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A model for the through-thickness elastic–plastic
behaviour of paper
Niclas Stenberg
*
STFI, Box 5604, SE-114 86 Stockholm, Sweden
Received 23 January 2003; received in revised form 1 September 2003
Abstract
An elastic–plastic material model for the out-of-plane mechanical behaviour of paper is presented. This model
enables simulation the elastic–plastic behaviour under high compressive loads in the through-thickness direction (ZD).
Paper does not exhibit a sharp transition from elastic to elastic–plastic behaviour. This makes it advantageous to define
critical stress states based on failure stresses rather than yield stresses. Moreover, the failure stress in out-of-plane shear
is strongly affected by previous plastic through-thickness compression. To cover these two features, a model based on
the idea of a bounding surface that grows in size with plastic compression is proposed. Here, both the bounding and the
yield surfaces are suggested as parabolas in stress space. While the bounding surface is open for compressive loads, the
yield surface is bordered by the maximum applied through-thickness compression.
Ó2003 Elsevier Ltd. All rights reserved.
Keywords: Paper; Paperboard; Biaxial; Tension; Compression; Shear; Z-direction; Out-of-plane; Model; Bounding surface; Yield
surface; Elastic; Plastic
1. Introduction
In understanding deformation of paper materials, the through-thickness direction is of utmost impor-
tance. There are several converting operations where the performance is mainly governed by the out-
of-plane behaviour. Examples of these converting operations are calandering, folding, cutting and creasing.
With the aid of computer simulations, the understanding of these processes could be increased and
products and processes optimised. However, such analyses have been limited by the fact that no satisfactory
material model has been available that describes the out-of-plane behaviour of paper and paperboard.
Paper materials are in general complex materials to model due to their highly anisotropic behaviour,
non-linear material response, rate sensitivity and dependency on the moisture content of the material.
*
Present address: Swedish EU/FoU-r
aadet, R&D Council, NCP, P.O. Box 7091, SE-10387 Stockholm, Sweden. Tel.: +46-8-454-
64-49; fax: +46-8-454-64-51.
E-mail address: niclas@eufou.se (N. Stenberg).
0020-7683/$ - see front matter Ó2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijsolstr.2003.09.003
International Journal of Solids and Structures 40 (2003) 7483–7498
www.elsevier.com/locate/ijsolstr
Yield in anisotropic materials is commonly modelled using for example the quadratic Hill criterion (Hill,
1948). Another yield criterion, based on the concept of a linear transformation of the anisotropic stress
state to an equivalent isotropic stress state, is presented by Karafillis et al. (1993). This criterion is of
particular interest here since it is used as the foundation for modelling of the in-plane elastic–plastic be-
haviour of paper materials (Tryding, 1996; M€
aakel€
aa, 2000). Recently Xia et al. (2002) presented an elastic–
plastic material model for the in-plane behaviour that also covers the anisotropic hardening properties of
paper materials.
To the knowledge of the author, there has not been any reports published dealing with the out-of-plane
elastic–plastic behaviour of paperboard under combined shear and normal loadings. However, the need for,
and usefulness of, such a model has been put forward (by among others Rodal (1993a,b)). The only
publications in this area are those of Stenberg et al. (2001a,b) who developed an elastic–plastic model for
combinations of out-of-plane shear and low compressive loads. When the out-of-plane compressive loads
are considered high, only isotropic yield criteria (Rodal, 1993a,b) seems to be used. Paper is highly an-
isotropic and consequently, isotropic material models do not in general give sufficiently good results for
most applications.
Being a porous material paperboard exhibits several properties similar to foam and soil. Yield in porous
materials is previously examined by, for example Schofield and Wroth (1968) for soils and Rusch (1970) for
foams. These types of material models have now matured to the stage where they already exist as options in
commercial software for solution of boundary value problems using the finite element method (ABAQUS,
1998).
In paper materials the mechanical properties are dependent of the elastic–plastic deformation history.
Especially, the failure stress in out-of-plane shear is strongly dependent on previous through-thickness
plastic compression of the material.
The actual yield point is hard to determine, it needs a definition rather than being physically clear
(Stenberg et al., 2001a,b), whereas the maximum load is easy to determine. This implies that using the peak
load instead of the yield point should be a more robust definition when defining a critical stress state. To
cover these features the concept of a bounding surface in stress space is proposed by Dafalias and Popov
(1982), and Krieg (1975). Bounding surface plasticity is based on the existence of a surface that encloses all
possible stress states. Inside the bounding surface plastic deformation is possible through a mapping of the
stresses onto its image on the bounding surface. The bounding surface can change size and position with
hardening parameters especially introduced for the bounding surface. The yield surface inside the bounding
surface is also allowed to change size and position. The material hardens while the yield surface expands
and approaches the bounding surface.
The model presented here is a model for combined shear and normal out-of-plane loadings. In this paper
an elastic–plastic model for quasi-static conditions and constant climate is developed. The change of
material behaviour due to changes in deformation rate and moisture content is handled elsewhere (Lif,
1997; R€
aatt€
oo, 2001). The model is based on the idea of a bounding surface that grows in size with plastic
compression. Here, both the bounding and the yield surfaces are suggested as parabolas in stress space. The
parabolic yield surface is in through-thickness compression bordered by the maximum applied through-
thickness compression.
2. Materials and methods
2.1. Paper materials
During the manufacturing process of paper and paperboard, a fibre suspension is drained through a net,
called a wire. The wire is traversed rapidly as the fibre–water suspension is sprayed onto the wire from a
7484 N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498
nozzle. Shear forces in the area where the suspension hits the wire ensure that the fibres in general are more
oriented in the direction of the paper machine than in the cross-machine direction. Another result of the
manufacturing process is that nearly no fibres are directed in the through-thickness direction. The direction
of the fibres combined with the drying process leads to anisotropy in the mechanical properties of paper.
The three main directions of the paper machine: the machine direction (MD), the cross-machine direction
(CD) and the through-thickness direction (ZD), can to a good approximation be used as the principal
directions of the paper material. Therefore, paper is often considered as an orthotropic material. The
stiffness in MD is usually 1–5 times greater than that in CD, and typically 100 times greater than that in ZD.
The elastic and elastic–plastic behaviour of different paper materials differ in detail, but the overall
characteristics can be captured in a fairly general model. As model material a five-layer paperboard was
used. This particular paperboard was chosen because other properties of the material are already known
(Stenberg et al., 2001a,b; Stenberg, 2002). The five-layer paperboard consists of three bulk layers of
chemical thermo mechanical pulp (CTMP) and two outer layers of bleached craft pulp. The outer chemical
pulp layers are stiffer than the bulk layers. However, these layers cannot be easily separated for individual
evaluation, and therefore the paperboard will be considered as homogenous just to illustrate the quality of
the model.
The through-thickness behaviour of the five-layer paperboard is experimentally investigated by Stenberg
et al. (2001a,b) and Stenberg (2002). In the latter a series of tests are performed to characterise the me-
chanical behaviour under compressive, shear and combined loadings. No major differences in the out-
of-plane behaviour, neither concerning the characteristic behaviour nor magnitudes, are noticed for shearing
in MD and CD and therefore shearing in MD is only considered when illustrating the general behaviour.
The mechanical behaviour under pure compression for one typical test piece is illustrated in Fig. 1. This
graph shows the compressive normal stress as function of the engineering strain for a load cycle consisting
of consecutive loadings and unloadings at an increasing overall load. The stress is here defined as recorded
force divided by initial cross-sectional area. The elastic–plastic behaviour in compression is well approxi-
mated as exponential. Such behaviour is previously reported by Pfeiffer (1981). The elastic response is non-
linear as indicated by the non-linear unloading stress–strain curves. Furthermore the elastic stiffness is an
increasing function of the plastic compression.
In Fig. 2 monotonous shear tests under different constant compressive loads are shown. In these test the
compressive load is first slowly applied, and then followed by shear loading. The material can withstand
1 0.8 0.6 0.4 0.2 0
-20
-10
0
Engineering strain
Normal stress / MPa
-30
-40
-50
-60
-----
Fig. 1. Typical stress–strain curve illustrating the non-linear elastic and elastic–plastic behaviour in compression of paperboard, under
consecutive through-thickness loadings and unloadings (Stenberg, 2002).
N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498 7485
higher shear loads if it is subjected to through-thickness compression. Moreover, for high compressive
loads the shear stress approaches an asymptote with no clearly visible peak in the form of a local maximum.
On the other hand, for shear tests under low compressive loads there is a distinct peak followed by sub-
sequent softening. This phenomenon is observed previously by Stenberg et al. (2001a,b) and Xia (2002).
When the peak loads are plotted in stress space, see Fig. 3, it is observed that for high compressive loads
there is a linear relation between normal and shear stresses, whereas for shear tests under low compressive
loads a parabola shaped relation is seen. Note, that it is shown by Stenberg et al. (2001a,b) that positive and
negative shear are of similar magnitude. Therefore, the failure stresses for negative shear in Fig. 3 are
obtained by mirroring the results for positive shear in the normal stress axis. The results in Fig. 4 illustrate
how the peak load in shear increases if the material first is plastically compressed. In these tests the test
pieces were first loaded to a normal stress of )10 or )30 MPa, and then unloaded to a normal stress of )3
MPa that was kept approximately constant as the shear load was applied.
Fig. 2. Typical stress–strain curves in shear in MD for different compressive loads; 1, 3, 20, 30 and 40 MPa (Stenberg, 2002).
–
30
–
20
–
10 0
–
10
–
5
0
5
10
Normal stress / MPa
Shear stress / MPa
Fig. 3. Failure stresses in shear (MD (}) and CD ()) under different compressive loadings. The failure stresses for low values of the
out-of-plane normal stress () are reported by Stenberg et al. (2001a,b).
7486 N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498
The relation between the out-of-plane normal and shear strains, i.e. the direction of the plastic defor-
mation, is dependent on the loading history. As shown in Fig. 5 the thickness of the material decrease under
shearing if no previous additional compaction has taken place. If the material has been plastically com-
pressed, it will increase in thickness when sheared.
2.2. Out-of-plane elastic–plastic response of paper
Paper materials are highly anisotropic with the thickness direction as the by far most compliant direc-
tion. The coupling between the in-plane and out-of-plane directions is mostly neglected when analysing
paper materials. The coupling itself is studied in detail by Stenberg and Fellers (2003), Baumgarten and
G€
oottsching (1976) and €
OOhrn (1965). It is found that the in-plane response due to out-of-plane deformation
Fig. 4. Typical stress–strain curves in shear in MD at a compressive load of 3 MPa. Two of the test pieces were first loaded to 10 and 30
MPa in compression, respectively, before being unloaded to a compressive stress of 3 MPa (Stenberg, 2002).
Fig. 5. Typical combinations of normal and shear (MD) strains for test pieces subjected to the same loading sequences as the test pieces
in Fig. 4 (Stenberg, 2002). Note that the normal strains are defined as zero at the start of the shear part of the loading.
N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498 7487
is very small and therefore in most cases negligible. Thus, the materials response is commonly separated
into in-plane and out-of-plane behaviours. The in-plane mechanical behaviour will here not be further
investigated. It has for example recently been thoroughly modelled by M€
aakel€
aa (2000) and Xia et al. (2002).
A theory for the out-of-plane mechanical behaviour of paperboard subjected to combined shear and
normal out-of-plane loadings will be presented. In omitting the Poisson effect, the problem is reduced to a
problem where the only non-zero components of the stress and strain tensors, rij and eij, are
r¼rzz
sxz
;e¼ezz
2exz
¼ezz
cxz
:ð1Þ
In the sequel any of the in-plane directions will be denoted by xand the out-of-plane direction by z.
The elastic–plastic theory presented here makes a distinction between elastic and plastic deformations.
The total strain is split into elastic and plastic parts according to
e¼eeþep:ð2Þ
The elastic through-thickness deformation is modelled as non-linear because of the porous structure of
paperboard and the high strains expected in applications. The elastic shear deformation is modelled as
linear. The plastic deformation will be modelled using a parabolic yield surface in rzz and sxz expanding
towards a bounding surface, and an associative flow rule.
2.3. Porous elasticity
For small ZD-strains in paper and paperboard the assumption of a linear elastic behaviour is appro-
priate (Stenberg et al., 2001a,b), but in general the porosity of the material has to be considered. As
mentioned above paper materials consist of fibres oriented in the plane of the sheet. Out-of-plane defor-
mation of the paper material will consist of deformation of the solid fibre-network, consisting of cellulose
fibres, and deformation of voids in the material. Hence, the elastic strain can be split in two parts, one
representing deformation of the voids, void ee, and one representing deformation of the solid fibre-structure,
solee.
ee¼voideeþsol ee:ð3Þ
To characterise porous materials a widely used property is the voids ratio,
r¼Vv
Vs
;ð4Þ
where Vvis the volume of voids and Vsis the volume of the solid fibre-structure. It is generally accepted that
for porous materials the change in elastic voids ratio, re,is
dre¼ldðlnðpþptÞÞ;ð5Þ
where lis a material parameter, pis the equivalent hydrostatic pressure and ptis the hydrostatic tensile
strength. Due to the small in-plane strain levels, compared to the out-of-plane strains, the in-plane stresses
are assumed to have negligible influence on the voids volume, thus only rzz is considered. Note, shear is
deviatoric and therefore do not affect the volume of the voids. When using logarithmic strains the elastic
volumetric strain is
voidee
vol ¼ln Je
void;ð6Þ
where the elastic volume change of the voids is
Je
void ¼1þre
1þr0
:ð7Þ
7488 N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498
In Eq. (7) r0is defined by,
r0¼rorig þrp;ð8Þ
where rorig is the original voids ratio and rpthe voids ratio induced by plastic deformation. Thus, rpis a
function of the plastic volumetric strain. Inserting Eq. (6) into Eq. (7) gives
rp¼ð1þrorigÞexp void ep
vol
1:ð9Þ
The strain in the through-thickness direction, ezz, is the only strain that can change the volume of the voids,
and consequently the strain in the thickness direction is identical to the volumetric strain due to the voids.
When integrating Eq. (5), using logarithmic strains, an expression for the stress related to the elastic
strain is obtained as
rzz ¼rt
z1
exp 1þr0
l1
exp voidee
zz;ð10Þ
where rzz is the stress in ZD, rt
zis the tensile strength in ZD and void ee
zz is the elastic strain in ZD due to
deformation of the voids.
The strain in the solid fibre-structure of the material is described by a modulus that is a function of the
density. The initial response from the network is Einitial
z. For a structure where the voids have been com-
pletely compressed the modulus is given by Eend
z. A linear dependency on rpwas applied on the moduli to
obtain the elastic response from the solid part of the material as
Esolid
z¼1þrp
rorig
Einitial
zrp
rorig Eend
z;
Gsolid
xz ¼1þrp
rorig
Ginitial
xz rp
rorig Gend
xz :
8
<
:ð11Þ
The part of the total strain related to the voids, void ee
zz, is determined by the total strain, the plastic strain and
the stress in ZD from the equation
voidee
zz ¼ezz ep
zz rzz
Esolid
z
:ð12Þ
According to the model presented here, the elastic deformation will be linear in out-of-plane shear,
whereas the deformation in the thickness direction will be non-linear with a stiffer response from a com-
pressed material than from a non-compressed one.
2.4. Bounding surface plasticity
Initiation of plastic deformation is governed by a yield function, f. Here, the response from in-plane
stresses is neglected, and there will be no in-plane dependency on the out-of-plane plastic behaviour. Thus,
the yield function is a function of the out-of-plane stress components, rzz and sxz, and hardening para-
meters, Kb(b¼1;2;...), according to
f¼fðrzz;sxz ;KbÞ:ð13Þ
Fig. 2 shows the results from shear tests performed at different compressive load levels. For all shear tests
an asymptote is reached at high shear strains. For shear tests under low compressive loads a distinct peak
load is seen prior to the asymptote, whereas for the shear tests under high compressive loads the asymptote
is reached with no previous peak stress. The points defined as being under high compressive load are the
points where the shear tests approach an asymptote monotonously with no prior peak load, while the other
tests are defined as being made under low compressive loads. The asymptotic stress observed for all shear
N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498 7489
tests resembles a frictional type of behaviour, and can be modelled as Coulomb friction with the internal
coefficient of friction, M.
As shown by Stenberg et al. (2001a,b) it is difficult to determine the load level where the deformation
changes from being elastic to elastic–plastic, whereas the maximum load is easy to determine. This implies
that using the peak load instead of the yield stress should be a more robust definition when defining a
critical stress state. Moreover, the failure load in shear is dependent on the through-thickness compressive
loading history. One way to approach this problem is to introduce a bounding surface that surrounds the
area in stress space where elastic–plastic deformation is allowed, Dafalias and Popov (1982) and Krieg
(1975). Using the failure surface as bounding surface makes it easy to find the shape of the bounding surface
and it is assumed that plastic deformation is allowed up to failure. The yield surface, f, is then projected to
the bounding surface, F, using a mapping, u,as
fðrzz;sxz ;KbÞ!
uFð^
rrzz;^
ssxz ;b
KKBÞ;ð14Þ
where the variables, ^
rrzz,^
ssxz and b
KKB, define the bounding surface. Thus, the mapping, u, transforms the
stress state at the yield surface, r, to the stress state at the bounding surface, ^
rr, with the hardening of
the material included in the transformation. We assume a linear mapping when projecting the stress state at
the yield surface onto the bounding surface (Borja et al., 2001). Since, no failure stress is observed in pure
compression the projection to the bounding surface will only be used for the shear stress. Because of the
symmetry in shear, the centre of the mapping is on the normal stress axis. Thus, here the mapping
u¼^
rr ¼r;
^
ss ¼sð1þjÞ
ð15Þ
is suggested, where jis a hardening parameter for the material in shear. Since only out-of-plane normal and
shear stresses are present in this formulation the indexes, zz and xz, are in the sequel omitted. In this
formulation of the problem, the shape and position of the bounding surface in stress space needs to be
found. Also, the evolution of the hardening parameter jhas to be determined. In Fig. 3 all experimentally
determined failure points in stress space are shown. The peak stresses in shear are linearly related to the
compressive stress for high compressive loads. The failure points at low compressive loads, on the other
hand, do form a distinct parabola. The bounding surface is assumed to be described by the equation
F¼^
rr ^
rrtþ^
ss2
^
ssp
¼0;ð16Þ
where ^
rrtis the tensile failure stress in ZD, and ^
sspis a parameter that describes the size of the parabola.
As shown in Fig. 4, the peak stress in shear increases with previous compression of the test piece. The
failure points form a failure surface in stress space, and the size of the failure surface is changed with
previous plastic deformation. To capture the increase in failure stress obtained by compressing the material,
the shape of the bounding surface will be a parabola that expands with compression following the internal
friction lines with slope Min the out-of-plane normal-shear stress space. If the normal yield stress, ra,is
beyond the initial bounding stress,
rra, the hardening parameter for the bounding surface, ^
ssp, will be a
function of the normal yield stress in compression. Otherwise, the bounding surface parabola keeps its
original size. To capture this behaviour, a new parameter, ^
rra, is introduced
^
rra¼ra;ra6
rra;
rra;ra>
rra:
ð17Þ
This parameter is illustrated in Fig. 6. Inserting Eq. (17) into Eq. (16) and using the definition of the internal
coefficient of friction gives
7490 N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498
^
ssp¼ M2^
rr2
a
^
rra^
rrt
:ð18Þ
In combining the mapping in Eq. (15) with Eq. (16) the bounding surface described by the stress state at the
yield surface is rendered:
F¼r^
rrtþð1þjÞ2s2
^
ssp
¼0:ð19Þ
With the chosen hardening parameters, the bounding surface described by the stresses at the yield surface
has the same properties as the yield function. Thus, the bounding surface expressed by the stresses at the
yield surface can be used as a yield function.
Introducing the yield stress in pure shear, s00 , an expression for jis obtained from Eq. (19) as
j¼ffiffiffiffiffiffiffiffi
^
rrt
ssp
ps00
1;ð20Þ
where
sspis ^
sspat the initial state f^
rra¼
rrag.
The condition for plastic deformation is controlled by the current stress state mapped onto the bounding
surface. The hardening of the material is defined by two hardening parameters, raand s00. The hardening
parameters are dependent on two independent internal variables; one is controlling the compressive plastic
strain and is denoted by ep
c. The other one controls the plastic shear strain and is denoted by cp. The
hardening parameters, as functions of the internal variables, are here proposed as
ra¼ArþBrexpðCrep
cÞ;
s00 ¼AsþBstanhðCscpÞ:ð21Þ
The yield surface is a parabola bordered by a straight vertical line at the normal yield stress, where the
normal to the yield surface is directed in the negative normal stress direction. Thus, the yield surface, with
the hardening parameters defined in Eq. (21), can be written as
Fig. 6. Yield and bounding surfaces in stress space. (- --) represents the initial bounding surface, (––) any yield surface inside the initial
bounding surface, and (– Æ–Æ–) the bounding surface when expanded to ^
rrafrom the initial bounding surface at
rra. The size of the
bounding surface is determined by ^
rraand the frictional behaviour represented by ().
N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498 7491
f¼vr
"rtþð1þjÞ2s2
^
ssp#þð1vÞ½rar;ð22Þ
where the switch, v, is a function of the normal stress according to
v¼1;r>ra;
0;r¼ra:
ð23Þ
2.5. Plastic deformation
Once the yield surface is reached, plastic deformation is introduced and will increase with further
loading. The direction of the plastic deformation is determined by a plastic potential, g, which gives the rate
of the plastic strain _
eep
ij,as
_
eep
ij ¼_
kk og
orij
;ð24Þ
where _
kk is a plastic multiplier. Here the dot notation means change of property rather than time derivative.
In associative flow the yield function is chosen as the plastic potential, and hence, the direction of the plastic
deformation is the normal to the yield surface, as shown in Fig. 7. For the theory presented here an as-
sociative flow is adopted. Eq. (24) then gives
_
eep
z
_
ccp
xz
"#
¼_
kk
of
or
of
os
"#
:ð25Þ
At the corners of the yield function the direction of the plastic flow is undetermined. Thus, the direction of
the flow vector has to be defined at these points. Here, the plastic flow at the sharp corners of the yield
surface is assumed to be
_
eep
z
_
ccp
xz
"#
¼_
kk
of
orv¼1
of
osv¼0
"#
:ð26Þ
–10 –8–6–4–2 0
–4
–3
–2
–1
0
1
2
3
4
Normal stress / MPa
Shear stress / MPa
1
2
3
Fig. 7. Definitions of the direction of the plastic flow at different positions on the yield surface. The normal to the yield surface is well
defined at the parabola, (1) and the cap, (2). At the corner, (3) the deformation direction is defined according to Eq. (26).
7492 N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498
2.6. Internal variables
Since the internal variables are chosen as the plastic deformations in the normal and shear directions
respectively, their evolution laws are assumed to follow the plastic flow as
_
eep
c
_
ccp
"#
¼_
kk
1
_
kk _
eep
c
of
osr¼0
^
ssp¼
ssp
2
6
43
7
5:ð27Þ
The evolution of ep
cfollows the plastic deformation in compression and cpfollows the plastic deformation in
shear for simple shear.
3. Simulations and results
3.1. Application of the theory
To illustrate the behaviour of the proposed model, simulations resembling the tests performed by
Stenberg (2002) were carried out. The model was implemented as a user-defined material subroutine in the
finite element software, ABAQUS (1998). Table 1 shows the numerical values of the model parameters used
for the simulations. The values of the parameters were chosen to be as close as possible to the true material
in order to illustrate how the model works.
The numerical values of the model parameters were determined from tests similar to those presented in
Figs. 1–5. The parameters related to the through-thickness compressive behaviour were obtained from tests
of the type shown in Fig. 1. An ordinary tensile test gave the failure stress in tension, and an out-of-plane
shear test gave the hardening parameter for shear. When doing several shear tests under different constant
compressive loads the internal coefficient of friction, M, and the initial bounding compressive stress,
rra,
were obtained.
A simulation of subsequent compressive loadings and unloadings, see Fig. 8, shows how the elastic
behaviour of the model changes with both elastic and plastic compression. Also shown is how the simulated
Table 1
Numerical values of the model parameters used in the simulations
Constant Value Description Introduced in equation
Einitial
z34 MPa Out-of-plane Youngs modulus (11)
Ginitial
xz 32 MPa Out-of-plane Shear modulus (11)
Eend
z5000 MPa Youngs modulus for fully compacted solid (11)
Gend
xz 1923 MPa Shear modulus for fully compacted solid (11)
M0.4 Internal coefficient of friction (20)
Ar0.4378 MPa Constant for hardening parameter (24)
Br)1 MPa Constant for hardening parameter (24)
Cr6.5 Constant for hardening parameter (24)
As0.3 MPa Initial yield point in shear (24)
Bs0.5192 MPa Constant for hardening parameter (24)
Cs100 Constant for hardening parameter (24)
rra)12 MPa Initial bounding compressive stress (19)
l0.0212 Constant for the elastic behaviour (5)
rorig 1.0 Initial voids ratio (9)
rt
z0.36 MPa Failure stress in tension (10)
N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498 7493
material stiffens when it has been previously compressed. In tension the model follows a typical stress–strain
curve of a through-thickness tensile test, as shown in Fig. 9, but the deformation in the model is elastic.
Moreover, simulations of shear tests under constant compressive load show that the bounding surface
theory is capable of modelling the increased shear strength at through-thickness compression, see Fig. 10.
The failure points from these simulations form the bounding surface, as shown in Fig. 11, where also the
experimentally determined failure stresses from Fig. 3 are included. In Fig. 12 simulations of three shear
tests under a compressive stress of 3 MPa is shown. In Fig. 13 the thickness change in these tests are shown.
The peak load in shear and the change of deformation are functions of previous initial through-thickness
compression. In Figs. 12 and 13 the material has been initially compressed to 10 and 28 MPa before un-
loaded to a compressive stress of 3 MPa.
–0.8 –0.6 –0.4 –0.2 0
–60
–50
–40
–30
–20
–10
0
Normal engineering strain
Normal stress / MPa
Fig. 8. Simulation of the out-of-plane compressive behaviour of paper with subsequent unloadings and loadings at different load levels.
00.02 0.04 0.06 0.08 0.1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Normal engineering strain
Normal stress / MPa
Fig. 9. Simulation of an out-of-plane tensile test.
7494 N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498
4. Discussion and conclusion
In combining non-linear elasticity and bounding surface plasticity the principle features of the out-
of-plane behaviour of paperboard are captured. Particularly the increasing out-of-plane stiffness, both in
compression and shear, and increasing shear strength after previous compression are well captured. Fur-
thermore, if plastically compressed, the direction of the plastic flow, see Fig. 7, changes in accordance with
experimental observations in Fig. 5.
The proposed model does not include plastic deformation in out-of-plane tension. The plastic defor-
mation in tension is relatively small compared to the plastic deformation in compression, (Stenberg et al.,
0 0.05 0.1 0.15 0.2 0.25 0.3
0
5
10
15
20
25
Shear engineering strain
Shear stress / MPa
60 MPa
40 MPa
30 MPa
20 MPa
10 MPa
3 MPa
Shear
Fig. 10. Simulations of the out-of-plane shear behaviour under different compressive loads; 3, 10, 20, 30, 40 and 60 MPa. Note the
different strain scale compared to Fig. 2.
–40 –30 –20 –10 0
-20
-15
-10
5
0
5
10
15
20
Normal stress / MPa
Shear stress / MPa
-
Fig. 11. The failure load in shear as a function of the normal load. The measured failure stresses (Stenberg et al., 2001a) are marked
with diamonds (}) for MD and circles () for CD, and the calculated failure stresses using the present model are marked with stars
().
N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498 7495
2001a,b; Stenberg, 2002). However, the load carrying capacity of the present model in tension is still intact,
and the non-linear elastic model resembles the initial part of the elastic–plastic stress–strain curve of a
tension test.
An important contribution to the softening observed in tension and shear, under low compressive loads,
is discrete damage in the form of delaminations (Stenberg et al., 2001a,b; Stenberg, 2002; Smith, 1999;
Dunn, 2000). However, the proposed model does not include delaminations. The influence of delaminations
on out-of-plane shear and out-of-plane tension has recently been modelled using an approach based on
interfaces with cohesive properties. In this model the material softens with increasing opening and tan-
gential displacement of the delamination surfaces, (Xia, 2002).
Fig. 12. Simulations that show the increasing peak load in shear if the material is plastically compressed over the initial bounding
compressive stress. The material has been loaded in compression to 10 and 25 MPa, and then unloaded to a compressive load of 3 MPa
prior to shear.
Fig. 13. Simulations that show the change in the direction of the deformation if the material is plastically compressed over the initial
bounding compressive stress. The material has been loaded in compression to 10 and 25 MPa, and then unloaded to a compressive load
of 3 MPa prior to shear.
7496 N. Stenberg / International Journal of Solids and Structures 40 (2003) 7483–7498
The consequence, of not including delaminations in the model is that the elastic–plastic behaviour in
shear at low compressive loads and tension does not encounter the post-peak behaviour experimentally
observed (Stenberg et al., 2001a,b; Stenberg, 2002). The peak load in shear will monotonously approach the
asymptote, related to the internal friction in the material, regardless of the magnitude of the compressive
load. For high compressive loads the softening is neutralised by the internal friction. It is believed that the
general out-of-plane behaviour of paperboard can be well captured by combining the present model with
the delamination model.
Due the generality of the model presented here, it can be calibrated for most paper materials. Calibration
possibilities are offered through changes of the material parameters and change of hardening parameters,
i.e. Eq. (21). Furthermore, the shape of the bounding surface can be modified. However, the model as it is
works satisfactorily and do show the principle features required from a model of the elastic–plastic out-of-
plane behaviour of paper materials. Moreover, the model presented here is shown for one out-of-plane
shear component, but the extension to both out-of-plane shear components is straightforward.
Acknowledgements
The financial support for this work from the Foundation for Strategic Research (SSF) through the
Forest Products Industry Research College (FPIRC) program is greatly acknowledged. The author also
would like to thank Mr. Petri M€
aakel€
aa for kind help in the development of the user-defined material sub-
routine for ABAQUS. Professor S€
ooren €
OOstlund is also acknowledged for valuable advice and comments on
the manuscript.
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