Cohesive fracture growth in a thermoelastic bimaterial medium

$: Cohesive fracture growth in a thermoelastic bimaterial medium$
Computers and Structures 82 (2004) 1875–1887

www.elsevier.com/locate/compstruc

Cohesive fracture growth in a thermoelastic bimaterial medium

S. Secchi a, L. Simoni b,*, B.A. Schrefler b

a CNR, Ladseb, Padova, Italyb Department of Structural and Transportation Engineering, University of Padova, Via F. Marzolo, 9, Padova 35131, Italy

Received 12 December 2002; accepted 24 March 2004

Available online 28 July 2004

Abstract

The paper presents a fully-coupled numerical model for the analysis of fracture initiation and propagation in a two

dimensional non-homogeneous elastic medium driven by mechanical loads and transient thermal fields. Cohesive crack

behaviour is assumed for the solid. The solution of the coupled problem is obtained by using the finite element method

without using special approximation techniques nor interface elements. Evolution of the process zone results in contin-

uous changes of the domain topology. This is accounted for by updating the boundary geometry and successive remesh-

ing of the domain. Optimality of the shape of the finite elements generated is controlled and the mesh density is adjusted

adaptively on the basis of an error estimator. Two numerical applications are presented, which demonstrate the effec-

tiveness of the proposed procedure. In the first, comparison is made with a laboratory experiment, whereas the second

handles a problem with crack path completely unknown.

� 2004 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved.

Keywords: Discrete crack models; Cohesive fracture; Thermal loads; Adaptive remeshing; Bimaterial sample; PMMA

1. Introduction

Many engineering structures experience thermal

loads during their life, i.e., imposed temperature along

the boundaries and/or heat production within their inte-

rior. Whereas the former situation is the consequence of

changes in the environment, the latter can be attributed

to different causes for instance to chemical reactions, as

in the hydration of cement for concrete structures, or to

internal transformation of mechanical work. For the lat-

ter case, we refer for instance to the significant tempera-

ture increase which has been observed at the tip of a

stationary crack subjected to dynamic loading [1].

0045-7949/$ - see front matter � 2004 Civil-Comp Ltd. and Elsevier

doi:10.1016/j.compstruc.2004.03.059

* Corresponding author.

E-mail address: [email protected] (L. Simoni).

The applied thermal loads may result in fracture enu-

cleation and growth, for instance due to the presence of

flaws along the boundary and/or within the domain and/

or simply due to the attainment of the tensile strength

for brittle materials. While on one hand, temperature

rise near a crack tip significantly affects the micromech-

anisms of fracture [2], on the other, the formation and

propagation of fractures involve changes in the bound-

ary and related conditions. For example, the spalling

of concrete during a tunnel fire suddenly exposes parts,

previously insulated by the broken off chips of concrete,

to very high temperatures. Similar problems arise in de-

lamination of composites and with expanding insulation

varnishes. All these problems exhibit a strong coupling

between mechanical and thermal effects. The thermal

field is further characterized by its variation in space

and time, hence numerical models assuming steady state

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1876 S. Secchi et al. / Computers and Structures 82 (2004) 1875–1887

conditions for temperature distribution suffer of strong

limitations in simulating the real physical phenomena.

Models of this type have been presented recently, e.g.,

[3–5] using boundary element method in a two and three

dimensional context, both for elastic and elastoplastic

materials. These models are further characterized by

fixed fracture configuration and are usually intended

for the calculation of the stress field and stress intensity

factors.

In the present approach the problem of fracture for-

mation and propagation is formulated within the frame

of cohesive fracture, which is very realistic for geoma-

terials, but is also widely applied to metallic materials

[6]. The medium is assumed as non-homogeneous and

the fracture(s) can enucleate and propagate according

to the maximum tensile stress criterion. At the same

time, the temperature field can evolve in space and time

according to energy balance equation, where heat

source terms and the effect of mechanical coupling

are accounted for. Internal heat production can depend

on thermoelastic effects, plastic work and friction be-

tween the fracture lips. The governing equations are

stated by using the phenomenological approach [7]

and the numerical solution for the mechanical problem

is limited to a 2-D context. Governing equations are

discretized in space by the finite element method and

in time by finite differences and monolithic solution

procedures are used. No special approximations are

used, for instance as in [3], to represent the field singu-

larities: bilinear interpolations are assumed for both

field variables and the solution is controlled and im-

proved by using an efficient error measure [8] with an

a posteriori refinement technique. Spatial discretization

is continuously updated as the phenomenon evolves

and the domain of definition changes. An efficient mesh

generator [9] is used to this scope. This technique, to-

gether with the assumption of non-homogeneous mate-

rial, differentiates the present approach from the one

used in [10]. As a result, the proposed model does

not suffer from the limits of those previously referred

to and the initial mesh can also be defined without spe-

cial care. In particular, cracks may enucleate every-

where depending only on the stress field and

propagate along paths and with a velocity of the tip

that is unknown a priori. The determination of the

crack path and the velocity of the tip propagation rep-

resent an important part of the solution, as the temper-

ature and stress fields and allows for correct updating

of the domain, boundaries and related conditions.

2. Field equations

Governing equations are obtained within the frame-

work of the phenomenological theory [7]. Small dis-

placements and displacement gradients are assumed.

2.1. Local balance equations

The local linear momentum balance at the generic

point of the domain X reads as

rji;j þ qgi ¼ q€ui in X; ð1Þ

where rij is the stress tensor, q the density of the mate-

rial, gi the gravity acceleration vector and ui the displace-

ment vector. Dots represent time derivatives. The

natural boundary conditions are

rjinj ¼ pi on C and rjinj ¼ ci on C0: ð2Þ

C is the external boundary and pi the forced traction, C0

the boundary of the process zone and ci the cohesive

force. Forced conditions represent fixed displacement

along the constrained boundary and completely define

the problem.

The local form of energy balance equation (first law

of thermodynamics), under the assumption of small

strains requires that

q _e ¼ rij _eij þ qr � qj;j in X; ð3Þ

where e is the specific internal energy, eij is the infinites-

imal strain tensor, r is the strength per unit mass of a

distributed internal heat source and qj the heat flux vec-

tor. Forced conditions fix the temperature along the

boundary, whereas natural conditions represent the im-

posed heat flux. The latter include also the convective

heat transfer towards the surrounding. Both the fluxes

will be defined by means of suitable constitutive mod-

els.

The topology of the domain X and boundary

change with the evolution of the fracture phenomenon.

In particular, the fracture path, the position of the

process zone and the cohesive forces are unknown

and must be determined during the analysis. After def-

inition of the constitutive models, the governing Eqs.

(1) and (3) are solved simultaneously to obtain the dis-

placement and temperature fields together with the

fracture path.

2.2. Constitutive models

The model must be completed by the constitutive

relations for solid mechanical behaviour, cohesive forces

as function of the crack opening, specific internal energy

and heat flux vectors.

2.2.1. Continuous medium: mechanical behaviour and heat

transfer

In the framework of discrete crack models, a hyper-

elastic material is assumed for the solid away from the

process zone. The stress tensor rij depends on effective

strain according to the relationship

S. Secchi et al. / Computers and Structures 82 (2004) 1875–1887 1877

rij ¼ cijrsðers � drseT Þ; eT ¼ a3T ; ð4Þ

where eij is the total strain, eT the strain associated to

temperature T changes, according to cubic expansion

coefficient a, drs the Kronecker symbol. The elastic coef-

ficients cijrs depend on the strain energy function W and

can be expressed in terms of the Lame constants k and las

cijrs ¼o2Woeijoers

¼ lðdisdjr þ dirdjsÞ þ kdijdrsdij ði; j; r; s ¼ 1; 2; 3Þð5Þ

and depend on strain level and temperature [2,7].

In the case of plane problems , as in the applications

presented, Eq. (5) is condensed in the standard way,

assuming the pertinent conditions for plane stress or

plane strain states.

As a constitutive assumption for heat flux, Fourier�slaw is assumed

qi ¼ �kijT j; ð6Þ

kij being the effective thermal conductivity tensor, which

can be dependent on temperature.

As far as convective flux is concerned, Newton�s lawis used

qconvi ¼ hðT � T1Þ; ð7Þ

being h the convective heat transfer coefficient and T1the temperature in the far field of the undisturbed sur-

roundings.

2.2.2. Cohesive fracture model: mode I crack opening

Let the complete domain be composed of a set of dif-

ferent homogeneous subdomains. Within a generic one

of these, a fracture can initiate or propagate under the

assumption of mode I crack opening, provided that

the tangential relative displacements of the fracture lips

are negligible. In the process zone cohesive forces are

transmitted, which are orthogonal to the fissure sides.

Following the Barenblatt–Dugdale model [11,12] and

cr

σ0

G

0δσ

δσ

σ

(a) (b

Fig. 1. Fracture energy (a) and loading/unloading

Hilleborg et al. proposals [13], the cohesive law is

(Fig. 1)

r ¼ r0 1� drdrcr

� �; ð8Þ

r0 being the maximum cohesive traction (closed crack),

dr the relative displacement normal to the crack, drcr themaximum opening with exchange of cohesive tractions

and G=r0·drcr/2 the fracture energy. Fig. 1 also pre-

sents the unloading/reloading paths represented by

r ¼ r0 1� dr1drcr

� �drdr1

; ð9Þ

where dr1 is the maximum attained opening in the previ-

ous loading process.

The individual homogeneous components differ only

due to different values attributed to the fundamental

parameters defining the mechanical behaviour (Eqs. (8)

and (9)). The aspect of the constitutive relationship re-

mains the same.

2.2.3. Cohesive fracture model: mixed mode crack opening

When tangential relative displacements of the sides of

a fracture in the process zone cannot be disregarded,

mixed mode crack opening takes place. This is usually

the case with a crack moving along an interface separat-

ing two solid components. Whereas the crack path in a

homogeneous medium is governed by the principal

stress direction, the interface has an orientation that is

generally different from the principal stress direction.

The mixed cohesive mechanical model involves the

simultaneous activation of normal and tangential dis-

placement discontinuity with respect to the crack and

corresponding tractions. The interaction between the

two cohesive mechanisms is treated as in Margolin [14]

and Dienes [15], by defining an equivalent or effective

opening displacement d and the scalar effective traction

t as

d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib�2d2s þ d2r

q; t ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2t2s þ t2r

q: ð10Þ

σ

σ

σ

0

1

2

δσ1 δσ2 δσcr

A

B

0δσ

σ

)

law (b) for each homogeneous component.


The resulting cohesive law is

t ¼ td

b2ds þ dr� �

: ð11Þ

b is a suitable material parameter and the cohesive law

takes the same aspect as in Fig. 1, by replacing displace-

ment and traction parameters with the corresponding

effective ones. For the choice of the value of parameter

b the original proposals of [14,15] are referred to, to-

gether with Ref. [10].

2.2.4. Internal energy and sources

In a generic thermodynamic phenomenon, for an

infinitesimal increment of deformation, the specific

internal energy can be assumed as a function of absolute

temperature, a set of observable variables (the displace-

ment field) and internal variables vi [7]. It is assumed for

the following applications that mechanical terms in en-

ergy balance equation are negligible compared with ther-

mal effects. This is not always the case. Sometimes, in

experimental investigations a significant temperature in-

crease has been observed in the zone of crack initiation

and propagation [1]. To account for these situations, dis-

sipation terms related to mechanical effects have to be

introduced in the energy balance equation, following

[16] for instance.

When mechanical terms are neglected, internal en-

ergy depends on temperature only and is related to heat

capacity at constant volume Cv. As a consequence, and

taking into account Eq. (6) for the flux term, the energy

balance equation takes the form

qðCv_T Þ ¼ qr � qj;j: ð12Þ

Volume heat sources of Eq. (12) can represent very dif-

ferent phenomena, e.g., heat production due to hydra-

tion of concrete, heat production due to plastic work

[16], hence sometimes are linked to coupling effects be-

tween stress and thermal fields. Source terms may also

arise along the boundary and represent frictional effects.

The rate at which heat is generated at the contact sur-

faces is

r0 ¼ t � kvk; ð13Þwhere t is the contact traction and ivi is the jump in

velocity across the contact. This effect can be incorpo-

rated in the weak form of the energy balance, which, fol-

lowing the usual approach takes the formZXqðCv

_T ÞhdXþZC0r0hdC0 þ

ZC0qconvhdCþ

ZC0qhdC

¼ZXqdivhdXþ

ZXshdX; ð14Þ

being h an admissible virtual temperature qconv and q the

convective and imposed heat flux normal to the bound-

ary. Except for the convective and boundary fluxes, this

equation is similar to the one used by Camacho and

Ortiz [10].

3. Discretized governing equations

Eqs. (1) and (3) (or (14)) are discretized in space by

the usual Galerkin residual method, incorporating the

constitutive equations at the same time. The values of

nodal displacements d and temperature T are the un-

knowns. This results in the following system (dot repre-

sents time derivative) at element level

KE �TUE

0 TSE

� � _dE_TE

" #þ

0 0

0 TRE

� �dE

TE

� �

¼_FE

0

" #þ

0

TGE

� �; ð15Þ

where adopting the usual symbols [17], the submatrices

of Eq. (15) are

KE ¼ZV E

BTDBdV ; ð15aÞ

TUE ¼ZV E

BTDma3NdV ; ð15bÞ

TSE ¼ZV E

NTqNdV ; ð15cÞ

TRE ¼ZV E

ðrNÞTkrNdV ; ð15dÞ

TGE ¼ZV E

NT sdV �ZCcrack

NT r0 dC�ZCNT qE dC

�ZCNT qconvE dC; ð15eÞ

_FE ¼ZV E

NT _fE dV þZCE

NT _tE dCþZCEcrack

NT _cE dC:

ð15fÞ

In Eq. (15f) �E represents the cohesive traction rate and is

applied along Ccrack of the generic element. In the

present formulation, non-linear terms arise through

cohesive forces in the process zone only. Under the

hypothesis of dropping the effects of stresses in the

energy equation, coupling comes from the boundary

terms defined along Ccrack.

Global equations are assembled and integrated in

time by means of the generalized trapezoidal rule

[17] and Newton–Raphson method is used for the

solution of the non-linear algebraic system of

equations. In a generic step, once the displacement

field is obtained, crack opening is calculated and, by

using Eqs. (10) and (11), cohesive tractions and

length process zone are determined. This is the part

of the crack where cohesive tractions are different

from zero.


4. Mesh generation and remeshing operations

Initial discretization and successive remeshing are car-

ried out using the procedure for unstructured Delaunay

mesh generation presented in detail in [9]. This algorithm

produces the initial mesh with element dimensions con-

trolled by a spacing function initially defined by the user

and then automatically updated to account for geometri-

cal domain singularities, improves mesh quality and

curved boundary representation. At each successive step

of the solution the domain is first updated, accounting

for fracture movements, then is triangulated by control-

ling the dimension of the elements using the a posteriori

error measure proposed by Zhu and Zienkiewicz [8]. The

refinement strategy suggested in [8] is followed and the

required element size is obtained by updating the current

spacing function values. The value of the spacing func-

tion represents the required dimension of each triangular

element. As far as the area surrounding the process zone

is concerned, this strategy is no more applicable. Near

the fracture tip, the element spacing is simply controlled

by locating a point source of elements [9], the weight of

which is determined in order to properly model the proc-

ess zone and the cohesive forces and to yield mesh inde-

pendent results. In the applications presented, this last

requirement always resulted the most strict one. The

meshing procedure proceeds then to regularize the mesh

size distribution. The discretization of the process zone

will be further discussed in Section 6.

A quadrilateral mesh, sometimes preferred in frac-

ture mechanics, can also be obtained, preserving mesh

quality control. In both cases, bilinear approximation

of the field variables is used.

To reduce computation time, the successive refine-

ment and/or derefinement operations can be limited to

suitable subdomains containing the singularities zones.

The dimensions of these areas and local density of dis-

cretization can be decided on the basis of the solution

at the actual and previous steps. This possibility is

strictly dependent on the used data structure [9], how-

ever, at present, the remeshing operations are made for

each crack increment in the whole domain.

Some changes to the procedure shown in [9] are re-

quired by the presence of non-homogeneity of the med-

ium [18]. Each homogeneous region is meshed

independently of the others, using different spacing func-

tion values within each one, with the only requirement

of assuming the same nodes along the interfaces. Fur-

ther, it is necessary to check that the triangulation com-

pletely represents the boundary of each region and the

complete domain. It should be remembered that Delau-

nay triangulation is a unique geometric construction for

an assigned set of points, hence it does not guarantee the

consistency of a generic contour defined through a sub-

set of points of the initial set. To this end, the discretiza-

tion procedure has been updated in this way:

� the nodes are inserted, beginning with those at the

boundary. The topological entities defining the

boundary handle the node distribution by accounting

for the spacing function at the end points. The

boundary entities recognize the regions connected

to them by means of non-zero pointers in the data

structure, not through searches in special lists and

without any orientation rules, and require the re-

gion/s topological entities to locate the new common

node. Each region then handles the generation of tri-

angles and contours separately, and checks conform-

ity, following the usual procedure (Bowyer–Watson

algorithm) [9];

� consistency check along each interface. Once the

node location phase has been accomplished, there is

no guarantee that the interface boundary is repre-

sented by the triangulation edges. The geometric con-

sistency is controlled and, if necessary, can be

reached by inserting new nodes as in [9]. In the pres-

ence of non-homogeneities, the region entities indi-

cate the existence of the anomaly and require the

boundary to locate a new node in a fixed position.

The boundary entity continues with the insertion of

the node, but informs all other regions linked to it

of this operation. This procedure is repeated until

the consistency of all boundaries is reached.

From the computational point of view and to im-

prove the efficiency of the generator, all above men-

tioned operations are performed by special Euler

operators which exploit the information contained in

the data structure representing the fundamental

entities, Vertices, Edges and Faces. Owing to the

data structure used, conditional loops are almost en-

tirely avoided, resulting in a highly efficient computer

code.

5. Nucleation and propagation of fractures

Let us consider a non-homogeneous two dimensional

domain under plane strain or stress assumptions. In a

generic solution step, once the displacement field has

been obtained, strain and stress fields are calculated.

The latter are constant within each triangular element,

given that a bilinear approximation of field variables is

used. Nodal values of stresses are calculated by applying

a smoothing procedure to elements of the same material

only. Stress discontinuities can, therefore, be present

only at nodes of an interface, whereas the smoothed

stress field is continuous within each homogeneous re-

gion. Assuming that unstructured meshes are used, each

node can be shared in general by n elements with nm dif-

ferent material properties. The topological entity node is

required to handle nm stress and strain tensors together

with fracture enucleation/propagation. This implies that,


at the same node, all cracks required by the stress field

can nucleate/propagate.

Even in the presence of non-homogeneous materials,

when the crack moves within a homogeneous subregion

it is assumed that nucleation and propagation take place

according to Barenblatt and Dugdale�s [11,12] cohesive

model for homogeneous and isotropic bodies under

mode I assumptions. Crack nucleation takes place when

the traction principal stress reaches the ultimate value rtof the material, the propagation direction being orthog-

onal to the same principal stress. The opened lips are as-

sumed as interacting and transmitting tractions as

dependent on crack opening, and vanishing when this

reaches the critical value drcr (Fig. 1). Once this value

is exceeded, the only possible tractions for the opposite

lips are in compression when the fracture reaches a con-

tact position when closing. When the fracture path ap-

proaches an interface between two different materials

the direction of propagation is different from the previ-

ous case. It is chosen to maximize the energy fracture re-

lease.

The computational procedure to find the point of

crack formation consists in finding the node Ni with the

highest principal positive stress. When this exceeds the

limiting value of the material, a new crack is created or

an existing one moves its tip. A detailed analysis of the

possible situations and related procedure can be found

in [18].

6. Applications

It is quite difficult to find a benchmark test for the

model presented. As previously pointed out, current

Fig. 2. Geometry of the three-point bend

Table 1

Mechanical and thermal properties of the materials at 25 �C

Young�s modulus

(GPa)

Poisson coefficient Thermal expan

coefficient (�C�

Al6061 69 0.33 2.36·10�5

PMMA 3.338 0.35 7.45·10�5

numerical models evaluate the stress intensity factors

for thermal loads in a stationary situation, both for

the crack geometry and thermal field, hence they

do not match the complexity of the present formula-

tion.

For comparison purposes, reference is made to the

laboratory experiment presented in [2]. A three-point

bending test is performed on a bimaterial specimen sub-

jected to a thermo-mechanical loading. One part of the

sample is made of aluminium 6061 and the other in poly-

methylmethacrylate (PMMA), bonded with methacry-

late adhesive. The geometry is presented in Fig. 2: the

sample has a notch with a sharp tip of 1 mm width

and 30 mm height shifted 3 mm from the interface in

the PMMA zone. The two materials present very differ-

ent Young�s moduli and thermal expansion coefficients,

so that, when the system is subjected to heat, stresses

arise near the interface as a result of the mismatch in

thermal expansion. Mechanical and thermal properties

of the two materials are presented in Table 1. Further,

a variation of PMMA Young�s modulus with tempera-

ture has been experimentally observed and values shown

in Table 2 were measured. Fracture energy was not de-

fined in [2], owing to the different approach to fracture

mechanics, hence it has been calculated in the present

application. Fracture energy has been obtained on the

basis of the stress intensity factors declared in [2] for

the two limiting cases (T=25 and 60 �C), whereas the

ultimate stress r0 is assumed of 25 MPa at the tempera-

ture of 60 �C. Linear variation with temperature of both

these parameters has been adopted. A second assump-

tion of the numerical simulation is the linear elasticity

of the PMMA. Following [19], a more suitable mechan-

ical model requires non-linear elasticity for this material.

ing test for a bimaterial specimen.

sion1)

Thermal

conductivity (W/mK)

Fracture energy

(MPam�1)

r0 (MPa)

167

0.2 320.0 50.0

Table 2

Dependence of PMMA Young�s modulus on temperature

25 �C 35 �C 45 �C 60 �C

Young�s modulus (GPa) 3.4 3.4 2.6 2.05


For the moment, however, the application is intended to

demonstrate the capability of the model presented, not

to closely match the laboratory results. For the same

reason no particular care is used in the determination

of parameter b (Eq. (10)), which value is assumed equal

to 0.866 according to [10]. Nevertheless, as it will be

shown in the following, the obtained results compare

quite well with the experimental ones also under the

assumption of linear elastic materials.

Two different experiments were performed in [2]. In

the first, at a room temperature of 25 �C, a load was ap-

plied 3 mm from the interface in the PMMA zone (Fig.

2) in order to trigger the fracture process. The loading

rate was very low and the resulting speed of crack prop-

agation at the initial stages was also quite slow, so that

quasi-static conditions can be assumed. The crack path

was individuated and stresses near the crack tip in the

PMMA were measured using a shearing interferometer.

In the second experiment the same operations were

performed when the temperature of the aluminium was

60 �C in steady state conditions. To reach these condi-

tions, a cartridge heater (Q in Fig. 2) was inserted into

the aluminium part near the external vertical side. The

Fig. 3. Zoom of the notch of the specimen with crack paths trajecto

numerical. Case A: uniform temperature (25 �C); case B: thermal load

with E=E(25 �C), r0=r0 (25 �C), drcr=drcr (25 �C).

variation in time of the PMMA temperature was

checked before the fracture test, which was performed

when steady state conditions were reached. The temper-

ature of PMMA was recorded at the crack tip location,

at 5 and 7 mm from the interface. Also in this case, the

crack path was spotted. From the differences between

the two situations, the authors gathered the thermal ef-

fects, which were independent of the magnitude of the

applied mechanical load.

In the two experiments the crack propagation trajec-

tories are remarkably different as shown in Fig. 3a and b

where a zoom of the fractured specimens in correspond-

ence of the notch is presented. In particular the crack

path is closer to the interface when the temperature is

higher. A strongly representative parameter is the crack

initiation angle, which experimental values are listed in

Table 3 for the two temperature levels.

Due to the fact that the numerical model used is 2-D,

whereas the real heat transfer problem is 3-D, the con-

vection effect on the lateral surfaces of the specimen is

accounted for through the pertinent terms in the heat

transfer equation. According to [20], convection matrix

Kc of a generic finite element is

ries: (a) and (b) experimental results (reproduced from [2]), (c)

with E, r0,drcr varying with temperature; case C: thermal load

Table 3

Comparison for crack initiation angle: Case A: uniform

temperature (25 �C); Case B: thermal load with E, r0, drcrvarying with temperature; Case C: thermal load with E=E (25

�C), r0=r0 (25 �C), drcr=drcr (25 �C)

Crack initiation angle (deg)

Case Experimental Numerical

A 24 25

B 13 12

C – 25

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0 200 400 600 800 1000 1200time [s]

Tem

pera

ture

[°C

]

2.5 3.5 5.010.017.0

Fig. 5. Time histories of temperature calculated at 2.5, 3.5, 5.0,

10.0, 17.0 mm distance from the interface, at 30 mm height

from the bottom level.


Kc ¼ZCkNTNdC; ð16Þ

being C the part of the boundary where convection takes

place, k the convection coefficient of air (25 W/m2K)

and N the shape functions of the boundary edge of the

element. In addition to this term, a second contribution

is introduced which accounts for the convection across

the element surface, hence is of the form

Kc ¼ 2

ZXkNTNdX: ð17Þ

In Eq. (17), X represents the element domain and N the

element shape functions. It is further assumed that heat

can propagate inside the notch. To this end, a material is

assumed to fill this zone which presents no mechanical

resistance with a fictious thermal conductivity of 0.1

W/m2K.

The calculated temperatures in PMMA near the

interface are shown in Fig. 4, together with the values

obtained in [2], both by an uncoupled finite element

analysis and by experiment. The results of the present

model compare very well with the experimental one.

Fig. 5 shows both for the steep gradient near the inter-

face and the numerically calculated time histories at

points located at different distance from the interface.

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0 5 10 15 20 25 30Distance from the interface [mm]

Tem

pera

ture

[°C

]

Present solutionExperimental results

Fig. 4. Temperature distribution of PMMA versus distance

from the interface.

Due to the different approach to the fracture prob-

lem, comparison of the present solution with the exper-

imental results is performed on the basis of crack path.

Fig. 3c presents calculated crack paths in three cases.

Case A considers uniform temperature (25 �C) in con-

junction with a mechanical load of 835 N. In case B

the effects of the thermal load due to heating and vertical

load of 650 N are analysed, assuming the PMMA

Young�s modulus, limiting stress r0 and critical opening

drcr varying with temperature. Case C is equal to the

previous one, except for parameters E, r0 and drc whichare assumed constant and corresponding to the ones atroom temperature. The obtained crack path configura-

tions are of the same nature of the experimental onesand compare quite well with them (Fig. 3a,b), in partic-ular showing that the path is closer to the interface whenthe temperature is higher. The agreement is corrobo-

rated by the crack initiation angle values listed in Table3: numerical and experimental results are in goodaccordance.

Limited to Case B, Fig. 6 shows the ultimate stress

and temperature at crack tip as it moves during the evo-

lution of the fracture phenomenon. The decrease of the

ultimate stress depends on the increase of the tempera-

ture while the fracture propagates. This is a physical jus-

tification for the moving of the crack path closer to the

interface, where temperature is higher and limit stress is

lower.

Fig. 7 shows the temperature map contours in the

whole sample at different times before the crack enucle-

ation. It is remarkable that these distributions soon be-

come nearly constant over the height of the specimen

and always exhibit a steep temperature gradient near

the interface. Further, the effect of the elements located

30.0

35.0

40.0

45.0

50.0

0 20 40 60 80s [mm]

σo [MPa]

T [°C]

Fig. 6. Case B Limit stress r0 and temperature T at crack tip

during fracture evolution (s representing the distance of the tip

from the notch).


in the notch is evident, which results in a continuous

temperature distribution.

Fig. 7. Temperature map contour at time t=60, 300 and 1200

Displacement and stress fields are on the contrary

discontinuous across the notch and the fracture path.

The map contour of vertical displacements is shown in

Fig. 8 together with the amplified deformed configura-

tion.

Fig. 9 presents the contour map of the maximum

principal stress and cohesive force distribution in the

process zone in an initial stage of fracture propagation.

The calculated length of the process zone is very limited.

The maximum length is nearly 0.8 mm at the beginning

and decreases during the evolution of the phenomenon.

Application of Barenblatt [11] theory for calculation

of characteristic cohesive zone size yields for PMMA

‘ ¼ pEGc

8ð1� m2Þr20

ffi 0:75 mm ð18Þ

and numerical results are in good accordance with this

value.

The knowledge of the length of the process zone al-

lows for important remarks on mesh discretization. To

s. Point heat source of 3.0804·105 J/s/m3 located in Q.

Fig. 8. Amplified deformed configuration with vertical displacement contour map for Case B.

Fig. 9. Contour map of maximum principal stress and cohesive forces at fracture tip (Case B).


obtain a mesh independent solution it is necessary to

properly discretize the process zone, hence severe limits

are introduced depending on the approximation of the

used elements. An heuristic assessment has been made

on the influence of the mesh size in the process zone:

using linear elements of decreasing size, the value of

the force P (Fig. 2), corresponding to an applied verti-

cal displacement on the same point, is calculated. Re-

sults are summarized in Fig. 10. As can be seen the

peak of the external load and the softening branch re-

sult independent of discretization when the process

zone is subdivided into at least five elements with edges

of 0.15 mm or smaller. This situation is handled by the

mesh generator simply by locating an element source

600

800

1000

1200

1400

0 0.2 0.4 0.6Mesh size [mm]

Max

F [N

]

0

200

400

600

800

1000

1200

1400

-1.40-1.20-1.00-0.80-0.60-0.40-0.200.00Vertical disp. [mm]

F [N

] < 0.15 mm

0.50 mm

1.00 mm

Fig. 10. External force versus vertical displacement and mesh

size.


[9] in correspondence of the crack tip. Its weight may

be a priori stated (using Eq. (18)) and/or can be a

Fig. 11. Geometry of the three-point bending test for a bimaterial

aluminium; Case 2: the upper part is PMMA with Young�s modulus

posteriori updated during the adaptive remeshing pro-

cedure once the length of the process zone has been

determined.

The second application is intended to demonstrate

the capabilities of the proposed method, in particular

as far as the study of crack path evolution is concerned.

To this end the bimaterial specimen is assumed as in

Fig. 11. The lower part of the sample is made of PMMA

having the mechanical characteristics shown in Tables 2

and 3. Three different cases are obtained by changing

the material in the upper part of the sample. In Case

1 the upper part is aluminium with mechanical proper-

ties as in Table 2; Case 2 presents in the upper part

PMMA with Young�s modulus three times larger than

in Table 2; in Case 3 an homogeneous sample of

PMMA with characteristics as in Tables 2 and 3 is as-

sumed.

The analysed sample is loaded by a concentrated

force acting along the axis of symmetry and a distributed

heat source is present in the upper region.

Fig. 11 presents the different crack paths obtained by

the application of the proposed procedure. Depending

on the mechanical characteristics the paths are differ-

ently influenced by the interface. Fig. 12 shows the max-

imum principal stress contour for Case 2 and in Fig. 13 a

zoom is presented containing the process zone where the

cohesive forces are different from zero.

It is interesting to remark that only the use of the

mesh generator proposed in [9] can handle such type

of problems. The unique alternative is a very fine

mesh in large areas, resulting in a huge numerical

model. On the contrary the use of interface elements

is prevented by the fact that crack path is a priori un-

known.

specimen and resulting crack paths. Case 1: the upper part is

three times larger; Case 3: homogeneous sample.

Fig. 13. Zoom of the maximum principal stress contour.

Fig. 12. Maximum principal stress contour.


7. Conclusions

The paper presents a fully-coupled numerical model

for the analysis of fracture initiation and propagation

in a two dimensional non-homogeneous elastic medium

driven by transient thermal and mechanical fields.

Cohesive crack behaviour is assumed for the solid.

The comparison with a laboratory experiment and

the solution of a purely numerical test validate the

model and demonstrate the capabilities of the proposed

procedure.

One of the key points to obtain meaningful numer-

ical simulation is the correct discretization of the

process zone, as demonstrated by the presented

mesh-independency test. In order to follow the evolu-

tion of the crack, the process zone and the continuous

changes of the domain topology a successive remesh-

ing of the domain is advisable on the basis of fracture

evolution and using an a posteriori refinement tech-

nique.

Acknowledgments

This research was partially supported by the Italian

Ministry of University and Scientific and Technological

Research (Grant MURST ex 60%) and (Cofinanzia-

mento MURST MM08161945_003).

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Cohesive fracture growth in a thermoelastic bimaterial medium

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