The shear properties and deformation mechanisms of porous metal fiber sintered sheets

Mechanics of Materials 70 (2014) 33–40

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The shear properties and deformation mechanisms of porous metal fiber sintered sheets T.F. Zhao a,b, C.Q. Chen a,⇑ a b

Department of Engineering Mechanics, CNMM & AML, Tsinghua University, Beijing 100084, PR China School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, PR China

a r t i c l e

i n f o

Article history: Received 27 August 2013 Received in revised form 21 November 2013 Available online 2 December 2013 Keywords: Metal fiber sintered sheets Cellular materials Shear modulus and strength Deformation mechanism Digital image correlation method

a b s t r a c t Porous metal fiber sintered sheets (MFSSs) are a type of layered transversely isotropic open cell materials with low relative density (i.e., volume fraction of fibers), high specific stiffness and strength, and controllable precision for functional and structural applications. Based on a non-contact optical full field strain measurement system, the in-plane and transverse shear properties of SMFFs with relative densities ranging from 15% to 34% are investigated. For the in-plane shear, the modulus and strength are found to depend linearly upon the relative density. The associated deformation is mainly due to fiber stretching, accompanied by the direction change of metal fibers. When the shear loading is applied in the transverse direction, the deformation of the material is mainly owing to fiber bending, followed by the separation failure of the fiber joints. Measured results show that the transverse shear modulus and strength have quartic and cubic dependence upon the relative density respectively and are much lower than their in-plane counterparts. Simple micromechanics models are proposed for the in-plane and transverse moduli and strengths of MFSSs in shear. The predicted relationships between the shear mechanical properties of MFSSs and their relative density are obtained and are in good agreement with the measured ones. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Metal fiber sintered sheets (MFSSs) are a type of layered open cell porous materials (Ducheyne et al., 1978; Markaki and Clyne, 2003a). Compared to other man-made porous materials such as two dimensional (2D) honeycomb materials or three dimensional (3D) foamed materials (e.g., Ashby et al., 2000; Gibson and Ashby, 1997; Grenestedt and Bassinet, 2000; Hohe and Beckmann, 2012; Meguid et al., 2004; Zhu and Chen, 2011), they are usually produced by the method of sequentially overlapping and sintering of randomly-laid fiber mats to get three-dimensional materials with fibrous-network structures (Xi et al., 2011). As a new filter material with high filter accuracy and large contaminant-holding capacity, MFSSs have been applied in ⇑ Corresponding author. Tel./fax: +86 10 62783488. E-mail address: [email protected] (C.Q. Chen). 0167-6636/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmat.2013.11.007

fluid filtration and dislocation in extreme environments of high temperature and corrosion (Chadwick, 2010), and gas purification and dust precipitation (Peukert, 1998). Besides, they have been widely applied in various fields, such as biomaterials (Jansen et al., 1992), catalytic reaction (Yuranov et al., 2003), heat transfer (Franco et al., 2006; Veyhl et al., 2012), fuel cells (Liu et al., 2004), etc. Because of its excellent acoustical property, they can also be employed in sound absorption and noise reduction (Zhang and Chen, 2009). MFSSs possess the features of low relative density, high specific stiffness and strength, and large surface area like other porous metal materials and are promising for applications as load bearing materials. Ducheyne et al. (1978) studied the elastic constants and yield strength of austenitic stainless steel fiber networks developed surface coatings of implants. Markaki and Clyne (2003a,b) investigated the mechanical properties of metal fiber networks.

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T.F. Zhao, C.Q. Chen / Mechanics of Materials 70 (2014) 33–40

Wan et al. (2012) and Zhou et al. (2009) measured the uniaxial and shear stress–strain responses of copper fiber sheets made by solid-state sintering. Based upon the Xray tomography images and the affine deformation assumption, Tsarouchas and Markaki (2011) formulated a micromechanics model for the elastic properties of MFSSs. Recently, Jin et al. (2013) developed a micromechanics random beam model for the in-plane uniaxial and bi-axial elastoplastic responses of MFSSs. Zhao et al. (2013) developed a phenomenological elastoplastic model for MFSSs. A set of characteristic stress and strain was defined and employed to formulate an elastoplastic constitutive model for the infinitesimal deformation of MFSSs subject to monotonic proportional loading. An important type of applications of MFSSs is as the core of sandwich structures. In such applications, the mechanical properties and failure mechanism of MFSSs under shear are usually of major concern. To facilitate their structural applications, a thorough understanding of the shear properties of MFSSs is essential. To this end, the shear stiffness and strength and their associated deformation mechanisms of MFSSs under shear are investigated by the method of experiment and micromechanics models are developed for the measured relative density dependent moduli and strengths. This paper is organized as follows. In Section 2, the microstructural feature of MFSSs and the employed shear test and strain measurement methods are introduced. The experimental results on the in-plane and transverse shear moduli and strengths are summarized in Section 3. Based upon the deformation mechanisms, analytical micromechanics models of MFSSs are proposed in Section 4. A few concluding remarks are given in Section 5.

2. Materials and test methods 2.1. Porous metal fiber sintered sheets Commercially available 316L stainless steel fibers of 12 lm in diameter are used to produce the MFSSs considered in this study. The fibers produced by the bundle drawing method were ordered from Xi’an Fiat Filter Company Ltd, China. The chemical composition of the fibers is given in Table 1. The as-received long fibers are cut into short fibers with length in the range between 10 mm and 20 mm and then overlapped into fiber layers by the air-laid web-forming technology. The thickness of a single fiber layer is approximately 0.1 mm, and its relative density is about 0.01. Within each layer, the fibers are randomly distributed. A number of single fiber layers are then sequentially overlapped and sintered in a vacuum furnace to get a metal fiber sheet. The solid state sintering technol-

ogy with proper environmental temperature is employed to ensure the bonding quality of the fiber joints and to ensure the final product to have desirable shape, size and relative density. The MFSSs produced by the aforementioned method consist of numerous layers of fibers and are transversely isotropic. Fig. 1 shows a MFSS with relative density of 23% and, schematically, its layer-by-layer nature. A coordinate system is introduced, with the XY plane being the isotropic in-plane of the layers and the Z direction denoting the transverse direction (i.e., the thickness direction of the layers). Relative density is employed to quantify the mechanical property of porous materials. For MFSSs, it can be expressed as:

q ¼

 

q d ¼C l qs

ð1Þ

where q and qs are the mass density of MFSSs and their constituent metal fibers, respectively, d is the diameter of the constituent metal fibers, l is the average pore size, and C is a constant to be determined. 2.2. Simple shear test and surface strain measurement The in-plane and transverse shear properties of MFSSs are measured by simple shear test, in accordance with ASTM C271M standard (ASTM, 2007). To explore the effects of relative density upon the material properties,  = 15%, MFSSs with 4 different relative densities (i.e., q 23%, 28%, 34%) are tested. Two typical in-plane and transverse specimens are shown in Fig. 2. For in-plane shear tests, specimens are of dimensions 10  120  10 mm (W  L  H), where W, L and H are along the x, y and z directions shown in Fig. 1, respectively, and shear load is imposed on the plane normal to the x direction. For the transverse shear tests, specimens are of dimensions 10  120  20 mm (W  L  H), with shear load applied on the plane normal to the z direction. Note that the specimen size is chosen to be much larger than the average pore size of MFSSs (about 50 lm) in order to minimize the size dependency (Tekog lu et al., 2011), if any, of the experimental results. All specimens are cut from bulk materials by electro-discharge marching (EDM). All tests are conducted at room temperature using a servo-hydraulic material testing machine (Model: MTS858) and the associated MTS TestStar control software. Displacement controlled loading is adopted, with a loading rate of 103/s in strain to mimic quasi-static loading. Fig. 3(a) shows the simple shear test setup, where the specimen is glued to the grippers with epoxy. Nominal global average shear stress is defined as the load divided by the cross section area.

Table 1 Chemical compositions of stainless steel 316L (wt.%). C

Cr

Mn

Mo

Ni

P

S

Si

60.03

16.0–18.0

62.0

2.0–3.0

12.0–15.0

60.035

60.030

61.0

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T.F. Zhao, C.Q. Chen / Mechanics of Materials 70 (2014) 33–40

(a)

(b)

x

x

z

y y

z

5mm

A layer with scattered short fibers Fig. 1. (a) A bulk metal fiber sintered sheet (MFSS); (b) schematic of its layer-by-layer feature. The xy plane refers to the isotropic in-plane and the z direction denotes the transverse (out-of-plane) direction.

120

τ x

z

τ

x

(a)

10

y

10

y

τ

120

y

In-plane shear loading

20

x

(b)

z

τ

Transversely shear loading

y

10

z

10

Fig. 2. Simple shear test specimens for metal fiber sintered sheets: (a) in-plane shearing and (b) transverse shearing.

(b)

Gripper

Specimen

(a)

Fig. 3. (a) Simple shear test setup, and (b) full field surface strain measurement with the digital image correlation method.

Due to the heterogonous microstructures of MFSSs, their surface strain is usually not uniform and, consequently, cannot be accurately measured by conventional

strain gauges. In ASTM C273M standard (ASTM, 2007), linear variable differential transducers (LVDT) are recommended to measure the relative displacement of the two grippers and the nominal global shear strain is defined as the ratio between the relative displacement and the specimen thickness. In this method, the deformation of the epoxy is assumed to be negligible. During the tests of MFSSs in this study, it is found that there are usually highly localized deformation zones between the specimen and test grippers. As a result, LVDT measured global shear strain may be significantly overestimated, if deformation the localized zone is not excluded. Here, a non-contact optical 3D deformation measurement system developed by GOM, Germany (Model: ARAMIS 4M) is adopted to measure the deformation. Before test, black and white sparks are sprayed on the front surface of the specimens monitored by the GOM system. During test, the load history is synchronously transmitted from MTS-858 to ARAMIS. A recording rate of 2 fps is adopted, as a result of balance between the accuracy and efficiency of measurement. With the help of the sparks, the recorded images are post-processed by the

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T.F. Zhao, C.Q. Chen / Mechanics of Materials 70 (2014) 33–40

digital image correlation method (DIC) to get the displacement and strain fields at each recorded loading step. The global shear strain is obtained by averaging the local strain over a predefined gauge area. Fig. 3(b) shows a typical shear strain field calculated by the DIC method, where a gauge area as enclosed by dashed lines is defined and used to calculate the average/global shear strain. It should be noted that, when define the gauge area, care must be taken to exclude the highly localized deformation zones between the test grippers and the specimen. The synchronized load and calculated global shear strain are then used to produce the macroscopic shear stress–strain curves. To get the shear modulus, each specimen is loaded and unloaded several times. The unloading modulus in the elastic region is taken as the shear modulus. As to the shear strength, a conventional offset strain definition of 0.2% is employed (Ashby et al., 2000; Wicklein and Thoma, 2005). The in-plane shear stress–strain curve of a MFSSs of relative density being 28% is shown in Fig. 4 as broken line. Also included is the in-plane uniaxial compression stress–strain curve of the same material from Zhao (2013), denoted by solid line. Engineering nominal stress and strain are employed in this paper. The inset in Fig. 4 shows the loading–unloading stress–strain curves in the initial deformations stage, with the in-plane shear and Young’s moduli associated with the corresponding unloading curves marked. Recall that MFSSs are isotropic in the XY-plane. The corresponding in-plane shear modulus GI and Young’s modulus E are related to each other by

GI ¼

E 2ð1 þ mÞ

ð2Þ

3. Experimental results 3.1. Shear moduli and strengths Fig. 5 shows the measured in-plane and transverse shear stress–strain curves of a MFSS with relative density being 23%. It is clear that three stages exist in the in-plane and transverse shear responses: an initial elastic stage, a plastic hardening stage, and a rapid softening stage due to material failure. The loading and unloading responses in the small strain regime are also included in the inset of Fig. 5. It can be seen that the in-plane unloading modulus and yield strength are about 4.1 GPa and 7.78 MPa, respectively and are much greater than the corresponding transverse ones (i.e., 0.41 GPa and 2.20 MPa). However, the in-plane failure strain (i.e., the strain associated with the peak strength) is less than the transverse failure stain (as 10% oppose to 16%). Systematical in-plane and transverse shear tests have also been conducted for MFSSs with different relative densities. For each relative density, at least four specimens are tested. The measured in-plane and transverse shear modulus and shear strength of MFSSs are respectively summarized in Fig. 6(a) and (b) as a function of relative density, where experimental results are denoted by symbols. Since the results in Fig. 6 are given in log–log scale, the slope corresponds to the index of power dependence (i.e., a slop of n means a power dependence of n). Curve fitting of the results in Fig. 6 gives

 GI ¼ 0:087Es q  sI ¼ 0:102rys q for the in-plane modulus (GI ) and strength (sI ) and

where m is the XY-plane Poisson’s ratio and subscript ‘‘I’’ refers to in-plane. Eq. (2) can be employed to check the consistency of the measured uniaxial and shear results. It is seen from Fig. 4 that the first unloading Young’s modulus is about 12.1 GPa and m is about 0.2 (Zhao, 2013). Accordingly, Eq. (2) gives GI = 5.04 GPa, which is very close to the experimental results of 5.05, 5.09, and 5.06 GPa. Such a good agreement provides additional support of the reliability of the designed test setup and the employed non-contact optical 3D deformation measurement method.

4 GT ¼ 0:078Es q

for the transverse modulus (GT ) and strength (sT ). In Eqs. (3) and (4), subscript ‘‘T’’ refers to the transverse properties, and Es and rys are the Young’s modulus and yield strength of the fibers. As can be seen from Fig. 6, the inplane modulus and strength are very different from their transverse counterparts. Both the in-plane modulus (GI ) and strength (sI ) depend linearly upon the relative density 20 In-plane Transverse

Shear stress, τ (MPa)

15

100 12.1 GPa

Stress, (MPa)

ð4Þ

sT ¼ 0:57rys q 3

120

10

80 60

5.06 GPa 5.05 GPa 5.09 GPa

0.1

40

0.2

20 0

ð3Þ

0.3

12.0 GPa5

0.4

0.5

Uniaxial compression (Zhao, 2013) Simple shear

0

10

20

30

40

50

Strain, (%) Fig. 4. Measured in-plane uniaxial and simple shear stress–strain curves of a MFSS with relative density of 28%. The inset shows the corresponding stress–strain curves in the small deformation region.

9 6

4.25 GPa 4.31 GPa 4.11 GPa 3 0.40GPa 0.41GPa 0.39GPa 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6

15

10

5

0 0

5

10

15

20

25

30

Shear strain, γ (%) Fig. 5. Measured in-plane and transverse shear stress–strain responses of a MFSS with relative density of 23%. The inset shows the corresponding stress–strain curves in the small deformation region.

Normalized transversely shear properties

Normalized In-plane shear properties

T.F. Zhao, C.Q. Chen / Mechanics of Materials 70 (2014) 33–40

0.1

(a)

Shear modulus Shear strength 1

τ*I /σs= C2ρ

G*I /Es= C1ρ 0.01 0.1

Relative density, ρ

1

1

(b)

Shear modulus Shear strength

0.1

τ*T /σyz=C4ρ3

0.01

1E-3

3

4

G*T /Es=C3ρ4

1E-4 0.1

Relative density, ρ

1

Fig. 6. Normalized modulus and yield strength of MFSSs as a function of relative density: (a) in-plane shear modulus and shear strength; (b) transverse shear modulus and shear strength.

whilst the transverse modulus (GT ) and strength (sT ) show higher order dependences on the relative density, with  4 and sT / q  3 . Since the relative density is usually GT / q much less than 1, the transverse properties are much less than the in-plane ones. In addition to the moduli and strengths, it is also important to uncover the deformation and failure modes in order to properly understand the mechanical behavior of MFSSs under shear loading. 3.2. In-plane local shear deformation and failure mode Fig. 7 shows the local in-plane shear strain contours at different global shear strain levels for a MFSS with relative

τ

τ

γ xy = 0% 0.2% 1%

5%

15%

Fig. 7. Measured local shear strain contours of a MFSS subject to in-plane shearing. The contours are shown for global in-plane shear strains of 0, 0.2%, 1%, 5% and 15%.The relative density of the MFSS is 23%.

37

density 23%. Results are shown for the global in-plane shear stains of 0, 0.2%, 1%, 5% and 15%. The strain fields without deformation are included for reference. It can be seen that, when subject to in-plane shearing, the local in-plane shear strain cxy is uniform for small deformations (i.e., 61%). With the load increasing, localized deformation zones start to appear. However, the overall deformation is still more or less uniform, as is evident by the uniformly distributed localized zones along the vertical direction. A zoomed-in photograph of the indicated area in Fig. 7 at global in-plane shear strain of 15% is given Fig. 8(a) can be seen from the SEM image in Fig. 8(b). Recall that Fig. 8 shows linear dependence of the in-plane shear modulus and strength on the relative density. A linear dependence of the modulus and strength of a cellular materials upon its relative density indicates that the dominant deformation mechanism of cell wall/edges stretching (Gibson and Ashby, 1997). It can thus be inferred from Fig. 6 that fiber stretching is the dominant deformation mechanism of MFSSs upon in-plane shearing. Accordingly, a constituent fiber in the MFSS specimen subject to in-plane shearing may either undergo tension or compression, depending on its orientation relative to the shear direction. This is illustrated in Fig. 8(b): fibers along about 45° measured anti-clockwise from the x direction deform mainly by axial stretching whist those along 45° clockwise from the x direction are subject to axial compression. For fibers undergoing compression, they may be buckled in an outof-plane manner and render the surface corrugated, giving rise to localization zones under in-plane shearing (see, Figs. 7 and 8(a)).

3.3. Transverse local shear deformation and failure mode The local transverse shear strain contours of a MFSS with relative density 23% under transverse shearing are plotted in Fig. 9. Similar to Fig. 7, results are also shown for several different global transverse shear strains (i.e., 0, 0.2%, 1%, 5% and 15%). It can be seen from Fig. 9 that the local transverse shear strain cyz starts to be localized even when the global shear strain is as small as 0.2%, much earlier than the in-plane strain localization shown in Fig. 7. The deformation localization zone becomes more pronounced with the load increasing (see, the strips close to the interface between the specimen and the left gripper). A zoomed-in photograph of the marked strain localization zone in Fig. 9 is given in Fig. 10, with the global transverse shear strain being 15%. It is clear that the transverse strain localization is mainly due to layer delamination. This implies that, under transverse shearing, the layer-by layer MFSSs most likely fail by interfacial failure between layers.

4. Micromechanics models In this section, two micromechanics models will be employed to quantitatively explain the measured relative density dependent shear properties (Eqs. (3) and (4)).

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T.F. Zhao, C.Q. Chen / Mechanics of Materials 70 (2014) 33–40

Compression

y

Tension +45o

τ

0o

τ

x y

(a)

−45o

x

(b)

Fig. 8. (a) A photograph of the deformed area marked in Fig. 7; (b) an SEM image showing the deformed fibers within highlighted region in (a).

Although the relationship between the shear modulus and relative density expression is not given in Jin et al. (2013), it can be inferred as follows. Note that in 2D isotropic materials, the shear modulus G is related to Young’s modulus E and bulk modulus j by

G ¼

τ

τ

4K  E 4K   E

By substituting Eq. (3) into (4), the macroscopic inplane shear modulus can be obtained as

 G ¼ 0:1Es q

γ yz = 0%

0.2% 1%

5%

15%

Fig. 9. Measured local shear strain contours of a MFSS subject to transverse shearing. The contours are shown for global transverse shear strains of 0, 0.2%, 1%, 5% and 15%. The relative density of the MFSS is 23%.

τ

1mm

τ

y x

2mm

ð7Þ

It can be seen that the micromechanics prediction (7) is in good agreement with the measured in-plane shear modulus of MFSSs (3). The slight over-estimation of Eq. (7) is believed to be due to the fact that various defects in the specimens are not accounted for in the micromechanics model. As to the in-plane shear strength, a direct comparison of the random beam model prediction and the experimental results is not possible because Zhao et al. (2013) only calculated the uniaxial and equal-biaxial yield strengths. Nevertheless, they reported that both the uniaxial and equal-biaxial tensile yield strengths have a linear dependence upon the relative density. Thus, it is highly possible that the shear strength predicted by the micromechanics model also depend linearly on the relative density, as shown by the experimental results. 4.2. Out-of-plane shear

Fig. 10. A photograph of the deformed area marked in Fig. 9, showing that the localization zone is associated with layer delamination.

4.1. In-plane shear Jin et al. (2013) developed a micromechanics two dimensional random beam model to simulate the isotopic in-plane properties of MFSSs under biaxial loading. A linear dependence of the in-plane Young’s modulus E and bulk modulus j on the relative density is predicted to be

 E ¼ 0:259Es q j ¼ 0:183Es q

ð6Þ

ð5Þ

By noting the layer-by-layer feature of MFSSs, a simplified micromechanics model is proposed to interpret the higher order dependence of the measured transverse shear properties upon the relative density. In the model (see Fig. 11), three layers of fibers are shown and marked by A, B, and C, respectively. Upon transverse shear, the load F is transmitted from one layer to another through the sintered joints, giving rise to a bending deflection d to accommodate the shear deformation. In accordance with the standard beam theory (Timoshenko and Goodier, 1970), d 3 4 is proportion to Fl =ðEs JÞ, where J ¼ pd =64 is the moment of inertia of the circular fibers. Note that the transverse

39

T.F. Zhao, C.Q. Chen / Mechanics of Materials 70 (2014) 33–40

F x

Layer B

F

Layer A

δ Layer A

y

Layer B z

Layer C

F

Simplified micromechanics

Transverse shear Loading

model Fig. 11. A simplified micromechanics model showing (a) the layered feature of MFSSs and (b) the deformation mode upon transverse shearing.

shear stress s and strain c are related to the shear load and 2 deflection by s / F=l and c / d=l, respectively. Therefore, the macroscopic transverse shear modulus GT can be obtained as

GT ¼

 4

s d / Es l c

ð8Þ

With Eqs. (1) and (8) can be expressed as

GT

4 / Es q

ð9Þ

Using the model in Fig. 11, the transverse shear strength can also be estimated. Assuming the collapse of the model by the forming of the four plastic hinges marked in Fig. 11, the collapse load can be obtained as

F ¼ 8M p =l

ð10Þ 3

where M p ¼ rys d =6 is the plastic bending moment of cir2 cular fibers. Substituting s / F=l and Eq. (1) into (10), the transverse shear strength can be shown to have the form of

sT / rys

 3 d 3 / rys q l

ð11Þ

sured by a non-contact optical full field strain measurement method. Experimental results show that, under in-plane shear, the deformation of MFSSs is mainly due to fiber axial stretching/compression, giving rise to linear dependence of the shear modulus and strength upon the relative density. Upon further shearing, the fibers undertaking axial compression tend to buckle in an out-of-plane manner. Consequently, the specimen surface becomes corrugated. For the transverse shear, the deformation of MFSSs is different from the in-plane counterpart and is dominated by fiber bending and the failure is owing to the separation of the fiber joints and delamination of fiber layers. Experimental results show that the transverse shear modulus and shear strength have quartic and cubic dependence upon the relative density respectively and are much less than the in-plane ones. A micromechanics model is developed for transverse shear and faithfully capture the higher order dependence of the transverse shear modulus and strength of MFSSs. Acknowledgements

As one can see, the predicted transverse shear modulus and strength in Eqs. (9) and (11) by the simple model shown in Fig. 11 are consistent with experimental results (4). In summary, the in-plane and transverse shear moduli and strengths of MFSSs can be expressed as

The authors are grateful for the financial support of this work by the National Natural Science Foundation of China (No. 11072127), the National Basic Research Program of China (No. 2011CB610305), and the Ph.D. Program of Ministry of Education of China (20110002110069).

 GI ¼ C 1 Es q  sI ¼ C 2 rys q

References

GT  T

4

¼ C 3 Es q s ¼ C 4 rys q 3

ð12Þ

where C 1  C 4 are material constants and are in general dependent on the bonding quality of the fiber joints. 5. Conclusions Porous metal fiber sintered sheets are transversely isotropic. Their in-plane and transverse shear properties are measured by simple shear tests, with the shear strain mea-

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