Creep in Timber
It is possible to quantify creep by a number of time dependent parameters, of which the two most common are creep compliance (known also as specific creep) and relative creep (known also as the creep coefficient); both parameters are a function of temperature.
Creep compliance (cc) is the ratio of increasing strain with time to the applied constant stress, i.e.:
cc(t,T) = strain variation/applied constant stress
while relative creep (cr) is defined as either the deflection or, more usually, the increase in deflection, expressed in terms of the initial elastic deflection, i.e.:
cr(t,T) = εt/εo or (εt – εo)/εo
where εt is the deflection at time t, and ε0 is the initial deflection.
Relative creep has also been defined as the change in compliance during the test expressed in terms of the original compliance.
In both timber and timber products such as plywood or particleboard (chipboard), the rate of deflection or creep slows down progressively with time (Fig. 1); creep is frequently plotted against log(time) and the graph assumes an exponential shape.
The results of creep tests can also be plotted as relative creep against log(time) or as creep compliance against stress as a percentage of the ultimate short time stress. The degree of elasticity varies considerably between the horizontal and longitudinal planes. Creep, as one particular manifestation of viscoelastic behaviour, is also directionally dependent.
In tensile stressing of longitudinal sections produced with the grain running at different angles, it was found that relative creep was greater in the direction perpendicular to the grain than it was parallel to the grain. Timber and wood-based panels, therefore, are viscoelastic materials, the time dependent properties of which are directionally dependent. The next important question is whether they are linearly viscoelastic in behaviour.
For viscoelastic behaviour to be defined as linear, the instantaneous, recoverable and non-recoverable components of the deformation must vary directly with the applied stress. An alternative definition is that the creep compliance or relative creep must be independent of stress and not a function of it.
Timber and wood-based panels exhibit linear viscoelastic behaviour at lower levels of stressing, but at higher stress levels this behaviour reverts to being non-linear. Examples of this transition in behaviour are illustrated in Figs 2 and 3, where for both redwood timber and UF bonded particleboard, respectively, the change from linear to non-linear behaviour occurs between the 45 and 60% stress levels.
The linear limit for the relationship between creep and applied stress varies with mode of testing, with species of timber or type of panel, and with both temperature and moisture content. In tension parallel to the grain at constant temperature and moisture content, timber has been found to behave as a linear viscoelastic material up to about 75% of the ultimate tensile strength, though some workers have found considerable variability and have indicated a range of from 36 to 84%.
In compression parallel to the grain, the onset of non-linearity appears to occur at about 70%, though the level of actual stress will be much lower than in the case of tensile strength, since the ultimate compression strength is only one third that of the tensile strength. In bending, non-linearity seems to develop very much earlier – at about 50–60% (Figs 2 and 3); the actual stress levels will be very similar to those for compression.
In both compression and bending, the divergence from linearity is usually greater than in the case of tensile stressing; much of the increased deformation occurs in the non-recoverable component of creep and is associated with progressive structural changes, including the development of incipient failure.
Increases not only in stress level, but also in temperature to a limited extent, and in moisture content to a considerable degree, result in an earlier onset of non-linearity and a more marked departure from linearity. For most practical purposes, however, working stresses are only a small percentage of the ultimate, rarely approaching even 50%, and it can be safely assumed that timber, like concrete, will behave as a linear viscoelastic material under normal service conditions.
Principle of Superposition
Since timber behaves as a linear viscoelastic material under conditions of normal temperature and humidity and at low to moderate levels of stressing, it is possible to apply the Boltzmann’s principle of superposition to predict the response of timber to complex or extended loading sequences. This principle states that the creep occurring under a sequence of stress increments is taken as the superposed sum of the responses to the individual increments.
This can be expressed mathematically in a number of forms, one of which for linear materials is:
where n is the number of load increments, Δσi is the stress increment, cci is the creep compliance for the individual stress increments applied for differing times, τ – τ 1, τ – τ 2, . . . , τ – τ n and εc(t) is the total creep at time t; or in integrated form:
In experiments on timber it has been found that in the comparison of deflections in beams loaded either continuously or repeatedly for two or seven days in every fourteen, for six months at four levels of stress, the applicability of the Boltzmann’s principle of superposition was confirmed for stress levels up to 50% (Nakai and Grossman, 1983). The superposition principle has been found to be applicable even at high stresses in both shear and tension in dry samples.
However, at high moisture contents, the limits of linear behaviour in shear and tension appear to be considerably lower, thereby confirming views expressed earlier on the non-linear behaviour of timber subjected to high levels of stressing and/or high moisture content.
Mathematical Modelling of Steady-State Creep
The relationship between creep and time has been expressed mathematically using a wide range of equations. It should be appreciated that such expressions are purely empirical, none of them possessing any sound theoretical basis. Their relative merits depend on how easily their constants can be determined and how well they fit the experimental results.
One of the most successful mathematical descriptions for creep in timber under constant relative humidity and temperature appears to be the power law, of general form:
ε(t) = eo + atm
where ε (t) is the timedependent strain, e0 is the initial deformation, a and m are material-specific parameters to be determined experimentally (m = 0.33 for timber), and t is the elapsed time.
The prime advantage of using a power function to describe creep is its representation as a straight line on a log/log plot, thereby making onward prediction on a time basis that much easier than using other models. The shape of the viscoelastic creep curve was predicted by Van der Put (1989) based on deformation kinetic theory.
Alternatively, creep behaviour in timber, like that of many other high polymers, can be interpreted with the aid of mechanical (rheological) models comprising different combinations of springs and dashpots (piston in a cylinder containing a viscous fluid). The springs act as a mechanical analogue of the elastic component of deformation, while the dashpot simulates the viscous or flow component.
When more than a single member of each type is used, these components can be combined in a wide variety of ways, though only one or two will be able to describe adequately the creep and relaxation behaviour of the material.
The simplest linear model that successfully describes the time-dependent behaviour of timber under constant humidity and temperature for short periods of time is the fourelement model illustrated in Fig. 4; the central part of the model will be recognised as a Kelvin element.
To this unit has been added in series a second spring and dashpot. The strain at any time t under a constant load is given by the equation:
where Y is the strain at time t, E1 is the elasticity of spring 1, E2 is the elasticity of spring 2, σ is the stress applied, ɳ2 is the viscosity of dashpot 2 and ɳ3 is the viscosity of dashpot 3.
The first term on the right-hand side of the above equation represents the instantaneous deformation, while the second term describes the delayed elasticity and the third term the plastic flow component.
Thus, the first term describes the elastic behaviour while the combination of the second and third terms accounts for the viscoelastic or creep behaviour. The response of this particular model will be linear and it will obey the Boltzmann superposition principle.
The degree of fit between the behaviour described by the model and experimentally derived values can be exceedingly good; an example is illustrated in Fig. 1, where the degree of correlation between the fitted line and experimental results for creep in the bending of urea–formaldehyde particleboard (chipboard) beams was as high as 0.941.
A much more demanding test of any model is the prediction of long-term performance from shortterm data. For timber and the various board materials, it has been found necessary to make the viscous term non-linear in these models where accurate predictions of creep (±10%) are required for long periods of time (> 10 years) from short-term data (6–9 months) (Dinwoodie et al., 1990a). The deformation of this non-linear mathematical model is given by the equation:
Y = β1 + β2[1 – exp (-β3t)] + β4tβs
where β1 = σ/E1, β2 = σ/E2, β3 = E2/ɳ2, β4 = σ/ɳ3, β5 is the viscous modification factor, with a value 0 < b < 1.
An example of the successful application of this model to predict the deflection of a sample of cement-bonded particleboard after ten years from the first nine months of data is given in Dinwoodie et al. (1990a). Various non-linear viscoelastic models have been developed and tested over the years, ranging from the fairly simple early approach by Ylinen (1965) – in which a spring and a dashpot in this rheological model are replaced by non-linear elements – to the much more sophisticated model by Tong and Ödeen (1989a), in which the linear viscoelastic equation is modified by the introduction of a non-linear function either in the form of a simple power function, or by using the sum of an exponential series corresponding to ten Kelvin elements in series with a single spring.
All this modelling provides further confirmation of the non-linear viscoelastic behaviour of timber and wood-based panels when subjected to high levels of stressing, or to lower levels at high moisture contents.
Reversible and Irreversible Components of Creep
In timber and many of the high polymers, creep under load can be subdivided into reversible and irreversible components.
The relative proportions of these two components of total creep appear to be related to stress level and to prevailing conditions of temperature and moisture content. The influence of level of stress is clearly illustrated in Fig. 5, where the total compliance at 70% and 80% of the ultimate stress for hoop pine in compression is sub-divided into the separate components. At 70%, the irreversible creep compliance accounts for about 45% of the total creep compliance, while at 80% of the ultimate, the irreversible creep compliance has increased appreciably, to 70% of the total creep compliance at the longer periods of time, though not at the shorter durations.
Increased moisture content and increased temperature will also result in an enlargement of the irreversible component of total creep. Reversible creep is frequently referred to in the literature as delayed elastic or primary creep and in the early days was ascribed to either polymeric uncoiling or the existence of a creeping matrix.
Owing to the close longitudinal association of the molecules of the various components in the amorphous regions, it appears unlikely that uncoiling of the polymers under stress can account for much of the reversible component of creep.
The second explanation of reversible creep utilises the concept of timedependent twostage molecular motions of the cellulose, hemicellulose and lignin constituents. The pattern of molecular motion for each component is dependent on that of the other constituents, and it has been shown that the difference in directional movement of the lignin and non-lignin molecules results in considerable molecular interference, such that stresses set up in loading can be transferred from one component (a creeping matrix) to another component (an attached, but non-creeping structure).
It is postulated that the lignin network could act as an energy sink, maintaining and controlling the energy set up by stressing (Chow, 1973). Irreversible creep, also referred to as viscous, plastic or secondary creep, has been related to either timedependent changes in the active number of hydrogen bonds, or to the loosening and subsequent remaking of hydrogen bonds as moisture diffuses through timber with the passage of time (Gibson, 1965).
Such diffusion can result directly from stressing; thus early work indicated that when timber was stressed in tension it gained in moisture content, and conversely when stressed in compression its moisture content was lowered. It is argued, though certainly not proven, that the movement of moisture by diffusion occurs in a series of steps from one adsorption site to the next, necessitating the rupture and subsequent re-formation of hydrogen bonds.
The process is viewed as resulting in loss of elastic modulus and/or strength, possibly through slippage at the molecular level. Recently, however, it has been demonstrated that moisture movement, while affecting creep, can account for only part of the total creep obtained, and this explanation of creep at the molecular level warrants more investigation; certainly not all the observed phenomena support the hypothesis that creep is due to the breaking and remaking of hydrogen bonds under a stress bias.
At moderate to high levels of stressing, particularly in bending and compression parallel to the grain, the amount of irreversible creep is closely associated with the development of kinks in the cell wall (Hoffmeyer and Davidson, 1989). Boyd (1982) in a lengthy paper demonstrates how creep under both constant and variable relative humidity can be explained quite simply in terms of stress-induced physical interactions between the crystalline and non-crystalline components of the cell wall.
Justification of his viewpoint relies heavily on the concept that the basic structural units develop a lenticular trellis format containing a water sensitive gel, which changes shape during moisture changes and load applications, thereby explaining creep strains.
Attempts have been made to describe creep in terms of the fine structure of timber, and it has been demonstrated that creep in the short term is highly correlated with the angle of the microfibrils in the S2 layer of the cell wall, and inversely with the degree of crystallinity.
However, such correlations do not necessarily prove any causal relationship, and it is possible to explain these correlations in terms of the presence or absence of moisture that would be closely associated with these particular variables.
Environmental Effects on Rate of Creep
Temperature: Steady-state. In common with many other materials, especially the high polymers, the effect of increasing temperature on timber under stress is to increase both the rate and the total amount of creep.
Figure 6 illustrates a two and a half-fold increase in the amount of creep as the temperature is raised from 20 to 54oC; there is a marked increase in the irreversible component of creep at the higher temperatures. Various workers have examined the applicability to wood of the time/ temperature superposition principle; results have been inconclusive and variable and it would appear that caution must be exercised in the use of this principle.
Temperature: Unsteady-state. Cycling between low and high temperatures will induce in stressed timber and panel products a higher creep response than would occur if the temperature was held constant at the higher level; however, this effect is most likely due to changing moisture content as temperature is changed, rather than to the effect of temperature itself.
Moisture content: Steady-state. The rate and amount of creep in timber of high moisture content are appreciably higher than those of dry timber; an increase in moisture content from 6 to 12% increases the deflection in timber at any given stress level by about 20%. It is interesting to note the occurrence of a similar increase in creep in nylon when wet. Hunt (1999) recorded that the effect of humidity can be treated by the use of an empirical humidity-shift factor curve to be used with an empirical master-creep curve.
At high moisture contents this logarithmic shift factor increases rapidly; creep at 22% moisture content compared with that at 10% was found to be 101.5 or 32 times as fast.
Moisture content: Unsteady-state. If the moisture content of small timber beams under load is cycled from dry to wet and back to dry again, the deformation will also follow a cyclical pattern.
However, the recovery in each cycle is only partial and over a number of cycles the total amount of creep is very large; the greater the moisture differential in each cycle, the higher the amount of creep (Armstrong and Kingston, 1960; Hearmon and Paton, 1964).
Figure 7 illustrates the deflection that occurs with time in matched test pieces loaded to 3/8 ultimate shortterm load where one test piece is maintained in an atmosphere of 93% relative humidity, while the other is cycled between 0 and 93% relative humidity.
After 14 complete cycles the latter test piece had broken after increasing its initial deflection by 25 times; the former test piece was still intact, having increased its deflection by only twice its initial value. Failure of the first test piece occurred, therefore, after only a short period of time and at a stress equivalent to only 3/8 of its ultimate (Hearmon and Paton, 1964).
The figure also shows the effect of loading a matched test piece to 1/8 its maximum load. It should be appreciated that creep increased during the drying cycle and decreased during the wetting cycle with the exception of the initial wetting when creep increased. It was not possible to explain the negative deflection observed during absorption, though the energy for the change is probably provided by the heat of absorption.
The net change at the end of a complete cycle of wetting and drying was considered to be a redistribution of hydrogen bonds, which manifests itself as an increase in deformation of the stressed sample (Gibson, 1965). More early work showed that the rate of moisture change affects the rate of creep, but not the amount of creep; this appears to be proportional to the total change in moisture content (Armstrong and Kingston, 1962).
This complex behaviour of creep in timber when loaded under either cyclic or variable changes in relative humidity has been confirmed by a large number of research workers (e.g. Schniewind, 1968; Ranta-Maunus, 1973; Hunt, 1982; Mohager, 1987). However, in board materials the cyclic effect appears to be somewhat reduced (Dinwoodie et al., 1990b). Later test work, covering longitudinal compression and tension stressing as well as bending, indicates that the relationship between creep behaviour and moisture change was more complex than first thought.
The results of this work (e.g. Ranta-Maunus 1973, 1975; Hunt, 1982) indicate that there are three separate components to this form of creep; these are illustrated in Fig. 8 and are:
- An increase in creep, which follows a decrease in moisture content of the sample, as described previously.
- An increase in creep, which follows any increase in moisture content above the previous highest level reached after loading; three examples can be seen in the middle of the graph in Fig. 7.
- A decrease in creep, which follows an increase in moisture content below the previous highest level reached after loading, as described previously.
It follows from 1 and 2 above that there will always be an initial increase in creep during the initial change in moisture content, irrespective of whether adsorption or desorption is taking place.
Further experimentation has established that the amount of creep that occurs depends on the size and rate of the moisture change, and is little affected by its duration or by whether such change is brought about in one or more steps (Armstrong and Kingston, 1962). These findings were to cast doubt on the previous interpretation that such behaviour constituted true creep.
Reinforcement of that doubt occurred with the publication of work by Arima and Grossman (1978). Small beams of Pinus radiata, 680 × 15 × 15 mm, were cut from green timber and stressed to about 25% of their short-term bending strength.
While held in their deformed condition, the beams were allowed to dry for 15 days, after which the retaining clamps were removed and the initial recovery measured. The unstressed beams were then subjected to changes in relative humidity followed by water immersion, and Fig. 9 shows the changes in recovery with changing humidity.
Most important is the fact that total recovery was almost achieved; what was thought to have been viscous deformation in the post-drying and clamping stage turned out to be reversible.
These two phenomena – that creep is related to the magnitude of the moisture change and not to time, and that deformation is reversible under moisture change – cast very serious doubts on whether the deformation under changes in moisture content constituted true creep.
It was considered necessary to separate these two very different types of deformation, namely the true viscoelastic creep that occurs under constant moisture content and is directly a function of time, from that deformation that is directly related to the interaction of change in moisture content and mechanical stressing, which is a function of the history of moisture change and is relatively uninfluenced by time.
A term of convenience was derived to describe this latter type of deformation, namely mechano-sorptive behaviour (Grossman, 1976). Changing levels of moisture content, however, will result in changes in the dimensions of timber and an allowance for this must be taken into account in the calculation of mechano-sorptive deformation. Thus:
εm = εvc + εms + εs
where εm = total measured strain, εvc is the normal time (constant moisture content) viscoelastic creep, εms is the mechano-sorptive strain under changing moisture content and εs is the swelling or shrinkage strain of a matched, zero-loaded control test piece.
The swelling/shrinkage strain (εs) (referred to as pseudo-creep by some workers), which is manifest during moisture cycling by an increase in deflection during desorption and a decrease during adsorption, has been ascribed to differences in the normal longitudinal swelling and shrinkage of wood – a tensile strain resulting in a smaller shrinkage coefficient and a compression strain resulting in a larger one (Hunt and Shelton, 1988).
Mechano-sorptive behaviour is linear only at low levels of stress; in the case of compression and bending the upper limit of linear behaviour is of the order of 10%, while in tension it is slightly higher (Hunt, 1980). The most accepted theoretical explanation of mechano-sorptive behaviour is that changes in moisture content result in the rupture of bonds with consequent slippage between the two new surfaces under stress, and the re-formation of bonds at different locations (Grossman, 1976).
Susceptibility to mechano-sorptive behaviour is positively correlated with elastic compliance, microfibrillar angle of the cell wall and dimensional change rates (Hunt, 1994). Thus, both juvenile wood and compression wood have been shown to creep much more (up to five times more) than adult wood.