Experimental Mechanics (2014) 54:1395–1410 DOI 10.1007/s11340-014-9903-1 General Anisotropy Identification of Paperboard with Virtual Fields Method J.M. Considine & F. Pierron & K.T. Turner & D.W. Vahey Received: 10 December 2013 /Accepted: 8 May 2014 /Published online: 1 August 2014 # Society for Experimental Mechanics (outside the USA) 2014 Abstract This work extends previous efforts in plate bending papermaking is more than 4,000 years old and modern paper- of Virtual Fields Method (VFM) parameter identification to making is 200 years old, analysis of paper’s engineering include a general 2-D anisotropic material. Such an extension properties remains a significant research area. was needed for instances in which material principal direc- Production of paper and paperboard includes the separation tions are unknown or when specimen orientation is not aligned of cellulose fibers, which are themselves anisotropic [1], from with material principal directions. A new fixture with a multi- wood through a pulping process, which may be mechanical, axial force configuration is introduced to provide full-field as in newsprint, or chemical, as in sack paper. Resulting fiber strain data for identification of the six anisotropic stiffnesses. flexibility and inter-fiber bonding are improved through an Two paper materials were tested and their Qij compared favor- additional mechanical action called beating. Absent require- ably with those determined by ultrasonic and tensile tests. ments for fiber bleaching, the fibers are dispersed at low Accuracy of VFM identification was also quantified by vari- concentrations, usually less than 1 % fiber/water,prior to being ance of stiffnesses. The load fixture and VFM provide an sprayed on a moving screen. Travel direction of the screen is alternative stiffness identification tool for a wide variety of called the machine direction (MD) while the in-plane direction thin materials to more accurately determine Q12 and Q66. perpendicular to the travel direction is called the cross- machine direction (CD). A combination of the spraying action and screen travel tend to orient the fibers in the MD, which Keywords Virtual fields method (VFM) . Digital image usually corresponds with the 1-direction of material proper- correlation (DIC) . Stiffness . Paperboard . Anisotropy ties. Depending on the type of paper produced paper machines can operate at speeds of 1,500 m/min or higher. While ratios vary, the typical ratio for E11/E22 is near 2. Offline stiffness Introduction measurement is used in process control [2]. Paper and paper- board are frequently sold on a strength/weight or stiffness/ Paper, paperboard, and other cellulose fiber composites have weight basis and so reduction of property variability and received significant attention for use in materials and struc- mechanical property improvement are persistent goals of pa- tures where biocompatibility is an important consideration permakers, even though costs associated with variability are because cellulose fiber composites are renewable, recyclable rarely acknowledged [3]. and compostable. However, even though single-sheet Paper and paperboard are hydrogen-bonded polymers : which exhibit hygro-thermal, viscoelastic and plastic behav- J. Considine (*) D. Vahey iors. Below their plastic limit their mechanical properties are US Forest Service, Forest Products Laboratory, One Gifford Pinchot considered to be linear elastic. Yeh et al. [4] made a compre- Drive, Madison, WI 53726, USA e-mail: [email protected] hensive examination of the effect of moisture on the orthotropy, elastic moduli, Poisson’s ratios, shear moduli and F. Pierron strain energy density. The materials in that work were linear University of Southampton, Southampton SO17 1BJ, UK elastic for MD and CD strains below ~1 %. A comprehensive K. Turner review of properties related to moisture and viscoelasticity is University of Pennsylvania, Philadelphia, PA 19104-6315, USA found in [5]. Castro and Ostoja-Starzewski [6] examined 1396 Exp Mech (2014) 54:1395–1410 paper plasticity at a single moisture level and found linear resolution than DIC include grip displacement [31] and mark- elastic behavior below 0.5 % strain. er tracking [32, 33]. The objective of this work was to develop a load fixture Parameter identification from full-field heterogeneous and analysis method to identify the anisotropic stiffnesses of strains is accomplished by the use of an inverse method. paper, paperboard, and other thin materials. Whereas the paper VFM [34] was chosen for this work because it is general, industry considers paper to be an orthotropic material, general flexible, and faster than other methods which include FEM- anisotropy is developed if the MD is not aligned with the 1- Updating [25, 35], energy-based [36] and equilibrium gap direction and is called ‘rotated orthotropy.’ The assumption of [37] methods. VFM requires no additional programs, such orthotropy requires confirmation, as the papermaking process as an FEM-solver, and analysis scripts can be easily written has many variables affecting sheet mechanical properties in- in Matlab®. cluding fiber properties, fiber orientation [7], fiber length Ultrasonic techniques have also been used to determine the distributions, sheet density, and drying restraint [8, 9]. Qij of paper [38, 39]. Work by Habeger [39] appears to be the Finally, identification of Qij, the in-plane stiffnesses, is impor- first attempt to determine Q16 and Q26 in paper materials. tant for process control [10] and for structures made from Three difficulties are associated with ultrasonic examination. paper and paperboard, e.g., corrugated fiberboard [11], wound First, significant wave attenuation occurs that requires sophis- cores [12] and paper/foil composites [13]. Additionally, an- ticated analysis to determine time-of-flight. Second, transmit- isotropic Qij identification is important for other thin materials, ted waves combine effects of all Qij and so relative scale of including surgical meshes [14], biological tissues [15], textiles individual Qij makes it difficult to identify smaller parameters, [16], and rubber [17], among many others. such as Q16 and Q26. Finally, in rate-dependent materials, Identification of a fully populated Qij matrix requires a ultrasonic properties depend on excitation frequency. specimen and load configuration in which heterogeneous Two cellulose-fiber webs were examined: a filter paper and strain fields of each ε1, ε2,andε6, are developed. Several a paperboard, a packaging grade known in the industry as methods have been proposed for creating those heterogeneous linerboard. Details of the load fixture and analysis method, fields, such as uniaxial tensile coupons cut in different orien- along with quantification of parameter identification accuracy tations [18], uniaxial tensile coupons with a central hole [19], are provided. The VFM analysis was extended to include cruciform [20], bulge tests [21], and thin-walled cylindrical identification of Q16 and Q26, along with associated methods tubes [22]. Each of these specimens and load configurations to reduce effect of strain measurement noise. Whereas the had some aspect which made them unsuitable for the current analysis assumes that the materials are linear elastic and work. For example, identification using uniaxial tensile spec- homogeneous, extension of the analysis to more general be- imens would require many specimens to develop statistical havior is straight-forward. Comparison of Qij identification certainty of identification; other configurations would cause from VFM, ultrasonic and tensile coupons is included. specimen wrinkling. Some specimens, such as cruciform specimens with and without central holes have stress concen- trations at corners and/or holes where nonlinear constitutive Material behavior may be present. Small aspect-ratio tensile and bulge tests are incapable of producing different principal stress Two materials were examined. The first material was ratios. Tube configurations are complicated by a joining seam Whatman® Chromatography Paper 3MM CHR and will be and the difficulty of full-field examination. referred to as filter paper. It had nominal physical properties: While application of VFM to thin materials is new, other grammage 180 g/m2, thickness 0.28 mm, and density cellulose-fiber based materials have been examined with 635 kg/m3. Filter papers are used in a variety of household, VFM. Xavier et al. [23, 24] used the unnotched Iosipescu commercial, and scientific applications to capture particulate geometry to identify the orthotropic Qij in maritime pine matter. The material was 100 % cellulose, as it is entirely wood. Le Magorou et al. [25] and Xavier et al. [26]identified comprised of cotton linters, a cellulose fiber that is typically plate bending stiffnesses in plywood and medium density 5–10 mm long. fiberboard, respectively. The second material was a commercial unbleached, kraft Most of the these geometries require some type of full-field single-ply linerboard that had nominal physical properties: measurement of displacements in order to determine strains. grammage 209 g/m2, thickness 0.30 mm, and density Digital image correlation (DIC) [27] is the most common 688 kg/m3. Fiber composition of this material is unknown, choice and is used here, given its general ease of use and but likely contains both virgin and recycled fibers. This ma- extensive development of analysis algorithms. In some cases, terial is commonly used in structural products such as corru-