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Proyecto Fin de CarreraIngeniería de Telecomunicación

Formato de Publicación de la Escuela TécnicaSuperior de Ingeniería

Autor: F. Javier Payán Somet

Tutor: Juan José Murillo Fuentes

Dep. Teoría de la Señal y ComunicacionesEscuela Técnica Superior de Ingeniería

Universidad de Sevilla

Sevilla, 2013

Proyecto Fin de MasterMaster en Diseño Avanzado en Ingeniería Mecánica

Coupled Criterion of the Finite FractureMechanics with Equivalent ConstraintModel for modelling transverse crackingin composite laminatesAutor: Ana Maria Pereira BrunTutor: Vladislav Mantic, Isarel García

Departamento de Mecánica de Medios Continuos yTeoría de Estructuras. Grupo de Elasticidad y

Resistencia de Materiales.Escuela Técnica Superior de Ingeniería

Universidad de Sevilla

Sevilla, 2017

Proyecto Fin de MasterMaster en Diseño Avanzado en Ingeniería Mecánica

Coupled Criterion of the Finite FractureMechanics with Equivalent Constraint Modelfor modelling transverse cracking in composite

laminates

Autor:

Ana Maria Pereira Brun

Tutor:

Vladislav Mantic, Isarel GarcíaCatedrático de Universidad, Profesor Ayudante Doctor

Departamento de Mecánica de Medios Continuos y Teoría de Estructuras.Grupo de Elasticidad y Resistencia de Materiales.

Escuela Técnica Superior de IngenieríaUniversidad de Sevilla

Sevilla, 2017

Proyecto Fin de Master: Coupled Criterion of the Finite Fracture Mechanics with EquivalentConstraint Model for modelling transverse cracking in composite lami-nates

Autor: Ana Maria Pereira BrunTutor: Vladislav Mantic, Isarel García

El tribunal nombrado para juzgar el trabajo arriba indicado, compuesto por los siguientes profesores:

Presidente:

Vocal/es:

Secretario:

acuerdan otorgarle la calificación de:

El Secretario del Tribunal

Fecha:

Acknowledgements

I would like to express my very great appreciation to Professor Mantic and Professor Kashtalyan for givingme the opportunity of getting involved in this work and discovering the wide world of fracture mechanics.I am particularly grateful for the assistance given by Israel Garcia without whom this wouldn’t have beenpossible.

Ana María Pereira BrunSevilla, 2017

I

Abstract

Previous works show how matrix cracking is the first damage mode in composite laminates under tensileloading. In these cases cracks parallel to the fibers appear causing stiffness reduction but not necessarily atotal failure of the laminate. Crack density will determine new properties of the damaged laminate. The aimof this paper is to predict the evolution of crack density as a function of applied strain in [θ/90]s laminatesby using the Equivalent Constraint Model (ECM). In this model the damaged 90° layer is replaced with anequivalent layer with reduced stiffness properties which leads to an equivalent laminate with homogeneousundamaged layers. To define the critical strain where a new crack appears Coupled Criterion of FiniteFracture Mechanics (CC-FFM) is applied. CC-FFM establishes that a finite fracture event occurs if both,stress criterion and energy criterion are fulfilled.

III

Index

Abstract III

1 Introduction 1

2 Model and assumptions. The Equivalent Constraint Model 32.1 Equivalent Constraint Model 4

3 Coupled criterion (CC) 73.1 Stress criterion 73.2 Energy criterion 7

4 Critical stress-strain analysis 94.1 Definition of geometry and materials 94.2 Calculation of stiffness and compliance matrices 94.3 Transformation into global coordinate system 104.4 Equivalent stiffness matrix of damaged lamina 104.5 Equivalent stiffness matrix of laminate 114.6 Critical stress and strain 11

5 Results and discussion 13

6 Concluding remarks 19

Apendix A MATLAB codes 21A.1 Master 21A.2 Definition of laminate 24A.3 Stress criterion 25A.4 Energy criterion 27

List of Figures 29Bibliografy 31

V

1 Introduction

Previous works[11][10] show how matrix cracking is the first damage mode in composite laminates undertensile loading. In these cases cracks parallel to the fibers appear causing stiffness reduction but notnecessarily a total failure of the laminate. This leads to a change in perspective as a damaged laminate stillmay have many cycles of life and it is necessary to study new models to predict its behavior as the standardmodels for undamaged laminates are no longer suitable for that purpose.The most common methods used to calculate stress distribution in cracked laminates are based on shear-lagparameters[8] so is the Equivalent Constraint Method (ECM)[2] that is applied in this paper. However, thereare another methods such as variational approach, internal variable models, self consistent method,continuum damage mechanics and many others that should lead to a similar result.A crack lamina behaves inside a laminate in a different way than an infinite effective medium containingmany cracks would do. That is mainly because the second approach assumes that interaction with theneighborhood layers is limited to the extreme of the crack tips. Shear lag models, in contrast, consider that thestiffness of the damaged lamina is influenced by the laminate where it is contained and to take into accountthis effect these models define the shear lag parametes that will be explained in the ECM dedicated chapter.Shear lag based models started by doing 1-D shear lags approaches[15] suitable for cross ply laminates.Most of the models developed at the beginning were restricted to cross ply laminates under uni-axial tensileloading. They are being modified and improved to guarantee its application to other types of laminates. Themodel used in this paper uses a 2-D shear lag analysis[16] where the out of plane shear stresses vary linearlyacross the whole thickness of constraining layer. 3-D shear lag based models are also appearing in the lastfew years[3].Structure of this paper has been designed from a theoretical point of view to a practical one: first, theory ofmodel and criterion are explained to allow the reader to follow the simulation with MATLAB made in thelast part of this paper with given materials and geometry.Assumptions and simplifications made during this work are defined in Section 2, where the ECM is alsoexplained. Once the equivalent homogeneous laminate is defined, stress analysis could be performed.Without a criteria to consider, a stress analysis wouldn’t give any information about evolution of damage, so,in order to do that the Coupled Criterion (CC) is explained in Section 3. Stress analysis over ECM laminatecombined with CC provides the critical stress for a new crack to appear.Reached these point all the theory applied in the MATLAB simulation should be known by the reader.Though, in the last part of this work a step by step procedure in Section 4 is showed starting with thedefinition of geometry and materials of a given laminate and ending in Section 5 with conclusions andgraphs showing crack density evolution versus critical strain.

1

2 Model and assumptions. The EquivalentConstraint Model

From this point on, different notation to name properties, stresses, strains and matrices of laminas andlaminate will be used. In order to have a clear reference through the reading, notation rules are explainedhere: horizontal bar is used for laminate properties (σ ), tilde is used for lamina properties (ε), superscriptbetween brackets points out which lamina is it referred to (ε(2)) and hat is used to talk about lamina beforeassuming any damage (Q). Laminate is referred to the global coordinate system x1,x2,x3 with x1 parallel tothe 90 ply fibers while laminas are referred to local coordinate system x(n)1 ,x(n)2 ,x(n)3 with x(n)1 parallel to θ plyfibers and where n indicates the ply. Laminate is formed by a 90 layer of 2h2 thickness (lamina 2) between

x₂

x

3

x1Cracks

Lamina 2

Lamina 1

Figure 2.1 Global axis.

two θ layers of h1 thickness (lamina 1). Perfect bonding between plies is assumed. The analysis is performedover a representative segment of the laminate between two consecutive cracks. Cracks are assumed to spanthe whole width of the laminate and be uniformly spaced a 2s distance in lamina 2. As the representativesegment is symmetric with respect to the mid pane and material and geometry are uniform the analysis canbe reduced to one quarter of the representative segment. As cracks are in lamina 2, relative crack density(Dm) is defined as Dm = h2s and stress distribution in the laminate will depend on this parameter. There aredifferent methods used to calculate how Dm affects the stiffness properties but the most common method isthe Equivalent Constraint Model.

3

4 Chapter 2. Model and assumptions. The Equivalent Constraint Model

x₂

x

3

2s

θº

90º

θº

Figure 2.2 Representative segment.

2.1 Equivalent Constraint Model

ECM[16], developed by Zhang and Fan, predicts the reduction of in-plane stiffness properties due to matrixcracking by a two-dimensional shear-lag analysis approach and the definition of In-Situ Damage EffectiveFunctions. The origins of the IDEF concept are discussed in detail in [8]. It was proved[4], using selfconsistent method, that the compliance tensor of a linearly elastic brittle anisotropic solid containingmicrocracks can be represented as a sum of two tensors: a tensor of elastic compliance of un undamagedsolid and a tensor of additional compliances, dependent on the configuration and distribution of microcracks.This tensor of additional compliances, could be reduced into one with two non-zero components assumingload parallel to the fibers, and those non-zero components result ti be IDEF defined in the equation below.The in plane microstresses in the damaged layer can be used to evaluate the reduction of the laminatestiffness properties due to damage. Instead of the damaged laminate ECM is considered, in which thedamaged ply is replaced with an equivalent homogeneous layer with reduced stiffness properties[9].

The equivalent stiffness matrix of a damaged lamina is given by:

[Q] = [Q]−R (2.1) Q11 Q12 0Q12 Q22 0

0 0 Q66

=

Q11 Q12 0Q12 Q22 0

0 0 Q66

− (Q12)

2

Q22Λ22 Q12Λ22 0

Q12Λ22 Q22Λ22 00 0 Q66Λ66

Here Λ22 and Λ66 are called in situ damage effective functions (IDEF) and describe the stiffness loss due tomatrix cracking. IDEF depend of Dm, K1 and K2, the shear lag parameters and a series of constantsdepending on compliance matrix of the original material and geometry (χ = h1/h2).

Λ22 = 1− φ1 +φ2Dm tanh(λ1/Dm)

φ1 +φ3Dm tanh(λ1/Dm)(2.2)

Λ66 = 1− Γ1 +Γ2Dm tanh(λ2/Dm)

Γ1 +Γ3Dm tanh(λ2/Dm)(2.3)

Where constants can be calculated as:

φ1 = Ω1/L1 φ2 =−Ω1/(L1λ1)

φ3 =φ1

χλ1[(S(1)22 +a1S(1)12 )Q

(2)22 +(S(1)12 +a1S(1)11 )Q

(2)12 Γ1 = Ω2/L2

Γ2 =−Ω2/(L2λ2) Γ3 =Γ1

χλ2Q(2)

66 S(1)66

L1 =K2h1

[S(1)22 +a1S(1)12 +χ(ˆ

S(2)22 +a1ˆ

S(2)12 )] L2 =K1h1

(S(1)66 +χˆ

S(2)66 )

2.1 Equivalent Constraint Model 5

λ1 = h2√

L1 λ2 = h2√

L2

Ω1 =K2h1

(1+χ)(S(1)22 +a1S(1)12 ) Ω2 =K1h1

(1+χ)(S(1)66 )

a1 =−S(1)1,2 +χS(2)1,2

S(1)1,1 +χS(2)1,1

And K1 and K2 are defined as:

K1 =3G(1)

13 G(2)13

h2G(1)13 +(1+(1−η)/2)ηh1G(2)

13 )(2.4)

K2 =3G(1)

23 G(2)23

h2G(1)23 +(1+(1−η)/2)ηh1G(2)

23 )(2.5)

With η = hs/h1.Many different definitions for K1 and K2 have been developed through the years based on differentassumptions, some analytical and some experimental: Highsmith and Reifsnider[1], Laws and Dvorak[13],and others.In this paper the definition given by Zhang and Fan [16] is adopted. Partial linear variation of shear stressesacross the thickness of the constraining layer is considered. After a crack in lamina 2 additional load istransferred into constraining layers through the out of plane shear deformation in the interface between plies.

3 Coupled criterion (CC)

The Finite Fracture Method (FFM) concept assumes the instantaneous formation of cracks of finite size atinitiation. In this framework, Leguillon [7] proposed a coupled stress and energy criterion to identify thecritical loading. CC establishes that a finite fracture event occurs if both, stress criterion and energy criterionare fulfilled [12]. In this paper Leguillon’s CC of FFM is extended to multiple transverse cracking incomposite laminates.

3.1 Stress criterion

Garret and Bailey [6] were the first to propose maximun stress criterion to predict the initiation of matrixcracking in cross ply laminates.The stress criterion assumes the existence of a critical value of tension required to originate fracture at acertain plane of the lamina. This value depends on the ply orientation with respect to the load directionbecause of the material microstructure of the ply, given by approximately parallel long fibers embedded inmatrix. [5]According to stress criterion, new cracks appear if:

σ(2)22 ≥ Yt (3.1)

Where Yt is transverse tensile strength of the lamina.The limitation of this criterion is that it does not agree well with experimental data which show that onset ofmatrix cracking strongly depends on the laminate structure.

3.2 Energy criterion

In order to surpass the limitation of stress criterion the use of energy criterion to predict crack densityevolution was proposed. According to this criterion the first crack appears when the energy release rateassociated with its formation exceed a critical value. The energy criterion adopted here is based on theincremental Griffith criterion proposed by several authors as Garrett and Bailey[6], Hashin[17] orLeguillon[7]. In the present paper, a constant fracture toughness Gc during the crack onset and growth can beassumed, because the transverse crack in the 90 ply grows in pure mode 1.According to energy criterion, new cracks appear if:

U1−U2 ≥ Gc∆Am (3.2)

Here, U is the total strain energy stored in a laminate with length L and width w, ∆Am is the total fracture areaof multiple transverse cracks and Gc is the critical fracture toughness associated with matrix cracking[14].

7

4 Critical stress-strain analysis

To analyze the loads under which a new crack may appear next steps have been followed:

• Definition of the geometry of the laminate and material of each ply• Calculation of stiffness and compliance matrices of undamaged plies in local coordinate system.• Transformation into global coordinate system of the previous.• Calculate equivalent stiffness matrix of the damaged lamina (using D as a variable)• Calculate equivalent stiffness matrix of the damaged laminate• Calculate the critical stress or strain for a fracture event to occur under Coupled Criterion

4.1 Definition of geometry and materials

Laminate is formed by a “lamina 1” of 2h1 thickness and angle 0 referred to x1,x2,x3 between two θ “layers2” of h2 thickness. hs = 0.05 Properties of material of lamina 1 are :

Lamina 1 definition.

Material 1 h1 (mm) E1 (GPa) E2 (GPa) E3 (GPa) ν12 ν13 G23 (GPa) G31 (GPa) G12 (GPa)GFRP 0.15 40000 10000 10000 0.31 0.31 3520 5000 5000

And properties of material of lamina 2 are:



4.2 Calculation of stiffness and compliance matrices

The constitutive equations in terms of ply strain and stresses are: ε(1)11

ε(1)22

γ(1)12

=

S(1)11 S(1)12 0S(1)12 S(1)22 0

0 0 S(1)66

σ

(1)11

σ(1)22

σ(1)12

,

ε(2)11

ε(2)22

γ(2)12

=

S(1)11 S(2)12 0S(1)12 S(2)22 0

0 0 S(2)66

σ

(2)11

σ(2)22

σ(2)12

(4.1)

WhereS(1)11 = 1/E(1)

1 , S(1)12 = nu(1)12 /E(1)2 , S(1)22 = 1/E(1)

2 , S(1)66 = 1/G(1)12 (4.2)

9

10 Chapter 4. Critical stress-strain analysis

S(2)11 = 1/E(2)1 , S(2)12 = nu(2)12 /E(2)

2 , S(2)22 = 1/E(1)2 , S(2)66 = 1/G(2)

12 (4.3)

And by doing the inverse calculation: σ(1)11

σ(1)22

σ(1)12

=

Q(1)11 Q(1)

12 0Q(1)

12 Q(1)22 0

0 0 Q(1)66

ε

(1)11

ε(1)22

γ(1)12

,

σ(2)11

σ(2)22

σ(2)12

=

Q(2)11 Q(2)

12 0Q(2)

12 Q(2)22 0

0 0 Q(2)66

ε

(2)11

ε(2)22

γ(2)12

(4.4)

Where[Q] = [S](−1) (4.5)

4.3 Transformation into global coordinate system

To operate with the different plies of the laminate it is necessary to refer all of them to the defined globalcoordinate system. The matrix [T] provides the relation between stresses in local coordinate system andstresses in global coordinate system.

[T ] =

cos2(θ) sin2(θ) −2sin(θ)cos(θ)sin2(θ) cos2(θ) 2sin(θ)cos(θ)

sin(θ)cos(θ) −sin(θ)cos(θ) cos2(θ)− sin2(θ)

(4.6)

σxyz = [T ]−1

σxyz(n) (4.7)

The same relationship can be establised between strais in local coordinate system and strains in globalcoordinate system. In this case is also needed to take into acount that γ

(n)12 = 2ε

(n)12 :

[P] =

1 0 00 1 10 0 2

(4.8)

εxyz = [P][T ][P]−1

εxyz(n) (4.9)

Components of compliance matrix of each play can be transformed by aplying:

Qxyz = [T ]−1Qxyz(n) [P][T ][P]−1 (4.10)

4.4 Equivalent stiffness matrix of damaged lamina

The reduced stiffness matrix of the equivalent homogeneous layer is related to the stiffness matrix of theundamaged layer via ECM. In the case of study, the damaged layer is “lamina 2” so the new stiffness matrixcan be calculated as: Q(2)

11 Q(2)12 0

Q(2)12 Q(2)

22 00 0 Q(2)

66

=

Q(2)11 Q(2)

12 0Q(2)

12 Q(2)22 0

0 0 Q(2)66

−

(Q(2)12 )2

Q(2)22

Λ22 Q(2)12 Λ22 0

Q(2)12 Λ22 Q(2)

22 Λ22 00 0 Q(2)

66 Λ66

(4.11)

As “lamina 1” is not damaged:Q(1) = Q(1) (4.12)

4.5 Equivalent stiffness matrix of laminate 11

4.5 Equivalent stiffness matrix of laminate

As equivalent stiffness matrices of each lamina have been obtained, the stiffness matrix of the laminate canbe calculated as:

[A] =2[Q(1)]h1 +2([Q(2)]−R]h2

2(h1 +h2)(4.13)

And in consequence, the relation between homogenized stresses and strains can be calculated using thefollowing expression: σ11

σ22σ12

=

A11 A12 A13A21 A22 A23A31 A32 A33

ε11ε22γ12

(4.14)

4.6 Critical stress and strain

Once this point is reached a complete description of the equivalent homogeneous laminate is available. Todefine critical strain or stress for a fracture event to happen CC is applied in this section. As was alreadymentioned stress criterion sets that new cracks appear if Equation (3.1) if fulfilled. This expressionestablished a condition based on σ22 values. To extrapolate this condition to the whole laminate it isnecessary to substitute σ22 of the “lamina 2” by σ22 of the equivalent laminate. This equivalency is madeusing a combination of equilibrium equations, constitutive equations and the assumption of perfect bondingbetween plies. The full development can be found in [16], resulting:

Ω22L1

(1− cosh(√

L1x2)

cosh(√

L1)s)σ22 ≥ Yt (4.15)

Where Yt us the transverse tensile strength of lamina 2, L1 and Ω22 are constants depending of compliancematrices of both plies, shear lag parameters and layer thickness ratio. This expression if fulfilled for everyx2 < s value. As the interest of this paper is to find critical stress values for a new crack to appear it isnecessary to use a value for x2 where stress is maximun. As proved in Figure (5.1) this value is x2 = 0. Inexperiments it is easier to measure strain, thus stress criterion condition can be put in terms of strain solvingthe following system of equations and solving σ22 assuming σ11=0 and σ12=0. σ11

σ22σ12

=

A11 A12 A13A21 A22 A23A31 A32 A33

ε11ε22γ12

(4.16)

With this, critical strain for a new crack to appear according to stress criterion is obtained. Same procedurewith energy criterion must be followed as CC established that both of them have to be fulfilled. The Energycriterion sets that a new crack appears if Equation (3.2) is fulfilled. Defining strain energy in the laminate asU = 1

2 [ε]T [A][ε], and taking into account Equation (4.13) the condition for a fracture event to occur may be

rewritten as:1

Dm2 −Dm

1t2[ε]

T [R(Dm2 )−R(Dm

1 )][ε]≥ Gc (4.17)

Note that energy criterion condition is expressed in different terms than stress criterion condition was. Sinceboth of them need to be compared according to CC it is convenient to rewrite this expression as:

1

γ

√Dm

2 −Dm1

[ε]T [R(Dm2 )[ε]− [ε]T [R(Dm

1 )][ε]

≥ Yt (4.18)

Where γ is a dimensionless brittleness number[14]:

γ =1Yt

√GcE(2)

2h2

(4.19)

12 Chapter 4. Critical stress-strain analysis

Two formulations of energy criterion can be made. Equation4.18 shows a discrete formulation where thenumber of cracks evolves from N to N +1 cracks and the relative crack density evolves from D1 to D2. Inorder to formulate a continuous formulation of energy criterion small increments between D1 and D2 shouldbe replaced by derivative yields.

5 Results and discussion

Results showed in this chapter have been calculated for a [45/90]s laminate with the following properties:





Stress criterion expressed in Equation(4.15) is represented in Figure5.1 and Figure5.2. Since this criterion isformulated as a function of x2, prior to any other analysis it is necessary to determine which value of x2 < s,being s the distance between two consecutive cracks, may lead to a maximum stress.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x2/s

Yt/(

ε 22E

2)

D=0.1D=0.2D=0.3D=0.4D=0.5D=0.6D=0.7D=0.8D=0.9D= 1

Figure 5.1 Stresses between cracks.

Figure5.1 shows that for any relative crack density the maximum is located at x2 = 0.

13

14 Chapter 5. Results and discussion

Once this point is determinated, following graphics and comparisons using stress criterion will be assumedto be performed at it.

Figure5.2 shows the relation between the normalized applied strain and the initial and final crack densities,assuming that crack density is doubled when conditions for damage increase are fulfilled. As expected crackdensity grows with the applied strain.

Energy criterion expressed in Equation(4.18) is represented in Figure5.2 for a range of brittleness numbers.Brittleness number is a structural parameter which combines stiffness, strength fracture toughness andtransverse layer thickness under an dimensionless parameter.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ε22

/(Yt/E

2)

D1,D

2

Critical strain for damage release

Function of D

1

Function of D2

Figure 5.2 Critical strain for damage release (stress criterion).

With increasing γ critical stress values are higher. This is, the critical strain in inversely proportional to h22

from the definition of the brittleness number: γ = 1Yt

√GcE(2)

2h2

.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ε22

/(Yt/E

2)

D


γ= 0γ=0.2γ=0.4γ=0.6γ=0.8γ= 1γ=1.2γ=1.4

Figure 5.3 Critical strain for damage release (energy criterion continuous formulation).

15

Comparison between energy criterion continuous formulation and energy criterion discrete formulation hasbeen made. It is showed in Figures 5.4 5.7.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ε22

/(Yt/E

2)

D,D

1,D2


Continuous formulationDiscrete formulation f(D

1)

Discrete formulation f(D2)

Figure 5.4 Critical strain for damage release (energy criterion). γ = 0.4.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ε22

/(Yt/E

2)

D,D

1,D2



1)



As expected it can be seen an increase in the transverse crack density with applied strain. In discreteformulation doubling of the relative crack density has been assumed, this is, for any given crack densitystress criterion discrete formulation estimates that in next state will be double of crack density. In contrast, inenergy criterion continuous formulation for any given crack density just one crack would appear.It can be seen how continuous formulation curve remains between the two curves for discrete formulation.This explains why continuous formulation usually has a better agreement with experiments. The two curvesof discrete formulation draw a region where possible crack density progression is confined.Since the aim of this paper is to define the critical strain for a new crack to appear applying CoupledCriterion, it is necessary to show energy criterion and stress criterion on the same graph.

16 Chapter 5. Results and discussion

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ε22

/(Yt/E

2)

D,D

1,D2



1)



0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ε22

/(Yt/E

2)

D,D

1,D2



1)


Figure 5.7 Critical strain for damage release (energy criterion). γ = 1.

In Figure 5.8 this comparison can be seen. For example, considering γ = 1.5, the transition point isapproximately located at D = 0.5. For high values of brittleness number Coupled Criterion is dominated byenergy criterion.Figure 5.9 show how the critical strain to reach a relative crack density Dm = 1 evolve with a changing angleof lamina 1.According to this graph for angles of lamina 1 between 0º and 30º (remember that angle is referred to globalcoordinate system where x2 is parallel to lamina 2 fibers) the critical strain to reach Dm = 1 is higher than forany other angle. In contrast, angles between50º and 80º show a critical strain lower than for any other angle.

17

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ε22

/(Yt/E

2))

D, D

1


Energy Criterion γ=0.5Energy Criterion γ=1Energy Criterion γ=1.5Energy Criterion γ=2Stress Criterion

Figure 5.8 Comparison between stress criterion and energy criterion.

0 10 20 30 40 50 60 70 80 903

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4Critical strain for D=1

angle of lamina 1

ε 22/(

Yt/E

2))

Figure 5.9 Critical strain to get D=1 as a function of angle of lamina 1.

6 Concluding remarks

A model to predict crack density evolution in [θ ,90]s laminates is presented.The basis of theoretic aspects has been set and codes for MATLAB have been developed to simulate thiscriteria involving a combination of the Equivalent Constraint Model and the Coupled Criterion of finiteFracture Mechanics. The coding was performed in a modular way in order to allow future students to use itfor further works.A review through the different results obtained using energy criterion continuous and stress criterion discretehave been made. Also the comparison between the two curves of stress criterion discrete have been plotted.Finally, a comparison between stress criterion and energy criterion continuous allows the reader evaluate theCoupled Criterion as one single criterion.

19

Apendix AMATLAB codes

The MATLAB codes written in this paper were developed for this purpose by the author and are original.Based on the analytical expressions explained through this work allow the user to get all the results alreadyshowed by introducing a pack of defined parameters from the materials of the laminate and its geometry.The code is structured in modules which are called from the “master” code in Apendix A.1. The modules are“definition of the laminate” (in Apendix A.2), “stress criterion” (in Apendix A.3) and “energy criterion” (inApendix A.4). Here a brief summary of each of them:

• Master code contains the information of laminate materials and geometry. It calls “definition oflaminate” module which, using these information, gives back a structure.

• Definition of laminate code operates with laminate information to give back a laminate-structurecontaining stiffness and compliance matrices of laminate in local and global coordinate system plus allthe input parameters associated to its lamina-substructure.

• Stress criterion code is codified to operate using laminate-structure defined, relative crack density, andx2. These function return the normalized critical strain for a new crack to appear according to stresscriterion.

• Energy Criterion code is codified to operate using laminate-structure defined, relative crack density,and the brittleness number γ . These function return the normalized critical strain for a new crack toappear according to stress criterion.

A.1 Master

clearclc%Inputs ’ definitionoflaminate ’CarboneExpoxy=struct(’E1’,144800.0, ’E2’ ,11380.0, ’E3’ ,11380.0, ’nu12’ ,0.3, ’nu13’ ,0.3, ’nu23’ ,0.3, ’G23’,3450.0, ’G31’

,6480.0, ’G12’,6480.0);AS4=struct(’E1’,139000,’E2’,11100,’E3’,11100,’nu12’ ,0.3, ’nu13’ ,0.3, ’nu23’ ,0.42, ’G23’,4200,’G31’,4790,’G12’,4790);CFRP=struct(’E1’,145000,’E2’,9500, ’E3’,9500, ’nu12’ ,0.31, ’nu13’ ,0.31, ’nu23’ ,0.42, ’G23’,3350,’G31’,5600,’G12’,5600);GFRP=struct(’E1’,40000,’E2’,10000,’E3’,10000,’nu12’ ,0.31, ’nu13’ ,0.31, ’nu23’ ,0.42, ’G23’,3520,’G31’,5000,’G12’,5000);%geometryh1=0.15; % mmh2=0.15; % mmhs=0.05; %mmmaterial1=GFRP;material2=GFRP;angle1=45;angle2=0;

%definition of laminategeometry=struct ( ’h1’ ,h1, ’h2’ ,h2, ’hs’ ,hs) ;laminate= definitionoflaminate ( material1 , material2 ,geometry,angle1 ,angle2) ;

%stresses between cracks (sigma22_2/e22/E2) ( stress criterion )D_interval =0.1:0.1:1;

21

22 Capítulo A. MATLAB codes

i=0;x2_s_interval =0.0000000000000000000000000001:0.1:1;for D=D_interval

i=i+1;j=0;for x2_s=x2_s_interval

j=j+1;sigma22_2_e22_E2_vec(i,j)= stresscriterion ( laminate ,D,x2_s); %matrix of results

endfigure (1)plot ( x2_s_interval ,sigma22_2_e22_E2_vec(i,:) , ’ color ’ ,rand(1,3))legend( strcat ( ’D=’,num2str(D_interval’)) )%title (’ Stresses between craks ( stress criterion ) ’)xlabel ( ’x_2/s’ )ylabel ( ’Y_t /(\ epsilon_2_2E_2)’)hold on

end%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%critical strain for damage release ( stress criterion ) For D_1 and D_2D_interval =0.0001:0.2:1.5;i=0x2_s=0for D=D_interval

i=i+1;e22_sigma22_2_E2_vec(i)=1/ stresscriterion ( laminate ,D,x2_s);

end

figure (2)plot (e22_sigma22_2_E2_vec(:),D_interval , ’ color ’ ,rand(1,3) , ’DisplayName’,’Function of D_1’)legend(’−DynamicLegend’)axis ([0,4,0,1])title ( ’ Critical strain for damage release ’ )xlabel ( ’ \ epsilon_2_2 /(Y_t/E_2))’ )ylabel ( ’D’)hold on

%critical strain for damage release ( stress criterion )D_interval =0.0001:0.2:1.5;i=0x2_s=0for D=2∗D_interval


end

figure (2)plot (e22_sigma22_2_E2_vec(:),D_interval , ’ color ’ ,rand(1,3) , ’DisplayName’,’Function of D_2’)legend(’−DynamicLegend’)axis ([0,4,0,1])title ( ’ Critical strain for damage release ’ )xlabel ( ’ \ epsilon_2_2 /(Y_t/E_2)’)ylabel ( ’D_1,D_2’)hold on

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%critical strain for damage release (energy criterion continuum) fora range of gamma

D_interval =0.0001:0.2:1.5;gamma_interval=0:0.2:1.4i=0factor =1.0001%gamma=sqrt(laminate.material2.E2∗Gc/h2)/Ytfor gamma=gamma_interval

i=i+1;j=0for D=D_interval

j=j+1e22_sigma22_2_E2_edisc(i,j)=1/ energycriterion ( laminate ,D, factor ,gamma)

endfigure (3)plot (e22_sigma22_2_E2_edisc(i ,:) , D_interval , ’ color ’ ,rand(1,3))

A.1 Master 23

axis ([0,4,0,1])legend( strcat ( ’ \gamma=’,num2str(gamma_interval’)))title ( ’ Critical strain for damage release ’ )xlabel ( ’ \ epsilon_2_2 /(Y_t/E_2)’)ylabel ( ’D’)hold on

end

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%critical strain for damage release comparison between energycriterion discrete and continuous

D_interval =0.0001:0.2:1.5;gamma_interval=2i=0factor =1.0001%gamma=sqrt(laminate.material2.E2∗Gc/h2)/Ytfor gamma=gamma_interval



endfigure (4)plot (e22_sigma22_2_E2_edisc(i ,:) , D_interval , ’ color ’ ,rand(1,3) , ’DisplayName’,’Continuous formulation ’ )axis ([0,4,0,1])legend(’−DynamicLegend’)title ( ’ Critical strain for damage release ’ )xlabel ( ’ \ epsilon_2_2 /(Y_t/E_2)’)ylabel ( ’D’)hold on

endi=0factor =2%gamma=sqrt(laminate.material2.E2∗Gc/h2)/Ytfor gamma=gamma_interval



endfigure (4)plot (e22_sigma22_2_E2_edisc(i ,:) , D_interval , ’ color ’ ,rand(1,3) , ’DisplayName’,’Discrete formulation f (D_1)’)axis ([0,4,0,1])legend(’−DynamicLegend’)title ( ’ Critical strain for damage release ’ )xlabel ( ’ \ epsilon_2_2 /(Y_t/E_2)’)ylabel ( ’D,D_1’)hold on

end%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−% comparison between stress criterion and energy criterion (energy criterion continuous + stress criterion )

D_interval =0.0001:0.2:1.5;gamma_interval=0.5:0.5:2x2_s=0i=0factor =1.0001

for gamma=gamma_intervali=i+1;j=0for D=D_interval


endfigure (5)plot (e22_sigma22_2_E2_edisc(i ,:) , D_interval , ’−−’,’color’ ,rand(1,3) , ’DisplayName’, strcat ( ’Energy Criterion \

gamma=’,num2str(gamma’)))axis ([0,4,0,1])


legend(’−DynamicLegend’)title ( ’ Critical strain for damage release ’ )xlabel ( ’ \ epsilon_2_2 /(Y_t/E_2)’)ylabel ( ’D’)hold on

end

i=0for D=D_interval


endfigure (5)plot (e22_sigma22_2_E2_vec(:),D_interval , ’ color ’ ,rand(1,3) , ’DisplayName’,’ Stress Criterion ’ )legend(’−DynamicLegend’)axis ([0,4,0,1])title ( ’ Critical strain for damage release ’ )xlabel ( ’ \ epsilon_2_2 /(Y_t/E_2))’ )ylabel ( ’D, D_1’)hold on

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%influence of angle of lamina 1 in laminateangle1_interval =0.1:1:90;D=1i=0x2_s=0for angle1= angle1_interval

%definition of laminate as function of angle 1laminate= definitionoflaminate ( material1 , material2 ,geometry,angle1 ,angle2) ;i=i+1;e22_sigma22_2_E2_vec(i)=1/ stresscriterion ( laminate ,D,x2_s);

end

figure (6)plot ( angle1_interval ,e22_sigma22_2_E2_vec(:),’color ’ ,rand(1,3))axis ([0,90,3,4])title ( ’ Critical strain for D=1’)xlabel ( ’angle of lamina 1’)ylabel ( ’ \ epsilon_2_2 /(Y_t/E_2))’ )hold on

A.2 Definition of laminate

function [ laminate]= definitionoflaminate ( material1 , material2 ,geometry,angle1 ,angle2)

%%%Description of the Geometry

geometry.chi=geometry.h1/geometry.h2;geometry.eta=geometry.hs/geometry.h1;

%%%Description of the material 1 =f(E1 ....)

material1 .G13=material1.G31;material1 .nu21=material1.E2/material1 .E1∗material1.nu12;material1 .nu31=material1.E3/material1 .E1∗material1.nu13;material1 .nu32=material1.E3/material1 .E2∗material1.nu23;

%Definition of compliance matrix 1 =f(material )material1 .S11=1/material1 .E1;material1 .S12=−material1.nu21/material1 .E2;material1 .S22=1/material1 .E2;material1 .S66=1/material1 .G12;material1 .compliancematrix=[material1 .S11,material1 .S12,0; material1 .S12,material1 .S22 ,0;0,0, material1 .S66];

%Definition of stiffness matrix 1

material1 . stiffnessmatrix =inv(material1 .compliancematrix) ;

A.3 Stress criterion 25

%definition of compliance matrix and global axis 1t1=pi∗angle1/180;

material1 .S11_Glob=material1.S11∗(cos(t1)^4)+(2∗material1 .S12+material1.S66)∗(sin ( t1 )^2)∗(cos( t1 )^2)+material1 .S22∗(sin( t1 )^4) ;

material1 .S12_Glob=(material1.S11+material1.S22−material1.S66)∗(sin( t1 )^2)∗(cos( t1 )^2)+material1 .S12∗(sin( t1 )^4+cos(t1)^4) ;

material1 .S22_Glob=material1.S11∗(sin(t1)^4)+(2∗material1 .S12+material1.S66)∗(sin ( t1 )^2)∗(cos( t1 )^2)+material1 .S22∗(cos(t1)^4) ;

material1 .S16_Glob=(2∗material1.S11−2∗material1.S12−material1.S66)∗sin(t1)∗(cos( t1 )^3)+(2∗material1 .S12−2∗material1.S22+material1.S66)∗(sin( t1 )^3)∗(cos( t1 ) ) ;

material1 .S26_Glob=(2∗material1.S11−2∗material1.S12−material1.S66)∗(sin(t1)^3)∗(cos( t1 ) )+(2∗material1 .S12−2∗material1.S22+material1.S66)∗(sin( t1 ) )∗(cos( t1 )^3) ;

material1 .S66_Glob=2∗(2∗material1.S11+2∗material1.S22−4∗material1.S12−material1.S66)∗(sin(t1)^2)∗(cos( t1 )^2)+material1 .S66∗(sin( t1 )^4+cos(t1)^4) ;

material1 .compliancematrix_Glob=[material1.S11_Glob,material1.S12_Glob,material1.S16_Glob;material1.S12_Glob,material1.S22_Glob,material1.S26_Glob;material1.S16_Glob,material1.S26_Glob,material1.S66_Glob];

%definition of stifness matrix and global axis 1material1 . stiffnessmatrix_Glob =inv(material1 .compliancematrix_Glob);

%%%Description of the material 2 =f(E1 ....)

material2 .G13=material2.G31;material2 .nu21=material2.E2/material2 .E1∗material2.nu12;material2 .nu31=material2.E3/material2 .E1∗material2.nu13;material2 .nu32=material2.E3/material2 .E2∗material2.nu23;

%Definition of compliance matrix 2 =f(material )material2 .S11=1/material2 .E1;material2 .S12=−material2.nu21/material2 .E2;material2 .S22=1/material2 .E2;material2 .S66=1/material2 .G12;material2 .compliancematrix=[material2 .S11,material2 .S12,0; material2 .S12,material2 .S22 ,0;0,0, material2 .S66];

%Definition of stiffness mtrix 2

material2 . stiffnessmatrix =inv(material2 .compliancematrix) ;%definition of compliance matrix and global axis 2t2=pi∗angle2/180;

material2 .S11_Glob=material2.S11∗(cos(t2)^4)+(2∗material2 .S12+material2.S66)∗(sin ( t2 )^2)∗(cos( t2 )^2)+material2 .S22∗(sin( t2 )^4) ;

material2 .S12_Glob=(material2.S11+material2.S22−material2.S66)∗(sin( t2 )^2)∗(cos( t2 )^2)+material2 .S12∗(sin( t2 )^4+cos(t2)^4) ;

material2 .S22_Glob=material2.S11∗(sin(t2)^4)+(2∗material2 .S12+material2.S66)∗(sin ( t2 )^2)∗(cos( t2 )^2)+material2 .S22∗(cos(t2)^4) ;

material2 .S16_Glob=(2∗material2.S11−2∗material2.S12−material2.S66)∗sin(t2)∗(cos( t2 )^3)+(2∗material2 .S12−2∗material2.S22+material2.S66)∗(sin( t2 )^3)∗(cos( t2 ) ) ;

material2 .S26_Glob=(2∗material2.S11−2∗material2.S12−material2.S66)∗(sin(t2)^3)∗(cos( t2 ) )+(2∗material2 .S12−2∗material2.S22+material2.S66)∗(sin( t2 ) )∗(cos( t2 )^3) ;

material2 .S66_Glob=2∗(2∗material2.S11+2∗material2.S22−4∗material2.S12−material2.S66)∗(sin(t2)^2)∗(cos( t2 )^2)+material2 .S66∗(sin( t2 )^4+cos(t2)^4) ;

material2 .compliancematrix_Glob=[material2.S11_Glob,material2.S12_Glob,material2.S16_Glob;material2.S12_Glob,material2.S22_Glob,material2.S26_Glob;material2.S16_Glob,material2.S26_Glob,material2.S66_Glob];

%definition of stifness matrix and global axis 2material2 . stiffnessmatrix_Glob =inv(material2 .compliancematrix_Glob);

%%%Definition of the laminate =f(geometria , material1 , material2 )

laminate= struct ( ’ material1 ’ , material1 , ’ material2 ’ , material2 , ’geometry’,geometry);

end

A.3 Stress criterion


function [ stresscriterion ]= stresscriterion ( laminate ,D,x2_s)

%Calculating the compliances matrices in global axisS1=laminate. material1 .compliancematrix_Glob;S2=laminate. material2 .compliancematrix_Glob;

Q1=laminate.material1 . stiffnessmatrix_Glob ;Q2=laminate.material2 . stiffnessmatrix_Glob ;

%Calculating the dimensionless ratio defining the stacking sequencechi=laminate .geometry.chi ;eta=laminate .geometry.eta ;

%Layer thicknessh1=laminate.geometry.h1;h2=laminate.geometry.h2;

%Parameter s=h2/Ds=h2/D;

%asignation of properties of the material to its position inside structureG23_1=laminate.material1 .G23;G23_2=laminate.material2 .G23;G13_1=laminate.material1 .G13;G13_2=laminate.material2 .G13;

%Shear lag parametersK2=(3∗G23_1∗G23_2)/(h2∗G23_1+(1+(1−eta)/2)∗eta∗h1∗G23_2);K1=(3∗G13_1∗G13_2)/(h2∗G13_1+(1+(1−eta)/2)∗eta∗h1∗G13_2);

%Constantsa1=−(S1(1,2)+chi∗S2(1,2)) /( S1(1,1)+chi∗S2(1,1) ) ;L_1_2=(K2/h1)∗(S1(2,2)+chi∗S2(2,2)+a1∗(S1(1,2)+chi∗S2(1,2)) ) ;L_2_2=(K1/h1)∗(S1(3,3)+chi∗S2(3,3));alpha_1_2=(1/chi )∗(Q2(2,2)∗(S1(2,2)+a1∗S1(1,2))+Q2(1,2)∗(S1(1,2)+a1∗S1(1,1))) ;alpha_2_2=(1/chi )∗(Q2(3,3)∗S1(3,3) ) ;l_2_1=h2∗sqrt(L_1_2);l_2_2=h2∗sqrt(L_2_2);

%IDEFL_22_2=1−(1−(D/l_2_1)∗tanh(l_2_1/D))/(1+alpha_1_2∗(D/l_2_1)∗tanh(l_2_1/D));L_66_2=1−(1−(D/l_2_2)∗tanh(l_2_2/D))/(1+alpha_2_2∗(D/l_2_2)∗tanh(l_2_2/D));

Omega22_2=(K2/h1)∗(1+chi)∗(S1(2,2)+a1∗S1(1,2));%Components of the matrix R from ECMR11=(Q2(1,2)^2)∗L_22_2/Q2(2,2);R12=Q2(1,2)∗L_22_2;R22=Q2(2,2)∗L_22_2;R66=Q2(3,3)∗L_66_2;

R=[R11 R12 0;R12 R22 0;0 0 R66];

%calculation of compliance matrix of laminatesyms e11 e22 e12 sigma11 sigma22 sigma12 A11 A12 A13 A21 A22 A23 A31 A32 A33sigma11=0;sigma12=0;A=[A11 A12 A13; A21 A22 A23; A31 A32 A33];strain =[e11; e22; e12];stress =[sigma11; sigma22; sigma12];[e11,e12,sigma22]=solve( stress ==A∗strain,e11,e12,sigma22);A=(2∗Q1∗h1+2∗h2∗(Q2−R))/(2∗h1+2∗h2);

A11=A(1,1)A12=A(1,2);A13=A(1,3);A21=A(2,1);A22=A(2,2);A23=A(2,3);A31=A(3,1);

A.4 Energy criterion 27

A32=A(3,2);A33=A(3,3);

%critical stress and strain

sigma22=eval(sigma22);

sigma22_2=sigma22∗Omega22_2/L_1_2∗(1−(cosh(sqrt(L_1_2)∗x2_s∗s))/(cosh(sqrt(L_1_2)∗s)));sigma22_2_e22_E2=sigma22_2/e22/laminate.material2.E2;stresscriterion =sigma22_2_e22_E2;

end

A.4 Energy criterion

function [ energycriterion ]= energycriterion ( laminate ,D, factor ,gamma)

%Calculating the compliances matrices in global axisS1=laminate. material1 .compliancematrix_Glob;S2=laminate. material2 .compliancematrix_Glob;

Q1=laminate.material1 . stiffnessmatrix_Glob ;Q2=laminate.material2 . stiffnessmatrix_Glob ;

%Calculating the dimensionless ratio defining the stacking sequencechi=laminate .geometry.chi ;eta=laminate .geometry.eta ;

%Layer thicknessh1=laminate.geometry.h1;h2=laminate.geometry.h2;

%Parameter s=h2/Ds=h2/D;

%ConstantsG23_1=laminate.material1 .G23;G23_2=laminate.material2 .G23;G13_1=laminate.material1 .G13;G13_2=laminate.material2 .G13;K2=(3∗G23_1∗G23_2)/(h2∗G23_1+(1+(1−eta)/2)∗eta∗h1∗G23_2);K1=(3∗G13_1∗G13_2)/(h2∗G13_1+(1+(1−eta)/2)∗eta∗h1∗G13_2);a1=−(S1(1,2)+chi∗S2(1,2)) /( S1(1,1)+chi∗S2(1,1) ) ;L_1_2=(K2/h1)∗(S1(2,2)+chi∗S2(2,2)+a1∗(S1(1,2)+chi∗S2(1,2)) ) ;L_2_2=(K1/h1)∗(S1(3,3)+chi∗S2(3,3));alpha_1_2=(1/chi )∗(Q2(2,2)∗(S1(2,2)+a1∗S1(1,2))+Q2(1,2)∗(S1(1,2)+a1∗S1(1,1))) ;alpha_2_2=(1/chi )∗(Q2(3,3)∗S1(3,3) ) ;l_2_1=h2∗sqrt(L_1_2);l_2_2=h2∗sqrt(L_2_2);

%assuming doubling of cracksD1=D;D2=factor∗D;i=0;for D=[D1,D2]i=i+1;%IDEFL_22_2=1−(1−(D/l_2_1)∗tanh(l_2_1/D))/(1+alpha_1_2∗(D/l_2_1)∗tanh(l_2_1/D));L_66_2=1−(1−(D/l_2_2)∗tanh(l_2_2/D))/(1+alpha_2_2∗(D/l_2_2)∗tanh(l_2_2/D));

Omega22_2=(K2/h1)∗(1+chi)∗(S1(2,2)+a1∗S1(1,2));%Components of the matrix RR11=(Q2(1,2)^2)/Q2(2,2)∗L_22_2;R22=Q2(2,2)∗L_22_2;R12=Q2(1,2)∗L_22_2;R66=Q2(3,3)∗L_66_2;

R=[R11 R12 0;R12 R22 0;0 0 R66];

28 Chapter A. MATLAB codes

syms e11 e22 e12 sigma11 sigma22 sigma12 A11 A12 A13 A21 A22 A23 A31 A32 A33sigma11=0;sigma12=0;A=[A11 A12 A13; A21 A22 A23; A31 A32 A33];strain =[e11; e22; e12];stress =[sigma11; sigma22; sigma12];[e11,e12,sigma22]=solve( stress ==A∗strain,e11,e12,sigma22);

A=(2∗Q1∗h1+2∗h2∗(Q2−R))/(2∗h1+2∗h2);A11=A(1,1); A12=A(1,2); A13=A(1,3);A21=A(2,1); A22=A(2,2); A23=A(2,3);A31=A(3,1); A32=A(3,2); A33=A(3,3);

R;e22=1;e11=eval(e11);e12=eval(e12);

o=[e11;e22;e12];b( i )=o’∗R∗o;end

numerador=(D2−D1);denominador=(b(2)−b(1))/laminate . material2 .E2;sigma22_2_e22_E2=1/(gamma∗sqrt(numerador/denominador));

energycriterion =sigma22_2_e22_E2;end

List of Figures

2.1 Global axis 32.2 Representative segment 4

5.1 Stresses between cracks 135.2 Critical strain for damage release (stress criterion) 145.3 Critical strain for damage release (energy criterion continuous formulation) 145.4 Critical strain for damage release (energy criterion). γ = 0.4 155.5 Critical strain for damage release (energy criterion). γ = 0.6 155.6 Critical strain for damage release (energy criterion). γ = 0.8 165.7 Critical strain for damage release (energy criterion). γ = 1 165.8 Comparison between stress criterion and energy criterion 175.9 Critical strain to get D=1 as a function of angle of lamina 1 17

29

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