The general rheological model of paddy soils in south China

Journal ofTerramechanics, Vol. 23, No. 2, pp. 59-68, 1986. Printed in Great Britain.

0022--4898/8653.00+0.00 Pergamon Journals Ltd. © 1986 ISTVS

THE G E N E R A L R H E O L O G I C A L MODEL OF P A D D Y SOILS IN SOUTH CHINA PAN JUN-ZHENG*

Summary--Analyses and experiments show that the Burgers model can be accepted as a general rbeological model of paddy soils in South China, among which most are clays. Orthogonal tests and variance analyses show that the effect of soil conditions (mechanical composition, water content, bulk density) on rheological parameters is particularly significant, while that of loading conditions (shape and size of bearing surface, load) is practically not significant.

NOMENCLATURE o e K s e G ~7" f0 R,S,M,N u cf P A w w0 u0 E Ex,Eu,hx,hu ue u, ua Er KI,K2,K3 S

stress strain bulk modulus viscosity deviator stress deviator strain shear modulus apparent plastic viscosity yield stress operators displacement shape factor load area unit displacement initial unit displacement initial displacement modulus of elasticity

rheologicalparameters elastic deformation viscous deformation delayed deformation test error sums of parameters corresponding to levels of factors sum of squares of differences

INTRODUCTION MOST o f the p a d d y soils in S o u t h C h i n a are clays. In the r i c e - g r o w i n g s e a s o n f r o m t r a n s p l a n t i n g to h a r v e s t i n g , w h e n m o s t o f the a g r i c u l t u r a l m a c h i n e s a r e o p e r a t i n g in the fields, t h e w a t e r c o n t e n t o f the soils r e a c h e s a fairly h i g h v a l u e , c o m m o n l y n e a r o r h i g h e r

*Professor, Agricultural Engineering College, Nanjing Agricultural University, Nanjing, Jiangsu, People's Republic of China. 59

60

PAN J U N - Z H E N G

than the liquid limit, and the soils show outstanding rheological properties. That is, they show the properties of fluids in addition to those of solids. This paper recommends an acceptable general rheological model for these soils, according to both theoretical analyses based on soil mechanics and the experiments on several soil samples collected in South China. T H E R H E O L O G I C A L BEHAVIOR OF P A D D Y SOILS U N D E R VEHICLE L O A D I N G

In engineering practice, the behavior of clays is generally restricted to isotropic materials which follow linear laws with small strain. The stress-strain-time relations are for a homogeneous continuum, although sometimes considerations of clays as multi-phased systems will be inherent in the analysis. According to Schiffman [1], the fundamental volume change theory of clays is the Terzaghi theory of consolidation. This theory considers the clays to be simplified saturated structures of incompressible particles and incompressible fluids. From a rheological point of view, the Terzaghi theory develops volume change as a Kelvin, or Voigt, material. The stress-strain-time behavior is as follows: 6 eii oii = 3 K eii + 3~" 5--/-

(1)

where K = bulk modulus and ~" = viscosity. The deviator behavior of clays in terms of rheological properties has been described by Geuze and Tan [2]: (
si/ = 2Geij or

si] = ~rl* -

(>fo)

5ei/ 6t

Since the paddy soils in rice-growing season are much softer than ordinary surface soils, it is practically impossible to keep the deviator stress below f0, the yield stress of the Bingham material, when these soils are under vehicle loading. Therefore, the deviator stress-strain-time differential equation will be expressed as follows: si! = 2 77*

8eij ~t

(si/ > f o )

(2)

where r/* = apparent plastic velosity. The general linear viscoelastic stress-strain equation is: n ~, A n

n=0

6noi/ ~t n

-

m Y~ B m

m=0

6meii ~t m

(3)

This equation can be written in operator form of volume change and deviatoric components: R (Oil) = S (eli)

(4a)

M O D E L O F P A D D Y SOILS IN S O U T H C H I N A

M(si/) = N (ei/) .

61

(4b)

For clayey soils, the viscoelastic stress-strain law is:

ai/=

MS-NR 3RM

N ekkSi] + M el~ "

(5)

The displacement of a loaded plate on the surface of an elastic half-space can be expressed by the Schleicher formula [3]:

u =

aP

3K+4G

x/ff

4G(3K+G)

(6)

Where u = displacement of plate; a = shape factor; P = load on plate and A = area of plate. Since the operators R, S, M, N are linear, the elastic surface displacement equation is readily converted to a viscoelastic equation:

off, u = --

M (MS + 2NR ) (7a)

N(2MS+NR)

or

N (2MS+NR)w = M ( M S + 2 N R ) P where

w=

(7b)

ot

When a light standing vehicle is resting on the surface of a soft soil, which is equivalent to a loaded plate on the surface of a viscoelastic half-space, the volume change of the soil will be considered to act as for a Kelvin material, while the deviator to be for a viscous material. Thus, as stated previously,

Oii = 3Keii + 3~" - -

eii

(1)

6t

sii = 27/* 6el~

~t

(si/>fo).

Comparing the above equations with equations (4a) and (4b), we have R=I /i S = 3K + 3 { ' - 6t

(2)

62

PAN JUN-ZHENG M = 1

6 N = 27/* 6t

The viscoelastic equation (7b) becomes: 62 w

~t 2

3K

+

3K

6w

3~" + r/*

~t

-

P 4rl* + (3~'+ rT*)

(8a)

w ( 0 ) = Wo.

(8b)

The solution o f the differential equation (8) is as follows:

w = Wo + - -

+

(9)

1-e -3kt/(3¢+n*)

4r/*

4K

or

+ u = u°

(10)

( 1 - e -3kt/(3~+n*)

4~*

+ -~

4K

According to the Terzaghi consolidation theory, there will be no volume change of clays at the instant o f loading. That is, K = ,0, when t = 0. This leads to c~P Uo

=

~

1 ( l 1)

4G

The m o d u l u s o f elasticity E of an elastic material can be expressed in terms of bulk m o d u l u s K and shear m o d u l u s G, according to the theory of elasticity, as follows: 9KG E =

(12) 3K +G

When there is no volume change, equation (12) becomes: E = 3G.

(13)

By means of the above equation, equation (1 1) becomes: 3 u0 .

.

4

.

~P . . . x/,4

1

4

Let

XM=4r~*,Ek=4K,

(14)

E

~.k = - - ( 3 ~ ' + r / * ) , a n d E M 3

4

3

E.

MODEL OF PADDY SOILS IN SOUTH CHINA

63

equation (10) becomes:

U =

~r~

+

+

xz

--

(1--e-Ekt/hk)

.

(15)

ek

For the load-deformation relation stated in equation (15), the load-deformation model is a Burgers model, as shown in Fig. 1. The deformation in equation (15) consists of three parts, as shown in Fig. 2, namely: (1) Elastic deformation u,, which is the instantaneous deformation at the instant of loading. a~P

1

•

(2) Viscous deformation u~, which increases linearly with time. Uv=

~.p

1

~ o

xM (3) Delayed deformation ud, which approaches a certain definite value as a limit. ttP Ud

----

vo

l • ~

(l_e-Ekt/hk).

It is evident that equation (15) holds true only when the loaded plate is laid on the surface of a viscoelastic half-space, in which case the plate will sink continuously without limit as time elapses. Actually the displacement of the plate has a limit, especially when there is a hard bottom underneath. This is the case when a light vehicle is resting on or passing steadily through the paddy field which has a plow bottom. Consequently equation (15)should be modified when this happens. However, equation (15) practically holds true for the first few seconds, which corresponds to the time of contact of track plate or tyre on vehicles with the surface of the field.

+

J kK

t,,v u~ O

FIG. I.

Burgers model.

FIG. 2.

Deformation curve.

64

PAN JUN-ZHENG EXPERIMENTAL STUDIES

E x p e r i m e n t a l studies on the r h e o l o g i c a l characteristics o f soils, including p a d d y soils, have been m a d e b y G e u z e , T a n , G u p t a , S u d o , Y o s h i d a a n d others [2, 4-8]. H o w e v e r , for the soils in p a d d y fields, especially d u r i n g the rice-growing season, c o m p r e s s i o n tests o r torsion tests are p r a c t i c a l l y i m p o s s i b l e , because the soils are so soft t h a t no cylindrical specimens can be p r e p a r e d as usual. E x p e r i m e n t s on soft soils have been m a d e successfully by Yoshida. The m e t h o d p r o p o s e d by him has been followed by the a u t h o r a n d his colleagues, t h o u g h the a p p a r a t u s a n d i n s t r u m e n t s used are quite different. The principle o f this m e t h o d will be d e s c r i b e d as follows: A p l a t e is laid on the surface o f a p r e p a r e d soil s a m p l e in a soil bin. A weight is then a p p l i e d on the plate. T h e sinkage o f the plate with time is r e c o r d e d by m e a n s o f a differential t r a n s f o r m e r a n d an oscilloscope. Ten soil s a m p l e s collected in J i a n g s u , H u p e i , H u n a n , G u i z h o u , Sichuan a n d Y u n n a n P r o v i n c e s have been tested a n d the r h e o l o g i c a l p a r a m e t e r s o f these soils have been c a l c u l a t e d b y the a u t h o r a n d his colleagues [9]. One o f such e x p e r i m e n t s is s h o w n b e l o w as an example. The p r o p e r t i e s o f the soil s a m p l e s in this e x p e r i m e n t are listed in Table 1.

TAaLE 1. PROPERTIESOF SOILSAMPLES NO.

Soil type

Liquid limit (%)

Plastic limit (%)

I 2 3 4 5

Clay Clay Clay Heavy clay Clay

37.2 40.3 51.2 44.1 56.2

19.3 20.5 28.8 23.1 31.9

TABLE2. FACTORSWITItLEVELS Level I D: Soil conditions

A: Shape of bearing surface B: Area

C: Load Note:

Level 2

Level 3

Guanyun Clay CP: 68.3% WC: 47.5% BD: 1.676 g/cm 3

Wujin Clay CP: 50.3% WC: 46.7% BD: 1.677 g/cm 3

Yueyang Clay CP: 32.1% WC: 39.8% BD: 1.770 g/cm 3

Round 6 sq cm 9.81 Pa

Square 10 sq cm 14.71 Pa

Rectangle (1:3) 15 sq cm 19.61 Pa

CP, clay particle; WC, water content;

BD, bulk density.

M O D E L OF P A D D Y SOILS IN SOUTH CHINA

~c

E

o e.

m

X

S

Z O

I

i-

O ,,4

I

G x I

i

I

65

66

PAN JUN-ZHENG

The water contents of each sample in the above table are of four levels, which lie between a value higher than the plastic limit but lower than the liquid limit, and a value a little higher than the liquid limit. The loads in each test are of five levels, namely: 5.88, 7.85, 9.81, 11.77, 14.71 N. The area of the round disc plate is 6 sq cm. These are quite consistent with the soil conditions in paddy fields during the rice-growing season in South China. The constitutive equation (15) of the Burgers model is adopted to fit the test data with satisfactory results. The coefficient of correlation for eight typical curves chosen are R = 0.987, 0.985, 0.991, 0.961,0.992, 0.972, 0.957, 0.973 respectively, each of which approaches 1 very nearly. This means that equation (15) fits the test data well, and that the Burgers model can be accepted as a general model for the paddy soils in South China. Orthogonal tests and variance analyses are used to identify the effect of the soil conditions and the loading conditions on rheological parameters. The orthogonal tests are carried out according to the orthogonal list L27(313). The factors showing the soil conditions and the loading conditions with their respective levels are shown in Table 2. The tests made and the values of the parameters calculated are shown in Table 3. The values of Kl, /(2 and K3, which are the sums of the values of each parameter corresponding to the levels 1,2 and 3 of each source, and the value of S, which is the sum of squares of the differences, are then calculated. These values derived from the test data for factor A are shown in Table 4 as an example.

TABLE 4.

VALUESOF Kt. K:, K~ ANDS

A (Shape of bearing surface)

K, K2 K3 S

EM

XM

EM

Ag"

116.9 89.8 98.4 42.6

12305 19835 7160 9030650

314.7 301.2 231.0 448.7

1226 1168 758 14489

Additional variance analyses are then made, which is shown in Table 5. The variance estimate for each source, which is equal to the sum of squares divided by degrees of freedom, is divided by that from the test error to get the Snedecor's value F. From a "percentage points of the F-distribution table" one can easily find the degree of significance of each source. Table 5 shows that the effect of the soil conditions (mechanical composition, water content, bulk density) on rheological parameters is particularly significant, while that of the loading conditions (shape and size of bearing surface, load) is practically not significant.

MODEL OF PADDY SOILS IN SOUTH CHINA TABLE 5.

67

VARIANCEANALYSES

Parameter

Source

Sum of squares

Degrees of freedom

Variance estimate

F

E#t

A B AXB C AXC BXC D

42.6 71.5 71.5 10.8 99.1 26.9 2473.7

2 2 4 2 4 4 2

21.3 35.8 17.9 5.4 24.8 6.7 1236.9

0.6 1.1 0.5 0.2 0.7 0.2 36.7

Error A B AXB C AXC BXC D

202.4 9030650 3127058 7623925 6864887 2777167 8261735 63753301

6 2 2 4 2 4 4 2

33.7 4515325 1563529 1905981 3432444 694292 2065434 31876651

1.6 0.6 0.7 1.2 0.3 0.7 11.3

Error A B AXB C AXC BXC D

16984936 448.7 1571.2 579.9 1437.1 2304.1 890.6 19932.5

6 2 2 4 2 4 4 2

2830823 224.4 765.6 145.0 718.6 576.0 222.7 9966.3

0.4 1.5 0.3 1.4 1.1 0.4 18.8

Error A B AXB C AXC BXC D

3184.8 14489 21528 17232 50988 48536 20771 303842

6 2 2 4 2 4 4 2

530.8 7245 10764 4308 25494 12134 5193 151921

1.0 1.5 0.6 3.5 1.7 0.7 20.8

Error

43746

6

7291

Xu

Ex

Degree of significance

Particularly significant

Particularly significant

Particularly significant

Particularly significant

68

PAN JUN-ZHENG CONCLUDING REMARKS

The a n a l y t i c a l a n d e x p e r i m e n t a l studies on the r h e o l o g i c a l b e h a v i o r o f the p a d d y soils in S o u t h China, a m o n g which m o s t are clays, have been m a d e b y the a u t h o r a n d his colleagues in recent years. The following conclusions have been reached: (1) The v o l u m e change o f p a d d y soils, a c c o r d i n g to the Terzaghi t h e o r y o f c o n s o l i d a t i o n , is c o n s i d e r e d to act as for a Kelvin m a t e r i a l , while the d e v i a t o r b e h a v i o r to be for a viscous m a t e r i a l . The constitutive e q u a t i o n is r e a d i l y o b t a i n e d , which is the l o a d - d e f o r m a t i o n r e l a t i o n o f the Burgers model. (2) E x p e r i m e n t s s h o w that the Burgers m o d e l can be a c c e p t e d as a general m o d e l o f the p a d d y soils in S o u t h C h i n a d u r i n g the rice-growing season f r o m t r a n s p l a n t i n g to harvesting, when m o s t o f the m a c h i n e s are o p e r a t i n g in the fields. (3) O r t h o g o n a l tests a n d variance analyses s h o w t h a t the effect o f the soil c o n d i t i o n s on r h e o l o g i c a l p a r a m e t e r s is p a r t i c u l a r l y significant, while that o f the l o a d i n g c o n d i t i o n s is p r a c t i c a l l y n o t significant.

REFERENCES [1] R. L. SCflI~'M^N, Analysis of the displacements on the ground surface due to a moving vehicle, Proc. 1st International Conference on the Mechanics o f Soil-Vehicle Systems, 1961. [2] E. C. W. A. G~uzE and TAN TJONG-KIE, The mechanical behavior of clays, Proc. 2nd International Conference on Rheology, Oxford, 1953. [3] F. SCHLEICHER,Bauingenieur, Bd.7, 1926; Bd. 14, 1933. [4] C. P. GuPrA and A. C. PAr~OVA,Rheological behavior of soil under static loading, Trans. ASAEg(5) (1966). [5] R.B. RAMand C. P. GuPrA, Relationship between rheological coefficients and soil parameters in compression test, Trans. A S A E 15(6) (1972). [6] S. SuDo, R. YASUrOMIand F. YAMAZAKI,The mechanical behavior of soil and its state of stress, J. Soc. Material Science, Japan 17, 175 (1968). [7] ISAO YosltiD^, Experimental study of the trafficability of agricultural vehicle (I). J. Sot'. Agricultural Machinery, Japan, 32(I) (1970). [8] ISAOYOSrlIDA, Experimental study of the trafficability of agricultural vehicle (4). d. Soc. Agricultural Machinery, Japan, 34(3) (1972). [9] Lu ZE-JIAN,OI^N YAN-QIAO,PANJuN-ZHENG, Rheological characteristics of paddy-field soils in China (1), Trans. Chinese Soc. Agricultural Machinery (2) (1982). [10] PAN JUI~-ZHEraGet al., Rheological characteristics of paddy-field soils in China (4). Trans. Chinese Soc. Agricultural Machinery (3) (1983).