina in the (ex, ey)-coordinateframe, the stress-strain relationship for the laminate is then given by the rule ofmixtures:

xy

yy

xx

tot

xy

yy

xx

Q̂

iQ̂

with

N

i

i

tot

itot

t

t

1

ˆˆ QQ

ttot①②③④⑤⑥

Thickness fraction



Fiber volume fraction

The fiber volume fraction Vf is a first basic characteristic of the two-phasecomposite microstructure.

volumetotal

fiberofvolumeV f

91.032

)max(

fV

The hexagonally closed-packed array provides a theoretical upper limit for thefiber volume fraction.



Material directions

A unidirectional composite laminae features two in-plane directions and oneout-of-plane direction. The latter is also referred to as thickness direction. Thelongitudinal direction coincides with the fiber direction, while in-plane directionthat is perpendicular to the fibers is called the transverse direction.

Thickness direction

Transversedirection

Longitudinaldirection



Longitudinal modulus

The longitudinal modulus corresponds to the slope of the stress-strain curvewhen the lamina is subject to uniaxial tension along the longitudinal direction.

1111 )1( mfff AEVAEVAE

mff EVEE )1(1

)1( 1L

1 1

MATRIX

FIBER



Poisson’s ratio

Using a simple layer model of the fiber reinforced composite, the Poisson’s ratiocan be estimated.

fffm WVVWW 11112 )1(

)1( 112W

)1( 1L

)1()1( 1mf WV

)1( 1 ffWV

fffm VV )1(12

1 1

MATRIX

FIBER



In-plane shear modulus

Using a simple layer model of the fiber reinforced composite, the in-plane shearmodulus can be estimated.

WVG

WVG

WG

f

m

f

f

)1(1212

12

12

WVG

f

f

12WV

Gf

m

)1(12

WG12

12

12

m

f

f

fG

VG

VG

1)1(

11

12

MATRIX

FIBER



Transverse Modulus

)1( 2W

2

Using a simple layer model of the fiber reinforced composite, the transversemodulus can be estimated.

m

f

f

fE

WVE

WVE

W 22

2

2 )1(

m

f

f

fE

VE

VE

1)1(

11

2

2

MATRIX

FIBER



Elastic lamina property estimates: summary

• Longitudinal modulus

mff EVEE )1(1

• Transverse modulus

m

f

f

fE

VE

VE

1)1(

11

2

• In-plane shear modulus

m

f

f

fG

VG

VG

1)1(

11

12

• In-plane Poisson’s ratios

fffm VV )1(12

1

21221

E

E and

)1(2 f

f

f

EG

)1(2 m

mm

EG

with



Elastic lamina property estimates: example

INPUT:Phase properties

GPaE f 75

GPaEm 5

22.0f

35.0m

7.0fV

OUTPUT:Lamina properties

GPaE 541

GPaE 4.142

26.012

07.021

Vf Ef Em nue_f nue_m Gf Gm E1 E2 nue_12 nue_21 G12

[MPa] [MPa] [-] [-] [MPa] [Mpa] [MPa] [MPa] [-] [-] [MPa]

0.7 75,000 5,000 0.22 0.35 30,738 1,852 54,000 14,423 0.26 0.07 5,412

GPaG 4.512

Evaluation of the above formulas for an E-glass composite with afiber volume fraction of 70% yields:




OUTPUT:Lamina properties

GPaE 541

GPaE 4.142

26.012

07.021

GPaG 4.512

As an alternative to using the material constants {E1,E2,G12,12},the elastic lamina properties may be described through {Q11, Q12,Q22, Q66}.

12

22

11

66

2212

1211

12

22

11

00

0

0

Q

QQ

QQ

GPaQ 5511 GPaQ 1522

GPaQ 8.312 GPaQ 1166

E1 E2 nue_12 nue_21 G12 Q11 Q22 Q12 Q66

[MPa] [MPa] [-] [-] [MPa] [MPa] [MPa] [MPa] [MPa]

54,000 14,423 0.26 0.07 5,412 54,985 14,686 3,804 10,824




12

22

11

66

22

1211

12

22

11

0

0

Q

Q

QQ

GPaQ 5511 GPaQ 1522

GPaQ 8.312 GPaQ 1166

GPaQ 37ˆ11 GPaQ 17ˆ

22


66

xy

yy

xx

xy

yy

xx

Q

QQ

QQQ

66

2622

161211

ˆ

ˆˆ

ˆˆˆ

Q11 Q22 Q12 Q66 alpha QH11 QH12 QH16 QH22 QH26 QH66

[MPa] [MPa] [MPa] [MPa] [deg] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa]

54,985 14,686 3,804 10,824 30 37,332 11,382 26,200 17,183 8,700 25,980


26



“Black aluminum”

The properties of a carbon fiber composite laminate can beadjusted such that it provides similar stiffness properties asaluminum. Starting point is a [0°, 60°, -60°] lay-up which featuresisotropic in-plane properties (hexagonal symmetry).

GPaE f 300 0

60

60



“Black aluminum”

The calculation shows that for a fiber volume fraction of Vf=0.62of carbon fibers of fiber modulus Ef=300GPa, an isotropic [0,60,-60] laminate exhibits the same in-plane modulus of 70 GPa asaluminum.

Vf Ef Em nue_f nue_m Gf Gm E1 E2 nue_12 nue_21 G12 Q11 Q22 Q12 Q66 alpha QH11 QH22 QH12 QH66 QH16 QH26

[MPa] [MPa] [-] [-] [MPa] [Mpa] [MPa] [MPa] [-] [-] [MPa] [MPa] [MPa] [MPa] [MPa] [deg] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa]

0.62 300,000 5,000 0.22 0.35 122,951 1,852 187,155 12,729 0.27 0.02 4,726 188,085 12,792 3,450 9,452 0 188,085 12,792 3,450 9,452 0 0

0.62 300,000 5,000 0.22 0.35 122,951 1,852 187,155 12,729 0.27 0.02 4,726 188,085 12,792 3,450 9,452 -60 23,789 111,436 36,276 75,104 -38,000 -113,808

0.62 300,000 5,000 0.22 0.35 122,951 1,852 187,155 12,729 0.27 0.02 4,726 188,085 12,792 3,450 9,452 60 23,789 111,436 36,276 75,104 38,000 113,808

LAMINATE: 78,555 78,555 25,334 53,220 0 0

1 70,000 70,000 0.33 0.33 26,316 26,316 70,000 70,000 0.33 0.33 26,316 78,555 78,555 25,923 52,632 0 78,555 78,555 25,923 52,632 0 0

ALUMINUM 78,555 78,555 25,923 52,632 0 0

The mass density of this laminate would be about 1.7 g/cm3 andthus about 40% lighter than aluminum.

The weight savings can be increased to about 50% through the use of high stiffness carbon fiberswith a modulus of 800GPa. However, in engineering practice, even higher weight savings areachieved through the use of anisotropic laminates.



Lamina Failure



Selected Failure Mechanisms

Source: Stefan Hartmann, DYNAmore GmbH, Composite Berechnung in LS-DYNA, Stuttgart (2013)



Uniaxial tension of unidirectional carbon fiber composite

Source: https://www.youtube.com/watch?feature=player_detailpage&v=NMiZQxBKu-E



Uniaxial tension of Carbon fiber Fabric Composite

Source: https://www.youtube.com/watch?feature=player_detailpage&v=aH9vcV7jzG0



Intra-laminar failure mechanisms

A laminate may fail due to delamination of the constituentlaminae, which is an inter-laminar failure mechanism. Inter-laminar failure is often preceded by intra-laminar failure whichincludes four basic failure mechanisms:

(1) Fiber tension failure

(2) Matrix failure under combined compression and shear

(3) Matrix failure under combined tension and shear

(4) Fiber compression failure (kinking)



(1) Fiber tension failure

11

11

In a first approximation, fiber tension failure is predicted using a maximum stress failurecriterion. In other words, the stress along the fiber direction of a lamina may not exceedthe tensile strength Xt,

tX11

As an alternative, a maximum strain criterion may be used:

t 11



(2) Matrix compression-shear failure

2

22 )(cos)( nσnn

sin

cos

0

n

000

0

0

2212

1211

σ

cos

sin

0

Tt

0

0

1

Lt

In a plane stress model, the stresses acting on plane which is inclined at an angle withrespect to the thickness-direction are

cos)( 12 LL tσn

1

2

3

Source: Pinho et al. (Composites A, 2006)

22

22

12

12




1

22

nLL

L

nTT

T

SS

The Puck’s semi-empirical matrix failure criterion for transverse compression reads

Source: Pinho et al. (Composites A, 2006)

with the longitudinal and transverse shear strengths ST and SL, and the frictionparameters T and L. The orientation of the critical plane is found defined by the angle for which the criterion is met first.

0nif

1

2

322

22

12

12




The four parameters of Puck model may be determined based on

Reference: Pinho et al. (Composites A, 2006)

• The strength Yc, and failure plane orientation 0 measured in a transversecompression test

• The in-plane shear strength SL

using the relationships

]2tan[

1

0 T

]2tan[2 0c

T

YS T

T

LL

S

S

1

2

322

22

12

12

(estimate after Puck)



(3) Matrix tension-shear failure

1

2

12

2

22

LT SY

022 if

Experimental data suggests a quadratic interaction of thetransverse stress and the in-plane shear stress,

1

222

L

L

T

T

T

n

SSY

0nif

22

22

12

12

A modified version providing also the fracture plane for tension-dominatedmatrix failure has been advocated by Pinho et al. (2006),



Experimental validation of matrix failure model(unidirectional E-glass/LY556 lamina)

Pinho et al. (2006)

22

12

Yc [MPa] Yt [MPa] SL [MPa] 0 [deg]

130.3 37.5 66.5 53

Model parameters:

Results from compression/tension-torsionexperiments by Hutter et al. (1974) oncircumferentially wound tubes (60mm diam.,2mm thick)



(4) Fiber compression failure

Under compression-dominated loading along the fiber direction,unidirectional laminae fail through fiber kinking.

Source: Pinho et al. (2006)

Due to initial fiber misalignment, a portion of the axial load isredistributed onto the matrix material. Fiber compression failureis thus expected to occur when the matrix material in regions ofmisalignment fails.




To illustrate the concept, we assume that the matrix fails when the longitudinalshear stress reaches a critical value SL (Argon’s model). Assuming that themacroscopic stress along the fiber axis 11 is redistributed from the fiber to thematrix due to a small fiber misalignment of angle qi, we have the longitudinalshear stress

1111

iq

nt

iii

i

i

i

iqqq

q

q

q

q 1111

11cossin

sin

cos

cos

sin

00

0)(

tσn




1111

iq

nt

And hence, according to this simple model, the relationship between thecompression failure stress Xc and the longitudinal shear failure stress SL:

icL XS q

Following the same concept, more advanced fiber kinking failure models (e.g.Pinho et al., 2006) can be developed, in particular through the use of moreadvanced matrix failure models such as the one present before.



Hashin’s damage initiation model

1

2

12

2

11

Lt SX

Another intra-laminar failure model is due to Hashin (1980). Inclose analogy with the above, it provides separate criteria for thefailure mechanisms:

• Fiber tension

1

2

11

cX

• Fiber compression

1

2

12

2

22

Lt SY

• Matrix tension

1122

2

1222

22

22

LcT

c

T SYS

Y

S

• Matrix compression



Single function failure models

Instead of using different criteria accounting for different intra-laminar failuremodes, an attempt was made to provide a single analytical form to predict thefailure of laminae. One of the first models of this type is the Tsai-Hill failurecriterion. In close mathematical analogy with the anisotropic Hill’48 yieldfunction, the Tsai-Hill criterion reads

12

12221112

2

222

2

111 SFFFF

The main shortcoming of the Tsai-Hill criterion is its inability to differentiatebetween compression and tension, i.e. it predicts the same strength forcompression and tension along the fiber direction.

This issue has been addressed by the Tsai-Wu failure criterion



Tsai-Wu failure criterion

The Tsai-Wu failure criterion (1971) is in essence a particular parametric formthat accounts for the interaction of different stress tensor components in a waythat both anisotropy and tension-compression asymmetry can be taken intoaccount:

111112

2

2

1222112211

12

2

22

2

11

SYYXXYYXX

F

YYXX ctctctctctct

Its five parameters are:

• Xt … tensile strength along longitudinal direction• Xc … compressive strength along longitudinal direction• Yt … tensile strength along transverse direction• Yc … compressive strength along transverse direction• S … in-plane shear strength• F12 … interaction coefficient (ideally to be determined from equi-biaxial

tension experiments)



Modeling Choices & Limitations

Source: Stefan Hartmann, DYNAmore GmbH, Composite Berechnung in LS-DYNA, Stuttgart (2013)



Next: Damage modeling

Source: R. Talreja, C. Veer Singh, Damage and Failure of Composite Materials



Reading Materials for Lecture #12

• E.J. Barbero (2011), Introduction to Composite Materials Design

• R. Talreja, C. Veer Singh (2012), Damage and Failure of Composite Materials

• S.T. Pinho, L. Iannucci, P. Robinson (2006), Physically-based failure models and criteria for laminated fibre-reinforced composites with emphasis on fibrekinking: Part I: Development, Composites Part A 37, 63-73.

• S.T. Pinho, L. Iannucci, P. Robinson (2006), Physically-based failure models and criteria for laminated fibre-reinforced composites with emphasis on fibrekinking: Part II: FE Implementation, Composites Part A 37, 766-777.

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