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Science & Technology Development Journal – Engineering and Technology 2023, 5(SI2):24-32
Open Access Full Text Article Research Article
1Department of Engineering Mechanics,
Faculty of Applied Sciences, Ho Chi
Minh City University of Technology
(HCMUT), 268 Ly uong Kiet Street,
District 10, Ho Chi Minh City, Vietnam.
2Vietnam National University Ho Chi
Minh City, Linh Trung Ward, u Duc
City, Ho Chi Minh City, Vietnam.
Correspondence
Nha Thanh Nguyen, Department of
Engineering Mechanics, Faculty of
Applied Sciences, Ho Chi Minh City
University of Technology (HCMUT), 268
Ly Thuong Kiet Street, District 10, Ho
Chi Minh City, Vietnam.
Vietnam National University Ho Chi Minh
City, Linh Trung Ward, Thu Duc City, Ho
Chi Minh City, Vietnam.
Email: nhanguyen@hcmut.edu.vn
History
•Received: 16-12-2022
•Accepted: 24-8-2023
•Published Online: 31-12-2023
DOI :
https://doi.org/10.32508/stdjet.v6iSI2.1066
Analysis of cracked Reissner-Mindlin plate using an extended
meshfree method
Vay Siu Lo1,2, Thien Tich Truong1,2, Nha Thanh Nguyen1,2,*
ABSTRACT
An extended meshfree method for analyzing cracked plates based on Reissner-Mindlin theory is
presented in this paper. Among a variety of meshfree formulations, the radial point interpolation
method (RPIM) is chosen in this study due to the satisfaction of the Kronecker delta property. Thees-
sential boundary conditions, therefore, are easily imposed in the RPIM. The shape function derived
from RPIM is employed to interpolate the eld variables. An extended RPIM formulation is used to
model the crack segment without explicitly dening it in the discretized domain. The discontinuity
due to the crack is dened by extrinsic enriched functions, particularly, the jump in the displace-
ment eld on two sides of the crack is modelled by the Heaviside function, and the stress singularity
near the crack tip is described by the asymptotic enriched function. In this study, the stress resul-
tant intensity factors (SRIFs) are evaluated through the interaction integral approach. The obtained
SRIFs are shown in the paper through many numerical examples for comparison purposes. The
trending variation of SRIFs is also observed from the numerical results. It can be remarked that the
SRIFs depend on many factors: the number of cracks, crack orientation, load type and boundary
conditions. The numerical examples show the accuracy of the present approach. The obtained
results are compared with analytical solutions and other numerical methods.
Key words: cracked plate, Reissner-Mindlin plate, extended meshfree method, XRPIM
INTRODUCTION
Many engineering applications involve thin plate
structures, such as civil, automobile, ship and other
mechanical systems. Hence, it is essential to analyze
the behavior of plate structures. For the numerical
computation of plate structures, using a plate formu-
lation, such as Kirchho-Love theory1–4, Reissner-
Mindlin theory 5–8, higher-order shear deformation
theory (HSDT) 9–11 and so on requires less number of
degrees of freedom (DOFs) than considering it as a 3D
solid model. Hence, the computational cost for mod-
elling plate structures is reduced when using a plate
formulation. e Reissner-Mindlin theory is a sim-
ple plate formulation that considers rst-order shear
deformation and is appropriate for moderately thick
plates. With a C0continuity formulation, it does not
require higher-order shape functions.
Fracture analysis is also a crucial aspect besides the
analysis of plate structures since it aects the durabil-
ity of the structures. erefore, more studies on the
modelling of plates with through thickness crack are
necessary. Many approaches have already been de-
veloped for solving cracked plate bending problems
using Reissner-Mindlin theory12–14, Kirchho-Love
theory 15,16 and HSDT 17,18 .
e eXtended Finite Element Method (XFEM) is a
powerful technique for modeling crack discontinuity
without explicitly dening it in the problem geome-
try and simulating crack growth without remeshing,
which was rst proposed by19 and has been widely
applied. In the XFEM formulation, the enrichment
functions are used to describe the discontinuity in dis-
placement elds across the crack faces and the singu-
larity in stress elds near the crack tip.
Besides the conventional mesh-based nite element
method, the meshfree method is a developing branch
of computational method. e Radial Point Inter-
polation Method (RPIM)20,21 is a meshfree method
that has a special property called the Kronecker delta.
is makes it easy to apply the essential boundary
conditions in the RPIM, unlike many other meshfree
methods. For that reason, the RPIM is chosen in this
study. In the same way as formulating XFEM, Nguyen
et al. combined RPIM and the extended formula-
tion with enrichment functions to introduce XPRIM,
which they used for 2D fracture problems22–24 . How-
ever, there are still few studies on fracture analysis
of cracked plates using XRPIM. In the scope of this
study, cracked Reissner-Mindlin plates is analyzed by
using the XRPIM.
e stress resultant intensity factors (SRIFs) of
Reissner-Mindlin plates witht hrough-thickness crack
Cite this article : Lo V S, Truong T T, Nguyen N T. Analysis of cracked Reissner-Mindlin plate using an
extended meshfree method.Sci. Tech. Dev. J. – Engineering and Technology 2024; 5(SI2):24-32.
24
Copyright
© VNUHCM Press. This is an open-
access article distributed under the
terms of the Creative Commons
Attribution 4.0 International license.
Science & Technology Development Journal – Engineering and Technology 2023, 5(SI2):24-32
are evaluated in this paper. e XRPIM is used for
modeling, and the radial basis to construct the shape
function is the in Plate Spline (TPS) function. e
accuracy of the proposed method is veried by vari-
ous numerical tests, showing the eectiveness of the
approach.
METHODOLOGY
Reissner-Mindlin plate
e rst-order transverse shear deformation is con-
sidered in the Reissner-Mindlin theory, so the plate
cross-section the aer deformation is still straight but
not normal to the mid-surface25. In the Cartesian co-
ordinate (see Fig. 1), the displacement components
are dened as the following equations
u1(x1,x2,x3) = x3
φ
1(x1,x2)
u2(x1,x2,x3) = x3
φ
2(x1,x2)
u3(x1,x2,x3) = w(x1,x2)
(1)
where (x1,x2,x3)is the position of the interest point,
w is the deection,
φ
1and
φ
2, in that order, are the
rotation angle about x2and x1axes. e sign conven-
tion for
φ
1and
φ
2is depicted in Fig. 1.
Figure 1: Deection, rotations and resultants of a
plate
In plate formulation, the constitutive relation for an
isotropic homogenous material is given as12
M=
M11
M22
M33
=Et 3
12 1−v2×
1v0
v1 0
0 0 1−v
2
φ
1,1
φ
2,2
φ
1,2+
φ
2,1
=t3
12 Db
ε
b
(2)
Q=Q1
Q2=
κ
t
µ
0
0
µ
φ
1+w,1
φ
2+w,2=
κ
tDs
ε
s(3)
where M denotes the moment resultants and Q stands
for the shear force resultants, E is the Young’s modu-
lus,
µ
is the shear modulus, v is the Poisson ratio, t is
plate thickness and tis the shear correction factor and
takes the value as 5/612.
ε
band
ε
sindicate the strain
due to bending and shear, respectively.
In the above equations, Dband Ds, in that order, are
the bending stiness tensor and shear stiness tensor,
and dened as
Db=E
1−v2
1v0
v1 0
0 0 1−v
2
(4)
Ds=
µ
0
0
µ
(5)
e stiness matrix of the Reissner-Mindlin plate is
derived from the weak formulation and expressed
as26
K=t3
12 ΩBT
bDbBbdΩ+ΩBT
sDsBsdΩ(6)
As seen in the equation, the stiness matrix is made
up of two parts: bending (the rst term on the right-
hand side) and shear components (the second term).
B-operators are dened as follow
BI
b=
0
ϕ
I,10
0 0
ϕ
I,2
0
ϕ
I,2
ϕ
I,1
(7)
BI
s=
ϕ
I,1
ϕ
I0
ϕ
I,20
ϕ
I(8)
here
ϕ
Idenotes the shape function. And in this pa-
per, as indicated in the Introductionsection, the shape
function is derived from the RPIM. More detail of the
construction of RPIM shape function can be found in
the reference27 .
For brevity, the RPIM shape function consists of two
constituents: a radial basis and a polynomial basis. In
this study, the in Plate Spline (TPS) function is em-
ployed as the radial basis to form the shape function
and is dened below
Ri(x1,x2) = r
η
i(9)
where
η
is the shape parameter, riis the distance be-
tween the interest point x and the node xi, and dened
as
Extended Radial Point Interpolation
Method
Fig. 2 shows dierent sets of nodes in a cracked plate
problem. W contains all nodes in the support domain
(includes grey, blue and red dots in Fig. 2). Wscon-
tains nodes whose support domain is cut by the crack
(blue dots in Fig. 2). And Wtis the set of nodes in
which the support domain containscrack tip (red dots
in Fig. 2).
25
Science & Technology Development Journal – Engineering and Technology 2023, 5(SI2):24-32
Figure 2: Type of enriched nodes and their support
domains
In the “extended” concept, the interpolation function
for an interest point x is incorporated with the en-
riched function and expressed as below
wh(x) = ∑i∈W
ϕ
i(x)wi+∑j∈Ws
ϕ
j(x)H−Hjbw
j
+∑k∈Wt
ϕ
k(x)∑4
l=1(Gl−Glk )cw
lk (10)
φ
h(x) = ∑i∈W
ϕ
i(x)
φ
i+∑j∈Ws
ϕ
j(x)H−Hjb
φ
j
+∑k∈Wt
ϕ
k(x)∑4
l=1(Fl−Flk )c
φ
lk (11)
in which Hdenotes the Heaviside function and is de-
ned as the following equation
H(f(x)) = +1i f f (x)>0
−1i f f (x)<0(12)
In Eqs. (10) and (11), the asymptotic enrichment
functions Fland Glare dened as12
Fl= (r,
θ
) = {√rsin
θ
2,√rcos
θ
2,
√rsin
θ
2sin
θ
,√rcos
θ
2sin
θ
}
(13)
Gl= (r,
θ
) = {r3/2sin
θ
2,r3/2cos
θ
2,
r3/2sin 3
θ
2,r3/2cos 3
θ
2}
(14)
where r is the distance from the interest point x to
the crack tip xT IP ,
θ
denotes the angle made up of
the crack segment and the line connecting the interest
point x and the crack tip xTI P.
As can be seen from Eqs. (13) and (14), Flis the en-
richment functions for the bending components ap-
pearing in the interpolation equation of the rotation
angle. Flcontains terms proportional to r1/2.Glis
the enrichment functions for the shear components
appearing in the equation of deection. Glcontains
terms proportional to r3/2.
In order to compute the stiness matrix as in Eq
(6), the B-operator is now including the standard
Bstan dard and enriched Benriched components, namely
B=Bstan dard ,Benriched (15)
e enriched B-operators for bending component are
given as
Bspit enr
b=
0[
ϕ
I(H−HI)],10
0 0 [
ϕ
I(H−HI)],2
0[
ϕ
I(H−HI)],2[
ϕ
I(H−HI)],1
(16)
BT IP enr,l
b=
0[
ϕ
I(F−FI)],10
0 0 [
ϕ
I(F−FI)],2
0[
ϕ
I(F−FI)],2[
ϕ
I(F−FI)],1
(17)
And for shear component
Bspit enr
S=
[
ϕ
I(H−HI)],1[
ϕ
I(H−HI)],2
ϕ
I(H−HI)0
0
ϕ
I(H−HI)
T
(18)
Bti p enr,l
s=
[
ϕ
I(Gl−GlI )],1[
ϕ
I(Gl−GlI )],2
ϕ
I(Fl−FlI )0
0
ϕ
I(Fl−FlI )
(19)
Stress resultant intensity factors
To evaluate the fracture behavior of cracked plates, the
factors of stress resultant intensity (SRIFs) are impor-
tant characteristics that need to be dened. SRIFs are
dened as the following equation25
K1=lim
r→0
√2rM22 (r,0)
K2=lim
r→0
√2rM12 (r,0)
K3=lim
r→0
√2rQ2(r,0)
(20)
where K1and K2are the factors of moment intensity
and K3is the factor of shear force intensity.
e stress intensity factors (SIFs) in 3D elasticity are
derived by the following relation [25]
k1(x3) = 12x3
t3K1
k2(x3) = 12x3
t3K2
k3(x3) = 3
2t1−2x3
t2K3
(21)
e values of SRIFs are calculated from the inter-
action integral, more detail can be found in refer-
ences28. e interaction integral for the cracked plate
is dened as the following expression
I=A(Mi j
φ
aux
i,1+Maux
i j
φ
i,1+Qjwaux
i,1+Qaux
jw,1
−Wint
δ
1j)q,jdA +A[Maux
i j,j−Qaux
i
φ
i,1+
Qi
φ
aux
i,1+waux
,i1+
ε
aux
si,1]qdA −Apwaux
,1qdA
(22)
where “aux” is the abbreviation of the “auxiliary” state,
weight function qis dened as
q=1−2|x1−xti p
1|
c1−2|x2−xti p
2|
c(23)
26
Science & Technology Development Journal – Engineering and Technology 2023, 5(SI2):24-32
Figure 3: Interaction integral domain
In Eq. (23), cis the side length of integral domain as
shown in Fig. 3.
Aer computing the interaction integral I, the SRIFs
are derived by using the following relation
I(1,2,3)=24
π
Et 3(K1Kaux
1+K2Kaux
2)+
12
π
10
µ
tK3Kaux
3
(24)
For example, K1is obtained by setting Kaux
1=
1,Kaux
2=Kaux
3=0.
NUMERICAL RESULTS
ere are two numerical examples in this section, and
each example contains two problems, particularly:
• Square plate with two edge-cracks.
• Square plate with inclined central crack.
In each example, there are two types of loads consid-
ered: distributed moment and uniform pressure. e
stress resultant intensity factors are computed in each
problem, and the variation of SRIFs with respect to
dierent crack orientations and crack lengths is also
examined.
Square plate with two edge-cracks
Distributed bending moment on opposite
edges
In this example, a square plate with the dimensions
of 2b= 2 m and thickness tis examined. e plate
contains two cracks on two opposite edges and is sub-
jected to distributed bending moment Mon the other
two edges (see Fig. 4). e crack length is 2a, and the
distance between the two crack tips is 2c= 2b- 4a. e
material properties are: Young’s modulus E=1000 Pa
and Poisson ratio v=0.3. e discretized model with
40 ×40 nodes is used in this example.
Figure 5 presents the variation of the normalized SRIF
K1/(M√a)versus various ratios c/b. e results are
Figure 4: Plate with two cracks on opposite edges.
Figure 5: Normalized K1with dierent c/b ratios of
two cases: b/t = 2 and b/t = 6
shown for two cases: b/t = 2 and b/t = 6. e cur-
rent XRPIM results are compared with the analytical
solution29 , showing good agreement.
Several observations can be drawn from Fig. 5:
thicker plate has higher SRIF, the SRIF in both cases
tends to decrease as the c/b ratio increases from 0.1 to
0.7, and aer that the SRIF slightly increases when the
c/b ratio is in the range of 0.7 – 0.9. It should be noted
that a decrease in c/b ratio means an increase in the
crack length, so the result can be interpreted as when
the crack length increases, the SRIF also increases.
Transverse uniform pressure
In the second load case, the plate is subjected to trans-
verse uniform pressure p. e plate is now simply sup-
ported on top and bottom edges (two edges without
cracks), see Fig. 6. e crack length is a, and the dis-
tance between the two crack tips is 2c= 2b- 2a. e
material properties are the same as the rst problem:
E=1000 Pa and v=0.3. is model is also discretized
into a set of 40 ×40 scattered nodes.
27
Science & Technology Development Journal – Engineering and Technology 2023, 5(SI2):24-32
Figure 6: Simply supported plate with two edge
cracks subjected to uniform pressure.
Figure 7 presents the variation of the normalized SRIF
K1/pb2√awith respect to dierent c/b ratios. e
results are shown for two cases: b/t = 2 and b/t =
6. e current XRPIM results are compared with
DBEM30, showing good agreement.
Figure 7: Normalized K1with dierent c/b ratios
Similar to the rst example, some observations can be
drawn from Fig. 7: the smaller the b/t ratio, the higher
the SRIF. e SRIF in both cases tends to decrease as
the c/b ratio increases from 0.1 to 0.7, and aer that
the SRIF slightly increases.
Square plate with inclined central crack
Distributed bending moment on two oppo-
site edges
is example analyzes a square plate with dimension
2bcontaining an inclined crack at the center, see Fig.
8. e crack length is 2a. e plate is subject to dis-
tributed moment Mon top and bottom edges. e
material properties are given as follows: E = 1000
Pa and v=0.3. e discretized model with 50 ×50
nodes is used.
e moment intensity factors K1and K2are com-
puted in the XRPIM approach and compared with the
analytical solutions31
K1=
ϕ
(t/a)M√acos2
β
,
K2=
ψ
(t/a)M√acos
β
sin
β
.(25)
where
β
is the inclination angle (see Fig. 8). e
values of coecients
ϕ
,
ψ
can be referred to32. For
2a= 2 m, b/t = 5 and t/a = 1, these coecients are
ϕ
=0.7475 and
ψ
=0.5218.
Figure 8: Square plate with a central slant crack un-
der bending
Figure 9 shows the change of the intensity factors K1
and K2due to the crack angle
β
. e gure shows that
K1decreases when the orientation angle
β
increases,
while K2increases and decreases symmetrically. It is
also observed that the opening mode is dominant in
the range of
β
from 00to 400, and aer that the mag-
nitude of both SRIFs is approximate.
Figs 10 – 12 illustrate the distributions of normal
stress
σ
11 and
σ
22, and shear stress
σ
12. e particu-
lar case for these gures is the inclined angle
β
=300.
e stress components are evaluated on the top sur-
face of the plate. e stress singularity is clearly ob-
tained at the crack tips.
28
Science & Technology Development Journal – Engineering and Technology 2023, 5(SI2):24-32
Figure 9: Variation of K1and K2versus the crack an-
gle
β
Figure 10: Normal stress distribution
σ
11 (Pa) on the
top surface,
β
=300.
Figure 11: Distribution of normal stress
σ
22 (Pa) on
the top surface,
β
=300.
Figure 12: Distribution of shear stress
σ
12 (Pa) on
the top surface,
β
=300.
Transverse uniform pressure
e square plate with a horizontal central crack is sub-
jected to transverse uniform pressure p. e bound-
ary condition is simply supported on all edges, see
Fig. 13. e material properties are: Young’s mod-
ulus E=1000 Pa and Poisson ratio v=0.3. e crack
length is 2a, and the ratio b/t = 6.
Figure 13: Simply supported square plate with cen-
tral horizontal crack subjected to uniform pressure.
For horizontal crack, as shown in Fig. 9, K2is zero.
erefore, only K1is considered in this example. Fig-
ure 14 illustrates the variation of the normalized SRIF
K1/pb2√awith respect to ratios a/b. e obtained
results by XRPIM are veried with the solution from
the conventional FEM25 , showing good agreement.
It can be observed from Fig. 14 that the SRIF tends to
decrease as the a/b ratio increases. is trend as well
29
Science & Technology Development Journal – Engineering and Technology 2023, 5(SI2):24-32
Figure 14: Normalized K1with dierent a/b ratios
as the slope of the curve is dierent from the exam-
ple in Section 3.1. ese two examples both have the
same load, so the cause of this dierence could be the
boundary conditions. For the two cracks case, only
two edges are simply supported while in this problem,
all four edges are simply supported.
DISCUSSIONS
As presented in Section 3, the obtained results given
by XRPIM are in good agreement with analytical so-
lutions, FEM and DBEM. e trending variation of
SRIFs is also observed from the numerical results. It
can be remarked that the SRIFs depend on many fac-
tors: the number of cracks, crack orientation, load
type and boundary conditions.
For the case of two cracks, the following conclusion
can be drawn: the plate with higher thickness has
higher SRIF, the SRIF decreases when the crack dis-
tance to plate length ratio c/b increases from 0.1 to
0.7, and aer that the SRIF slightly increases when the
c/b ratio is in the range of 0.7 – 0.9. It should be noted
that a decrease in c/b ratio means an increase in the
crack length, so the result can be interpreted as when
the crack length increases, the SRIF also increases.
For the case of inclined crack, the result shows that
K1decreases when the orientation angle
β
increases,
while K2tends to increase and decrease symmetri-
cally. It is also observed that the opening mode is
dominant in the range of
β
from 00to 400, and aer
that the magnitude of both SRIFs is approximate.
And for the case of horizontal crack under uniform
pressure, the SRIF values decrease as the crack length
to plate length ratio a/b increases. is decreasing
trend is dierent from the two cracks example. ese
two examples both have the same uniform pressure,
so the cause of this dierence could be the boundary
conditions. For the two cracks case, only two edges
are simply supported while in the single horizontal
crack problem, all four edges are simply supported.
CONCLUSIONS
Cracked plate problems are investigated in this study
with the help of the extended meshfree XRPIM asso-
ciated with the Reissner-Mindlin plate theory. e
RPIM is dierent from other meshfree methods be-
cause it has the Kronecker delta property of shape
functions. is property makes it easy to impose the
essential boundary conditions in the RPIM as in the
conventional FEM. e Reissner-Mindlin theory is
suitable for the relatively thick plates due to the as-
sumtion of rst-order shear deformation. With a C0
continuity formulation, it does not require higher-
order shape functions. e present meshfree method
would be possible to extend for complex cracked plate
problems such as dynamic fracture, crack growth and
nonlinear (geometry and material) analysis in future
works.
e present approach is shown to be accurate in the
evaluation of the stress resultant intensity factors.
Many numerical examples show good agreement with
analytical solutions and numerical solutions of other
methods. e trend of SRIF in dierent cases is also
observed.
ACKNOWLEDGMENT
is research is funded by Vietnam National Uni-
versity Ho Chi Minh City (VNU-HCM) under grant
number B2022-20-02. We acknowledge the support
of time and facilities from Ho Chi Minh City Uni-
versity of Technology (HCMUT), VNU-HCM for this
study.
ABBREVIATIONS
DOF: Degree of freedom.
FEM: Finite Element Method.
FSDT: First-order shear deformation theory.
HSDT: Higher-order shear deformation theory.
RPIM: Radial Point Interpolation Method.
SIF: Stress intensity factor.
SRIF: Stress resultant intensity factor.
TPS: in Plate Spline.
XFEM: eXtended Finite Element Method.
XRPIM: eXtended Radial Point Interpolation
Method.
CONFLICT OF INTEREST
Group of authors declare that this manuscript is origi-
nal, has not been published before and there is no con-
ict of interest in publishing the paper.
30
Science & Technology Development Journal – Engineering and Technology 2023, 5(SI2):24-32
AUTHORS’ CONTRIBUTION
Vay Siu Lo is the main developer of the method and
edits the manuscript.
ien Tich Truong plays the role of the supervisor,
he also contributes overall ideas for the proposed
method.
Nha anh Nguyen contributes key idea for the pro-
posed method and also checking the manuscript.
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Tạp chí Phát triển Khoa học và Công nghệ – Engineering and Technology 2023, 5(SI2):24-32
Open Access Full Text Article Bài nghiên cứu
1Bộ môn Cơ kỹ thuật, Khoa Khoa học
ứng dụng, Trường Đại học Bách khoa
Tp. HCM, Việt Nam
2Đại học Quốc gia ành phố Hồ Chí
Minh, Việt Nam
Liên hệ
Nguyễn Thanh Nhã, Bộ môn Cơ kỹ thuật,
Khoa Khoa học ứng dụng, Trường Đại học
Bách khoa Tp. HCM, Việt Nam
Đại học Quốc gia Thành phố Hồ Chí Minh,
Việt Nam
Email: nhanguyen@hcmut.edu.vn
Lịch sử
•Ngày nhận: 16-12-2022
•Ngày chấp nhận: 24-8-2023
•Ngày đăng: 31-12-2023
DOI : https://doi.org/10.32508/stdjet.v6iSI2.1066
Bản quyền
© ĐHQG Tp.HCM. Đây là bài báo công bố
mở được phát hành theo các điều khoản của
the Creative Commons Attribution 4.0
International license.
Phân tích tấm nứt Reissner-Mindlin bằng một phương pháp không
lưới mở rộng
Lồ Sìu Vẫy1,2, Trương Tích Thiện1,2, Nguyễn Thanh Nhã1,2,*
TÓM TẮT
Bài báo này trình bày một phương pháp không lưới mở rộng để phân tích các tấm nứt dựa trên
lý thuyết tấm Reissner-Mindlin. Trong các phương pháp không lưới, phương pháp nội suy điểm
hướng kính (RPIM) được chọn trong nghiên cứu này do sự thỏa mãn thuộc tính Kronecker delta.
Do đó, các điều kiện biên cần thiết có thể được áp đặt dễ dàng trong RPIM. Hàm dạng RPIM được
sử dụng để nội suy chuyển vị và các biến môi trường trong bài toán. Phương pháp RPIM mở rộng
được sử dụng để mô hình vết nứt mà không cần mô tả tường minh nó trong miền rời rạc của bài
toán. Sự bất liên tục gây ra bởi vết nứt được xác định bằng các hàm làm giàu, cụ thể là, bước nhảy
trong trường chuyển vị trên hai mặt của vết nứt được mô hình hóa bằng hàm Heaviside và sự suy
biến ứng suất gần đỉnh vết nứt được mô tả bằng hàm làm giàu tiệm cận đỉnh vết nứt. Trong nghiên
cứu này, các hệ số cường độ ứng suất tổng hợp (SRIFs) được đánh giá thông qua phương pháp
tích phân tương tác. Xu hướng biến thiên của SRIFs cũng được xem xét trong các kết quả tính toán
số. Có thể nhận xét rằng SRIFs phụ thuộc vào nhiều yếu tố: số lượng vết nứt, định hướng của vết
nứt, loại tải trọng và điều kiện biên. Các kết quả SRIFs thu được trong bài báo được trình bày thông
qua nhiều ví dụ số nhằm mục đích so sánh và kiểm chứng độ chính xác của phương pháp. Kết quả
thu được được so sánh với các kết quả giải tích và các phương pháp số khác.
Từ khoá: tấm nứt, tấm Reissner-Mindlin, phương pháp không lưới mở rộng, XRPIM
Trích dẫn bài báo này: Vẫy L S, Thiện T T, Nhã N T. Phân tích tấm nứt Reissner-Mindlin bằng một
phương pháp không lưới mở rộng . Sci. Tech. Dev. J. - Eng. Tech. 2023, 5(SI2):24-32.
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