Bread dough rheology and recoil

J. Non-Newtonian Fluid Mech. 148 (2008) 33–40

Bread dough rheology and recoil I. Rheology Roger I. Tanner ∗ , Fuzhong Qi, Shao-Cong Dai School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, Sydney NSW 2006, Australia Dedicated to the memory of Arthur S. Lodge (1922–2005).

Abstract A new set of experiments on a bread dough includes small-strain oscillatory behaviour, larger-strain oscillatory behaviour, simple shearing beginning from rest, uniaxial elongation beginning from rest, relaxation after sudden shear and recoil from elongation. We believe this is the most complete set of rheological data yet reported for a bread dough. Analysis of these soft-solid experiments proceeds from a Lodge-type rubberlike material with a power-law memory function. The model suggests that the response to steady shear and elongational flows may be described as a product of (strain rate)p times a function of strain; the exponent p is found to be about 0.2–0.3 from small-strain oscillatory measurements. Experiments confirm this finding. The model overestimates stresses, and in order to improve predictions, the use of a KBKZ model and a damage function model are investigated. Due to the eventual fracture of the soft-solid material, the idea of a “damage function” was adopted to produce a simple accurate, integral-type constitutive model for small-strain oscillations, simple shearing and elongation. Further analysis of reversing strains, for example, larger-strain oscillatory flows and recoil, is needed. © 2007 Elsevier B.V. All rights reserved. Keywords: Bread dough; Soft-solid; Experiments; Recoil; Integral model; Damage function

1. Introduction to bread dough modelling A mixture of wheat flour and water plus a small amount of salt and possibly other materials such as preservatives or yeast, constitutes bread dough. Dough rheology plays an important role in the quality of baking products [1] and moreover poses many intriguing questions about mechanical behaviour. There is, however, no general consensus as to what set of constitutive equations should be used to describe dough rheology. In particular, there seems to be no basic set of experiments including recoil after stress release, despite the fact that some processes (e.g. sheeting and pressing of dough) could be described by using this information. The present paper therefore describes new experiments, suggests a new approach to constitutive modelling (Part 1) and also applies the model to new experiments on recoil from elongation (Part 2, forthcoming). The present paper is restricted to unyeasted dough; a recent Ph.D. thesis [2] shows that yeasted dough behaves rheologically

∗

Corresponding author. Tel.: +61 2 9351 7153; fax: +61 2 9351 7060. E-mail address: [email protected] (R.I. Tanner).

0377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2007.04.006

in a similar manner to unyeasted dough, so we believe that many of the ideas can be carried over to the yeasted case, at least for processes that are quick compared to the yeast development time. The rheology of dough is sensitive [3] to changes in water content, starch content, wheat genetics and mixing procedure, as well as temperature. In our experiments, we have used a single flour, mixed to the same degree with the same water content; temperature was always at room temperature (24 ◦ C). This enables us to concentrate on the general mechanical behaviour of the material, leaving the other variables for possible future exploration. The investigation of dough rheology goes back a long way; the early work (1932–1937) of Schofield and Scott Blair [4] established the solid-like behaviour of dough, and since then there have been many investigations [1]. Bread dough is a soft-solid, which may be regarded as a filled elastomeric network. Starch particles of two kinds (lenticular particles of about 14 ␮m in size and smaller spherical particles of about 4 ␮m diameter) make up the filler, which comprises about 60% of the volume in natural doughs [1]. Electron microscope pictures of dough show clearly the network and the starch particles. Manipulation of the starch content to change dough properties has also been discussed [3]. In dough, the filler

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particles are not strongly attached to the rubbery gluten network, and the starch can easily be washed out of the dough with water. Nevertheless, an increase of about 20 times the small-strain storage modulus (G ) above that for gluten alone (∼103 Pa at 1 Hz) to around 2 × 104 Pa, is observed, and hence starch makes a very important contribution to dough rheology. Various mechanical analogue models have been suggested; a complex model due to Lerchenthal and Muller [5] is shown in Fig. 1a. Fig. 1b shows a model of Bloksma [6]. While these models can be fitted to uniaxial elongational behaviour they do not describe the small-strain behaviour of dough well, since they contain only one or two relaxation modes. In any case, it is very difficult to generalize these uniaxial models to a complete three-dimensional system. From another viewpoint, attempts have been made to simply use models devised for unfilled noncrosslinked polymers [7,8] but these models do not reflect the solid-like behaviour of dough. Phan-Thien et al. [9] showed that a model of the type shown in Fig. 1(c) could be used for shear behaviour, including large amplitude sinusoidal shearing. However, there are difficulties [2] when one attempts to describe both elongation and shear behaviour with this model; in any case, one can see immediately from Fig. 1(c) that predictions of recoil from elongation will always give perfect (100%) recoil so it cannot be applied to recoil experiments, where recovery is usually less than perfect. Charalambides et al. [10] used a purely rubber–elastic model of the Mooney-Rivlin type to discuss biaxial stretching with some success, but the model clearly cannot describe partial recoil or small-strain oscillatory behaviour. Leonard et al. [11] have also tried to use a Mooney-Rivlin model, but the results are not completely successful. A review by Dobraszczyk and Morgenstern [12] also discusses the use of several models, but clearly the description of recoil is not satisfactory.

Hence, we believe that better models covering a wider range of experimental conditions are needed. In the present paper, we give data for a single dough type in (1) (2) (3) (4)

Small-strain oscillatory shear. Oscillatory shear for strains up to 10%. Simple steady shearing beginning from rest. Constant-rate elongation beginning from rest.

In a subsequent paper, we shall give data for shear relaxation and recoil from constant-rate elongation. There do not seem to be any previously published data covering all of the above tests for the same dough, although there are many reports covering some of the tests. In Part 1, we discuss analysis of small-strain behaviour, shear behaviour and elongation. Part 2 will discuss the analysis of larger amplitude oscillatory flows, relaxation and recoil. 2. The material and experimental methods used 2.1. Dough preparation The material that was used for this study was a brand of commercial Australian flour. The flour sample was variety JANZ wheat, grown in 2001 at Narrabri, NSW, milled on a Buhler experimental mill. It is a benchmark Australian hard kernel wheat, said to be of medium dough strength. The dough was produced in a 10 g mixograph by mixing 200 mg of salt, 6.0 g of distilled water and 9.5 g of flour, as determined by using a ® Sartorious digital high precision scale. The sample was mixed by four planetary pins on the head revolving round three stationary pins on the bottom of the mixing bowl. The rotation speed was measured to be 71 rpm. The mixing operation was conducted at a temperature of 24 ◦ C and under ambient humidity in an air-conditioned laboratory. The mixing time to peak dough development was determined from the mixing curves [13] and took about 7 min. When the signal peaked, the dough was judged to have been developed [14] and the processing was stopped. 2.2. Elongation measurements

Fig. 1. Bread dough models. (a) 1967 model of Lerchental and Muller [5]. Note the springs, dashpots, shear pins and yield elements. (b) 1960 model of Bloksma [6]. Simpler than (a), it contains one yield element. (c) Phan-Thien model of 2000 [9], successful in unsteady shear, unsatisfactory in elongation and recoil.

For the elongation measurements, the sample after mixing was first formed in an aluminium cylinder with an inside diameter of 30.6 mm, and stored in a sealed bag to relax for 45 min [14,15]. The sample was then transferred to an Instron 5564 rheometer to perform the elongation tests. The measuring geometries used were two parallel plates: one is a fixed lower plate with a diameter of 31.0 mm, the other is a moving upper plate with a diameter of 30.3 mm. The sensitive load cell used has a measuring range of 10 N. Before testing, the rheometer was calibrated without any loading. The aluminum cylinder containing the dough was fitted on the upper plate, and slowly moved up over the upper plate. Simultaneously, the upper plate was brought down gently until the sample was compressed properly between the plates. However, this compression could not guarantee that the dough would not partially peel from the

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plates during the elongation, even if sandpaper were used. Therefore, the excess dough after compressing was wiped upwards or downwards evenly to wrap the upper and lower plates, respectively, so that the sample would be well attached to the plates during the elongation. In addition, to prevent moisture loss, a thin layer of Shell® petroleum jelly was applied to the edges of the sample. Then the mounted sample was compressed to the set gap of 8 mm, and allowed to relax for a further 45 min to ensure decay of any built-up residual stress [14,15], this was confirmed by monitoring the load. The tests were conducted at elongation rates of 0.001, 0.01 and 0.1 s−1 , respectively. The samples were stretched until they were physically broken. At the same time, a digital camera was used to capture the diameters of the cylindrical dough at a rate of 25 fps, and the results were downloaded to a computer as a movie. The circularity of the cross-section was generally excellent, as judged from specimens after cutting in half (This aspect, and the axial symmetry of the specimens, will be shown in the second part of the paper.). Hence, the elongational stress σ = F/A could be calculated, where F is the elongation force measured by the load cell and A is the minimum cross-sectional area of the sample. The actual rate of extension at the centre of the sample could also be found from the variation of diameter in the sample with time. 2.3. Recoil and relaxation measurements Recoil measurements were conducted to measure the recoil of the dough from a stress state induced by an elongation. The tests were carried out at elongation rates of 0.001, 0.01 and 0.1 s−1 , respectively, based on the experimental procedures of the elongation measurements, but the sample was cut at the total strain (elongation rate × time) of 0.2–2.5. At the higher strains, the sample was stretched to a right cylinder, and the lower section of the cylindrical dough after cutting was cylindrical. The recoiled lengths were measured on the upper and lower sections of the cylindrical dough, respectively. Then they were averaged so that the small effect of gravity on the total recoiled length was compensated for. Data for relaxation in shear and more details of the recoil measurements will be given in Part 2 of the paper (forthcoming).

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by moving down the upper plate to a set gap. Excess dough was trimmed, and the edge of the sample was coated with Shell® petroleum jelly to prevent moisture loss. After that, the sample was allowed to relax for a further period of 45 min. For oscillatory shear measurements, the frequency sweeps in the frequency range of 0.01–30 Hz were conducted at the strain amplitudes of 0.1%, 1%, 5% and 10%, respectively. After testing, the sample was unloaded, and a new sample was loaded for the next frequency sweep. Figs. 9 and 10 show the measured results of the storage modulus G and the loss modulus G . The G and G simply increase with frequency at set strain amplitudes; they decrease with the strain amplitude at a fixed frequency. In the reduction of these data, following Phan-Thien et al. [9], standard procedures were used. 3. Small oscillatory strain behaviour The range of maximum strain that results in a linear response is very small for doughs [3]. Typically, for wheat flour, the maximum shear strain permitted is around 0.001 (0.1%) and in our oscillatory tests we have used this figure. Here, the strain is set at the outer radius of the parallel plates. By contrast, the gluten network has a linear range of around 3–4% and in a pure starch dough the range drops to ∼0.04% [3]. Results for the storage and loss moduli G (ω), G (ω) as functions of the applied frequency ω (rad/s) are shown in Fig. 2. The results are clearly adequately described by a power-law relation: G (ω) = G (1)ωp

(1)

with a similar relation for G ; p is a constant. These results are in accord with many others in the literature, for example, see refs. [2,3,9]. In all cases, the exponent p is in the range 0.2–0.3 and G > G , indicating soft-solid behaviour. Some small deviations from constant loss angle were noticed by Newberry [2], but despite this, we shall use the form (1) here. For the data shown in Fig. 2, the exponent p estimated from G is 0.26, while that based on G is 0.28, so we have assumed that p = 0.27 for this material.

2.4. Oscillatory and steady shear measurements The dough used for the oscillatory and steady shear measurements was stored in a sealed bag after mixing, and allowed to relax for 45 min. The shear experiments were carried out on a Paar Physica MCR 300 rheometer. The parallel plates with a diameter of 25 mm were used, and the gap was set to 2 mm for the measurements. Slippage during testing was prevented by two pieces of sandpaper that had been glued to the parallel plates. This important issue had previously been extensively investigated by us [2], and no slip was encountered. Since 2-mm thick specimens were used, errors of not more than 2–3% were expected due to roughness of the sandpaper. Before testing, calibration of the rheometer was performed. Then the sample was mounted on the lower plate, and compressed between the plates

Fig. 2. Small-strain (0.1% strain) data for a medium strength Australian dough. The slope of the power-law fitted curves is 0.27, so G (ω) = G (1)ω0.27 . tan δ is 0.452, indicating solid-like behaviour. G (1) = 12.2 kPa.

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Table 1 Linear viscoelastic relations for power-law models p 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3

δ (◦ )

G(1)/G (1)

18.0 18.9 19.8 20.7 21.6 22.5 23.4 24.3 25.2 26.1 27.0

0.903 0.899 0.895 0.891 0.887 0.883 0.880 0.876 0.872 0.868 0.865

tan δ ≡ G /G 0.325 0.342 0.360 0.378 0.396 0.414 0.433 0.452 0.471 0.490 0.510

pπ G = tan G 2

m(t) = pG(1)t −(p+1)

(2)

and so is defined by the exponent p. Hence, the approximation (1) is attractive, as it only involves two constants (p, G (1)) as opposed to the multitude of discrete time constants and moduli needed to describe the very wide-spectrum linear behaviour with a set of Maxwell elements. In Fig. 2, p = 0.27, G (1) = 12.2 kPa. For the usual representation of linear viscoelasticity, the shear stress (τ) shear strain (γ) behaviour is described in terms of the Boltzmann integral:  t ˙  ) dt  τ= (3) G(t − t  )γ(t where G(t) is the relaxation function and γ˙ is the shear rate. It is known [17] that the form of G corresponding to Eq. (1) is (4)

The constants G(1) and G (1) are related [17] by G(1) =

2G (1)(p!) pπ sin pπ 2

(5)

(7)

Winter and Mours [23] have studied (6) and (7) for shearing motion, in another context. Note that the dimensions of G(1) in Eq. (7) are actually Pa − sp ; however, the numerical value of G(1) is the same as G(t) (a stress) at t = 1 s. The form (6) assumes that the dough is incompressible. Because of entrained air, however, dough is somewhat compressible, and this has been discussed by Wang et al. [18]. However, in our elongation tests, where it might be important, the bulk modulus was measured as approximately 2 MPa [18] (7.5% air by volume) and the underpressure was about 0.01 MPa, and hence volume changes were less than 1%, which has been ignored in the modelling. If we now use (6) and (7) to compute the response to a steady elongation (˙ε constant) imposed at t = 0, we find  t  −(p+1) 2˙ε(t−t  ) σ = σ11 − σ22 =pG(1) (t − t  ) {e −e−˙ε(t−t ) }dt  0

+ pG(1){e2˙εt − e−˙εt }



0

−∞

−(p+1)

(t − t  )

dt 

(8)

The quantity ε˙ (t − t  ) is a Hencky strain εH , and by changing variables we find  εH p σ(t) = p˙ε G(1) z−(p+1) (e2z − e−z )dz o

p

−p

+ ε˙ G(1)εH (e2εH − e−εH )

−∞

G(t) = G(1)t −p

(6)

−∞

where ␴ is the stress tensor, P the pressure, I the unit tensor and C−1 is the Finger tensor relative to the present time t as in ref. [16]. The memory function m is related to G (Eq. (4)) by m = −dG/dt, and so in the power-law case

The solid lines in Fig. 2 show the fitting obtained. For the data given by Phan-Thien et al. [9] the slope p is taken as 0.22 (G data). Uthayakumaran et al. [3] have shown that the slope p is not appreciably affected by changes in water content for a Strong flour dough, although the level of G changes. It is also known [16,17] that the phase angle δ is given, for a power-law material, by tan δ =

power-law relaxation function. Let us assume  t m(t − t  )C−1 (t  ) dt  ␴ + PI =

(9)

Thus, the model predicts, for a given strain εH , that the stress σ is proportional to ε˙ p . The stress function can be evaluated for various p values (Fig. 3 shows σ/p˙εp G(1) as a function of Hencky strain); it is not very sensitive to p in the range shown. For a shearing of shear rate γ˙ beginning at t = 0, one can evaluate the integrals explicitly, finding the viscometric functions with the same form of shear rate and strain dependence as in elongation [23]: G(1) p 1−p γ˙ (γ) , 1−p

where p! is the factorial function. Various relations are shown in Table 1 over the range of p from 0.2 to 0.3; this is the range of interest for the doughs studied. Thus, we believe that the representation of the small-strain behaviour is accurate and economical.

τ=

4. A Lodge model for elongation and shear

From our oscillatory data, Fig. 2, G (1) ∼ 12.2 kPa, and hence using p = 0.27, from Table 1 we can find G(1) = 10.7 kPa. From (10), if γ˙ = 0.1 s−1 , and γ = 2, we find τ ∼ 13 kPa, instead of about 1.1 kPa as measured. Similarly, the elongational stress

Given the elastic nature of the gluten matrix, it is natural to look at Lodge’s rubberlike models [19] in conjunction with a

˙ where γ = γt

(10)

N2 = 0.

(11)

and N1 =

2G(1)γ˙ p 2−p γ ; (2 − p)

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Fig. 3. The h-function (σ/p˙εp G(1)) as a function of elongational Hencky strain in Eq. (9) for various p values. This function is insensitive to changes of p in the range shown. The curves are very well approximated by the function −p h = (1 + 0.35/1 + 2εH )p−1 εH (e2εH − e−εH ) (points plot).

Fig. 5. Shear stress data for JANZ dough plotted as τ/γ˙ p as a function of the Hencky strain εH (see Eq. (12)). Fracture occurs at εH ∼ 3+; p = 0.27. The mean line through the data is used for damage analysis.

is overestimated by Eq. (9). Notice there are no steady-state stresses in this formulation. These results are not surprising; if one adds a “damping” ˙ to the integral form (6) either function of the shear γ(≡ γt) in the KBKZ/Wagner way [16], or outside the integral, giving results similar to the Phan-Thien et al. differential form [9], then it still appears that the model yields a result for shear of the form ε˙ p times a function of strain. This conclusion can also be arrived at by simple dimensional analysis reasoning, by supposing from Eqs. (6) and (7) that the stress σ is a function of p, G(1), ε˙ and t. Since G(1) has the dimension of Pa − sp , then one finds that σ/G(1)˙εp is a function of ε˙ t and p only, at least when ε˙ is a constant. One can test if this is reasonably concordant with experiment by plotting σ/˙εp , τ/γ˙ p as a function of strain. The results are shown in Figs. 4–6. Fig. 4 shows the results from the Phan-Thien et al. [9] data—the plot shows that τ/γ˙ p may be regarded as a function of shear strain γ(≡ γ˙ t) right up to fracture at γ ∼ 20. Bearing in mind that each of the five shear

rates used a new sample, with some inevitable variability, the agreement is reasonable. Similarly, we show in Fig. 5 new shear data for the JANZ dough; instead of plotting against γ we use the Hencky strain, which is related to γ by [20]

Fig. 4. Shear stress (τ) data of Phan-Thien et al. [9] replotted to show coincidence ˙ as a of curves of τ/γ˙ p over a range of nearly four decades of shear rate (γ), function of strain (γ˙ t ≡ γ). p = 0.22. Fracture occurs at γ ∼ 20 corresponding to a Hencky strain of ∼3.0.

⎡ ⎤  1 ⎣ γ2 ⎦ γ2 εH = ln 1 + +γ 1+ 2 2 4

(12)

Thus, this material fractures when εH ∼ 3, corresponding to γ ∼ 20 as before. In Fig. 5, the stress and strain rates were evaluated at 3/4 of the plate radius in the torsional flow. Finally, in Fig. 6 we show the elongational stress/˙εp as a function of εH = ε˙ t for our measurements. Again, the superposition is reasonable and fracture occurs at εH ∼ 3. Hence, there is some justification of the type of model used and we now consider it further.

Fig. 6. Elongational stress data for JANZ dough plotted as σ/˙εp as a function of the Hencky strain εH . p = 0.27 here. Fracture occurs when εH ∼ 3.2. The solid mean line is used in damage function discussions.

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5. KBKZ model To improve the model used above, we can introduce a ‘damping function’ of strain, so that Eq. (9) changes to  εH σ = pG(1) z−(p+1) F (z)(e2z − e−z )dz ε˙ p o −p

+ G(1)εH F (εH )(e2εH − e−εH )

(13)

where F(z) is a damping function in the sense used by Wagner [16]. For large εH , the second term dominates, and also it is possible to find a damping function that enables Eq. (13) to describe the experimental data. However, because of the fracture that occurs, we shall explore the use of damage mechanics as an alternative. Fig. 8. Reconstruction of elongational data using damage reduction factor of Fig. 7 in terms of Hencky strain εH .

6. Damage mechanics Frequently in solid mechanics, the idea that strain damages a material is used [21]. Simply, the stress is reduced by a damage function D, so that ␴ = (1 − D)␴0

(14)

where ␴ is now a damaged stress for the material. We will suppose the ‘undamaged’ stress ␴0 is given by Eqs. (6)–(11) above (the Lodge model). The damage function D can take various forms [21], here it will be assumed to be a function of the strain; D varies from zero at small-strains up to 1 at rupture. For the steady elongation and shear discussed above, we can then find (1 − D) (≡f, say) for elongation and shear by using the results of Figs. 4–6 and Table 1. The damage reduction factors f (≡1 − D) can be plotted as a function of the Hencky strain εH , Fig. 7, for both the shear and elongation data of Figs. 5 and 6. Clearly, the factors for shear and elongation are nearly the same, so it is not necessary to depart

Fig. 7. Showing the damage reduction function f (≡1 − D) as a function of Hencky strain for shear and elongation of the JANZ dough. Points are experimental data; curves are fitted using f = (1/(1.0953 + (111.74εH )1.1026 ) − 30 0.0225εH + 0.087)e−(0.32εH ) .

from the simple dependence on εH . Note that the drop from f = 1 at εH = 0 is very fast. The reconstruction of the data in elongation is seen to be satisfactory (Fig. 8). Notice the extremely rapid reduction from the linear case stress for quite small-strains; this has been noted previously [9]. This appears to be a sort of exaggerated Mullins effect [22]. In order to demonstrate this more clearly, we did the larger-strain oscillatory tests discussed below; it turns out that reversing stresses behave in an unexpected manner. 7. The Mullins effect in oscillatory shear strain We have conducted G and G measurements at strain amplitudes (at the rim) of 0.001, 0.01, 0.05 and 0.1 (Figs. 9 and 10). There is a considerable change in the response: (1) The reduction of G , G is very noticeable (Figs. 9 and 10). (2) There is a change in slope of G from ∼0.27 to ∼0.15, while still being fairly well-described as a power-law material. However, G now behaves like ω0.2 —a considerable change and there is a change in slope with frequency.

Fig. 9. G vs. frequency measurements at four strain amplitudes (0.1%, 1%, 5%, 10%) for JANZ dough.

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Fig. 10. G vs. frequency at the four strain amplitudes for the same dough. Note the rapid drop in the moduli for modest strain amplitudes.

If we ignore all harmonics except the first, then no slope change is predicted from the type of formulation described above, so a finite amplitude analysis does not seem to yield the results shown in the figures. (However, since the strain and strain rate vary across the plates, careful analysis of the experimental data may sometimes be needed, and this is discussed in Part 2 of this paper. Errors of at most 1% in G , G are expected). If we look at Fig. 9, we see a drop of 82% in G at 10% strain amplitude and 1 rad/s frequency; this is similar to that predicted by using the results shown in Fig. 7. Similar conclusions apply to the data of Phan-Thien et al. [9], and hence any errors of data reduction are not believed to be vital. What we seem to be seeing is a kind of exaggerated Mullins effect—strain softening plus a more viscous behaviour at higher frequencies. For example, at ω = 1 rad/s G goes from ∼6 kPa at an amplitude of 0.1% to only ∼2 kPa at 5% amplitude; G goes from ∼12 to ∼3 kPa. This behaviour is similar to the classic Mullins effect—it does not affect the simple shear and elongation measurements, and so it is an effect additional to the damage softening due to stress reversal. There do not seem to be any very complete theories of the Mullins effect for loosely bound particles in a viscoelastic medium; obviously, from our experiments there are rate effects. For the more tightly bound carbon black-rubber system there are many papers; perhaps the work of Govindjee and Simo [22] is nearest to our interests, but their results show only about a 1% drop in modulus G at 10% strain. Hence, further work on loosely bound fillers seems to be needed. 8. Conclusion This paper explores the use of an integral model of the Lodgetype, based on the idea that bread dough is a rubbery, somewhat viscoelastic matrix (gluten) containing around 60% by volume of hard starch particles. The very wide relaxation spectrum is modelled by a power-law function, which is economical (only two constants) relative to a Prony series (Maxwell-type) spectrum. The model indicates that stresses are proportional to (strain rate)p times strain. While the Lodge model overpredicts stresses,

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we find that addition of the customary KBKZ/Wagner “damping function” to the integral model indicates that the damping function must be zero near the specimen fracture point. Fracture occurs at a Hencky strain ∼3 in both shear and elongation. We have preferred to use a “damage function”, external to the integral as an alternative to the KBKZ model, and in this way we reconcile the predicted and measured stresses. While the damage function concept is common in various parts of solid mechanics, we have not so far encountered it in dough analysis. With the use of the Hencky strain, we are able to describe both small-strain, shear and elongational behaviour. In the latter two cases, there is no stress reversal. Eventually, this limitation has to be removed to predict behaviour in reversing strains. The extraordinary drop in moduli G and G with strain amplitude is described and clearly needs further study, although the present model gives a rough description of the data. Experiments on recoil and relaxation have been completed, but need further analysis. Further tests would be desirable—for example, do simple shear and pure shear have the same damage function? We believe that the present model is an improvement over our previous model [9], but more work is necessary. Acknowledgement This work was supported by a Discovery Grant from the Australian Research Council, and this support is gratefully acknowledged. References [1] A.H. Bloksma, Dough structure, dough rheology, and baking quality, Cereal Foods World 35 (1990) 237–244. [2] M. Newberry, The rheological properties of non-yeasted and yeasted wheat flour doughs, Ph.D. Thesis, University of Sydney, 2003. [3] S. Uthayakumaran, M. Newberry, N. Phan-Thien, R.I. Tanner, Small and large strain rheology of wheat gluten, Rheol. Acta 41 (2002) 162– 172. [4] R.K. Schofield, G.W. Scott Blair, The relationship between viscosity, elasticity and plastic strength of soft materials as illustrated by some mechanical properties of flour doughs, I, Proc. R. Soc. Lond. A138 (1932) 707– 719; R.K. Schofield, G.W. Scott Blair, The relationship between viscosity, elasticity and plastic strength of soft materials as illustrated by some mechanical properties of flour doughs, II, Proc. R. Soc. Lond. A139 (1933) 557–566; R.K. Schofield, G.W. Scott Blair, The relationship between viscosity, elasticity and plastic strength of soft materials as illustrated by some mechanical properties of flour doughs, III, Proc. R. Soc. Lond. A141 (1933) 72–85; R.K. Schofield, G.W. Scott Blair, The relationship between viscosity, elasticity and plastic strength of soft materials as illustrated by some mechanical properties of flour doughs, IV, Proc. R. Soc. Lond. A160 (1937) 87–94. [5] C.H. Lerchental, H.G. Muller, Research in dough rheology at the Israel Institute of Technology, Cereal Sci. Today 12 (1967) 185–187, 190– 192. [6] A.H. Bloksma, Bakers’ dough, in: R. Houwink, H.K. de Decker (Eds.), Elasticity, Plasticity and Structure of Matter, third ed., Cambridge University Press, 1971, p. 395, Chapter 14. [7] E.B. Bagley, D.D. Christianson, Response of chemically leavened doughs to uniaxial compression, in: H. Faridi, J. Faubion (Eds.), Fundamentals of Dough Rheology, Am. Assoc. Cereal Chem. St. Paul, Minnesota, 1986, pp. 27–36.

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