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An analytical description of the mechanical hysteresis of entangled materials during loading–unloading in uniaxial compression
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Scripta Materialia 64 (2011) 107–109 www.elsevier.com/locate/scriptamat
An analytical description of the mechanical hysteresis of entangled materials during loading–unloading in uniaxial compression O. Bouaziz,a,b,⇑ J.P. Massea and Y. Bre´chetc a
ArcelorMittal Research, Voie Romaine-BP30320, 57283 Maizie`res-le`s-Metz Cedex, France Centre des Mate´riaux, Ecole des Mines de Paris, CNRS UMR 7633, BP 87, 91003 Evry Cedex, France c SIMAP, Domaine Universitaire de Grenoble BP75, F-38402, Saint Martin d’He`res, France
b
Received 18 May 2010; revised 30 August 2010; accepted 6 September 2010 Available online 15 September 2010
An extension to the non-monotonic behaviour of previous analytical models of the scaling law between the density and the properties of entangled materials is proposed and validated using new experimental data and recent intensive computer simulations. This extension captures in particular the role that the friction between fibres plays in the hysteresis. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Entangled material; Behaviour; Hysteresis; Modelling
Entangled materials are used in thermal insulation, mechanical reinforcement and filtration. Given their low density and discrete architecture, these materials can be regarded as being close to cellular materials. The current state of knowledge about metal foams is well developed, from both experimental and modelling viewpoints [1,2]. Comparatively entangled materials have been much less investigated. Compression experiments have been performed on several types of entangled materials, both natural (e.g. animal wool and human hair) and synthetic (e.g. carbon nanotubes) [3]. Models to understand the mechanical behaviour of non-bonded fibrous material have been proposed [4–6]; essentially based on dimensional analysis, they established a power-law relationship between stress and fibre volume fraction during compression testing. Recently, this scaling law has been confirmed by discrete threedimensional (3-D) simulations [7]. A strong hysteresis is observed both experimentally [3,8–10] and numerically [11] during the loading–unloading transition. Numerically it has been shown that the hysteresis increases with the friction coefficient [11]; however, no analytical development is yet available to capture this effect.
The aim of this work is to extend these previous analytical models in order to capture the very important hysteresis observed during the loading–unloading sequence as a function of the density of the wool and of the friction between fibres. One of the main conclusions of previous analytical models [4,5] developed for the monotonic loading of entangled materials is that the flow stress r and the relative density q follow a power-law relationship, which can be derived assuming that the fibres are deformed in bending during the compression: r ¼ k E qn
ð1Þ
where E is the Young’s modulus of the fibres, k is a constant, and n = 3 for a random 3-D structure and n = 5 for random 2-D planar structure [4,5] This kind of relationship is consistent with some experimental data [8,9]. More generally, this type of relation holds as long as the number of contacts per unit volume scales with the power of the current wool density. More recently in the case of steel wool it has been reported using X-ray tomography measurements that N indeed follows a power-law of the density [12]: N qb
ð2Þ
and that the flow stress can be expressed as [12]:
⇑ Corresponding
author at: ArcelorMittal Research, Voie RomaineBP30320, 57283 Maizie`res-le`s-Metz Cedex, France. Tel.: +33 3 87 70 47 81; fax: +33 3 87 70 47 12; e-mail: [email protected]
r ¼ k Eq4b3 if a 2-D planar structure is assumed.
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.09.007
ð3Þ
O. Bouaziz et al. / Scripta Materialia 64 (2011) 107–109
Using Eqs. (2) and (3) the flow stress can be also expressed as a function of N as: N E r¼ ð4Þ N0 q 3 where N0 is a constant. Concerning the unloading behaviour where the important hysteresis is experimentally observed [3,8], it has been noticed that the slope of flow stress at the beginning of the unloading (a sort of unloading modulus) exhibits the same power-law relationship as Eq. (1) [8] for the tangent loading modulus, and that the hysteresis is independent of the loading rate (i.e. it cannot be attributed to viscous mechanisms). In addition, recent intensive numerical simulations have shown that the hysteresis is mainly due to the instability of the contacts between fibres built up during the loading when the unloading of the sample is performed [11]. In order to describe the unloading curve, it is reasonable to consider the kinetics of accumulation and disappearance of fibreto-fibre contacts. The simplest expression to describe the changing evolution of the number of contacts during the loading and the unloading is: dN ¼ aðqÞ dq bðqÞ jdqj; ð5Þ where a and b are two positive functions of the relative density. dN ¼ ðaðqÞ bðqÞÞ dq
ð6Þ
(9) (see Table 1) in order to mimic the behaviour illustrated in Figure 1. Except for the last step of the unloading, the agreement between experiments and modelling is satisfactory especially for the amplitude of the significant hysteresis. Barbier et al. [11] have also reported that the evolution of the hysteresis is mainly affected by the friction coefficient between fibres [11]. In our approach the hysteresis is mainly controlled by the parameter g, which expresses the proportion of fibre-to-fibre contact which can be destroyed during unloading. Figure 2 shows the evolution of the hysteresis for different values of g assuming the same loading curve. Hence g should be
Table 1. Identified parameters of the model to capture the behaviour of the steel wool. h (mm3)
g
N0 (mm3)
834
0.30
3941
a 0.24 Stress (Mpa)
108
if , dq P 0, and ð7Þ
If dq 6 0. Taking into account Eq. (2) and the same power-law relation between stress and density during the loading and the beginning of the discharge, the functions a and b in Eq. (5) have to be proportional to qb1: aðqÞ1bðqÞ1b qb1
ð8Þ
This amounts to assuming that, at least for the first unloading steps, a constant proportion g = b/a of recently created contacts is destroyed. Finally, Eq. (5) is written as: jdqj dq ð9Þ dN ¼ h b qb1 1 g dq where h and g are two constant material parameters with 0 6 g 6 1. This means that the hysteresis increases with g and that during the loading the number of contacts is given by (integration of Eq. (9)):
experiment modelling
0
0.06 0.08 Relative density
0.1
0.12
0.06 0.08 Relative density
0.1
0.12
0.08 0.06 0.04 0.02 0 0
0.02
0.04
Figure 1. Comparison between modelling and experiments for the loading–unloading of the steel wool after different loading stresses.
0.25
ð10Þ
0.2 Stress (Mpa)
In order to assess the proposed approach, the modelling has been identified on experimental results related to a steel wool [12]. The number of the contacts has been determined experimentally as:
0.04
experiment modelling
0.1
0.225
N ¼ h ð1 gÞ qb
0.02
b 0.12 Stress (Mpa)
dN ¼ ðaðqÞ þ bðqÞÞ dq
0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
g=0.1 g=0.5
0.175 0.15 0.125 0.1 0.075 0.05
N ¼ 1190 q1:5
ð11Þ
which is not far from the theoretical exponent given by the tube model assuming straight fibres [13]. Using the loading–unloading curves, it has been possible to identify the parameters involved in Eqs. (4) and
0.025 0 0
0.02
0.04
0.06 0.08 Relative density
0.1
0.12
Figure 2. Predicted evolution of the loading–unloading curve for different values of g (b = 1.5).
O. Bouaziz et al. / Scripta Materialia 64 (2011) 107–109
an increasing function of the friction coefficient. The larger the friction coefficient, the larger the proportion of contacts made during loading will be destroyed during unloading, which is consistent with the idea of a threshold local stress for fibre locking. Moreover it seems that when unloading proceeds, the system behaves as if g became smaller. This is in agreement with the idea that it is increasingly difficult to destroy the contacts during unloading: “older locks” are more “entangled” than the fresher ones. It therefore seems that the g parameter used in the present phenomenological description can capture the expected influence of friction and contact building history. Experimental (via in situ X-ray tomography) and theoretical (via computer simulation) treatments of this behaviour are planned for the future. [1] M.F. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, H.N.G. Wadley, Metal Foams: A Design Guide, Butterworth–Heinemann, Boston, MA, 2000. [2] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, 2nd ed., Cambridge University Press, Cambridge, 1999.
109
[3] D. Poquillon, B. Viguier, E. Andrieu, J. Mater. Sci. 40 (2005) 5963. [4] C.M. Van Wyk, J. Text. Inst. 37 (1946) 285. [5] S. Toll Polym, Eng. Sci. 38 (1998) 1337. [6] M. Baudequin, G. Ryschenkow, S. Roux, Eur. Phys. J. B 12 (1999) 157. [7] D. Rodney, M. Fivel, R. Dendievel, Phy. Rev. Lett. 95 (2005) 108004. [8] J.P. Masse, L. Salvo, D. Rodney, Y. Bre´chet, O. Bouaziz, Scr. Mater. 54 (2006) 1379. [9] A. Janghorban, D. Poquillon, B. Viguier, E. Andrieu, MATERIAUX 2006, 13–17 November 2006, Dijon. [10] J.M. Haeffelin, F. Bos, P. Castera 2002 Mode´lisation du comportement d’un matelas de fibres cellulosiques au cours de sa consolidation, in Mate´riaux 2002, Tours. [11] C. Barbier, R. Dendievel, D. Rodney, Phys. Rev. E 80 (2009) 16115. [12] J.P. Masse, Y. Bre´chet, L. Salvo, O. Bouaziz, in: Y. Brechet, J.D. Embury, P.R. Onck (Eds.), Architectured Multifunctional Materials, Mater. Res. Soc. Symp. Proc. Vol. 1188, paper 5. [13] S. Raganathan, S.G. Advani, J. Rheol. 35 (1991) 1499.
Scripta Materialia 64 (2011) 107–109 www.elsevier.com/locate/scriptamat
An analytical description of the mechanical hysteresis of entangled materials during loading–unloading in uniaxial compression O. Bouaziz,a,b,⇑ J.P. Massea and Y. Bre´chetc a
ArcelorMittal Research, Voie Romaine-BP30320, 57283 Maizie`res-le`s-Metz Cedex, France Centre des Mate´riaux, Ecole des Mines de Paris, CNRS UMR 7633, BP 87, 91003 Evry Cedex, France c SIMAP, Domaine Universitaire de Grenoble BP75, F-38402, Saint Martin d’He`res, France
b
Received 18 May 2010; revised 30 August 2010; accepted 6 September 2010 Available online 15 September 2010
An extension to the non-monotonic behaviour of previous analytical models of the scaling law between the density and the properties of entangled materials is proposed and validated using new experimental data and recent intensive computer simulations. This extension captures in particular the role that the friction between fibres plays in the hysteresis. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Entangled material; Behaviour; Hysteresis; Modelling
Entangled materials are used in thermal insulation, mechanical reinforcement and filtration. Given their low density and discrete architecture, these materials can be regarded as being close to cellular materials. The current state of knowledge about metal foams is well developed, from both experimental and modelling viewpoints [1,2]. Comparatively entangled materials have been much less investigated. Compression experiments have been performed on several types of entangled materials, both natural (e.g. animal wool and human hair) and synthetic (e.g. carbon nanotubes) [3]. Models to understand the mechanical behaviour of non-bonded fibrous material have been proposed [4–6]; essentially based on dimensional analysis, they established a power-law relationship between stress and fibre volume fraction during compression testing. Recently, this scaling law has been confirmed by discrete threedimensional (3-D) simulations [7]. A strong hysteresis is observed both experimentally [3,8–10] and numerically [11] during the loading–unloading transition. Numerically it has been shown that the hysteresis increases with the friction coefficient [11]; however, no analytical development is yet available to capture this effect.
The aim of this work is to extend these previous analytical models in order to capture the very important hysteresis observed during the loading–unloading sequence as a function of the density of the wool and of the friction between fibres. One of the main conclusions of previous analytical models [4,5] developed for the monotonic loading of entangled materials is that the flow stress r and the relative density q follow a power-law relationship, which can be derived assuming that the fibres are deformed in bending during the compression: r ¼ k E qn
ð1Þ
where E is the Young’s modulus of the fibres, k is a constant, and n = 3 for a random 3-D structure and n = 5 for random 2-D planar structure [4,5] This kind of relationship is consistent with some experimental data [8,9]. More generally, this type of relation holds as long as the number of contacts per unit volume scales with the power of the current wool density. More recently in the case of steel wool it has been reported using X-ray tomography measurements that N indeed follows a power-law of the density [12]: N qb
ð2Þ
and that the flow stress can be expressed as [12]:
⇑ Corresponding
author at: ArcelorMittal Research, Voie RomaineBP30320, 57283 Maizie`res-le`s-Metz Cedex, France. Tel.: +33 3 87 70 47 81; fax: +33 3 87 70 47 12; e-mail: [email protected]
r ¼ k Eq4b3 if a 2-D planar structure is assumed.
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.09.007
ð3Þ
O. Bouaziz et al. / Scripta Materialia 64 (2011) 107–109
Using Eqs. (2) and (3) the flow stress can be also expressed as a function of N as: N E r¼ ð4Þ N0 q 3 where N0 is a constant. Concerning the unloading behaviour where the important hysteresis is experimentally observed [3,8], it has been noticed that the slope of flow stress at the beginning of the unloading (a sort of unloading modulus) exhibits the same power-law relationship as Eq. (1) [8] for the tangent loading modulus, and that the hysteresis is independent of the loading rate (i.e. it cannot be attributed to viscous mechanisms). In addition, recent intensive numerical simulations have shown that the hysteresis is mainly due to the instability of the contacts between fibres built up during the loading when the unloading of the sample is performed [11]. In order to describe the unloading curve, it is reasonable to consider the kinetics of accumulation and disappearance of fibreto-fibre contacts. The simplest expression to describe the changing evolution of the number of contacts during the loading and the unloading is: dN ¼ aðqÞ dq bðqÞ jdqj; ð5Þ where a and b are two positive functions of the relative density. dN ¼ ðaðqÞ bðqÞÞ dq
ð6Þ
(9) (see Table 1) in order to mimic the behaviour illustrated in Figure 1. Except for the last step of the unloading, the agreement between experiments and modelling is satisfactory especially for the amplitude of the significant hysteresis. Barbier et al. [11] have also reported that the evolution of the hysteresis is mainly affected by the friction coefficient between fibres [11]. In our approach the hysteresis is mainly controlled by the parameter g, which expresses the proportion of fibre-to-fibre contact which can be destroyed during unloading. Figure 2 shows the evolution of the hysteresis for different values of g assuming the same loading curve. Hence g should be
Table 1. Identified parameters of the model to capture the behaviour of the steel wool. h (mm3)
g
N0 (mm3)
834
0.30
3941
a 0.24 Stress (Mpa)
108
if , dq P 0, and ð7Þ
If dq 6 0. Taking into account Eq. (2) and the same power-law relation between stress and density during the loading and the beginning of the discharge, the functions a and b in Eq. (5) have to be proportional to qb1: aðqÞ1bðqÞ1b qb1
ð8Þ
This amounts to assuming that, at least for the first unloading steps, a constant proportion g = b/a of recently created contacts is destroyed. Finally, Eq. (5) is written as: jdqj dq ð9Þ dN ¼ h b qb1 1 g dq where h and g are two constant material parameters with 0 6 g 6 1. This means that the hysteresis increases with g and that during the loading the number of contacts is given by (integration of Eq. (9)):
experiment modelling
0
0.06 0.08 Relative density
0.1
0.12
0.06 0.08 Relative density
0.1
0.12
0.08 0.06 0.04 0.02 0 0
0.02
0.04
Figure 1. Comparison between modelling and experiments for the loading–unloading of the steel wool after different loading stresses.
0.25
ð10Þ
0.2 Stress (Mpa)
In order to assess the proposed approach, the modelling has been identified on experimental results related to a steel wool [12]. The number of the contacts has been determined experimentally as:
0.04
experiment modelling
0.1
0.225
N ¼ h ð1 gÞ qb
0.02
b 0.12 Stress (Mpa)
dN ¼ ðaðqÞ þ bðqÞÞ dq
0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
g=0.1 g=0.5
0.175 0.15 0.125 0.1 0.075 0.05
N ¼ 1190 q1:5
ð11Þ
which is not far from the theoretical exponent given by the tube model assuming straight fibres [13]. Using the loading–unloading curves, it has been possible to identify the parameters involved in Eqs. (4) and
0.025 0 0
0.02
0.04
0.06 0.08 Relative density
0.1
0.12
Figure 2. Predicted evolution of the loading–unloading curve for different values of g (b = 1.5).
O. Bouaziz et al. / Scripta Materialia 64 (2011) 107–109
an increasing function of the friction coefficient. The larger the friction coefficient, the larger the proportion of contacts made during loading will be destroyed during unloading, which is consistent with the idea of a threshold local stress for fibre locking. Moreover it seems that when unloading proceeds, the system behaves as if g became smaller. This is in agreement with the idea that it is increasingly difficult to destroy the contacts during unloading: “older locks” are more “entangled” than the fresher ones. It therefore seems that the g parameter used in the present phenomenological description can capture the expected influence of friction and contact building history. Experimental (via in situ X-ray tomography) and theoretical (via computer simulation) treatments of this behaviour are planned for the future. [1] M.F. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, H.N.G. Wadley, Metal Foams: A Design Guide, Butterworth–Heinemann, Boston, MA, 2000. [2] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, 2nd ed., Cambridge University Press, Cambridge, 1999.
109
[3] D. Poquillon, B. Viguier, E. Andrieu, J. Mater. Sci. 40 (2005) 5963. [4] C.M. Van Wyk, J. Text. Inst. 37 (1946) 285. [5] S. Toll Polym, Eng. Sci. 38 (1998) 1337. [6] M. Baudequin, G. Ryschenkow, S. Roux, Eur. Phys. J. B 12 (1999) 157. [7] D. Rodney, M. Fivel, R. Dendievel, Phy. Rev. Lett. 95 (2005) 108004. [8] J.P. Masse, L. Salvo, D. Rodney, Y. Bre´chet, O. Bouaziz, Scr. Mater. 54 (2006) 1379. [9] A. Janghorban, D. Poquillon, B. Viguier, E. Andrieu, MATERIAUX 2006, 13–17 November 2006, Dijon. [10] J.M. Haeffelin, F. Bos, P. Castera 2002 Mode´lisation du comportement d’un matelas de fibres cellulosiques au cours de sa consolidation, in Mate´riaux 2002, Tours. [11] C. Barbier, R. Dendievel, D. Rodney, Phys. Rev. E 80 (2009) 16115. [12] J.P. Masse, Y. Bre´chet, L. Salvo, O. Bouaziz, in: Y. Brechet, J.D. Embury, P.R. Onck (Eds.), Architectured Multifunctional Materials, Mater. Res. Soc. Symp. Proc. Vol. 1188, paper 5. [13] S. Raganathan, S.G. Advani, J. Rheol. 35 (1991) 1499.