Micro-mechanical mechanisms for deformation in polymer-material structures

Jessica Strömbro

Doctoral thesis no. 68, 2008

KTH School of Engineering Sciences

Department of Solid Mechanics Royal Institute of Technology SE-100 44 Stockholm, Sweden

TRITA HFL-0442 ISSN 1654-1472 ISRN KTH/HFL/R-08/02-SE

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknisk doktorsexamen fredagen den 22 februari kl 10.15 i sal F3, Kungliga Tekniska Högskolan, Stockholm.

Preface The work presented in this doctoral thesis was carried out during my time as a graduate student at the department of Solid Mechanics, Royal Institute of Technology, Stockholm, Sweden, between January 2003 and January 2008. The project has partly been funded by Biofibre Materials Centre (BiMaC) and the Paper Mechanics cluster at STFI-Packforsk, which is greatly acknowledged. First of all I would like to thank my supervisor Professor Peter Gudmundson for giving me the opportunity to become a graduate student at the department. I would also like to express my gratitude for your excellent guidance and advices, but especially for knowing the fine line between giving enough directions and letting me work independent and evolve. Further thanks to my assistant supervisor during the first years, assistant Professor Kristofer Gamstedt, for sharing your knowledge on experimental work and for improving the papers. Thanks to Messrs. Bertil Dolk, Johan Wikström and Bengt Möllerberg, for manufacturing the moulds to the specimens used in the two experimental series when I examined length-scale effects in polymers, presented in Paper A and B, and for building up the experimental setup used in Paper B, but also for contributing with good ideas of how to solve experimental problems. I would also like to thank Tech. Lic. Petri Mäkelä at STFI-Packforsk for providing experimental results for my micro-mechanical paper model presented in Papers C and D. Thanks to Tech. Lic. Petri Mäkelä and Dr. Johan Alfthan at STFI-Packforsk for valuable discussions during my work on the micro-mechanical paper model. I would also like to express my gratitude to my colleagues at KTH Solid Mechanics for making the workplace enjoyable. Finally I would like to thank my family for your unconditional love and encouragement. A special thanks to my mother, Maria Agde, for being a great parent all my life. You have always supported me and given me the confidence I needed to go my own way and do what I wanted to. You always listen and give great advices. My biggest thanks go to my husband, Jörgen Strömbro, for your great love, understanding and support; it is wonderful to have you in my life. I am also grateful for your excellent graphics knowledge and for your help with many of the figures in the papers. I would also like to throw in a small thanks to my loving ferrets, Gaia, Nova, Leo, Nemo, Grim, Liot, Elric and Enso, for lighting up my days by waiting for me when I get home, sleeping in my arms, showing affection and doing all kinds of fun tricks that makes me laugh.

Stockholm, January 2008 Jessica Strömbro

List of appended papers This thesis consists of an introduction and the following appended papers: Paper A: Agde Tjernlund*, J., Gamstedt, E.K., Xu, Z.-H., 2004. Influence of

molecular weight on strain-gradient yielding in polystyrene. Polymer Engineering and Science 44 (10), 1987-1997.

Paper B: Agde Tjernlund*, J., Gamstedt, E.K., Gudmundson, P., 2005. Length-

scale effects on damage development in tensile loading of glass-sphere filled epoxy. International Journal of Solids and Structures 43 (24), 7337-7357.

Paper C: Strömbro*, J., Gudmundson, P., 2008. Mechano-sorptive creep under

compressive loading – a micromechanical model. International Journal of Solids and Structures, doi:10.1016/j.ijsolstr.2007.12.002.

Paper D: Strömbro*, J., Gudmundson, P., 2008. An anisotropic fibre-network

model for mechano-sorptive creep in paper. Report 441/1, KTH Solid Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden. Submitted for international publication.

In addition to this thesis, the work has resulted in the following publication: Agde Tjernlund*, J., Gamstedt, E.K., Gudmundson, P. An experimental investigation of the relationship between molecular structure and length scales in inelastic deformation of an amorphous thermoplastic. Presented at the IUTAM Symposium on “Size effects on material and structural behavior at micron- and nano-scales”, Hong Kong, 2004. To appear in the IUTAM book series, Kluwer Academic Publishers. * The author has during the work to this thesis changed last name from Agde Tjernlund to Strömbro.

Abstract In this thesis, the focus has been on micro-mechanical mechanisms in polymer-based materials and structures. The first part of the thesis treats length-scale effects on polymer materials. Experiments have showed that the smaller the specimen, the stronger is the material. The length-scale effect was examined experimentally in two different polymers materials, polystyrene and epoxy. First micro-indentations to various depths were made on polystyrene. The experiments showed that length-scale effects in inelastic deformations exist in polystyrene. It was also possible to show a connection between the experimental findings and the molecular length. The second experimental study was performed on glass-sphere filled epoxy, where the damage development for tensile loading was investigated. It could be showed that the debond stresses increased with decreasing sphere diameter. The debonding grew along the interface and eventually these cracks kinked out into the matrix. It was found that the length to diameter ratio of the matrix cracks increased with increasing diameter. The experimental findings may be explained by a length-scale effect in the yield process which depends on the strain gradients. The second part of the thesis treats mechano-sorptive creep in paper, i.e. the acceleration of creep by moisture content changes. Paper can be seen as a polymer based composite that consists of a network of wood fibres, which in its turn are natural polymer composites. A simplified network model for mechano-sorptive creep has been developed. It is assumed that the anisotropic hygroexpansion of the fibres leads to large stresses at the fibre-fibre bonds when the moisture content changes. The resulting stress state will accelerate creep if the fibre material obeys a constitutive law that is non-linear in stress. Fibre kinks are included in order to capture experimental observations of larger mechano-sorptive creep effects in compression than in tension. Furthermore, moisture dependent material parameters and anisotropy are taken into account. Theoretical predictions based on the developed model are compared to experimental results for anisotropic paper both under tensile and compressive loading at varying moisture content. The important features in the experiments are captured by the model. Different kinds of drying conditions have also been examined. Keywords: Length-scale effects; Strain gradient; Polymer; Micro-indentation; Particle composite; Interfacial debonding; Matrix cracking; Paper; Mechano-sorptive creep; Accelerated creep; Moisture change; Humidity change; Network model; Fibre model; Mathematical model

Contents Introduction............................................................................................................... 1 Length-scale effects in polymer materials.................................................................. 2

Polymer materials.................................................................................................. 2 Length-scale effects............................................................................................... 3 Suggestions for future research.............................................................................. 4

Mechano-sorptive creep in paper and other materials ................................................ 6 Observations of mechano-sorptive creep................................................................ 6

Paper................................................................................................................. 7 Other materials................................................................................................ 12 Differences between tensile and compressive loading ...................................... 17

Models for mechano-sorptive creep ..................................................................... 18 Empirical models............................................................................................. 18 Physically based models .................................................................................. 24

Suggestions for future research............................................................................ 31 Summary of appended papers.................................................................................. 32

Paper A: Influence of molecular weight on strain-gradient yielding in polystyrene............................................................................................................................ 32 Paper B: Length-scale effects on damage development in tensile loading of glass-sphere filled epoxy .............................................................................................. 33 Paper C: Mechano-sorptive creep under compressive loading – a micromechanical model .................................................................................................................. 34 Paper D: An anisotropic fibre-network model for mechano-sorptive creep in paper............................................................................................................................ 35

References............................................................................................................... 36

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Introduction The first step in examining a new phenomenon in a material is to experimentally quantify and observe the effect. Later a model of the phenomenon can be developed from the experimental findings, hopefully a physically based model and not only a phenomenological model. In all constitutive modelling, an appropriate length scale must be chosen. From the molecular scale to micro scale, for example molecular chains, fibrils and fibres and gradually up to product size, for example electronic components and boxes. In this work the focus has been on the micro scale and micro-mechanical mechanisms in polymer-based materials and structures. Length-scale effects in yielding and fracture of metallic materials have been studied quite extensively. Experiments show that the smaller the specimen, the stronger is the material. Length-scale effects in yielding of polymers, as well as in related fracture and damage processes, have not been studied to any great extent although the phenomenon has been observed also in these materials. Polymers of different kinds are used in applications such as electronic components, micro-electromechanical systems (MEMS), composite materials and thin films, where large strain gradients can be present. The set of underlying mechanisms for length-scale effects in various polymers is still unknown. In the first part of this work, together with Paper A and Paper B, the length-scale effect in polymers is examined. Packages of paper often have to withstand loads for a long time, but in addition to that the humidity often varies, during for example transport and storing. The time dependent deformation, creep, must therefore be considered, but also the fact that the creep is dependent on the moisture content, and thereby also the humidity. In addition, creep is accelerated by varying humidity, a phenomenon that is called mechano-sorptive creep or accelerated creep. In the second part of this work, together with Paper C and Paper D, the mechano-sorptive creep is examined with special interest to paper. Paper usually consists of a network of hydrogen bonded wood fibres. Wood fibres are natural polymer composite fibres, consisting of cellulose microfibrils and a surrounding hemicellulose-lignin matrix. Mechano-sorptive creep in paper has been studied since the 1970’s, but so far no generally excepted model has been developed. In this study both experiments and models are considered, and in Paper C and Paper D a micromechanical model, i.e. a model based on the fibre properties, is developed.

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Length-scale effects in polymer materials Experimental observations have indicated that the geometrical size of components has an influence on the inelastic behaviour of polymers, as well as in metals e.g. [1] and ceramics e.g. [2]. There are indications that the smaller the specimen, the stronger is the material. Length-scale effects in yielding and fracture of metallic materials have been studied quite extensively. This field of research has been driven by miniaturization of e.g. electronic devices. Another driving force is the need for understanding of microstructure-property relations in composite materials and micro-electromechanical systems (MEMS) in the presence of large strain gradients. Polymers of different kinds are also used in applications such as electronic components, MEMS, composite materials and thin films. Despite the widespread use of polymers in these applications, the length-scale effect in yielding of polymers, as well as in related fracture and damage processes, has not been studied to any great extent. The set of underlying mechanisms for length-scale effects in various polymers is still unknown. Due to lack of knowledge, the higher strength at small scales can not be fully utilized without extensive component testing. More experiments and more realistic models for deformations in small scale components could open possibilities for improved design. In crystalline metallic materials, geometrically necessary dislocations are of importance for the length-scale effect. The density of geometrically necessary dislocations is determined from gradients of the plastic strains, which introduce a length scale in a natural way [3]. Microstructural dimensions like typical distances between dislocations and sizes of dislocation cells are also of importance [4]. Several classical experiments on different metallic materials have confirmed that the plastic yield stress increases with decreasing structural dimensions [1, 3, 5]. The length-scale effect is also qualitatively understood in ceramics. For example, when ceramics deform inelastically during indentation, shear bands consisting of cracks form. The typical distance between these shear bands is the controlling length-scale parameter [6, 7]. Polymer materials Polymers can be divided into two groups, thermosets and thermoplastics. The structure of thermosets is a densely cross-linked network of polymer chains (see Fig. 1a), while thermoplastics are either amorphous (see Fig. 1b) or semicrystalline (see Fig. 1c), which essentially means a composite of stiffer crystallites mixed with the more compliant amorphous phase. Thermosets are generally relatively homogenous and isotropic due to a high and uniform cross-link density, whereas thermoplastics can show a considerable anisotropy due to molecular orientation. The only apparent structural length scale for an amorphous thermoplastic is the molecular length, whereas for a thermoset it is the average distance between adjacent cross-links. Lam and Chong [8] have shown that a length-scale parameter associated with strain-gradient yielding is influenced by the cross-link distance in epoxies.

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(a) (b) (c) Fig. 1. Schematic illustration of (a) a cross-linked thermoset, (b) an amorphous thermoplastic, (c) a semicrystalline thermoplastic. Length-scale effects As for metals, most experimental studies on strain-gradient yielding in polymers have been conducted by micro-indentation. Experiments on both thermosets and thermoplastics [9-12] have shown that the hardness evaluated from shallow indents is larger than that resulting from a deeper one. This is due to a higher plastic yield stress in the presence of larger strain gradients. This method has generally, for simplicity reasons, been preferred in comparison to other test methods for quantification of length-scale effects. Other test methods than indentation at a free surface would however be desirable for independent quantifications of length-scale effects. Particle-filled polymers constitute a material of widespread use. Fillers of various dimensions are frequently added to make polymers stiffer, cheaper and improve on properties such as heat resistance, dimensional stability, fracture toughness and colour control. If a suitable interfacial adhesion can be assured and if strengthening length-scale effects can be invoked from small particles, a stronger material could be produced. The yielding process in the matrix is known to precede other damage processes in polymer-matrix composites. Debonding, matrix cracking, crazing and cavitation are other damage mechanisms that can set in prior to ultimate failure. These mechanisms may also show length-scale effects. For instance, Zhuk et al. [13] showed that the debonding stress increases for decreasing glass particle size in polyethylene and polypropylene matrices and Pukánszky et al. [14] found the same behaviour for polypropylene filled with calcium carbonate particles. Previous works have also shown that finer particles (but the same volume fraction particles) can lead to improvements in strength properties, see Fig. 2. Mitsuishi et al. [15] and Massam and Pinnavaia [16] showed that the yield stress increases if the particles have smaller dimensions in polypropylene filled with calcium carbonate and epoxy-clay nanocomposites. Indications of the same behaviour with increasing yield stress for decreasing particle size have also been observed in polyester filled with glass spheres [17]. The tensile strength in solid- and hollow-glass filled rigid polyurethane composites [18] and in amine-epoxy-layered silicate nanocomposites [19] have been shown to increase for composites with smaller filler sizes. Also the strain to failure in polypropylene with glass beads [20] showed the same trend.

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Fig. 2. Schematic illustration of a stress-strain curve for a particle-filled polymer material. The experimental observations with increasing yield stress, tensile strength and strain to failure for decreasing particle size at constant volume fraction can be seen in the figure.

Zhandarov et al. [21] measured increasing interfacial fracture toughness in pull-out of glass fibres from polyamide matrices with decreasing fibre diameters. A possible explanation for the experimental results above is a size effect. Since the particle/fibre diameters were small, the plastic deformation in the matrix occurred in a small volume with large strain gradients. Several experiments have indirectly shown that there is a length-scale effect by measuring yield stresses for thin polymer films [22] or interfacial shear strength in single-fibre composites with polymer matrix [23-25] that were higher than the bulk values. These findings promote understanding that could be used in materials development to suppress deleterious damage mechanism by e.g. controlling the size distribution of reinforcing particles and interfacial adhesion. Further insight into the damage mechanisms and their length-scale dependence is however necessary. Suggestions for future research To be able to make use of the advantages with length-scale effects, for example in improved design, these effects have to be better understood by further examination. The first step is to do more experimental work with different polymer materials to quantify the effect. The underlying mechanisms for length-scale effects in various polymers have to be examined. A key question is: What controls the effect in different polymer materials at the molecular level? The time and temperature dependence of the length-scale effect could be useful to examine, since these dependencies are

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known to be significant for polymers. Realistic models that describe the phenomenon, preferably mechanism based, should be developed. More experimental work could be necessary to calibrate parameters of the models. Development and use of different and complementing characterization methods to quantify the length scale effects would be valuable in the formulation of general models. The models could be implemented with finite element method (FEM) to predict the behaviour in arbitrary geometries and loading cases.

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Mechano-sorptive creep in paper and other materials The response of paper to mechanical load is time-dependent, i.e. the paper creeps, and even moderate stress levels lead to non-elastic deformation. The creep is affected by humidity conditions. High humidity, and therefore high moisture content in the paper, means faster creep compared to low humidity [26-29]. Moreover, creep is accelerated by varying humidity, so that the creep during cycling between low and high humidity is likely to exceed the creep at constant humidity even at the highest level, a phenomenon known as mechano-sorptive creep or accelerated creep [27-32]. Strength properties of paper are not affected as much by change in temperature as by change in relative humidity [33]. It has also been observed that the mechano-sorptive effects are larger in compression than in tension. Mechano-sorptive creep is an important time-dependent behaviour which creates practical problems. Packages of paper often have to withstand loads for long times, and therefore creep is an important factor in package design. In addition, in storage or transportation of corrugated board boxes or other packages of paper, the relative humidity tends to vary during time, causing collapses. Mechano-sorptive creep makes it difficult to predict deformation and life-time of packages stored under varying humidity conditions. Accelerated creep can also cause problems with the control of paper-making machines. Mechano-sorptive creep was independently discovered for wool [34, 35] and wood [36, 37] in the late 1950s and early 1960s. A decade later accelerated creep was discovered in paper [30, 31]. A similar behaviour has also been found in concrete [38] and in the 1990s mechano-sorptive creep was found in some synthetic fibres [39-41]. The most important load cases in packages are compression and bending. There are many experimental observations that indicate a different behaviour in compression in comparison to tension [27, 28, 30, 31]. Observations of mechano-sorptive creep Mechano-sorptive creep have been observed in many different kinds of materials; in paper and corrugated board, wood and wood based materials, for example hardboard, some synthetic fibres such as aramid fibres, in polymers, for example water-blown polyurethane foam, and in concrete. Mechano-sorptive creep is not limited to cellulose and cellulose-based composites, although most experimental studies have been on paper and wood. A common property of these materials is that they are hygroscopic materials and that they contain hydrogen bonds. Mechano-sorptive behaviour has mostly been studied and found in tension, compression and bending, but it has also been observed in shear

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Paper Mechano-sorptive creep in paper was first reported in 1972 by Byrd [30, 31]. He conducted tensile creep tests on handsheet papers exposed to constant and changing humidity conditions [30]. Later the same year Byrd also examined compressive creep response of corrugated boards when exposed to edgewise loading, in both cyclic and constant humidity [31]. He found that the specimens under cyclic humidity conditions have much larger creep rates and shorter lifetimes than those under constant humidity conditions, see Fig. 3. As the stress level increased, the difference between creep response in cyclic and constant humidity increased. In 1978 Byrd and Koning [42] conducted more edgewise compression creep tests on corrugated fibreboard specimens. Creep rates for cyclic humidity, 90 – 35 % RH, were 3-14 times higher than at constant humidity, 90 % RH. They also observed slightly higher creep rates for specimens exposed to 24 – hr cycles than for specimens exposed to 3 – hr cycles, when the same load and total time is considered. Leake and Wojcik [43] has examined the influence of humidity cycling rates on the lifetime of corrugated containers. They showed that boxes subjected to the shortest cycle time have the longest average lifetime and vice versa. Furthermore, they concluded that rapid cycles result in smaller moisture changes, and therefore longer lifetimes. It was found that boxes subjected to a constant environment have significantly longer lifetimes than those subjected to cyclic conditions.

(a)

(b)

Fig. 3. Creep response of paper under tensile loading in cyclic and constant humidity in (a) tension [30] and (b) compression [31]. Back et al. [44, 45] made tensile experiments on sack paper, and found a reduction in the tensile stiffness during sorption compared with the properties at the same moisture content under equilibrium conditions. Unlike Byrd and Koning [42] they found that the creep increases during moisture changes may be reduced by reducing the sorption rate. Gunderson and Tobey [46], on the other hand, who carried out creep tests in cyclic relative humidity on paperboard in tension, found that the mechano-sorptive creep was independent of the rate at which humidity changes. They compared

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different cyclic humidity conditions, and found that the deformation after a given number of cycles was almost the same, independent of the ramp time and hold time, see Fig. 4 (a). However, this implies that rapid cycling, i.e. short cycle periods, causes an earlier specimen failure because of more creep events occurs per interval time. They also found that if the upper limit of the relative humidity was held constant, creep is worsened by trying to reduce the mean relative humidity, i.e. by lowering the lower limit, see Fig. 4 (b). They concluded that the creep is a function of the magnitude of the moisture change.

(a)

(b)

Fig. 4. Mechano-sorptive creep curves for (a) cycles with different ramp and hold times and (b) different relative humidity minimum [46]. Padanyi [47, 48] examined linerboard, i.e. the face sheets used in corrugated boards, by using dynamic mechanical analysis (DMA). In DMA a cyclic varying stress is applied to the test sample. For a viscoelastic material the strain response lags behind the stress by some angle �, and a loss coefficient or loss factor is defined as tan(�). Padanyi found transient increase in the loss factor every time the moisture was changed, which is equivalent to an increase in creep compliance. Other researchers have showed similar experimental results in paper and wood [49-52]. Salmén and Fellers [52] claimed however that moisture-induced transient effects do not reflect a mechano-sorptive creep phenomenon. They drew this conclusion from experimental results on paper and nylon, where they found transient increases of the loss factor in both materials, even though nylon does not exhibit mechano-sorptive creep. They raised the question to whether this kind of testing truly reflects the material properties. This subject has been discussed in a number of papers [53, 54]. Haslach et al. [29, 55] examined mechano-sorptive creep in paper loaded in tension and tried to divide the creep into an elastic part, a creep part and a hygroexpansive part. If the suggested separation was used the hygroexpansion appeared to be dependent on the load, and not only on the relative humidity. Söremark et al. [27, 28] also found that the load changes the hygroexpansion. They got more hygroexpansion in compression than in tension. Urbanik [56] has examined mechano-sorptive creep in corrugated board in compression, and he also tried to divide deformation into a

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hygroexpansion part and a mechano-sorptive creep. Clearly there is no straightforward method to separate the different parts. Söremark et al. [27, 28] carried out creep tests on corrugated board under cyclic humidity, loaded in bending, see Fig. 5. It can be noted that the deflection increased when the moisture decreased, for example between point C and D, and the deflection decreased when the moisture increased, for example between point D and E. Creep tests on the linerboard in tension and compression were also carried out by substituting on of the linerboards with steel ribbons. It was found that the creep rate was higher in compression than in tension. They also found that a freely dried paper exhibits more mechano-sorptive creep than a paper dried under restraint.

Fig. 5. Creep curves for corrugated board in bending at 40 % RH, 80 % RH and during humidity cycling [27].

Considine et al. [57] examined the mechano-sorptive behaviour of two paperboards loaded with an edgewise compressive load. They found that cyclic humidities caused faster creep rates, extensional stiffness loss, greater failure strain and earlier failures. Later Considine et al. [58] found that the compressive strength, failure strain, stiffness and energy absorption do not predict compressive creep performance of corrugated board components in cyclic humidity. However, it is claimed that the hygroexpansive strain does provide adequate prediction of creep performance. In the 1990´s Gunderson and Laufenberg [59] examined mechano-sorptive creep in compressive loading of corrugated board and found among other things that the mechano-sorptive creep differed between two corrugated boards even though the specimens had the same strength at standard conditions. Leake and Wojcik [60, 61] showed a similar behaviour in small corrugated board samples and in corrugated containers. Panek et al. [62] found that data from just three humidity cycles can be used to predict the strain after many cycles. Urbanik and Lee [63] examined the influence of various RH cyclic periods on the creep of corrugated fibreboard. They performed experiments on corrugated tubes loaded under static compression and exposed to swept sine RH schedules in which frequency varied logarithmically with time. They found that increasing RH periods

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resulted in increasing creep rate and increasing hygroexpansion amplitude. They claimed that high frequencies does not allow enough time for moisture to diffuse and build up in the material, and if the humidity is cycled faster than the ability for moisture to diffuse in and out of the material the creep can terminate. They also stated that since a constant RH can be seen as a cyclic RH with an infinite cycle period, there should be a long cycle period beyond which the creep rate would decrease. Coffin et al. [64-66] also examined the influence of humidity cycle period. They performed experiments with tensile load and varying humidity on paper, where they varied the cycle time. In contrast to results by Urbanik and Lee [63], they found that if the cycle time was increased the accelerated creep decreased. They concluded that this may be because moisture gradients and stress concentrations will be established for only small portions of the total creep time. But they also stated that if the cycle time is shorter than the sorption time, equilibrium will not be achieved and most likely no accelerated creep either. They thought that there is an optimum cycle time that would maximize the mechano-sorptive effect; this optimum appeared to be when the moisture cycle is of the order of sorption time. This is in accordance with the assumption by Urbanik and Lee [63] that there should be a long cycle period beyond which the creep rate would decrease. This might also explain why quit many contradictory results have been presented whether increased cycle time gives increased or decreased accelerated creep [42, 44-46, 63-67]. Coffin and Habeger [64] also carried out tensile creep experiments on paper where they cycled the relative humidity between 10 % RH and 50 % RH. They drew the conclusion that accelerated creep does not require a certain level of moisture to be initiated. Coffin and Habeger [65] also found that more hygroexpansive paper sheets exhibit more accelerated creep. Experiments also showed that heterogeneity due to blending of different fibre types does not produce a noticeable influence on accelerated creep. This observation supports the idea that the major part of the influence of fibre heterogeneity comes from fibre anisotropy rather than from fibre variety. Olsson and Salmén [68] studied mechano-sorptive creep in paper from a molecular point of view. They carried out experiments using IR-spectra to determine molecular changes occurring with time. They found that different parts of the molecules were activated during moisture cycling compared to constant moisture. Demaio and Patterson [69] has performed experiments on paper with different bond strength between the fibres. The results showed that, as long as there is a sufficient bonding between the fibres, changing bonding does not influence constant humidity creep or mechano-sorptive creep. It also showed that changes in bonding and bond structure are not contributors to mechano-sorptive creep. However, bonds are believed to be necessary for mechano-sorptive creep since experiments have shown that neither pure cellulose (cellophane, rayon) [29, 54, 70] nor pulp fibres [71, 72] exhibits a mechano-sorptive creep behaviour under the same conditions as paper and wood. Coffin and Boese [72] conducted creep tests in tension under cyclic humidity of single fibres and hand sheets made of made of fibres from the same source. They found that the single fibres did not exhibit mechano-sorptive creep, while the hand sheets showed substantial accelerated creep, see Fig. 6. There have however been experimental findings that show accelerated creep in single fibres, if the moisture cycle time and the sorption time are close, so that moisture gradients will rise in the

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specimen at each sorption, for example in cellulose fibres (ramie fibres, lyocell fibres) [66] and pure cellulose (cellophane) [32, 64, 66, 73], see Fig. 7. Recent experiments by Olsson et al. [74] also showed mechano-sorptive creep behaviour in single wood fibres. It might also here be the case that the rate of change in moisture content is related to the rate of sorption, i.e. moisture gradient driven accelerated creep, but that has not been investigated.

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(b)

Fig. 6. Creep curves for (a) fibres and (b) hand sheets under first constant and then varying humidity [72]. The ratios of the curve slopes during the two phases are used to quantify the possible acceleration of creep.

Fig. 7. Accelerated creep for a lyocell fibre [66].

Mechano-sorptive creep in packaging loaded in compression has also been found [60, 61, 75, 76]. Experiments by Byrd have showed that the creep of a corrugated board is 2-5 times faster than indicated by the measurements of the constituents (linerboard and corrugating medium/fluting) [76]. This indicates that the increased creep rate under cyclic humidity of corrugated board not only relates to the material characteristics of the constituents but also to other factors, such as panel micro failure,

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the way the specimens are supported and the use of water-soluble adhesives in manufacturing of the corrugated board. Later on Byrd [77] conducted an investigation on corrugated fibreboards in order to better understand the influence of the adhesive on mechano-sorptive creep. It was found that the sensitivity to moisture of the adhesive severely affects the ability of the corrugated fibreboard to resist creep deformation in a changing humidity. A water-sensitive adhesive absorbs water in a cyclic relative humidity environment, triggering additional deformation in the corrugated board. Corrugated board made with water-resistant adhesive resists cyclic RH creep deformation better and it was suggested that the bonding was responsible for the increased creep rate compared to the constituents showed in the previous study [76]. Other materials Wood was one of the first materials for which mechano-sorptive creep was observed, and it has been extensively studied since that. In the beginning of the 1960s Armstrong and Kingston [36] reported the effect in wood. During previous studies on creep in wood under various types of loading they had found that changes in the moisture content under load markedly influenced both creep rate and total creep. To explore this further they studied creep in bending of small wooden beams at three different moisture conditions, namely, never dried wood at constant moisture content, never dried wood allowed to dry during the test and dried wood at constant moisture content. They found that the creep in a wood specimen allowed to dry during loading was at least twice that of specimen kept at constant moisture content.

Fig. 8. Effect of changes in moisture content on bending of wood specimens under loading [37].

This observation was followed by a more extensive investigation carried out by Armstrong and Christensen [37]. In this study bending of both very small and large wooden beams were studied during cyclic changes in moisture content between a

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completely dry state and a wet state. If the specimens were loaded in the dry state very little creep occurred, after the initial elastic deformation, see Fig. 8. During the first adsorption the deflection increased rapidly, see A-B in Fig. 8. The deflection increased further during the following desorption, see B-C in Fig. 8. However, during the subsequent absorptions the deflection decreased, while the deflection increased during the following desorptions, see C-D in Fig. 8. The final deflection was several times larger than the initial deflection, see Fig. 8. If the deformed specimen was unloaded in the dry condition, little recovery other than the initial elastic response occurred, see D-E in Fig. 8. However, if the moisture content was increased to the wet condition after the unloading, a large recovery occurred; see E-F in Fig. 8. Further moisture cycling after unloading slightly enhanced the recovery; see F-G in Fig. 8. They also discovered that the sorption rate did not affect final deflection. However, if the change in moisture content was made smaller, the total deflection decreased, but the principal behaviour remained.

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(b)

Fig. 9. (a) Deflection ratio – time curve and moisture content – time curve for beams subjected to moisture content cycling and bending [78]. (b) Compressive strain ratio – time curves and moisture content – time curve for specimens subjected to moisture content cycling and compression [78].

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Armstrong and Kingston [78] continued their work with bending of wooden beams (both hardwood and softwood), but also added tensile and compressive tests. They found that the effect of moisture content changes was most pronounced in wood under compressive stress. The total fractional deformations (total deformations as fraction of initial deformations) in compression were greater than those in bending which, in turn, were greater than those in tension. They found that the creep would increase both under adsorption and desorption. Their studies confirmed that the total change in deformation is dependent on the range of the moisture content change, but unaffected by the rate of change. When cycling the moisture content, the final deformation at the end of each cycle showed an increased value, see Fig. 9. When subjected to compression the compressive strain increased during desorption and decreased during adsorption, while it was the other way around in tension. They also found compression failures in the beams subjected to cyclic humidity, which did not occur at constant humidity.

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Fig. 10. Deflection as a function of time from bending tests (a) and shear tests (b), with constant and varying relative humidity [85].

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Many experimental studies on mechano-sorptive creep in wood have been carried out and reported in the literature, since it was discovered and first reported by Armstrong and co-workers. Mechano-sorptive effects has also been found in wood-based materials, or processed wood, such as particleboard [79, 80], hardboard [80-82], plywood [83], flakeboard [83] and chipboard [84]. Hearmon and Paton [85] investigated mechano-sorptive creep in bending and shear. In both cases there is a marked increase in deflection as a result of humidity cycling, see Fig. 10. The curve in shear differs a bit from that obtained in bending, the adsorption areas is not so clearly defined, and is preceded by a short period in which the deflection increases. In bending they found a linear relationship between the moisture content change and the additional deflection caused by moisture variations. In addition they found that their specimens broke at lower loads if the humidity was cycled. The increase in deformation during each humidity cycle decreases as long as no failure takes place [85, 86-88]. Hunt and Shelton [86-88] claimed that the deformation eventually will reach a limiting value, a creep limit. They performed testes in bending, tension and compression. Experiments by Armstrong [89] showed that a constant moisture gradient does not produce mechano-sorptive creep, i.e. a moisture content change is needed. The moisture gradient was obtained by using a steady state flow of water through the hollow core of a wood specimen, in that way producing a different humidity on the inside and the outside of the specimen. A compressive load was applied to the specimen during testing. Back et al. [44, 45] have showed similar results in hardboard, i.e. moisture gradients by themselves do not produce mechano-sorptive creep. Hunt [90] has found a correlation between mechano-sorptive creep and the microfibrillar angle, and also a correlation to the elastic compliance. The microfibrillar angle directly controls the magnitude of the tensile and shear stresses applied to the inter-microfibril boundaries and hence the inter-molecular bonds. Hoffmeyer and Davidson [91, 92] found that the number of microscopical failures, or slip planes, in the fibre walls increases during mechano-sorptive creep in bending or compression. Liu [93] performed extensive bending creep experiments and analysed the results. He found that the mechano-sorptive effect was stronger for heavily loaded wood than for lightly loaded. The mechano-sorptive effect was also stronger for wood with lower elastic modulus and vice versa. At the same time as mechano-sorptive creep was discovered in wood, i.e. in the late 1950s and early 1960s, it was also independently discovered for wool [34, 35]. In the 1990s mechano-sorptive creep was found in some synthetic fibres; aramid (Kevlar) fibres [32, 39-41, 66, 94], see Fig. 11, Kevlar fibre reinforced epoxy composites [39] and cellulose acetate butyrate (CAB) fibres. Mechano-sorptive creep was also found in some polymers; cellophane, polyamide 6 (PA6) [73] and water-blown polyurethane foams [73, 95], see Fig. 12. A similar behaviour has also been found in concrete [38].

16

Fig. 11. Accelerated creep for a Kevlar fibre [32].

Fig. 12. Mechano-sorptive behaviour for a flexible water-blown slabstock polyurethane foam under constant load and (a) constant 20 % RH, (b) constant 98 % RH and (c) cyclic moisture conditions [95].

Most researchers claim that nylon fibres do not exhibit mechano-sorptive creep [39, 41, 52, 96], which can be explained by the absence of anisotropic swelling in nylon [49]. However, a small effect has been obtained if the moisture cycles are adjusted to the sorption time [66]. Haslach and Wu [70] claimed that moisture accelerated creep is observed in hydrophilic fibrous materials such as wood pulp paper, cotton paper, wool and Kevlar, but not in hydrophilic materials such as cellophane and nylon. They suggested that the fibrous nature is a significant factor in the phenomenon of accelerated creep. They also stated that those materials that do undergo mechano-sorptive creep also exhibit anisotropic swelling.

17

Differences between tensile and compressive loading The most important load cases in packages are compression and bending. There are many experimental observations that indicate a different behaviour in compression in comparison to tension [27, 28, 30, 31, 78, 82]. Armstrong and Kingston [78] was the first who noticed the difference in the 1960s. They found that the effects of moisture content changes were most pronounced in wood under compressive stress. The deformations in wood in compression were greater than those in bending which, in turn, were greater than those in tension. Similar results have been found by Mårtensson and Thelandersson [82] in hardboard who also examined the mechano-sorptive behaviour in the different loading modes. They found that the mechano-sorptive effect was larger in compression than in tension. The effect in bending was also larger than in tension. In the 1970s Byrd [30, 31] performed creep experiments on hand sheets loaded in tension and in fibreboards loaded in compression and showed a larger mechano-sorptive effect in compression than in tension, see Fig. 3.

(a)

(b)

Fig. 13. Mechano-sorptive creep of a linerboard in (a) tension and (b) compression [27]. A couple of decade’s later Söremark et al. [27, 28] examined the mechano-sorptive effects in linerboard under tensile and compressive loading. They found that the creep and the hygroexpansion were higher in compression than in tension during cyclic humidity, i.e. the mechano-sorptive effect was larger in compression than in tension, see Fig. 13. It was also found that under a cyclic relative humidity the compression creep stiffness index (defined as the specific stress divided by the strain at a certain time) was less than the tensile creep stiffness index, i.e. the creep rate was higher in compression than in tension. However, no difference could be seen between tensile and compression creep stiffness indices at constant relative humidity, see Fig. 14.

18

Fig. 14. Isochrones for tension and compression loads, at constant climate (cycle No 0) and under cycled climate (cycle No 5) [28].

Models for mechano-sorptive creep There is no generally accepted model for mechano-sorptive creep. The attempts made for developing models can be divided into two groups, empirical models and physically based models. The empirical models have often been developed in connection with experimental investigations, based on experimental data. These models have limited use since they are not based on actual physical phenomena, and are furthermore often not generally applicable, but limited to a specific application or material. By these models, mechano-sorptive effects can be seen, but they do not provide any explanation for the behaviour. The empirical models are often used for comparison, for example between different materials or humidity conditions. Mechanisms that explain the mechano-sorptive creep have also been proposed, but the effect of these mechanisms are rarely quantified and implemented into models. Empirical models The first empirical model was developed by Leicester in 1971 [97]. It was an attempt to model drying of wooden beams under load in bending. This model is a modified Maxwell model for viscoelasticity, with an elastic and a mechano-sorptive element in series, see Fig. 15. The mechano-sorptive element has no time dependent response, but rather deforms only under the combined influence of load and change in moisture content. The total deflection � was taken as the sum of an elastic component �e and a mechano-sorptive component �m, i.e.

me ∆+∆=∆ . (1) The two components are related to the load parameter P as follows,

KP=∆ e (2)

19

and

( )mPfm

=∆−d

d m , (3)

where K is a constant, m is the moisture content and ( )mf is a function of moisture content. Comparisons to experimental results showed that this model could explain about 85 % of the total deformation. The model was also validated in two other articles [98, 99], which showed that the model could be used to describe the behaviour of drying wood.

Fig. 15. Schematic representation of Leicester’s rheological model [97].

Ranta-Maunus [100] developed a generalized viscoelastic model for wood that includes moisture changes. In the one-dimensional case the constitutive equation between strain and stress is given by

( ) ( ) ( )τστε d0� −=t

tJt

( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]{ } ( )τττσσττστ uututtLtKt

d0� −−−+−+ (4)

where � is the strain, � is the stress, u is the moisture content, t and � are time variables, J is the creep compliance (under equilibrium conditions of temperature and moisture content), K and L are kernel functions describing the moisture dependence. To describe mechano-sorptive creep, K has to take on different values during absorption and desorption. The function L is a recovery parameter, i.e. the last term in Eq. (4) is the mechano-sorptive recovery strain. Later Mårtensson and Thelandersson [81, 82] followed Ranta-Maunus idea and developed an incremental strain-stress relationship for wood. The constitutive relation is given in rate form, and a uniaxial stress in the longitudinal direction is assumed. The total strain rate ε� is given as the sum of linear elastic strain rate, creep strain rate and stress-dependent moisture induced strain rate. They took into consideration the experimental discovery that the mechano-sorptive effect is more significant in compression than in tension. Mårtensson [101] continued the development of the model some years later. The total strain rate ε� was now assumed to consist of four parts, one elastic part eε� , one part describing creep under constant conditions cε� , one

20

part describing the shrinkage-swelling behaviour stε� and one mechano-sorptive part msε� , i.e.

msstce εεεεε ����� +++= . (5) Mårtensson proposed a relation between the mechano-sorptive creep strain, the stress � and moisture content u, um �� σε =ms , (6) where m is the mechano-sorptive parameter, given by

��

��

�

≤≤��

�

�

−−

><=

∞maxmin

mstms

mst0

maxmin0

ifexp

or if

uuum

uuuumm

εεε (7)

where minu and maxu are the moisture limits, 0m is a constant, ∞msε is the mechano-sorptive creep limit and

�=t

t0

msmst dεε � . (8)

A resembling one-dimensional constitutive model was developed by Mårtensson and Svensson [102] a few years later for wood under conditions of drying. The model was verified and extended to two dimensions in a second article [103]. Salin [104] divided up the deformation in the same way as Mårtensson et al. [81, 82, 102], i.e. as given by Eq. (5), but the time derivative of the mechano-sorptive creep was given by

( )t

uEm

t ∂∂−= bms

ms

ms

dd εσε

(9)

where m is the coefficient of mechano-sorptive creep, msE is a constant related to the maximal mechano-sorptive deformation achievable for a given stress level and bu is the bound water content. Eq. (9) is analogous to the Kelvin model for viscoelastic creep. Fridley et al. [105] extended a four-element viscoelastic model, a so called Burger model, to model the creep response of lumber. A fifth element was added to account for mechano-sorptive creep effects, see Fig. 16. Written in differential form, the constitutive relationship is given by msvke εεεεε ����� +++= (10) where ε� is the total time rate if strain, eε� is the elastic time rate of strain, kε� is the viscoelastic time rate of strain, vε� is the viscoplastic time rate and msε� is the mechano-sorptive strain rate, which is defined as

21

msms

ms µωσ

µσε

′==

�� , (11)

where msµ is the viscosity of the mechano-sorptive element, msµ′ is a constant and ω� is the rate of change in the relative moisture content factor.

Fig. 16. Five –element model with mechano-sorptive element [105].

Chassagne et al. [106] suggested a generalised Maxwell model with n elements, the first m Maxwell units representing normal creep and the other (n-m) units model mechano-sorptive creep. The total strain consists of the sum of an elastic strain, a viscoelastic strain, a mechano-sorptive strain and a hygro-expansive strain. The activation of the (n-m) units depends on the rate of relative moisture changes, the accumulated moisture history and the stress level. The delayed strain generated by these units corresponds to the mechano-sorptive effect. Their viscosities are expressed as

( ) ( )κη

ωη χω

χχ

bdbec

′−

= sinh1

hist0

� (12)

where χ is one of the (n-m) Maxwell units, i.e. nmn ,,�−=χ , ω� is the rate of relative moisture changes, histω represents the accumulated adsorption or desorption,

0χη , χb′ , b, c and d are constants.

Haslach et al. [55] described a model where the mechano-sorptive creep is separated into three parts; an elastic part, a creep part and a hygroexpansive part. The initial (elastic) part is first removed from the data. It is assumed that the hygroexpansion is zero at the reference humidity level, so the values from the curve where the humidity is at reference level is used as purely creep strains. The values from that creep strain was then fitted to the following creep law

22

( ) ( )n

C tct βε += 1ln (13) where t is the time and c, � and n are fitting parameters. When the creep strain has been determined by Eq. (13) it can be subtracted from the curve and the hygroexpansion is then left. For the hygroexpansion the following law was suggested,

( ) ( ) ( )traBtS 0exp σε = (14) where 0σ is the applied stress, ( )tr is the relative humidity and a, B are fitting parameters. It can be noticed that by separating the strains in this way the hygroexpansion is not only dependent on the relative humidity, but also the applied load. Ormarsson et al. [107] developed a three-dimensional model for mechano-sorptive creep in wood where the total strain rate was assumed to be the sum of the elastic strain rate e�

� , the moisture strain rate w�� and the mechano-sorptive strain rate w��

� , i.e. w�we ����

���� ++= . (15) The mechano-sorptive strain rate was given by aw�� �m�w� = (16) where aw� denotes the absolute value of the rate of change in moisture content. The

matrix � is the stress matrix and m is the mechano-sorption property matrix containing mechano-sorptive coefficients in the longitudinal, radial and tangential directions of wood and coefficients that describes the coupling of the mechano-sorptive strains in the different directions. The mechano-sorption matrix is assumed to be symmetric. Urbanik [56] suggested a model for corrugated boards subjected to a constant compression load and humidity cycling. In this model the total deformation, ( )tX , is the sum of the hygroexpansion, ( )tX h , and the mechano-sorptive creep. ( )tX c . The creep rate of mechano-sorption was assumed to be proportional to the magnitude of the rate of change of the relative moisture content. It was also assumed that the hygroexpansion was linearly dependent on the moisture content. These two assumptions lead to the following equation,

( )dt

dXdt

tdX hµ=c (17)

where � is a mechano-sorptive creep constant. The model was found to fit experimental data well, although viscoelasticity is not included in the model.

23

Hanhijärvi [108, 109] started with a physical based creep model for wood, based on the molecular deformation kinetics theory. The moisture dependence was assumed to add an extra internal microscopic stress. The creep equation then became

( )

��

�

� +⋅��

�

� ∆−=kT

ffVkTG

hkT

VV

2sinhexp

2 h�h

m

hε� (18)

where hV is the volume that is swept by the motion, mV is the volume of the moving segment, k is the Boltzmann constant, T is the temperature, h is Planck´s constant,

G∆ is activation energi, �f is the microscopic stress due to macroscopic stress and

hf is the microscopic stress due to moisture changes. To obtain a creep equation with macroscopic quantities he did some assumptions and finally ended up with ( )( ) hhDBA h

��� αφσε ++= tanhsinh (19) where h� is the relative vapour pressure change rate and A, φ , B and D are temperature dependent constants. He used this creep law as a non-linear dashpot in a generalised Maxwell model, with a linear spring. The last term in Eq. (19) represent an extra swelling/shrinkage component included in the model to get a good fit to experimental measurements, see Fig. 17. The larger creep rate in compression than in tension is modelled by assuming strain dependent parameters in the strain rate equation [108]. The coefficients A and hα in Eq. (19) are assumed to be dependent on mechanical strain (total strain with hygroexpansion excluded).

Fig. 17. Generalised Maxwell model for wood [108].

24

Physically based models To explain the mechano-sorptive effect it is not sufficient with an empirical model, the mechanisms behind the behaviour have to be determined. Several ideas of the physical mechanism of mechano-sorptive creep have been presented in the literature. Ideas of the mechanisms behind the mechano-sorptive creep effect were published basically at the same time as the experimental discoveries of the effect in the late 1950s and early 1960s. Both Mackay and Downes [34] who had found the effect in wool and Armstrong and Christensen [37] that examined wood suggested that transient stresses are introduced in the material when the moisture content changes, adding to the stress state caused by external loads, in that way giving an increased creep. This early idea with transient stresses that are created during changes in moisture content (stress redistribution) have later been supported and developed by other researchers [27, 28, 32, 44, 45, 64, 110-113]. Back et al. [44, 45] had an idea that moisture gradients always are obtained during sorption. Differences in swelling in different parts of the material would then give an uneven stress state, which may produce accelerated creep. The differences in stress relaxation rates between areas with different stresses will eventually even out the stresses and a new moisture change is needed to maintain the accelerated creep. Gunderson and Tobey [46], on the other hand, found experimental results that the creep strain was independent of the rate at which the humidity changed. They concluded that a stress gradient caused by a moisture gradient within the material can not explain the mechano-sorptive effect, since different rates of humidity changes should have produced different gradients. They did not however take into account that they used different creep times (the same number of cycles), and therefore was the total creep time much larger during the slow cycle experiments. Nordon [35] claimed that interchain hydrogen bonds are broken during moisture sorption. This will form a much weaker and more flexible link between chains, which will increase the molecular mobility. He also assumed that there is an equilibrium chain configuration for each moisture content. When the moisture is changed the increased mobility will lead to accelerated creep, but the unstable state will change continually so as to increase the rigidity by molecular rearrangement until the equilibrium state is reached, and than the creep slows down again. Similar models, based on bond breaking and molecular mobility, have been proposed by other researchers [29, 33, 106, 114, 115-118]. Gibson [114] suggested that during moisture content changes a non-equilibrium state occurs, in which hydrogen bonds will be broken and recreated by the interaction with migrating water molecules. During the phase of breaking and recreation of bonds the material, in this case wood, will be more compliant as the molecular mobility increases. The hydrogen bond model was questioned by Armstrong [89] since experiments showed that a constant moisture gradient caused by steady state flow does not produce mechano-sorptive creep. If hydrogen bond mechanism causes mechano-sorptive creep the moisture transport in steady state flow would be enough to give the effect also in this case. He suggested that volume changes during sorption of water could give rise to changes in the hydrogen bond configurations, and if an external force is applied, relative displacements may occur between the cellulose

25

chains in the wood substance, i.e. the mechano-sorptive effect could be explained by changes in the macromolecular configurations of the wood cell wall, but only if volume changes occur in the wood substance during water movement. Other developments of the hydrogen bond theory have also been presented to take the observation that a constant moisture gradient caused by steady state flow does not produce mechano-sorptive creep into account. Bažant [115] stated that two common properties of materials that exhibit mechano-sorptive creep are that they are porous with a broad range of pore sizes and the material forming the pore walls is strongly hydrophilic, with strong adsorption properties. In the very small micropores water molecules are strongly bonded to the pore walls, and the water molecules in these micropores are capable of transmitting significant load. The relatively large pores, macropores, on the other hand do not participate in the transmission of load. If moisture is transported through the micropores, by absorption or desorption, the bonds are disrupted, leading to accelerated creep. During steady-state flow the moisture transports through the macropores only, and therefore no disruptions of hydrogen bonds take place, explaining the lack of accelerated creep. Another explanation has been presented by Wang et al. [116]. During moisture transport without a change in moisture content hydrogen bonds are disrupted and formed at the same rate, i.e. the water molecules are in equilibrium inside the material, which explains that no mechano-sorptive creep takes place. If instead the moisture content is changed a disturbance in the exchange rate between breaking and reforming bonds between the wood molecules is caused, which leads to accelerated creep. Schaffer [117] and van der Put [118] also claimed that moving water will cause hydrogen bond breaking and thereby increasing the compliance of the material. The model by Schaffer [117] is based on molecular kinetics and the creep rate sε� is given by ( ) ( )RTQuBuS /exp/1 ss −++= γψρε �� (20) where u is the moisture content, � is the dry material density, � is the stress per (unbroken) bond, Qs is an average energy of activation for position changes of molecular segments, R is the gas constant, T is the absolute temperature and S, B, are material parameters. The model by van der Put [118] is based on a kinetic model derived from chemical reaction equations. He derived equations for the stress during relaxation. For a thin specimen, the stress after a step change in moisture is given by

tAK ϕωσσ ′−=��

�

�1

0ln , (21)

where � is the stress at time t, is the change in moisture, K1 is the modulus of elasticity and A′ , � are constants. This equation applies for very thin specimens, when the moisture content changes almost instantaneously. For thicker specimens the moisture has to diffuse into the material and van der Put then suggests another equation for the stress. Amorphous polymers are characterised by a glass transition temperature, below which the mobility of main-chain polymer segments is reduced enormously. This means

26

that, if a material is cooled rapidly, the increase in molecular packing, which proceeds readily as the temperature falls, is severely impeded. This causes the polymer to be in a thermodynamically non-equilibrium state. An excess volume, called the free volume, is locked in the material, increasing the molecular mobility and thereby also the creep compliance of the material. The polymer chains will then continuously readjust itself by conformational segment motions, reducing the free volume at an increasingly slow rate in a process towards equilibrium, simultaneously the compliance will decrease. This process is called aging. The material can be de-aged by raising the temperature above the glass transition temperature and then lowered again. The aging process then starts all over again. Paper is a partially amorphous polymeric material, it therefore has a glass transition. Padanyi [47, 48] stated that rapid moisture content changes also create free volume, and that mechano-sorptive creep is the same phenomenon as physical aging. He found that exposure of paper to high humidity had an effect similar to the de-aging procedure. The glass transition temperature is lowered by high moisture content, so it is possible to de-age paper by raising the moisture content instead of the temperature. It is possible to get the same effect when the humidity is lowered, which Padanyi also stated was a de-aging effect. Hunt and Gril [119] also suggested that physical ageing can explain part or all of mechano-sorptive creep. Habeger et al. [32, 66] pointed out that the desorption effect cannot be rationalized by appeal to a glass transition explanation, but they agreed that accelerated creep and moisture deaging are closely related phenomena. They stated that there is nothing in the physical-aging mechanism that would push beyond the highest humidity creep compliance.

Fig. 18. A generalized Voigt model [120].

27

Nakano [120] has derived a model for drying wood based on the free volume idea. A generalized mechanical model consisting of Kelvin elements and a Maxwell element in series, see Fig. 18, was used. The viscosity of each dashpot element is considered to be equal to the viscosity during water desorption,

( ) ( )tnFt

+=

1eηη (22)

where eη is the viscosity in the steady state of moisture, ( )tF is the rate of water desorption and n is a constant. By using this viscosity, the creep behaviour for wood during drying can approximately be represented by the simple formulation ( ) ( ) ( )tKMtJtJ += 0 , (23) where ( )tJ is the creep compliance during water desorption, ( )tJ 0 is the creep compliance at constant moisture content, ( )tM is the moisture change and K is a constant. Other models explain the mechano-sorptive effect by dislocations, for example kinks and slip planes, which are created and affected by stresses caused by moisture content changes. Compressive loading would lead to an increased number of dislocations and more dislocations lead to a higher hygroexpansion and lower stiffness [27, 28, 91, 92]. Hoffmeyer and Davidson [91, 92] proposed a slip plane mechanism for mechano-sorptive creep in wood. During mechano-sorptive creep in bending and compression local microscopical compression failures take place in the fibre wall. The compression failures form slip planes, which increase the compliance of the material. The number of slip planes is a function of stress level, moisture content and duration of load, and suggested in these articles, also moisture content change. It is not possible to explain mechano-sorptive creep in tension with this model. Chassagne et al. [106] explained the mechano-sorptive effect in wood by microcompressions. When the moisture content is reduced, below the fibre saturation point, transverse contractions of crossing fibres induce compressive stresses in the fibre-fibre bonds and causes buckling, or microcompressions, in the amorphous regions located between crystalline cellulose fibrils. Through the microcompressions the compressive stresses are partially reduced. A partial recovery of those deformations may be possible by increasing the moisture content. When the cell wall is subjected to tensile stresses, the microcompressions are partially released by an elastic deformation and a delayed deformation reflecting viscous behaviour. It is assumed that water diffusion disturbs moisture content equilibrium and amplifies delayed strains of amorphous regions until microfibrils reach an almost stable state. If external compressions stresses are applied parallel to wood fibres, external forces counter the release of microcompressions. Thus the rate of delayed strains could be greater than in the case of external tension stresses. In addition, during drying, releasing bound water molecules causes hydrogen bond raptures between wood fibrils, increasing delayed deformations, in particular by lowering the stress level activating the buckling of the amorphous areas. Hydrogen bonds tend to significantly reduce relative slippages between cellulose chains. The stable state is troubled when a load is applied, thus hydrogen bonds break and enable partial slidings between

28

microfibrils. New hydrogen links are recomposed in other sites, but the relative movements of cellulose chains cause other hydrogen bonds to break, so the process continues to cause creep deformations. Söremark et al. [27, 28] has suggested a model for paper that combines dislocations with transient stresses. In the papermaking process and during drying dislocations or kinks are created in the fibres. The consequence is a higher hygroexpansion and a lower stiffness of the paper. If paper is subjected to a static load during changing humidity, anisotropic swelling or deswelling occurs in the structure. These changes in swelling leads to a redistribution of stresses in the paper, on the molecular and fibre level. The already existing dislocations will be affected to different degrees. In compression the number of dislocations will increase during moisture variations, leading to length decrease in the loading direction and increased creep rate, which together lead to a fast creep rate and a large decrease in the stiffness. In tension the dislocations will decrease, which causes length increase, but also decreases the creep rate. These two effects counteract each other, but results in a slighter increased creep rate and decrease in the stiffness. Thus the model explains mechano-sorptive creep and the difference between compressive and tensile loading. Coffin and Boese [72] suggested a very similar model as the model described above by Söremark et al. [27, 28], to explain the difference in creep that they observed between single fibres and hand sheets. The accelerated creep behaviour that was observed in paper, was absent in the single fibres. Single fibres were dried flat and straight, and had few defects or microcompression dried into the fibre. This was in contrast to the fibres in the sheet which was influenced by the radial shrinkage of neighbouring fibres, which will form microcompressions at or near the bonds. They suggested that it might be these microcompressions or defects that lead to accelerated creep in paper. A model combining the mechanisms explained above, i.e. creation of free volume, bond breaking and formation of dislocations or microcompressions, has been proposed by Haslach and co-workers [29, 33, 70]. They discussed how the anisotropic hygroexpansion will cause additional stresses at the fibre bonds. They also states that moisture changes generate free volume, which increases the hygroexpansion due to the fibres lamellar structure. This will further increase the stresses at the bonds. The large stresses at the bonds will lead to disruption of the bonds, which in turn will lead to a weaker material and higher creep compliance. Lamellae slip may also take place due to fibre swelling. Wood and paper consists of long, hollow fibres. In wood the fibres are arranged parallel to each other, joined together by a middle lamella mainly consisting of lignin. In paper the wood fibres has been separated from each other and rearranged into a fibre network. The fibre walls are made up by microfibrils, i.e. collections of cellulose chains, in a matrix of hemicellulose and lignin. Within the fibre wall there are a number of layers with different microfibril orientation, the intercellular layer/middle lamella, the primary wall and the secondary wall which consists of three layers, the outer layer (S1), the middle layer (S2) and the inner layer (S3). The mechanical properties of the wood are primarily determined by the thickest layer, the S2 layer. Boyd [121] proposed a model for mechano-sorptive creep in wood based on the anatomy of the S2 layer. Mukudai and Yata [122-124] have also developed a model

29

for wood that are based on the inherent microstructure of the wood cell wall. Their hypothesis was that mechano-sorptive behaviour resulted from redistribution of applied stress in the cell wall because of change of friction between the S1 and S2 layers induced by moisture content changes. They have also tried to verify their model by comparing computer simulations with experiments and also by observing the specimens fracture with a scanning electron microscope and confirming that splits at interfaces of the S1 and S2 layers could be observed but also some other types of fractures [124]. In some recent investigations [32, 64, 110-113] it has been suggested that mechano-sorptive creep in paper is an effect of non-linear creep of the individual fibres in combination with large transient stresses created during moisture content changes which give rise to redistribution of stresses. The principal controlling mechanism for accelerated creep in paper at varying moisture contents is the hygroscopic nature and the anisotropic properties of the individual fibres in combination with their non-linear creep behaviour. One particular kind of material heterogeneity in paper is the difference in hygroexpansion along and across the fibres. The fibre swelling and shrinkage due to moisture content changes are much larger transverse to the fibre axis in comparison to the axial direction of the fibres. When moisture is changed, the fibres will expand (or shrink). At the bonds, the expansion is restricted by the crossing fibres because of the anisotropic hygroexpansive properties. Instead large stresses are created. In resemblance with the material heterogeneities, moisture gradients also cause heterogeneous hygroexpansive strains, which introduce similar internal stresses because of the restriction by the surrounding material. The internal stresses will be added to the stress state caused by external mechanical loads. This stress redistribution gives rise to an increased creep rate, driven by the nonlinear creep of the fibres. The stresses in the network relax with time during creep and the internal stress state will even out. After a while a new change in moisture content is required to create a new redistribution of stresses and to further accelerate the creep. The advantage of this kind of model is that accelerated creep turns out to be a natural consequence of regular creep, and not a completely new phenomenon. This model is basically a further development of Pickett’s ideas from 1942 [38], where he stated that a moisture change will produce non-uniform shrinkage or swelling that produces transient stresses. The transient stresses combined with a non-linear stress-strain relation would produce accelerated creep. This mechanism was first developed by Habeger and Coffin [32, 64]. They proposed a simple model, two Maxwell elements in parallel, to illustrate the mechanism, see Fig 19. The model can demonstrate accelerated creep if it includes a moisture gradient or material heterogeneity, which will produce stress gradients. The two Maxwell elements are restrained to deform together and are each composed of an elastic element, a hygroexpansive element and a creep element. The strain rate of each Maxwell element is given by

( )c,1

dd εσβσε

fdtdm

dtd

Et++= (24)

where E is the modulus of elasticity, � is the hygroexpansion coefficient, m is the moisture content and ( )c,εσf is the creep rate. A requirement for accelerated creep is

30

that the function ( )c,εσf is non-linear in stress �. Habeger and Coffin suggested three different creep laws. Another requirement is that the hygroexpansive strains �∆m are different in the two sections. Material heterogeneities would give different amplitudes while a moisture gradient would give a phase difference. With their model, Habeger and Coffin were able to simulate the accelerated creep of paper.

Fig. 19. The simple mechanical model developed by Habeger and Coffin [32].

Alfthan et al. [110-113] also developed models for mechano-sorptive creep based on large transient stresses caused by anisotropic swelling during sorption in combination with non-linear creep of the fibres. They developed a simplified network model which included effects from the elastic, hygroexpansive and creep behaviour of the fibres, but also the influence of bonds were included [111, 112]. The strains at fibre level are assumed to be the sum of elastic, creep and hygroexpansive strains

mE

∆++= βεσε c (25)

where m∆ is the moisture content change. The material parameters E, the modulus of elasticity and �, the hygroexpansion coefficients, have different values along and transverse to the fibre. Three different creep laws were suggested, and it was suggested that the creep law ( )( )cc

22c sinh εσε Eba −=� (26)

will give a creep response resembling those of paper and fibres. In Eq. (26) 2a , 2b and cE are different material parameters. Later they linearized the non-linear network model to formulate a continuum model [113]. As a result, the model can be used to describe material in a finite element program and solve problems with complicated geometries.

31

Suggestions for future research Since quite many experimental results are available for mechano-sorptive creep in paper the next step is to develop a model that can capture the effect in different loading conditions and also the different features that have been discovered during the experimental studies. It would then be appropriate to formulate a finite element implementation in order to predict the behaviour for arbitrary geometries and load cases. If it is possible, the model should be linearized, and from that a continuum model can be derived. In that way the number of equations can be reduced and the computation speed can be increased. A continuum model can be used as a constitutive law in problems with complex geometries for example corrugated board boxes. This development would make it possible to predict deformation and life-time of packages stored under varying humidity.

32

Summary of appended papers Paper A: Influence of molecular weight on strain-gradient yielding in polystyrene The length-scale effects in inelastic deformations of the thermoplastic polystyrene (PS) have been experimentally quantified with respect to the molecular length. The experimental technique used was nano-indentation to various depths with a Berkovich indenter. The hardness has been determined with the method by Oliver and Pharr [125], and also by direct measurements of the indentation area from atomic force microscopy. The experiments showed that length-scale effects in inelastic deformations exist in polystyrene at ambient conditions, see Fig. 20. The direct method gave a smaller hardness than the Oliver-Pharr method. It was also showed that the length-scale parameter according to Nix and Gao [3] increases with increasing molecular weight, see Fig. 21. For high molecular weights, above a critical value of entanglement, there was no pertinent increase in the length-scale parameter.

Fig. 20. The hardness, H, plotted against the maximum indentation depth, hmax, for the sample with molecular weight 52 220 g/mol with least-squares fits of the function in Nix and Gao (1998).

Fig. 21. The length-scale parameter (Nix and Gao, 1998), h*, for polystyrene plotted against their respective molecular weights of 4 920, 10 050, 19 880 and 52 220 g/mol.

33

Paper B: Length-scale effects on damage development in tensile loading of glass-sphere filled epoxy In this study, the effect of glass-sphere size on the damage development in tensile loaded epoxy has been investigated. The first type of damage observed was debonding at the sphere poles, which subsequently grew along the interface between the glass spheres and epoxy matrix. These cracks were observed to kink out into the matrix in the radial direction perpendicular to the applied load. The debond stresses increased with decreasing sphere diameter, shown in Fig. 22, whereas the length to diameter ratio of the matrix cracks increased with increasing diameter, shown in Fig. 23. These effects could not be explained by elastic stress analysis and linear-elastic fracture mechanics. A possible explanation is that there is a length-scale effect in the yield process which depends on the strain gradients. Cohesive fracture processes can also contribute to the influence of sphere size on matrix crack length.

Fig. 22. The diameter of the smallest particle with damaged matrix (*) and the largest sphere without damaged matrix (�) are indicated for both the applied stress �� and the stress concentration at the pole �d. To the left of the line the matrix was undamaged and to the right it was damaged.

Fig. 23. The normalized crack length plotted against the sphere diameter with a quadratic equation fitted to the data. The crack lengths were measured after fracture. The applied stress before fracture was 73 MPa.

34

Paper C: Mechano-sorptive creep under compressive loading – a micromechanical model In this paper a simplified network model for mechano-sorptive creep is presented. It is assumed that the anisotropic hygroexpansion of the fibres leads to large stresses at the fibre-fibre bonds when the moisture content changes. The resulting stress state will accelerate creep if the fibre material obeys constitutive laws that are non-linear in stress. Geometrical fibre effects are included in the model in order to capture experimental observations of the differences between paper loaded in tension and compression, i.e. larger mechano-sorptive creep effects in compression. Theoretical predictions based on the developed model are compared to experimental results for paper both under tensile and compressive loading at varying moisture content. The important features in the experiments are captured by the model, i.e. the creep is accelerated by the moisture cycling and the mechano-sorptive effects are larger in compression than in tension, see Fig. 24 and 25.

Fig. 24. Mechano-sorptive creep curves from the model, where the same stress has been used in tension and compression, 4.7 kNm/kg.

Fig. 25. Isocyclic data from experiments (*) and from the model (—) after three cycles ( s102.7 4⋅ ).

35

Paper D: An anisotropic fibre-network model for mechano-sorptive creep in paper In this paper the simplified network model for mechano-sorptive creep presented in Paper C is further developed. As in Paper C it is assumed that the anisotropic hygroexpansion of the fibres leads to large stresses at the fibre bonds when the moisture content changes. The resulting stress state will accelerate creep if the fibre material obeys a constitutive law that is non-linear in stress. Fibre kinks are included in the model in order to capture experimental observations of larger mechano-sorptive creep effects in compression than in tension. In this paper moisture dependent material parameters and anisotropy in the fibre distribution have been introduced. Theoretical predictions based on the developed model are compared to experimental results for an anisotropic paper both under tensile and compressive loading at varying moisture content. The important features in the experiments are captured by the model, i.e. the creep is accelerated by the moisture cycling and the mechano-sorptive effect is larger in compression than in tension, see Fig. 26 and 27. Different kinds of drying conditions have also been examined.

Fig. 26. Mechano-sorptive creep curves from experiments the model, where the same stress has been used in tension and compression, 4.7 kNm/kg.

Fig. 27. Isocyclic data from experiments (*) and from the model (—) after 3 cycles ( s102.7 4⋅ ).

36

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