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Simulation of Progressive Failure in Composite Laminates

and, consequently

+ G

. (47)

Substitution of Eq. (44) and

= 1 in Eq. (40) gives the relation between the

initial shear traction,

, and

, which may be written as:

| = F

, (48)

with

= F

+ F

(49)

and

1 −

, (50)

. (51)

Similarly, substitution of (45) in (40) with

gives the relation between the

shear crack opening that corresponds with complete failure,

, and

| =

. (52)

With Eqs. (48) and (52), the energy release in mode II can be computed as a

function of

(considering that the sign of t

will always be equal to that of

. (53)

Finally, with Eq. (47), the total energy release as a function of

is:

= G

+ G

. (54)

Now,

can be deﬁned as a function of

such that G

in (54) equals G

(42):

+ (G

IIc

−G

)

). (55)

Because equilibrium was assumed for this derivation, while equilibrium is only

weakly met, the amount of energy dissipated in mixed mode conditions does not

satisfy Eq. (42) exactly, but it does approach the correct value upon mesh reﬁnement.

365

F.P. van der Meer and L.J. Sluys

3.4 Consistent tangent

The element matrix for the two overlapping elements together is







bulk

0 K

bulk





coh

−K

coh

−K

coh





coh

−K

coh

−K

coh



, (56)

with

bulk

Ω

DB d

Ω

, (57)

bulk

Ω

DB d

Ω

, (58)

coh

= T

AQN d

, (59)

coh

AQHDB d

. (60)

The nonlinearity of the system is in matrix A, which is deﬁned as

A =

∂

= I −Ω −

t⊗

∂ω

∂

, (61)

with

∂ω

∂

∂ω

∂κ

∂

∂ω

∂κ

∂β

∂

. (62)

3.5 Off-axis tensile test

The phantom node method with the introduced constitutive relation is applied to

the 10

◦

off-axis tensile test. The same elastic parameters and ply strength parameters

as in Section 2.5 are used. Additional parameters are: G

= G

IIc

= 3 N/mm and

= 1. The length of the specimen is 44 mm, apart from which, dimensions are the

same as in Section 2.5. There is no weak zone to trigger the initialization, but the

location where the crack will initiate is speciﬁed in advance. Notably, the geometry

is such, that the stress ﬁeld is homogeneous until nonlinearities occur, which means

that the failure criterion is exactly satisﬁed in the entire specimen when the crack is

initiated. As a consequence a sharp snap-back occurs. An arc-length method with

energy release control [15] is employed to follow this behavior.

In Fig. 19, the load displacement relation is plotted for two different meshes, one

with an average element size 1.1 ×0.5 mm and the other with average element size

0.8 ×0.25 mm. It can be observed that the discontinuous approach does not require

366

Simulation of Progressive Failure in Composite Laminates

0 0.2 0.4 0.6 0.8 1

F (kN)

(%)

Coarse

Fine

In Fig. 20

Fig. 19 Load displacement data for off-axis test with phantom node method

any precautions to obtain mesh independent results as opposed to the continuum

damage model.

Fig. 20 Off-axis tensile test results with phantom node method (ﬁne mesh, unscaled deformations.

367

F.P. van der Meer and L.J. Sluys

Figure 20 shows the deformed mesh from the ﬁne mesh solution for different

stages. The normalized stress quantity that is used for the shading is the failure

criterion in Eq. (31). In Fig. 19, the corresponding points on the load displacement

curve are indicated. It can be observed that in a sudden event at the peak load level,

a cohesive crack of considerable length appears, after which the load drops without

much additional crack growth. Then the cohesive zone, i.e. the part of the crack in

which tractions are present, becomes smaller and the crack ﬁnds its way toward the

opposite boundary of the specimen. Contrary to the analysis discussed in Section 2,

the crack grows in the correct direction, not surprisingly since the growth direction

is ﬁxed. In this example not only the direction of propagation, but also the location

of initiation was predeﬁned, which renders it a rather trivial problem that could also

be tackled by meshing the crack with interface elements. However, the presented

model can also be applied to more complex geometries and cases with multiple

cracking, where the advantage that the crack is represented in a mesh-independent

fashion becomes signiﬁcant. The use of interface elements would require complex

discretized conﬁgurations.

4 Conclusions

A continuum damage model with viscous regularization has been introduced. With

this regularization method mesh independent results can be obtained, which is an

issue of concern in the computational modeling of failure in materials.

It has been shown that the regularized continuum damage model is not always

adequate to predict the correct failure pattern in composite laminates. A relevant

microstructural property is lost in the homogenization procedure, namely the pref-

erence of matrix damage to propagate in ﬁber direction. Still bands with matrix

failure may be obtained that are oriented correctly, especially because orthotropic

elasticity properties still promote this behavior, but bands with different orientation

may also arise. This is not only the case for the presented model, but also for other

continuum damage models presented in literature in recent years. The predictive

quality of continuum models for localized matrix failure in isolated plies should be

doubted.

One way to overcome this is to abandon the continuum approach for the speciﬁc

mechanism of splitting. For this purpose, a phantom node method (equivalent to

XFEM) with crack propagation in ﬁxed direction is proposed. This method allows

for mesh-independent modeling of splitting cracks, which is of particular impor-

tance when location and number of cracks is not known a priori. A new cohesive

law has been applied, which can deal with varying mixed mode conditions robustly.

In combination with an energy arc-length method, it is possible to simulate crack-

ing in a 10

◦

off-axis tensile test. After combination of this splitting formulation with

models for other ply failure mechanisms and delamination, simulation of complex

laminate failure mechanisms on the mesolevel may become feasible.

368

Simulation of Progressive Failure in Composite Laminates

Acknowledgments

and the Ministry of Public Works and Water Management, The Netherlands.

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371

Numerical Modeling of Wave Propagation,

Breaking and Run-Up on a Beach

G.S. Stelling and M. Zijlema

Abstract A numerical method for free-surface ﬂow is presented to study water

waves in coastal areas. The method builds on the nonlinear shallow water equa-

tions and utilizes a non-hydrostatic pressure term to describe short waves. A vertical

boundary-ﬁtted grid is used with the water depth divided into a number of layers. A

compact ﬁnite difference scheme is employed that takes into account the effect of

non-hydrostatic pressure with a small number of vertical layers. As a result, the pro-

posed technique is capable of simulating relatively short wave propagation, where

both frequency dispersion and nonlinear shoaling play an important role, in an ac-

curate and efﬁcient manner. Mass and momentum are strictly conserved at discrete

level while the method only dissipates energy in the case of wave breaking. A simple

wet-dry algorithm is applied for a proper calculation of wave run-up on the beach.

The computed results show good agreement with analytical and laboratory data for

wave propagation, transformation, breaking and run-up within the surf zone.

1 Introduction

Wave transformation in the surf zone plays an important role in coastal engineer-

ing since, nearshore waves are the driving forces for many nearshore phenomena,

such as longshore currents, water level set-up, sedimentation, erosion, and coastal-

structure loading. Surf zones are characterised by the irreversible transformation of

organized wave motion of the incident short, wind-generated waves into motions of

different types and scales e.g., turbulence and low-frequency motion (well-known

G.S. Stelling

Delft University of Technology, Faculty of Civil Engineering and Geosciences, P.O. Box 5048,

2600 GA Delft, The Netherlands, e-mail: g.s.stelling@tudelft.nl

M. Zijlema

Delft University of Technology, Faculty of Civil Engineering and Geosciences, P.O. Box 5048,

2600 GA Delft, The Netherlands, e-mail: m.zijlema@tudelft.nl

373

Lecture Notes in Computational Science and Engineering 71,

DOI 10.1007/978-3-642-03344-5_13,

B. Koren and C. Vuik (eds.), Advanced Computational Methods in Science and Engineering,

G.S. Stelling and M. Zijlema

as the ”surf-beat”). The main features associated with the transformation of coastal

waves across a surf zone are illustrated in Fig. 1. (SWL denotes the still water level.)

In the pre-breaking region, wave transformation is described by the effects of wave

set−down

set−up

SHOALING BREAKER ZONE RUN−UP

SWL

Fig. 1 Wave transformation across a typical surf zone.

steepness due to the nonlinearity (or amplitude dispersion) and wave shortness due

to the frequency dispersion. The front face of a wave will steepen continuously un-

til the front becomes vertical. Therefore, the frequency and amplitude dispersion

effects must balance each other, so that waves of ﬁnite amplitude and permanent

form are possible [31]. The bathymetric variations in shallow water distort this bal-

ance and cause instabilities and subsequent wave breaking. Once the wave breaks,

turbulence is generated and becomes a dominating feature of the ﬂow ﬁeld. The

wavebreaker-generated turbulence balances the steepening of the front and stabi-

lizes the surface proﬁle [31]. The broken waves propagate with a gradual change of

form and resemble steady bores. At the end, they become relatively long and run up

on the beach. The run-up starts when the bore reaches the shoreline corresponding

to a stage of the motion with no water in front of the bore.

The simulation of broken waves and wave run-up amounts to the solution of the

nonlinear shallow water (NLSW hereinafter) equations for free-surface ﬂow with-

out viscosity terms in a depth-integrated form [16, 21, 32, 5, 19]. These hyperbolic

equations are mathematically equivalent to the Euler equations for compressible

ﬂows. Discontinuities are admitted through the weak form of these equations and

can take the form of bores which are the hydraulic equivalent of shock waves in

aerodynamics. The conservation of energy does not hold across the discontinuities

but the conservation of mass and momentum remains valid. By considering the sim-

ilarity between broken waves and steady bores, energy dissipation due to turbulence

generated by wave breaking is inherently accounted for [16, 5].

In the pre-breaking region, however, the NLSW equations do not hold since,

they assume a hydrostatic pressure distribution. These equations prohibit a correct

calculation of frequency-dispersive or short waves. Moreover, they predict that the

front face of any wave or bore will steepen continuously until a vertical front is

formed. Only the deviations from hydrostatic pressure can balance the steepening

of the front and stabilize the surface proﬁle before it becomes vertical. In this study,

we discuss an extension of the NLSW equations to include the effect of vertical

374

Numerical Modeling of Wave Propagation, Breaking and Run-Up on a Beach

acceleration so that the propagation of short, nonlinear waves with ﬁnite amplitudes

can be simulated.

A main difﬁculty occurring in the simulation of free-surface ﬂows is the proper

handling of a moving free surface since this is part of the solution itself. Many

methods for the treatment of the free surface are described in the literature. The

most well-known are the Marker-and-Cell (MAC) method [15], the Volume-of-Fluid

(VOF) method [18] and the level-set method [30]. An overview of these methods

can be found in [25]. Although, these techniques can describe wave overturning in a

very accurate manner, they yield more detailed information than necessary for many

coastal engineering applications. Moreover, they are too computing intensive when

applied to the large-scale wave evolution in the surf zone.

A much simpler approach is the one in which the free-surface motion is tracked

using a single-valued function of the horizontal plane as done in the NLSW method-

ology. Recently, the development of so-called non-hydrostatic models using this

approach has been a popular topic of many ocean and coastal modeling activities.

Well-known papers on this subject are Casulli and Stelling [9] and Stansby and Zhou

[26]. The models in these papers consist of the NLSW equations with the addition

of a vertical momentum equation and non-hydrostatic pressure in horizontal mo-

mentum equations. As such, the total pressure is decomposed into hydrostatic and

non-hydrostatic components. The underlying motivation for this approach is that

existing shallow water packages need to be adapted slightly only, since the correc-

tion to the hydrostatic pressure is done after the NLSW equations have been solved.

As a consequence, this reduces the effort of software extension and maintenance

to a minimum. Also, Mahadevan et al. [22] have shown that this technique leads

to a more stable and efﬁcient non-hydrostatic calculation than in the case without

splitting the pressure into hydrostatic and non-hydrostatic parts. Moreover, the non-

hydrostatic models require much less grid cells in the vertical direction than the

MAC and VOF methods. These beneﬁts make simulations of wave transformation

in coastal waters much more feasible and efﬁcient. There is, however, still ongoing

research on this approach. The choice of an appropriate numerical approach for a

non-hydrostatic free-surface ﬂow model appears to be non-trivial. Several different

solution procedures (fractional step versus pressure-correction approach, modeling

of the free-surface boundary condition, Cartesian versus

−coordinates, etc.) have

been proposed by different authors; for an overview, see [35] and the references

therein.

Another issue that remains to be discussed is wave breaking. In principle, non-

hydrostatic models represent a good balance between nonlinearity (enables wave

shoaling) and frequency dispersion (corrects celerity of shoaling wave) so that initi-

ation of the wave breaking process and the associated energy losses can be described

adequately by these models. However, most of the well-established non-hydrostatic

models, [22, 9, 26], are by no means momentum-conservative. It is evident that the

numerical schemes involved must treat shock propagation adequately in order to

model broken waves in the surf zone. Traditionally, the shock-capturing schemes

applied to shallow water ﬂows at collocated grids are based on the Godunov-type

approach, where a discontinuity in the unknown variables (water depth and dis-

375