Lecture #13: Damage of fiber-reinforced composites · 151-0735: Dynamic behavior of materials and...

2/15/2016 1 1Lecture #13 – 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 #13:

• Damage of fiber-reinforced composites

© 2015



Modeling Damage of Fiber-reinforced Composites



Recall from last lecture

In the last lecture, we established criteria to estimate themechanical loads under which a single lamina fails. For this weconsidered four failure modes:

(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)

In the context of laminates, such criteria are useful to predict theso-called first ply failure load, i.e. the load at which the firstlamina fails.



Continuum Damage Mechanics (CDM)

The first ply failure criteria also serve as damage initiationcriteria. It is often assumed that a lamina does not loose its entireload carrying capacity instantaneously. Instead it is assumed thatit undergoes a damage process throughout which it looses itsload carrying capacity gradually.

u0

damage phase(dissipation)

Loading phase (fully elastic)

damage initiation

0



Continuum Damage Mechanics (CDM)

In the elastic loading phase, the entire internal energy (work performed bystress) is recovered upon unloading. The particular feature of the damagephase is that it is dissipative. At the instant of damage initiation, 100% of theinternal energy can still be recovered upon unloading. This percentagedecreases gradually to 0% as the material is strained from 0 to f.

u0

0

u0

0

u0

0

100% dissipated

100% recovered 60%

recovered

40% dissipated



Introduction of a scalar damage variable

A scalar damage variable d is introduced to represent material damage. Forexample, consider the reduction of the effective load carrying cross-sectiondue to voids.

F

F

The presence of voids is usually neglected when defining theaxial stress in tension experiments,

A

F

However, if d denotes the volume fraction of isotropicallydistributed voids, the effective cross-section is only (1-d)A.Consequently, the local stresses are higher,

)1()1( dAd

Floc

A



Reduced modulus due to damage

The elastic stress-strain relationship then applies for the local stresses:

F

F

The macroscopic stress-strain relationship then reads

In other words, damage is associated with a modulusreduction.

EdAd

Floc

)1()1(

A

Ed )1(

EdEd )1(

which corresponds to a “damaged modulus” of



Elastic strain energy

The repartition of the internal energy into elastic strain energy and dissipationcan be calculated after introducing the internal damage variable. Note that atany instant of loading the recoverable elastic strain energy is defined as

u0

0

Elastic loading (undamaged material)

Loading (damage phase)

Elastic unloading(damaged material)

E

Ed

22 )1(2

1

2

1 EdEde

u0

0

recovered



Damage evolution

If we assume a linearly decreasing relationship in the damage phase, we haveu0

0

Elastic loading (undamaged material)

Loading (damage phase)

Elastic unloading(damaged material)

E

Ed

0

00 1

u 0

0

u

ud EEand thus

According to the definition of the damage variable, we then have the damageevolution under monotonic loading:

0

01

u

ud

E

Ed



Incremental damage evolution law

To account for the irreversibility of the damage process, the damage evolutionlaw is written as

2

0

0u

ud 0 0 andif

0d 0 0 orif

u0

1

u0

0

d



To account for the irreversibility of the damage process, the damage evolutionlaw is written as

2

0

0u

ud 0 0 andif

0d 0 0 orif

u0

1

u0

0

d

Incremental damage evolution law



Failure Mode Initiation criterion Equivalent strain

Fibertension

Fiber compression

Matrix tension & shear

Matrix comp. & shear

Multiple sources of damageFor each failure mode, we define:

)(i

t

ftX

f

11

c

fcX

f

11

2

1222

22

22 122

LcT

c

T

mtSYS

Y

Sf

• a stress-based damage initiation criterion of the form f(i)=1

• an effective strain measure

2

12

2

22

Lt

mtSY

f

11 ft

11 fc

2

12

2

22 )2( mt

2

12

2

22 )2( mc

Note: the Macauley brackets <..> are defined as:

0 0

0

xif

xifxx



Multiple sources of damage

Subsequently, a damage variable is defined for each failure mode using thesame generic damage evolution law. For example, for monotonic fiber tension,we have

ftft

u

ft

ft

ft

ft

uftd

0

0

with the damage parameters },{ 0

ftft

u :0

ft

ft For

Note that is a dependent parameter which can be calculated from theelastic constitutive equation and the tensile strength Xt.

ft

uft

0

11

tX

ft

0



Interaction of failure modes

In close analogy, we can then define the damage variables for all other failuremodes:

fcfc

u

fc

fc

fc

fc

ufcd

0

0

mtmt

u

mt

mt

mt

mt

umtd

0

0

mcmc

u

mc

mc

mc

mc

umcd

0

0

The stiffness of the lamina is affected by all failure modes. For example, thestiffness along the fiber direction is reduced through both fiber tensile failureand fiber compression failure. This accumulation of damage is then taken intoaccount through three “global damage variables”:

)1)(1(1 fcftf ddd

)1)(1(1 mcmtm ddd

)1)(1)(1)(1(1 mcmtfcfts ddddd

• Fiber damage:

• Matrix damage:

• Shear damage:



Damaged compliance matrix

Recall the undamaged compliance matrix for an orthotropic lamina

12

22

11

12

21

12

2

21

1

12

22

11

2

100

01

01

G

EE

EE

With the help of the global damage variables, the corresponding damagedcompliance matrix is then defined as

12

22

11

12

21

12

2

21

1

12

22

11

)1(2

100

0)1(

1

0)1(

1

Gd

EdE

EEd

s

m

f



Damaged stiffness matrix

The damaged stiffness matrix is then given by the inverse of thedamaged compliance matrix,

12

22

11

12

2212

1211

12

22

11

)1(200

0)1()1)(1(

0)1)(1()1(1

GdD

EdEdd

EddEd

Ds

mmf

mff

with

2112)1)(1(1 mf ddD



Localization of deformation

Up to the point of onset of damage, there is a uniform solution to theboundary value problem for a bar subject to an axial load. However, beyondthe point of onset of damage, two types of domains emerge:

• Damage band in which the axial strain increases rapidly and wheredissipation takes place

• Elastically unloaded domains in which the axial strain decreases and wherethe material remains undamaged

uF ,

Damageband

Elasticallyunloadedmaterial

Elasticallyunloadedmaterial

F

u



Localization of deformation

In finite element simulations, the deformation will localize in the row ofelements where the largest imperfections (of physical or numerical origin)prevail. As a result, the width of the damage band is set by the element size.Consequently, the numerical solutions are mesh-size dependent:

uF ,

Source: I. Lapczyk, J. Hurtardo (Simulia, ppt, 2006)



Fracture energy

The energy dissipated in a uniformly-strained single cubic element of edgelength L for a given failure mode is

u0

0

100% dissipated

ud

LdL

u

0

3

0

3

2

L

L

L



Fracture energy

Assuming that the damage localizes is a single row of elements, the fractureenergy per unit area of the created crack would be

ud L

LG

02 2

From a physical point of view, this fracture energy per unit area may beassociated with micro cracks that form within a narrow band around the crack(fracture process zone). It is a material property which is independent of theelement size.

L



Regularization

To ensure the independence of the predicted fracture energy per unit area fromthe element size, we define the damage model parameter as a function ofthe element size

0

2][

L

GL

f

uu

L

while the fracture energy per unit area, Gf, is introduced as additional materialmodel parameter.

u

Large strains are shown for visualization purposes, small strains are expected to prevail in reality



Regularization

As a result of this regularization, the width of the band oflocalization still corresponds to a single row of elements (and ishence element size dependent). However, the displacement tofracture Duf associated with the deformation in this band is nowelement size independent:

0

2

f

uf

GLu D

fuD

fuD




Regularization

0.31.0

0.6

1.3

2.0

3.0L

33.213.0

0.1u

7.0D Lu uf

6.0L

17.116.0

3.1u

7.0D Lu uf

6.0L

33.216.0

0.2u

4.1D Lu uf

Coarse meshFine mesh

0.7




Solution after regularization

uF ,

Source: I. Lapczyk, J. Hurtardo (Simulia, ppt, 2006)



Summary: Lamina Material Parameters

• Modulus along fiber direction• Modulus along transverse

direction• In-plane Poisson’s ratio• In-plane shear modulus

Undamaged

Elasticity Damage Initiation Ultimate Failure

• Tensile strength along fiber direction

• Compressive strength along fiber direction

• Tensile strength along transverse direction

• Compression strength along transverse direction

• In-plane shear strength• Out-of-plane shear strength

• Fracture energy per unit area for fiber tensile failure

• Fracture energy per unit area for fiber compression failure

• Fracture energy per unit area for matrix tensile failure

• Fracture energy per unit area for matrix compression failure

TLctct SSYYXX ,,,,,121221 ,,, GEE mcmtfcft GGGG ,,,

Aside from basic characteristics such as the lamina orientation,density and thickness, the following material properties must bespecified:



Modeling Choices & Limitations

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



Composite Modeling Summary

Source: https://en.wikipedia.org/wiki/Micro-mechanics_of_failure#/

• Material properties need to be provided by the user at the lamina level

• Laminate properties are computed automatically by the FE code (after user specifies the lay-up)

structure

laminate

lamina

Micromechanics



Reading Materials for Lecture #13

• Z.P. Bazant, B.H. Oh (1983), Crack band theory of fracture of concrete, Materials and Structures 16, 155-177.

• A. Matzenmiller, J. Lubliner, R.L. Taylor (1995), A constitutive model for anisotropic damage in fiber-composites, Mechanics of Materials 20, 125-152

• I. Lapczyk, J. Hurtardo (2006), Progressive Damage Modeling In Fiber-Reinforced Materials, http://paginas.fe.up.pt/~comptest/proc/files/presentations/lapczyk.pdf

• S.J. Hiermaier (2008), Structures under crash and Impact, Springer.

http://paginas.fe.up.pt/~comptest/proc/files/presentations/lapczyk.pdf

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Lecture #13: Damage of fiber-reinforced composites · 151-0735: Dynamic behavior of materials and...

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