chapter 5 numerical analysis -...

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CHAPTER 5

NUMERICAL ANALYSIS

5.1 INTRODUCTION

The failure process of high performance composite laminates is

quite complex, involving damage mechanisms such as matrix cracking, fibre

fracture, interlaminar damage and delamination. Though there are many

failure modes, the delamination gives the measure of the damage area. Some

progress has been made lately in the development of accurate analytical tools

for the prediction of intralaminar damage growth, yet similar efficient tools

for delamination are not available, and thus delamination is generally not

considered in damage growth analysis. Without the delamination failure

mode, the predictive capabilities of progressive failure analysis will remain

limited. However delamination is one of the predominant forms of failure in

laminated composites when there is no reinforcement in the thickness

direction. It is a typical interlaminar failure mode of laminated composite

materials that may arise due to the low resistance of the thin resin-rich

interface existing between adjacent layers under the action of impacts,

transversal loads or free-edge stresses. As a result, impact can cause a

significant reduction in the load-carrying capacity of the structure.

The analysis of delamination is commonly divided into the study of

the initiation and the analysis of the propagation of an already initiated area.

Delamination initiation analysis is usually based on stress and the use of

criteria such as the quadratic interaction of the interlaminar stresses in

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conjunction with a characteristic distance. This distance is a function of

specimen geometry and material properties and its determination usually

requires extensive testing. Crack propagation, on the other hand, is usually

predicted using the Fracture Mechanics approach. The Fracture Mechanics

approach avoids the difficulties associated with the stress singularity at a

crack front but requires the presence of a pre-existing delamination whose

exact location may be difficult to determine. When used in isolation, either

the strength-based approach or the Fracture Mechanics approach is not

adequate for a comprehensive analysis of progressive delamination failure.

However, most analysis of delamination growth apply a fracture

mechanics approach and evaluates the energy release rates for self-similar

delamination growth. The energy release rates are usually evaluated using the

virtual crack closure technique (VCCT) (Rybicki and Kanninen, 1997). The

VCCT technique is based on the assumption that when a crack extends by a

small amount, the energy released in the process is equal to the work required

to close the crack to its original length. The Mode I, Mode II, and Mode III

energy release rates, GI , GII and GIII respectively, can then be computed

from the nodal forces and displacements obtained from the solution of a finite

element model. The approach is computationally effective since the energy

release rates can be obtained from only one analysis. Although valuable

information concerning the onset and the stability of delamination can be

obtained using the VCCT, its use in the simulation of delamination growth

may require complex moving mesh techniques to advance the crack front

when the local energy release rates reach a critical value. Furthermore, an

initial delamination must be defined; and for certain geometries and load

cases, the location of the delamination front might be difficult to determine.

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In recent years, a growing interest has emerged in the application of

cohesive zone models to study delamination fractures in composite materials.

Cohesive damage models presume the presence in front of the physical crack

tip of a process zone delimited by cohesive surfaces that are held together by

cohesive tractions. The cohesive tractions are related to the relative

displacements of the cohesive surfaces by a constitutive law that simulates the

accumulation of damage through progressive decohesion in the process zone.

The use of finite elements based on a cohesive damage approach allow to

overcome the main limitations of crack closure schemes. In particular, the

initiation and progression of damage are explicitly incorporated in the

formulation of the element and do not necessitate external node release

routines while requirements on mesh geometry and mesh density are less

stringent than those associated with crack closure techniques.

The delamination incurred to composite plates of various

thicknesses and ply orientation subjected to medium velocity impact with the

use of cohesive zone model is attempted. A numerical simulation was

performed using Abaqus/Explicit, transient dynamic finite element analysis

code. Finite Element models with interface elements were first calibrated and

validated by simulation of simple fracture mechanics tests and subsequently

employed to predict the delamination of (0/90) and (+45/-45) unidirectional

glass fibre/vinyl ester composite laminates subjected to impact.

5.2 FINITE ELEMENT MODEL

ABAQUS is a well regarded finite element commercial code, and it

has a number of material models but all of them are not supported with an

appropriate failure model for its progressive damage which is extremely

important in a fully penetrating impact process considered in the present

study. As the structure has symmetry in both the directions, a quarter of the

panel (Figure 5.1) has been modeled using C3D8R solid elements. An

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element size of 0.5 mm by 0.5 mm (on the laminate plane) was used in the

highest density region of the mesh. The impactor has been modelled like the

laminated panel and the two separate bodies have been connected through

contact mechanism. In simulation, the impactor has been idealized as a rigid

body, as the deformation of the impactor is always found to be insignificant in

the experiment. The impactor has been modeled as 3D rigid surface using the

4 node 3D quadrilateral element R3D4. The movement of the impactor has

been restrained in all the directions except translation along its axis. The

initial/incidental velocity of the impactor has been specified as the field (a

feature available in ABAQUS) at a reference point (an ABAQUS feature)

defined at the centroid of the impactor. The mass of the impactor has also

been assigned at the reference point. In order to have a simple and

computationally efficient model, the fibre reinforced composite laminates

have been modelled with the options of reduced integration, hourglass control

and finite membrane strains.

Figure 5.1 Finite element model of the (0/90) composite plate

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Figure 5.2 Finite element model of the (+45/-45) composite plate

The interaction between the plate and the projectile was simulated

by surface-to-surface contact elements connecting the projectile and laminate

surfaces. Cohesive elements are placed at the interfaces between lamina to

simulate the delaminated area. Cohesive elements are based on a Dudgale-

Barenblatt cohesive zone approach which can be related to Griffith’s theory of

fracture when the cohesive zone size is negligible when compared with

characteristic dimensions regardless of the shape of the constitutive equation.

These elements use failure criteria that combine aspects of strength based

analysis to predict the onset of the softening process at the interface and

fracture mechanics to predict delamination propagation. A main advantage of

the use of cohesive elements is its capability to predict both onset and

propagation of delamination without previous knowledge of the crack

location and propagation direction. The zero-thickness C3H3D8 cohesive

elements (Figure 5.3) are used to simulate the resin-rich layer connecting the

several lamina of a composite laminate. In all the cases, it is noticed that the

bottom most layer interface is subjected to maximum delamination.

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Figure 5.3 Cohesive element with eight nodes

5.3 SIMULATION OF DELAMINATION

In structural applications of composites, delamination growth is likely

to occur under mixed-mode loading. Therefore, a general formulation for

decohesive elements dealing with mixed-mode delamination onset and

propagation is required. Many attempts have been made at describing the

mixed-mode delamination failure response of composite materials. Failure

criterions have been based on stress or strain near the crack tip, crack opening

displacement, stress intensity factor, or strain energy release rate. Strain

energy release rate is a good measure of a materials resistance to delamination

extension and most of the failure criteria that have been suggested can be

written in terms of critical strain energy release rate or fracture toughness.

Delamination in structures is often subjected to mixed-mode loading and

therefore mixed-mode fracture toughness becomes important. Since

delamination will often be subjected to mixed-mode loading and because the

mixed-mode failure response cannot be determined from the pure-mode

toughness’s, it is important that mixed-mode toughness testing be included

during the characterization of a material.

5.3.1 Delamination Initiation

The onset of damage at the interface can be determined simply by

comparing the traction components with their respective allowable under pure

Mode I, II or III loading. However, under mixed-mode loading damage onset

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and the corresponding softening behavior may occur before any of the

traction components involved reach their respective allowable. A mixed-mode

criterion accounting for the effect of the interaction of the traction

components in the onset of delamination is used. The initiation of the

softening process is predicted using the quadratic failure criterion

(equation 5.1) considering that the compressive normal tractions do not affect

delamination onset.

2 2 2

3 2 1 1N S T

(5.1)

A cohesive element was implemented to improve the delamination

prediction over stress-based criteria. Cohesive elements allow displacement

discontinuities associated with cracks to be explicitly represented. These

elements relate the displacement discontinuity across a crack to tractions that

act across the crack which may be sustained by fibrils in cracked regions,

bridging fibres, friction and other crack bridging mechanisms. Failure of the

cohesive elements is initiated by a mixed-mode quadratic stress-based

criterion involving the normal and two shear stress components (Figure 5.4).

Figure 5.4 Mixed mode traction separation

2 2 2

3 2 1 1N S T

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This implies that damage will most likely initiate at a stress smaller than the

maximum stress for failure for the individual mode.

5.3.2 Delamination Propagation

Once the damage initiation criterion is reached, the traction

displacement is degraded linearly. The softening behaviour and the traction

separation can be either linear or exponential in ABAQUS and for validation

of a simpler model, a bilinear case was chosen. The softening nature of the

decohesive element constitutive equation causes convergence difficulties in

the solution of the finite element analysis. It is found that when using the

Newton- Raphson method, under load (with the arc-length method) or

displacement control, the iterative solutions often oscillated when a positive

slope of the total potential energy was found, and therefore failed to converge.

Hence, the cylindrical arc-length method is used to obtain converged

solutions. This technique chooses between the alternative roots to the arc-

length constraint, selecting the one involving the minimum residual.

However, even with this technique, the solution sometimes enters a cycle and

oscillates between two points. The oscillation must be detected so that the

increment size can be reduced.

It was also found that when coarse meshes are used, a 'snap-back'

type of behavior might occur. Therefore, in order to obtain a relatively smooth

solution, the mesh should be fine enough to include at least two decohesive

elements in the cohesive zone located at the crack tip. The convergence

difficulties were also overcome by reducing the interlaminar tensile strength

to 40% of the original value whilst maintaining constant fracture energy. The

Mixed-Mode Bending (MMB) test specimen is used to find the mixed-mode I

and II interlaminar fracture toughness.

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The area under the traction displacement curve or the energy

dissipated at crack propagation is equal to the fracture toughness for the

particular fracture mode. The criteria used to predict delamination

propagation under mixed-mode loading conditions are usually established in

terms of the energy release rates and fracture toughness. For mixed-mode

loading the dependence of the fracture toughness on mode ratio must be

accounted for in the formulation of decohesive elements. In order to

accurately account for the variation of fracture toughness as a function of

mode ratio in thermoplastic composites, the mixed-mode criterion proposed

by Benzeggagh and Kenane (B-K criterion) is used here. The B-K criterion

gives the mixed mode fracture toughness over a comprehensive range of

mode mixity. This criterion is expressed as a function of the Mode I and

Mode II fracture toughness and a parameter obtained by simulating mixed

mode bending tests at different mode ratios. The B-K parameter = 1.87 is

calculated by applying the least-squares fit procedure (Camanho et al 2002) to

the experimental data.

IIIc IIc Ic c

I II

GG G G GG G (5.2)

The progression of damage is monitored by the scalar damage

indicator d, which ranges from the initial value of 0 for the undamaged

interface to the value of 1 which indicates the complete decohesive of the

interface and the physical separation of the upper and lower cohesive

surfaces. The evolution of the variable ‘d’ is controlled by the energy

dissipated as work of separation during the decohesion process.

The solution of the equilibrium equations by an explicit direct

integration method requires the adoption of a time increment smaller than the

critical time-step (which decreases with increasing mesh density and material

stiffness and with decreasing material mass density) in order to ensure the

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stability of the iterative time-stepping scheme. The size of the time increment

requires special consideration since an excessively small time-step may

render the analysis of a dynamic event extremely intensive in terms of

computing time if the total time-span of the phenomenon is many orders of

magnitude larger than the time interval adopted for temporal integration. It

should also be observed that these elements with reduced integration were

employed in the present study, stabilisation techniques were applied to reduce

hourglassing so as to avoid excessive distortion of elements or structure.

5.4 PREDICTION OF DELAMINATED AREA

The decohesive element numbers are given to user defined

subroutine written in FORTRAN to measure and plot the merged delaminated

area of the last two bottom interfaces in order to find the total delaminated

area of the composite plate. Impacts of different energies were simulated by

imposing the experimental ballistic limit velocities at the instant of contact for

different nose shaped projectiles. The size of the (0/90) and (+45/-45)

composite plates used in this numerical analysis are as shown in the

Figures 5.5 and 5.6.

Figure 5.5 Dimensions of the numerical model of (0/90) composite plate

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Figure 5.6 Dimensions of the numerical model of (+45/-45) composite

plate

The above procedure is repeated for composite plates of different

orientations impacted by different nose shaped projectiles with their ballistic

limit velocity found experimentally. The delaminated area of the (0/90) and

(+45/-45) composite plates impacted with different nose shaped projectiles is

shown in the Figures 5.7 to 5.12. The Ballistic limit and the corresponding

simulated damage area for (0/90) and (+45/-45) are listed in the Tables 5.1

and 5.2. This analysis indicates that the initial damage consists of a tensile

matrix crack that develops within the distal stack of 0 plies and quickly

propagates towards the boundary of the sample. It is observed that the

amount of delaminated area produced by different nose shaped projectiles is

different for the composite plate having same properties. Also, the orientation

of the fibre in the composite plates has considerable influence on the size of

the delaminated area. The obtained results are compared with the

experimental and analytical model results in Chapter 6.

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Ogival at 43.3 m/s Conical at 45.4 m/s

Hemispherical at 47.8 m/s Truncated ogival at 52 m/s

Truncated conical at 53.8 m/s

Figure 5.7 Delaminated area of the 3.64mm (0/90) plates at ballisticlimit velocity for different nose shaped projectiles

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Hemispherical at 58.4 m/s Truncated ogival at 59.8 m/s


Figure 5.8 Delaminated area of the 5.35mm (0/90) plates at ballisticlimit velocity for different nose shaped projectiles

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Figure 5.9 Delaminated area of the 7.1mm (0/90) plates at ballistic limitvelocity for different nose shaped projectiles

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Figure 5.10 Delaminated area of the 3.64mm (+45/-45) plates at ballisticlimit velocity for different nose shaped projectiles

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102





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Table 5.1 Numerical results for (0/90) composite plates

PlateThickness

(mm)

Projectilenose shape

Ballisticlimit, bv

(m/s)

Damage Area,damA

(mm2)

3.64

Ogival 43.3 8252

Conical 45.4 8955

Hemispherical 47.8 9227

Truncated Ogival 52.0 10014

Truncated Conical 53.8 11605

5.35

Ogival 53.7 8907

Conical 57.1 9147




7.1

Ogival 61.6 10667

Conical 65.8 11579




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Table 5.2 Numerical results for (+45/-45) composite plates

PlateThickness

(mm)

Projectilenose shape

Ballistic limit,bv (m/s)

Damage Area,damA (mm2)

3.64

Ogival 46.1 7197

Conical 51.7 7408




5.35

Ogival 60.4 7343

Conical 63.4 8658




7.1

Ogival 69.9 9609

Conical 73.2 11346




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5.5 SUMMARY

An efficient numerical model used for the simulation of the

delaminated area has been presented. The commercially available finite

element code ABAQUS has been used for this purpose in which a method for

the simulation of progressive delamination based on cohesive elements is

presented. Decohesive elements are placed between layers of solid elements

that open in response to the loading situation. The onset of damage and the

growth of delamination are simulated without previous knowledge about the

location, the size and the direction of propagation of the delamination. The

mixed-mode criterion is used to predict the strength of composite structures

that exhibit progressive delamination. The numerical model has been applied

to the simulation of glass fibre reinforced plastic laminates having different

orientations impacted by different nose shaped projectiles.

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