Numerical Investigation of Paperboard Forming - Investigation of Paperboard Forming . Hui Huang and Mikael Nygrds . ... forming mould was defined as exponential overclosure behavior

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  • Numerical Investigation of Paperboard Forming Hui Huang and Mikael Nygrds

    KEYWORDS: Paperboard, Forming, Numerical

    investigation, Mechanical properties

    SUMMARY: A three dimensional numerical investiga-

    tion of a commercial four-ply paperboard formed into a

    pear-shaped mould was presented. The numerical

    investigation included the effect of pressure, boundary

    conditions, material properties and different deformation

    and damage mechanisms such as delamination and

    plasticity. Simulations were done in both the MD and CD

    using different pressures. A paperboard model with a

    combination of anisotropic continuum model and a

    softening interface model had good deformation behavior

    during the forming simulations. Forming experiment that

    mimicked the simulations was performed. Numerical and

    experiment results were compared with good agreement.

    ADDRESSES OF THE AUTHORS: Hui Huang

    (huih@kth.se), Mikeal Nygrds (nygards@kth.se),

    KTH, Department of Solid Mechanics, SE-100 44,

    Stockholm.

    Corresponding author: Mikael Nygrds

    Forming of paperboard into double-curved surfaces has

    the potential to expand the utilization space for paper

    products. This since packages can be made more flexible

    and individualized; hence package design can bring

    significant advantages for producers, customers and also

    the society. However, many deformation and damage

    mechanisms within the paperboard during forming

    processes are still unknown; therefore parameters need to

    be identified and analyzed. Compared to e.g. sheet metals

    that often are formed, there are several complicating

    factors for paper materials, such as anisotropy, gradients

    in the thickness direction, moisture and temperature

    dependency etc, which need to be understood in order to

    enable forming of paperboard. Since paperboard is highly

    anisotropic and has complex mechanical properties it will

    crack and wrinkle differently than e.g. metals and

    plastics, that normally is formed into double-curved

    structures. Moreover, commercial paperboard is usually

    composed of three to five plies. Pulp fibers are generally

    lying along one direction on each ply, which direction is

    called machine direction (MD). The direction

    perpendicular to machine direction is called cross

    machine direction (CD). The third direction is out-of-

    plane direction (ZD), as Figure 1 shows.

    Nowadays, two main strategies can be used to form

    double-curved paper surfaces. First, a fiber suspension

    can be sprayed onto a mould with desired shape. This is

    how egg box like packages are made. Second, a paper

    sheet can be formed into a mould with desired shape by

    applying load on an initial flat surface. This can be done

    by having hard male and female mould, which is

    typically used to form plastic containers. A strategy of

    this kind has been presented by Morris and Siegel (1997)

    for deep drawing of a food container. Alternatively, the

    load can be applied by a membrane using pressurized air

    or a liquid on the concave side of the package, or a

    vacuum on the convex side of the package (stlund et al.

    2011). To maintain high stiffness and strength that

    paperboard sheets have, it is advantageous if any of the

    latter methods can be used, rather than spraying fiber

    suspensions.

    The finite element method is advantageous since it can

    complement experimental studies in order to study

    deformation and damage mechanisms in paper material in

    more detail. The method has been used to study the

    mechanical response of paperboard at relatively large

    scale deformation, such as analyzing paperboard in-tack

    delamination (Hallbck et al. 2006), compression loading

    of paperboard package (Beldie et al. 2001), creasing of

    paperboard (Nygrds et al. 2008, Beex et al. 2008),

    folding of paperboard (Huang, Nygrds, 2010), paper

    webs during printing (Kulachenko et al, 2005).

    This paper is aiming to use the finite element method

    to investigate the behavior of paperboard during forming

    of complex shaped surfaces. First, the choice of element

    type and interfaces in the finite element model was

    investigated. Second, the forming performance due to

    pressure, boundary conditions and material properties

    was investigated. Lastly, forming experiments were done

    and its results were compared with numerical results.

    Material and material model

    The paperboard that was used in this work was a multiply

    paperboard, with a top ply, two middle plies and a bottom

    ply. The elastic-plastic material properties of the different

    plies in this paperboard have been characterized by

    Nygrds (2008). In addition, the shear strength profiles

    have been characterized by Huang and Nygrds (2011).

    This paperboard has relatively strong bottom ply and

    relatively weak middle plies.

    The creasing and folding behaviour of paperboard has

    previously been shown to be well predicted by a

    combination of continuum and delamination models, as

    shown in Fig 1 (Huang, Nygrds, 2010, 2011). In the

    finite element simulation, the plies in the paperboard

    were represented by an elastic-plastic continuum model,

    and the interfaces were represented by an elastic-plastic

    interface model. In the forming operation there are no

    excessive opening of interfaces, therefore each individual

    ply was assumed to have uniform properties in the

    thickness direction, i.e. the material mapping suggested

    by Huang and Nygrds (2011) was not utilized. Another

    purpose for this study was to study the strain distribution

    during double-curve forming so that the results can be

    used to better understand how the paperboard properties

    influence the forming behaviour. Since damage is not

    desired during forming, in-plane damage was not

    considered in the modelling. The finite element

    simulations were performed by using Abaqus/Explicit

    (ABAQUS, 2010). Alternatively, Abaqus/Standard or

    Abaqus/Dynamic implicit could have been used. Initial

    forming simulations did however show that

    Abaqus/Explicit was the fastest and most reliable solver.

    PAPER PHYSICS

    Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 211

  • Fig 1. Illustration of the paperboard structure and coordination system.

    Fig 2. 3D view of the forming mould with dimensions.

    [1]

    The forming mould used in the simulations is seen in

    Fig 2. The mould was pear-shaped and has three different

    curvatures: r1, r2, and r3, the largest depth in the mould

    was 25 mm. Simulations were done with both the

    machine direction and cross machine direction along the

    centreline. MD forming indicates that the centreline of

    the mould was parallel with the paperboard MD, and CD

    forming indicates the centreline of the mould was parallel

    with paperboard CD.

    Continuum model

    The continuum model was used to represent the plies of

    the paperboard, which was an elastic-plastic model with

    orthotropic linear elasticity, Hills yield criterion and

    isotropic hardening, available in ABAQUS (2010). Hills

    yield criterion is defined in Eq 1 (see above) where

    represent the initial yield stresses in the different material

    directions, and the constants F, G, and M are defined as

    ,

    , [2]

    .

    In the model, the hardening was assumed to be linear

    with hardening modulus H.

    Delamination model

    The delamination model was used to represent the

    interfaces in the paperboard such that different plies

    could be combined together to represent the paperboard.

    The delamination model was an orthotropic elastic-plastic

    cohesive law which relates the interface tractions to the

    opening and sliding of the interface. Eq 3 was defined to

    govern the change in the traction vector t across the

    interface due to incremental relative displacements.

    , , [3] , where is the stiffness in the interface normal

    direction, i.e. ZD, and are the shear stiffness components in MD and CD respectively. Damage

    initiation was governed by

    , [4]

    where ,

    , are the maximum stresses needed to

    initiate damage in the respective directions. indicated

    that purely compressive stress (i.e., a contact penetration)

    does not initiate damage. Softening behaviour was

    defined by exponential damage evolution, which means

    when the initiation criterion was reached, the cohesive

    strength decrease with exponential behaviour. The

    damage evolution D was defined as:

    Mould

    r2

    r3 r1 L

    L=94 mm r1=35 mm r2=95 mm r3=15 mm

    PAPER PHYSICS

    212 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012

  • , [5]

    where is a non-dimensional parameter that defines the rate of damage evolution,

    is the maximum value of

    effective displacement at during loading history, and

    is the effective separation at complete failure. The

    effective displacement is defined as

    , [6]

    which

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