A method of determining effective elastic properties of honeycomb cores based on equal strain energy 1 3 4 5 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22 Chinese Journal of Aeronautics, (2017), xxx(xx[.]
Trang 16 School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China
7 Received 25 January 2016; revised 11 August 2016; accepted 29 December 2016
8
11
12 Elastic properties;
Abstract A computational homogenization technique (CHT) based on the finite element method (FEM) is discussed to predict the effective elastic properties of honeycomb structures The need of periodic boundary conditions (BCs) is revealed through the analysis for in-plane and out-of-plane shear moduli of models with different cell numbers After applying periodic BCs on the represen-tative volume element (RVE), comparison between the volume-average stress method and the boundary stress method is performed, and a new method based on the equality of strain energy
to obtain all non-zero components of the stiffness tensor is proposed Results of finite element (FE) analysis show that the volume-average stress and the boundary stress keep a consistency over different cell geometries and forms The strain energy method obtains values that differ from those
of the volume-average method for non-diagonal terms in the stiffness matrix Analysis has been done on numerical results for thin-wall honeycombs and different geometries of angles between oblique and vertical walls The inaccuracy of the volume-average method in terms of the strain energy is shown by numerical benchmarks
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17
18 1 Introduction
19 For the past several decades, sandwich plates with a
honey-20 comb core have been widely used in the field of aviation In
21 understanding the behavior of sandwich structures under
dif-22 ferent types of load, the honeycomb core is often regarded as
23 homogeneous solid with orthotropic elastic properties.1 As a
24 result, research on the effective elastic properties of the
honey-25 comb core is of great essence for the calculation and design of
26 honeycomb sandwich structures
27
A computational homogenization technique (CHT) has
28 been found to be a powerful method to predict the effective
29 properties of structures with periodic media In order to obtain
30 the effective stiffness tensor, which relates to the equivalent
31 strain and stress, this process is divided into solving six
elemen-32 tary boundary value problems, which refer to uniaxial tensile
33 and shear in three directions.2–5The equivalent strain is
deter-34 mined after applying the unit displacement boundary
condi-35 tions (BCs) on the representative volume element (RVE) cell
36 corresponding to one of the six elementary problems Different
* Corresponding author.
E-mail address: 07343@buaa.edu.cn (Z Guan).
Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
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Chinese Journal of Aeronautics
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http://dx.doi.org/10.1016/j.cja.2017.02.016
Trang 237 methods have been used when dealing with the equivalent
38 stress Volume-average stress is often used to determine
effec-39 tive properties in the literature.2–6Catapano and Montemurro2
40 investigated the elastic behavior of a honeycomb with
double-41 thickness vertical walls over a wide range of relative densities
42 and cell geometries The strain-energy based numerical
43 homogenization technique was also used by Catapano and
44 Jumel3 in determining the elastic properties of
particulate-45 polymer composites Montemurro et al.4 performed an
opti-46 mization procedure at both meso and macro scales to obtain
47 a true global optimum configuration of sandwich panels by
48 using the NURBS curves to describe the shape of the unit cell
49 Malek and Gibson5got numerical results of a thick-wall
hon-50 eycomb closer to their analytical solutions by considering
51 nodes at the intersections of vertical and inclined walls Shi
52 and Tong6focused on the transverse shear stiffness of
honey-53 comb cores by the two-scale method of homogenization for
54 periodic media Many researchers also seek the stress on the
55 boundary of the RVE cell Li et al.7used the sum of the node
56 force on the boundary of the RVE cell to obtain the equivalent
57 stress Papka and Kyriakides8set plates on the top and bottom
58 of the RVE cell to exert BCs However, regarding honeycomb
59 structures as a combination of cell walls and air, the stress
vari-60 ations on the boundary cause the boundary stress inaccurate to
61 calculate effective properties Some divergence still exists in
62 numerical results of regular hexagonal honeycomb structures
63 with analytical solutions, especially for the in–plane and
out-64 of-plane shear moduli From the definitions of effective elastic
65 properties expressed by Yu and Tang,9the equivalent stress is
66 required to make sure that the RVE cell and the corresponding
67 unit volume of the homogeneous solid undergo the same strain
68 energy Hence, the whole honeycomb structure containing a
69 finite number of RVE cells have the same strain energy as that
70 of the whole volume of the homogeneous solid The
mathe-71 matical homogenization theory (MHT) has proven that the
72 strain energy in the RVE can be determined by the
volume-73 average stress and strain.10However, it is not always suitable
74 for the calculation of the volume-average stress method in
75 the CHT The volume-average method cannot get all precise
76 values in the stiffness matrix, and it is found to get larger strain
77 energy than that obtained from direct analysis in
two-78 dimensional porous composites by Hollister and Kikuchi.11
79 Therefore, we focus on the total strain energy of the RVE cell
80 and propose a new method to determine all the components of
81 the stiffness tensor more accurately in terms of the strain
82 energy
83 In Section2, the differences between the proposed energy
84 method and previous methods are analyzed A process to
85 obtain 9 components of the effective stiffness tensor based
86 on the energy method is introduced Then, finite element
87 (FE) models are discussed in Section3 Convergence analysis
88 has been done over material properties, mesh sizes, and BCs
89 applied on the whole model In addition, two models are
pro-90 posed to acquire in–plane and out-of-plane shear moduli
91 according to the different deformations of a single RVE cell
92 and a finite number of RVE cells under the same loading After
93 establishing appropriate models for honeycomb structures,
94 numerical results over a range of cell geometries are compared
95 to analytical solutions in literature in the next section Finally,
96 Section5ends the paper with some conclusions
97
2 Prediction method
98 2.1 Introduction of a computational homogeneous technique
99 Previous experimental data and theory have proven that a
100 honeycomb core can be classified as an orthotropic material.12
101 Under this assumption, a honeycomb core conforms to
gener-102 alized Hook’s law13as
103
r11
r22
r33
r23
r13
r12
2 6 6 6 6 4
3 7 7 7 7 5
¼
2 6 6 6 6 4
3 7 7 7 7 5
e11
e22
e33
c23
c13
c12
2 6 6 6 6 4
3 7 7 7 7 5 ð1Þ
105 106 where r and e are, respectively, the equivalent stress and strain
107 tensors for the whole geometry of an RVE cell Cijis one of the
108 components of the stiffness tensor C which is symmetric as
109
Cij= Cji.In addition, the shear strain relates the components
110
of the strain tensor as follows,
111
e11! e11¼@u
@x
e22! e22¼@v
@y
e33! e33¼@w
@z
c23! 2e23¼@w
@z
c13! 2e13¼@w
@z
c12! 2e12¼@v
@y
8
>
>
>
<
>
>
>
:
ð2Þ
113 114 where u, v, and w represent the displacements in the x, y, and z
115 directions
116
To determine the effective stiffness matrix of the RVE cell,
117 six elementary BCs are applied on the RVE cell, which refer to
118 three uniaxial extensions and three shear deformations For
119 each load case, only one component of the strain tensor is
120 not zero Then the relative stiffness component is determined
121
by the equivalent stress Take C11for example,
122
C11¼ r11
e11
124 125 After obtaining all the 9 independent components in the
126 stiffness matrix, engineering constants can be derived from
127 the compliance matrix which is the inverse of the stiffness
128 matrix
129 2.2 Energy method
130 Assuming that an elementary shear boundary displacement is
131 applied on the RVE cell (ckl– 0), Eq.(4)is tenable since the
132 boundary of the RVE cell has an identical displacement
133 1
V
Z
135 136 where V represents the total volume of the RVE and subscript
137
‘‘kl” stands for the certain BC
138 The strain energy of the RVE cell under certain loading can
139
be determined by the FE result as
140
U¼1 2
Z
142
Trang 3143 To guarantee the RVE cell and the corresponding volume
144 of the homogeneous material having the same strain energy,
145 the equivalent stress in this method is
146
rkl¼ U
ð1=2ÞcklV¼
R
rijeijdv
148
149 Hence, the effective modulus through this energy method is
150
G¼
R
rijeijdv
c2
152
153 In Section2.1, a CHT has been introduced which obtains
154 components of the stiffness tensor by solving six elementary
155 BC problems However, in this energy method expressed in
156 Eq (7), only one component of the equivalent strain tensor
157 is non-zero, which means that only one component of the
158 equivalent stress tensor can be calculated through Eq (6)
159 (i.e., only one component of the equivalent stress rkl
con-160 tributes to the strain energy) Thus, only diagonal components
161 in the stiffness matrix can be acquired by the six elementary
162 BC problems In order to get all the 9 independent elastic
con-163 stants of the corresponding homogeneous solid, a bi-axial
164 strain field is applied to obtain the value of Cij(i– j) in Eq.(1)
165 Taking C12for example, the BC is set as e11¼ e22¼ e (two
166 uniaxial tensions applied simultaneously) while the
displace-167 ments in other directions are zero According to the equality
168 of the strain energy,
169
1
2r11e11Vþ1
171
172
r11þ r22¼2U
174
175 From Eq.(1),
176
178
179
181
182 Adding Eqs.(10) and (11), and considering the symmetry of
183 the stiffness matrix,
184
C12¼1
2
r11þ r22
e C11 C22
¼1 2
2U
e2V C11 C22
ð12Þ 186
187 C13and C23can also be acquired by exerting similar BCs on
188 the RVE cell With the diagonal components obtained by the
189 six elementary load cases, the entire stiffness matrix is
deter-190 mined and 9 engineering constants are then calculated from
191 the compliance matrix
192 2.3 Comparative study
193 As mentioned in Section 1, different methods have been
194 applied to determine the equivalent stress In this section, a
195 comparative study is done among the volume-average method,
196 the boundary method, and the energy method
197 (1) Volume-average method
198
199 The equivalent stress in this method is calculated as follows:
200
rkl¼ 1
V
Z
202
203 where rkl is the corresponding stress component in the stress
204 field obtained from the FE analysis.The related shear modulus
205 can be written as
206
G1¼V1
R
rkldv
ckl
¼2
R
rkldvR
ekldv
V2
c2 kl
ð14Þ 208 209
212 The equivalent stress is obtained by summing up node
213 forces on the boundary as
214
rkl¼
P
Fkl
217 where Fklis the corresponding node force on the boundary of
218 the RVE cell and S0means the area of the section on which the
219 displacement is applied
220 Thus, the effective shear modulus is determined by
221
G2¼
P
Fkl
S0ckl
ð16Þ 223 224
227 The effective property obtained by the energy method is
228 shown in Eq.(7)
229
We review Eqs (14) and (16) to compare the
volume-230 average method and the boundary stress method Without loss
231
of generality, the volume-average stress in the RVE cell14is
232 1
V
Z
V
rijdv¼ 1 V
Z
V
rikdkjdv¼ 1
V
Z
V
rik
@xj
@xk
dv
¼ 1 V
Z
V
½ðrikxjÞ;k rik;kxjdv ¼1
V
Z
234 235 where i; j; k 2 f1; 2; 3g
236
Eq.(17)shows that the average stress depends uniquely on
237 the surface loading Here, a further proof is given to show that
238 the average stress only relates to the average stress on the
sur-239 face where the unit displacement is applied
240 For a rectangular RVE cell, as shown in Fig 1, the six
241 boundary surfaces are named A to F respectively
242 Z
@vriknkxjds¼
Z
riknkxjdsAþ
Z
riknkxjdsF
¼
Z
ðri2xjÞAdsAþ
Z
ðri2xjÞBdsB
þ
Z
ðri3xjÞCdsC
Z
ðri3xjÞDdsD þ
Z
ðri1xjÞEdsE
Z
ðri1xjÞFdsF ð18Þ
244
Fig 1 A rectangular RVE
Trang 4245 Consider the six elementary displacement BCs:
246 (1) e11– 0
247
248 In this periodic media, for the two points having the same
249 local load on Surfaces E and F,
250
252
253 where uEFmeans the height of the RVE
254 As a result of the symmetry of both the honeycomb RVE
255 and the applied BCs, the stress also distributes symmetrically,
256 i.e.,
257
ðr12ÞA;B¼ 0
ðr13ÞC;D¼ 0
(
ð20Þ 259
260 Considering the correspondence between Surfaces E and F
261 in the periodicity of the RVE, Eq.(18)can be written as
262 Z
@vriknkxjds¼
Z
ðr11x1ÞEdsE
Z
ðr11x1ÞFdsF
¼
Z
r11ð1 þ e11ÞuEFdsE ð21Þ 264
265 Then under this BC, the volume-average stress can be
deter-266 mined from Eqs.(17) and (21)as
267
1
V
Z
V
r11dv¼uEF
V
Z
r11ð1 þ e11ÞdsE 1
sE
Z
r11dsE ð22Þ 269
270 Similar results can be acquired for uniaxial tensions in
271 other two directions (e22–0 and e33–0)
272 (2) e12– 0
273
274 In this periodic media, for the two points having the same
275 local load on Surfaces E and F,
276
278
279 As a result of the symmetry of both the honeycomb RVE
280 and the applied BCs, the stress also distributes symmetrically,
281 i.e.,
282
ðr13ÞC;D¼ 0
ðr11ÞE;F¼ 0
(
ð24Þ 284
285 Considering the correspondence between Surfaces E and F
286 in the periodicity of the RVE, Eq.(18)can be written as
287 Z
@vriknkxjds¼
Z
ðr12x2ÞEdsE
Z
ðr12x2ÞFdsF
¼
Z
r12ð1 þ e12ÞuEFdsE ð25Þ 289
290 Then under this BC, the volume-average stress can be
deter-291 mined from Eqs.(17) and (25)as
292
1
V
Z
V
r12dv¼uEF
V
Z
r12ð1 þ e12ÞdsE 1
sE
Z
r12dsE ð26Þ 294
295 Similar results can be acquired for shear deformations in
296 other two directions (e13–0 and e23–0)
297 The above analysis shows that under both the tensions and
298 shear deformations, the volume-average stress only depends on
299 the average stress on the surface where the unit displacement is
300 applied Eqs (22) and (26) show an equality of the
volume-301 average stress and the boundary stress, which will be discussed
302 later in Section4
303
We review Eqs.(7) and (14)to compare the energy method
304 and the volume-average method The difference lies in
305
R
rijdvR
eijdv and 2VR
rklekldv
306 Without loss of generality, for the energy method,
307 V
Z
V
rijeijdv¼ V
Z
Z
@Vrijeikxknjds
¼ Veik
Z
@Vrijxknjds¼ Veikdkj
Z
V
rijdv
¼ Z
V
eijdv Z
V
309 310 where i; j; k 2 f1; 2; 3g
311
Eq (27) indicates an equality between the
volume-312 average method and the energy method when calculating
313 the components in the stiffness matrix However, as
314 mentioned in Section 2.2 for the operating process of the
315 energy method, the equilibrium shown in Eq.(27)only exists
316 for the diagonal elements in the stiffness matrix, because
317 non-diagonal components can’t be calculated directly by
318
Eq (7) In other words, the volume-average method in the
319 calculation of the equivalent stress is only acceptable for
320 diagonal elements like C11, C22, and so on, while divergence
321 exists at non-diagonal elements
322 Taking the calculation of C12for example, in the
volume-323 average method, C12is calculated as
324
C012¼ 1
Ve
Z
326 327 where r011stands for the local stress when e22¼ e is applied
328
Eq.(12)shows the calculation in the energy method, which
329 considers the total strain energy when applied to e11¼ e22¼ e
330
On one hand, by using C012acquired by the volume-average
331 method as Eq.(28), the total strain energy of the RVE under
332 bi-axial BCs e11¼ e22¼ e is written as
333
U0¼1
2Vðr11e11þ r22e22Þ ¼Ve2
2 ðC11þ C22þ 2C0
12Þ
¼Ve2
2 ðC11þ C22Þ þ e
Z
335 336 where r011means the stress field inside the RVE when a
mono-337 axial strain e22¼ e is employed, which is the process of the
338 volume-average method On the other hand, similar to the
339 problem for the work and energy under several loads (which
340
is often used for the introduction of Maxwell’s reciprocal
the-341 orem15), the total strain energy can be written as
342
U¼Ve2
2 ðC11þ C22Þ þ
Z
344 345
In this equation, r011 also means the stress field inside the
346 RVE when a mono-axial strain e22¼ e is employed
347 Comparison between Eqs.(29) and (30) shows that
diver-348 gence lies in eR
r011dv andR
r011e11dv
349 Set d1¼R
r011dvR
e11dv and d2¼ VR
r011e11dv
350 For a homogeneous material (with a stiffness component
351
C12), d1¼ C12
R
e22dvR
e11dv and d2¼ C12VR
e11e22dv
352
In FE software ABAQUS, strain and stress fields are
353 present in every integral point inside a single element, which
Trang 5354 form the entire mesh Therefore, the integrals of d1 and d2
355 transform into the summation of each element Assuming that
356 in the FE model, the corresponding strain in each element is
357 e22: x1; x2; ; xn and e11: y1; y2; ; yn, and the volume in
358 each element as v: z1; z2; ; zn, then
359
d1¼ C12ðx1z1þ x2z2þ þ xnznÞðy1z1þ y2z2þ þ ynznÞ
¼ C12
Xn i¼1
xiy2
iz2
i ;j¼1;2; ;n i<j
ðxiyjþ xjyiÞzizj
0
B
1
361
362
d2¼ C12ðx1y1z1þ x2y2z2þ þ xnynznÞðz1þ z2þ þ znÞ
¼ C12
Xn i¼1
xiy2
iz2
i;j¼1;2; ;n
i <j
ðxiyiþ xjyjÞzizj
0
B
1
364
365 Comparison between Eqs (31) and (32) shows that the
366 inhomogeneity of the strain distribution is the main reason
367 that causes the difference between the volume-average method
368 and the energy method Numerical results indicate that a
rela-369 tively greater strain exists in the air zone than that in the wall,
370 and the impact of the variation of the strain within the wall
371 material is insignificant mainly due to its small volume
372 fraction
373 According to all the above analysis, the difference of the
374 acquired effective properties between the volume-average
375 method and the energy method, especially for the in-plane
376 moduli, depends on the volume fraction of the air zone and
377 the inhomogeneity of the strain field in the RVE cell
More-378 over, all these factors mentioned relate to the wall thickness
379 and cell geometries
380
3 FE model
381 3.1 Convergence
382 The RVE cell chosen for a single-wall-thickness honeycomb
383 structure is shown inFig 2 As the vertical and oblique walls
384 have the same thickness in the single-wall-thickness
honey-385 comb, the cell geometric parameters in Fig 2(b) can be
386
t1¼ t, t2¼ t=2 Three parameters including t=l, h=l and h
387 determine the geometric configuration of the RVE cell In later
388 analysis, we focus on the most commonly used honeycombs,
389 whose l and h remain the same as l = h = 15 This absolute
390 value is not significant as geometries only provide
non-391 dimensional coefficients for the effective properties Moreover,
392 the core height is set to hc= 50 and it is in the range where the
393 core height has little influence on effective properties according
394
to Ref.2
395 Regarding the honeycomb structure as a two-phase mixture
396
of the core material and air as shown inFig 2(c), aluminum
397 alloy is the common material for honeycombs, whose modulus
398
E¼ 70 GPa and Poisson’s ratio t ¼ 0:3 The so-called ‘‘elastic
399 air” is endowed to the air zone to get the strain field in the air
400 whose modulus and Poisson’s ratio are set as Eair¼ 0:001 MPa
401 and tair¼ 0 In order to evaluate whether the elastic constant
402
of elastic air is appropriate, the in-plane shear modulus is
cho-403 sen owing to its relatively small value compared to other
404 properties
405
Fig 3(a) presents different values of the calculated in-plane
406 shear modulus at different properties of elastic air, in which Gc
407
is the converged value The deviation of the shear modulus
408 with Eair¼ 1 MPa from the shear modulus with
409
Eair¼ 104MPa is about 0.8%, while the deviation of
410
Eair¼ 103MPa from 104MPa is less than 0.001% This
Fig 2 Single-wall-thickness honeycomb structure
Trang 6411 shows that a smaller modulus of elastic air does not lead to a
412 significant change of the numerical result Therefore, it is
413 appropriate to choose 103MPa as the elastic property of
414 the air zone
415 Bending deformation dominates in the honeycomb
struc-416 ture under in-plane loading.16The contributions of axial and
417 shear deformations are also included for the analysis of the
418 out-of-plane properties,17 so a model composed of 3D solid
419 elements is established to take into account all the
three-420 dimensional deformations to get strain and stress fields that
421 are more precise As a result of the hourglass phenomenon
422 existing in the reduced-integration linear element C3D8R (with
423 enhanced hourglass control) in ABAQUS,18 the number of
424 mesh divisions along the cell wall thickness is studied in the
425 bending problem, as shown in Fig 3(b) When the number
426 of divisions n¼ 1, the numerical result of the effective shear
427 modulus shows the hourglass phenomenon as the
‘‘zero-428 energy mode” which makes the stress field in the wall material
429 almost zero, thus resulting in a very small effective shear
mod-430 ulus obtained by this mesh size As the number of divisions
431 increases, hourglass is suppressed and a converged modulus
432 is approached While the deviation between the results of
433 n¼ 5 and n ¼ 7 is less than 1%, n ¼ 5 is chosen for the
num-434 ber of divisions along the wall thickness
435 According to Eq.(3), elementary BCs will be applied to get
436 an equivalent strain for the whole RVE cell The essence of this
437 computational homogeneous method is that the stiffness of the
438 honeycomb structure is replaced by the stiffness of equivalent
439 solids Therefore, it is of great significance to simulate the
440 deformation of the whole structure accurately
441 Hence, nonlinear effects brought by the possible large
442 deformation of the wall are taken into account Based on the
443 above analysis, numerical results considering nonlinear effects
444 at varying strain are shown inFig 3(c) The results indicate
445 that a nonlinear effect happens at a strain of 101, whose
effec-446 tive modulus is above 50% higher than that at a 103strain
447 The difference between results at 102and 103 strains is
448 within 1% as well as the difference between those at 103
449 and 104 strains Thus, a strain of 103 is the chosen value
450 for the elementary BCs In the meanwhile, the maximum stress
451 of the honeycomb wall under a 103 strain BC is 21.1 MPa
452 from the FE results, which is lower than the yield stress of
453 5052 aluminum alloy commonly used in commercial
honey-454 comb structures.19,20Therefore, the honeycomb wall stays in
455 the linear elastic stage under this loading, and it is appropriate
456 to use this model to calculate the elastic properties
457 3.2 Boundary conditions
458 Catapano and Montemurro2 gave detailed BCs when taking
459 into account the symmetries of the unit cell As discussed by
460 Hori and Nemat-Nasser,21 the homogeneous stress BC and
461 the homogeneous strain BC only provide the lower and upper
462 bounds of effective moduli The plane-remains-plane
homoge-463 neous BCs (or unit-displacement BCs) not only over-constrain
464 the boundaries, but also violate the stress periodicity
condi-465 tions In theoretical analysis of the FE model proposed by
Cat-466 apano and Montemurro2 in the calculation for the in-plane
467 shear modulus G12 and the out-of-plane shear modulus G13,
468
we found that the calculated value differs from different cell
469 numbers
470 3.2.1 G12under unit-displacement BCs
471
In this part, the accuracy of the RVE cell selected inFig 2(b)
472
to simulate the entire honeycomb structure under in-plane
473 shear loading is discussed The effect of cell numbers on the
474 effective shear modulus is analyzed, based on which a new
475 model (The new model is called Model 2, as the model shown
476
inFig 2(c) is called Model 1) is proposed to get more precise
477 properties of in-plane shear under unit-displacement BCs
478 When the honeycomb structure is subject to a shear loading
479
in the 1-2 direction as shown inFig 4, as mentioned before,
480 bending deformation of the walls dominates in the honeycomb
481 core The deflection of the vertical walls and the rotation of the
482 oblique walls together cause a deformation for the whole
hon-483 eycomb core under 1-2 shear loading From the previous
the-484 ory in literature,22 the proportion of the deflection of the
485 vertical walls in the whole shear deformation on the boundary
486 is
487
F 1 h 3 24EI
F 1 h 2 ðhþlÞ
489 490 Therefore, the deflection of the vertical walls plays an
491 important role in determining the effective in-plane shear
492 modulus
493
Fig 4(a) and (b) illustrate the difference between a single
494 RVE cell and a finite number of RVE cells under the same
495 in-plane shear loading, which show that the local load on the
496 vertical walls is not the same, thus causing different deflections
497 For a single RVE cell, the deflection of the vertical Wall AE
498 (CF) is
499
Fig 3 Convergence analysis of FE model
Trang 7501
502 While for the honeycomb core consisting of a finite number
503 of RVE cells, the deflection of Wall AE (CF) is
504
w2¼ Fh3
506
507 Load conditions on other parts of the RVE cell are the
508 same, so there is no difference in the caused shear deformation
509 From the above analysis, it can be seen that the discrepancy
510 in the deflection of the vertical Wall AE (CF) will result in a
511 difference of the calculated effective shear modulus, and the
512 result of a single RVE cell will be lower than that of the
hon-513 eycomb core containing a finite number of RVE cells The
514 essence of this phenomenon is that the process of the
homoge-515 neous technique in this loading condition is replacing the
516 shearing stiffness of the honeycomb core by the bending
stiff-517 ness of the wall The vertical Wall AE (CF) in the chosen RVE
518 cell has a half-wall thickness, which leads to a considerable
519 reduction in the bending stiffness However, the halved wall
520 causes little section change of the RVE cell due to the very
521 small volume fraction of the wall in the whole honeycomb
522 core For these reasons, the effective shear modulus of a single
523 RVE cell is different from that of the whole honeycomb core
524 To make a single RVE cell precisely simulate the
deforma-525 tion of a real honeycomb core with respect to in-plane shear,
526 We introduce a new model named Model 2 on the basis of
527 the FE model (Model 1) shown in Fig 2 Only a geometric
528 change is done in Model 2 as t2= 0.795t from t2= t/2 in
529 Model 1 For Model 2, the deflection of Wall AE (CF) under
530 the same in-plane loading is
531
w01¼ ðF=2Þh3
533
534 This result has the same value of the deflection as in a finite
535 number of RVE cells
536
FE models containing different numbers of cells as
respec-537 tively n = 1, 2, 4, 8, 14, 16 are established to evaluate the effect
538
of the cell number on the effective in-plane shear modulus, and
539 these models have the same BCs as those in Ref.2
540 Calculations have been conducted on honeycombs of three
541 geometries:
542 (1)t ¼ 0:2; h ¼ 30o; l ¼ h ¼ 15
543 (2)t ¼ 0:4; h ¼ 30o; l ¼ h ¼ 15
544 (3)t ¼ 0:2; h ¼ 45o; l ¼ h ¼ 15
545 546 The FE results inFig 5indicate that for all the geometric
547 situations, the effective in-plane shear modulus increases as
548 more cells are included in the FE models, which is consistent
549 with the previous analysis Moreover, all the curves have a
ten-550 dency of approaching a converged value that is closer to the
551 modulus obtained by Model 2 Therefore, the results presented
552
inFig 5can be a validation for the accuracy of Model 2 to
553 simulate a real honeycomb core under in-plane shear loading
554 3.2.2 G13under unit-displacement BCs
555 Similar to Section3.2.1, the effect of the cell number on the
556 effective out-of-plane shear modulus is analyzed, on the basis
557
of which another new model called Model 3 is proposed to
558 get more precise properties of the out-of-plane shear modulus
559
in the 1-3 direction for the whole honeycomb core
560 When the honeycomb structure is subject to a shear loading
561
in the 1-3 direction as shown inFig 6, the overall deformation
562
of the honeycomb core is governed by the shear deformations
563
of the vertical and oblique walls We conduct an analysis on
564 the shear flows inside each wall in a similar way used by Kelsey
565
et al.23 Fig 4 Honeycomb structure under shear loading along 1-2 direction
Trang 8566 For the single RVE cell presented in Fig 6(a) under 1-3
567 shear loading (with a shear stress s), the following equation
568 is acquired by equilibrium conditions:
569
qb¼ qd
qc¼ qe
qaþ qc¼ qb
sðh þ l sin hÞ 2l cos h ¼ ðqbþ qcÞl cos h
qah
2þ ðqb qcÞl sin h þ ðqd qeÞ h
8
>
>
>
>
>
>
>
>
ð37Þ
571
572 From Eq.(37),
573
qa¼ 0
qb¼ qc¼ qd¼ qe¼ sðh þ l sin hÞ
ð38Þ 575
576 For the real honeycomb core consisting of a finite number
577 of RVE cells, under the consideration of periodicity,
578
580
581 From equilibrium conditions,
582
qb¼ qcþ qd
qaþ qc¼ qb sðh þ l sin hÞ 2l cos h ¼ ðqbþ qcÞl cos h
qah
2þ ðqb qcÞl sin h þ ðqdþ qeÞ h
8
>
<
>
:
ð40Þ 584 585 From Eqs.(39) and (40), we get
586
qa¼ qd¼ qe¼ 0
qb¼ qc¼ sðh þ l sin hÞ
ð41Þ 588 589 Comparison between Eqs.(38) and (41) shows that under
590 the same shear loading in the 1-3 direction, the shear flow in
591 Wall AE as well as CF varies in a single RVE cell and a finite
592 number of RVE cells In a single RVE cell, similar to the
situ-593 ation in the simulation of in-plane shear loading, Walls AE
594 and CF suffer higher shear flows than those in a finite number
595
of RVE cells, leading to differences in the deformation for the
596 entire honeycomb structure The higher shear flows qdand qe
597 cause the in-plane bending deformation of the oblique walls
598
as shown inFig 7from the FE analysis of a single RVE cell
599
It can be seen that the extra bending deformation of the
obli-600 que walls is in a direction contributing to the 1-3 shear
defor-601 mation, and FE results illustrate that this bending deformation
602
is weakening as the cell number increases For these reasons, a
603 single RVE cell has a larger deformation than that of a finite
604 number of RVE cells under the same loading owing to the
605 bending deformation of the oblique walls, thus having a lower
606 effective shear modulus in the 1-3 direction than that of a finite
607 number of RVE cells
608
In order to find an appropriate RVE model to simulate the
609 deformation of the real honeycomb structure under the 1-3
610 direction shearing precisely, the thickness of the oblique walls
611
is adjusted to suppress the extra bending deformation The
612 geometry of the proposed new model called Model 3 is
deter-613 mined according to the FE results shown inFig 8
614
Fig 8 shows the effective shear modulus along the 1-3
615 direction of a RVE cell with a geometry of t = 0.2, h ¼ 30,
616 and l = h = 15 at different thicknesses of the oblique walls
Fig 6 Honeycomb structure under shear loading along 1-3 direction
Fig 5 Effective shear modulus vs number of RVE cells (Model 1
t2= t/2, Model 2 t2= 0.795t)
Trang 9617 The horizontal dotted line in Fig 8 represents the
618 converged value of the finite number of RVE cells It can be
619 seen that when the thickness of the oblique walls
620 t1= 0.243 = 1.215t, we get a value very close to the converged
621 value for G13 According to these results, t1= 1.215t is chosen
622 as the geometric change of Model 3 from Model 1, and the
623 accuracy of Model 3 is going to be evaluated later inFig 9
624 Similar to the in-plane shear loading, FE models containing
625 different numbers of cells as respectively n = 1, 2, 4, 8, 14, 16
626 are also used to evaluate the effect of the cell number on the
627 effective out-of-plane shear modulus along the 1-3 direction,
628 and honeycombs of three geometries are also taken into
con-629 sideration, as shown in Fig 9 With the number of cells
630 increasing, the effective shear modulus G13 grows and
631 approaches a converged value, as analyzed before In addition,
632 the converged value is very close to the result of Model 3 for all
633 the three cell geometries Therefore, this new model can
pro-634 vide a relatively more accurate value of G13than that of Model
635 1 under this shear loading
636 3.2.3 Periodic boundary conditions
637 As stated before, a single RVE cell cannot provide accurate
638 G12 and G13 as those of the whole honeycomb core with
639 unit-displacement BCs However, when applied to periodic
640 BCs, the effective properties remain constant with the cell
641 number changing Moreover, periodic BCs are required in
642 determining the elastic properties of periodic media to ensure
643 the periodicity of displacements and tractions on the boundary
644 of the RVE In order to generate a symmetrical mesh for the
645 convenience of prescribing the periodic BCs, a whole unit cell
646 is remained for FE analysis
647 Mathematical expressions can be found in the literature by
648 Whitcomb,24Xia,25Li and Wongsto,26which have been used
649
in periodic media like unidirectional composites and plane
650 and satin weave composites Constraint equations (CEs) are
651 utilized for periodic BCs of the rectangular solid RVE shown
652
inFig 1 These equations can be sorted into three categories,
653 i.e., equations for surfaces, edges, and vertices
654 (1) Equations for surfaces
655 656 Under three uniaxial tensions and shear deformations
657 (e0
11; e0
22; e0
33; c0
12; c0
13; c0
23), CEs can be applied to three pairs of
658 surfaces
659 For surfaces perpendicular to 1-axis, i.e., Surfaces E and F:
660
ux¼W 1 ux¼0¼ e0
vx¼W1 vx¼0¼ 0
wx¼W1 wx¼0¼ 0
8
>
662 663 For surfaces perpendicular to 2-axis, i.e., Surfaces A and B:
664
uy¼W2 uy¼0¼ c0
vy¼W 2 vy¼0¼ e0
wy¼W 2 wy¼0¼ 0
8
>
666 667 For surfaces perpendicular to 3-axis, i.e., Surfaces C and D:
668
uz¼W 3 uz¼0¼ c0
vz¼W 3 vz¼0¼ c0
wz¼W 3 wz¼0¼ e0
8
>
670 671 672 (2) Equations for edges
673 675 Two or three in Eqs.(42)(44)are satisfied for nodes on the
676 edges of the RVE As these constraints are not independent,
677
FE analysis cannot function properly if the CEs for edges
678 are not considered separately as well as the CEs for vertices
679 For edges parallel to the 1-axis, i.e., Lines hd, ed, fb, and gc:
680
uea uhd¼ e0
vea vhd¼ 0
wea whd¼ 0
8
>
682 683
ufb uhd¼ e0
vfb vhd¼ e0
wfb whd¼ 0
8
>
685
Fig 9 Effective shear modulus vs number of RVE cells (Model 1
t1= t, Model 3 t1= 1.215t)
Fig 8 Effective shear modulus along 1-3 direction vs thickness
of oblique wall AB (BC)
Fig 7 In-plane bending deformation of oblique walls under
shear loading along 1-3 direction
Trang 10ugc uhd¼ c0
vgc vhd¼ e0
wgc whd¼ 0
8
>
688
689 Similar equations can be given for edges parallel to the
690 2-axis and 3-axis
691 (3) Equations for vertices
692
693 Point d is fixed to avoid rigid-body motions of the RVE
694 Then seven CEs are defined for other vertices with the
refer-695 ence of Point d Equations for Points f and g are given here
696 as examples:
697
uf ud¼ e0
vf vd¼ e0
wf wd¼ e0
8
>
699
700
ug ud¼ c0
vg vd¼ e0
wg wd¼ e0
8
>
702
703 Periodic BCs are prescribed in the ABAQUS software
com-704 bined with Python scripts Python scripts are used to find
705 nodes on surfaces, edges, and vertices, generate CEs for
corre-706 sponding nodes in the mesh according to Eqs (42)(49), and
707 submit jobs in the ABAQUS environment After jobs are
fin-708 ished, post-processing Python scripts are executed to calculate
709 effective moduli by both the volume-average method and the
710 energy method
711 4 Results
712 4.1 Volume-average stress and boundary stress
713 A total of six cases as listed inTable 1are analyzed
Control-714 ling parameters include the form of the cell (single-wall
715 thickness and double-wall thickness), the thickness of the wall
716 (t = 0.2 and t = 0.4), and the angle between vertical and
717 oblique walls (h ¼ 30 and h ¼ 45) The reason for choosing
718 these parameters is that later analysis has shown that the
719 energy method and the volume-average method have larger
720 divergences under the chosen geometries For each case, the
721 volume-average stress and the boundary stress are both used
722 to get the effective modulus E1=E0
1inTable 1means the ratio
723 of the modulus obtained by the boundary stress to that by the
724 volume-average stress
725 An inspection ofTable 1 shows that the values of E=E0,
726 G=G0
, and t=t0 in all directions are very close to unity for all
727 cases Therefore, it can be concluded that the
volume-728 average stress is equal to the boundary stress regardless of
729 changes in cell geometries and forms
730
As discussed in Section 2.3, Eqs (22) and (24) show an
731 equality of the volume-average stress and the boundary stress,
732 which is validated by the results in Table 1 Understanding
733 such an equality, only the energy method and the
volume-734 average method are operated in later FE analysis
735 4.2 Results by energy method
736
In this part, we have compared the numerical results by the
737 volume-average method and the energy method Different cell
738 geometries such as the wall thickness and the angle between
739 vertical and oblique walls are taken into consideration to
eval-740 uate the supposed discrepancy between the two methods
741
Fig 10shows the numerical results of the energy method
742 and the volume-average method at different wall thicknesses
743 which range from 0.2 to 1.0 as the RVE cell has a dimension
744
of l = h = 15
745
It can be seen inFig 10that the three shear moduli G12,
746
G13, and G23 calculated by the volume-average method and
747 the energy method are all nearly the same with a maximum
dif-748 ference of less than 3% Since the shear moduli only relate to
749 the diagonal elements in the stiffness matrix while calculated
750 from the compliance matrix, this equilibrium of these three
751 shear moduli can be a validation for Eq (27) Nevertheless,
752 for the elastic properties relate to non-diagonal components
753 (i.e E1, E2, E3, t12, t13, and t23), divergences exist between
754 the volume-average method and the energy method
755
Fig 10(d) shows that the volume-average method gets the
756 same results as those of Malek & Gibson’s model, which
vali-757 dates our proper use of the volume-average method
758
As stated in Section2.3, the main motivation that we put
759 forward this energy method is the supposed discrepancy in
760 the calculation of C12 It has been proven in Section2.3that
761
an inaccurate calculation of C12in the volume-average method
762 leads to an inaccurate strain energy under bi-axial BCs The
763 discrepancy of the strain energy in each model of unit cells is
764 presented in Fig 11 It can be seen that the discrepancy
765 remains 1.3–1.6% within our computing range Although the
766 relative error remains nearly constant, the absolute value
767 increases as the wall thickness increases
768 For the in-plane elastic properties E1 and E2, within the
769 range of calculation, the absolute value of the discrepancy
770 between the two methods varies with the cell wall thickness
771 increasing However, in these geometries, as the wall thickness
772 increases, the relative error between these two methods is
773 slightly changed from nearly 8% to 6% This follows from
Table 1 Comparison between boundary stress method and volume-average stress method
E01
E 2
E02
E 3
E03
t 12
G 13
G013
G 23
G023