Force prediction models for brittle and flow failure of soil by draught tillage tools

J. agric. Engng Res. (1984) 29, 51-60

Force Prediction Models for Brittle and Flow Failure Soil by Draught Tillage Tools

of

J. V. STAFFORD*

Models have been developed to predict the forces acting on tillage tools of simple shape. The models are a useful advance on previous prediction models because they take account of the different types of soil disturbance or failure that may be caused by a tillage tool. The effect of tool speed on forces, which can be very significant, is taken into account. It is shown that, in brittle failure, the effect of speed is due to inertial forces and rate effects on soil-metal friction. With flow failure, inertial effects are negligible and the large effect of strain rate on soil shear strength dominates. The models have been verified with force data from soil tank experiments.

1.

Introduction

Cultivator tines are used very widely in arable crop production for primary tillage, seedbed production, harrowing and weed control. In soil management, they are essential to many subsoiling and drainage operations. For efficient operation, tines should be designed specifically for a given function. Rational design must be based on a knowledge of tine performance, the significant tine and soil parameters and the means to predict performance, Since the rigid chisel tine was introduced as a cultivation tool there have been many experimental studies of its performance, exploring the effects of both soil and implement parameters. There have been some attempts to model the behaviour mathematically so that the soil forces generated by tine movement may be predicted. The nature and extent of soil disruption or failure caused by the movement has received less attention and, currently, quantitative prediction models are lacking. This paper considers force prediction models and their limitations. It presents two new models which take account of the soil failure mode and implement speed, and are therefore of wider application than previous models. 2.

Performance 2.1.

of tines

Soil failure

As a tine moves, it causes permanknt deformation of the soil, which will be referred to as soil failure. In his classic work on narrow tines, Payne’ showed that a locahzed shear plane emanated from the base of the tine and formed a crescent-shaped block of failed soil. This type of failure has often been reported; soil may, however, fail in other ways. Elijah and Weber* reported four types of failure in front of flat cutting blades; shear-plane, flow, bending and tensile. Olson and Weber3 and Stafford4 showed that the type of failure can change from “shear” to “flow” as the implement speed is increased. Stafford4 identified shear failure with conditions of low confining stress around the tine and flow failure with high confining stresses. He then described the failure of soil by tines in terms of the critical state model of soil mechanics (Atkinson and Bransbys). *Sod Mechamcs. Received Paper

National

25 August

presented

lnst~tute

1983; accepted

at AG ENG

of Agricultural m revved

84. Cambridge.

Engmeermg,

form 29 October U.K.. l-5 Aprd

Silsoe, Bedford

MK45

4HS

1983 1984 51

0021 8634.841010051

+ IO $03.00/O

0

1984 The British

Soaety

for Research

in Agncultural

Englnecnng

52

SOIL

FAILURE

MODELS

NOTATION

;: C

c. F

F, FH’ FV

g K,, k N,, N,, IV., N, 4 r, s, r V Z

c( :

cross-sectional area of furrow formed by interface width of interface soil cohesion soil-interface adhesion force on interface of unit width force due to acceleration of loosened soil horizontal and vertical force, respectively, on interface acceleration due to gravity nondimensional coefficients in force models non-dimensional coefficients in Eqn (1) surcharge on soil surface constants horizontal velocity of interface depth of interface rake angle (to forward horizontal) of interface soil specific weight soil-interface friction angle strain rate soil internal friction angle

2.2. Prediction offorces A number of force prediction models have been developed directly or indirectly from theories of soil failure associated with civil engineering structures. They have been based primarily on passive earth pressure theories for movement of vertical retaining walls and bearing loads for footings where a dead wedge of soil is formed. The models of Payne’ and O’Callaghan and Farrellye were of this kind. Reece7 proposed a simple additive equation of the form F = yz2N, + czNc + cazNa+ qzN4,

. ..(l)

representing the gravitational, cohesive, adhesive and surcharge contributions to the force, F, acting on an interface of depth, z. This equation was applied to a wide (i.e. two-dimensional failure) cutting blade by Hettiaratchi et al! with the N factors determined on the basis of a failure plane of logarithmic spiral shape. A later, more rigorous solution (Hettiaratchi and Reeces) used the methods of Sokolovskil” to solve the equations of equilibrium for the coefficients of the four terms in Eqn (1). Where the interface is narrow compared with its depth, a three-dimensional analysis is necessary and semi-empirical models have been proposed by Hettiaratchi and Reece” and Godwin and Spoor.12 However, the more rigorous two-dimensional models have been shown to predict interface forces more accurately in some three-dimensional cases.13 The critical soil property in all these models is shear strength (Staffordla) and force prediction is much less sensitive to self-weight, interface adhesion or the assumed shape of the failure surface.14 Accurate measurement of shear strength parameters is thus a prerequisite to the use of the models described. These models all make the three assumptions. (1) Yielding of soil in shear obeys the Mohr-Coulomb criterion.

J. V. STAFFORD

53

(2) A distinct rupture surface forms in front of the tine, bounding a volume of soil in a state of plastic equilibrium. (3) Rate effects on the relevant soil parameters are negligible. Assumption (1) has been shown to be approximately true for all soils, except under some extreme conditions such as very high normal stress. The assumption of a distinct localized failure surface is, however, often not justified. A distinct failure surface is associated with brittle failure occurring in dry, compact soils. Where soil is wet or confining stresses are high, then the soil yields plastically with the degree of deformation falling off with increasing distance from the interface. Where a distinct failure surface is formed, soil within the failure surface is observed experimentally to undergo very little plastic yielding. Initial failure leads to movement of the block of soil, bounded by the failure surface, as a whole. This tendency for “dead” zones to form within the soil failure boundary has been recognized by Hettiaratchi and Reece.‘s They modified their earlier model&g to take account of these boundary wedges. Similarly, the assumption that rate effects are negligible is not reasonable for prediction of tine forces at practical cultivator speeds, The draught force of mouldboard ploughs is known to increase in proportion to the square of speed due primarily to soil inertial forces. The draught forces of narrow tines has also been shown to increase significantly with speed,‘3 but the relationship depends on soil and implement parameters. Of equal importance is the fact that the mode of soil failure may change with increasing speed.sp 4

3.

Effect of deformation rate 3.1. Draught force

A number of studies have been reported on the effect of tine speed on draught forces and these have been summarised by Stafford. 13 Draught force generally increases with speed but some studies (see, for example, Payne’) have observed no or very little speed dependance. Stafford*13 reported a systematic study of the effect of speed under controlled conditions in a soil tank, and showed that both the draught force magnitude and the relationship between speed and force depended on the mode of failure. With brittle failure, a second-order polynomial could be fitted to the force (Ftspeed (u) relationship, i.e. force increases at an increasing rate with speed. With flow failure, the relationship could be satisfactorily approximated by an exponential expression : F =

Y +

s

exp ( - tu),

. . .(2)

i.e. force increases at a decreasing rate with speed and tends towards an asymptotic value. [The two types of curve are typified by Figs 2 (left) and 3, respectively, in section 5.1 The type of failure depended on confining stresses around the tine4 and so varied with such soil and tool parameters as initial density, moisture content, rake angle and speed. The forcespeed relationship was therefore discontinuous where a change in failure mode occurred at a speed within the range of observation (5 mm/s to 5 m/s). It appeared unlikely that one force prediction model would cope with the two very different failure conditions. 3.2. Soil strength Considering Eqn (l), it is reasonable to expect that the effect of speed on draught force is due to rate dependence of one or more of the soil properties. Stafford and Tanner14 l7 have shown that both soil shear strength and soil-metal friction are rate-dependent. Their experiments showed that a Mohr-Coulomb model could be applied to both properties and they further showed that it

54

SOIL

FAILURE

MODELS

was cohesion and soil-metal friction angle that were rate-dependent. For both cohesion (c) and soil-metal friction angle (6) the relationship with strain rate (i) was of a logarithmic form: cor6=r+slOg(l

+ti).

. . .(3)

Theexperiments on which the cohesion model was based were carried out under conditions of high coniining stress in the soil under shear. It was shown in the same paper that a reduction in confining stress led to lower values of cohesion. Results also suggested that the effect of strain rate was less at the lower confining stress. Aref et al ?* showed in dynamic compression tests that strain to failure increased with strain rate under confined conditions. There was no effect of strain rate on strain to failure or on failure stress in unconfined tests.

4.

Proposed force prediction models

4.1.

Brittlefailure

Where brittle failure occurs in the soil around the tine to the full depth of operation then conditions are well-defined up to the point of failure and it is quite acceptable to carry out an equilibrium analysis of forces after the manner of Hettiaratchi and ReeceP A shear plane is assumed to run from the base of the tine to the soil surface ahead of the tine, meeting the soil surface at an angle of (45 - 4/2) such as to minimize the horizontal force (Fig. 1). A succession of shear planes are formed as blocks of soil separate from the soil mass and so the forces on the tine are of a periodic nature. It has been shown13 that the cohesive term of Reece’s additive equation7 accounts for !N”/, or more of the peak force exerted on a tine, and so the quasi-static forces may be approximately predicted by horizontal: F, = b(czK,) sin (M+ 6)

. ..(4a)

vertical : F v = b(czK,) cos (a + 6).

. . .(4b)

The coefficient, K,, is a function of CC, 6 and 4 and has been computed for a range of values by Hettiaratchi and Reece.s It has been shown in the last section that there is some evidence that soil shear strength is not strain-rate-dependent under unconfined conditions, i.e. in the soil conditions that prevail in brittle failure. Soil forces on a tine would then vary with speed due to the effect of speed on soilmetal friction angle (6) and soil inertia forces. 6 would then appear in Eqns (4a) and (4b) as a logarithmic functionJ7 The inertial forces may be calculated from consideration of the movement of soil blocks along the shear plane (Fig. 1). If the blocks of soil are accelerated in a direction

-rr__:1:: Fig. I. Brittlefailure geometry. (Left) Soil interjace at initial failure;

(right) periodic creation of soil blocks

J. V.

55

STAFFORD

of (45 - +/2)” to the horizontal then, by consideration of the rate of change of momentum, it may be shown that the horizontal component of the inertial force is F _‘Q y2 sin ci cos (45 - 412) (1-sin (a + 45 - 4/2) ’ g

. . .(5)

Stafford13 has shown that the cross-sectional area of the furrow formed by a tine increases with speed and so a in Eqn (5) is also a function of speed. The prediction equations for forces acting on a tine are then sin a cos (45 - 4/2) horizontal : F, = b(czK,) sin (N+ 6) + :‘a vz sin (U+ 45 - d/2) ’ g sin tl sin (45 - d/2) vertical: F, = b(czK,) cos (u + 6) + ra v2 sin(cc+45-4/2)’ g

”

”

44

where K,, 6 and a are functions of tine speed. 4.2. Flow failure In flow failure, no distinct shear planes are evident in the soil around the interface. Deformation of soil is accommodated primarily by compression into the sides and base of the furrow and by generalized flow nearer to the free surface. Plastic yielding occurs throughout the soil mass, with soil displacement decreasing with distance from the tine. Boundary conditions can only be defmed around the tine and at the free soil surface. The soil forces acting on the interface are proportional to the yield stress of the soil and to the width (b) and working depth (z) of the interface. Cohesion is the yield stress of the soil in shear at zero normal stress. The resultant force again acts at an angle, 6, to the interface. In flow failure, the tine cuts a furrow the size of its projected cross-sectional area or causes soil to flow around the tine. As soil blocks are not being formed and accelerated to the speed of the tine, inertia forces are much lower than for brittle failure. However, the soil-metal friction angle, 6, and soil cohesion are highly strain-rate-dependent16p ” and, hence, tine forces may vary with speed. The approximate force prediction equations for flow failure are then horizontal: F, = bz[r + s log( 1 + tv)]k sin(a + 6) vertical: F, = bz[r + s log(1 + tv)]k

COS(LY+ 6).

. ..(7a) . .(7b)

6 is a function of tine speed, v. The coefficient, k, cannot be determined analytically at present, as in the case of K, in brittle failure, because the failure boundary geometry is indeterminate. Eqn (7) is similar in form to the lateral failure equation of Godwin and Spoor.12 However, they considered the particular case of the part of a tine working below the “critical depth”, and assumed that the failed soil was bounded by a shear surface of logarithmic spiral shape. 5.

Verification of models

In order to test the validity of the models, experimental data from a comprehensive study of the performance of chisel tines, reported by Stafford,‘3 were used. Rigid plane tines (40 mm wide) of 45” and 90” rake angle and 150 mm depth were used. The experiments were carried out in a laboratory soil tank in two contrasting soils over a range of moisture content. The soil properties required as input data for the models were measured in the soil tank. The type of soil failure, whether brittle or flow, was noted for each experiment.

SOIL

56

FAILURE

MODELS

L

Speed,

m/s

Fig. 2. Predicted and measured draught forces in brittle failure. Tine rake angle: @, - 45”; A, -90” (Left) Sandy clay loam; (right) clay

TABLE 1

Values of k [EAp (7)] for various soil conditions

I

Clay

I

28.8 38.7

I

90 45 90

10.28 7.25 8.18

The model for predicting forces in brittle failure was applied to a sandy clay loam soil at 12.6% w/w moisture content (speed range 5 mm/s to 5 m/s) and to a clay soil at 18.2% w/w moisture content (speed range 5 mm/s to 2 m/s). The experimental points and predicted curves for draught force are shown in Fig. 2 (left) and (right). The small “down-turn” at low speed on all predicted curves was due to the rapid decrease in soil-metal friction angle. The experimental data appeared to follow the same trend, although there were insuthcient data to be conclusive. At higher speeds, inertial forces dominated and the curves turned up. Overall, the prediction agreed well with the experimental data The model for predicting forces in flow failure was applied to results for a sandy clay loam at 17% w/w moisture content and to a clay soil at 28.8% and 38.7% w/w moisture content. Soil strength parameters for the model were based on results given by Stafford and Tanner14 17for the effect of strain rate on cohesion and angle of soil-metal friction. Cohesion was scaled by the quasi-static value measured in the soil tank by a torsional shear annulus. The coefficient, k, in Eqn (7) was determined by matching the predicted force to a best-fit curve through the experimental points at a speed of 5 m/s. The values of k are given in Table 1. The predicted force curves for the sandy clay loam are shown in Fig. 3. The failure mode for CI= 45” was not clearly of the flow type. Cine film of the soil around the tine indicated some brittle and some flow failure whilst the draught forc+speed curve was typical of flow failure conditions. The value of k was different from that for c(= 90”. The prediction was good for c( = W, where the failure mode was clearly of the flow type.

51

J. V. STAFFORD

,

Speed, m/s

Fig. 3. Predicted and measured draught force in pow failure in sandy clay loam. Tine rake angle: @, - 45”; A, - 90”

. .

.

.

.

Ah 8.

2-

OO

I

2

3

4

5

0

I

2

3

4

5

Speed, m/s

Fig. 4. Predicted and measured draught force in flow failure in clay. Moisture content: (lejt) 2880/, w/w (right) 3X7% w/w. Tine rake angle: @, - 45”; A, - 90”

The clay soil at 28.8% w/w moisture content was also in a region of transition in failure mode. Cine film of the soil failure and draught force-speed curves%‘3 indicated that brittle failure occurred at a = 45”. At a = 90“, failure was mainly of the flow type but there was some evidence of brittle failure. It was not surprising then that the force curve for a = 90” [Fig. 4 (left)], predicted from the flow failure model, did not fit the experimental points very well. A notable feature of the experimental results for clay at 38.7% w/w moisture content (and also at 48-9%13)was that the draught force for a vertical tine was lower than for the forward raked tine, as shown in Fig. 4 (right). With brittle failure, draught force invariably increases with rake angle and is well modelled by Eqn (6). As a check that the effect of rake angle in flow failure was

SOIL

58

FAILURE

MODELS

not due to an experimental artefact, a small blade was pulled through a block of Plasticine in a model box. For rake angles of 45”, 90” and 135” (at constant depth) the draught forces were 110, 90 and 70 (arbitrary) units, respectively. When the Plasticine was replaced with hard-packed soil that failed in brittle mode, the forces for CI= 45” and 90” were 23 and 43 units, respectively. Eqn (7) in fact, allows for a rising or falling draught force-rake angle relationship, depending on the value of 6. The prediction curves [Fig. 4 (right)] fit the experimental data very well with similar values of k for tl = 45” and 90”. Because of the low quasi-static value of 6, predicted draught force is similar for both rake angles at low speed. As speed increased, the value of 6 increased and the force curves for c(= 45” and 90” diverged. For the three cases where failure was distinctly of the flow type [Fig. 3, CI = 90” and Fig. 4 (right)], the values of k were similar. This (empirically determined) value may be substituted for k in Eqn (7). 6.

Discussion

Overall, the predictions of the proposed brittle failure and flow failure models were in good agreement with the experimental data. Prediction was less accurate in mixed failure conditions, but still better than with previous models. The models are clearly sensitive to soil parameters, the most important being c and 6 and, in the case of brittle failure, 4. Whilst 6 can be measured accurately, the apparent values of c and $J vary with the measurement method19 and with the degree of soil confinement in the measuring equipment ?8 Empirical determination of the coefficient k in Eqn (7) is not very satisfactory and the indication that its value may lie between 7 and 8 (Table 1) is based on insufficient evidence. Analytical determination must resort to the theory of plasticity but, as in metal cutting theory, this is not possible at present. In the meantime, the model provides a good guide to the draught forces in flow failure and their variation with interface speed. The models predict forces over a wider range of conditions than previous quasi-static models and are therefore of more general use. Under brittle failure, quasi-static conditions, any of the previous models mentioned give reasonable predictions. However, even with modest cultivation speeds, the models presented in this paper give improved prediction. In most cultivation operations, brittle failure is required but, in practice, both types of failure frequently occur. In certain operations, such as creating mole drains, flow failure is required. Under these conditions, use of one or both models should give improved prediction over previous models. The models were tested successfully using force measurements in remoulded soils. Because of the heterogeneity of field soils and the consequent variation in force measurements, field data were not suitable for this initial validation. In principle, the models are applicable to undisturbed field soils provided that the failure mode is known. In practice, though, soil heterogenity will frequently lead to mixed failure conditions where the models do not predict so reliably. Because of the cemented nature of undisturbed soils, the extent of each failure mode will be different from that under remoulded conditions. To use the correct model in a given situation, the failure mode must be known. As shown in Reference (4) the type of failure is determined by the confining stresses in the soil around the tine. These stresses depend on implement speed, geometry and depth and soil moisture content and initial density. Whilst it can be stated that flow failure is likely to occur at high moisture content (above the plastic limit) or high speed, the transistion point is determined by a combination of the above parameters. It is not possible, at present, to predict the transition point accurately, but Reference (4) gives guidelines for two particular contrasting soils. 7.

Calculation procedure

In order to estimate the forces acting on a simple implement under practical conditions, the

J.

V.

59

STAFFORD

likely failure mode must be determined from Reference (4) or by visual observation. may then be calculated as follows.

Forces

Brittle failure

1. Determine soil parameters y, c, 4 and 6 (quasi-static values) and implement parameters v, a, z and b. 2. Calculate value of 6 at speed v from Reference (17). 3. Calculate value of a at speed v from Reference (13). 4. Calculate K, using c(,4 and dynamic value of 6 from Reference (9). 5. Calculate forces on implement by Eqn (6). Flow failure

1. Determine soil parameters y, c, 4 and 6 (quasi-static values) and implement parameters v, x, z and b. 2. Calculate value of c and 6 at speed v from References (16) and (17), respectively. 3. Assume a value of 7-8 for k until better data or analysis procedure are available. 4. Calculate forces on implement by Eqn (7). 8.

Conclusions

Improved models have been developed to predict the forces acting on simple tillage tools for application in brittle and flow failure conditions. The models take into account interface geometry, speed and soil strength. Validation of the models has been successfully achieved using soil tank force measurements where soil failure was distinctly of brittle or of flow type. Prediction under conditions of mixed failure was less accurate, but still gave much better than an order of magnitude prediction of forces. The models are a useful advance on previously published models, which do not take account of interface speed and only consider brittle failure conditions. REFERENCES

6 7 * g lo ” I2 l3 l4 l5

Payne, P. C. J. The relationship between the mechanical properties of soil to the performance of simple cultivation implements. J. agric. Engng Res., 1956 1 (1) 23-50 Elijah, D. L.; Weber, J. A. Soil failure and pressure patterns for flat cutting blades. Trans. ASAE, 1971 14 (4) 781-785 Olson, D. J.; Weber, J. A. Effect of speed on soil failure patterns in front of model tillage tools. S.A.E. Paper No. 650691, 1965 Stafford, J. V. An application of critical state soil mechanics: the performance of rigid tines. J. agric. Engng Res., 1981 26 (5) 387-l Atkinson, J. H.; Bransby, P. L. The Mechanics of Soil. Maidenhead: McGraw-Hill, 1978 O’Callaghan, J. R.; Farrelly, K. M. Cleavage of soil by tined implements. J. agric. Engng Res., 1964 9 (3) 259-270 Reece, A. R. Thefundamental equation of earth-moving mechanics. Proc. Symp. Earth-moving Machinery. I.Mech.E., 1965 179 (3F) 8-14 Hettiaratchi, D. R. P.; Witney, B. D.; Reece, A. R. The calculation of passive pressure in 2-D soil failure. J. agric. Engng Res., 1966 11 (2) 89-107 Hettiaratchi, D. R. P.; Reece, A. R. The calculation of passive soil resistance. Gtotechnique, 1974 24 (3) 289-3 10 Sokolovski, V. V. Statics of Soil Media. Oxford : Pergamon Press, 1965 Hettiaratchi, D. R. P.; Reece, A. R. Symmetrical 3-D soil failure. J. Terramech., 1967 4 (3) 45-67 Godwio, R. J.; Spoor, G. Soilfailure with narrow tines. J. agric. Engng Res., 1977 22 (4) 213-228 Stafford, J. V. 7’he performance of a rigid tine in relation to soil properties and speed. J. agric. Engng Res., 1979 24 (1) 41-56 McKyes, E.; Ali, 0. S. The cutting of soil by narrow blades. J. Terramech., 1977 14 (2) 43-58 Hettiaratchi, D. R. P.; Reece, A. R. Boundary wedges in 2-D passive soil failure. Gkotechnique, 25 (2) 197-220

60 ‘6 Stafford,

SOIL

FAILURE

MODELS

J. V.; Tanner, D. W. Effect of rate on soil shear strength and soil metal friction.

1. Shear

strength. Soil Tillage Res., 1983 3 (3) 245-260 l7 Stafford, J. V.; Tanner, D. W. Eflect of rate on soil shear strength and soil metal friction. II. Soil metal friction. Soil Tillage Res., 1983 3 (4) 321-330 l* Aref, K. S.; Cbaacellor, W. J.; Nielson, D. R. Dynamic shear strength properties of unsaturated soils. Trans. ASAE, 1975 18 818-823 lS Stafford, J. V.; Tanner, D. W. Field measurements of soil shear strength. Proc. 9th Cod. I.S.T.R.O., Osijek, Jugoslavia, June 1982,656-661