Received: 10 October 2018
Revised: 17 April 2019
Accepted: 15 July 2019
DOI: 10.1002/nme.6170
RESEARCH ARTICLE
Removing mesh bias in mixed-mode cohesive fracture
simulation with stress recovery and domain integral
Habeun Choi
Kyoungsoo Park
Department of Civil and Environmental
Engineering, Yonsei University, Seoul,
South Korea
Correspondence
Kyoungsoo Park, Department of Civil and
Environmental Engineering, Yonsei
University, 50 Yonsei-ro Seodaemun-gu,
Seoul 120-749, South Korea.
Email: k-park@yonsei.ac.kr
Funding information
Ministry of Science, ICT & Future
Planning, Grant/Award Number:
NRF-2018R1A2B6007054; Ministry of
Trade, Industry & Energy, Grant/Award
Number: KETEP 20174030201480
Summary
To remove mesh bias and provide an accurate crack path representation in
mixed-mode investigation, a novel stress recovery technique is proposed in conjunction with a domain integral and element splits. Based on a domain integral
and stress recovery technique, a maximum strain energy release rate is estimated
to determine a crack path direction. Then, for a given crack path direction, continuum elements are split, and a cohesive surface element is adaptively inserted.
One notes that the proposed stress recovery technique provides a more accurate
stress field than a standard stress evaluation procedure. The proposed computational framework is verified and validated by solving mode-I and mixed-mode
examples. Computational results demonstrate that the domain integral with
the stress recovery accurately evaluates a crack path, even with a lower-quality
mesh and under a biaxial stress state. Furthermore, the cohesive surface element
approach, with the element split in conjunction with the stress recovery and
the domain integral, predicts mixed-mode fracture behaviors while removing
mesh bias in the crack path representation. Additionally, the condition numbers
of stiffness matrices are within the same order of magnitude during cohesive
fracture simulation.
K E Y WO R D S
domain integral, element split, extrinsic cohesive zone model, mesh bias, mixed-mode fracture,
stress recovery
1
I N T RO DU CT ION
Cohesive zone models have been widely utilized to characterize nonlinear crack propagation and failure of materials
and structures in various engineering applications. One challenge in applying the cohesive zone model is representing
arbitrary discontinuities in space and time. Thus, to describe strong discontinuities, such as cracks in a domain, various
computational methods have been proposed. For example, cohesive surface elements were introduced within a standard finite element framework to investigate crack propagation problems.1-4 However, in the cohesive surface element
approach, the critical limitation is associated with mesh bias and dependency. This is because a crack length cannot be
converged for an arbitrary path if one inserts surface elements between existing continuum elements.5
To remove mesh bias and dependency, cracks are alternatively described by employing discontinuous enrichment functions in conjunction with the partition-of-unity method. This leads to a generalized/extended finite element method
(GFEM/XFEM).6-8 For example, Daux et al6 utilized trigonometric functions to describe a crack tip singularity for linear
elastic fracture mechanics, and Pereira et al9 employed a face-offsetting method to represent crack surfaces in a threedimensional domain. Remmers et al10 employed a Heaviside function to investigate cohesive dynamic crack propagation,
Int J Numer Methods Eng. 2019;120:1047–1070.
wileyonlinelibrary.com/journal/nme
© 2019 John Wiley & Sons, Ltd.
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CHOI AND PARK
whereas Kim and Duarte11 used global-local enrichments to simulate cohesive crack propagation. Similarly, displacement discontinuities were described by employing enrichments within an element,12-14 and an extended cohesive damage
model was developed using an equivalent stiffness matrix rather than additional enriched degrees of freedom.15
However, spurious traction oscillations in a precracking phase were observed, which can result in inaccurate crack patterns in the XFEM/GFEM.16 Then, a symmetrical structured mesh was generally utilized for mode-I cohesive fracture
investigations. To provide a smoother stress field, discontinuities can be described using an isogeometric approach.17,18
Furthermore, when the ratio of area (or volumes) on either side of the discontinuity in an element is very small, an
ill-conditioned system of equations for the GFEM/XFEM can be expected, which is associated with the accuracy and
convergence of a problem.19 To improve conditioning and accuracy of the GFEM/XFEM, a preconditioning scheme
was developed for a Heaviside enrichment function,20 whereas a stable GFEM/XFEM was proposed to solve fracture
problems.21-23 Additionally, when a special enrichment function is utilized, the numerical integration should be performed with great caution.6,24 Recently, a phase-field model was also utilized for cohesive zone modeling, whereas fracture
properties, such as fracture energy and cohesive strength, should be calibrated in conjunction with internal length-scale
parameters,25,26 and some form of numerical regularization was needed to prevent a loss of ellipticity.27,28
Among various computational methods, the present study employs the cohesive surface element approach while removing mesh bias. Previously, to reduce mesh bias in 4k structured meshes, Paulino et al29 and Park et al30 employed nodal
perturbation and edge-split operators with the adaptive mesh refinement and coarsening schemes. Note that a 4k structured mesh consists of sets of four isosceles right triangles within a rectangular domain. Rimoli and Rojas31 addressed
bias in 4k structured meshes and proposed a strategy to alleviate mesh dependence using barycentric subdivision with
k-means meshes, named as a conjugate-directions mesh. Alternatively, polygonal elements were utilized to solve dynamic
fracture problems.32 Leon et al33 and Spring et al34 employed local mesh modifications, eg, element splitting and refinement, to reduce mesh dependency and computational cost. Although several approaches have been employed to manage
mesh bias in the surface element approach, an accurate crack path cannot be achieved.
In this study, to eliminate mesh bias and dependency in the cohesive fracture computation, a computational framework with a novel stress recovery technique and a domain integral is proposed in conjunction with the element split. For
a crack initiation criterion, a maximum strain energy release rate is evaluated using the proposed computational framework, which provides an accurate crack propagation direction even with a lower-quality mesh and under a biaxial stress
state. One notes that the proposed stress recovery technique provides a better approximation of a stress field with lower
mesh quality than a general stress evaluation procedure. The proposed computational framework provides smooth crack
paths in a domain, equivalent to the GFEM/XFEM, whereas the condition numbers of the stiffness matrices remain of
the same order of magnitude. Furthermore, computational results capture the experimental results of crack path and
load-displacement relations without having a calibration process of material parameters.
The remainder of this paper is organized as follows. The element splitting procedure is illustrated in Section 2. In
Section 3, the crack initiation criteria are explained in conjunction with a stress recovery scheme and a domain integral.
Then, the computational implementation of the cohesive zone model is briefly explained in Section 4. The proposed
computational framework is verified and validated in Sections 5 and 6, respectively. Finally, key findings are summarized
in Section 7.
2
ELEMENT SPLITTING SCHEME
To describe an arbitrary crack in the cohesive surface element approach, an element splitting scheme is proposed in conjunction with a novel stress recovery technique and a domain integral. The element splitting scheme consists of four steps:
(1) identification of a crack path, (2) splitting a continuum element, (3) treatment of a hanging node, and (4) insertion of
a cohesive surface element. First, a potential crack path, eg, the dashed line in Figure 1A, is identified on the basis of a
crack initiation criterion. Next, one searches an element, which corresponds to a potential crack path, eg, the light-gray
element (see Figure 1B), and splits the element along the given crack path. A new node that corresponds to a new crack
tip position is created on an edge shared by light- and dark-gray elements. Then, a hanging node can be created in the
dark-gray element, as shown in Figure 1B. To avoid a hanging node being retained, the dark-gray element is split, as
shown in Figure 1C. After splitting two continuum elements, a cohesive surface element is inserted along the edge that
is identical to a given crack propagation direction, and new fracture surface is created (Figure 1D).
The element splitting scheme is not employed for the following two cases. If the crack propagation direction is the same
as a facet (or edge) of elements, a cohesive surface element is inserted along the facet. Second, a new crack tip position
CHOI AND PARK
(A)
(C)
(A)
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(B)
(D)
FIGURE 1 Schematics of the element
splitting procedure. A, Identifying a crack
path; B, Continuum element split; C,
Treatment of a hanging node; D, Insertion
of a surface element [Colour figure can be
viewed at wileyonlinelibrary.com]
(B)
FIGURE 2 Schematics of the node
relocation. A, Identifying a crack path; B,
Relocating a node and inserting a surface
element [Colour figure can be viewed at
wileyonlinelibrary.com]
can be close to an existing finite element node while a potential crack path is within an element (Figure 2A). In this case,
an existing node is relocated to the position corresponding to a potential crack path, and a surface element is inserted
along that path (Figure 2B). In this study, when the distance between a new crack tip and an existing node is less than
20% of the element edge length, the existing node is relocated, and no element split occurs.
When a potential crack path is within a continuum element, either the creation of a node or the relocation of a node
is expected. For a new or relocated node, nodal quantities, such as displacements, are interpolated from adjacent nodal
quantities. In this study, the Lagrange basis shape functions are used in conjunction with an existing finite element mesh.
It should be noted that, for an element split of linear triangular elements, the strain energy is conserved when the Lagrange
basis shape functions are utilized.30 This is because two split elements reproduce the displacement field of the original element. However, strain energy conservation is not guaranteed when node relocation is utilized. Additionally, in this study,
constant strain triangular elements are utilized to avoid numerical integration error associated with an ill-conditioned
Jacobian matrix due to the element split.
During the element splitting procedure, an element, which has a large aspect ratio, may be generated, especially when
a nodal reposition is introduced. For example, a potential crack path is illustrated as a dashed line in Figure 3A, and a
node is repositioned to a white circle (Figure 3B) based on the nodal repositioning criterion, which results in an element
with a large aspect ratio. To avoid an element with a large aspect ratio, local remeshing is conducted when the angle of a
triangular element, which shares a new crack tip node, is greater than a certain value, ie, 130◦ . If one identifies an element
with a large aspect ratio, eg, the light-gray element in Figure 3B, one or two adjacent elements are selected, which share
the new crack tip node, eg, the dark-gray elements in Figure 3B. The number of adjacent elements is determined on the
basis of an angle of new elements. Then, the element with a large aspect ratio and the adjacent elements are removed, and
a new node is inserted at the average point of the nodes of the removed domain, eg, the gray node in Figure 3C. Finally,
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FIGURE 3 Schematics of the local
element remeshing procedure. A,
Identifying a crack path; B, Relocating a
node and recognizing a skinny element, ie,
the light-gray element; C, Remeshing
skinny and its opposite elements; D,
Insertion of a surface element [Colour figure
can be viewed at wileyonlinelibrary.com]
(A)
(B)
(C)
(D)
new elements are created using the new node and the nodes of the removed domain, and a cohesive surface element is
inserted along the potential crack path (Figure 3D).
During the element splitting and insertion of surface elements, new nodes are created in a finite element mesh, and
thus, the element connectivity is locally updated. To efficiently handle operations associated with adjacent information,
a topology-based data structure, named TopS, is utilized.35,36 TopS is based on topological entities, such as node, element,
vertex, edge, and facet. Nodes and elements are explicitly represented, whereas vertices, edges, and facets are implicitly
derived from the basis of element templates. Additionally, a client-server approach is employed to maintain a consistent
data structure when modification events occur.29
3
CRAC K INITIATION CRITERION
New fracture surface is created in a continuum domain when a crack initiation criterion is satisfied along a certain direction. For the evaluation of a crack propagation direction, two types of criteria are utilized, ie, the maximum principal
stress37 and the maximum strain energy release rate.38 Then, when a tensile stress along the evaluated crack propagation
direction is greater than the cohesive strength of a material, a cohesive crack is initiated by adaptively inserting surface
elements. Because the crack initiation criteria are based on a stress state in this study, the stress around a crack tip should
be evaluated precisely. However, because of the element split, mesh quality around a crack tip region can decrease, which
adversely impacts the stress evaluation and crack path prediction. In this section, to improve the accuracy of a stress (or
strain) field, a stress recovery technique is first proposed by introducing a virtual mesh grid. Then, the evaluation of a
crack propagating direction is explained in the following subsections.
3.1
Stress recovery technique
For accurate stress evaluation, a virtual mesh grid is generated around a crack tip region. Displacements (u) of virtual
nodes in a virtual mesh grid are approximated using the solution of the finite element analysis. Based on the approximated
displacements of virtual nodes, strain and stress are evaluated. It should be noted that a virtual mesh grid is introduced
only for strain and stress evaluations. For instance, a finite element mesh with a crack is shown in Figure 4A. To evaluate
the stress around a crack tip region, a virtual mesh grid of 6 × 6 is generated, as shown in Figure 4B. A virtual grid size
is selected as the element size of the original finite element mesh around the crack tip. The nodal locations of the virtual
mesh grid are determined according to a crack trajectory and a crack tip position. The center of the virtual mesh grid corresponds to the crack tip position. The front part of the virtual mesh grid (eg, gray nodes in Figure 4B) is generated parallel
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(A)
(B)
(C)
(D)
FIGURE 4 Stress recovery scheme. A, Original finite element mesh with low quality; B, Creation of a virtual grid; C, Virtual nodal
displacement interpolation from nodal displacements of the original finite element mesh; D, Stress evaluation at the Gauss points and nodes
of a virtual grid [Colour figure can be viewed at wileyonlinelibrary.com]
and perpendicular to the crack propagation direction. The rear part of the grid (white nodes in Figure 4B) is generated
based on the crack trajectory, ie, perpendicular and parallel to the cohesive crack. Next, the virtual nodal displacements
(u) are interpolated from the nodal displacements (u) of the original finite element mesh with the Lagrange basis shape
functions (Figure 4C). Based on the nodal displacements of the virtual mesh grid, the stresses at the Gauss points of a
virtual square element are evaluated using the shape functions of the bilinear quadrilateral (Q4) element, as shown in
Figure 4D. One notes that the proposed stress recovery scheme may be limited especially when a crack tip approaches the
domain boundary because a part of the virtual mesh grid can go beyond the domain.
For the interpolation of the virtual nodal displacement, one first searches an element in which a virtual node exists. The
element is simply identified by evaluating areas of triangles. For example, one denotes three nodal points of a triangular
element as N1 , N2 , and N3 , whereas the position of a virtual node is indicated as Np , as shown in Figure 5. Then, the
areas of triangles Np N2 N3 , Np N3 N1 , and Np N1 N2 are evaluated. When the sum of the three areas, eg, A1 , A2 , and A3 (see
Figure 5A), is equal to the area of the triangular element, a virtual node is inside or at the boundary of the triangular
element. If the sum of the three triangle areas is greater than the area of the triangular element, a virtual node is outside of
the triangular element, as shown in Figure 5B. After searching the element where a virtual node exists, the virtual nodal
displacement is evaluated using the Lagrange basis shape functions (see Figure 5C).
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(A)
(B)
FIGURE 5 Schematics of the interpolation procedure for a virtual node. Identification
of an element where a virtual node N p exists (A) inside or (B) outside as well as an (C)
evaluation of the displacement for the virtual node
3.2
(C)
Maximum principal stress
A crack propagation direction is identified on the basis of the maximum principal stress. A cohesive surface element is
adaptively inserted along the normal to the principal tensile stress direction. To accurately evaluate the principal tensile
stress direction, a 6 × 6 virtual mesh grid is first generated, which provides a more accurate stress field than an actual finite
element mesh. Then, the stress at each Gauss point of the virtual square elements is obtained, and the stress state around
a crack tip is determined by averaging the stresses at all the Gauss points within the virtual mesh grid. Finally, the crack
propagation angle is determined by evaluating the principal stress direction using the stress state around a crack tip.
3.3
Maximum strain energy release rate
Based on the Griffith energy theory, Hussain et al38 showed that a crack propagates along the direction where the strain
energy release rate is maximized, called the maximum strain energy release rate criterion. To evaluate a strain energy
release rate, the J-integral39 is utilized in conjunction with a domain integral and the proposed stress recovery. The domain
integral expression for the strain energy release rate around a crack tip (see Figure 6) is given as40
(
)
𝜕u𝑗
𝜕qk
𝜕u
J=
− W𝛿ki
dA −
t i q dC,
(1)
𝜎ij
∫A
∫C+ +C− i 𝜕xk k
𝜕xk
𝜕xi
where 𝜎 ij , ti , and ui are the components of the stress, traction, and displacement, respectively. W is the strain energy, and
𝛿 ij is the Kronecker delta. Additionally, qk can be defined as
{
q sin 𝜃,
if k = 1
qk =
(2)
q cos 𝜃,
if k = 2,
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FIGURE 6 Contour C = C1 − 𝛤 + C+ + C− surrounding a crack tip for the domain integral.
Note that n is the outward normal vector to the contour 𝛤 and m is a unit vector outward normal
to the closed domain A
Inner contour ( )
Outer contour ( )
FIGURE 7 Plateau function for the domain integral [Colour figure can
be viewed at wileyonlinelibrary.com]
where q is a sufficiently smooth function. The calculated domain integral is insensitive to the shape of a q function, and
thus, a simple plateau function is employed for q,40-42 as shown in Figure 7. For the computational evaluation of the
domain integral, a 12 × 12 virtual mesh grid is first introduced so that the region of the domain integral corresponds to
the virtual grid domain. Based on the stress obtained from the virtual grid, the domain integral is estimated using the
numerical integration for each element in the virtual mesh grid.
4
CO MPUTAT IONAL IMPLEMENTATION
For cohesive fracture simulation, the weak form of the governing equation is based on the principle of virtual work. The
internal virtual work within a domain (Ω0 ) is equal to the sum of the virtual work done by the external traction (Text ) on
boundary surface (Γ0 ) and the cohesive traction (Tcoh ) on fracture surface (Γcoh ) in the reference configuration, given as
∫Ω0
𝛿E ∶ SdΩ0 =
∫Γ0
𝛿u · Text dΓ0 +
∫Γcoh
𝛿𝚫 · Tcoh dΓ0 ,
(3)
where 𝛿E and S are the virtual Lagrangian strain and second Piola-Kirchhoff stress, respectively. The stress-and-strain
relationship is defined by a compressible neo-Hookean material, and the corresponding strain-energy function is given as
W(F) =
𝜇0
1
𝜆0 log2 (det(F)) − 𝜇0 log (det(F)) + tr(FT F − 3),
2
2
(4)
where 𝜆0 and 𝜇0 are undeformed Lamé's constants, and F is the deformation gradient. Additionally, 𝛿u is the virtual
displacement, and 𝛿𝚫 is the virtual separation on fracture surface. Then, based on the Galerkin approximation, one obtains
a system of nonlinear equations from the weak form, ie, fint = fext + fcoh , where fint , fext , and fcoh are the internal force,
external force, and cohesive force vectors, respectively.
A system of nonlinear equations is solved using the generalized displacement control method43-45 to capture softening
behaviors during crack propagation. In the generalized displacement control method, the sign of a load increment factor
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FIGURE 8 Bilinear softening model for fracture of plain concrete
is based on structural stiffness. Then, a load increment factor at the ith loading step and the jth iteration is given as
√
Δup 11 ·Δup 11
⎧±𝜆1
i−1
i ,
1
Δu
⎪
|
p 1 ·Δup 1 |
𝜆i𝑗 = ⎨
i
i−1
⎪− Δup 1 ·Δur 𝑗 ,
i−1
⎩ Δu ·Δu i
p1
p𝑗
for 𝑗 = 1
(5)
for 𝑗 ≥ 2,
where Δup i𝑗 and Δur i𝑗 are the displacement increments associated with a reference load and a residual force, respectively,
i
and 𝜆11 is a prescribed initial control factor. We note that the sign of Δup i−1
1 ·Δup 1 only changes immediately after load limit
points. Therefore, the generalized displacement control method captures both load and displacement limit points.43,45
For the cohesive traction-separation relationship, in this study, a bilinear softening model is utilized because it has
been widely used to predict fracture behaviors of plain concrete.46,47 A bilinear softening model can be defined using four
fracture parameters: cohesive strength (𝜎 max ), total fracture energy (GF ), initial fracture energy (Gf ), and kink point ratio
(𝜓),48 as shown in Figure 8. We note that the cohesive strength and the total fracture energy are obtained directly from
literature examples. The initial fracture energy corresponds to the area under the initial descending slope, which leads to
the horizontal axis intercept (𝛿 1 ) of 2Gf /𝜎 max . Then, Gf is determined by defining the ratio of the total fracture energy to
the initial fracture energy, where the ratio is generally selected as 2.5.49 The ratio of kink point stress to cohesive strength
(𝜓) is assumed as 0.33.50 The complete failure separation (𝛿 f ) is evaluated by equating GF to the area under the bilinear
softening model, given as
𝛿𝑓 =
2(GF − G𝑓 )
+ 𝛿1 .
𝜓𝜎max
(6)
Additionally, the bilinear softening is only employed for the normal traction-separation (Tn − Δn ) relationship. For the
tangential traction-separation (Tt − Δt ) relation, a simple linear traction-separation relation is employed without softening
behavior, ie, Tt = Dtt Δt , where Dtt is tangent stiffness. In this study, the value of tangent stiffness is selected as 1 N/mm3 51 ,
and thus, the contribution of tangential resistance is small.
5
VERIFICAT ION
To verify the proposed computational framework, two computational examples are employed, ie, strain approximation
with the stress recovery technique and mode-I fracture example. One notes that, for a given displacement field, the approximation of a strain field is equivalent to the evaluation of a stress field. Furthermore, a mode-I example is solved with an
unstructured mesh, whereas a straight crack path is reproduced using the proposed computational framework.
5.1
Strain error for the stress recovery technique
To verify the proposed stress recovery technique, an arbitrary displacement field is introduced to a square domain, and
strain error is estimated. The domain is discretized into two types of meshes, ie, high- and low-quality meshes, with
constant strain triangular elements. Then, the Lagrangian strain is evaluated for the following three cases: (1) general
strain evaluation with a high-quality mesh, (2) general strain evaluation with a low-quality mesh, and (3) proposed strain
recovery with a low-quality mesh. For a high-quality mesh, a 4k structured mesh is employed in this study, as shown in
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(A)
(B)
FIGURE 9 Finite element discretization.
A, 4k structured mesh (high-quality mesh);
B, 4k perturbed mesh (low-quality mesh)
1.19E-03
1.18E-03
1.17E-03
1.16E-03
1.15E-03
1.14E-03
1.13E-03
1.12E-03
1.11E-03
1.09E-03
1.08E-03
1.07E-03
1.06E-03
1.05E-03
1.04E-03
1.03E-03
1.02E-03
1.01E-03
(A)
1.19E-03
1.18E-03
1.17E-03
1.16E-03
1.15E-03
1.14E-03
1.13E-03
1.12E-03
1.11E-03
1.09E-03
1.08E-03
1.07E-03
1.06E-03
1.05E-03
1.04E-03
1.03E-03
1.02E-03
1.01E-03
(B)
1.19E-03
1.18E-03
1.17E-03
1.16E-03
1.15E-03
1.14E-03
1.13E-03
1.12E-03
1.11E-03
1.09E-03
1.08E-03
1.07E-03
1.06E-03
1.05E-03
1.04E-03
1.03E-03
1.02E-03
1.01E-03
(C)
FIGURE 10 Estimated strain field. A, General strain evaluation with the 4k
structured mesh; B, General strain evaluation with the 4k perturbed mesh; C, Strain
recovery technique with the 4k perturbed mesh
Figure 9A. A 4k mesh grid of 15 × 15 is generated within a 0.1 × 0.1 domain, where the numbers of nodes and elements
are 481 and 900, respectively. For a low-quality mesh, the internal nodes of the 4k structured mesh are perturbed with
random directions and magnitudes, denoted as a 4k perturbed mesh (see Figure 9B). In this example, random magnitudes
are limited to half the adjacent edge lengths.
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CHOI AND PARK
An arbitrary quadratic displacement field is employed along the horizontal direction, ie, ux (x,y) = 10−3 (x2 +x), and a
zero displacement is imposed along the vertical direction, ie, uy (x,y) = 0. The displacement fields are imposed to the
nodes of the 4k structured mesh (Figure 9A) and 4k perturbed mesh (Figure 9B). Then, the Lagrangian strain along the
horizontal direction is evaluated at the nodes of the 4k structured and perturbed meshes. The strain at a node is evaluated
by averaging strains of adjacent elements, which corresponds to a general strain evaluation procedure in the finite element
analysis. Note that strain is constant within an element when a constant strain triangular element is used. Additionally,
the proposed strain (or stress) recovery technique is employed to evaluate a strain field in the 4k perturbed mesh. A virtual
5.00E–03
4.75E–03
4.50E–03
4.25E–03
4.00E–03
3.75E–03
3.50E–03
3.25E–03
3.00E–03
2.75E–03
2.50E–03
2.25E–03
2.00E–03
1.75E–03
1.50E–03
1.25E–03
1.00E–03
7.50E–04
5.00E–04
2.50E–04
0.00E+00
(A)
5.00E–03
4.75E–03
4.50E–03
4.25E–03
4.00E–03
3.75E–03
3.50E–03
3.25E–03
3.00E–03
2.75E–03
2.50E–03
2.25E–03
2.00E–03
1.75E–03
1.50E–03
1.25E–03
1.00E–03
7.50E–04
5.00E–04
2.50E–04
0.00E+00
(B)
FIGURE 11 Relative strain error. A, General strain evaluation with the
4k structured mesh; B, General strain evaluation with the 4k perturbed
mesh; C, Strain recovery technique with the 4k perturbed mesh
5.00E–03
4.75E–03
4.50E–03
4.25E–03
4.00E–03
3.75E–03
3.50E–03
3.25E–03
3.00E–03
2.75E–03
2.50E–03
2.25E–03
2.00E–03
1.75E–03
1.50E–03
1.25E–03
1.00E–03
7.50E–04
5.00E–04
2.50E–04
0.00E+00
(C)
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P
150 mm
50 mm
FIGURE 12 Geometry and boundary conditions of a three-point bending
600 mm
specimen
mesh grid of 15 × 15 is generated, and displacements at the virtual nodes are interpolated from the nodal displacements
of the 4k perturbed mesh. Then, the strains at the Gauss points are evaluated using the Lagrange basis shape functions of
the bilinear quadrilateral (Q4) element. Finally, the strain at a virtual node is evaluated by extrapolating from the strain
at the four Gauss points and averaging the nodal strains obtained from adjacent elements.
The estimated strain fields for the 4k structured and perturbed meshes are illustrated in Figure 10. For the strain field of
the 4k structured mesh, uniformly spaced straight contour lines are observed, as shown in Figure 10A, which accurately
captures the analytical linear strain field along the horizontal direction. When the 4k perturbed mesh is utilized with the
general strain evaluation procedure, the contour lines of the strain field are not straight but zigzag (see Figure 10B). This is
because the low-quality mesh results in a less accurate strain field, as expected. When the virtual mesh grid is introduced
for strain recovery, the corresponding contour lines are almost straight, which accurately describes the analytical solution,
as shown in Figure 10C.
Additionally, the relative strain error within the domain is plotted in Figure 11. For the 4k structured mesh, the relative
error for interior nodes is less than 0.001 (see Figure 11A). The error on the boundary nodes is approximately five times
higher than the error on the interior nodes. When the 4k perturbed mesh is employed, the relative error of the approximated strain increases, as expected (see Figure 11B). If the proposed strain recovery technique is utilized with the 4k
perturbed mesh, the error level of the interior nodes is almost equivalent to the error of the 4k structure mesh, as shown
in Figure 11C. In summary, an inaccurate stress field due to low mesh quality can be improved by utilizing the proposed
stress recovery technique.
5.2
Mode-I example: three-point bending test
For a mode-I example, a three-point bending test is employed. The span of a beam is 600 mm with the cross section of
150 mm × 100 mm. An initial notch is placed at the center of the beam, as shown in Figure 12. The elastic modulus and
Poisson's ratio of concrete are 32 GPa and 0.2, respectively. For fracture parameters, the cohesive strength is 3.14 MPa,
and the total fracture energy is 120 N/m.
The domain is discretized into two types of finite element meshes, ie, structured mesh and unstructured mesh, as shown
in Figure 13. The numbers of elements for the structured and unstructured meshes are 13 098 and 10 826, respectively,
whereas the element size is approximately 1 mm along the potential crack path for the two meshes. For the structured
mesh (Figure 13A), a potential crack path, ie, vertical direction, corresponds to the element boundary, and thus, no element split is expected. For the unstructured mesh (Figure 13B), element splits are expected, because the crack path does
not match the element boundaries.
The crack propagation direction is evaluated using three different approaches, ie, the principal stress with the general
stress evaluation, the principal stress with the proposed stress recovery, and the maximum strain energy release rate with
the proposed stress recovery. Note that the general stress evaluation procedure calculates stress based on the original finite
element mesh with split elements, whereas the proposed stress recovery technique introduces the virtual mesh grid, as
discussed in Section 3.
For the structured mesh, a crack propagates along the vertical direction in a straight line, as expected. For the unstructured meshes, the crack patterns are affected by the crack initiation criterion and the stress evaluation procedure, as shown
in Figure 14, because of the error associated with discretization. When the principal stress criterion is utilized with the
general stress evaluation procedure, a crack initially propagates along the vertical direction, whereas a zigzag crack pattern is observed as shown in Figure 14A. If the principal stress is estimated using the proposed stress recovery technique,
the crack path is almost straight up to a certain point and is deviated from the center line (see Figure 14B). Although
the stress recovery technique improves the accuracy of the crack path, a straight line cannot be achieved with the principal stress criterion. Finally, when the maximum strain energy release rate criterion is utilized with the stress recovery
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CHOI AND PARK
(A)
FIGURE 13 Finite element
discretization along the potential crack
path for (A) a structured mesh and (B) an
unstructured mesh
(B)
technique, an almost straight line is obtained as shown in Figure 14C. The relative error of the crack propagation angle
is plotted in Figure 15 for the three cases of the crack initiation criterion. For the maximum strain energy release rate
criterion, the relative error is approximately 10−3 , which can be considered as a straight line. However, the error for the
principal stress criterion with the stress recovery technique suddenly increases when a crack tip position reaches about
40 ∼ 50 mm from the initial crack tip.
The larger error for the principal stress criterion is resulted from the error associated with element quality and a biaxial
stress state in this example. One notes that the angle (𝜃 p ) of the principal stress is given as tan 2𝜃 p = 2𝜎 12 /(𝜎 11 − 𝜎 22 ).
The shear stress at the crack tip region is zero when the structured mesh is employed, and thus, the perfect straight line
is obtained for the crack path. When the unstructured mesh is utilized, the shear stress at the crack tip region is not zero
but a small value, which can lead to a large error of 𝜃 p under a biaxial stress state (𝜎 11 = 𝜎 22 ). Then, the absolute values of
shear stress are plotted in Figure 15A according to the crack tip position. Although the stress recovery technique provides
more accurate shear stress than the general stress evaluation, the evaluated shear stress tends to increase for the case of the
principal stress criterion. The increased shear stress is resulted from the accumulated error associated with a biaxial stress
state. The magnitude of |𝜎 11 − 𝜎 22 | becomes small when the crack tip position reaches 30 ∼ 40 mm, as shown in Figure 15B.
Therefore, a higher error may be expected for the principal stress criterion when a biaxial stress state occurs. However,
CHOI AND PARK
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(A)
(B)
(C)
FIGURE 14 Crack patterns of the unstructured mesh according to the crack initiation criterion. A, The maximum principal stress
criterion with the general stress evaluation; B, The maximum principal stress criterion with the stress recovery technique; C, The maximum
strain energy release rate criterion with the stress recovery technique
100
Relative error
10-2
10-4
10-6
Principal stress
Principal stress with stress recovery
Strain energy release rate with stress recovery
0
20
40
60
Crack tip position (mm)
80
FIGURE 15 Relative error of a crack angle for the unstructured mesh
[Colour figure can be viewed at wileyonlinelibrary.com]
for the maximum strain energy release rate criterion, the domain integral is utilized with the stress recovery technique,
and thus, the crack path is less sensitive to the finite element discretization and the biaxial stress state. Therefore, the
evaluated stress with the stress recovery and the maximum strain energy release rate criterion is more accurate than the
stress for the cases of the principal stress criterion, as shown in Figure 16.
The computational results of the load–crack mouth opening displacement (CMOD) curves are plotted in Figure 17. The
result of the aligned structured mesh is on top of the results of the unstructured mesh with the maximum strain energy
release rate criterion. When the maximum principal stress criterion is utilized, the result of the unstructured mesh with
the general stress evaluation procedure provides a higher post-peak load than the other results. The higher post-peak
load results from the inaccurate crack path prediction, ie, the zigzag crack (see Figure 14A). When the stress recovery
technique is used for the evaluation of principal stress, it gives almost similar results with the structured mesh. However,
it provides a slightly overestimated post-peak load at the end of the analysis because the crack path is deviated from the
vertical direction under the biaxial stress state.
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4
Structured mesh: principal stress
Unstructured mesh (UM): principal stress
UM: principal stress with stress recovery
UM: Max. strain energy release rate
3.5
3
| (MPa)
2.5
12
2
|
1.5
1
0.5
0
0
20
(A)
40
60
Crack tip position (mm)
80
1
-1
11
-
22
(MPa)
0
-2
Structured mesh: principal stress
Unstructured mesh (UM): principal stress
UM: principal stress with stress recovery
UM: Max. strain energy release rate
-3
-4
FIGURE 16 Stress states for the structured and unstructured meshes. A,
Absolute values of shear stress; B, Difference between horizontal and vertical
stresses [Colour figure can be viewed at wileyonlinelibrary.com]
0
20
(B)
40
60
Crack tip position (mm)
80
5
Load (kN)
4
3
2
Structured mesh: principal stress
Unstructured mesh (UM): principal stress
UM: principal stress with stress recovery
UM: Max. strain energy release rate
1
FIGURE 17 Load–crack mouth opening displacement (CMOD) response for
the three-point bending test [Colour figure can be viewed at
wileyonlinelibrary.com]
0
0
0.05
0.1
0.15
0.2
CMOD (mm)
0.25
0.3
0.35
The effect of the virtual grid size on the computational results is investigated. Three virtual grid sizes are utilized, ie,
2 × 2, 6 × 6, and 12 × 12, whereas the crack propagation direction is determined by either the maximum principal stress or
the maximum strain energy release rate. For the maximum principal stress criterion, the grid size is too small (eg, 2 × 2),
and the relative error of the crack propagation angle rapidly increases and reaches up to 50% (see Figure 18A). For the
maximum strain energy release rate criterion, the straight crack path is well captured, and the relative error is less than
4% for the three cases, as shown in Figure 18B. In summary, according to the change of the virtual grid size, the crack
CHOI AND PARK
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100
Relative error
10-1
10-2
10-3
10-4
Grid size of 2 2
Grid size of 6 6
Grid size of 12 12
0
20
(A)
40
60
Crack tip position (mm)
80
100
10-1
Relative error
10-2
10-3
10-4
Grid size of 2 2
Grid size of 6 6
Grid size of 12 12
10-5
0
20
(B)
40
60
Crack tip position (mm)
80
FIGURE 18 Relative error of a crack angle according to the virtual grid
100
size. A, The maximum principal stress criterion; B, The maximum strain
energy release rate criterion [Colour figure can be viewed at
wileyonlinelibrary.com]
108
5
Structured mesh
Unstructured mesh
4.5
Condition number
4
3.5
3
2.5
2
1.5
1
0.5
FIGURE 19 Condition numbers of global stiffness matrices. CMOD, crack
0
0.02
0.04
0.06
CMOD (mm)
0.08
0.1
mouth opening displacement [Colour figure can be viewed at
wileyonlinelibrary.com]
propagation angle obtained from the maximum strain energy release rate is less sensitive than the angle obtained from
the principal stress.
Furthermore, the variations in the condition numbers of the global stiffness matrices are shown in Figure 19 for the
structured mesh and the unstructured mesh with the maximum strain energy release rate criterion. The condition numbers approximately start from about 0.58 × 108 in both cases. When the CMOD reaches about 0.04 mm, ie, the peak load,
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CHOI AND PARK
the condition number sharply increases and decreases. The peak value of the condition number for the unstructured
mesh is almost the same as that for the structured mesh. It shows that the element splitting procedure has little effect on
the variation of the condition number and solution quality.
6
MIXED-MODE EXAMPLES
Two mixed-mode fracture examples of plain concrete are illustrated to validate the proposed computational framework.
The bilinear softening model is used to define the traction-separation relationship, and the corresponding fracture parameters, ie, total fracture energy and cohesive strength, are directly obtained from each experimental work.52,53 In the bilinear
softening model, the ratio of total fracture energy to initial fracture energy is 2.5,49 and the kink point ratio (𝜓) is selected
to be 0.3350 in this study. Thus, neither calibration of material properties nor predefined crack path is introduced in this
FIGURE 20 Geometry and boundary conditions of a four-point
shear specimen
(A)
(B)
FIGURE 21 Finite element meshes of the four-point shear specimen. A, 4k structured mesh; B, Zoomed-in view of the aligned mesh to a
crack path
CHOI AND PARK
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study. For the evaluation of a crack propagation direction, the maximum strain energy release rate is utilized for numerical
simulations.
6.1
Four-point shear test
A four-point shear test performed by Schlangen52 is utilized to predict mixed-mode fracture behaviors of concrete. The
geometry and boundary conditions of the four-point shear test specimen are shown in Figure 20. The length of the specimen is 440 mm with a cross section of 100 × 100 mm. An initial notch of 20 mm is located at the center of the specimen. A
load of 10P/11 was applied on a steel plate, placed 20 mm away from the center of the specimen, and a concentrated force
of P/11 was applied at the lower-left side of the specimen (see Figure 20). Then, two relative displacements were measured, ie, CMOD and crack mouth sliding displacement (CMSD). We note that CMOD is the relative horizontal (opening)
displacement at the mouth of the initial notch, whereas CMSD is the relative vertical (sliding) displacement at the mouth
of the initial notch, as shown in Figure 20.
The material properties of concrete are obtained from the experimental work by Schlangen.52 The elastic modulus
was 30 GPa, and Poisson's ratio was 0.2. The plane strain condition is employed for the computational simulation. The
cohesive strength and the fracture energy were 3 MPa and 115 N/m, respectively. The elastic modulus and Poisson's ratio
of the steel loading plates were 210 GPa and 0.3, respectively. Additionally, the initial fracture energy is calculated as
46 N/m, which results in a GF /Gf value of 2.5, whereas the kink point ratio is selected as 0.33 in the bilinear softening
(A)
(B)
FIGURE 22 A, Predicted crack path for the four-point shear test; B, Comparison between the experimental and computational results
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CHOI AND PARK
model, as discussed previously. Then, the test specimen is initially discretized into two types of finite element meshes,
ie, a 4k structured mesh and an aligned mesh to a crack path. Because a crack path is unknown, a 4k structured mesh is
initially utilized, and thus, the boundary of a 4k structured mesh does not correspond to a potential crack path, as shown
in Figure 21A. The element size around the potential crack path region is 1 mm, which is small enough to capture the
nonlinear fracture process zone.47,54 For the 4k structured mesh, the number of nodes is 9256, and the number of elements
is 18 207. Next, to investigate the effects of the element splits and the stress recovery on computational results, a finite
element mesh is generated so that element boundaries are aligned to a potential crack path, which results in an aligned
mesh (Figure 21B). A potential crack path is assumed as a crack path obtained from the computational result with the 4k
structured mesh. The numbers of nodes and elements of the aligned mesh are 5519 and 10 780, respectively. One notes
that the aligned mesh is utilized to investigate the effects of mesh quality on load-displacement results.
The crack path of the computational result with the 4k structured mesh is plotted in Figure 22A. The crack initially
propagates from the bottom-right corner of the initial notch with an angle of 42◦ . Then, the direction of the crack path
gradually changes along the vertical direction, whereas continuum elements are split. The predicted crack path is within
the experimental results, as shown in Figure 22B. If the principal stress criterion is employed for the crack initiation criterion, a crack kinking phenomenon may be expected at the final stage, as in the previous computational results,55,56 because
of the biaxial stress condition. Next, the computational results of the load-displacement relationships are compared to
the experimental results. The force (P) versus CMOD relationships are plotted in Figure 23A, whereas the force-CMSD
curves are illustrated in Figure 23B. These comparisons demonstrate that the computational results accurately capture
the experimental results, including peak load and post-peak load behaviors. However, the post-peak load of the computational results is slightly lower than that of the experimental results for the force-CMSD relationships. Additionally, when
the aligned mesh is employed, neither the element split nor the node relocation is introduced. Thus, the estimated crack
45
4k structured mesh
Aligned mesh
Experimental results
40
35
Load, kN
30
25
20
15
10
5
0
0
50
100
150
200
CMOD, m
250
300
(A)
4k structured mesh
Aligned mesh
Experimental results
50
Load, kN
40
30
20
10
FIGURE 23 Comparison between the experimental and computational
results for (A) the load–crack mouth opening displacement (CMOD)
relationship and (B) the load-crack mouth sliding displacement (CMSD)
relationship [Colour figure can be viewed at wileyonlinelibrary.com]
0
(B)
0
20
40
60
CMSD, m
80
100
CHOI AND PARK
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path is identical to the element boundary, as expected. Furthermore, the computational results with the aligned mesh are
on top of the results with the 4k structured mesh, as shown in Figure 23.
6.2
Double-edge-notched specimen test
A double-edge-notched fracture test is also employed to validate the proposed computational framework. The geometry of
a test configuration is illustrated in Figure 24. The mixed-mode fracture experiment was performed by Nooru-Mohamed.53
The specimen size was 200 mm × 200 mm with a thickness of 50 mm, and two initial notches of 25 mm were created on the
specimen. Then, a horizontal force (PS ) of 10 kN was initially applied on the upper-left edge of the specimen, whereas the
bottom and lower-right edges were fixed. Next, while maintaining the horizontal force of 10 kN, a vertical displacement
(uN ) was applied along the top surface of the specimen, as shown in Figure 24. Two relative vertical displacements between
points A and A′ and between points B and B′ (uAA′ , uBB′ ) were measured using linear variable differential transformers.
The material properties of the concrete specimen are obtained from the work by Nooru-Mohamed.53 The elastic modulus and Poisson's ratio are 32.8 GPa and 0.2, respectively, and the plane strain condition is assumed for the computational
simulation. The cohesive strength is 3 MPa, and the total fracture energy is equal to 110 N/m. The kink point ratio and
initial fracture energy are selected as 0.33 and 44 N/m, respectively. Then, the domain is discretized into a finite element
mesh, as shown in Figure 25. The numbers of elements and nodes are 8399 and 16 720, respectively. The element size
around the potential crack path region is 1 mm, which is the same size as in the previous examples.
The predicted crack path of the computational result is illustrated in Figure 26A. Two cracks initially propagate from
the initial notches with an approximated angle of 60◦ while the horizontal force increases to 10 kN. When the vertical
displacement increases, the crack propagation direction gradually changes. The predicted crack path agrees well with the
u ,P
P
10 kN
B
P
25 mm
200 mm
A
30 mm
5 mm
30 mm
A'
t
time
t
time
u
B'
200 mm
FIGURE 24 Geometry and boundary conditions of a
double-edge-notched specimen
FIGURE 25 Finite element mesh of the double-edge-notched
specimen
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CHOI AND PARK
(A)
: Experimental results (front face)
: Experimental results (rear face)
: Numerical simulation results
FIGURE 26 A, Predicted crack path for
the double-edge-notched simulation; B,
Comparison between the experimental and
computational results
(B)
25
Computational results
Microplane model (Pivonka et al., 2004)
XFEM (Cox, 2009)
Experimental results
PN (kN)
20
15
10
5
FIGURE 27 Comparison between experimental and computational results
for the double-edge-notched specimen test. XFEM, extended finite element
method [Colour figure can be viewed at wileyonlinelibrary.com]
0
0
0.01
0.02
0.03
0.04
0.05
Average relative displacement (mm)
0.06
experimental results, as shown in Figure 26B. The relationship between the vertical load (PN ) and averaged relative vertical
displacement, ie, (uAA′ + uBB′ )/2, is compared with the computational result. Figure 27 shows that the computational
result accurately predicts the experimental results, although it slightly overestimates the peak load. We note that the
double-edge-notched specimen test was also investigated using the XFEM57 and a microplane model.58 The present study
provides better approximations for the peak and post-peak load behaviors than the previous computational results.
The condition numbers obtained from the current computational results are compared with the condition numbers for
the XFEM,22 as shown in Figure 28. Note that Wu and Li22 investigated the double-edge-notched specimen test using the
XFEM. They used two types of meshes, and the numbers of elements are 594 and 1660, which lead to the approximated
Condition number
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1010
105
Computational results: n ele = 8399
XFEM: nele = 1660 (Wu and Li, 2015)
XFEM: nele = 594 (Wu and Li, 2015)
FIGURE 28 Conditioning numbers for the current computational results
0
0.01
0.02
0.03
0.04
Vertical displacement, u N (mm)
0.05
and the previous extended finite element method (XFEM) results [Colour
figure can be viewed at wileyonlinelibrary.com]
element sizes of 8 and 5 mm, respectively. Because of the relatively larger element sizes in the XFEM, the condition
numbers for the XFEM are lower than those for the current study, especially at the initial stage. While a crack propagates,
the condition numbers for the XFEM increase by a few orders of magnitude, whereas the condition numbers for the
present study remain within the same order of magnitude.
7
CO N C LU S I O N
To remove mesh bias and represent an accurate crack path for mixed-mode cohesive fracture simulation, the present study
introduces the element split in conjunction with the novel stress recovery and domain integral. The crack propagation
direction is determined by the maximum strain energy release rate, which is evaluated using the domain integral and
stress recovery technique. Based on the crack propagation criterion, continuum elements are split, and surface elements
are adaptively inserted along the specified direction. The proposed computational framework is verified and validated
by solving four examples: strain approximation, three-point bending test, four-point shear test, and double-edge-notched
test. Computational results demonstrate that the cohesive surface element approach, with the proposed computational
framework, computes mixed-mode fracture behaviors while removing mesh bias in the crack path representation. Finally,
the main contributions of this paper are summarized as follows.
• To remove mesh bias in the cohesive surface element approach, the use of the element split is proposed with the
novel stress recovery technique and domain integral. For any given direction, the proposed computational framework
represents smooth crack paths within a domain.
• To determine a crack propagation direction, the maximum strain energy release rate criterion is utilized. The criterion
provides an accurate crack path, although a finite element mesh has a bias on discretization. Furthermore, the present
study demonstrates that the maximum principal stress criterion can lead to a large error when a stress state around a
crack tip region corresponds to a biaxial stress state, whereas the maximum strain energy release rate criterion does
not suffer from the biaxial stress state.
• To achieve a more accurate stress field around a crack tip region, a stress recovery technique is proposed by creating
a virtual mesh grid. For a lower-quality mesh, the recovery technique provides better stress and strain approximation
than the general evaluation procedure.
• For the mode-I example, the crack propagates along a straight line even though element boundaries are not aligned to
a potential crack path at the initial discretization. The computational result with an unstructured mesh reproduces the
computational result with a structured mesh provided that the domain integral and stress recovery are utilized with
the element split. These computational results confirm the consistency and accuracy of the proposed computational
framework.
• While a crack propagates, the condition numbers of the stiffness matrix are within the same order of magnitude for
the proposed computational framework. However, we note that the GFEM/XFEM results demonstrated that condition
numbers at the post-peak load regions were a few orders of magnitude higher than the condition number at the initial
discretization.
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CHOI AND PARK
• To validate the proposed computational framework, four-point shear and double-edge-notched tests are investigated.
Material properties are obtained from the literature without a calibration procedure. The computational results capture
smooth crack paths, which agree well with experimental results. Furthermore, the computational results accurately
predict the load-versus-displacement relationships of the experimental results.
ACKNOWLEDGEMENTS
This research was supported by the National Research Foundation of Korea (NRF) through the Basic Science Research
Program funded by the Ministry of Science, ICT and Future Planning (NRF-2018R1A2B6007054) and by the Korea
Institute of Energy Technology Evaluation and Planning funded by the Ministry of Trade, Industry and Energy
(20174030201480). The information presented in this paper is the sole opinion of the authors and does not necessarily
reflect the views of the sponsoring agencies.
ORCID
Kyoungsoo Park
https://orcid.org/0000-0001-6243-5016
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How to cite this article: Choi H, Park K. Removing mesh bias in mixed-mode cohesive fracture simulation with stress recovery and domain integral. Int J Numer Methods Eng. 2019;120:1047–1070.
https://doi.org/10.1002/nme.6170