Role of inter-fibre bonds and their influence on sheet scale behaviour of paper fibre networks

2025-04-26 0 0 7.25MB 26 页 10玖币

侵权投诉

Role of inter-ﬁbre bonds and their inﬂuence on sheet scale

behaviour of paper ﬁbre networks?

P. Samantraya,b,c, R.H.J. Peerlingsa, T.J. Massartb, O. Rokoˇsa,∗

, M.G.D. Geersa

aMechanics of Materials, Department of Mechanical Engineering, Eindhoven University of Technology,

P.O. Box 513, 5600 MB Eindhoven, The Netherlands

bBuilding, Architecture and Town Planning, Universit´e Libre de Bruxelles, 50 Avenue F.D. Roosevelt, CP

194/2, B–1050 Brussels, Belgium

cMaterials innovation institute (M2i), P.O. Box 5008, 2600 GA Delft, The Netherlands

Abstract

In ﬁbrous paper materials, an exposure to a variation in moisture content causes changes in

the geometrical and mechanical properties. Such changes are strongly aﬀected by the inter-

ﬁbre bonds, which are responsible for the transfer of the hygro-mechanical response from

one ﬁbre to its neighbours in the network, resulting in sheet-scale deformation. Most models

developed in literature assume perfect bonding between ﬁbres. In the 3D reality, there is some

ﬂexibility in the bond region, even for the perfectly bonded ﬁbres, because of the possibility

of deformation gradients through the ﬁbre thickness. In earlier 2D idealizations, perfectly

bonded ﬁbres were assumed, implying full kinematic constraint through the entire thickness

of the sheet. The purpose of the present study is to assess the eﬀect of this assumption.

Using a homogenization approach, a random network of ﬁbres is generated with diﬀerent

coverages and modelled using ﬁnite elements. In order to understand the role of bonding

between ﬁbres on the hygro-expansive behaviour of a network, a bond model is developed.

In this model, the ﬁbres are modelled using 2D regular bulk ﬁnite elements and the bonds

are represented by interfacial elements of ﬁnite stiﬀness, which are introduced between each

pair of ﬁbres bonded in the network. These embedded interfacial elements form a connection

between two respective ﬁbres, allowing relative displacements between their mid-planes. The

hygro-elastic response of networks obtained with this bond model is investigated by varying

the bond stiﬀness and the network coverage under the application of mechanical loading and

changes in moisture content. Furthermore, the bond model is used to analyse the inﬂuence

of inter-ﬁbre bonds on the anisotropic response of the paper ﬁbre network.

Keywords: Fibrous network, hygro-expansion, inter-ﬁbre bond, interfacial elements,

coverage, homogenization

?The post-print version of this article is published in Int. J. Solids Struct.,10.1016/j.ijsolstr.2022.111990

∗Corresponding author.

Email addresses: priyam.samantray@gmail.com (P. Samantray), R.H.J.Peerlings@tue.nl

(R.H.J. Peerlings), Thierry.J.Massart@ulb.be (T.J. Massart), O.Rokos@tue.nl (O. Rokoˇs),

M.G.D.Geers@tue.nl (M.G.D. Geers)

Preprint submitted to Int. J. Solids Struct. October 13, 2022

arXiv:2210.05736v1 [cond-mat.soft] 11 Oct 2022

1. Introduction

Fibrous materials like paper consist of natural ﬁbres that are bonded in overlapping

regions, as shown in Fig. 1a. In paper in particular, individual ﬁbres are hollow layered

structures made of wood pulp, which consist of a primary cell wall on the outer side and a

secondary cell wall S. The secondary cell wall further consists of an outer layer S1, middle

layer S2, and an inner layer S3, as indicated in Fig. 2a. Individual cell walls are furthermore

composed of three types of polymers (in particular cellulose, hemicellulose, and lignin),

having the ultrastructural composition shown in Fig. 2b. The distribution of the polymers

varies across the primary and secondary walls of the ﬁbre, where the secondary wall layer

contains higher amounts of cellulose and hemicellulose as compared to the primary layer,

whereas lignin content is nearly the same among all the secondary layers, cf. (Neagu and

Gamstedt,2007).

During the manufacturing of paper, the ﬁbres are preferentially oriented in the machine

direction, which entails anisotropy in the mechanical behaviour of the paper network (Lars-

son and W˚agberg,2008). The principal directions of the paper material are denoted as the

machine direction (MD) and cross direction (CD), as shown in the random ﬁbre network

in Fig. 1b (Bosco et al.,2015b). Here, individual ﬁbres are interconnected to each other

in overlapping regions, called bonds, complemented with freely standing parts in between

bonds. In a paper ﬁbre network, the ﬁbres in inter-ﬁbre bonds are connected together by

hydrogen bonding and van der Waal’s forces (Henriksson et al.,2008). The anisotropic be-

haviour, originating at the scale of individual ﬁbres, is transferred from one region in the

network to another through these inter-ﬁbre bonds. The coeﬃcient of transverse hygro-

expansion of individual ﬁbres is nearly 20 times the corresponding value in the longitudinal

direction (Niskanen,1998). This is because at the ﬁbril scale, the polymers (i.e., hemi-

cellulose, cellulose, and lignin) exhibit hygro-expansion that varies depending on multiple

factors, which in combination with the complex ﬁbre microstructure (recall Fig. 2) results

in high anisotropy. For example, the hygro-expansion of the hemicellulose has been found

to be highest among all three polymers due to high solubility of its side groups (Hansen

and Bj¨okman,1998), whereas cellulose (in crystalline form) has negligible hygro-expansion,

as shown by several studies (Lindner,2017;Marklund and Varna,2009;Wang et al.,2014;

Cave,1978). The water uptake by cellulose decreases with an increase in crystallinity below

75% relative humidity (Mihranyan et al.,1978), although the wood ﬁbre has 60–70% cellu-

lose chains in crystalline form and only the rest is in water-absorbing amorphous form and

hence contributing to hygro-expansion (Alince,2002). Apart from crystallinity, the water

absorption in cellulose depends on the temperature at which the ﬁbres are processed, on the

solvents around cellulose, the surrounding cellulose molecules, and the processing of the ﬁ-

bres (Fahl`en and Salm`en,2003;Uetani and Yano,2011;Berthold et al.,1994). For lignin, the

moisture content varies depending on the type, i.e., 3–5% for dioxane lignin and 10–15% for

Klason and periodate lignin. Other parameters aﬀecting hygro-expansivity of lignin include

softening temperature (Eriksson et al.,1991) and microﬁbril angle (Wang et al.,2014).

(a) a micrograph of paper ﬁbres (b) a random paper ﬁbre network

Figure 1: (a) A micrograph of paper showing random orientation of individual ﬁbres, and (b) an artiﬁcially

generated network of random paper ﬁbres subjected to hygroscopic strain. The preferential directions are

the Machine Direction (MD), oriented along ~ex, and the Cross Direction (CD), oriented along ~ey.

(a) wall layer structure

of a wood ﬁbre

(b) ultrastructural organization of cellulose

within the wood cell wall

Figure 2: (a) A schematic diagram of diﬀerent cell wall layers within a wood ﬁbre, adapted from (Br¨andstr¨om,

2002;Gilani,2006). (b) Organisation of cellulose polymers at the ultrastructural level within a wood cell

wall, adapted from (Neagu and Gamstedt,2007).

The degree of bonding between ﬁbres and the number of ﬁbres in the network have a

crucial inﬂuence on the hygro-mechanical response of the network. Earlier studies demon-

strated that the eﬀective stiﬀness of the network is mainly dependent on the density of

the network, ﬁbre orientations, ﬁbre properties, and of the inter-ﬁbre connectivity resulting

from the bonds (Mao et al.,2017;Karako¸c et al.,2017;Heyden,2000;Bergstr¨om,2018;

Cox,1952). It was furthermore revealed that when subjected to moisture changes, the ef-

fective hygro-expansivity of the paper network is governed mainly by the bonded regions

of the network (Bosco et al.,2015a;Samantray et al.,2021b;Uesaka,1994;Uesaka and

Qi,1994;Vainio and Paulapuro,2007;Niskanen,1998;Larsson and W˚agberg,2008;Erkkila

et al.,2015;Nordman,1958;Salm´en et al.,1987;Schulgasser and Page,1988). In another

study (Thorpe et al.,1976), the properties of the bonded ﬁbres were computed from load-

displacement measurements. The hygro-mechanical properties of the paper, which are highly

dependent on the inter-ﬁbre connectivity, thus hold signiﬁcance in understanding the dimen-

sional stability of paper products like packaging materials, printed papers, corrugated boxes,

tissue papers, or paper boards. Instabilities appearing as curls or waviness at the paper

sheet scale are moreover signiﬁcantly dependent on bond stiﬀness. In this study, the main

objective is therefore to examine the eﬀect of the bond stiﬀness on the sheet-scale response

of paper network, and to critically review the assumption often made in earlier works (i.e.,

full kinematic tying in the inter-ﬁbre bonds).

Several earlier works consider bonds between the ﬁbres to be rigid, for both 2D and 3D

conﬁgurations (Bronkhorst,2003;Mao et al.,2017;Bosco et al.,2017;Ostoja-Starzewski

and Stahl,2000;Lee and Jasiuk,2013;Stahl and Cramer,1998). A limitation of such a

bond description is its inability to describe the inﬂuence of inter-ﬁbre connectivity on the

sheet scale deformations, as well as the extent to which certain ﬁbre network properties (e.g.,

bond area or bonding stiﬀness) aﬀect the macroscale response that can be modiﬁed in the

manufacturing process. Some other works, on the other hand, described the bonds as ﬂexible

and not rigid, using for instance springs (Heyden,2000). The ﬁbres were partially bonded in

regions of overlap (Koh and Oyen,2012), and were modelled based on a non-linear contact

law with bond failure (Kulachenko and Uesaka,2012), or as two-node line elements (Liu

et al.,2011). Studies by Ramasubramanian and Perkins (1988) and Perkins (1990) consider

the shear stresses and inelastic behaviour to be present in the bonds, hence assuming their

deformability. However, most of the literature that incorporates the deformation in bonds

relates only to the response of networks under mechanical loading. Analysis and modelling

of the hygro-mechanical behaviour and the anisotropic response of the network due to non-

rigid bonds are not available, which especially in 3D entail costly computations that prevent

assessing parametric variations of the bond eﬀect. Even though for hygro-mechanics only

perfect bonding is relevant, a perfect bonding in 3D reality is not equivalent to full compat-

ibility in 2D. There is therefore a need to understand the hygro-mechanical response of the

networks due to varying bonding properties (stiﬀness) connecting the ﬁbres, along with its

inﬂuence on the anisotropy of the network using a computationally eﬃcient 2D modelling

framework.

In this contribution, such issues are addressed by describing the inter-ﬁbre connectivity

through an appropriate 2D bond model, starting with a periodic unit cell of randomly

generated rectangular strips (Samantray et al.,2020) that represent ribbon-shaped paper

ﬁbres. Each ﬁbre in the unit cell network is discretized with standard triangular ﬁnite

elements. At the regions of inter-ﬁbre bonds, additional triangular elastic interfacial elements

with a suitable stiﬀness, connecting two successive ﬁbres (in the thickness direction), are

inserted in the model. These ﬁnite stiﬀness elements render the bonds to be no longer

rigid and the connected ﬁbres can therefore have relative displacements through induced

elasticity. Note that, although possible, no dissipation within the interfacial layer (plasticity

nor damage) is assumed within the inserted layer, and hence modelling of degradation of the

bond is not adopted in this work. With this model, the macroscopic behaviour of the paper

network with diﬀerent values of the bond stiﬀness can be computed when subjected to an

external mechanical load or a uniform moisture variation. This provides clear insights in the

sheet level response using a 2D network model with moderate computational eﬀorts only,

while incorporating key elements of the full 3D reality. The inﬂuence of the connectivity

between ﬁbres at the bonds on the anisotropy of the paper network at the sheet scale can

furthermore be assessed.

The contents of this manuscript is organized as follows. In Section 2, the hygro-mechanical

model for the ﬁbre and the network is described. This includes the generation of random

networks, the bond model, and the adopted numerical discretization together with consid-

ered stacking of ﬁbres in the network. Obtained results, illustrating the inﬂuence of the bond

stiﬀness at the sheet level response of the network along with its anisotropy, are detailed in

Section 3. The conclusions are ﬁnally drawn in Section 4.

Throughout this contribution, the following notation is used for operations on Cartesian

tensors and for tensor products. Scalars, vectors, and tensors are denoted as a,~a, and

A. For tensor and vector operations, the following equivalent notations are used, with

Einstein’s summation convention on repeating indices: A:B=Aij Bji, where i=x, y, z,

for the global reference system, and i=l, t, z, for the ﬁbre local reference system. The Voigt

notation employed to represent tensors and tensor operations in a matrix format uses a

and Ato denote a column matrix and a matrix of scalars; matrix multiplication then reads

A b =Aij bj.

2. Hygro-mechanical constitutive model and numerical discretization

2.1. Fibre model

Consider a 2D conﬁguration with plane stress assumption in the z-direction (with z

normal to the paper sheet). This assumption is adopted because a fully-complex 3D model

may easily become computationally too expensive, especially for parametric studies. The

3D reality is hence simpliﬁed by considering only the mid-planes of the ﬁbres, see Fig. 3b

where a pair of bonded ﬁbres is depicted. Such assumption is adopted because the main

objective of this work is to seek information about the role of stiﬀness in the bond on the

hygro-expansion and anisotropy of the paper networks, with aﬀordable computational costs.

Since computational eﬀorts are limiting in full 3D simulations, the herein proposed model

is formulated in two dimensions while accounting for the key mechanisms of a full three-

dimensional reality through bond elasticity. Earlier-mentioned instabilities, such as curls or

waviness, could be incorporated within the proposed model by introducing shell elements

instead of plane stress elements. Under such assumptions, the hygroscopic strain tensor hεf

and the stress tensor in a ﬁbre fare given by the constitutive relations

hεf=βf∆χ, (1)

σf=4Cf: (εf−hεf),(2)

where σf,4Cf,εf, and βfare the stress tensor, the elastic stiﬀness tensor, the strain tensor,

and the hygroscopic expansion tensor of the ﬁbre. The variable ∆χ=χ−χ0then denotes

change in the moisture content relative to the reference value χ0, where the moisture content

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Roleofinter-brebondsandtheirinuenceonsheetscalebehaviourofpaperbrenetworks?P.Samantraya,b,c,R.H.J.Peerlingsa,T.J.Massartb,O.Rokosa,,M.G.D.GeersaaMechanicsofMaterials,DepartmentofMechanicalEngineering,EindhovenUniversityofTechnology,P.O.Box513,5600MBEindhoven,TheNetherlandsbBuilding,Architecturea...

展开>> 收起<<

Role of inter-fibre bonds and their influence on sheet scale behaviour of paper fibre networks.pdf

共26页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Role of inter-fibre bonds and their influence on sheet scale behaviour of paper fibre networks

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: