Mathematics in paper - from fiber suspension fluid dynamics to solid state paper mechanics
© Hämäläinen et al.; licensee Springer. 2014
Received: 28 February 2014
Accepted: 25 October 2014
Published: 11 November 2014
In papermaking process the base paper is formed by starting from low concentration suspension. Depending on process stage different models can be utilized for better understanding and performance of the process. A short general overview of fluid dynamics related to papermaking process is given as well as new approaches including fiber network modeling and continuum mechanical fracture simulations. The fluid dynamical results presented illustrated some paper machine wet end phenomena while the solid state modeling approaches can be exploited from drying section of a paper machine to the several converting processes.
The paper is organized as follows: The second section deals with specific features of fiber suspension flows, namely fiber orientation and flocculation and how they can be modeled. The third section is devoted to the microscopic level solid mechanics of paper as a fiber network. Finally, the last section discusses the rheology of paper and its macroscopical modeling.
2 Fluid dynamics of fiber suspension flows
2.1 Fiber orientation and flocculation
Both fiber orientation and fiber flocculation are characteristic fluid dynamical phenomena of fiber suspension flows at low concentrations in the headbox and the forming section. The main purpose of a headbox is to create a thin and even jet which then enters the forming section. In the headbox, the suspension is mixed in the turbulence generator and then accelerated in the contracting slice channel. After the slice opening, the jet enters into free air and hits the moving fabrics called “wires”, where the water removal and the formation of the fiber network are initiated. Subsequent water removal processes by pressing and drying make paper more solid, but the basic fibrous structure of paper does not change noticeably.
2.2 Formulation of the fiber orientation probability distribution model
When multiphase flows are concerned, there are namely two different methods to model the particles in such a flow. The first option is the Lagrangian approach, which models the motion of individual fibers. However, as accurate as it may be, this approach is computationally very expensive and thus it is not practical to be used in such a large scale flow as takes place in the headbox. The other option is the Eulerian approach, which takes into account an undefined number of fibers and predicts their statistical behavior. The latter one is the basic idea behind the fiber orientation probability distribution (FOPD) model.
The basis of the probability distribution modeling approach is the conservation of the probability flux and the use of diffusion-convection or the Fokker-Planck equation. This approach has been widely used (e.g. [2–5]) in studying the development of fiber orientation in contracting channel flows. As a result, the fiber orientation distribution may be obtained.
The earliest studies of non-spherical particles were conducted by Jeffery already in 1922  when he derived the equations of motion for a single ellipsoid, non-Brownian particle in Newtonian fluid with simple shearing and showed that the particles execute rotational motion in periodic, closed orbits. Jeffery’s work has been the basis for current modeling and has been widely applied. The model given above is based on the assumption that the fibers are rigid. However, it can also be used in describing the orientation of flexible fibers at least from a statistical point of view as discussed in  even though it does not take into account the flexibility of the fibers. Some studies have also taken into account the fiber-fluid coupling through an orientation tensor including momentum exchange between the fiber and the fluid phases . This type of approach to model the most probable orientation state of the fibers on different flow conditions has shown to be rather convenient when studying the flows of industrial scales. Naturally, there are some assumptions e.g. about the low enough concentration and rigidity of the fibers which does not necessarily meet the conditions in reality. The latter assumption is violated in the case of papermaking, where the wood fibers are long and flexible, thus causing additional complexity such as possible flocculation for the flow. However, the commonly taken approach gives practical information about the real-life phenomena present under the flows of papermaking and is of importance in trying to improve the papermaking process.
2.3 Modeling of fiber floc dynamics by using a population balance approach
Fiber flocculation has not been widely studied due to the challenges it creates both for measuring and computational methods. The simplest models treat fiber suspension as a one phase flow ([14–16]) although the interaction of the two fibers should be taken into account, as shown in  when comparing these results to measurements . More accurate models [19, 20] use the Eulerian two-phase flow approach and are based on the population balance model also taking into account the occurrence of flocs of different sizes.
where and represent the specific break-up rate and the specific coalescence rate, respectively. For the break-up and coalescence models, the equations developed for bubble flows [21, 22] can be used, if the model parameters are adequately tuned and verified with measurements.
where is the volumetric concentration, L is the average fiber length, d is the average fiber diameter, is the pulp concentration, and ω is the fiber coarseness. The crowding factor represents the number of fibers within the rotational sphere of influence of a single fiber. It was shown in  that the internal strength of flocs is related to this factor.
to the momentum equation of the dispersed phase. In Eq. (11) and are the reference elasticity modulus and the compaction modulus, respectively, and and represent the local volume fraction of the fibrous phase and the maximum packing parameter, respectively. Now by selecting suitable value for the coefficient , the solids pressure gradient is only activated in the regions close to the maximum packing.
The following sections continue from the paper web obtained in the forming section and further dewatering in the papermaking process.
3 Paper as fibrous material
3.1 Paper physics
Stress-strain curves yield important information on materials behavior. For a typical paper grade, such as newsprint, the stress-strain curve is characterized by an initial linear region which is followed by a non-linear region. The transition from the linear to the non-linear region takes place smoothly. In engineering literature, the transition point is known as the proportional limit. The proportional limit is commonly associated with the yield limit which is the point in the stress-strain curve after which plastic or permanent strains are developed. In the non-linear region gradient of the curve decreases but is still positive. The degree of non-linearity depends on humidity conditions. In dry conditions, the curve is rather linear even after the proportional limit. However, in humid conditions clear strain hardening is observed. In this sense, paper is brittle in dry conditions and ductile in humid conditions. The stress-strain curve depends also on the rate of loading. Both the elastic modulus and tensile strength increase with the strain rate. The viscous nature of paper can also be observed from a relaxation or creep test.
Since paper is a fibrous network in which fibers are connected by hydrogen bonds, we may ask what is the influence of fiber and bond properties on the mechanical properties of a paper sheet. It is intuitively clear that the elastic modulus of paper must depend on the elastic modulus of fibers. Due to the porous structure of paper and random orientation of fibers, the modulus is less than the corresponding modulus of fibers. In a seminal paper by Cox  a network theory for the tensile behavior of paper was presented. In the network model, long, straight and non-interacting fibers were considered. In Cox’s theory, the elastic modulus of a paper sheet is given by , where is the elastic modulus of fibers and ρ and are densities of the sheet and the fibers, respectively. For highly bonded networks, Cox’s theory seems to be in reasonable agreement with the experimental observations. However, for less dense networks the modulus can be lower than predicted by Cox’s theory .
The non-linear irreversible response of a paper sheet under elongation has been attributed to the progressive disruption of inter-fiber bonds and to intra-fiber mechanisms. Rance  argued that the breaking of bonds results in increased deformation due to more localized stress concentrations. However, according to Alava , the inelasticity of paper arises from plastic deformations that take place in the fibers when microfibrils slide relative each other.
It is also still unclear whether the time-dependence of paper arises from fiber properties or from the behavior of inter-fiber bonds. In an in-depth review by Haslach , it was concluded that most likely the time-dependency is controlled by a mechanism which occurs within the inter-fiber bonds, namely the release of microcompression in the inter-fiber bonds.
Although there is currently no comprehensive theory of the inelasticity deformation of paper, there is consensus between paper physicists that the fracture strength of paper is governed by the failure of bonds. Only in highly bonded networks is the sheet strength determined by the tensile failure of fibers.
3.2 Fiber network modeling
It is obviously in the interest of papermakers to gain information on the relationship between the mechanical properties of paper and the constituent parts of paper. Extensive efforts have been made to construct realistic fiber networks and to simulate their behavior. Heyden  classifies network modeling approaches into uniform strain models, semi-analytical models and to computer simulation models. In this paper, we are interested in the latter and especially in simulation models in which the finite element method (FEM) is utilized.
The first step in the fiber network modeling is to construct a realistic network, which is a challenging task especially if 3D structures are considered. The generation begins by placing fibers randomly inside a cell  or by depositing fibers on a rectangular surface one-by-one  until the desired network density or grammage is attained. Fiber parameters such as length, orientation and curvature may vary according to a probability distribution. In the network generation, geometric intersection or closest point problems are typically addressed in order to find possible inter-fiber bonding points. Free fibers or fiber clusters which are not connected to the rest of the network are typically removed from the network in order to obtain positive definite system matrix in finite element calculations. Network connectedness can be examined with aid of graph search algorithms . The graph nodes are the intersection and boundary points and free fiber segments represent the graph edges. It should be noted that network systems are large. In an area of one square millimeter there are roughly 100-200 fibers, and every fiber has approximately 10-40 inter-fiber bonds. Furthermore, the number of the fibers increases quadratically with the area of the network. The size of the network is still the limiting factor when considering industrially important problems. We are still in the millimeter scale when dense three-dimensional networks are considered. To increase the size of the networks, efficient algorithms and coding methods are needed.
Network generation is only one part of the problem. In order to perform simulations, a finite element model of the generated network must be created. Although it is possible to model fibers with solid continuum elements , beam elements are typically used [30, 31] since they reduce the size of the discrete finite element system. Fibers are commonly assumed to follow the linearly elastic constitutive law , although inelastic models  can and should be used if extensive deformations are considered. Considering the physical behavior of paper networks, it seems that the proper modeling of inter-fiber bonding is critical in order to model paper networks realistically. For small deformations, ideal rigid bonding, as used in Cox’s theory, may be adequate. However, to model the non-linear response or failure of a paper sheet, deformable bonding models which allow bond failure should be utilized. In Heyden’s  network model, fibers are bonded together with an elastic spring, the stiffness and strength of which reduces in a stepwise manner until complete failure occurs. There has been growing interest to take into account the subsequent sliding of fibers after bond failure has occurred [31, 33]. The inter-fiber frictional sliding may be particularly important for wet networks since water breaks hydrogen bonds so that fibers can more easily slide against one another.
The resulting discrete finite element problem of fiber networks is typically very large and non-linear due to possible material and geometric non-linearity and frictional sliding. Thus, tools of high performance computing must be utilized.
3.3 A fiber network model
For illustrative purposes we consider here the problem of simulating the behavior of a fiber network in an extension test. Recently, Lavrykov et al.  have presented an alternative way to generate fiber networks. In their model, the sheet structure is obtained by compressing randomly placed fibers between two rigid plates. The compressing was simulated with the finite element method. A similar approach is used here.
The network generation procedure was implemented in ABAQUS finite element software using Python scripting. In this way, we can combine finite element model making and simulations with useful numerical tools. In the initial intersection testing and in the search for fiber-to-fiber contact Gilbert-Johnson-Keerthi (GJK) collision detection algorithm  of convex objects is utilized. For a non-convex object, the same GJK algorithm can be used to detect the collision of the convex hulls of the objects. In the contact searching, bounding volume hierarchies and hierarchy traversal collision testing are used. In a bounding volume hierarchy, a geometric object is partitioned into a tree structure. The bounding volume hierarchy of a fiber is constructed using convex hulls. The root node is the convex hull of the fiber. At each node, the fiber part is subdivided into two nearly equal size parts and the convex hulls of the parts are stored in the child nodes. This procedure is continued until we end up in a structure in which the leaf nodes contain the convex hull of a single element. The fiber-to-fiber intersection tests are conducted using their bounding volume hierarchies. In the intersection tests, both hierarchies are descended simultaneously using a depth-first approach . After all contacts have been found, the network connectedness is studied using the breadth-first search (BFS) strategy.
Fiber network properties
Fiber basis weight
Elastic modulus of fibers
Number of fibers
4 Paper as continuous material
4.1 Continuum description of paper
Although the uniaxial inelasticity of paper has been long recognized  and is currently relatively well understood [29, 37], only very recently have well calibrated multiaxial inelastic constitutive models been proposed. Classical plasticity models have been proposed by Xia et al. , Mäkelä and Östlund  and more recently by Harryson and Ristinmaa . Integral based viscoelastic constitutive models have been proposed by Uesaka et al.  and Lif . Apparently, we are still missing a model which combines the viscous and plastic properties of paper.
The continuum description of paper is widely used in modeling applications. A typical application is FEM modeling related to papermaking or to printing processes. Examples include web runnability , calendering , winding  and printing nip mechanics .
4.2 Modeling the fracture of paper
The failure model applied is a simplification and generalization of Hashin’s theory [49, 50]. Hashin’s theory was created for fiber-reinforced materials. It requires that the behavior of undamaged material is linearly elastic. It takes into account four different failure modes: fiber tension, fiber compression, matrix tension, and matrix compression. When this theory is used, two different stages of the fracture process should be defined: damage initiation and damage evolution.
In the case of paper, the linear elasticity requirement in Hashin’s theory is a substantial weakness; the stress-strain behavior of paper is commonly nonlinear and inelastic, as stated earlier. From that point of view, the failure model is a generalization of Hashin’s theory since no restrictions are imposed for material behavior. Actually, if fracture simulations are considered, the essential issue is the determination of the tensile strength (and stress-strain behavior) dependencies (see Figure 13). Whereas in Hashin’s theory four different failure modes are considered, in the failure model applied only two failure modes are considered; the failure of fibers and matrix are not separated. From that point of view, the present model is a simplification compared to Hashin’s theory. Also the damage evolution is much simpler in the present model than in Hashin’s theory: when the stress level reaches the tensile strength, the stiffness of the material is decreased to a thousandth. The failure model was implemented in the ABAQUS user subroutine USDFLD. Numerical problems were addressed by a two step stabilization process. The first step is directly controlled by the user and the second step is adaptively process controlled by ABAQUS.
This paper aimed at giving a general view on mathematical modeling in papermaking. Many modeling challenges are excluded from this paper: Fluid-Structure Interactions (FSI) for the running paper web, coating of the paper web, the rheology of fluids and solids, heat transfer during the drying process, to mention a few. However, modeling approaches presented in this paper may be used separately or together in several paper production, converting or end-use situations. For example, by fluid dynamics the variation of fiber orientation and basis weight may be predicted and by solid mechanics the importance of these factors on paper behavior can be studied. The future work concentrates to the collaboration of the presented modeling approaches, for example, stress-strain behavior used in continuum mechanical approach could be obtained by fiber network modeling.
- Hämäläinen J: Mathematics in paper. J Math Ind 2012., 2: Article ID 2 Article ID 2Google Scholar
- Olson JA: The motion of fibres in turbulent flow, stochastic simulation of isotropic homogeneous turbulence. Int J Multiph Flow 2001, 27: 2083–2103. 10.1016/S0301-9322(01)00050-7MATHView ArticleGoogle Scholar
- Hyensjö M, Dahlkild A: Study of the rotational diffusivity coefficient of fibres in planar contracting flows with varying turbulence levels. Int J Multiph Flow 2008, 34: 894–903. 10.1016/j.ijmultiphaseflow.2008.02.005View ArticleGoogle Scholar
- Krochak PJ, Olson JA, Martinez DM: The orientation of semidilute rigid fiber suspensions in a linearly contracting channel. Phys Fluids 2008., 20: Article ID 073303 Article ID 073303Google Scholar
- Niskanen H, Eloranta H, Tuomela J, Hämäläinen J: On the orientation probability distribution of flexible fibres in a contracting channel flow. Int J Multiph Flow 2011, 37: 336–345. 10.1016/j.ijmultiphaseflow.2010.11.006View ArticleGoogle Scholar
- Olson JA, Kerekes RJ: The motion of fibres in turbulent flow. J Fluid Mech 1998, 377: 47–64.MATHView ArticleGoogle Scholar
- Olson JA, Frigaard I, Chan C, Hämäläinen J: Modelling turbulent fibre suspension flowing in a planar contraction: the one-dimensional headbox. Int J Multiph Flow 2004, 30: 51–66. 10.1016/j.ijmultiphaseflow.2003.10.006MATHView ArticleGoogle Scholar
- Krochak PJ, Olson JA, Martinez DM: Near-wall estimates of the concentration and orientation distribution of a semi-dilute rigid fibre suspension in Poiseuille flow. J Fluid Mech 2010, 653: 431–462.MATHView ArticleGoogle Scholar
- Shin M, Koch DL: Rotational and translational dispersion of fibres in isotropic turbulent flows. J Fluid Mech 2005, 540: 143–173. 10.1017/S0022112005005690MATHView ArticleGoogle Scholar
- Parsheh M, Brown M, Aidun C: On the orientation of stiff fibres suspended in turbulent flow in planar contraction. J Fluid Mech 2005, 545: 245–269. 10.1017/S0022112005006968MATHView ArticleGoogle Scholar
- Phelps JH, Tucker CL III: An anisotropic rotary diffusion model for fiber orientation in short- and long-fiber thermoplastics. J Non-Newton Fluid Mech 2009, 156: 165–176. 10.1016/j.jnnfm.2008.08.002MATHView ArticleGoogle Scholar
- Jeffery GB: The motion of ellipsoidal particles immersed in a viscous fluid. Proc R Soc Lond A 1922, 102: 161–179. 10.1098/rspa.1922.0078View ArticleGoogle Scholar
- Krochak PJ, Olson JA, Martinez M: Fiber suspension flow in a tapered channel: the effect of flow/fiber coupling. Int J Multiph Flow 2009, 35: 676–688. 10.1016/j.ijmultiphaseflow.2009.03.005View ArticleGoogle Scholar
- Steen M: Turbulence and flocculation in fibre suspensions. PhD thesis. University of Trondheim, Norway; 1990. Steen M: Turbulence and flocculation in fibre suspensions. PhD thesis. University of Trondheim, Norway; 1990.Google Scholar
- Kuhn DCS, Sullivan PE: Analysis and measurement of the flocculation intensity of flowing pulp suspensions. TAPPI Papermakers Conference 2001.Google Scholar
- Plikas A, Kuhn DCS, Sullivan PE: A numerical model of fibre suspensions in a grid generated turbulent flow. Proceedings of ASME FEDSM’00 2000, 991–996.Google Scholar
- Hyensjö M: On fibre suspension flow modelling: mechanical fibre flocculation and fibre orientation. PhLic thesis. Royal Institute of Technology, Stockholm, Sweden; 2005. Hyensjö M: On fibre suspension flow modelling: mechanical fibre flocculation and fibre orientation. PhLic thesis. Royal Institute of Technology, Stockholm, Sweden; 2005.Google Scholar
- Karema H, Salmela J, Tukiainen M, Lepomäki H: Prediction of paper formation by fluidisation and reflocculation experiments. 12th Fundamental Research Symposium 2001, 559–589.Google Scholar
- Hämäläinen T, Hämäläinen J: Fibre floc evolution model, part I: flocculation in a headbox. Nord Pulp Pap Res J 2010,25(1):39–47. 10.3183/NPPRJ-2010-25-01-p039-047View ArticleGoogle Scholar
- Hämäläinen T, Hämäläinen J: Fibre floc evolution model, part II: influence of the jet-to-wire ratio to resulting fibre floc evolution in the initial drainage zone. Nord Pulp Pap Res J 2010,25(1):48–55. 10.3183/NPPRJ-2010-25-01-p048-055View ArticleGoogle Scholar
- Luo H, Svendsen HF: Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J 1996,42(5):1225–1233. 10.1002/aic.690420505View ArticleGoogle Scholar
- Prince MJ, Blanch HW: Bubble coalescence and break-up in airsparged bubble columns. AIChE J 1990,36(10):1485–1499. 10.1002/aic.690361004View ArticleGoogle Scholar
- Kerekes RJ, Schell CJ: Characterization of fibre flocculation regimes by a crowding factor. J Pulp Pap Sci 1992,18(1):32–38.Google Scholar
- Gidaspow D: Multiphase Flow and Fluidization. Academic Press, London; 1994.MATHGoogle Scholar
- Cox HL: The elasticity and strength of paper and other fibrous materials. Br J Appl Phys 1952, 3: 72–79. 10.1088/0508-3443/3/3/302View ArticleGoogle Scholar
- Sampson WW: Materials properties of paper as influenced by its fibrous architecture. Int Mater Rev 2009, 54: 134–156. 10.1179/174328009X411154View ArticleGoogle Scholar
- Rance HF: The formulation of methods and objectives appropriate to the rheological study of paper. Tappi J 1956, 2: 104–115.Google Scholar
- Alava M, Niskanen K: The physics of paper. Rep Prog Phys 2006, 69: 669–723. 10.1088/0034-4885/69/3/R03View ArticleGoogle Scholar
- Haslach HW Jr.: The moisture and rate-dependent mechanical properties of paper: a review. Mech Time-Depend Mater 2000, 4: 169–210. 10.1023/A:1009833415827View ArticleGoogle Scholar
- Heyden S: Network modelling for the evaluation of mechanical properties of cellulose fluff. PhD thesis. Lund University, Department of Mechanics and Materials; 2000. Heyden S: Network modelling for the evaluation of mechanical properties of cellulose fluff. PhD thesis. Lund University, Department of Mechanics and Materials; 2000.Google Scholar
- Kulachenko A, Uesaka T: Direct simulations of fiber network deformation and failure. Mech Mater 2012, 51: 1–14.View ArticleGoogle Scholar
- Lavrykov S, Lindström SB, Singh KM, Ramarao BV: 3D network simulations of paper structure. Nord Pulp Pap Res J 2012, 27: 256–263. 10.3183/NPPRJ-2012-27-02-p256-263View ArticleGoogle Scholar
- Wilbrink DV, Beex LAA, Peerlings RHJ: A discrete network model for bond failure and frictional sliding in fibrous materials. Int J Solids Struct 2013, 50: 1354–1363. 10.1016/j.ijsolstr.2013.01.012View ArticleGoogle Scholar
- Gilbert EG, Johnson DW, Keerthi SS: A fast procedure for computing the distance between complex objects in three-dimensional space. IEEE J Robot Autom 1988, 4: 193–203. 10.1109/56.2083View ArticleGoogle Scholar
- Mukundan R: Advanced Methods in Computer Graphics: With Examples in OpenGL. Springer, London; 2012.View ArticleGoogle Scholar
- Steenberg B: Paper as a visco-elastic body. Sven Papperstidn 1947, 50: 127–140.Google Scholar
- Coffin DW: The creep response of paper. 13th Fundamental Research Symposium 2005, 651–747.Google Scholar
- Xia QS, Boyce MC, Parks DM: A constitutive model for the anisotropic elastic-plastic deformation of paper and paperboard. Int J Solids Struct 2002, 39: 4053–4071. 10.1016/S0020-7683(02)00238-XMATHView ArticleGoogle Scholar
- Mäkelä P, Östlund S: Orthotropic elastic-plastic material model for paper materials. Int J Solids Struct 2003, 40: 5599–5620. 10.1016/S0020-7683(03)00318-4MATHView ArticleGoogle Scholar
- Harryson A, Ristinmaa M: Large strain elasto-plastic model of paper and corrugated board. Int J Solids Struct 2008, 45: 3334–3352. 10.1016/j.ijsolstr.2008.01.031View ArticleGoogle Scholar
- Uesaka T, Kodaja I, Okushima S, Fukuchi R: History-dependent dimensional stability of paper. Rheol Acta 1989, 28: 238–245. 10.1007/BF01332856View ArticleGoogle Scholar
- Lif J: In-plane hygro-viscoelasticity of paper at small deformations. Nord Pulp Pap Res J 2005, 20: 139–149. 10.3183/NPPRJ-2005-20-02-p139-149View ArticleGoogle Scholar
- Lif J: Hygro-viscoelastic stress analysis in paper web offset printing. Finite Elem Anal Des 2006, 42: 341–366. 10.1016/j.finel.2005.08.002View ArticleGoogle Scholar
- Wikström M: Finite element modelling of calendering - some aspects of the effects of temperature gradients and structure inhomogeneities. J Mater Sci 1996, 31: 3159–3166. 10.1007/BF00354662View ArticleGoogle Scholar
- Ärölä K, von Hertzen R: An elastoplastic continuum model for a wound roll with interlayer slippage. Finite Elem Anal Des 2006, 42: 503–517. 10.1016/j.finel.2005.10.005View ArticleGoogle Scholar
- Kariniemi M, Parola M, Kulachenko A, Sorvari J, von Hertzen L: Effect of blanket properties on web tension in offset printing. Advances in Printing and Media Technology 2010.Google Scholar
- Lipponen P, Erkkilä A-L, Leppänen T, Hämäläinen J: On the importance of in-plane shrinkage and through-thickness moisture gradient during drying on cockling and curling phenomena. 14th Pulp and Paper Fundamental Research Symposium 2009, 389–436.Google Scholar
- Erkkilä A-L, Leppänen T, Hämäläinen J: Empirical plasticity models applied for paper sheets having different anisotropy and dry solids content levels. Int J Solids Struct 2013, 50: 2151–2179. 10.1016/j.ijsolstr.2013.03.004View ArticleGoogle Scholar
- Hashin Z, Rotem A: A fatigue criterion for fiber-reinforced materials. J Compos Mater 1973, 7: 448–464. 10.1177/002199837300700404View ArticleGoogle Scholar
- Hashin Z: Failure criteria for unidirectional fiber composites. J Appl Mech 1980, 47: 329–334. 10.1115/1.3153664View ArticleGoogle Scholar
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