This work is licensed under a Creative Commons
Attribution-NonCommercial-ShareAlike 4.0 International License.
21
C. C. P. Martins, B. R. Lemos, J. P. O Bone, A. G. S. Galdino, R. G. M. de Andrade,
“Numerical Modeling For Fracture Mechanics Problems Using The Open-source Fenics Platform”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024
Numerical Modeling
For Fracture Mechanics
Problems Using The
Open-source Fenics
Platform
ARTICLE HISTORY
Received 15 March 2023
Accepted 12 May 2023
Published 08 January 2024
Caio César Prates Martins
Federal Institute of Espírito Santo
Vitória - Brazil
caiocp.martins@gmail.com
ORCID: 0009-0009-0609-200X
Brenda Resende Lemos
Federal Institute of Espírito Sant
Vitória - Brazil
brendarlemos2020@gmail.com
Jean Pierre de Oliveira Bone
Federal Institute of Espírito Santo
Guarapari - Brazil
jean.bone@ifes.edu.br
André Gustavo de Souza Galdino
Federal Institute of Espírito Sant
Vitória - Brazil
andre.galdino@ifes.edu.br
ORCID: 0000-0002-5990-0287
Rodolfo Giacomim Mendes de Andrade
Federal Institute of Espírito Santo
Vitória - Brazil
rodolfo.andrade@ifes.edu.br
ORCID: 0000-0003-3956-5941
ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024
This work is licensed under a Creative Commons
Attribution-NonCommercial-ShareAlike 4.0 International License.
ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024
22
DOI:
LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024
10.5281/zenodo.10402112
LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January - June 2024
Numerical Modeling For Fracture Mechanics
Problems Using The Open-source Fenics Platform
Caio César Prates Martins
Federal Institute of Espírito Santo
Vitória - Brazil
caiocp.martins@gmail.com
André Gustavo de Souza Galdino
Federal Institute of Espírito Sant
Vitória - Brazil
andre.galdino@ifes.edu.br
Brenda Resende Lemos
Federal Institute of Espírito Sant
Vitória - Brazil
brendarlemos2020@gmail.com
Jean Pierre de Oliveira Bone
Federal Institute of Espírito Santo
Guarapari - Brazil
jean.bone@ifes.edu.br
Rodolfo Giacomim Mendes de Andrade
Federal Institute of Espírito Santo
Vitória - Brazil
rodolfo.andrade@ifes.edu.br
Abstract— Fracture mechanics is the mechanical approach to
fracture processes, which emerged due to limitations in applying
traditional concepts of Mechanics of Materials to predict the
behavior of cracked materials. Analytical problem solutions with
this approach may be unattainable, so it is necessary to use
numerical modeling, such as the finite element method. However,
the use of more advanced software that solves engineering problems
numerically is limited by its high cost. FEniCS is an open-source
computational platform that solves partial differential equations by
the finite element method. Thus, from a tutorial for this
computational platform, this work proposes to reproduce a classic
problem of linear elastic fracture mechanics, based on the validation
of a comparison of a linear elastic problem with the commercial
software ANSYS ®. With the help of the provided tutorial, a code
was built to model a three-point bending test. Implemented with the
aid of Gmsh and Paraview, it was possible to obtain satisfactory
results and to show that FeniCS is a powerful and accessible tool for
solving fracture mechanics problems.
Keywords— Fracture Mechanics, Numerical Modeling,
FEniCS, Finite Elements
I. INTRODUCTION
The Finite Element Method (FEM) is an approach to
solving partial differential equations using numerical
techniques in which a continuous domain is discretized into
finite elements called mesh. With the advancement of
technology and, consequently, computational power, more
advanced engineering problems have become simpler to
solve. This is due to the fact that analytical solutions can be
complex and even unreachable, and the error achieved in
numerical solutions is considerably acceptable [1].
In general, the use of advanced FEM-based software
is restricted to companies and some teaching institutions, as it
is the case with ANSYS ® [2]. On the other hand, open-source
programs, such as the FEniCS Software [3], which is free, are
used more widely due to their availability and ease of access.
However, they tend not to have a graphical interface, unlike
commercial software, which makes the use of pre- and post-
processors essential for visualizing the solution. Even for a
simple computational solution approach, the development of
the finite element method can be complex, as it requires
knowledge about tensor and variational calculus in some
cases. The FEniCS [3] software, for example, uses these
principles and, based on programming knowledge and the
library itself, it is possible to solve partial differential
equations using a variational approach. Therefore, this work
proposes to present a numerical modeling of a linear elastic
problem and two of fracture mechanics, presenting a
preliminary comparison with ANSYS® for the linear elastic
problem.
A. Fracture Mechanics
Fracture mechanics is the mechanical approach to fracture
processes that emerged due to limitations applying traditional
concepts of Mechanics of Materials to predict the behavior of
materials in the presence of cracks. It was developed and
founded after the 2nd World War [4] and is widely used in
structural contexts in the areas of civil, mechanical, and
metallurgical engineering [5-7]. For materials with brittle
behavior, the linear elastic fracture mechanics (LEFM)
approach is used, while for materials with ductile behavior,
the elastoplastic fracture mechanics (EPFM) is used [4,8].
For example, in industry, a component may have such a
high cost that, depending on the conditions and its integrity,
it is more viable to have knowledge about fracture mechanics
and continue using it with cracks to perform an exchange,
which results in a complete pause of an operation. Alan
Arnold Griffith studied the behavior of an elliptical hole
when external stress is applied and established a
thermodynamic model for crack propagation [8]. Griffith
concluded that the strength of a material is not only linked to
chemical bonding parameters but also to the existing defects.
Therefore, it was realized that defects in the material are
factors that intensify the applied stress, making it susceptible
to exceeding the yield strength of the material and causing a
rupture, which is the basis of fracture mechanics [4]. With
this, a good characterization of the material must also have
experimental parameters of the LEFM, such as the fracture
toughness (K IC) and the critical energy release rate (Gc), for
example. Fracture toughness is independent of size, geometry,
and loading levels for a material with a given microstructure
and is the main obtained experimentally properties related to
fracture mechanics [8].
II. METODOLOGY
A. Numerical Modeling
The main concept addressed in the FEM is the
discretization of a continuous domain into finite geometric
elements, in addition to the use of polynomial interpolation to
determine the results in the region inside the elements [1].
ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024
23
DOI:
LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024
10.5281/zenodo.10402112
C. C. P. Martins, B. R. Lemos, J. P. O Bone, A. G. S. Galdino, R. G. M. de Andrade
Numerical Modeling For Fracture Mechanics Problems Using The Open-source Fenics Platform,
Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024.
These elements form a mesh, and each node has a
displacement u and a stress σ , which are represented in a
linear system and determined through the variational calculus.
The stress and strain of the solid are expressed by tensors,
which are, by definition, mathematical entities that produce a
linear transformation in vectors, transforming them into
different vectors [9]. Tensors are represented in Equations (1)
and (2), where u , v , and are the horizontal, vertical, and
transverse components of infinitesimal displacement. The
strain tensor can also be defined as ε=symu , that is, the
symmetric gradient of u. Both mathematical entities σ and ε
have Cartesian x, y, and z components.
=
=









(1)
=


(


+


)
(


+


)
(


+


)


(


+


)
(


+


)
(


+


)


(2)
Mesh refinement generates more accurate results that are
closer to the analytical ones, but there is a computational limit
to be respected, which is analyzed through a convergence test,
where the best results are sought with the minimum possible
elements [9]. In the formulation involving the FEM applied to
fracture mechanics, some parameters must be provided to the
program, such as Young’s modulus (E), Poisson’s coefficient
(), and the critical energy release rate ( Gc). After providing
the input data, using concepts of variational calculus, it is
possible to obtain the results, which are observed through a
post-processing software.
Simulation allows engineers to use basic principles of
modeling, physics, mathematics, and computer science to
evaluate design performance in different scenarios. Thus, for
the development of Engineering, it is important to analyze
solutions via software to ensure that the result obtained is
adequate and that it meets the functional needs of a project
[10].
1. Numerical modeling for FEniCS software
a. Linear elasticity
Numerical modeling is used for a linear elasticity problem
in a plane strain state [9] shown in Fig. 1, based on the FEniCS
library [11]. The problem consists of a three-dimensional plate
200 mm long, 500 mm high and whit thick e = 10 mm
subjected to a load. Acting field forces are disregarded and a
plane strain state is defined. For the Young’s modulus of the
material, 200 GPa was adopted and, for Poisson's coefficient,
0.3. The problem has the following governing equations for a
Ω domain.
= , = 0 in
(1)
=

(
)
+ 2
(2)
=
(
+ ()
)
(3)
where T is the applied stress, represented by the ratio
between the uniformly distributed load at the base of the bar
and the thickness, Ⅰ is the three-dimensional identity matrix,
and μ and λ are the Lamé constants, which depend on the
Young’s modulus and the Poisson’s coefficient of the
material. Considering the principle of virtual work, one must
find values of u that satisfy the weak formulation [9].
(
)
:
(
)
=
, ,
(4)
where u and p are the trial and test functions, respectively,
and V is the vector field containing them. In this example,
second-degree Lagrange polynomials are defined for the
interpolation between nodes. The vertical face is fixed, and the
load F is uniformly applied in the negative y-direction. The
mesh was built using a function from the FEniCS library,
containing 48000 tetrahedral elements with 5 mm on each
side.
Fig. 1. Representation of geometry (a) and boundary conditions (b)
b. Fracture Mechanics
Numerical modeling for the fracture mechanics problem
was proposed by [12], using models by [13], with
contributions by [14]. It is considered an elasto-static body
with a discontinuity, which occupies a domain Ω R2. The
Dirichlet and Neumann boundary conditions [9] are imposed
by ΓD and ΓC. In the case of a discrete fracture mechanism,
the crack is represented by a discontinuous surface ΓC. The
variable that models crack propagation is ϕ∈[0,1]. When it
assumes a null value, the material is intact, and when it
assumes a unitary value, there is a complete fracture. The
crack size is controlled by a variable ℓ, a length scale
parameter inherent to the model and which depends on the
developed mesh refinement [12-15], called characteristic
length. The approximate crack surface energy is defined as:
󰇡

+
|

|
󰇢
(5)
Adding the Bulk energy to Eq. (5) the total potential energy
of the solid (Ψ) is obtained as:
=
(
1
)
(
)
+
󰇡

+
|

|
󰇢
(6)
where ψ(ε) is the strain energy density of the solid, in terms
of the Lamé parameters and the strain tensor, represented in
Eq. (2), whose mathematical expression is:
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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024
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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January – June 2024
(
)
=
(())
+ (
) (7)
Applying Gauss's theorem in Eq. (7) the following field
equations are obtained, with arbitrary values for the
kinematic variables δu and δф.
(
1
)
= ,
in
󰇡
󰇢2
(
1
)
() = 0 (8)
The natural boundary conditions for a traction T are:
(
1
)
= , on
= 0,
on (9)
where n is the normal vector to the surface Γ. With this, the
constitutive equations and the boundary conditions are given.
The procedure now consists of implementing the finite
element method. The main objective is the resolution of the
system of equations (8) with the boundary conditions
expressed by Eqs. (9.1) and (9.2). However, it is necessary to
use the finite element method, discretizing the continuous
domain. Equation (8.2) is modified to:
󰇡
󰇢2
(
1
)
() = 0 (10)
where H
+
is called the variable storage (or history) field,
which changes with time, expressed mathematically as:
(
)
= max
(
()
)
(11)
and ψ
+
is the variable strain energy density of the solid,
defined as:
(
)
=
(+
|
|
)
+ (
dev
:
dev
) (12)
where K is the Bulk modulus, which can be expressed in
terms of the Young’s modulus and the Poisson’s coefficient
[9]. Finite element modeling uses a weak, or variational,
formulation that uses dimensional trial (
,
) and test (ℒ,
)
spaces, which contain the trial (u, ) and test (p, q) functions,
respectively. A discrete space (
) is also defined around the
mesh that contains the phase field variable (ф) and the
displacement field (u). All spaces have a dimension d.
(
,
)
= {(, )
[
(
)]
: (, )
[
(
)]
[
(
)]
(
,
)
= {(, )
[
(
)]
: (, )
[
(
)]
[
(
)]
(13)
In the reformulation of the system of constitutive equations,
applying the Bubnov-Galerkin procedure, remote tractions
and field forces are disregarded, making it:
[
(
1
)
(
)
: ()] = 0
󰇣
+ 󰇡
+ 2
󰇢2
󰇤= 0
(14)
2. Numerical modeling for ANSYS ® software
In order to validate the results obtained by FEniCS for the
linear elasticity problem (Section II.1.a), a numerical model
was implemented in the Ansys ® software, version 2022. The
material used in the modeling has the same properties as in
Section II.1.a. As it is a three-dimensional model, a load of
0.1N/mm² ( Pressure type ) was applied to the lower face of
an xz plane of the structural element in the vertical direction
with a downward direction ( -y ). Opposite the load application
plane, all nodes were restricted to translation, which
represented a crimp. The mesh illustrated in Fig. 2 records the
geometry containing 45125 nodes and 8000 cubic elements.
Fig. 2. Mesh result obtained in ANSYS ® for linear elasticity problem
III. RESULTS
AND
DISCUSSION
A. FEniCS software validation
With the help of ANSYS ®, a kind of FEniCS validation
was carried out, solving the same linear elasticity problem
and comparing the results. Fig. 3 shows the bar displacements
obtained by ANSYS ®, and Fig. 4 exposes those obtained by
FEniCS, both in the y direction. Note that there is a
qualitative similarity regarding the vector field represented
by the scale. The maximum supported stresses are found on
the crimped face of the bar, opposite to the force application
face, and, for both cases, a value of 0.1MPa was obtained.
The maximum deformation obtained analytically is -
2.50010-4 mm. The comparison between FEniCS and
ANSYS ® for the linear elastic problem resulted in errors of
less than 0.6%, as shown in Table I. Therefore, free open-
source software can be operated safely.
Fig. 3. Result of the displacement field obtained in ANSYS ® for linear
elasticity problem
ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024
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DOI:
LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024
10.5281/zenodo.10402112
C. C. P. Martins, B. R. Lemos, J. P. O Bone, A. G. S. Galdino, R. G. M. de Andrade
Numerical Modeling For Fracture Mechanics Problems Using The Open-source Fenics Platform,
Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024.
Fig. 4. Result of the displacement field obtained in FEniCS for linear
elasticity problem
TABLE I. C
OMPARISON BETWEEN ANSYS AND
FE
NI
CS
MAXIMUM
DEFORMATION
Ansys vs FEniCS comparison
Method Value (mm) Relative Error
a
(%)
Ansys ®
-2.486⋅10
-4
0.560
FEniCS
-2.487⋅10
-4
0.520
Analytical
-2.500⋅10
-4
-
a.
Relative to the analytical value
B. Application of the phase field method using FEniCS
1. Tensile test with simple pre-crack
Elaborated by [15], the problem to be solved consists of a
representation of a fracture mechanics test of a plate
subjected to uniaxial tensile stress that has a pre-crack to
simulate a pure fracture in Mode 1, as illustrated in Fig. 5. In
order to reduce the computational time, small geometric
proportions were considered, being L=0.5 mm. The code
structure of this problem, implemented for this work,
followed the tutorial developed by [12].
The mesh was built using the Gmsh preprocessor [16]
and has 30546 triangular elements. The material has a
modulus of elasticity E = 210GPa, a Poisson coefficient =
0.3, and a critical energy release rate Gc = 2.7MPa mm. Thus,
the Lamé parameters λ =121153.8MPa and μ = 80769.2MPa
were obtained. A value of 0.011mm was also used for the
characteristic length ℓ.
Fig. 5. Tensile stress problem in a pre-cracked plate under uniaxial force,
adapted from [12,13]
The base of the mesh (y=0) is fixed, and a remote offset
of 0.007mm is used as the first iteration. To help with the
code, the value of the phase field variable ϕ for every pre-
crack was defined as 1. During the execution of the code, the
necessary number of iterations for convergence of the
solution during a given step is provided. The main results of
the analysis are shown in Fig. 6 and Fig. 7. Visualization of
crack propagation is easily observed using the post-processor
software Paraview [17]. Code execution stops at a value
determined as a maximum (t=1.0), at which there has already
been catastrophic failure of the material. A change was made
regarding the test loading rate, for reasons of computational
power. The red region represents the complete failure of the
material, and the blue shows the initial state (intact).
Fig. 6. Force-displacement curve for traction problem (a)
Fig. 7. Displacement crack propagation u=5,7x10
-3
mm
The results obtained are satisfactory and close to the
literature, as shown in Table II [12,13,15,18], although
changes have been made to the code and the mesh used has
been less refined.
TABLE II. C
OMPARISON BETWEEN THIS PAPER AND
L
ITERATURE OF
T
ENSILE
T
EST
’
S PEAK VALUES
Ansys vs FEniCS comparison
Reference Peak Value (kN) Relative Error
a
(%)
This Paper 0.6880 -
[12] 0.7162 3.93
[13] (Isotropic) 0.6807 1.07
[15] (ℓ=0.015) 0.6842 0.555
[18] (ℓ=0.011) 0.6476 6,23
a.
Relative to literature values
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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024
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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January – June 2024
2. Three-point bending test
The following problem consists of a three-point bending
test. More details can be seen in [15]. The geometry and
boundary conditions are given in Fig. 8. The mesh was built
using the Gmsh preprocessor and has 72768 triangular
elements, in which a refinement was performed in the center,
where the crack is expected to propagate [18]. The material
has a modulus of elasticity E = 20.8GPa, a Poisson coefficient
= 0.3 and a critical energy release rate Gc = 0.54MPa mm.
Thus, the Lamé parameters λ =12000MPa and μ = 8000MPa
were obtained. A value of 0.03mm was also used for the
characteristic length ℓ, similar to that used in the literature
[13,15].
We start with the same numerical modeling for the
traction problem but now with a point force. The point (0,0)
has zero nodal displacements in x and y. At the point (8,0),
the shift is restricted to y only. The force is applied punctually
at (4,2). An initial displacement of 0.005mm was defined at
the top of the geometry, changing to 0.00001mm when
approaching the failure and returning to 0.005mm after the
failure to follow the crack in greater detail. Fig. 9 shows the
vector field of the phase field variable as a ϕ function of the
load level, with representation like the previous problem. The
force-displacement curve is shown in Fig. 10. The results
obtained are satisfactory and close to the literature, as shown
in Table III, even with a variation of displacements different
from that used by the authors to reduce computational costs
[13,15,18].
Fig. 8. Geometry and boundary conditions for three-point bending
test, adapted from [13]
Fig. 9. Phase field for displacements u=0.04 mm (a), u=0.045 mm (b),
u=0.056 mm (c), u=0.071 mm (d)
Fig. 10. Force-displacement curve for three-point bending test
TABLE III. C
OMPARISON BETWEEN THIS PAPER AND
L
ITERATURE OF
T
HREE
-
POINT
B
ENDING
T
EST PEAK VALUES
Ansys vs FEniCS comparison
Reference Peak Value (kN) Relative Error
a
(%)
This Paper 0,0365 -
[13] (Hybrid) 0,0417 12,47
[15] 0,0385 5,19
[18] 0,0412 11,4
a.
Relative to literature values
IV. CONCLUSIONS
This work proposes the implementation of classic
linear elastic fracture mechanics problems based on a tutorial
and examples found in the literature, with the help of Gmsh
and Paraview. The comparison between the FeniCS and
ANSYS software for the elastic linear problem obtained a
satisfactory result, with an error, on average, of less than
0.6%. The traction problem presented an error, on average, of
approximately 2.9% for the peak value when compared to the
literature. The three-point bending problem, on the other hand,
presented an error, on average, of approximately 9.7% for the
peak value in comparison with the same source. Thus, it is
concluded that FEniCS can be used for academic purposes
both for solving classic problems of Strength of Materials and
MFLE. Furthermore, this tool, together with Gmsh and
Paraview, provides users with advanced approaches to
learning engineering problem-solving.
ACKNOWLEDGEMENTS
The work was developed under the 2021 Researcher
Productivity Notice, and the authors would like to thank Ifes
for the opportunity and infrastructure. To CNPq, for the
scholarship, allowing full dedication to Scientific Initiation.
To FAPES and CNPq - Support Program for Emerging
Centers - PRONEM - Notice 06/2019 and Grant Term -
502/2020 - SIAFEM: 220-765DF.
R
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ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024
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DOI:
LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024
10.5281/zenodo.10402112
C. C. P. Martins, B. R. Lemos, J. P. O Bone, A. G. S. Galdino, R. G. M. de Andrade
Numerical Modeling For Fracture Mechanics Problems Using The Open-source Fenics Platform,
Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024.
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ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024
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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024
AUTHORS
I am Caio César Prates Martins. I am 21 years old and live in Espírito
Santo - Brazil, with my girlfriend and my mother. I like to study, learn
new things, cook, gym, and hang out with my girlfriend and friends.
I am a Metallurgical Engineering student at the Federal Institute of
Espírito Santo and I am in the 9th period of the course. I have also
been an intern at Biancogres for 3 months. Since I was a kid I like
soccer and I support Flamengo. At the age of 14 I left my home state
and ventured to São Paulo in search of my dream of becoming a
soccer player. I learned a lot about interpersonal relationships and
how to manage on my own. Unfortunately, I was not successful, but
life went on.
At the age of 17 I entered college. Recently I did a scientific initiation
that introduced me to Research and Development, helping me to
advance my knowledge in Python programming, finite element
modeling, and fracture mechanics. At the company where I work, we
produce porcelain tiles, ceramic and vinyl tiles with excellence and
quality, and I am happy to contribute to this.
I'm Brenda Resende Lemos. I'm 23 years old and I live in Espírito Santo
- Brazil, with my parents and my brother. I like to study, play handball,
watch football games, go to church and hang out with friends.
At the age of 14, I won a scholarship to play handball at a school
that is a reference in the sport. That same year, the team in which I
participated won the regional, state and Brazilian. It was an amazing
experience, I learned about perseverance and that we should never
give up on our dreams. In addition, I am in love with Flamengo and I
collect cups and shirts for the team.
I'm a Civil Engineering student at the Federal Institute of Espírito Santo
and I'm in the 7th period of the course. I did a scientific initiation on
numerical modeling and I'm currently doing one on the use of coconut
fiber in cement matrices. I like it and I am very interested in following
in the area of research.
Caio César Prates Martins
Brenda Resende Lemos
C. C. P. Martins, B. R. Lemos, J. P. O Bone, A. G. S. Galdino, R. G. M. de Andrade,
“Numerical Modeling For Fracture Mechanics Problems Using The Open-source Fenics Platform”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024
ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024
29
LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024
AUTHORS
C. C. P. Martins, B. R. Lemos, J. P. O Bone, A. G. S. Galdino, R. G. M. de Andrade,
“Numerical Modeling For Fracture Mechanics Problems Using The Open-source Fenics Platform”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024
Possui graduação em Engenharia de Materiais pela Universidade Federal
da Paraíba (1997), mestrado em Engenharia e Ciência de Materiais pela
Universidade Federal do Ceará (2003) e doutorado em Engenharia
Mecânica pela Universidade Estadual de Campinas (2011). Atualmente
é professor no Instituto Federal do Espírito Santo, Campus Vitória,
Coordenadoria de Mecânica. Tem experiência na área de Engenharia de
Materiais e Metalúrgica, com ênfase em Materiais Metálicos e Cerâmicos,
atuando principalmente nos seguintes temas: cerâmicas porosas,
propriedades físicas e mecânicas, cerâmica vermelha, metalurgia do pó,
compósitos, biomateriais, cerâmicas refratárias e ensino de Materiais.
Além disso, é líder nos grupos de pesquisa "Materiais e Processos de
Fabricação" e "Materiais de Construção Civil" no Campus Vitória. É
professor associado do Instituto Nacional de Ciência e Tecnologia em
Biofabricação BIOFABRIS e pesquisador no Polo de Inovação EMBRAPII
IFES em Materiais e Metalurgia. É professor permanente no Programa de
Pós-Graduação em Engenharia Metalúrgica e de Materiais e professor
permanente no Mestrado Profissional em Tecnologias Sustentáveis.
André Gustavo de Souza Galdino
Born in Vitória, Espírito Santo state, has a degree in Mechanical
Engineering from Ufes - Federal University of Espírito Santo - ES,
completed in 2011. He is currently pursuing his Master's degree in
Metallurgical and Materials Engineering at Ifes - Federal Institute
of Espírito Santo at the Victory campus, where he researches the
influence of welding processes on the mechanical properties of
steels and teaches disciplines in the area of Materials Science and
Welding at the Guarapari campus of Ifes. Has courses in the areas
of non-destructive testing by Abend - Brazilian Association of Non-
Destructive Testing and in the area of defect diagnosis by vibration
analysis by Fupai - Foundation for Research and Advice to Industry of
Unifei - Federal University of Itajubá - MG. He is interested in the areas
of welding, heat treatment, fracture mechanics, and materials science.
He also has a background in other areas such as post-graduation in
Education from Faculdade Pitágoras - MG and a degree in Sociology
from Unicesumar - PR, to be concluded in 2023.
Jean Pierre de Oliveira Bone
ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024
30
LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024
AUTHORS
Possui doutorado em Estruturas e Materiais pela COPPE/Universidade
Federal do Rio de Janeiro (2020), mestrado em Estruturas pela Escola
Politécnica da Universidade de São Paulo (2012) e graduação em
Engenharia Civil pela Universidade Federal do Espírito Santo (2009).
Atualmente, é professor de ensino básico, técnico e tecnológico do
Instituto Federal de Educação Ciência e Tecnologia do Espírito Santo,
Campus Vitória. Tem experiência na área de Engenharia Civil, com ênfase
em Estruturas, nos seguintes temas: estruturas de concreto, modelagem
numérica do concreto e compósitos cimentícios reforçados com fibras
de aço
Rodolfo Giacomim Mendes de
Andrade
C. C. P. Martins, B. R. Lemos, J. P. O Bone, A. G. S. Galdino, R. G. M. de Andrade,
“Numerical Modeling For Fracture Mechanics Problems Using The Open-source Fenics Platform”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024