Development of Methods and Tools for the Analysis of Composite Aerospace Structures

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Universit`

a degli Studi di Pisa

Scuola di Ingegneria

Dipartimento di Ingegneria Civile e Industriale Sezione Aerospaziale

Development of Methods and

Tools for the Analysis of

Composite Aerospace

Structures

Tesi di Laurea Magistrale in Ingegneria Aerospaziale

Candidato:

Federico Bovecchi

Relatori: Prof. Daniele Fanteria Prof.ssa Luisa Boni Ing. Federico Danzi Maggio 

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Ai miei genitori. Avete sempre fatto di tutto per non farmi mancare niente.

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Abstract

Carbon fibre reinforced plastics, though carrying excellent specific proper-ties, present also new and not yet completely understood damaging pro-cesses. Delaminations, above all the damage mechanisms, pose a serious threat to aerospace structures, because of their difficult detectability and the associated strength reduction. Taken into account the prohibitive costs of extensive experimental campaigns, the development of reliable numer-ical tools, capable of correctly modelling the onset and propagation of delaminations, is the key for lighter and safer structures.

The aim of this work is to expand the knowledge on innovative techniques for the simulation of delaminations. The interlaminar behaviour is simu-lated using Abaqus cohesive elements endowed with a user written mate-rial subroutine (UMAT), that implements the constitutive cohesive zone model. Standard low velocity impact tests on multi-directional laminates are simulated and results in terms of forces and delaminated areas are assessed. Non-linear post-buckling analyses on damaged wing stiffened panels are performed in order to determine the effects of skin-to-stiffener debonding. A simple procedure for injecting artificial delaminations is pre-sented. Ultimately, it is shown that the adopted techniques well predict delaminations caused by impact events onset and their propagation under static compression loads.

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4.4.3 Model verification . . . 49 4.5 Model overview . . . 49 4.5.1 Skin . . . 49 4.5.2 Stringers . . . 51 4.5.3 Boundary conditions . . . 52 4.5.4 Loads . . . 52 4.5.5 Material allowables . . . 54 4.6 FE Model details . . . 54 4.6.1 A parametric model . . . 54 4.6.2 Scripts arrangement . . . 55 4.6.3 Parts . . . 56 4.7 Results . . . 60

4.8 Results: panel 19 with complete debonding . . . 62

4.8.1 Panel’s behaviour . . . 62

4.8.2 Stringer’s buckling . . . 62

4.9 Results: panel 19 with partial debonding . . . 66

4.9.1 Panel’s behaviour . . . 66

4.9.2 Stringer’s buckling . . . 66

4.10 Results: panel 29 with complete debonding . . . 67

4.11 Conclusions . . . 69

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List of Figures

2.1 Micrograph of a quasi-isotropic laminate section . . . 4

2.2 Micrograph of a unidirectional laminate section . . . 5

2.3 Modelling of plies and cohesive layers . . . 7

2.4 Cohesive bi-linear constitutive law for pure modes. . . 9

2.5 Cohesive bi-linear constitutive law at a given mode-mix. . . 13

3.1 Testing machine. . . 19

3.2 LVI FE model side view. . . 20

3.3 LVI FE model upper view. . . 21

3.4 Detail of the hinges connecting the point mass (green dot), representing the sleeve, to the frame. . . 22

3.5 FE model of a rubber clamp . . . 24

3.6 Tie and mesh transition influence: the models . . . 29

3.7 Tie and mesh transition influence: per-cent difference, in terms of stress component S11, between the two models, for all four layers. . . 30

3.8 Tie and mesh transition influence: per-cent difference, in terms of stress component S22, between the two models, for all four layers. . . 31

3.9 Mesh-sensitivity analysis: model overview . . . 32

3.10 Mesh-sensitivity analysis: impactor’s mesh . . . 33

3.11 Mesh-sensitivity analysis: plies’ mesh . . . 34

3.12 Mesh-sensitivity analysis: cohesive layers’ mesh . . . 34

3.13 LVI, force vs time as measured in the connector linking the dart to the rest of the impactor. . . 37

3.14 LVI, overview of delaminations when they’ve reached their maximum value during the impact. . . 37

3.15 LVI, delaminations in each of the fifteen cohesive layers . . 38

3.16 Damage each layer endured during the impact. Layer 15 is the closest to the impact side. . . 39

4.1 Overall isometric view of the 18-bays outer wing box (OWB) 41 4.2 View of the semi-wing box bays . . . 41

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LIST OF FIGURES vi

4.3 Top view of the OWB . . . 43

4.4 View of bay 19 panel . . . 44

4.5 View of bay 29 panel . . . 45

4.6 View of the hot zone (panel 19) . . . 45

4.7 Modelling details of the hot zone . . . 46

4.8 View of the “cold zone” (panel 29) shown without rendering shells’ thickness. . . 46

4.9 Detail of the cold zone (panel 19), shown with shells’ thick-ness rendering . . . 47

4.10 Comparison between first eigenvalues and eigenmodes of the all-shell model (left) and the “two-region” model (right) . . 50

4.11 Stringers . . . 52

4.12 Boundary conditions overview . . . 53

4.13 Cold skin geometry and partitions, upper view (panel 29) . 56 4.14 View of the hot skin sketch and subsequent extrusion (panel 29). . . 57

4.15 Detail of the hot stringer cross-section (panel 29) . . . 58

4.16 Upper view of the adhesive layer . . . 59

4.17 View of a “complete” debonding . . . 61

4.18 View of a “partial” debonding . . . 61

4.19 Compressive force vs. applied displacement for panel 19 with “complete” debonding. The relationship is practically linear because stringer collapses before large global buckling occurs. . . 62

4.20 Panel 19 with “complete” debonding. View of the panel deformation at the end of the simulation. . . 63

4.21 Panel 19 with “complete” debonding. View of the stringer’s flange buckling: black regions exceeded 3000 µ of strain. . 63

4.22 Panel 19 with “complete” debonding. Qualitative view of the stringer’s flange buckling. . . 64

4.23 Panel 19 with “complete” debonding. Longitudinal com-pressive strain of the more critical elements of the stringer’s flange. . . 64

4.24 Panel 19 with “complete” debonding. View of the stringer’s web buckling: black regions exceeded 3000 µ of strain. . . 65

4.25 Panel 19 with “complete” debonding. Delamination status at the end of the simulation. . . 65

4.26 Panel 19 with “partial” debonding. Compressive force vs. applied displacement relationship. . . 66

4.27 Panel 19 with “partial” debonding. Configuration of the panel at the end of the simulation. . . 67

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LIST OF FIGURES vii

4.28 Panel 19 with “partial” debonding. Collapse of the cen-tral stringer web and flange. Regions in black exceeded the allowable value of 3000 µ. . . 68 4.29 Panel 19 with “partial” debonding. ‘Strain - applied

dis-placement’ relationship of the more critical elements of the stringer’s web. . . 68 4.30 Panel 19 with “partial” debonding. Delamination front at

the end of the simulation. . . 69 4.31 Panel 29 with “complete” debonding. View of the buckled

configuration at the end of the simulation. . . 69 4.32 Panel 29 with “complete” debonding. View of the strain

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List of Tables

3.1 LVI material allowables . . . 25 3.2 LVI cohesive properties . . . 26 3.3 Scheme of the most important characteristics of the nine

simulations performed. . . 36 4.1 Fundamental properties of the material. . . 54 4.2 Material allowables. Elastic moduli E11 and E22 shown in

the table are the mean values between their respective trac-tion and compression counterparts. . . 54 4.3 Cohesive properties. These are the allowables that will be

implemented in the UMAT routine. Only shear strength is shown because of the assumption of equal shearing and tearing strengths (i.e. τ_s0 = τ_t0 = τ_shear0 ). . . 55

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Chapter 1

Introduction

Excellent specific properties and no fatigue or corrosion problems are among the many reasons carbon fibre reinforced plastics (CFRPs) have drawn so much interest from the aerospace industry and academic world. But this innovative high-performance material also brings new and not yet completely understood damage processes. A major problem of composite structures is damage tolerance, and of all the possible damaging mecha-nisms, delamination is the the most dangerous. Delaminations, that are partial detachments of two adjacent plies, are decidedly difficult to detect and they significantly decrease structural strength. Delaminations can be caused by impacts during operational service or by improper manufactur-ing processes.

The need to meet strict damage tolerance requirements, without a deep understanding of the damage mechanisms, would lead to unacceptably heavy structures. Experimental research is the accepted way to demon-strate compliance to damage tolerance regulations. Numerical simulations, however, can help contain the costs of test campaigns by giving precious information on the damage processes and by helping individuate the best possible design. Ultimately, taken into account the prohibitive price for extensive experimental campaigns, the development of reliable numerical tools, capable of correctly modelling the onset and propagation of delami-nations in CFRPs, is the key for lighter structures.

A valid approach to numerical simulation of delaminations can be de-veloped within the framework of damage mechanics, using cohesive damage zone models [2] that can be implemented in finite element codes. Cohesive elements, placed between layers of solids that model composite material plies, can effectively simulate delamination processes. In order to have direct control on the main cohesive parameters, especially in mixed-mode conditions, a user defined material routine (UMAT) endows Abaqus

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CHAPTER 1. INTRODUCTION 2

sive elements with a properly designed bilinear traction-separation law, as will be extensively discussed in Chapter 2. Several authors have assessed the capability of this method to predict delamination onset and propaga-tion [1][5][6][10][11].

The objective of the present thesis is to further increase the experience and the knowledge of this technique.

In Chapter 2 the cohesive zone model is presented, along with details on how the constitutive model is implemented in Abaqus cohesive elements via a UMAT subroutine. The bi-linear traction-separation law is extensively described, first for single-mode loading and then for the more complex mixed-mode case.

In Chapter 3 standard low velocity impacts (LVIs) [12] on multi-directional composite laminates are simulated. However, before the actual simulation, preliminary analyses assess the validity of the adopted modelling technique and mesh-sensitivity studies are performed in order to improve both con-vergence time and results reliability compared with other studies [1][13]. New approaches are developed in order to make modelling processes with cohesive elements faster, simpler, and more flexible. Contact force vs. time and in-plane delaminated areas are assessed and compared with ex-perimental data gathered during LVI tests.

In Chapter 4, with the objective of validating key technologies for a next generation regional aircraft, linear and non-linear buckling analyses are performed in order to assess skin-to-stiffener debonding effects on the outer wing box stiffened panels. A comprehensive parametric Python script, used to automate the modelling process, is presented. A new, simple, and reliable method of injecting artificial delaminations is described. Details are given on the way the FE models are built using innovative modelling approaches. Results are then presented and commented.

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Chapter 2

Simulation of delaminations

using FE codes

2.1 Introduction

We already mentioned in the introduction that delamination, or interfa-cial cracking between composite layers, is one of the most common types of damage in laminated fibre-reinforced composites due to their relatively weak interlaminar strengths. Delamination is often a significant contribu-tor to the collapse of a structure. The reason for such a “weak link” be-tween the plies is readily understood by looking at micrographs of CFRP sections (Fig. 2.1). Resin-rich strips are interposed between every ply and, since resin mechanical performances are notoriously lower than those of the plies, these regions constitutes the “nest” for delaminations onset and propagation. The width of these resin-rich strips vary with the ma-terial, the lamination processes, but also with the orientation of the two plies being laid down together. Strips between same-orientation plies are narrower than strips between plies with a high difference in orientation (see again Fig. 2.1). Between same-orientation plies resin-rich strips are absent or not easily identified (see Fig. 2.2): this is why in LVI simula-tions most of the delaminasimula-tions occur between plies with high difference in orientation, as we will see in Chapter 2.

Our goal, then, is to model the behaviour of a relatively thin cohesive layer in a composite laminate in order to predict delamination onset and growth. Methods such as Linear Elastic Fracture Mechanics (LEFM), Vir-tual Crack Closure Techniques (VCCT), J-integral method, virVir-tual crack extension and stiffness derivative prove to be effective. However difficul-ties are encountered when these techniques are implemented using finite element codes. Another approach to the numerical simulation of delamina-tions can be developed within the framework of Damage Mechanics, using cohesive damage zone models. This technique can be easily implemented in

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CHAPTER 2. DELAMINATIONS IN FE CODES 4

Figure 2.1: Micrograph of a quasi-isotropic laminate section. The laminate stacking sequence is [90, 45, -45, 0, 0, -45, 45, 90]. Ply thickness: 0.150 mm. Rich-resin “strips” between the plies are clearly visible. In the central-upper part of the micrograph a resin pocket is also visible, it probably generated during lamination between the two 0o central plies; defects like this one are considered normal. The material is: TENAX J HTA 5231 6K carbon fibre; CYCOM 985 epoxy resin.

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Figure 2.2: Micrograph of a unidirectional laminate section. Four 0.150 mm plies are shown. In case of a unidirectional laminate, resin-rich strips between plies are absent or not easily identified. The material is: TENAX J HTA 5231 6K carbon fibre; CYCOM 985 epoxy resin.

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finite element codes using cohesive elements (COH3D8) and will be exten-sively explained in the following paragraphs. One of the major advantages of cohesive elements over other techniques for modelling delaminations in composites is their capability to effectively capture the inception of new delamination fronts (onset) in a pristine laminate.

Cohesive elements, placed between composite material plies, can effec-tively simulate delamination processes in standard FE analyses. Indeed, the micro-structural configuration depicted in Figs. 2.1 and 2.2 are trans-lated in the FE model as shown in Fig. 2.3: a “thin” layer of cohesive elements (COH3D8) realizes the cohesive layer, tying together two plies, made up of solid (SC8R) or continuum shell (SC8R) elements (depending on the particular model).

In order to have a direct control on the main cohesive parameters, espe-cially in mixed mode conditions, a user defined material routine (UMAT) is used to assign a bi-linear traction-separation constitutive law to Abaqus cohesive elements. The definition of a dedicated traction-separation law permits to use state-of-the-art mixed-mode formulations. A thermody-namically consistent damage model for the simulation of progressive de-lamination under variable mode ratio is used.

In this chapter the cohesive zone model is first introduced. After that, traction-separation constitutive model, first for pure mode and then for mode-mix loading, are thoroughly explained.

2.2 Cohesive Zone Model

Accurate numerical simulations of delaminations can be challenging. In this context, the cohesive zone model (CZM) implemented in FE codes with appropriate elements (cohesive elements) and suitable material models is a powerful tool to predict both the onset and propagation of delaminations. Cohesive finite element formulation, based, as mentioned, on the Cohe-sive Zone Model (CZM) approach, is one of the most commonly used strat-egy to investigate interfacial fracture. CZM correlates the micro-structural failure mechanism to the continuum fields governing bulk deformations. It assumes that a process (softening) zone is located ahead of the crack tip. Within this zone interfacial stresses, τ (referred to as tractions) are correlated with relative displacements of the surfaces, δ (referred to as dis-placement jump) according to the “traction/disdis-placement jumps” relation, which is characterized by numerical parameters and functions to define both the onset and the evolution crack criteria.

In a general mixed mode load case, the length of this process zone, lcz, is defined as the distance from the crack tip to the point where the

maximum cohesive traction is reached. The actual length of the cohesive zone is not easily verifiable, but all the models available in literature have

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Figure 2.3: Modelling of plies and cohesive layers. A layer of solid (C3D8) or continuum shell (SC8R) elements realizes the ply (in blue). A layer of cohesive elements (COH3D8) realizes the cohesive layer (in red), tying together two plies. A thickness of 0.01 mm is chosen for the cohesive layers: small enough not to alter laminate compliance, but big enough to be handled using the graphical user interface.

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CHAPTER 2. DELAMINATIONS IN FE CODES 8 this form [6]: lcz = M E2 Gc (τ0)2 (2.1) where E2 is the transverse Young modulus of the material, Gcis the

criti-cal energy release rate, τ0 is the maximum interfacial strength, and M is a

parameter that depends on the particular model (by many chosen as ap-proximately one). Typically, for carbon-epoxy composite materials, lcz can

go down to a millimetre or less. Considering that at least three cohesive elements must fall into this zone1, it is clear the computational challenge one must face when dealing with particularly big models.

In conclusion, the CZM can be combined with FE methods to develop interfacial cohesive elements. Such elements, placed between composite material plies, can effectively simulate delamination processes in standard FE analyses. In order to have direct control on the main cohesive pa-rameters, especially in mixed mode conditions, a user defined material Fortran routine (UMAT) is used to assign a bi-linear traction-separation constitutive law to Abaqus cohesive elements. Another advantage of using a UMAT subroutine is the possibility to initialize the cohesive constitu-tive behaviour according to the procedures that will be explained in the following sections.

2.3 Single mode loading

2.3.1 Traction-separation law

The behaviour of cohesive layers can be accurately simulated by defining an appropriate separation law. The simplest possible traction-separation constitutive model is the bi-linear one [5] [6] [1], given its low number of easily defined parameters.

Focusing our attention on the elastic part of the constitutive model, the most general relationship between traction vector τ and the displacement jump vector δ can be written as follows:

τ =    τn τs τt    =   Knn Kns Knt Ksn Kss Kst Ktn Kts Ktt  =    δn δs δt    = δ (2.2)

where the subscripts correspond to the three modes: opening (n), shear-ing (s), and tearshear-ing (t), and the terms in matrix K are intended as the equivalent stiffness coefficients of the cohesive element material.

1

A proper traction distribution must be reproduced in the cohesive zone, thus the FE spatial discretization (cohesive element size) must be less than the cohesive zone length.

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Figure 2.4: Cohesive bi-linear constitutive law for pure modes.

However it is typical to consider the modes as non-interactive and thus set to zero the off-diagonal terms:

τ =    τn τs τt    =   Knn 0 0 0 Kss 0 0 0 Ktt  =    δn δs δt    = δ (2.3)

in which case K terms must be interpreted as element penalty stiffnesses for the three opening modes.

Furthermore, equal penalty stiffnesses can be assumed:

τ =    τn τs τt    =   K 0 0 0 K 0 0 0 K  =    δn δs δt    = δ (2.4)

This choice is justified by how these penalty stiffnesses are computed. Penalty stiffness The effective elastic properties of the whole laminate depend on the properties of both the cohesive surfaces and the bulk consti-tutive relations of the plies. To obtain a successful FEM simulation using CZM, one of the conditions that must be met is that the cohesive contri-bution to the global compliance before crack propagation should be small enough to avoid the introduction of a fictitious compliance to the model. For instance, considering the transverse direction of a laminate, as shown

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in [6], the effective elastic properties of the composite will not be affected by the cohesive surface whenever:

K > αE3 t M P a mm , α ≈ 50 (2.5)

where t is the thickness of the sub-laminates tied together by the cohesive layer. Typical values of K range from 200,000 to more than a million. However, large values of penalty stiffnesses may cause numerical problems, such as spurious oscillations of the tractions. Thus, the interface stiffness should be high enough to provide a reasonable stiffness but small enough to reduce the risk of numerical problems. In the context of this study a value of 300,000 [M P a/mm] has been chosen.

In conclusion, the choice of considering Knn = Kss = Ktt = K is

justified by the following reasoning. Initial stiffness coefficients must be so high, as we saw, that, if K identifies the highest of them, the model is not very sensitive to the differences between K and the real stiffness coefficients.

The relationship between tractions and displacement jumps, for the elastic part, then, can be simply written as:

τ =    τn τs τt    = K    δn δs δt    = δ (2.6)

We can say now that, while the material remains undamaged, traction-separation behaviour follows a linear elastic behaviour defined by the in-terlaminar stiffness K. Equation 2.6 represents the elastic linear loading that goes from point A to point B in Fig. 2.4.

2.3.2 Damage onset criterion

Traction in the cohesive layer grow linearly with displacement jump until a damage criterion is met. Damage initiates when traction reaches single mode interfacial strength, τ0 (point B in Fig. 2.4). The corresponding displacement is the onset displacement jump, δ0.

   τ_n0 τ_s0 τ_t0    = K    δ_n0 δ0_s δ0_t    (2.7) 2.3.3 Softening curve

If the displacement jump is increased beyond δ0 the cohesive traction lin-early decreases to zero (softening curve BD). A damage variable that ac-counts for damage state needs to be defined and it must keep track of

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damage state during the analysis. This scalar damage variable, d, pro-gressively reduces the initial penalty stiffness by a factor of (1 − d). In other words, the material experience a softening process. The higher the maximum displacement jump reached, the more compliant the cohesive element is. Taking also into account this softening part of our constitutive law, the constitutive equations become:

   τn τs τt    = (1 − d)K    δn δs δt    (2.8)

where d is assumed to be zero until δ_n,s,tmax≤ δ0

n,s,tand one if δn,s,tmax= δ f n,s,t.

Considering opening mode, let’s suppose that δ_nmax reached the dis-placement jump corresponding to point C in Fig. 2.4, δ_nC. Now let’s reduce the jump from δn= δnC to δn= 0: the traction will follow the straight line

C-A, corresponding to a penalty stiffness of (1 − d)K. Increasing again δn

will have the traction follow the straight line A-C.

When the displacement jump is increased beyond the failure displace-ment jump, δf, damage variable d becomes 1, interface failure occurs and, consequently, zero traction are transmitted, point D in Fig. 2.4, unless the element is loaded in compression. Indeed, in the case of opening mode, neg-ative values of δn implicates interpenetration, which does not occur, since

compressive stiffness remains active when cohesive elements are damaged. To account for material contact and to avoid the update of d when δn< 0,

the constitutive equations for single mode loading can be expressed as:    τn τs τt    = (1 − d)K    δn δs δt    + Kd    h−δ_ni 0 0    (2.9) where hxi = ( x, if x ≥ 0 0, if x < 0 (2.10)

is the Macauley brackets.

The area under the softening part of the traction-separation curve (line B-D in Fig. 2.4) represents the work needed to create a new delaminated surface, that is, the critical fracture toughness, Gn,s,tc of the material at

the considered pure mode:

Gc=

δfτ0

2 (2.11)

For opening and shearing modes standard methods exists for the assess-ment of fracture toughness [7] [8] [9]. Interlaminar failure is assumed to occur when the energy release rate G reaches the critical fracture toughness Gc.

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In conclusion, four parameters, of which three independent, are neces-sary to completely define the traction-separation law:

• penalty stiffness, K • interfacial strength, τ0

n,s,t

• failure displacement jump, δ_n,s,tf • fracture toughness, Gc

n,s,t.

2.3.4 Damage value storing

Since damage is an evolutionary irreversible process, damage rate must be non-negative. In cohesive elements damage remains constant (i.e. damage rate is zero) when displacement jumps diminish and tractions decrease (unloading). To keep track of damage amount, maximum displacement jump occurred is stored in an internal variable:

δ_nmax= max (δ_nmax, hδni) (2.12)

δmax_s,t = max (δ_s,tmax, |δs,t|) (2.13)

Now all the quantities necessary to define the damage variable have been introduced and d can be expressed as:

     d = 0, if δmax≤ δ0 d = δf(δmax− δ0) δmax(δf − δ0) , if δmax > δ0 (2.14)

2.4 Mixed mode behaviour

Composite delaminations occur at the thin cohesive layer interposed be-tween two plies, thus they are constrained to propagate in a specified plane, where toughness of the material is low with respect to that of ply material. For this reason in most cases propagation happens under a combination of the three basic modes: opening, shearing, and tearing. It is necessary, then, to consider mode mixity and to establish appropriate criteria to ac-count for it.

2.4.1 Traction-separation law

In mixed-mode conditions the traction-separation law is more conveniently expressed in terms of stress norm τ :

τ = q

τ2

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Figure 2.5: Cohesive bi-linear constitutive law at a given mode-mix.

and equivalent displacement jump, λ: λ = q hδni2+ δ_shear2 , δshear= q δ2 s+ δ2t (2.16)

The bi-linear cohesive law is shown in Fig. 2.5 for a fixed mixed-mode ratio.

2.4.2 Onset criterion

Assuming equal shearing and tearing strengths (i.e. τ0

s = τt0 = τshear0 ),

damage onset is predicted by means of a quadratic criterion: hτni τ0 n 2 + τshear τ0 shear 2 = 1 (2.17) with: τshear= p τ2

s + τt2. Using this definition of τshear, the expression of

the stress norm becomes: τ =

q τ2

n+ τshear2 (2.18)

From Eq. 2.18 and Eq. 2.17 it is possible to define the onset stress norm as: τ0 = r (τ2 n+ τshear2 ) _satisfy (2.17) (2.19)

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From equation 2.17 it’s possible to obtain the onset equivalent displace-ment jump, λ0 [10]: λ0=          δ_n0δ_s0 s 1 + β2 (δ0 s)2+ (βδn0)2 , if δn> 0 δ_shear0 , if δn≤ 0 (2.20) where: β = δshear δn (2.21)

2.4.3 Mixed mode propagation

For a given mode-mix we already defined the onset strength norm (Eq. 2.19) and the onset displacement jump norm (Eq. 2.20). Now a propaga-tion criterion needs to be established.

The criterion used to predict delamination propagation under mixed-mode loading conditions is usually established in terms of the energy re-lease rate and fracture toughness. It is assumed that when the equivalent energy release rate, Geq, exceeds a critical value (the equivalent critical

energy release rate, Gc_eq) delamination will grow. The criterion proposed by Benzeggagh and Kenane will be used here, under the assumption that Gc_t= Gc_s: Gc_eq(B, η) = Gc_n+ (Gc_s− Gc_n)Bη (2.22) with: B = Gs+ Gt Gn+ Gs+ Gt (2.23) and η being a mixed-mode interaction parameter. The values of toughness Gc_n, Gc_sand of η can be evaluated by means of standard tests such as Dou-ble Cantilever Beam (DCB) [7], End-Notched Flexure (ENF), and Mixed Mode Bending (MMB) [9].

The value of failure displacement jump norm, λf, is finally computed

observing that, for a given mixed-mode ratio, the area under the traction-separation curve represents the interlaminar critical fracture energy:

λf(B, η) =

2Gc_eq(B, η) Kλ0

(2.24) The traction-separation law for a generic mode-mix is now completely defined.

2.4.4 Mixed mode damage tracking

The same reasoning discussed in paragraph 2.3.4 applies here, with the only exception that the displacement jump norm, λ, is now used in place of the pure-mode displacement jump, δ.

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2.4.5 Normal and shear strengths correlation

The proposed bi-linear constitutive law may become thermodynamically inconsistent, for a generic, arbitrary, set of model parameters, when mode-mix changes during a simulation. According to [11], to avoid such a possi-bility, pure mode maximum tractions must respect the following relation-ship: τ_n0 = τ_shear0 s Gc_n Gc s (2.25) 2.4.6 Viscous regularization

Tangent stiffness matrix discontinuities may cause convergence problems with implicit integration schemes, especially when used in combination with algorithms that control time increments automatically. A mitigation of such problems is offered by numerical viscous regularization that may be applied to the damage evolution (see [5] for details). A small viscous regularization eliminates the discontinuity of the traction-separation rela-tionship (point B in Fig. 2.5) without compromising results. Therefore, this expedient will be used whenever possible to facilitate convergence.

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Chapter 3

Low Velocity Impacts (LVI)

3.1 Introduction

LVIs are a dangerous phenomenon when it comes to aerospace composite structures. Many authors showed that significant internal damage can be produced without any visible mark remaining on the outside of the com-ponent. LVIs trigger delaminations (which are prone to occur in interfaces between plies with different fibre orientation), remarkably degrading the tensile and the compressive strengths. LVIs can be caused by improper maintenance or inadequate operational practices such as a tool acciden-tally dropped on the structure during inspection.

Definition of LVI. Although impacts are classified into low-, high- and hyper-velocity impacts, the velocity alone cannot be used to exactly discern between these categories since other factors play a fundamental role in determining the effects produced by the impact:

• impactor’s mass; • impactor’s size; • target stiffness;

• target material properties; • target damping properties; • boundary conditions.

Rather than using impactor velocity to classify impacts, it is preferable to use a criterion based on damage occurred. Following this criterion one could say that LVIs are impacts that cause mostly delaminations and matrix cracking, often without leaving clearly visible indentations on the surface.

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CHAPTER 3. LVI 17

Such small indentations are particularly difficult to detect during inspec-tion and can fall in the category of BVID (Barely Visible Impact Damage). In terms of airworthiness requirements, BVIDs define the threshold for re-liable detection, hence, they must not reduce the strength of the structure below its ultimate load capability. For this reason safety factors are intro-duced to take BVIDs into account, leading to significant weight penalties and reduced structural efficiency of composite structures.

LVI simulations. LVIs simulations can provide a useful tool for study-ing the onset and propagation of delaminations. Sensitivity analyses can be performed in order to assess the effects of laminate stacking sequences, material properties, damping properties, and other parameters, providing precious information for the design process and helping reduce the number of costly experimental tests.

3.2 Testing machine specifics

In order to assess the accuracy of the FE simulations, results are compared with experimental data, available in [13]. Our FE model accurately repro-duces the experimental test configuration depicted in Fig. 3.1 and here briefly described. Impact tests have been conducted following the guide-lines depicted in the ASTM D7136 - Measuring the damage resistance of a fibre-reinforced polymer matrix composite to a drop-weight impact event [12]. The testing machine is composed by various parts.

• Impactor. It consists of a steel dart attached through a piezoelectric sensor to an aluminium frame. Two recirculating ball sleeves guide the impactor sliding down two guide rails. Guide rails axes are 400 mm apart and high enough to permit drop-weight testing for a wide range of impact energy levels. The total mass is 1.25 kg, most of which is concentrated in the lateral sleeves (0.5 kg each). The fact that most of the mass is far from the central impactor (see again Fig. 3.1) has tangible consequences on the impact dynamics. The correct, although simplified, modelling of the impactor in all its parts is essential in order to obtain accurate results.

The hemispherical part of the dart has a radius of 6.35 mm.

• Supporting fixture. It consists of a 20 mm thick steel plate with a 75 x 125 mm central hole and supported by four steel columns. The 100 x 150 mm specimen is placed over the hole; three guiding pins are properly located so that the specimen is centrally positioned over the cut-out (in Fig. 3.1 one of them is visible in the bottom-right corner of the specimen).

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CHAPTER 3. LVI 18

• Rubber clamps. They act on the top surface of the specimen and hold the specimen down with a specified force during the impact (in red in Fig. 3.1).

• A laser photoelectric sensor permits to measure the instants when the impactor enters in and exits the laser beam (both prior to the im-pact and after the rebound), allowing for the possibility to precisely compute the final speed of the striker tip.

3.3 FE model

Overviews of the FE model are shown in Figs. 3.2 and 3.3.

3.3.1 Impactor

Frame

The aluminium frame is modelled with 2-node linear beam elements (B31). The 8 x 8 mm square cross-section of the beams has been carefully chosen in order to recreate the correct weight distribution of the frame, though maintaining a simple geometry. Each lateral sleeve has been modelled as a point mass situated on the axis of the guiding rails (green point in Fig. 3.4); equivalent rotational inertia of the sleeves has been taken into account and assigned to the point masses.

Connection between sleeves and aluminium frame

First a rigid body between the three reference points shown in Fig. 3.4 is created. The “RP left” reference point is free to translate along the Z direction and to rotate around the Z axis; the other two RPs will act accordingly. Then, both “RP Frame” reference points have been connected to the respective nodes on the frame via a connector of the type hinge, to correctly reproduce the behaviour of the connection using bolts (Fig. 3.1).

Dart

As shown in Fig. 3.2, only the final part of the actual impactor is modelled. The dart is connected to the aluminium frame via a connector of the type translator. Such connector generates a cylindrical coupling: no relative rotations or translations between dart and frame are possible, with the exception of the translation along the linking axis. A spring and a dashpot operate along the translation direction: the spring has a stiffness of 880 kN/mm (extremely stiff) and the dashpot coefficient is 3.4. This modelling

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CHAPTER 3. LVI 19

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CHAPTER 3. LVI 20 Figure 3.2: L VI FE mo del side view.

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CHAPTER 3. LVI 21 Figure 3.3: L VI FE mo del upp er view.

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CHAPTER 3. LVI 22

Figure 3.4: Detail of the hinges connecting the point mass (green dot), representing the sleeve, to the frame.

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CHAPTER 3. LVI 23

artifice allows for the user to easily obtain the force interchanged between dart and specimen.

Drop mechanism

In the real test the impactor is taken to a predefined height and dropped. The FE simulation does not reproduce the impactor’s fall: it would be a waste of computational time and power. Instead, at the beginning of the analysis, the impactor is positioned 1/100 mm distant from the specimen surface and an initial velocity is assigned to it. The speed imposed on the impactor is the same speed of the real impactor prior to impact, as captured from the laser photoelectric sensor, which is 2.807 m/s (corresponding to a drop height of 0.5 m).

3.3.2 Steel support

Only the plate in contact with the specimen is modelled, not the four columns supporting it (Fig. 3.2). Hexahedral elements of type C3D8 have been used for this part. A surface-to-surface contact, with small sliding formulation1, has been chosen for the interaction between specimen and plate. Friction has been taken into account, activating the tangential behaviour with a friction coefficient of 0.3. Since the steel plate is stiffer, it is selected to be the master surface, while the bottom surface of the specimen is the slave2. Regarding boundary conditions, all of the six degrees of freedom of the plate’s lower surface are constrained.

3.3.3 Rubber clamps

For obvious reasons only the final part of the clamps are modelled, as shown Fig. 3.5. In the real test these clamps are pressed on the specimen with a specified force, this action is reproduced during the first step of the simulation. For this step a static, general solver is used, while for the impact step a dynamic, implicit solver is adopted.

1_{If only small tangential relative displacements are expected, the small sliding}

ap-proach can be used. The group of master nodes linked to each slave counterpart is established at the beginning of the analysis and remain the same for its whole duration.

2

Abaqus enforces contact condition imposing no penetration of the slave into the master. The choice of both the master and the slave surfaces depends on some practical guidelines: the surface on the stiffer body should be the master one and should have a coarser mesh.

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CHAPTER 3. LVI 24

Figure 3.5: FE model of a rubber clamp: in blue the final part of the cylindrical steel pin, in red the rubber “wrapping”.

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CHAPTER 3. LVI 25

Nominal Ext. Plies Int. Plies

E11 112,700 116,400 120,400 [M P a] E22 10,350 10,600 11,000 [M P a] E33 7,900 - - [M P a] G12 3,500 3,600 3,700 [M P a] G13 3,500 3,600 3,700 [M P a] G23 3,640 3,700 3,800 [M P a] ν12 0.32 - -ν13 0.32 - -ν23 0.42 -

-Table 3.1: Material allowables. Elastic moduli E11 and E22 shown in the

table are the mean values between their respective traction and compres-sion counterparts.

3.3.4 The specimen

Geometry

The in-plane dimensions are those required by the ASTM: 150 x 100 mm. The thickness is 2.49 mm.

Material properties

The specimen simulated is a 16 plies quasi-isotrpic laminate with the fol-lowing stacking sequence: [0◦/ ± 45◦/(90◦)2/ ∓ 45◦/0◦]s. Each ply is 0.156

mm thick. Material allowables are listed in table 3.1, while cohesive prop-erties in table 3.2.

A “two-region” model

Every FE model reaches for a compromise between results accuracy and computational efficiency. The best way to accomplish such objective is to create a “two-region” model.

• An outer region, here called cold zone, is modelled with continuum shell elements (SC8R) and a coarser mesh: two elements in the thick-ness and in-plane dimensions of 2 x 2 mm. In this region material properties are given assigning a continuum shell composite section, this choice being more flexible, since it leaves the user free to choose

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CHAPTER 3. LVI 26

Inter-laminar normal strength 30 [M P a]

Inter-laminar shear strength 56 [M P a]

Inter-laminar fracture toughness [GIC] 200 [J/m2]

Inter-laminar fracture toughness [GIIC] 700 [J/m2]

η 1.45

K 300,000 [M P a/mm]

Table 3.2: Cohesive properties. These are the allowables that will be implemented in the UMAT routine. Only shear strength is shown because of the assumption of equal shearing and tearing strengths (i.e. τ_s0 = τ_t0 = τ_shear0 ).

the number of elements in the thickness3. The cold zone only needs 6800 elements to be modelled and reliably simulates the behaviour of the laminate where no delaminations occur.

• A central region, here called hot zone. In this region every single ply is modelled with a layer of linear hexahedral elements of type C3D8 with in-plane dimensions of 1 x 1 mm. Cohesive layers, modelled with cohesive elements COH3D8, tie together the plies. Cohesive layers are 0.01 mm thick. Thus, in order to maintain the correct global laminate thickness, plies’ thickness is reduced accordingly: the first and last plies are 0.151 thick, while internal plies are 0.146 mm thick. As a consequence of this thickness reduction, plies material allowables need to be properly enhanced to restore original laminate stiffness (see table 3.1).

Hot zone in-plane dimensions (40 x 40 mm) are chosen in order to enclose all possible delaminations. Cohesive elements near the cold zone are not damageable, so that no spurious, unrealistic delamina-tion can be triggered by mesh-transidelamina-tion effects. The hot zone, given its complexity, is generated using a parametric Python script, which also automatically translates the impactor at the correct distance from the laminate top ply. This central region absorbs 49,600 out of the 63,999 total elements necessary for the entire model (77%). Cold and hot zones are tied together.

3_{Abaqus composite layup feature, instead, can be used if there is only one element in}

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CHAPTER 3. LVI 27

3.3.5 Critical review of the modelling choices

In previous paragraphs the characteristics of the FE model have been pre-sented, however a basic critical review of the main modelling choices must be made. In the following section 3.4, simulations have been performed in order to assess that an abrupt mesh transition has no significant influ-ence on the stress field. In section 3.5 mesh element’s sizes of three main components are justified.

3.4 Tie and mesh transition influence

“Two-region” modelling is a convenient strategy for obtaining precise re-sults while maintaining the lowest possible number of elements. Mesh transitions and ties, however, can influence final results. It is good prac-tice, then, to quantify these influences, at least on simple models, which don’t require excessive computational power. Indeed, for the following comparison study, a simple simulation with the following characteristics is considered:

• laminate stacking sequence: [0◦/90◦/90◦/0◦]; • same material allowables as depicted in tab. 3.1;

• four plies, each 0.204 mm thick, adding up to a total thickness of 0.816 mm;

• in-plane dimensions: 125 x 75 mm ;

• boundary conditions: simply supported on all four edges; • loads: distributed pressure as shown in fig 3.6.

The laminate is modelled using two techniques (see Fig. 3.6):

• a “two-region” model (like the one used for the LVI simulation) shown in Fig. 3.6 (top image); continuum shells (two elements in the thick-ness) for the outer region and linear hexahedral elements (four in the thickness) in the central region for a total of 10,312 elements; • a “single-region” model, generated using only linear hexahedral

ele-ments (four in the thickness), for a total of 37,800 eleele-ments, as shown in Fig. 3.6 (bottom image).

Results

Stress values reached in the 40x40 mm hot zones of the two models are compared. A Python script is used to extract stresses. Specifically, the script:

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CHAPTER 3. LVI 28

• extracts stresses as computed in the centroid of every element in the hot zones of both models; in particular it extracts the two principal stresses: in the X (S11) and Y (S22) directions;

• extracts also the centroid’s coordinates; • considers all four layers of the laminate;

• organizes a text output with all the stresses coupled with the coor-dinates of the centroid they were taken from.

We know, then, the two principal stresses’ value for both models, for all layers, and for each element. At this point it is possible to use a simple MATLAB routine that can plot the difference between the two models in terms of stress, element per element, layer per layer. For practical reasons differences are normalized with the maximum stress occurred at the considered layer and direction, as exemplified in the following equation:

∆S11 S₁₁max =     (S11)model A layer i element k     −     (S11)model B layer i element k     max (S11)model A layer i ! (3.1)

Results are shown in Figs. 3.7 and 3.8.

Displacements of the two models are almost identical. The difference in terms of maximum displacement (occurring at the center of the specimen) is less than 0.005 per cent.

Conclusions

The following conclusions can be drawn:

i. differences between the two models, although present, are small, al-ways within a few percent (6% maximum);

ii. the S22stress component manifests the biggest differences;

iii. great differences emerge at the borders, the most affected by the mesh transition and the tie enforcement;

iv. the “one-region” simulation required 55 minutes to complete, four times as much as the “two-region” simulation.

Given the negligible differences between the two models and the huge advantage, in terms of computational efficiency, of having lower element number, the choice of the “two-region” model is obvious.

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CHAPTER 3. LVI 29

Figure 3.6: Tie and mesh transition influence: two different models for the same laminate. On top, the “two-region” model (10,312 elements); beneath, the “one-region” model (37,800 elements). A central pressure (pink arrows) constitutes the loading condition.

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CHAPTER 3. LVI 30 Figure 3.7: Tie and mesh tran siti on influence: p er-cen t difference, in terms of stress comp onen t S11 , b et w e en the tw o mo dels, for all four la y ers.

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CHAPTER 3. LVI 31 Figure 3.8: Tie and mesh tran siti on influence: p er-cen t difference, in terms of stress comp onen t S22 , b et w e en the tw o mo dels, for all four la y ers.

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CHAPTER 3. LVI 32

Figure 3.9: Overview of the FE model used for mesh-sensitivity analyses.

3.5 Mesh sensitivity analysis

When it comes to FE simulations, it is of paramount importance to make sure that results are not affected by mesh size, necessity that calls for “mesh-sensitivity” analyses. The objective is to individuate the larger mesh size that is still capable of returning realistic results. When dealing with LVIs, there are three key components that could greatly affect final results, depending on the way they are meshed: the impactor, the plies and the cohesive layers. In this section a sensitivity analysis on these three components’ meshes will be presented. The outcomes of this study led to the elements’ sizes already introduced in section 3.3.

3.5.1 FE model description

Given our limited computational resources, we chose to perform this mesh-sensitivity analysis on the simple four-ply laminate already described in the previous section. The overview of the FE model is shown in Fig. 3.9. The simplified impactor is composed by the dart and a superior cylindrical part where all the impactor’s mass is concentrated (a fictitious density was used). The two components are connected via a connector and an initial velocity of 2.807 m/s is assigned to them. Boundary conditions consist of simple supports on all four edges of the laminate.

3.5.2 Mesh parameters

As already mentioned, this sensitivity analyses investigate how the mesh size of three main components affects the final result.

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CHAPTER 3. LVI 33

Figure 3.10: The three mesh sizes used to model the impactor. The nine elements actually coming into contact with the specimen have the size of 0.7 x 0.7 mm (left), 1.2 x 1.2 mm (centre) and 1.7 x 1.7 mm (right).

Impactor. Concerning the dart, only nine elements actually come into contact with the top ply of the laminate. The size of these nine elements influences the way the contact is enforced, thus altering the force exchanged between laminate and impactor. Three mesh sizes for this nine elements are considered (see Fig. 3.10): 0.7 x 0.7 mm, 1.2 x 1.2 mm, 1.7 x 1.7 mm. Plies. In the hot zone the laminate is modelled ply per ply, thus the plies mesh size greatly influences the total number of elements needed to model the specimen. Keeping a low number of elements is key for maintaining reasonable computational times. For this reasons only the central region of the plies is actually affected by the change in mesh size (see Fig. 3.11).

Three solutions are proposed:

i. homogeneous mesh size with in-plane dimensions of 1.0 x 1.0 mm; ii. central region, of the top ply only, with elements’ in-plane dimensions

of 0.5 x 0.5 mm;

iii. central region of all plies with elements’ in-plane dimensions of 0.5 x 0.5 mm.

Cohesive layers. Three mesh sizes are considered: 0.33 x 0.33 mm, 0.5 x 0.5 mm, and 1.0 x 1.0 mm (see Fig. 3.12).

3.5.3 Results

Results are summarized in Tab. 3.3. The outcomes of the simulations reveal four tendencies.

i. No matter the combination of the parameters, the main delamination (occurring between plies 2 and 3) remains almost unaltered.

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CHAPTER 3. LVI 34

Figure 3.11: The three meshing solutions adopted for the plies: no mesh refinement (left), central mesh refinement for the top ply only (central), central mesh refinement for all plies (right).

Figure 3.12: Three mesh sizes used for the mesh-sensitivity simulation: cohesive elements in-plane dimensions are 0.33 x 0.33 mm (left), 0.5 x 0.5 mm (centre), 1.0 x 1.0 mm (right).

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CHAPTER 3. LVI 35

ii. Cohesive layers’ and impactor’s mesh sizes do not have significant ef-fects on delaminations areas.

iii. Local ply mesh refinements reveal local delaminations right beneath the impactor. This information, though, comes at a great computa-tional cost (always more than four hours of simulation) and shows only small, predictable delaminations. It is expected, in fact, that right beneath the impact severe damage occurs.

iv. The fastest simulation, the one with the coarsest meshes (SIM 4 in Tab. 3.3) gives results in accordance with all the other simulations.

3.5.4 Conclusions

This study shows that ultra-refined meshes, despite having a high price in terms of computational time, do not bring real benefits to the results.

3.6 Results

In section 3.4 the “two-region” modelling has been validated. In section 3.5 element sizes of the three main components have been justified. Now that this basic critical review of the modelling choices is complete, it is possible to present the final results of the LVI simulation.

3.6.1 Force vs. Time

One of the most important result is the force exchanged between impactor and specimen. In the real impactor this force is measured with a piezo-electric sensor, in the FE model it is measured in the connector linking dart and aluminium frame. In figure 3.13 force vs. time is plotted for the experimental test and for the simulation. Good agreement is obtained both in terms of maximum force and impact duration. However, the simulation is not able to reproduce the high-frequency vibrations (around 5 kHz) visible in experimental results.

3.6.2 Delaminated areas

A correct prediction of delaminated areas is the main objective of the sim-ulation. An overview of delaminations is shown in Fig. 3.14, while Fig. 3.15 shows the damage layer per layer. As already mentioned the largest delaminations occur between plies with different orientations, namely co-hesive layers ‘c’ (between −45◦ and 90◦), ‘g’ (between 45◦ and 0◦) and ‘k’ (between −45◦ and 90◦) (see again Fig. 3.15 and Fig. 3.16).

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CHAPTER 3. LVI 36 SIM 1 SIM 2 SIM 3 SIM 4 SIM 5 SIM 6 SIM 7 SIM 8 S IM 9 P arameters Impactor mesh size [mm ] 1.7 1.2 1.7 1.7 1.7 1.2 1.2 0.7 0.7 Coh. elm. size (side length) 0.33 0.33 0.5 1.0 0.5 0.5 0.5 0.5 0.5 Ply cen tral refinemen t no no no n o y es (1 st ) y es (1 st ) y es (all) y e s (1 st ) y es (all ) Num b er of elemen ts 61,404 61,404 32,778 18,380 32,908 32,908 33,292 33,125 33,509 Delaminated areas [mm 2 ] Cohesiv e la y er 1-2 0 0 0 0 25 0 22 0 15 Cohesiv e la y er 2-3 127.5 128.4 128.5 138 128.5 129.3 122.5 130 122.7 Cohesiv e la y er 3-4 0 0 0 0 6 5.5 4.5 5.2 2 Sim ulation details Time [hr : min : sec ] 5: 03:09 5:33:07 2:16:45 1:16:40 4:18:39 4:16:30 4:20:31 4:00:00 4:36:7 T ot. n um. iterations 929 1001 648 633 1111 1026 1025 952 1097 T able 3.3: Sc heme of the most imp ortan t characte ristic s of the nine sim ulations p erformed.

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CHAPTER 3. LVI 37

Figure 3.13: LVI, force vs time as measured in the connector linking the dart to the rest of the impactor.

Figure 3.14: LVI, overview of delaminations when they’ve reached their maximum value during the impact.

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CHAPTER 3. LVI 38 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o)

Figure 3.15: LVI, delaminations in each of the fifteen cohesive layers. El-ements coloured in red are completely damaged; elEl-ements in blue did not undergo any damaging process. Cohesive layer (o) is the nearest to the impact side.

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CHAPTER 3. LVI 39

Figure 3.16: Damage each layer endured during the impact. Layer 15 is the closest to the impact side.

3.6.3 Performances

Simulations, performed on a 12-CPUs workstation with 32 GB of RAM, needed less than three hours to complete. Default Abaqus General So-lution Controls have been used. These accomplishments, remarkable for impact simulations, allow for more expedite and more reliable simulations, and constitute an important contribution to the research pursued by our department. Future researchers, for instance, will be able to complete “batches” of sensitivity studies in days, rather than in months.

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Chapter 4

Post-buckling

skin-to-stiffener debonding

of CFRP stiffened panels

4.1 Wing stiffened panels of a next generation

regional aircraft

In the last decades industries and academic research focused on developing innovative, cutting-edge technology aimed at reducing CO2, gas emissions and noise levels produced by aircraft. The objective of the present work is contributing in demonstrating and validating key technologies that will enable a 90-seat class turboprop aircraft to deliver breakthrough economic and environmental performance and superior passenger experience.

In order to decrease fuel consumption, lightness of the structure be-comes a primary objective, necessity that calls for the use of carbon fi-bre reinforced plastics (CFRP). This is the case of the outer wing box (OWB) (Fig. 4.1), which is made up of laminated composite parts. The work presented in this chapter verifies the structural strength of the OWB. Specifically, the focus is on the repercussions of embedded defects and low velocity impacts on the structure’s performances under compressive load-ing. In this chapter we converge our attention on the consequences of embedded defects.

4.2 Objectives

Skin-to-stringer debonding is a common defect that can critically reduce the performance of composite structures, especially during compressive loads. Such defects can be considered on all accounts as delaminations and compressive loading can induce their onset and unstable propagation,

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CHAPTER 4. STIFFENED PANEL: POST-BUCKLING 41

Figure 4.1: Overall isometric view of the 18-bays outer wing box (OWB) (spars are omitted for clarity of representation).

Figure 4.2: View of the semi-wing box bays. The outer wing box comprises bays 13 through 30. The inner part of the wing is realized in aluminium, while the OWB is made up of CFRP.

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further deteriorating the overall strength of the panel [1]. This chapter will focus on the skin-to-stringer debonding caused by embedded defects, i.e. lack of adhesion during fabrication due to inclusions of various nature. In most cases skin-to-stringer delaminations have the highest probability to propagate [5] and, therefore, are the most dangerous. Our objective is to study how these debondings evolve during loading and, in doing so, affect the compressive strength of the wing box panels.

4.3 Strategy

Our analyses concentrate on the upper panels of the wing box, the ones more often and more severely subjected to compressive loads. Two para-metric finite element (FE) models are generated: one for the three-stringers central panels and one for the two-stringers tip panels (see Fig. 4.3). A layer of ABAQUS cohesive elements simulates the bond between stringer and skin. A user defined traction-separation law is coded in a User-MATerial subroutine (UMAT) that endows such cohesive elements with damage initialization capabilities. An artificial delamination is injected in the adhesive layer via an initial conditions input command that pre-set the damage variable of the UMAT cohesive elements. Then, ABAQUS Standard (Dynamic Implicit solver) non-linear buckling analyses have been performed in order to assess the response of the panels.

Both models are generated using parametric Python scripts. This ap-proach allows the user to model every panel of the upper OWB within minutes. Our analyses, however, will concentrate on only two panels: three-stringer panel 19 and two-stringer panel 29 (Fig. 4.2). Panel 19 is representative of the whole OWB, since previous analyses comprising the whole wing determined that this is the first panel to show buckling instability. Panel 29, instead, has been selected among the tip panels to investigate the effects of such proximity to the spars.

4.4 Modelling choices

4.4.1 A “two-region” model

Every FE model is the result of a compromise between results’ accuracy and computational efficiency. When it comes to buckling, general purpose fully integrated S4 shell elements are often considered as a highly reliable and computationally efficient choice [3]. However, they have a major limi-tation: they are not stackable, which is quite a problem in the part of the model where delamination processes take place and a greater modelling precision is required.

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Figure 4.3: Top view of the OWB (upper skin and spars omitted for clarity of representation). It is evident the extremely low number of stringers: four in the first three root bays, three in the central ten bays and two in the tip five bays. Spars are so close that their influence cannot be disregarded.

Unlike conventional shells, continuum shell elements (SC8R) can be stacked to provide more refined through-thickness response. Stacking con-tinuum shell elements allows for a richer transverse shear stress and force prediction [4]. For greater accuracy every ply of the laminate will be mod-elled with one layer of continuum shell elements. This modelling strategy cannot be adopted for the entire panel: the number of elements would be too high, penalizing computational efficiency.

The obvious and generally adopted solution, then, is to create a model with two different regions.

• A “hot zone” (Fig. 4.6): the most comprehensive and elaborate region of the model. Here we consider every geometrical feature of the panel like, for instance, the curvature of the stringer plies linking web and flanges (Fig. 4.7a) or the tapering of the skin pad under the stringers (Fig. 4.7b). Every ply of the laminate is modelled by a layer of continuum shell elements. The adhesive layer is present only in this part of the model and not outside, due to the extremely high number of elements required to accurately simulate delamination processes. The hot zone absorbs more than 95% of the elements necessary to model the entire panel.

• A “cold zone” (Fig. 4.8): a highly computationally efficient area that embrace the hot zone. This region is made up of shell elements only

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Figure 4.4: View of bay 19 panel. In shades of blue the cold zone, made up of conventional shell elements, here shown without rendering the shell thickness. In shades of red the hot zone, where delamination processes be-tween skin and stringer are simulated. The panel is modelled with 223,323 elements, 97% of which are packed in the hot zone, and weighs 11.9 kg.

and presents a simplified geometry: no curved plies linking stringers’ webs and flanges and the skin’s pad tapering is discretized (Fig. 4.9). The result are the models shown in Fig. 4.4 for panel 19 and in Fig. 4.5 for panel 29.

4.4.2 Initial geometrical imperfection

Traditionally, the procedure for a non-linear buckling analysis contem-plates two phases. A linear eigenvalue buckling analysis is first performed for each model. Then the resulting first eigenmodes are added as an imper-fection to the geometry for the non-linear post-buckling analysis, in order to trigger the buckling of the panel. There are many ways to assign this geometrical imperfection, however the most accurate is to assign a linear combination of the first few eigenmodes. In absence of a real panel with its actual deformations, assigning an initial geometrical imperfection based on the panel’s eigenmodes is the most rational way to force the solution into a buckling behaviour.

Nevertheless, adopting such a strategy can be seen as “forcing” the solution one way or another. On this account three results need to be mentioned.

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Figure 4.5: Overview of bay 29 panel. In shades of blue the cold zone, made up of conventional shell elements, here shown without rendering the shell thickness. In shades of red the hot zone, where delamination processes between skin and stringer are simulated. The panel is modelled with 130,142 elements, 96% of which are packed in the hot zone, and weighs 3.7 kg.

Figure 4.6: View of the hot zone (panel 19). This is the most detailed and accurate part of the model: it absorbs 216,975 elements, 97% of the total number needed for the entire model.

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(a) (b)

Figure 4.7: Modelling details of the hot zone. (a) The most complex part of the model: the interaction between skin, adhesive, filler, and stringer. A precise modelling of the web-flange intersection is of paramount importance for a correct estimate of the delamination processes, since it determines the way forces between stringer and skin are exchanged. (b) A view of the skin pad’s tapering.

Figure 4.8: View of the “cold zone” (panel 29) shown without rendering shells’ thickness.

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Figure 4.9: Detail of the cold zone (panel 19), shown with shells’ thickness rendering. In the cold zone numerous geometrical simplifications have been adopted: no curved plies linking stringers’ webs with flanges and no ply-per-ply tapering of the skin pads under the stringers.

i. In every single non-linear post-buckling simulations performed, no matter the weight chosen for the eigenmodes, panels always deformed according to the first eigenmode.

ii. Analyses without initial imperfections were performed and they showed almost identical final results. In fact, even without the initial geomet-rical imperfection procedure described above, panel’s geometry is far from “perfect”: there are, indeed, at least two kinds of numerical im-perfections. First one: nodes can be placed in positions that are not precisely the ones indicated in the model (generally these rearrange-ments are quite small). Second one: nodes involved in a tie can be repositioned in order to improve performances (these relocations can be significant and must be checked in the .dat file to make sure they don’t alter the geometry).

iii. A series of analyses were performed giving as an initial imperfection only the second eigenmode, the most energetically convenient after the first one, increasing its weight at each simulation. It was proved that even with a high but reasonable weight the panels keep finding their way back to the first eigenmode-type of deformation. Only if the weight became unreasonably high (more than 0.5) the panel deforms according to the second eigenmode.

These results point out that, eventually, with a reasonable initial imper-fection, the panel always deforms in accordance with the first eigenmode

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and this certainly adds to the robustness of the solutions found. Nonethe-less these answers are not enough for a simulation where delamination processes depend on the deformation history. We assessed that the final deformation is the same, but the way the panels reach that configuration changes depending on the initial imperfection given. One “history” or another can potentially damage the adhesive more or less, changing the behaviour of the panel and eventually final results.

Although plausible, according to the analyses’ outcomes this eventual-ity does not occur. Considering all the models, each with its own initial imperfection, their histories differ only in the first few seconds of simula-tion (out of 120 total), while displacements, strains and tensions are still too small to induce delaminations. After these first few seconds all the simulations practically show the same deformation course and delamina-tions occur well after.

From a technical standpoint, geometrical imperfections are assigned adding a few lines of code to the .inp file, immediately before the steps definition. In these lines of code it’s possible to determine which eigen-modes are to be chosen for the linear combination and with which weight each must be accounted for. Eigenmodes are normalized in such a way that the maximum displacement, occurring at any one node, is equal to one millimetre. The weight assigned to each eigenmode, multiplies all the displacements: a weight of 0.1 means, then, that the maximum displace-ment will be 0.1 millimetres. The procedure is impledisplace-mented in the Python scripts and completely automated. Examples of the necessary code lines follow.

*NODE FILE, GLOBAL=YES, LAST MODE=10 U

These lines of code must be added in the .inp linear buck-ling file in order to write the first ten eigenmodes in an out-put file called .fil. This file will be read by the non-linear buckling analysis input file processor in order to generate initial geometrical imperfection of the panel.

Development of Methods and Tools for the Analysis of Composite Aerospace Structures

Universit`

a degli Studi di Pisa

Development of Methods and

Tools for the Analysis of

Composite Aerospace

Structures

Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Simulation of delaminations

using FE codes

2.1

Introduction

2.2

Cohesive Zone Model

2.3

Single mode loading

2.4

Mixed mode behaviour

Chapter 3

Low Velocity Impacts (LVI)

3.1

Introduction

3.2

Testing machine specifics

3.3

FE model

3.4

Tie and mesh transition influence

3.5

Mesh sensitivity analysis

3.6

Results

Chapter 4

Post-buckling

skin-to-stiffener debonding

of CFRP stiffened panels

4.1

Wing stiffened panels of a next generation

regional aircraft

4.2

Objectives

4.3

Strategy

4.4

Modelling choices