Modelling of crack formation and growth using FEM for selected

structural materials at static loading

VLADISLAV KOZÁK, JIŘÍ VALA

Institute of Mathematics and Descriptive Geometry

Brno University of Technology, Faculty of Civil Engineering

CZ - 602 00 Brno, Veveří 331/95

CZECH REPUBLIC

Abstract: The purpose of this paper is to show the results of a study focused on the occurrence of damage het-

erogeneous materials, especially on the issue of modelling crack formation and propagation. In the beginning

the attention is paid to the direct application of the finite element method to different types of materials in order

to find critical parameters determining behaviour of materials at damage process. The applications of damage

mechanics and possible approaches to model the origin of a crack propagation through modifications in FEM sys-

tems are presented and some practical applications are tested. Main effort is devoted to cement fibre composites

and the search for new methods for their more accurate modelling, especially close to the field stress concentra-

tor, respectively ahead of the crack tip. Modified XFEM method has been used as a suitable tool for numerical

modelling.

Key-Words: Computational damage; crack growth modelling; finite element method.

Received: March 23, 2023. Revised: August 24, 2023. Accepted: September 25, 2023. Published: November 1, 2023.

1 Introduction

The question of ensuring the safety of structural com-

ponents and predicting their service life is increas-

ingly associated with the development of new devices

and components. In the case of constructions, it may

be directly dependent on the occurrence of defects

that may arise during the production stage or through

their lifetime. One of the concepts used in construc-

tion and safety assessment is a set of theories and

methods known in fracture mechanics. This scien-

tific field, combining continuum mechanics with ma-

terial engineering, describes the behaviour of defects

in structures. It is a complex defect-stress-material

relationship. To understand the relationships end ex-

tend lifetime, it is necessary to modernize construc-

tion practices and also use new numerical methods.

The aim of fracture mechanics is to describe or

predict the behaviour of bodies containing defects.

In many cases, cracks can lead to total failure of the

structure due to fracture. There are two basic ap-

proaches for deriving the conditions in the moment

of initiation of unstable crack propagation. The first

one uses the weakest link theory, the second model

considers the accumulation of damage during load-

ing. Failure of structural materials is understood as a

continuous process in which the stages of plastic de-

formation, nucleation and initiation of cracks is inter-

mingled. The final stage in the development of failure

of bodies, which is the subject of investigation of frac-

ture mechanics, is the propagation of cracks (unstable

or stable). The goal of the presented works was how

to find out the mutual relations between physical reg-

ularities and the physical laws themselves the essence

of the breaking process on the one hand and the theory

of continuum mechanics on the other.

The promotion of non-traditional materials, con-

structions and technologies in the modern construc-

tion industry also requires new approaches to the in-

vestigation of their physical properties, where proven

ones cannot be relied upon. Simulation of the be-

haviour of material samples, structural elements and

buildings as complex structures becomes necessary.

After these motivational comments in Section 1 some

notes to fracture mechanics and to model problems

are presented in Section 2. The overview of classi-

cal approaches and crack growth modelling in Section

3, will be followed by some details of damage me-

chanics and cohesion approach in Section 4, with spe-

cial attention to the modified finite element method

in Section 5. The principal investigation as to build-

ing materials by Section 6 is supplied by references

to potential applications in Section 7 and concluding

remarks with future research priorities in Section 8.

2 Some notes about fracture

mechanics

Fracture mechanics deals with the problem of bodies

with cracks. The term fracture refers to the division

of a body into two or more parts. For the purpose of

fracture mechanics, we define a crack as a violation

of the cohesion of bodies along a surface bounded by

a curve that is either closed or it ends on the surface

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

243

Volume 18, 2023

of the body.

Figure 1: Contour integral definition.

With a simplified view of the mentioned problem,

it is crucial to define the stability criteria, which estab-

lish the conditions under which an existing or emerg-

ing crack will propagate and thus enable the calcula-

tion of the corresponding critical stress or the critical

length of the crack.

The concept of stability criterion can be extended

to the case of general stress concentrators. In such a

case, we are looking for conditions under which crack

will start to propagate from the considered concen-

trator. In general, any fracture-mechanical quantity

can be used for the formulation of stability criteria,

and the criteria can be formulated on the basis of en-

ergy, [1], [2], using the deformation energy density,

crack driving force, G,Jintegral, or based on stresses

and strains at the crack tip (concept of stress intensity

factor, crack opening). For linear fracture mechan-

ics, the more commonly used stability criterion is the

KIC criterion (the GIC criterion can be used equiv-

alently). For non-linear fracture shear it is then the

JIC criterion, [3], or the crack opening displacement

(CTOD) condition, with a number of criteria for com-

bined stress, [4], [5], [6].

The described procedure works quite reliably if

we limit ourselves to the first singular term when de-

scribing the stress around the crack front, so we are

talking about one-parameter fracture mechanics. It

is precisely in the case of multiaxial stresses, when

we switch to multi-parameter fracture mechanics and

work with concepts as are T-stress and Q-parameter,

[7]. The general form of the equation in case of static

loading for the overall decomposition in the classical

(differential) approach, is based on the decomposition

of the overall deformation, represented by the strain

tensor , as a square matrix of order 3, into four addi-

tional components, namely the elastic, plastic, creep

and thermal, [8], denoted as e,p,cand θhere.

All these quantities will considered as variable in the

time t, with values prescribed at t= 0 are prescribed

(which can be marked as Cauchy initial conditions).

Namely in the isotropic and homogeneous case,

taking i, j ∈ {1,2,3}, we can consider

e

ij = ((1 + ν)/E)σij −(ν/E)σ∗δij ;

here we need the Kronecker symbol δij = 1−sgn |i−

j|,σ∗= (σ11 +σ22 +σ33)/3 refers to the princi-

pal stress and E,νis the couple of material parame-

ters: the Young modulus and the Poisson ratio. The

contribution of pcan be expressed using the curve

of real stress versus real deformation during plastic

loading (nonlinear in general), as an additional mate-

rial characteristic; The increment of plastic deforma-

tion during creep, is evaluated using the finite element

method, using the initial deformation approach, typ-

ically. The contribution of θcomes (if no more de-

tailed thermodynamic analysis is available) from tem-

perature difference, applying the thermal expansion

factor. Consequently, very important is evaluation of

cfrom ˙c

ij = ((3˙c

ef )/(2σef )) σ0

where the lower index ef means certain scalar ef-

fective value, σ0being the deviatoric part of σ, i. e.

σ0

ij =σij −σ∗δij , with i, j ∈ {1,2,3}again.

Contour integrals play an important role in linear

and nonlinear fracture mechanics, see Fig. 1. Well

known is the J-integral, [3], [8]. An analogous in-

tegral for bodies subjected to creep deformation is

called C-integral, [9], [10], [11], [12]. To understand

the implementation of individual types of curve inte-

grals, the manual for the WARP3D software system

is recommended, [13].

3 Crack growth modelling using

classical FEM approaches

In the course of the 1960s, theories were gradually

created that were able to describe the behaviour of

bodies with cracks by considering plastic zones of

larger scale, and thus get closer to more realistic con-

ditions arising during loading of bodies with a crack.

Currently, there are several procedures that can be

used to solve the problem of simulating crack propa-

gation using FEM. Among the first and oldest are the

modelling of stable crack growth using the node re-

lease method, where starting criterion is necessary to

define. Modelling of crack propagation by the node

release method is possible only in the case of a 2D

problem, see Fig. 2, where effective plastic deforma-

tion (ef ) is displayed.

The first criterion is based on knowledge of the

crack length relationship as a function of time, which

is assigned to the time period required to solve the

modelled problem. The second type of criterion uses

the stress definition for crack propagation. The third

type of criterion uses the critical value of crack open-

ing, which is the deformation characteristic, as a

boundary condition for crack propagation. The prin-

ciple is to separate the deformable surface of the body

from the analytically defined non-deformable surface.

During the release of the nodes, the stresses between

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

244

Volume 18, 2023

Figure 2: Node release approach for 2D.

Figure 3: Remeshing technique.

the two contact elements are gradually reduced to

zero. Other important methods are based on constant

remeshing, [14], see Fig. 3.

The second option is the method of “disappearing”

elements, within its framework are also included the

latest approaches using cohesive elements. In fact,

they are actually a generalized contact. Both methods

mentioned above will be discussed in more detail in

the following session.

4 Damage mechanics and cohesion

approach

The prediction of crack propagation through interface

elements based on fracture mechanics has been se-

lected for crack growth modelling. Damage is be-

ing implemented in constitutive models and accumu-

lation of damage is processed, [15]. As an example of

the use of damage mechanics for ductile failure, the

Gurson-Tvergaard-Needleman model can be pointed

out, [16], [17]. Based on the evaluation of the ex-

periments performed, it is recommended introducing

two (or three) optional parameters q1, q2, q3, usually

q1= 1.5,q2= 1,q3=q2

1. Analysis of the behaviour

of these parameters was performed, [18]. Here, σY S

is the yield stress of the matrix material, σmis the

principal (hydrostatic) stress and fis the volume frac-

tion of voids. The difference between the original

model and the modified one is shown in Fig. 4, the

schematic situation in front of the crack front in Fig. 5.

The energy balance of such so-called complete model,

[19], implemented as a user procedure in Abaqus soft-

ware, [20], reads equation for plastic potential de-

scribing plastic flow in porous materials

i,j=1

2Sij Sij

3σ2

Y S

+ 2q1fcosh 3q2

2σm

2σY S

−q3f2= 1 .(1)

For practical calculations, f(volume fraction of cavi-

ties) in (1) is usually replaced by certain effective vol-

ume fraction f∗, introduced as

f∗=fc−((f∗

u−fc)/(fF−fc))(f−fc) ;

here fcis the critical volume at which voids (cavities)

coalesce, fFis the volume of voids at ultimate dam-

age and f∗

u= 1/q1. This effective value f∗applies to

condition when fcis less than f.

Figure 4: Definition of the Gurson-Tvergaard (GT)

and Gurson-Tvergaard-Needleman (GTN) damage

model.

Figure 5: Schematic illustration of damage models in

front of the crack.

Another type of elements for modelling crack

propagation are cohesive elements. These elements

are originally evolved from contact elements and they

are based on the idea of material separation with the

creation of new surfaces, [21]. Practically, it is a cer-

tain phenomenological description that characterizes

the behaviour of the material using the so-called trac-

tion separation law, thanks to which we can then pre-

dict local violations; schematic shape of cohesive el-

ement for the simplified 2-dimensional geometry is

presented by Fig. 6.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

245

Volume 18, 2023

Figure 6: Schematic shape of cohesive element for

2D in the local coordinates.

Figure 7: Traction separation law for fibre composites

optimized for the numerical stability of the iterative

algorithm.

There are several options for implementing the

cohesion law (traction separation law) into a com-

mercial FEM system. The presented work is based

on long-term experience with the commercial system

Abaqus, which makes it relatively easy to write cus-

tom user procedures for some special types of dam-

age, new types of elements, or user control of some

system options. Just the possibility of writing a user

procedure UEL (User’s Element) became the basis for

the creation and implementation of the procedure for

traction separation law, [22], [23], [24]. The origi-

nal version is shown in Fig. 7, this traction separation

law was further modified. The following Fig. 8 shows

comparison of these two methods for a specific ma-

terial. As crucial in this real case, it is necessary to

point out the exact modelling of the dependence of

actual stress versus actual deformation.

A numerical implementation of both the material

model for ductile GTN material including the con-

struction of a special element into the Abaqus soft-

ware system for the cohesive model shown in Fig. 7

was documented, [25]. The relevant equations and

the derived procedure for the realization of special el-

ements are presented here. At the same time, a nu-

Figure 8: Determination of the J - R curve, showing

the applied integral Jversus crack extension, for both

methods, material forged steel.

merical analysis is performed and some aspects of

practical calculations are pointed out. An example

for application of the cohesive approach for ceramics

with long fibres is described, [26], for the viscoelastic

model, [27].

5 Crack growth via XFEM modelling

There is no doubt that the FEM is widely used mainly

in the area of solving differential equations, however,

as already indicated in the article, the FEM mesh

may not always be ideal for modelling crack propa-

gation. And this is one of the most significant inter-

est in solid mechanics problems. First models were

based on the weak (strain) discontinuity that could

pass through finite element mesh using variational

principle, [28]. Other authors and investigators con-

sidered strong (displacement) discontinuity by mod-

ifying the principle of virtual work statement (which

is also the case for models with the traction separation

law), [29], [30], [31], including stability and conver-

gence of such problems, [32], and improvement of the

precision of numerical procedure, [33], [34], [35].

In the strong discontinuity approach, the dis-

placement consist of regular and enhanced compo-

nents, where the enhanced components yield via jump

across discontinuity surface, [30]. There is a modi-

fication of the basic equation, well-known from the

standard finite element method (FEM), for the rela-

tionship between displacements ue(x)on particular

points xof the e-th element (3-dimensional vectors

of functions in general) and displacements at selected

element nodes ui, utilizing some standard shape func-

tions Ni(x), i. e.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

246

Volume 18, 2023

ue(x) = X

i∈EA

Ni(x)ui;(2)

here EAdenotes the set of nodes corresponding to the

standard e-th element.

For the extended finite element method (XFEM),

some more terms must be added to slightly modi-

fied displacements inside element end displacements

at element nodes of (2). The second term realizes

the technique of penetration into the element and the

movement of the crack, the third criterion the fail-

ure (decohesion) and the possible direction of move-

ment according to the preferred criteria. So the pre-

vious equation for the case of crack growth for 2-

dimensional modelling (not limited to one element)

is modified into the form

uh(x) = X

i∈EA

Ni(x)ui(3)

j∈CB

Nj(x)Hj(x) + X

k∈CC

Nk(x)

m=1

Φm

k(x)cm

including certain specialized shape functions Ni(x),

Nj(x)and Nk(x)for intrinsic version of XFEM,

same shape function for extrinsic version, where CA,

CB,CCare the sets of points corresponding to Fig. 9

and H(x)is the Heaviside function. The first term in

(3) corresponds to the standard method of finite ele-

ments, the second realizes the formation of a crack

and the third appropriate criterion of its formation,

while Φm

k(x)characterizes the local situation in front

of the crack, mtakes on values from 1 to 4, which

corresponds to the basic possible directions of crack

propagation in 2D.

Figure 9: Illustration of enrichment function; do-

mains A, B, C, D, E under the influence of crack

growth.

Fig. 9 also characterizes the crack movement.

Area Ais affected by a crack, areas Dand Eare at risk

of a crack creation and the situation here is already

affected by the crack. In area Bthe crack moves and

the second term of the (3) applies, in area Cthere is a

crack tip and the third term of the equation (3) applies.

For practical calculations, the time discretization

method and the extended finite element method can

Figure 10: Crack propagation scheme for XFEM, HE

stands for Heaviside Enrichment, CTE for Crack Tip

Enrichment.

be used (chart can be seen in Fig. 10,) working with

adaptive enrichment of the set of basis functions near

singularities. This method (including a number of its

modifications with their own names and designations)

it already has quite a rich history with the remarkable

progress in recent years, [36], [37], [39], [40], [41],

[42], [43].

The FEM software AbaqusTM utilizes just (3)

where the third term represents a simple criterion

based on the minimum deformation energy, [5], [44],

[45]. The mentioned equation, however, represents

only one class of XFEM-based methods, the so-called

extrinsic formulation. In this case, the number of de-

grees of freedom (DOF) increases, but the standard

shape function from (2) does not change.

For the second group of XFEM methods, we use

the label intrinsic method. In this case, there is no

change in the number of degrees of freedom, but in the

shape functions, which is numerically much more de-

manding and procedures based on the method of least

squares are used. It should be noted that the third term

of (3), implementing the criterion of initialization and

crack growth, does not change. The second term of

the equation will therefore change significantly and

may look like

uh(x) = X

j∈CN

Nj(x)˜a;

here CNis complete set of nodes, whereas ˜arefers to a

set of cautiously designed parameters, whose relation

to uihas to be evaluated from certain non-trivial aux-

iliary problem, e. g. using the least squares technique,

[37], [38]. This entry into the history of XFEM and

its division into extrinsic and intrinsic methods, [46],

is only informative and serves to understand the dam-

age modelling of fibre composites, in particular in the

following considerations; cf. concurrent approaches,

[47], [48], [49], too.

6 Damage modelling of structral

fibre composites

Fibre cement composites belong to the class of per-

spective concretes with higher mechanical resistance

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

247

Volume 18, 2023

against the formation of cracks. This allows for a finer

and more economical construction; therefore, a new

perspective on the creation of building structures or

replacement is necessary steel constructions. These

structures subjected to loads can result in stresses in

the body exceeding the strength of the material, and

thus lead to gradual failure. Such failures are often

initiated by surface or near-surface cracks, which re-

duces the strength of the material.

A separate serious problem is the setting of ma-

terial parameters at the macroscopic level, supported

by appropriate experiments, if there is at least some

information about the structure of the material, e.g.

about random or intentionally preferred fibre direc-

tions; problems of this kind with an emphasis on

non-destructive or minimally invasive ones test meth-

ods (especially x-ray, tomographic and electromag-

netic, working with stationary magnetic or harmonic

electromagnetic field) is covered, [50], as well as

temperature-dependent problems, [51], [52]. A more

general procedure for randomly oriented fibres was

developed, [53], [54]; an example of a real distribu-

tion in reinforced concrete, [55], is in Fig. 11.

Figure 11: Radiographic image of fibre distribution

in wire composite.

The formation of microcracks, [50], can be taken

into account by introducing a failure factor based on

the approach modifying the stress field and working

with the non-local Eringen model, [56], a schematic

diagram describing the stress calculation ahead the

crack tip is shown in Fig. 12. About the interfaces be-

tween the matrix and the fibres, but also inside the ma-

trix, possibly also the fibres, depending on the gradu-

ally activated macrocracks, it is usually assumed that

they can be described by a cohesion model according

to [57], [58], or, [59].

The XFEM application is able to suppress the dis-

advantages in the simulation of cohesive crack prop-

agation; however, it must handle the absence of a

sharp singularity at the crack tip with a more com-

plex derivation of the required stresses from displace-

Figure 12: Calculating the stress in front of the crack

tip: a non-local approach.

ments. A complete computational model should gen-

erally include the initiation and propagation of cracks,

their bridging by fibres, loss of cohesion between fi-

bres and matrix, their mutual sliding by friction and

destruction of fibres; special functions are required,

e.g., for stress singularities in case opening and clos-

ing of cracks. The scheme of crack propagation

through the original FEM elements is represented in

Fig. 10. A unified scale-spanning approach cover-

ing elastic and plastic behaviour along with fracture

a with other defects leads to the concept of structured

deformation, [60].

By considering models based on micromechan-

ics, the macro-constitutive equations of unidirectional

or randomly distributed fibres of reinforced materi-

als, taking into account the possibility of forming and

spreading cracks in the matrix, as well as to the sep-

aration and breakage of fibres. The computational

model is finally used in numerical simulations in or-

der to outline its reliability in evaluating both the

fibre-matrix interaction phenomenon, so the ability to

predict fracture failure of fibre composites. Mechan-

ical behaviour of fibre-reinforced brittle matrix com-

posites, with an emphasis on cementitious compos-

ites, will be investigated based on both discontinuous

approach and modified approaches based on FEM.

XFEM is mesh-independent as to internal bound-

ary such as material interfaces and cracks. These in-

ternal boundaries usually cause weak or strong field

discontinuities variables that will be taken into ac-

count in XFEM by incorporating enrichment func-

tions into the standard FEM approximation.

7 Application of the XFEM approach

For simplicity, let us start with the small-strain static

formulation of the problem of deformation of lin-

ear elastic body, as the first step for the implemen-

tation of XFEM-based computational approach to the

analysis of both micro- and macrocracking in quasi-

brittle composites, with obvious applications to ce-

mentitious composites with various types of rein-

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

248

Volume 18, 2023

forcement. Such weak formulation is usually based

on the conservations principles from classical ther-

momechanics, namely of mass, momentum and en-

ergy, supplied by appropriate constitutive equations,

whose material parameters come from experimental

identification procedures. In our case the conserva-

tion of linear momentum on a deformable body Ωin

the 3-dimensional Euclidean space, supplied by some

Cartesian coordinate system x= (x1, x2, x3), reads

((ε(w), σ)) = (w, f ) + hw, gi(4)

for all virtual displacements w= (w1, w2, w3), re-

lated to an initial configuration, whereas (for any

fixed x)σis a symmetric matrix with 3×3elements, f

denotes prescribed body forces on Ω,grepresents pre-

scribed surface forces on certain part Γof the bound-

ary ∂Ωof Ω, whereas Θ = ∂Ω\Γcorresponds to the

supported part of ∂Ω;ε(w)then means a strain ten-

sor, introduced as εij (w) = (∂wi/∂xj+∂wj/∂xi)/2

with i, j ∈ {1,2,3}. All virtual displacements can be

restricted to those with zero values on Θ; the same is

expected from (still unknown) real displacements u.

In (4) the simplified notation of integrals

((ε(w), σ)) =

i,j=1 ZΩ

εij (w)σij dx ,

(w, f) =

i,j=1 ZΩ

wifidx ,

hw, gi=

i,j=1 ZΓ

wigids(x)

is used; moreover from the Hooke law we need to

evaluate σ=Cε(u)using an order 4 symmetric ten-

sor with 21 different material characterisics in gen-

eral, which can be reduced to 2 characteristics E, µ

again (cf. Section 2). Thus, we obtain a weak for-

mulation of an elliptic mixed (Dirichlet and Neu-

mann) boundary value problem for a partial differ-

ential equation of elliptic type, whose solvability can

rely on the classical Lax-Milgram theorem in spe-

cial Sobolev spaces. Its numerical analysis can be

preformed using the standard FEM techniques effec-

tively: for the unknown values of uin discrete points

xwe come to the sparse system of linear algebraic

equations.

However, such interpretation of (4) covers the pure

elastic static case only, without implementation of de-

velopment of any fracture. As certain remedy, [61],

[62], we can rewrite (4) into its quasi-static form,

introducing, due to the Kelvin parallel viscoelastic

model, σ=C(ε(u) + βε(v)), understanding vas

the displacement rate, i. e. the time derivative of u,

utilizing the homogeneous Cauchy initial condition

(all zero values of uin the initial zero time), αbe-

ing a structural damping factor, as an additional ma-

terial characteristic. Consequently, (4) can be un-

derstood as a weak formulation of a parabolic mixed

boundary value problem, with an unknown displace-

ment u(t)developed in time t≥0, whose solvability

can rely on the method of discretization in time and

on the properties of special Rothe sequences (namely

piecewise simple abstract functions and linear La-

grange splines composed of such functions) in special

Bochner-Sobolev spaces. Their practical construction

is easy: taking usfor any t=sh approximately, hbe-

ing certain time step, s∈ {1,2, . . .}, and vssimilarly,

too, we are able to rewrite (4) as

((ε(w), αCε(vs))) (5)

+ ((ε(w), Cε(us))) = (w, f ) + hw, gi

for hvs=us−us−1, starting from the zero-valued u0

everywhere on Ω. Therefore we have to analyse, step-

by-step, linear elliptic problems; the standard FEM

techniques are available again, with the similar results

as above.

Unfortunately, this is the end of simple linear com-

putations. Both i) the formation of microscopic frac-

tured zones and ii) the initiation, opening and closing

of macroscopic cracks needs certain modifications of

(5), disturbing its linearity substantially. The case

i) can be handled by careful introducing of certain

non-local regularizing damage factor Dwith values

between 0and 1, working with some stress invari-

ants and with the Eringen model typically, respecting

namely the specific material behaviour under tension

and compression. This results in some stiffness loss,

expressed by the replacement of Cby (1 −D(t)) C.

The case ii), at least for the a priori known poten-

tial positions of cracks, forces an additional right-

hand-side term of (5): similarly to hw, giwe need

hw, γ(Du)on Λinstead of Γ, with Λrepresenting all

internal interfaces, where Du are differences in dis-

placements on Λand γ(.)must be seen as a non-trivial

new material characteristics, including the cohesive

properties of Λ. Denoting still by the upper index s

our approximations for t=sh, for any such time we

can implement the iterative procedure to

((ε(w),(1 −Ds

×)αCε(vs))) (6)

+ ((ε(w),(1 −Ds

×)Cε(us))) = (w, f ) + hw, gi

+hw, γ(Dus

×)iΛ

where (if no better information is available) us−1can

be taken as an initial guess of us

×, then replaced by

us, obtained from (6), for the next iterative step, etc.

Whereas the main difficulty of i) is the compli-

cated design and evaluation of D(.), for the standard

FEM applications ii) suffers for the unpleasant duty of

repeated remeshing, to cover potential creation of still

new parts of Λ, which can be very expensive. This

highlights the priority of XFEM (and similar meth-

ods), thanks to the advanced choice of nodal func-

tions. Nevertheless, the basic form of (6) stays un-

changed.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

249

Volume 18, 2023

Let us also mention the need of further improve-

ments for numerous engineering applications, ex-

ceeding the scope of this article, namely: a) the im-

plementation of finite strains, b) the replacement of

empiric linearized constitutive relations as Hooke,

Kelvin, etc., ones, by proper relations coming from

thermodynamic considerations, c) the passage from a

quasi-static to a dynamic formulation, adding the in-

ertia forces, mass damping and a more general dissi-

pative mechanism in general. Although a), b), c) are

not avoided by recent software packages, the related

mathematical existence and convergence theory con-

tains still open questions because no system of propo-

sitions, theorems, etc., comparable with that for linear

and quasilinear problems, is available; consequently

one must rely on experimental validation of compu-

tational results only.

Figure 13: The application of Mazars exponential

model.

Asample with a cement matrix and steel fibres was

selected for computational modelling. Numerical re-

sults show the surface propagation of cracks in the

damaged body depending on the location fibre and

material properties. The reinforcing effect of the fi-

bres plays a significant role in the direction of crack

propagation. A sample with a cement matrix and steel

fibres was selected for computational modelling. The

attention is paid in particular to Eringen’s model for

Figure 14: The application of non-local approaches.

generating the multiplicative damage factor, related

quasi-static analysis, the existence of weak solution

of the corresponding boundary value and initial value

problem with a parabolic partial system differential

equations.

The proposed procedure thus combines the pos-

sibilities of several approaches for modelling crack

propagation in fibre composites. The XFEM method

is primary, the stress in front of the crack front is recal-

culated according to the non-local approach, in the en-

tire body according to the exponential or power law of

violation, [63], [64], [65], [66], which implies some

degree of averaging (especially as to stress) ahead of

the crack front. The following Fig. 13 and Fig. 14

present some results, due to the well-tried so-called

Mazars model, [67], [68], [69], [70].

Mazars damage model is based on a strain formu-

lation and generally is used for the physically nonlin-

ear analysis of concrete structures. The main objec-

tive is modification of damage model, in which both

tension and compression damage evolution laws are

regularized using a classical fracture energy method-

ology. Its hypothesis is founded on the base of an

elastic damage isotropic behaviour. This model as-

sumes the following premises: i) damage evolves/oc-

curs only due to positive strains in the principal di-

rections, which indirectly promotes “smeared crack

grow”; ii) only one scalar damage variable is defined,

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

250

Volume 18, 2023

this is due to the damage model being isotropic; iii)

represents the material as totally damaged, and it is

limited in the interval; iv) no permanent strains are

admitted, and the unloading path is linear, therefore,

no hysteresis loops.

8 Conclusions

The practical application of this work is to show some

techniques suitable for the crack growth modelling in

case of heterogeneous materials, in order to find crit-

ical parameters determining the behaviour of materi-

als and damage. The basic step that conditions the

realistic modelling of the behaviour of the material is

finding or using a suitable criterion that determines

the formation of a new crack and then the subsequent

growth of resulting crack. It stands to reason that in

searching for a suitable criterion (often based on frac-

ture mechanics), the first approximation a model with

an a priori crack or stress concentrator is used.

Special attention is concentrated on modelling

building fibre composites, especially on the use of a

modified FEM and on the presentation of some tech-

niques that can improve XFEM. However, the prob-

lem of local stress determination for these composites

is more general for the entire group of fibre compos-

ites regardless to understanding of the size effect. The

field of application of XFEM is still in the forefront of

interest of the mentioned group of authors, especially

the solution of crack propagation in fibre composites.

The combination of the Mazars model, which is

very popular and is implemented in numerous sys-

tems for FEM, with the refinement and determination

of the stress distribution in front of the crack tip is

a promising solution for structural composites mod-

elling. Generally speaking, the numerical stability

of crack propagation modelling in the case of some

stress averaging in front of the crack tip is much bet-

ter than in the case of sharp concentrators, especially

their effect on XFEM.

References:

[1] A. A. Griffith: The phenomena of rupture and

flow in solids. Phil. Trans. Roy Soc. Lond. A221

(1920), 163–198.

[2] G. C. Sih: Strain-energy density factor applied

to mixed mode crack problems. Int. J. Fract. 10

(1974), 304–321.

[3] J. R. Rice: A path independent integral and the

approximate analysis of strain concentration by

notches and cracks. J. Appl. Mech. 35 (1968)

379–386.

[4] G. I. Barenblatt: The mathematical theory of

equilibrium of cracks in brittle fracture. Adv.

Appl. Mech. 7 (1962), 55–129.

[5] F. Erdogan, G. C. Sih: On the crack extension in

plates under plane loading and transverse shear.

J. Basic Eng. 85 (1963), 519–527.

[6] I. Enescu: Some researches regarding stress

intensity factors in crack closure problems.

WSEAS Trans. Appl. Theor. Mech. 13 (2018),

187–192.

[7] C. Betegon, J. J. W. Hancock: Two-parameters

characterization of elastic-plastic crack tip field.

J. Appl. Mech. 58 1991, 104–110.

[8] S. Taira, R. Ohtani, T. Kitamura: Application

of J-integral to high-temperature crack propaga-

tion, Part I – Creep crack propagation. Journal

Engineering Materials Technology 101 (1979),

154–161.

[9] J. D. Landes, J. A. Begley: Mechanics of

Crack Growth. ASTM International, West Con-

shohocken, 1976.

[10] H. Riedel: Fracture at High Temperatures,

Springer, Berlin, 1987.

[11] O. Kolednik, R. Schöngrundner, F. D. Fischer:

A new view on J-integrals in elastic–plastic ma-

terials. Int. J. Fract. (2014), 77–107.

[12] J. Scheel, A. Schlosser, A. Ricoeur: The J-

integral for mixed-mode loaded cracks with co-

hesive zones. Int. J. Fract. 227 (2021), 79–94.

[13] B. Healy, A. Gullerund, K. Koppenhoefer,

A. Roy, S. Roychowdhury, J. Petti, M. Walters,

B. Bichon, K. Cochran, A. Carlyle, J. Sobotka,

M. Messner, T. Truster, R. Dodds: WARP3D Re-

lease 18.3.6. User Manual, 3-D Dynamic Non-

linear Fracture Analysis of Solids. University of

Illinois, 2023.

[14] A. Hermosillo, M. Romo, R. Magaña, J. Carrera:

Automatic remeshing algorithm of triangular el-

ements during finite element analyses. Rev. Int.

Metodos Numer. Calc. 34 (2018).

[15] L. M. Kachanov: Introduction to Continuum

Damage Mechanics. Martinus Nijhoff, Dor-

drecht, 1986.

[16] A. L. Gurson: Continuum theory of ductile rup-

ture by void nucleation and growth: Part I –

Yield criteria and flow rules for porous ductile

media. J. Eng. Mater. Technol. 99 (1977), 2–15.

[17] V. Tvergaard, A. Needleman: Analysis of the

Cup-Cone Fracture in a Round Tensile Bar. Acta

Metall. 32 1984, 157–169.

[18] L. Vlček: Numerical Analysis of the Bodies with

Cracks. Ph.D. thesis, Brno University of Tech-

nology, 2004.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

251

Volume 18, 2023

[19] W. Brocks, D. Klingbeil, G. Künecke, D. Z. Sun:

Application of the Gurson model to ductile

tearing resistance. In: Constraint Effects in

Fracture: Theory and Applications (M. Kirk,

A. Bakker, Eds.), 232–252. ASTM, Dallas

(Texas), 1995.

[20] Z. L. Zhang: A complete Gurson model. In:

Nonlinear Fracture and Damage Mechanics

(M. H. Alibadi, Ed.), 223–248. WIT Press,

Southampton, 2001.

[21] B. F. Sorensen, T. K. Jacobsen: Composites:

Part A 29 A (1998), 1443–1451.

[22] O. Rabinowitch: Eng. Fract. Mech. 75 2008,

2842–2859.

[23] Q. R. Barani, A. R. Khoei, M. Mofid: Modelling

of cohesive crack growth in partially saturated

porous media: A study on the permeability of

cohesive fracture. Int. J. Fract. 167 2011, 15–31.

[24] A. Hillerborg, M. Modeer, P. E. Peterson: Cem.

Concr. Res. 6 (1976), 773–782.

[25] V. Kozák: Ductile crack growth modelling using

cohesive zone approach. In: Composites with

Micro- and Nano-Structure (V. Kompiš, Ed.),

191–208. Springer, Berlin, 2008.

[26] V. Kozák, Z. Chlup, P. Padělek, I. Dlouhý: Pre-

diction of traction separation law of ceramics us-

ing iterative finite element method. Solid State

Phenom. 258 (2017), 186–189.

[27] Z. P. Bazant, Y.-N. Li: Cohesive crack model

with rate-dependent opening and viscoelasticity:

I. Mathematical model and scaling. Int. J. Fract.

86 1997, 247–265.

[28] M. Ortis, Y. Leroy, A. Needleman: A finite

element method for localized failure analysis.

Comput. Methods Appl. Mech. Eng. 61 (1987),

189–214.

[29] A. Hansbo, P. Hansbo: A finite element method

for simulation of strong and weak discontinu-

ities in solid mechanics. Comput. Methods Appl.

Mech. Eng. 193 (2004), 3523–3540.

[30] P. M. A. Areias, T. Belytschko: Two-scale shear

band evolution by local partition of unity. Int. J.

Numer. Methods Eng. 66 (2006), 878–910.

[31] A. R. Khoei: Extended Finite Element Method:

Theory and Applications. J. Wiley & Sons,

Hoboken 2015.

[32] Y. Shen, A. Lew: Stability and convergence

proofs for a discontinuous-Galerkin-based ex-

tended finite element method for fracture me-

chanics. Comput. Methods Appl. Mech. Eng. 199

(2010), 2360–2382.

[33] M. Stolarska, D. L. Chopp, N. Moës N, T. Be-

lytschko: Modelling of crack growth by level

sets in the extended finite element method. Int.

J. Numer. Methods Eng. 51 (2001), 943–960.

[34] Q. Z. Xiao, B. L. Karihaloo: Improving the ac-

curacy of XFEM crack tip fields using higher

order quadrature and statically admissible stress

recovery. Int. J. Numer. Methods Eng. 66 (2006),

1378–1410.

[35] C. Xiao, L. Wen, R. Tian: Arbitrary 3D crack

propagation with improved XFEM: accurate and

efficient crack geometries. Comput. Methods

Appl. Mech. Eng. 377 (2021), 113959 / 1–32.

[36] I. Babuška, J. M. Melenk: The partition of unity

method. Int. J. Numer. Methods Eng. 40 (1997),

727–758.

[37] T. P. Fries, T. Belytschko: The intrinsic XFEM: a

method for arbitrary discontinuities without ad-

ditional unknowns. Int. J. Numer. Methods Eng.

68 (2006) 1358–1385.

[38] R. Tian, L. Wen, L. Wang: Three-dimensional

improved XFEM (IXFEM) for static crack prob-

lems. Comput. Methods Appl. Mech. Eng. 343

(2019), 339–367.

[39] T. P. Fries, T. Belytschko: The extended/gener-

alized finite element method: an overview of the

method and its applications. Int. J. Numer. Meth-

ods Eng. 84 (2010), 253–304.

[40] M. Jirásek: Damage and smeared crack models.

In: Numerical Modelling of Concrete Crack-

ing (G. Hofstetter, G. Meschke, Eds.), 1–49

Springer, Vienna, 2011.

[41] T. P. Fries, M. Baydoun: Crack propagation with

the extended finite element method and a hybrid

explicit-implicit crack description. Int. J. Numer.

Methods Eng. 89 (2012), 1527–1558.

[42] F. Shi, D. Wang, Q. Yang: An XFEM-based

numerical strategy to model three-dimensional

fracture propagation regarding crack front seg-

mentation. Theor. Appl. Fract. Mech. 118

(2022), 103250 / 1–17.

[43] V. B. Panday, I. V. Singh, B. K. Mishra: A

new creep-fatigue interaction damage model

and CDM-XFEM framework for creep-fatigue

crack growth simulations. Theor. Appl. Fract.

Mech. 124 (2023), 103740 / 1–13.

[44] M. H. Aliabadi: Boundary elements methods in

linear elastic fracture mechanics. Comprehen-

sive Structural Integrity 5 (2003), 89–125.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

252

Volume 18, 2023

[45] G. Molnár, A. Gravouil, R. Seghir, J. Réthoré:

An open-source Abaqus implementation of the

phase-field method to study the effect of plas-

ticity on the instantaneous fracture toughness in

dynamic crack propagation. Comput. Methods

Appl. Mech. Eng. 365 (2020), 113004 / 1–27.

[46] T. Belytschko, R. Gracie, G. Ventura: A review

of extended/generalized finite element methods

for material modelling. Modell. Simul. Mater.

Sci. Eng. 17 (2009).

[47] L. Saucedo-Mora, T. James Marrow: FEMME:

A multi-scale finite element microstructure

meshfree fracture model for quasi-brittle mate-

rials with complex microstructures. Eng. Fract.

Mech. 147 (2015), 355–372.

[48] A. Ortega, E. Roubin, Y. Malecot, L. Daudeville:

Amixed-mode E-FEM approach for the study of

local fracture processes in heterogeneous quasi-

brittle materials. Materials and Structures 55

(2022), 222 / 1–23.

[49] M. Puccia, A. Spada, G. Giambanco: Finite el-

ements with embedded interphases for strain lo-

calization in quasi-brittle materials. Eng. Fract.

Mech. 277 (2023) 108956 / 1–26.

[50] L. Bouhala, A. Makradi, S. Belouettar,

H. Kiefer-Kamal, P. Fréres: Modelling of

failure in long fibres reinforced composites by

X-FEM and cohesive zone model. Composites,

Part B 55 (2013), 352–361.

[51] T. T. Yu, Z. W. Gong: Numerical simulation of

temperature field in heterogeneous material with

the XFEM. Archives of Civil and Mechanical

Engineering 13 (2013), 199–208.

[52] Y. Yosibash: Numerical thermo-elastic analysis

of singularities in two-dimensions. Int. J. Fract.

74 (1995), 341–361.

[53] R. Brighenti, D. Scorza: Numerical modelling

of the fracture behaviour of brittle materi-

als reinforced with unidirectional or randomly

distributed fibres. Mechanics of Materials 52

(2012), 12–27.

[54] C. Ye, J. Shi, G. J. Cheng: An extended finite el-

ement method (XFEM) study on the effect of re-

inforcing particles on the crack propagation be-

haviour in a metal-matrix composite. Int. J. Fa-

tigue 44 (2012), 151–156.

[55] J. Vala, L. Hobst, V. Kozák: Detection of metal

fibres in cementitious composites based on sig-

nal and image processing approaches. WSEAS

Trans. Appl. Theor. Mech. 10 (2015), 39–46.

[56] A. C. Eringen: Theory of Nonlocal Elastic-

ity and Some Applications. Technical report 62,

University Press Princenton, 1984.

[57] K. Park, G. H. Paulino, J. R. Roesler: Cohesive

fracture model for functionally graded fibre re-

inforced concrete. Cem. Concr. Res. 40 (2010),

956–965.

[58] M. G. Pike, C. Oskay: XFEM modelling of short

microfibre reinforced composites with cohesive

interfaces. Finite Elem. Anal. Des. 106 (2005),

16–31.

[59] X. Li, J. Chen: An extensive cohesive damage

model for simulation arbitrary damage propaga-

tion in engineering materials. Comput. Methods

Appl. Mech. Eng. 315 (2017), 744–759.

[60] M. Morandotti: Structured deformations of con-

tinua: theory and applications. In: Mathemat-

ical Analysis of Continuum Mechanics and In-

dustrial Applications II (Proc. Int. Conf. CoM-

FoS (Continuum Mechanics Focusing on Sin-

gularities) 2016 in Fukuoka, P. van Meurs,

M. Kimura, H. Notsu, Eds.), 125–138. Springer,

Tokyo, 2016.

[61] J. Vala, V. Kozák: Computational analysis of

quasi-brittle fracture in fibre reinforced cementi-

tious composites. Theor. Appl. Fract. Mech. 107

(2020), 102486 / 1–8.

[62] J. Vala: On a computational smeared damage ap-

proach to the analysis of strength of quasi-brittle

materials. WSEAS Trans. Appl. Theor. Mech. 16

(2021), 283–292.

[63] H. H. Zhang, G. Rong, L. X. Li: Numerical

study on deformation in a cracked viscoelas-

tic body with the XFEM. Eng. Anal. Boundary

Elem. 34 (2010), 619–624.

[64] A. Afshar, A. Daneshyar, S. Mohammadi:

XFEM analysis of fiber bridging in mixed-

mode crack propagation in composites. Compos.

Struct. 125 (2015), 314–327.

[65] S. H. Ebrahimi: Singularity analysis of cracks

in hybrid CNT reinforced carbon fiber compos-

ites using finite element asymptotic expansion

and XFEM. Int. J. Solids Struct. 214-215 (2021),

1–17.

[66] V. Kozák, J. Vala: Crack growth modelling in

cementitious composites using XFEM. Proce-

dia Struct. Integrity, 43 (2023), 47–52.

[67] J. Mazars: A description of micro- and

macroscale damage of concrete structures. Eng.

Fract. Mech. 25 (1986), 729–737.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

253

Volume 18, 2023

[68] J. Mazars, F. Hamon, S. Grange: Anew 3D dam-

age model for concrete under monotonic, cyclic

and dynamic loadings. Mater. Struct. 48 (2015),

3779–3793.

[69] M. R. T. Arruda, J. Pacheco, L. M. S. Castro,

E. Julio: A modified Mazars damage model with

energy regularization. Eng. Fract. Mech. 259

(2022), 108129 / 1–22.

[70] Y. Wang: A 3D stochastic damage model for

concrete under monotonic and cyclic loadings.

Cem. Concr. Res. 171 (2023), 107208 / 1–20.

Contribution of individual authors

Vladislav Kozák elaborated both for the physical

formulations and the computational simulations. Jiří

Vala was responsible for the mathematical analysis.

Conflicts of interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Sources of funding for research presented in

a scientific article or scientific article itself

This research has been supported from the project

of specific university research at Brno University of

Technology No. FAST-S-22-7867.

Creative Commons Attribution License 4.0

This article is published under the terms of Creative

Commons Attribution License (Attribution 4.0 Inter-

national, CC BY 4.0).

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.23

Vladislav Kozák, Jiří Vala

E-ISSN: 2224-3429

254

Volume 18, 2023