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Consolidation and flow of cohesive bulk solids
Chemical Engineering Science 57 (2002) 287–294
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Consolidation and ow of cohesive bulk solids J&org Schwedes ∗ Institute of Mechanical Process Engineering, Technical University Braunschweig, Volkmaroder Stra e 4=5, 38104 Braunschweig, Germany Received 14 May 2001; received in revised form 11 September 2001; accepted 20 September 2001
Abstract The strength and the ow properties of bulk solids can be determined by performing shear tests. A shear test always consists of two parts, consolidation and measurement of strength. The measured strength depends on the way and degree of consolidation. During consolidation a stress history is impressed on the bulk solid sample which only vanishes after steady state ow. Since the consolidation is normally done nonhydrostatically, the strength will depend on the direction of stress application (anisotropy). Thus, the strength of a bulk solid sample strongly depends on its stress history and it can show anisotropic behaviour. These e1ects have to be considered when comparing available shear testers. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Particulate materials; Consolidation; Flow; Anisotropy; Shear tester
1. Introduction Today many testers, methods and ideas exist to measure the strength and the ow properties of bulk solids. It is often tried to compare the testers and to declare which tester can be best used for which application. For those, who have not to deal regularly with bulk solids and their characterisation it is hardly possible to decide which tester is the best for a special application and why it is like that. If a bulk solid sample has to be tested, it 8rst has to be consolidated in some way (8rst part) and afterwards it can be tested regarding its strength or ow properties (second part). What many people do not want to accept is the fact that the second part of the test very much depends on the 8rst part. Only if the 8rst part is identical, it can be expected that the second part gives similar results, even if di1erent testers are used. If the 8rst part is di1erent, di1erent results have to be expected even if the same tester is used. This paper will concentrate on what is going on during consolidation and how this in uences the subsequent failure test. Only with those knowledges available testers can be judged regarding their capabilities and suggestions are possible, which tester to use for which application. 2. Consolidation and steady state ow For a complete description of both parts of a test, consolidation and failure, the complete state of stress and the ∗
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complete state of strain have to be known. In bulk solids technology small stresses prevail as compared with the stresses being of interest in soil mechanics. For the low stress region the true biaxial shear tester in Fig. 1 best ful8ls the requirements mentioned above (Harder, 1986; Schwedes & Schulze, 1990; Nowak, 1993; Feise, 1996,1998; Schwedes, 1996, 2000). In the biaxial shear tester the bulk solid sample is constrained in lateral x- and y-direction by four steel plates. Vertical deformations of the sample are restricted by rigid top and bottom plates. The sample can be loaded by the four lateral plates which are linked by guides so that the horizontal cross-section of the sample may take di1erent rectangular shapes. In deforming the sample, the stresses x and y can be applied independently of each other in x- and y-directions. To avoid friction between the plates and the sample, the plates are covered with thin rubber membranes. Silicon grease is applied between the steel plates and the rubber membranes. Thus, it is ensured, that no shear strains or shear stresses can develop on x-, y- and z-planes. Therefore, all measured normal stresses and normal strains must be principal stresses and principal strains. This means that the experimental results can fully be described in the principal stress and strain space. The line connecting all measured states in the principal strain space is referred to as a “strain path”. Similarly, a “stress path” is de8ned in the principal stress space. As the biaxial shear tester allows deformations in the x–y-plane only, the strain path must lie completely in the x –y -plane. The stress path is, however, a general three-dimensional curve in the x –y –z -plane. Nevertheless, for convenience the diagrams will be limited to two dimensions.
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 3 7 8 - 5
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Fig. 1. Biaxial shear tester.
Experiments with a linear strain path belong to the simplest deformations possible in the biaxial shear tester. The experiment starts with = 0 and progresses with a constant deformation rate in x- and y-directions. As the ratio of ˙y = ˙x
remains constant, these strain paths are referred to as proportional strain paths. For proportional strain paths the ratio ˙y = ˙x equals to the ratio y =x . Therefore, in Fig. 2 the y =x -ratio is used to characterise the depicted strain paths. The resulting stress path is linear and proportional. For each strain path the measured stress paths from two independent experiments are shown. One can clearly see that a distinct direction of the stress path can be correlated to each direction of the strain path. These paths are therefore called “associated stress and strain path”, their directions “associated directions”. The broken straight line in Fig. 2 (left) with an inclination of “−1” connects states of strain where the samples have undergone a volume reduction of 15%. Depending on the selected strain path, di1erent stresses x , y and (not shown) z are necessary to reach this volume change. In Fig. 3 the results from experiments are shown, in which the strain path shows a sharp bend. In the 8rst part of the experiment the sample was consolidated with y =x =1:0. After 10%, 12.5% and 15% of volume change the y-plates were stopped and deformation commenced only in the x-direction (curves C0 –C2). The measured stress paths are made up
Fig. 2. Consolidation under proportional deformation (limestone: x50:3 = 4:8 m).
Fig. 3. Asymptotic behaviour (limestone: x50:3 = 4:8 m).
J. Schwedes / Chemical Engineering Science 57 (2002) 287–294
Fig. 4. Bulk solids response under pure shear loading (limestone: x50:3 = 4:8 m).
of two parts as well. Initially the stress paths follow the associated direction already known from Fig. 2. After the change in the respective strain path, however, y decreases rapidly and the stress paths 8nally evolve parallel to the stress path A2, which is associated with the strain path with y =x =0. The ordering of the stress path C0 –C2 with respect to A2 follows the amount of volume change during the 8rst part of the strain path. This behaviour is called “asymptotic behaviour”. Again, di1erent stresses are necessary to get the same volume reduction of 15% (broken straight line). But additionally to Fig. 2 the 8nal stresses depend on both parts of the strain path. Many di1erent strain paths consisting of several (one, two and even more) parts with different strain rate ratios are possible. The resulting state of stress at the end of consolidation, e.g. to get a volume reduction of 15%, depends very much on the strain and=or stress history. An experiment leading to steady state ow has to guarantee a deformation at constant volume. In the biaxial shear tester this requires ˙y = ˙x = y =x = −1, and this deformation is called “pure shearing”. To perform a pure shearing test, the sample always has to be consolidated before it can be subjected to pure shearing. Fig. 4 shows the stress and strain paths of six experiments with pure shear loading. Starting from the same initial bulk density, the samples have been subjected to consolidation up to two stress levels under three initial strain path directions. One can see that the stress paths initially follow the directions known from Fig. 2. As soon as the shear part of the strain path begins, the stresses decrease down to some stationary level. Subsequent shearing results in scatter around this point but no significant change in stress. This behaviour—deformation without change in volume and stress—is considered as steady state ow. From Fig. 4 one can also see the in uence of the loading history. While six di1erent consolidation paths have been used, only two levels of steady state ow arise. Stress paths, which belong to strain paths with the same level of volumetric strain, end at the same steady state stress level. Further experiments have shown that all steady state stresses lie on a line through the origin, the “critical state line”.
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Fig. 5. Jenike’s shear cell.
Fig. 6. Yield locus and e1ective yield locus.
In steady state ow the sample deforms at constant bulk density and constant stresses. As can be seen in Fig. 4, the 8nal state (stresses x and y ) is dependent only on the bulk density and is independent of the stress history, i.e. independent of the strain and stress paths. Thus, in steady state ow the sample has lost its memory of the stress history, whereas in all other stress states prior to steady state ow the state of stress depends on the stress and=or strain history. 3. Incipient failure and strength Before reporting on the second part of the test in the biaxial shear tester it shall shortly be described how a shear test is performed with Jenike’s shear tester, being the most common tester in bulk solids technology for practical application (Jenike, 1964; SSTT, 1989). The shear cell (Fig. 5) consists of a base (A), a ring (B) resting on the top of the base and a lid (C). Base and ring are 8lled with a bulk solid sample. A vertical force is applied to the lid. A horizontal shearing force is applied on a bracket attached to the lid. Running shear tests with identically “preconsolidated” samples under di1erent normal loads results in maximum shear forces S for every normal load N . Division of N and S by the cross-sectional area of the shear cell leads to the normal stress and the shear stress , respectively. Fig. 6 shows a , -diagram. The curve represents the maximum shear stress the sample can
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Fig. 7. Procedure to get a yield locus with Jenike’s shear tester.
support under a certain normal stress and is called the yield locus. The parameter of a yield locus is the bulk density b at preconsolidation. With higher preconsolidation loads, the bulk density b increases and the yield loci move upwards. Each yield locus terminates at point E in the direction of increasing normal stresses . Point E characterises the steady state ow, which is the ow with no change in stresses and bulk density. Two Mohr stress circles are shown. The major principal stresses of the two Mohr stress circles are characteristic of a yield locus, that is, 1 is the major principal stress at steady state ow, called major consolidation stress, and c is the uncon8ned yield strength. Each yield locus gives one pair of values of the uncon8ned yield strength c and the major consolidation stress 1 . Plotting c versus 1 leads to the ow function (see later, Fig. 10). The angle ’e between the -axis and the tangent to the larger Mohr stress circle, called e1ective yield locus, is a measure for the inner friction at steady state ow and is very important in the design of silos for ow (Feise, 1996; Schwedes, 2000). To get the yield locus in Fig. 6, the bulk solid sample is sheared by two steps (Jenike, 1964; SSTT, 1989): In the 8rst, often called “preshear”, the sample is sheared under a normal stress sf until steady state ow with = sf =const. prevails, thus leading to the larger Mohr stress circle in Fig. 6. After steady state ow has been obtained, which is indicated by a constant shear stress sf , the shear stress is reduced to zero, the normal stress is reduced to ¡ sf and the second step of the shear process, often called “shear”, is performed by applying again a shear force on the bracket (attached to the lid C in Fig. 5). The maximum in the path of shear stress versus shear strain gives one value of the yield locus. To get further points the procedure has to be repeated with new samples, the same normal stress sf during “preshear”, but with di1erent normal stresses ¡ sf during “shear”. The procedure is shown schematically in Fig. 7. To get further yield loci the normal force during preconsolidation and “preshear” has to be increased or decreased and the procedure has to be repeated. The biaxial shear tester can also be used to obtain a yield locus corresponding to Figs. 6 and 7: The minor principal
Fig. 8. Shear at constant 2 in the biaxial shear tester.
Fig. 9. Yield locus gained with the biaxial shear tester.
stress 2 , e.g. acting in y-direction (Fig. 1), is kept constant during a test (Fig. 8). In x-direction, a positive strain rate ˙1 is applied resulting in an increasing major principal stress 1 . 1 is increased continuously up to the point of steady state ow, where constant values of 1 and b are reached. After this “preshear” the state of stress is reduced to a smaller 2 value (2 = 2; b in Fig. 9) and the second step of the shear process is performed by applying again a positive strain rate ˙1 in x-direction resulting in increasing 1; b -values. 2; b and the maximum value of 1; b de8ne the Mohr stress circle of incipient failure, thus belonging
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Fig. 11. Sample consolidation in the biaxial shear tester.
Fig. 10. Flow function (limestone: x50:3 = 4:8 m).
to the yield locus. Furthermore, Mohr stress circles of failure can be obtained by repeating the procedure with identical values of 1 and 2 during “preshear” and “shear” with di1erent values of 2; b ¡ 2 . By setting 2; a = 0; the uncon8ned yield strength c = 1; a can be measured directly (Fig. 9). Comparative tests with the same 8ne limestone powder were performed with Jenike’s shear tester and the biaxial shear tester following the described procedures with “preshear” and “shear” (Harder, 1986; Schwedes & Schulze, 1990; Schwedes, 1996). In Fig. 10 the ow function is given, which is the plot of uncon8ned yield strength c versus major consolidating stress 1 at steady state ow. Although two di1erent kinds of shear testers were used, the measurements are in agreement. It was also shown that with a ring shear tester identical results can be obtained, when the tests are performed according to Fig. 7 (Schwedes & Schulze, 1990; Schulze, 1996; Schwedes, 2000). 4. Inuence of stress history, anisotropy For investigating the in uence of di1erent consolidation procedures on the resulting strength samples were consolidated in the biaxial shear tester di1erently up to the same overall bulk density (Harder, 1986; Schwedes & Schulze, 1990; Nowak, 1993; Schwedes, 1996, 2000; Feise, 1996,1998). Fig. 11 demonstrates three di1erent possibilities (I,II,III) of consolidating the sample in order to get the same sample volume and, hence, the same bulk density. In the case of procedure I the x-axis and in the case of procedure III the y-axis coincide with the direction of the major principal stress 1; c at consolidation. In the case of procedure II the major principal stress 1; c is acting in both directions. (With respect to Fig. 2 the strain rate ratio has been y =x = 0 (I), 1 (II) and ∞ (III).) After consolidation the samples were sheared in the following way (see
Fig. 12. Flow function from Fig. 10 and uncon8ned yield strength c versus major principle stress at consolidation 1; c (limestone: x50:3 =4:8 m).
Fig. 8): 2 in y-direction was kept constant at 2 = 0 and 1 was increased up to the point of failure, leading to the uncon8ned yield strength c . The results are plotted in Fig. 12 as c versus 1; c , with 1; c being the major principal stress at consolidation. The functions c = f(1; c ) corresponding to procedures I–III are below the ow function c =f(1 ), being identical with the results already shown in Fig. 10. It can be clearly seen that the three functions c = f(1; c ) underestimate the uncon8ned yield strength c which a bulk solid sample gains after steady state ow ( ow function). Procedure I is comparable to the procedure realised in uniaxial testers. In an uniaxial tester (Fig. 13) (Schwedes & Schulze, 1990; Maltby & Enstad, 1993; Schwedes, 2000) a sample is 8lled into a cylinder with frictionless walls and is consolidated under a normal stress 1; c . After removing the cylinder, the sample is loaded with an increasing normal stress up to the point of failure. The stress at failure is the uncon8ned yield strength c . Contrary to shear tests following the procedure shown in Fig. 7 steady state ow cannot be reached during consolidation, i.e. the Mohr stress circle will be smaller (Fig. 14) at the end of
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But isostatic consolidation is a rare event in industrial applications. 5. Judgement of available shear testers
Fig. 13. Uniaxial tester.
Fig. 14. Mohr stress circle for steady state ow (SF) and uniaxial consolidation (UC).
consolidation (Schulze, 1995; Schwedes, 2000). As a result, bulk density b and uncon8ned yield strength c will also be smaller compared to results gained with the procedure of Fig. 7. The test of Fig. 13 is often used to explain the ow function c = f(1 ). As can be clearly seen in Figs. 12 and 14 a uniaxial tester always underestimates the ow function. The function c = f(1; c ) of procedure II (Figs. 11 and 12) leads to even smaller uncon8ned yield strengths. Steady state ow ( ow function), uniaxial consolidation (procedure I) and the biaxial consolidation of procedure II thus produce di1erent strengths, although identical major principal stresses were used during consolidation. Consolidation procedures I and III are basically identical. Only the direction of measuring the uncon8ned yield strength is di1erent. Since the results deviate strongly, a remarkable anisotropic e1ect exists. Thus, the strength of a bulk solid sample depends on its “stress history” and even if this stress history is identical, it can show anisotropic behaviour. In Fig. 4 it was shown that a sample is loosing its memory of the stress history only in steady state ow. However, this does not imply that it behaves isotropically in the following failure test, and typically the anisotropic behaviour after steady state ow is most distinct. This can be explained with the stress ratio of minor to major principal stresses 2 =1 being the smallest after steady state ow consolidation (Fig. 14). Only when all three principal stresses are identical (iso- or hydrostatic consolidation), an isotropic behaviour might be expected.
From the foregoing it follows that identical results of shear tests in di1erent testers can only be obtained when the consolidation procedure (8rst step of a shear test) and the failure test (second step of a shear test) are absolutely identical. Di1erent consolidation procedures are not associated with any in uence on the failure test when at the end of consolidation the sample has reached steady state ow. The reported measuring procedures for Jenike’s shear tester (SSTT, 1989) argued that “preshear” up to steady state ow has to be performed as a check for the correct preconsolidation. This preconsolidation is supported by a twisting motion of the cover of the shear cell under a normal stress . Thus, shear stresses are induced in the sample acting tangentially. These combined normal and shear stresses might locally lead to a steady state ow, but the complete cross section of the shear cell is not in an identical steady state ow condition. This is because the directions of the major principal stresses during the twisting motion are di1erent at di1erent positions within the cell. Also it is not sure if steady state ow is really obtained at all positions. Thus, the 8rst step of a shear test — “preshear” to steady state ow (Fig. 7) — is not a check for a correct preconsolidation, but a necessary step to get steady state ow across the complete cross-section and in the same direction. Only this “preshear” guarantees a reproducible and a clearly de8ned consolidation where all in uences of the stress history are eliminated. The ow function (relationship between uncon8ned yield strength c and major principal stress 1 at steady state ow) can only be determined with testers in which both stress states can be realised and both stresses 1 and c act in the same direction — atleast nearly (Schwedes, 2000). Steady state ow can be achieved in Jenike’s shear tester, in ring shear testers, in a torsional shear cell, in biaxial shear testers and in a very specialised triaxial cell (Schwedes, 1996). The uncon8ned yield strength c can be determined by running tests in Jenike’s shear tester, in ring shear testers, in uniaxial testers and in biaxial shear testers. Therefore, only Jenike’s shear tester, ring shear testers and biaxial shear testers can guarantee the measurement of ow functions c = f(1 ) without further assumptions. All other procedures to get a dependence of the uncon8ned yield strength c on the major principal stress at consolidation 1; c (without reaching steady state ow) lead to smaller uncon8ned yield strengths. Those relationships can only be used as estimates of the ow function. But, when using these estimates for some applications (i.e. design of silos for ow), it has to be kept in mind that the predictions for the bulk solids behaviour are on the “unsafe” side.
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All other testers, which are discussed in detail in the paper (Schwedes, 2000), have de8ciencies. Either the state of stress in the sample is not homogeneous, not known or the state of failure can hardly be used to get quantitative values for strength or ow properties. 6. Application of measured ow properties 6.1. Design of silos for 6ow The best known and the most applied method to design silos for ow is the method developed by Jenike (1964). He distinguishes two ow patterns, mass ow and funnel ow, the border lines of which depend on the inclination or apex angle of the hopper, the angle ’e of the e1ective yield locus (Fig. 6) and the wall friction angle ’w between the bulk solid and the hopper wall. The angle ’w can easily be tested with Jenike’s shear tester, but also with other direct shear testers. To get reliable ow in and out of the silo it has to be designed in such a way that the strength of the stored bulk solid at no time and in no place is suRcient to form stable domes or stable ratholes. The most critical area in a silo is the area directly above the outlet. The bulk solid owing in the silo from top to bottom is subjected to di1erent stresses. According to these stresses it is consolidated and gets its strength. Above the outlet the ow is converging with decreasing stresses towards the outlet. Due to these decreasing stresses the strength of the bulk solid also decreases towards the outlet. A passive plastic state of stress prevails in this converging ow. At each point within this region steady state ow exists. As already explained only at steady state ow a bulk solid sample looses its memory of the stress history, i.e. any bulk solid element at a certain location in the convergent geometry is always exposed to the identical state of stress (in steady state ow) and hence has an identical strength independent of the stress history. This is independent of all the possible ways to reach this point. Closing the outlet and opening it again will decide if the strength of the bulk solid is large enough for doming to occur. The dome is stable if the stress in the dome (parallel to the bottom surface) is smaller than the uncon8ned yield strength which the bulk solid developed during steady state ow before closing the outlet. The uncon8ned yield strength can be gotten in shear tests if these are performed according to Fig. 7: The consolidation in the shear tester must ensure steady state ow, because only steady state ow characterises the stress history representing the state of stress in converging ow. Having received this stress history (“preshear”), samples have to be sheared to failure (“shear”) to get yield loci. The Mohr stress circle being tangential to the yield locus and going through the origin yields the uncon8ned yield strength as the major principal stress of this Mohr stress circle. The minor principal stress
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is zero, because the stress normal to the surface of a dome is zero. From this explanation it follows that a shear tester, used for the design of silos for ow, has to ensure steady state ow at “preshear” and reliable , -values in the low stress region at “shear” to get a yield locus, from which the uncon8ned yield strength can be derived. The anisotropic behaviour of bulk solids mentioned in Section 4 is of no in uence in the design of silos for ow. It was explained that for getting the yield loci and the ow function the directions of the major principle stresses during consolidation to steady state ow and during shear to failure have to be identical. During steady state ow in the hopper the major principle stress is normal to the hopper axis. In a stable dome above the outlet the uncon8ned yield strength also acts laterally to the hopper axis. Therefore, the ow function re ects reality in the hopper area. 6.2. Other applications The design of silos for ow is the only known application, where the stress history and its in uence on strength and failure is exactly known and can be simulated in shear testers. Shear tests yield the ow function — or time ow functions — describing the (uncon8ned yield) strength of a bulk solid as a result of consolidation. Consolidation is achieved at steady state ow with the major principal (consolidation) stress 1 . The minor principal stress follows from the Mohr stress circle being tangential to a yield locus at its endpoint. If a sample is stressed by an identical major principal stress 1 but without reaching steady state ow, the minor principal stress will be larger compared to the one at steady state ow. This Mohr stress circle is smaller and the sample is less consolidated and gains less strength, see Fig. 14. Thus, the highest strength which a bulk solid sample can achieve when it is stressed by a consolidation stress 1 results from steady state ow at consolidation. This highest strength is represented by the ow function. The strength a bulk solid sample gets in a speci8c application depends on the stress history, i.e. on the way the bulk solid sample is consolidated. If during consolidation steady state ow was not achieved, which is the most probable situation, besides silo ow, the strength will be smaller than indicated by the ow function. The strength of a consolidated sample that was not subjected to steady state ow depends not only on the 8nal stress state of consolidation but also on the stress path, i.e. on the way how the stress was applied. Many stress paths are possible and exist in true applications: uniaxial, biaxial, stepwise, cyclic, mixtures, etc. Only if the stress path in an application is known and this stress path is repeated or simulated in a shear tester, a shear test can predict the correct strength of the sample. Many situations exist in which bulk solids are compacted under their own weight, but without reaching steady state
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ow. The storage in sacks or bags, the storage in open storage piles for blending and the storage in silos having a at bottom have to be mentioned. The compaction in a silo with a at bottom is similar to the compaction in an uniaxial tester. Thus, the uniaxial tester (Figs. 13 and 14) might yield a more realistic value of the uncon8ned yield strength. The compaction of a bulk solid in a bag is di1erent, because the bag can expand in the horizontal direction. As a result the horizontal minor principal stress decreases and the size of the Mohr stress circle increases which results in larger values of bulk density and strength (see Fig. 14). For this application the uniaxial tester underestimates the strength. The strength of the bulk solid in the bag is somewhere in between the strength documented by the ow function and the strength following from an uniaxial test. In open piles for blending an expansion in horizontal direction is also not prohibited. Thus, a uniaxial test will also underestimate the strength. Basing a design on the ow function is considered to be on the safe side. Additionally, a strong in uence of anisotropy exists. Coming back to the above-mentioned storage in a at bottom silo, in bags and in open piles for blending, the prediction of a strength depends on the direction. For the at bottom silo the strength in vertical direction is larger than the strength in horizontal direction, following the dependencies of procedures I and III in Fig. 12. For the strength of a bulk solid in a bag compacted by bags lying on top, it was already explained that due to a horizontal expansion the strength in the vertical direction is larger as compared with procedure I but lower than the strength according to the ow function. It can be argued that the strength of the bulk solid in this bag in the horizontal direction is even smaller than the prediction from procedure III. In conclusion, stress history and anisotropic behaviour have a strong in uence on the strength of a bulk solid. Only if the stress history and the directions of the major principal stresses during consolidation and failure are known for the application and can be simulated in a shear tester, a reliable prediction of the strength is possible. Possible tendencies can be derived from plots like Fig. 12. The ow function describes the maximum strength a bulk solid can develop during consolidation. Therefore, a design for ow is always on the safe side, if it is based on the ow function. The extent of safety can only be checked, if the stress history and the directions of stress application at consolidation and failure are known. Design of silos for strength, calibration of constitutive models, quality control and qualitative comparisons of the ow properties of di1erent bulk solids are further applications of measured ow properties. These applications are
extensively discussed in (Schwedes, 1996, 2000) and shall not be repeated here. 7. Conclusion Identical results of shear tests in di1erent testers can only be obtained when the consolidation procedure (8rst step of a shear test) and the failure test (second step of a shear test) are absolutely identical. Di1erent consolidation procedures are only without any in uence on the failure test when at the end of consolidation the sample has reached steady state ow. For all other consolidation procedures an in uence of the stress history exists. Additionally bulk solids show anisotropic behaviour, i.e. the strength a bulk solid sample gains during consolidation depends on the direction of stress application. Thus, stress history and anisotropic behaviour have a strong in uence on the strength of a bulk solid. Only if the stress history and the directions of the major principal stresses during consolidation and failure are known for the application and can be simulated in a shear tester, a reliable prediction of the strength is possible. Such a correlation exists for the design of silos for ow. References Feise, H. (1996). Modellierung des mechanischen Verhaltens von Sch9uttg9utern. Ph.D. thesis, Technical University Braunschweig, Germany. Feise, H. (1998). A review of induced anisotropy and steady-state ow in powders. Powder Technology, 98, 191–200. Harder, J. (1986). Ermittlung der Flie eigenschaften koh9asiver Sch9uttg9uter mit einer Zweiaxialbox. Ph.D. thesis, Technical University Braunschweig, Germany. Jenike, A. W. (1964). Storage and 6ow of solids. Bulletin 123, Engineering and Experiment Station, University of Utah, USA. Maltby, L. P., & Enstad, G. G. (1993). Uniaxial tester for quality control and ow property characterization of powders. Bulk Solids Handling, 13(1), 135–139. Nowak, M. (1993). Spannungs-=Dehnungsverhalten von Kalkstein in der Zweiaxial-box. Ph.D. thesis, Technical University Braunschweig, Germany. Schulze, D. (1995). Zur FlieUf&ahigkeit von Sch&uttg&utern — De8nition und MeUverfahen. Chemie-Ingenieur-Technik, 67(1), 60–68. Schulze, D. (1996). Flowability and time consolidation measurements using a ring shear tester. Powder Handling & Processing, 8(3), 221–226. Schwedes, J. (1996). Measurement of ow properties of bulk solids. Powder Technology, 88, 285–290. Schwedes, J. (2000). Testers for measuring ow properties of particulate solids. Powder Handling & Processing, 12(4), 337–354. Schwedes, J., & Schulze, D. (1990). Measurement of ow properties of bulk solids. Powder Technology, 61, 59–68. SSTT, (1989). Standard shear testing technique for particulate solids using the Jenike shear cell. UK: Institution of Chemical Engineers.
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Consolidation and ow of cohesive bulk solids J&org Schwedes ∗ Institute of Mechanical Process Engineering, Technical University Braunschweig, Volkmaroder Stra e 4=5, 38104 Braunschweig, Germany Received 14 May 2001; received in revised form 11 September 2001; accepted 20 September 2001
Abstract The strength and the ow properties of bulk solids can be determined by performing shear tests. A shear test always consists of two parts, consolidation and measurement of strength. The measured strength depends on the way and degree of consolidation. During consolidation a stress history is impressed on the bulk solid sample which only vanishes after steady state ow. Since the consolidation is normally done nonhydrostatically, the strength will depend on the direction of stress application (anisotropy). Thus, the strength of a bulk solid sample strongly depends on its stress history and it can show anisotropic behaviour. These e1ects have to be considered when comparing available shear testers. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Particulate materials; Consolidation; Flow; Anisotropy; Shear tester
1. Introduction Today many testers, methods and ideas exist to measure the strength and the ow properties of bulk solids. It is often tried to compare the testers and to declare which tester can be best used for which application. For those, who have not to deal regularly with bulk solids and their characterisation it is hardly possible to decide which tester is the best for a special application and why it is like that. If a bulk solid sample has to be tested, it 8rst has to be consolidated in some way (8rst part) and afterwards it can be tested regarding its strength or ow properties (second part). What many people do not want to accept is the fact that the second part of the test very much depends on the 8rst part. Only if the 8rst part is identical, it can be expected that the second part gives similar results, even if di1erent testers are used. If the 8rst part is di1erent, di1erent results have to be expected even if the same tester is used. This paper will concentrate on what is going on during consolidation and how this in uences the subsequent failure test. Only with those knowledges available testers can be judged regarding their capabilities and suggestions are possible, which tester to use for which application. 2. Consolidation and steady state ow For a complete description of both parts of a test, consolidation and failure, the complete state of stress and the ∗
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complete state of strain have to be known. In bulk solids technology small stresses prevail as compared with the stresses being of interest in soil mechanics. For the low stress region the true biaxial shear tester in Fig. 1 best ful8ls the requirements mentioned above (Harder, 1986; Schwedes & Schulze, 1990; Nowak, 1993; Feise, 1996,1998; Schwedes, 1996, 2000). In the biaxial shear tester the bulk solid sample is constrained in lateral x- and y-direction by four steel plates. Vertical deformations of the sample are restricted by rigid top and bottom plates. The sample can be loaded by the four lateral plates which are linked by guides so that the horizontal cross-section of the sample may take di1erent rectangular shapes. In deforming the sample, the stresses x and y can be applied independently of each other in x- and y-directions. To avoid friction between the plates and the sample, the plates are covered with thin rubber membranes. Silicon grease is applied between the steel plates and the rubber membranes. Thus, it is ensured, that no shear strains or shear stresses can develop on x-, y- and z-planes. Therefore, all measured normal stresses and normal strains must be principal stresses and principal strains. This means that the experimental results can fully be described in the principal stress and strain space. The line connecting all measured states in the principal strain space is referred to as a “strain path”. Similarly, a “stress path” is de8ned in the principal stress space. As the biaxial shear tester allows deformations in the x–y-plane only, the strain path must lie completely in the x –y -plane. The stress path is, however, a general three-dimensional curve in the x –y –z -plane. Nevertheless, for convenience the diagrams will be limited to two dimensions.
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 3 7 8 - 5
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Fig. 1. Biaxial shear tester.
Experiments with a linear strain path belong to the simplest deformations possible in the biaxial shear tester. The experiment starts with = 0 and progresses with a constant deformation rate in x- and y-directions. As the ratio of ˙y = ˙x
remains constant, these strain paths are referred to as proportional strain paths. For proportional strain paths the ratio ˙y = ˙x equals to the ratio y =x . Therefore, in Fig. 2 the y =x -ratio is used to characterise the depicted strain paths. The resulting stress path is linear and proportional. For each strain path the measured stress paths from two independent experiments are shown. One can clearly see that a distinct direction of the stress path can be correlated to each direction of the strain path. These paths are therefore called “associated stress and strain path”, their directions “associated directions”. The broken straight line in Fig. 2 (left) with an inclination of “−1” connects states of strain where the samples have undergone a volume reduction of 15%. Depending on the selected strain path, di1erent stresses x , y and (not shown) z are necessary to reach this volume change. In Fig. 3 the results from experiments are shown, in which the strain path shows a sharp bend. In the 8rst part of the experiment the sample was consolidated with y =x =1:0. After 10%, 12.5% and 15% of volume change the y-plates were stopped and deformation commenced only in the x-direction (curves C0 –C2). The measured stress paths are made up
Fig. 2. Consolidation under proportional deformation (limestone: x50:3 = 4:8 m).
Fig. 3. Asymptotic behaviour (limestone: x50:3 = 4:8 m).
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Fig. 4. Bulk solids response under pure shear loading (limestone: x50:3 = 4:8 m).
of two parts as well. Initially the stress paths follow the associated direction already known from Fig. 2. After the change in the respective strain path, however, y decreases rapidly and the stress paths 8nally evolve parallel to the stress path A2, which is associated with the strain path with y =x =0. The ordering of the stress path C0 –C2 with respect to A2 follows the amount of volume change during the 8rst part of the strain path. This behaviour is called “asymptotic behaviour”. Again, di1erent stresses are necessary to get the same volume reduction of 15% (broken straight line). But additionally to Fig. 2 the 8nal stresses depend on both parts of the strain path. Many di1erent strain paths consisting of several (one, two and even more) parts with different strain rate ratios are possible. The resulting state of stress at the end of consolidation, e.g. to get a volume reduction of 15%, depends very much on the strain and=or stress history. An experiment leading to steady state ow has to guarantee a deformation at constant volume. In the biaxial shear tester this requires ˙y = ˙x = y =x = −1, and this deformation is called “pure shearing”. To perform a pure shearing test, the sample always has to be consolidated before it can be subjected to pure shearing. Fig. 4 shows the stress and strain paths of six experiments with pure shear loading. Starting from the same initial bulk density, the samples have been subjected to consolidation up to two stress levels under three initial strain path directions. One can see that the stress paths initially follow the directions known from Fig. 2. As soon as the shear part of the strain path begins, the stresses decrease down to some stationary level. Subsequent shearing results in scatter around this point but no significant change in stress. This behaviour—deformation without change in volume and stress—is considered as steady state ow. From Fig. 4 one can also see the in uence of the loading history. While six di1erent consolidation paths have been used, only two levels of steady state ow arise. Stress paths, which belong to strain paths with the same level of volumetric strain, end at the same steady state stress level. Further experiments have shown that all steady state stresses lie on a line through the origin, the “critical state line”.
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Fig. 5. Jenike’s shear cell.
Fig. 6. Yield locus and e1ective yield locus.
In steady state ow the sample deforms at constant bulk density and constant stresses. As can be seen in Fig. 4, the 8nal state (stresses x and y ) is dependent only on the bulk density and is independent of the stress history, i.e. independent of the strain and stress paths. Thus, in steady state ow the sample has lost its memory of the stress history, whereas in all other stress states prior to steady state ow the state of stress depends on the stress and=or strain history. 3. Incipient failure and strength Before reporting on the second part of the test in the biaxial shear tester it shall shortly be described how a shear test is performed with Jenike’s shear tester, being the most common tester in bulk solids technology for practical application (Jenike, 1964; SSTT, 1989). The shear cell (Fig. 5) consists of a base (A), a ring (B) resting on the top of the base and a lid (C). Base and ring are 8lled with a bulk solid sample. A vertical force is applied to the lid. A horizontal shearing force is applied on a bracket attached to the lid. Running shear tests with identically “preconsolidated” samples under di1erent normal loads results in maximum shear forces S for every normal load N . Division of N and S by the cross-sectional area of the shear cell leads to the normal stress and the shear stress , respectively. Fig. 6 shows a , -diagram. The curve represents the maximum shear stress the sample can
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Fig. 7. Procedure to get a yield locus with Jenike’s shear tester.
support under a certain normal stress and is called the yield locus. The parameter of a yield locus is the bulk density b at preconsolidation. With higher preconsolidation loads, the bulk density b increases and the yield loci move upwards. Each yield locus terminates at point E in the direction of increasing normal stresses . Point E characterises the steady state ow, which is the ow with no change in stresses and bulk density. Two Mohr stress circles are shown. The major principal stresses of the two Mohr stress circles are characteristic of a yield locus, that is, 1 is the major principal stress at steady state ow, called major consolidation stress, and c is the uncon8ned yield strength. Each yield locus gives one pair of values of the uncon8ned yield strength c and the major consolidation stress 1 . Plotting c versus 1 leads to the ow function (see later, Fig. 10). The angle ’e between the -axis and the tangent to the larger Mohr stress circle, called e1ective yield locus, is a measure for the inner friction at steady state ow and is very important in the design of silos for ow (Feise, 1996; Schwedes, 2000). To get the yield locus in Fig. 6, the bulk solid sample is sheared by two steps (Jenike, 1964; SSTT, 1989): In the 8rst, often called “preshear”, the sample is sheared under a normal stress sf until steady state ow with = sf =const. prevails, thus leading to the larger Mohr stress circle in Fig. 6. After steady state ow has been obtained, which is indicated by a constant shear stress sf , the shear stress is reduced to zero, the normal stress is reduced to ¡ sf and the second step of the shear process, often called “shear”, is performed by applying again a shear force on the bracket (attached to the lid C in Fig. 5). The maximum in the path of shear stress versus shear strain gives one value of the yield locus. To get further points the procedure has to be repeated with new samples, the same normal stress sf during “preshear”, but with di1erent normal stresses ¡ sf during “shear”. The procedure is shown schematically in Fig. 7. To get further yield loci the normal force during preconsolidation and “preshear” has to be increased or decreased and the procedure has to be repeated. The biaxial shear tester can also be used to obtain a yield locus corresponding to Figs. 6 and 7: The minor principal
Fig. 8. Shear at constant 2 in the biaxial shear tester.
Fig. 9. Yield locus gained with the biaxial shear tester.
stress 2 , e.g. acting in y-direction (Fig. 1), is kept constant during a test (Fig. 8). In x-direction, a positive strain rate ˙1 is applied resulting in an increasing major principal stress 1 . 1 is increased continuously up to the point of steady state ow, where constant values of 1 and b are reached. After this “preshear” the state of stress is reduced to a smaller 2 value (2 = 2; b in Fig. 9) and the second step of the shear process is performed by applying again a positive strain rate ˙1 in x-direction resulting in increasing 1; b -values. 2; b and the maximum value of 1; b de8ne the Mohr stress circle of incipient failure, thus belonging
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Fig. 11. Sample consolidation in the biaxial shear tester.
Fig. 10. Flow function (limestone: x50:3 = 4:8 m).
to the yield locus. Furthermore, Mohr stress circles of failure can be obtained by repeating the procedure with identical values of 1 and 2 during “preshear” and “shear” with di1erent values of 2; b ¡ 2 . By setting 2; a = 0; the uncon8ned yield strength c = 1; a can be measured directly (Fig. 9). Comparative tests with the same 8ne limestone powder were performed with Jenike’s shear tester and the biaxial shear tester following the described procedures with “preshear” and “shear” (Harder, 1986; Schwedes & Schulze, 1990; Schwedes, 1996). In Fig. 10 the ow function is given, which is the plot of uncon8ned yield strength c versus major consolidating stress 1 at steady state ow. Although two di1erent kinds of shear testers were used, the measurements are in agreement. It was also shown that with a ring shear tester identical results can be obtained, when the tests are performed according to Fig. 7 (Schwedes & Schulze, 1990; Schulze, 1996; Schwedes, 2000). 4. Inuence of stress history, anisotropy For investigating the in uence of di1erent consolidation procedures on the resulting strength samples were consolidated in the biaxial shear tester di1erently up to the same overall bulk density (Harder, 1986; Schwedes & Schulze, 1990; Nowak, 1993; Schwedes, 1996, 2000; Feise, 1996,1998). Fig. 11 demonstrates three di1erent possibilities (I,II,III) of consolidating the sample in order to get the same sample volume and, hence, the same bulk density. In the case of procedure I the x-axis and in the case of procedure III the y-axis coincide with the direction of the major principal stress 1; c at consolidation. In the case of procedure II the major principal stress 1; c is acting in both directions. (With respect to Fig. 2 the strain rate ratio has been y =x = 0 (I), 1 (II) and ∞ (III).) After consolidation the samples were sheared in the following way (see
Fig. 12. Flow function from Fig. 10 and uncon8ned yield strength c versus major principle stress at consolidation 1; c (limestone: x50:3 =4:8 m).
Fig. 8): 2 in y-direction was kept constant at 2 = 0 and 1 was increased up to the point of failure, leading to the uncon8ned yield strength c . The results are plotted in Fig. 12 as c versus 1; c , with 1; c being the major principal stress at consolidation. The functions c = f(1; c ) corresponding to procedures I–III are below the ow function c =f(1 ), being identical with the results already shown in Fig. 10. It can be clearly seen that the three functions c = f(1; c ) underestimate the uncon8ned yield strength c which a bulk solid sample gains after steady state ow ( ow function). Procedure I is comparable to the procedure realised in uniaxial testers. In an uniaxial tester (Fig. 13) (Schwedes & Schulze, 1990; Maltby & Enstad, 1993; Schwedes, 2000) a sample is 8lled into a cylinder with frictionless walls and is consolidated under a normal stress 1; c . After removing the cylinder, the sample is loaded with an increasing normal stress up to the point of failure. The stress at failure is the uncon8ned yield strength c . Contrary to shear tests following the procedure shown in Fig. 7 steady state ow cannot be reached during consolidation, i.e. the Mohr stress circle will be smaller (Fig. 14) at the end of
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But isostatic consolidation is a rare event in industrial applications. 5. Judgement of available shear testers
Fig. 13. Uniaxial tester.
Fig. 14. Mohr stress circle for steady state ow (SF) and uniaxial consolidation (UC).
consolidation (Schulze, 1995; Schwedes, 2000). As a result, bulk density b and uncon8ned yield strength c will also be smaller compared to results gained with the procedure of Fig. 7. The test of Fig. 13 is often used to explain the ow function c = f(1 ). As can be clearly seen in Figs. 12 and 14 a uniaxial tester always underestimates the ow function. The function c = f(1; c ) of procedure II (Figs. 11 and 12) leads to even smaller uncon8ned yield strengths. Steady state ow ( ow function), uniaxial consolidation (procedure I) and the biaxial consolidation of procedure II thus produce di1erent strengths, although identical major principal stresses were used during consolidation. Consolidation procedures I and III are basically identical. Only the direction of measuring the uncon8ned yield strength is di1erent. Since the results deviate strongly, a remarkable anisotropic e1ect exists. Thus, the strength of a bulk solid sample depends on its “stress history” and even if this stress history is identical, it can show anisotropic behaviour. In Fig. 4 it was shown that a sample is loosing its memory of the stress history only in steady state ow. However, this does not imply that it behaves isotropically in the following failure test, and typically the anisotropic behaviour after steady state ow is most distinct. This can be explained with the stress ratio of minor to major principal stresses 2 =1 being the smallest after steady state ow consolidation (Fig. 14). Only when all three principal stresses are identical (iso- or hydrostatic consolidation), an isotropic behaviour might be expected.
From the foregoing it follows that identical results of shear tests in di1erent testers can only be obtained when the consolidation procedure (8rst step of a shear test) and the failure test (second step of a shear test) are absolutely identical. Di1erent consolidation procedures are not associated with any in uence on the failure test when at the end of consolidation the sample has reached steady state ow. The reported measuring procedures for Jenike’s shear tester (SSTT, 1989) argued that “preshear” up to steady state ow has to be performed as a check for the correct preconsolidation. This preconsolidation is supported by a twisting motion of the cover of the shear cell under a normal stress . Thus, shear stresses are induced in the sample acting tangentially. These combined normal and shear stresses might locally lead to a steady state ow, but the complete cross section of the shear cell is not in an identical steady state ow condition. This is because the directions of the major principal stresses during the twisting motion are di1erent at di1erent positions within the cell. Also it is not sure if steady state ow is really obtained at all positions. Thus, the 8rst step of a shear test — “preshear” to steady state ow (Fig. 7) — is not a check for a correct preconsolidation, but a necessary step to get steady state ow across the complete cross-section and in the same direction. Only this “preshear” guarantees a reproducible and a clearly de8ned consolidation where all in uences of the stress history are eliminated. The ow function (relationship between uncon8ned yield strength c and major principal stress 1 at steady state ow) can only be determined with testers in which both stress states can be realised and both stresses 1 and c act in the same direction — atleast nearly (Schwedes, 2000). Steady state ow can be achieved in Jenike’s shear tester, in ring shear testers, in a torsional shear cell, in biaxial shear testers and in a very specialised triaxial cell (Schwedes, 1996). The uncon8ned yield strength c can be determined by running tests in Jenike’s shear tester, in ring shear testers, in uniaxial testers and in biaxial shear testers. Therefore, only Jenike’s shear tester, ring shear testers and biaxial shear testers can guarantee the measurement of ow functions c = f(1 ) without further assumptions. All other procedures to get a dependence of the uncon8ned yield strength c on the major principal stress at consolidation 1; c (without reaching steady state ow) lead to smaller uncon8ned yield strengths. Those relationships can only be used as estimates of the ow function. But, when using these estimates for some applications (i.e. design of silos for ow), it has to be kept in mind that the predictions for the bulk solids behaviour are on the “unsafe” side.
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All other testers, which are discussed in detail in the paper (Schwedes, 2000), have de8ciencies. Either the state of stress in the sample is not homogeneous, not known or the state of failure can hardly be used to get quantitative values for strength or ow properties. 6. Application of measured ow properties 6.1. Design of silos for 6ow The best known and the most applied method to design silos for ow is the method developed by Jenike (1964). He distinguishes two ow patterns, mass ow and funnel ow, the border lines of which depend on the inclination or apex angle of the hopper, the angle ’e of the e1ective yield locus (Fig. 6) and the wall friction angle ’w between the bulk solid and the hopper wall. The angle ’w can easily be tested with Jenike’s shear tester, but also with other direct shear testers. To get reliable ow in and out of the silo it has to be designed in such a way that the strength of the stored bulk solid at no time and in no place is suRcient to form stable domes or stable ratholes. The most critical area in a silo is the area directly above the outlet. The bulk solid owing in the silo from top to bottom is subjected to di1erent stresses. According to these stresses it is consolidated and gets its strength. Above the outlet the ow is converging with decreasing stresses towards the outlet. Due to these decreasing stresses the strength of the bulk solid also decreases towards the outlet. A passive plastic state of stress prevails in this converging ow. At each point within this region steady state ow exists. As already explained only at steady state ow a bulk solid sample looses its memory of the stress history, i.e. any bulk solid element at a certain location in the convergent geometry is always exposed to the identical state of stress (in steady state ow) and hence has an identical strength independent of the stress history. This is independent of all the possible ways to reach this point. Closing the outlet and opening it again will decide if the strength of the bulk solid is large enough for doming to occur. The dome is stable if the stress in the dome (parallel to the bottom surface) is smaller than the uncon8ned yield strength which the bulk solid developed during steady state ow before closing the outlet. The uncon8ned yield strength can be gotten in shear tests if these are performed according to Fig. 7: The consolidation in the shear tester must ensure steady state ow, because only steady state ow characterises the stress history representing the state of stress in converging ow. Having received this stress history (“preshear”), samples have to be sheared to failure (“shear”) to get yield loci. The Mohr stress circle being tangential to the yield locus and going through the origin yields the uncon8ned yield strength as the major principal stress of this Mohr stress circle. The minor principal stress
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is zero, because the stress normal to the surface of a dome is zero. From this explanation it follows that a shear tester, used for the design of silos for ow, has to ensure steady state ow at “preshear” and reliable , -values in the low stress region at “shear” to get a yield locus, from which the uncon8ned yield strength can be derived. The anisotropic behaviour of bulk solids mentioned in Section 4 is of no in uence in the design of silos for ow. It was explained that for getting the yield loci and the ow function the directions of the major principle stresses during consolidation to steady state ow and during shear to failure have to be identical. During steady state ow in the hopper the major principle stress is normal to the hopper axis. In a stable dome above the outlet the uncon8ned yield strength also acts laterally to the hopper axis. Therefore, the ow function re ects reality in the hopper area. 6.2. Other applications The design of silos for ow is the only known application, where the stress history and its in uence on strength and failure is exactly known and can be simulated in shear testers. Shear tests yield the ow function — or time ow functions — describing the (uncon8ned yield) strength of a bulk solid as a result of consolidation. Consolidation is achieved at steady state ow with the major principal (consolidation) stress 1 . The minor principal stress follows from the Mohr stress circle being tangential to a yield locus at its endpoint. If a sample is stressed by an identical major principal stress 1 but without reaching steady state ow, the minor principal stress will be larger compared to the one at steady state ow. This Mohr stress circle is smaller and the sample is less consolidated and gains less strength, see Fig. 14. Thus, the highest strength which a bulk solid sample can achieve when it is stressed by a consolidation stress 1 results from steady state ow at consolidation. This highest strength is represented by the ow function. The strength a bulk solid sample gets in a speci8c application depends on the stress history, i.e. on the way the bulk solid sample is consolidated. If during consolidation steady state ow was not achieved, which is the most probable situation, besides silo ow, the strength will be smaller than indicated by the ow function. The strength of a consolidated sample that was not subjected to steady state ow depends not only on the 8nal stress state of consolidation but also on the stress path, i.e. on the way how the stress was applied. Many stress paths are possible and exist in true applications: uniaxial, biaxial, stepwise, cyclic, mixtures, etc. Only if the stress path in an application is known and this stress path is repeated or simulated in a shear tester, a shear test can predict the correct strength of the sample. Many situations exist in which bulk solids are compacted under their own weight, but without reaching steady state
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ow. The storage in sacks or bags, the storage in open storage piles for blending and the storage in silos having a at bottom have to be mentioned. The compaction in a silo with a at bottom is similar to the compaction in an uniaxial tester. Thus, the uniaxial tester (Figs. 13 and 14) might yield a more realistic value of the uncon8ned yield strength. The compaction of a bulk solid in a bag is di1erent, because the bag can expand in the horizontal direction. As a result the horizontal minor principal stress decreases and the size of the Mohr stress circle increases which results in larger values of bulk density and strength (see Fig. 14). For this application the uniaxial tester underestimates the strength. The strength of the bulk solid in the bag is somewhere in between the strength documented by the ow function and the strength following from an uniaxial test. In open piles for blending an expansion in horizontal direction is also not prohibited. Thus, a uniaxial test will also underestimate the strength. Basing a design on the ow function is considered to be on the safe side. Additionally, a strong in uence of anisotropy exists. Coming back to the above-mentioned storage in a at bottom silo, in bags and in open piles for blending, the prediction of a strength depends on the direction. For the at bottom silo the strength in vertical direction is larger than the strength in horizontal direction, following the dependencies of procedures I and III in Fig. 12. For the strength of a bulk solid in a bag compacted by bags lying on top, it was already explained that due to a horizontal expansion the strength in the vertical direction is larger as compared with procedure I but lower than the strength according to the ow function. It can be argued that the strength of the bulk solid in this bag in the horizontal direction is even smaller than the prediction from procedure III. In conclusion, stress history and anisotropic behaviour have a strong in uence on the strength of a bulk solid. Only if the stress history and the directions of the major principal stresses during consolidation and failure are known for the application and can be simulated in a shear tester, a reliable prediction of the strength is possible. Possible tendencies can be derived from plots like Fig. 12. The ow function describes the maximum strength a bulk solid can develop during consolidation. Therefore, a design for ow is always on the safe side, if it is based on the ow function. The extent of safety can only be checked, if the stress history and the directions of stress application at consolidation and failure are known. Design of silos for strength, calibration of constitutive models, quality control and qualitative comparisons of the ow properties of di1erent bulk solids are further applications of measured ow properties. These applications are
extensively discussed in (Schwedes, 1996, 2000) and shall not be repeated here. 7. Conclusion Identical results of shear tests in di1erent testers can only be obtained when the consolidation procedure (8rst step of a shear test) and the failure test (second step of a shear test) are absolutely identical. Di1erent consolidation procedures are only without any in uence on the failure test when at the end of consolidation the sample has reached steady state ow. For all other consolidation procedures an in uence of the stress history exists. Additionally bulk solids show anisotropic behaviour, i.e. the strength a bulk solid sample gains during consolidation depends on the direction of stress application. Thus, stress history and anisotropic behaviour have a strong in uence on the strength of a bulk solid. Only if the stress history and the directions of the major principal stresses during consolidation and failure are known for the application and can be simulated in a shear tester, a reliable prediction of the strength is possible. Such a correlation exists for the design of silos for ow. References Feise, H. (1996). Modellierung des mechanischen Verhaltens von Sch9uttg9utern. Ph.D. thesis, Technical University Braunschweig, Germany. Feise, H. (1998). A review of induced anisotropy and steady-state ow in powders. Powder Technology, 98, 191–200. Harder, J. (1986). Ermittlung der Flie eigenschaften koh9asiver Sch9uttg9uter mit einer Zweiaxialbox. Ph.D. thesis, Technical University Braunschweig, Germany. Jenike, A. W. (1964). Storage and 6ow of solids. Bulletin 123, Engineering and Experiment Station, University of Utah, USA. Maltby, L. P., & Enstad, G. G. (1993). Uniaxial tester for quality control and ow property characterization of powders. Bulk Solids Handling, 13(1), 135–139. Nowak, M. (1993). Spannungs-=Dehnungsverhalten von Kalkstein in der Zweiaxial-box. Ph.D. thesis, Technical University Braunschweig, Germany. Schulze, D. (1995). Zur FlieUf&ahigkeit von Sch&uttg&utern — De8nition und MeUverfahen. Chemie-Ingenieur-Technik, 67(1), 60–68. Schulze, D. (1996). Flowability and time consolidation measurements using a ring shear tester. Powder Handling & Processing, 8(3), 221–226. Schwedes, J. (1996). Measurement of ow properties of bulk solids. Powder Technology, 88, 285–290. Schwedes, J. (2000). Testers for measuring ow properties of particulate solids. Powder Handling & Processing, 12(4), 337–354. Schwedes, J., & Schulze, D. (1990). Measurement of ow properties of bulk solids. Powder Technology, 61, 59–68. SSTT, (1989). Standard shear testing technique for particulate solids using the Jenike shear cell. UK: Institution of Chemical Engineers.