An analytical model to predict the equivalent creep strain rate of a lattice truss panel structure

Author’s Accepted Manuscript An analytical model to predict the equivalent creep strain rate of a lattice truss panel structure Wenchun Jiang, Shaohua Li, Yun Luo, Shugen Xu, Jianming Gong, Shan-Tung Tu www.elsevier.com/locate/msea

PII: DOI: Reference:

S0921-5093(16)30230-1 http://dx.doi.org/10.1016/j.msea.2016.03.028 MSA33423

To appear in: Materials Science & Engineering A Received date: 14 January 2016 Revised date: 3 March 2016 Accepted date: 4 March 2016 Cite this article as: Wenchun Jiang, Shaohua Li, Yun Luo, Shugen Xu, Jianming Gong and Shan-Tung Tu, An analytical model to predict the equivalent creep strain rate of a lattice truss panel structure, Materials Science & Engineering A, http://dx.doi.org/10.1016/j.msea.2016.03.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

An analytical model to predict the equivalent creep strain rate of a lattice truss panel structure Wenchun Jianga, Shaohua Lia, Yun Luoa, Shugen Xua, Jianming Gongb, Shan-Tung Tuc a

State Key Laboratory of Heavy Oil Processing, China University of Petroleum (East China), Qingdao, 266580, PR

China b

School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 210009, PR China

c

Key Laboratory of Pressure System and Safety (MOE), School of Mechanical and Power Engineering, East China

University of Science and Technology, Shanghai 200237, PR China 

Corresponding author. Tel.: +86 532 86980609, fax: +86 532 86980609. Jiang)

E-mail address: [email protected] (Wenchun

Abstract We developed an analytical model to predict the equivalent creep strain rate of a lattice truss panel structure. The model, which takes into account the effects of the bonded node and the intersection node of the trusses, is well validated by finite element analysis. Compared with Hodge and Dunand model, this model obtains a more accurate prediction result. The creep deformation of the panel structure is controlled by the creep of vertical trusses parallel to the applied load. The equivalent creep strain rate is determined by five key parameters including punching angle, cutting angle, truss thickness, width and length. A slight change of truss dimension can lead to a big variation of the creep rate by orders of magnitude. With the increase of punching angle and cutting angle, the relative density decreases and the stresses in the trusses increase, leading to an increase of creep rate. With the increase of truss thickness and width, the creep rate decreases because the relative density increases and the stresses in the truss decrease. As the truss length increases, the creep rate increases due to the decrease of relative density and the increase of stresses in the truss.

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Key words: Lattice truss panel structure; Equivalent creep strain rate; Analytical model

1. Introduction Lightweight lattice truss panel structures are currently designed as compact heat exchangers at high temperature gas reactor (HTGR) and modern steam turbine, because they have high efficiency of heat transfer and high strength [1-6]. At high temperature, the time-dependent creep deformation is the main potential failure mode, and the design procedure for creep strength should be developed before their applications [7]. It is a cellular material and the creep strength will be different from the bulk homogeneous material. In recent years, the equivalent creep strength of cellular or foam materials has been extensively studied. The first model was developed by Gibson and Ashby [8] (GA model) which assumes that the creep deformation is controlled by the bending of struts perpendicular to the applied stress. Andrews et al. [9] and Huang [10] found that the creep-rupture of foams can be described by the well-known Monkman-Grant relationship. The creep strain rate is affected significantly by the relative density, microstructural imperfection and the creep parameters of solid cell struts [11]. Huang and Gibson [12] studied the power-law creep of open-cell foams by using finite element method (FEM). They found that removal of only a few percent of the struts can increase the creep rate by one to two orders of magnitude. Gibson et al. [13] also found that the creep of sandwich beams with metallic foam cores depends on the geometry of the sandwich beam, the creep behavior of the foam core and the loading conditions. Diologent et al. [14] found that GA model can give a satisfactory result with experimental for aluminum-magnesium foam. Taking the mass at strut nodes into

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account, Fan et al. [15] developed a modified GA model to calculate the creep rate of low density foams, which was well verified by FEM. A second model, developed by Hodge and Dunand (HD model), assumes that the compression of the struts are the main creep failure mode [16] and provides a very closed result with FEM and within a factor of 2 of the experimental creep data. Basing on GA and HD models, Boonyongmaneerat and Dunand [17, 18] developed a set of analytical models based on engineering beam analysis to predict creep behavior of cellular materials over a broad range of relative density. As the porosity decreases, the controlling creep mechanism changes from strut bending, to strut shearing, and ultimately to strut compression. Assuming the metal foam as elastic foundation, Zhou and Tu [19] developed another model to predict the creep deformation of plate-foam structure, which was also verified by FEM. To date the creep models are mainly focused on low-density foams with or without cell walls. Lattice truss panel structure is a sandwich structure made of two metallic thin face sheet and lattice truss material, which leads to a different creep behavior from the foam material. The previous work is concentrated on the mechanical strength and fabrication technology [20-22], and little attention has been paid on the creep strength. Therefore, it is essential to develop an analytical model to calculate the equivalent creep rate. The geometrical dimensions of the core have a great effect on residual stress [23, 24], but how they affect the creep strength is still unclear. In the present work, an analytical model is developed to calculate the equivalent creep strain rate of lattice truss panel structure, which is verified by FEM. In addition, the effects of geometrical parameters on equivalent creep rate have been investigated, which gives a reference for the structure optimization. 2. Experimental

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Figure 1a shows the schematic of a lattice truss panel structure which consists of two face sheets and lattice truss structure. The lattice truss structure is formed by punching continuously at the middle of the edge of a diamond sheet [21]. It is a periodic sandwich structure made of Hastelloy C276 material, and the unit cell is shown in Fig.1b. The chemical compositions are listed in Table 1. The bulk material obeys the Norton creep equation: .

c  B n

(1)

where  c is the creep strain rate (h-1); σ is stress (MPa)；B and n are material constants. Uniaixial creep test of bulk material was performed to get the creep parameters B and n. The test temperature is 600°C. Fig. 2 shows the geometry of the specimen, and the experiments were carried out under constant stress levels of 270, 290, 310 and 330 MPa. The obtained creep curve is shown in Fig. 3a. Taking logs of both sides of Eq. (1) gives .

log( c )  n log(  )  log B

(2)

Using experimental uniaxial creep data to plot vs. and fitting a straight line of best fit through this data can get n from the gradient and B from the y-axis intercept as shown in Fig.4. The fitted creep constants B and n are 1.26×10-28 MP-nh-1 and 9.63. Fig.3b plots the creep rate versus time, which clearly shows that there are typical three creep stages. The creep rate decreases very quickly at the primary stage; then almost keeps stable at secondary stage and increases sharply at the tertiary stage. 3. Analytical model 3.1 HD model 4

For lattice truss panel structure, the trusses mainly bear the compressive or tensile load, which also obey the HD model proposed by Hodge and Dunand [16]. They studied the creep rate of foam with hollow struts, and found that the creep deformation takes place mostly in the vertical struts parallel to the uniaxial applied stress, while the horizontal struts carry very little stress. In their model, the uniaxial deformation of the cell is given by the average creep rate of four vertical struts, representing one-third of the cell volume. Therefore, the creep rate of the cell with relative density (ρ) can be approximated by the creep rate of a cell with relative density of ρ/3 containing only vertical struts: n

     B n 3

(3)

where ρ and ε are the relative density and the equivalent creep strain of the unit cell, 1/3 is the ratio of the vertical truss volume and unit cell volume. The relative density ρ is calculated by



m M / Vm Vt   t M / Vt Vm

(4)

where ρm and ρt are the density of the unit cell and solid truss; M is the total mass; Vm and Vt are the volume of the unit cell and solid truss. Vm is calculated by:

           Vm   2l sin  2w / cos  2l cos sin  4w  2d cot(45  )  l cos cos  d  (5) 2 2  2 2 4  2 2   where l, w and d are length, width and thickness of truss, α and β are punching angle and cutting angle, respectively (see Fig.1b). The volume of the vertical truss V is calculated by

V  4  l  cos   cos   d  wd

5

(7)

The ratio V / Vm , which is similar to ρ/3 in Eq. (3), is calculated by     4  l cos cos  d  wd V 2 2   (8)           Vm    2l sin  2w / cos  2l cos sin  4w  2d cot(45  )  l cos cos  d  2 2  2 2 4  2 2  

Substituting Eq. (8) into Eq. (3), we obtain the equivalent creep rate: n

 V  n   V  m

  B

n

      4  l cos cos  d  wd   2 2    n  B   2l sin   2 w / cos   2l cos  sin   4 w  2d cot(45   )  l cos  cos   d       2 2  2 2 4  2 2   (9) n

    wd  n  B   l sin   w / cos   l cos  sin   2 w  d cot(45   )      2 2  2 2 4   

3.2 The present analytical model According to Hodge and Dunand [16], we assume that the creep deformation is mainly induced by the vertical struts. We simplified the stress model in Fig.5 by the following assumptions: (1) The truss is decomposed to the horizontal and vertical directions, respectively, and the vertical component is defined as vertical truss. The creep is mainly induced by the vertical trusses parallel to the load, and the horizontal part vertical to the load is ignored. Similarly, the bonded node (see Fig.1b) is also vertical to the load and its effect is also ignored. (2) The intersection node (see Fig.1b) has little effect on creep, because it has strengthened the mechanical behavior [25]. (3) The effect of face sheet on creep is ignored because the trusses have great constraint and support on the face sheet.

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By the above assumptions, it deduces that each truss bears the same stress and the stresses along the truss length (σ’) are uniform. Thus the creep deformation of each truss is equal to the equivalent creep deformation of the panel structure. Substituting σ’ into Norton Equation Eq. (1), we can obtain the equivalent creep rate. In the vertical direction, there is a force balance between the total stress F applied to the top surface of face sheet and the total force F applied on the vertical truss:

F  F

(10)

S    S   '

(11)

' 

S  S

(12)

' 

S  h   Vm    S  h V

(13)

' 

 

(14)

where S and S  are the cross-section area of the face sheet and vertical trusses, V and   are the volume and relative density of the vertical trusses, respectively; σ’ is the stress of the vertical truss. Similar to the ρ/3 in HD model, the ratio of the volume between vertical trusses ( V ) and the total volume of trusses ( Vt ) is：     d w    (2 w) 2  ( ) 2  wd cos  cos l    2 2  cos  cos  cos  V  2 2 2    Vt  lwd   w  2w / cos  d 2 

  is calculated by:

7

(15)

  

V Vt V V   Vt Vm Vt Vm

   d w 2   (16)  (2w) 2  ( )  wd cos  cos l     2 2  cos cos  cos 2 2 2                l sin  w / cos  l cos sin  2 w  d cot(45  )  l cos cos  d  2 2  2 2 4  2 2  

Substituting Eq. (16) into Eq. (14), the stress in the vertical truss σ’ is calculated. Then substituting σ’ into Eq. (1), the equivalent creep rate is obtained: n

        d w    (2w) 2  ( ) 2  wd cos  cos   l     2 2    cos cos  cos   2 2 2   'n   B  B  n              l sin  w / cos  l cos sin  2 w  d cot(45  )  l cos cos  d   2 2  2 2 4  2 2        

(17)

Eq. (17) has very important significance, because it can calculate the creep rate with different geometrical dimensions at different load. In order to verify the present model, we perform a solid finite element creep analysis to a unit cell of lattice truss panel structure in Section 4. 4. Finite element analysis The finite element analysis code ABAQUS is adopted to simulate the creep behavior of the lattice truss panel structure. As shown in Fig.1, the lattice truss panel structure is a periodic structure, and thus we build a three dimensional finite element model of a unit cell. The truss length l, width w, thickness d, punching angle α and cutting angle β are 22mm, 1.5mm, 1mm, 70° and 60°, respectively. A uniaxial creep test for 50000h is simulated. The tensile load is 1.5MPa. Norton creep law Eq. (1) is adopted for the solid

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material and the creep constant is obtained by the experimental in section 2. The finite element meshing and boundary conditions are shown in Fig.6. It notes that a rigid plate is added on the top surface of face-sheet in order to get a uniform creep deformation. The distributed tensile load was applied on the rigid plate. The element type is C3D8. The effect of number of elements on calculation results has been examined. In total, 54466 nodes and 43928 elements were meshed. The symmetric boundary conditions were applied on the left and front face of the model, and the bottom face was constrained in Y-direction. The simulated equivalent creep rate of the lattice truss panel structure  eq is calculated by:

 eq 

h ht

(18)

where h is creep deformation of the panel structure, h is the initial total thickness of the lattice truss material, as shown in Fig.5, and t is the creep time. 5. Results and discussion 5.1 Comparison between the present analytical model and HD model Figure 7 shows the equivalent creep rate with different load by the analytical model, HD model and FEM. The creep rate increases almost linearly as the load increases from 0.9 to 1.3MPa, as similarly found by Hodge and Dunand [16]. The result by the present analytical model has a good agreement with that of FEM, while the result by HD model is smaller by a factor of 2 as compared to FEM data. This means that the present analytical model is right and could be used to predict the equivalent creep rate of the lattice truss panel structure. The results prove that the vertical truss is dominant for the

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creep deformation. In the lattice truss panel structure, the effects of intersection node and the bonded node should be considered in the creep calculation. As clearly shown in Fig.7, when the intersection node and bonded node have not been considered, the creep rate has been underestimated by half and one order, respectively. In other word, it indicates that the intersection node is useful to decrease the creep rate, proving that the X-type structure developed by Zhang et al. [25] is superior to the pyramid lattice structure in terms of creep strength. It concludes here that the creep deformation of the lattice truss panel structure is mainly induced by the tensile creep deformation of the vertical truss parallel to the tensile load. Figure 8 shows the contours of creep stress and strain at a load of 1.35MPa for 50000h. As shown in Fig.8a, the creep stresses in the bonded node and the intersection node are around 100MPa, while the stresses in the other part of the trusses are about 263MPa. Therefore, the creep strains in the bonded node and the intersection node are very small as shown in Fig.8b, which proves the assumption that the bonded node and the intersection node have little effect on creep deformation is right. In addition, the creep stresses and the strains in the face sheet are very small, which also proves that the assumption that the face sheet has little effect on creep is also right. 5.2 Discussion Eq. (17) is very meaningful because it can determine the equivalent creep rate with different dimensions. The equivalent creep rate is influenced by five key parameters: punching angle, cutting angle, truss length, width and thickness. How they influence the equivalent creep rate is discussed fully here. Figure 9 shows the effects of the punching angle and cutting angle on the equivalent creep rate and relative density. Obviously, the creep rate increases as the punching angle

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and cutting angle increase, because the relative density decreases and the stresses in the trusses increase according to Eq. (4). As the punching angle is 25°, the creep strain rate is 4.01e-9; as the punching angle increases to 60°(2.4 times), the creep strain rate increases by three order (2.04e-6). Similarly, the creep strain rate increases from 4.88e-9 to 2.03e-6 as the cutting angle increases from 30° to 70°. This means that too big punching angle and cutting angle can lead to very big creep strain rate. Figure 10 shows the effects of truss thickness, width and length on equivalent creep strain rate and relative density, as the punching angle and cutting angle are 30° and 25°, respectively. Obviously, the equivalent creep rate is very sensitive to the truss geometrical conditions. As shown in Fig.10a, the creep rate decreases with the increase of truss thickness. This is because when the truss thickness increases, the relative density increases, and the stresses in the truss decrease according to Eq. (4), resulting a decrease of the equivalent creep rate. As the thickness is 0.1mm, the creep strain rate is very big (0.13); it decreases rapidly to 1.64e-4 when the thickness slightly increases to 0.2 mm; as the thickness further increases to 5mm, the creep rate decreases greatly to 4.23e-18. As shown in Fig.10b, the equivalent creep rate decreases with the increase of truss width, due to that the relative density increases and the stress in the truss decreases. As the truss width is 0.1mm, the creep strain rate is also very big (0.002), which means that too small truss width is not proposed to be used; when the truss width increases to 2 mm, the equivalent creep strain rate decreases to 2.71e-11. As shown in Fig.10c, the equivalent creep strain rate increases with the increase of truss length because of the decrease of the relative density. As the truss length increases from 10mm to 80 mm, the equivalent creep strain rate increases from 9.58e-14 to 0.005, indicating that shorter trusses are proposed in order to increase the creep strength.

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Based on the above analysis, it finds that the equivalent creep strain is very sensitive to the dimension changing. As punching angle or cutting angle increases from 25° to 70°, the equivalent creep strain increases by near four orders. Therefore, it suggests that small punching angle and cutting angle should be used during the fabrication in order to get a longer creep life. As shown in Fig.10, a slight change of truss dimension can lead to a big variation of creep strain rate on the order of magnitude. As shown in Fig.10a and b, too small truss thickness and width leads to very big creep strain rate, and decreasing the truss length can increase the creep strain rate. In this paper, the plastic behavior has not been considered. Fan et al. [26] studied the nonlinear mechanical properties of lattice truss materials by increment method. It is a good method to study the plasticity of strain hardening solids as the strut property was nonlinear, which is worthy to be considered for the present model in the future. In addition, the validation of the present model by experimental should be performed in the future as a supplement. Due to that the lattice truss panel structure is porous structure, we are developing a strength design method by homogeneous method [27]. This method treats the porous structure as homogeneous-solid plate, and it is essential to find a relation between the stress-strain state of the lattice truss structure and the stress-strain state of the truss members. Therefore, a study of the effective elastic modulus should also be performed by the equivalent continuum method [28] in the future. 6. Conclusions This paper studies the equivalent creep strain rate of a lattice truss panel structure by analytical analysis and finite element method. The effects of geometrical conditions on creep strain rate have been investigated. Based on this study the following conclusions

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can be drawn. (1) The equivalent creep strain rate developed by the analytical method has a good agreement with the result by FEM. The bonded node and intersection node have a great effect on prediction result and should be considered in the creep analysis. (2) The creep deformation of the lattice truss panel structure is mainly induced by the creep deformation of the vertical component of the trusses parallel to the applied load. (3) As the increase of punching angle and cutting angle, the creep strain rate increases because the relative density decreases and the stresses in the trusses increase. Too big punching angle and cutting angle should be avoided. (4) The dimensions of the trusses have a great effect on creep strain rate. A slight change of truss dimension can lead to a big variation of the equivalent creep strain rate on the order of magnitude. The equivalent creep strain rate decreases as the truss thickness and width increase, while it increases as the truss length increases.

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Acknowledgments The authors gratefully acknowledge the support provided by Taishan Scholar Construction Funding (ts201511018) and Natural Science Foundation for Distinguished Young Scholars (JQ201417) of Shandong Province, National Natural Science Foundation of China (11372359) and Fundamental Research Funds for the Central Universities (14CX05036A and 15CX08006A).

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Table 1 Chemical compositions of Hastelloy C276 (in wt.%). C

Si

Mn

P

S

Fe

Cr

Mo

V

W

Ni

Co

0.02

0.08

1.00

0.03

0.03

5.5

15.5

16

0.35

3.5

55.5

2.5

18

Fig.1 Schematic of a lattice truss panel structure (a) and the unit cell (b)

19

Fig.2 The geometry of the creep test specimen

20

Fig. 3 Uniaxial creep data for Hastelloy C276 steel at 600°C 21

(a)

0.250 270MPa 290MPa 310MPa 330MPa

0.225 0.200

Creep strain

0.175 0.150 0.125 0.100 0.075 0.050 0.025 0.000

0

100 200 300 400 500 600 700 800 900 1000 Time (h)

(b) 0.006 270MPa 290MPa 310MPa 330MPa

Creep strain rate

0.005 0.004 0.003 0.002 0.001 0.000 0

200

400

600

800

1000

Time (h)

Fig.4 Linear fit to creep strain rate vs. σ on a log-log scale for C276 steel

22

log (minimun creep strain rate/h)

-3.4 -3.6

Linear fit

-3.8 -4.0 -4.2 -4.4 -4.6 2.42

2.44

2.46

2.48

2.50

2.52

2.54

Log ( [MPa])

Fig.5 The simplified stress model for the present method 23

Fig.6 Finite element meshing and boundary conditions 24

25

Fig.7 The equivalent creep strain rate by the present analytical model, HD model and FEM -4

-1 Equivaelnt creep strain rate (h )

10

FEM The present analytical model Not considering the brazed node Not considering the intersection node HD model

-5

10

-6

10

-7

10

-8

10

-9

10

0.9

1

1.1

1.2

Applied load (MPa)

26

1.3

1.4

Fig.8 Contours of creep stress (a) and strain (b)

27

Fig. 9 Effect of punching angle (a) and cutting angle (b) on the equivalent creep strain rate

(a)

-5

1.3

10

-1

1.2

-6

10

1.1 1.0

-7

10

0.9 0.8

-8

10

Relative density (%)

Equivalent creep strain rate (h )

Equivalent creep strain rate Relative density

0.7 -9

10

25

30

35

40

45

50

55

60

65

70

0.6 75

Punching angle ()

(b)

-5

1.3

10

-1

1.2

-6

10

1.1 1.0

-7

10

0.9 0.8

-8

10

0.7 -9

10

20

25

30

35

40

45

Cut angle ()

28

50

55

60

0.6 65

Relative density (%)

Equivalent creep strain rate (h )

Equivalent creep strain rate Relative density

Fig.10 Effects of truss thickness (a), width (b) and length (c) on the equivalent creep strain rate (a)

0

14

10

Equivalent creep strain rate Relative density

12

-1

Equivalent creep strain rate (h )

-3

10

10 8 -9

10

6

-12

4

10

Relative density (%)

-6

10

2

-15

10

0 -18

10

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 Truss thickness (mm)

(b)

-2

3.0

10

Equivalent creep strain rate Relative density (%)

-3

-1

2.5

-4

10

-5

2.0

10

-6

10

1.5 -7

10

-8

1.0

10

-9

Relative density (%)

Equivalent creep strain rate (h )

10

10

0.5 -10

10

-11

0.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

10

Truss width (mm)

(c)

-1

4.5

10

Equivalent creep strain rate Relative density

-2

10 -1

4.0 3.5

-4

10

-5

10

3.0

-6

10

2.5

-7

10

-8

10

2.0

-9

10

1.5

-10

10

-11

10

1.0

-12

10

0.5

-13

10

-14

10

0

10

20

30

40

50

60

Truss length (mm)

29

70

80

0.0 90

Relative density (%)

Equivalent creep strain rate (h )

-3

10