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Delayed elastic strain criterion for first cracks in ice
Sinha, N. K.
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DELAYED ELASTIC STRAIN CRITERION FOR
FIRST CRACKS I N ICE
by
N.K. Sinha
ANALYZED
Reprinted from
International Union of Theoretical and
Applied Mechanics
Symposium on Deformation and Failure of
N R C
-
CBLDC.
RES.
L I B R A R Y
03-
03-
? 1
B I B ~ I O T H ~ Q ~ E
Reeh. 5,iEim.
I'.
3 t $Granular Materials
Delft / 3 1 August
-
3 September 1982
p. 323 -
330
r
DBR Paper No. 1079
Division of Building Research
&suG
La f o r m a t i o n d e s f i s s u r e s pendant l e s e s s a i s de f l u a g e ( c o n t r a i n t e c o n s t a n t e ) e t l e s e s s a i s d e r e s i s t a n c e ( c o n t r a i n t e v a r i a b l e ) f a i t l ' o b j e t de c e t t e Ctude r C a l i s 6 eA
p a r t i r d ' u n modele r h e o l o g i q u e c o u r a n t . Les r C s u l t a t s d e s e s s a i s e f f e c t u i 5 s s u r d e s 6 c h a n t i l l o n s de g l a c e e t l e s e q u a t i o n s d e d s f o r m a t i o n proposi5es p o u r d i v e r s e s c o n d i t i o n s d e chargement o n t s t 6 u t i l i s C s pour m o n t r e r que l a f o r m a t i o n d e s f i s s u r e s p r o v i e n t d ' u n e d 6 f o r m a t i o n c r i t i q u e d i f f B r 8 e d a n s l e domaine Q l a s t i q u e . Ces o b s e r v a t i o n s c o n f i r m e n t l e s h y p o t h P s e s s e l o n l e s q u e l l e s l a d C f o r m a t i o n Q l a s t i q u e d i f f b r e e e s t l i 6 e a u g l i s s e m e n t e n t r e c r i s t a u x . On montre que l a c o n t r a i n t e depend du temps au d e b u t d e l a f i s s u r a t i o n e t que l a c o n t r a i n t e minimum r e q u i s e pour c e t t e f i s s u r a t i o n p e u t t t r e p r e v u e .INTERNATIONAL UNION OF THEORETICAL AND APPLIED MECHANICS
SYMPOSIUM ON DEFORMATION AND FAILURE OF GRANULAR MATERIALS
DELFT
/
3
1
AUGUST
-
3
SEPTEMBER
1982
Deformation and Failure
of Granular Materials
Edited by
P.A.VERMEER
Delft University of Technology
H.
J.LUGER
Delft Soil Mechanics Laboratory
lUTAM Conference on Deformation and Failure of Granular Materials /Delft /31 Aug.
-
3
Sept 1982Delayed elastic
strain criterion
for first cracks in
ice
N.K.SPNHA
Nafioml Research Council Canada, Ottawa
1 INTRODUCTION
I n e m p i r i c a l l y developed c o n s t i t u t i v e e q u a t i o n s p a r t i c u l a r c o n s i d e r a t i o n i s not u s u a l l y given t o t h e t e x t u r e o r f a b r i c of m a t e r i a l s o r t o t h e micromechanism involved i n t h e deformation and f a i l u r e
processes. Sinha (19791, however, r e c e n t l y
proposed such a p h y s i c a l meaning i n a n e m p i r i c a l c r e e p e q u a t i o n (Sinha 1978a,b) developed f o r p o l y c r y s t a l l i n e i c e i n compression, t r y i n g t o r e l a t e t h e e q u a t i o n t o p r o c e s s e s a c c r u i n g i n t h e s o l i d . T h i s approach a l l o w s f o r t h e i n t r o d u c t i o n of t h e e f f e c t of g r a i n s i z e . The a n a l y s i s showed t h a t t h e u n i a x i a l c r e e p e q u a t i o n can be a p p l i e d t o t e n s i l e s t r e s s , provided t h e s t r a i n i s small. The most important f a c t , however, i s t h a t p o l y c r y s t a l l i n e i c e does n o t behave i n any e x c e p t i o n a l manner i f due c o n s i d e r a t i o n i s given t o t h e e f f e c t of g r a i n s i z e and temperature. For t h i s r e a s o n t h e equation'developed was thought t o b e g e n e r a l l y a p p l i c a b l e . I n ' t h e p r e s e n t paper t h i s c o n s t i t u t i v e e q u a t i o n i s a p p l i e d t o f a i l u r e p r o c e s s e s i n t h e l i g h t of a p a r t i c u l a r mechanism of c r a c k n u c l e a t i o n . Because g r a i n boundary s h e a r can cause s t r e s s c o n c e n t r a t i o n s a t t r i p l e p o i n t s and jogs i n t h e i n t e r g r a n u l a r boundaries, and because g r a i n boundary s h e a r o r s l i d i n g h a s been a s s o c i a t e d (Sinha 1979) w i t h delayed e l a s t i c e f f e c t , i t i s hypothesized t h a t delayed e l a s t i c i t y can be l i n k e d t o t h e n u c l e a t i o n of c r a c k s i n i c e and, t h e r e f o r e , i n o t h e r g r a n u l a r m a t e r i a l s . 2 CONSTITUTIVE EQUATIONS I r r e s p e c t i v e of l o a d i n g c o n d i t i o n s , deformation, E , i n any m a t e r i a l can be
d e s c r i b e d i n terms of e l a s t i c o r i n s t a n t l y ( i n a p r a c t i c a l s e n s e ) r e c o v e r a b l e s t r a i n , E ~ delayed e l a s t i c o r time-dependent , s t r a i n , ~ d , and a v i s c o u s o r permanent
deformation, E,
Constant load c r e e p r e s u l t s on i c e were used by t h e a u t h o r t o formulate an e m p i r i c a l u n i a x i a l c r e e p e q u a t i o n (Sinha 1978a,b) t o d e s c r i b e t h e above r e l a t i o n e x p l i c i t l y i n terms of s t r e s s , a , t i m e , t , and temperature, T , f o r a g i v e n g r a i n s i z e , d. T h i s e m p i r i c a l e q u a t i o n was l a t e r modified (Sinha 1979) t o i n t r o d u c e t h e e f f e c t of g r a i n s i z e and was p r e s e n t e d a s
where E i s Young's modulus, E i s t h e
v1
v i s c o u s s t r a i n r a t e f o r u n i t o r r e f e r e n c e s t r e s s ol, c1 i s a c o n s t a n t corresponding t o t h e u n i t o r r e f e r e n c e g r a i n s i z e , d l , b, n and s a r e c o n s t a n t s , and aT i s t h e
i n v e r s e r e l a x a t i o n time. Both 6 and a ~ v1 were shown t o v a r y with temperature w i t h
t h e same a c t i v a t i o n energy.
U n i a x i a l c o n s t a n t s t r e s s ( u s u a l l y c o n s t a n t l o a d ) experiments a r e one of t h e two t y p e s of commonly a p p l i e d mechanical t e s t s used t o determine m a t e r i a l p r o p e r t i e s f o r e n g i n e e r i n g a p p l i c a t i o n s . The o t h e r s a r e t h e u n i a x i a l c o n s t a n t s t r a i n r a t e t e s t s
are favoured because they provide a measure of strength.
In order to develop a rheological model that will apply to both tests Sinha (1981a) derived a deformation equation for non- steady stress history on the basis of equation (2). This was done by assuming
(a) the applicability of the principle of superposition associated with a standard linear solid to the delayed elastic term for the class of materials having s = 1, and (b) the applicability of the commuta- tive law of creep for viscous flow. It was shown that the strain due to an arbitrarily increasing stress path, commonly encoun- tered during strength tests, can be represented by
Equation (3) was developed from equation (2) by representing the stress path as a series of positive stress values each acting during an equal interval of time, At, so that the total loading time, t, equals NAt. Average discrete stresses were represented by Ao1, A02
.
.
. .
Thefirst term in equation (3) represents elastic strain, E ~ ; the second and third terms represent delayed elastic, ~ d , and viscous, cV, components, respectively.
Equation (3) was developed from equation (2) by representing the stress path as a series of positive stress values each acting during an equal interval of time, At, so that the total loading time, t, equals NAt. Average discrete incremental stresses were represented by Ao Ao2
....'
The first term in equation (3) represents elastic strain, E ; the second and thirdterms represent dglayed elastic, cd, and viscous, cV, components, respectively.
3 MICROMECHANISM OF DEFORMATION
different lattice planes of a single crystal, provided the aggregate is made up of only one material.
The third term in equations (2) and
(3)
-
viscous strain-
is governed by the stress-induced and thermally activated mobility of vacancies, interstitials, dislocations, etc., in the intragranular space and the action of the grain boundaries as barriers, sources or sinks for these defects. For conditions, o/E > where grain boundary diffusional processes do not play a dominating role in the accommodation processes and where the microstructure has not deteriorated by the formation of internal voids and cracks, viscous creep can be controlled by the intragranular mobility of dislocations. The effect of grain size may therefore not be significant during the early creep period of most concern in engineering. As the velocity of dislocations depends on applied stress and temperature, the stress and the temperature dependencies of the viscous term are physically definable. Grain sizeinsensitivity of E, during uniaxial creep of ice of grain size usually encountered in nature, 1 to 10 mm, has been shown
experimentally by Duval and Le Gac (1980). This supports the speculation made (Sinha 1979) for grain size independence of the third term in equation (2). . .
No direct experimental evidence of the effect of grain size on deformation
behaviour has been found in constant strain rate experiments. Insensitivity of upper yield stress to grain size, at least at one nominally constant strain rate, has
recently been reported by Jones and Chew (1981) for granular snow ice. As well, the strain rate sensitivity of upper yield stress for columnar-grained S-2 ice of average grain size of 2 mm has been determined by-the author to be the same as that of coarse-grained ice of
4
to 5 mm reported earlier (Sinha 1981b).In developing the generalized creep equation (eq. 2) it was hypothesized
(Sinha 1979) that shear or sliding in the grain boundary regions could induce a delayed elastic effect in polycrystals. Some indirect experimental support for this assumption came from grain boundary sliding (gbs).investigations of metals, alloys and ceramics. Direct evidence of one-to-one correspondence between delayed elasticity and gbs is, however, still missing. The first term in equations (2) and (3)
describes the time-independent, pure
elastic effect. It can be related, 4 CRITERION FOR CRACK INITIATION approximately, to the average value of a Grain boundary sliding could result in number of elastic moduli corresponding to elastic stress concentrations at
irregularities in the grain boundaries and grain boundary junctions or triple points. These stress concentrations could be sufficient to initiate cracks (Zener 1948;
Gifkins 1959). The degree of stress
concentration will, of course, depend on the amount of grain boundary sliding and the stress relaxation processes by grain boundary diffusion or other creep
mechanisms. For low stresses, diffusional accommodation could be sufficient to relax the stress concentrations. At high stresses, crack initiation due to pile-up of dislocations or elastic stress
concentrations around existing flaws could dominate the failure processes. There could be an intermediate region where initiation of cracks due to grain boundary sliding could play a leading role. If this possibility is accepted, then a critical grain boundary displacement for crack initiation might be expected.
The strain, cgbs, induced by grain boundary sliding can be given by (Gifkins 1959),
in which
x
is the average grain boundarydisplacement and K is a constant (~1). If
zC
is the average critical grain boundarydisplacement for the nucleation of cracks, then equation (4) gives the critical gbs strain,
The assumption made by Sinha (1979) in developing the generalized creep equation
(eq. 2) was that
Equations (5) and (6) then givecthe critical delayed elastic strain ~ d , corresponding to €ibs, as
If the concept of a critical grain boundary displacement, or the gbs strain for the initiation of cracking activity, is valid, and if the assumption expressed in
equation (6) is-not far from reality, one
would expect for a given grain size a critical delayed elastic strain for cracking. This will be explored, using experimental observations in conjunction with the constitutive equations.
5 CONSTANT STRESS CRACKING
Gold (1967, 1972) investigated the onset of cracking activity in transversely
isotropic, columnar-grained S-2 ice
subjected to constant load applied
perpendicular to the long direction of the grains. In his first report on this subject Gold(1967) noted that the formation of the first cracks is a reasonably well defined event in previously undeformed specimens. He decided later (Gold 1972) to record the time of formation of the first three cracks to obtain a better measure of the beginning of cracking activity and further experiments were carried out. All the experimental results at -lO°C are shown in Fig. 1.
Figure 2 shows the delayed elastic strains calculated for all observations in Fig. 1. Computations were made on the basis of the second term in equation (2) and the values of the material constants in Table 1, assuming a grain size, d, of 4.5 mm. It appears that the first cracks form when cd reaches a critical value, E:,
irrespective of the applied level of stress
in the load range studied. For the chosen -
grain size the numerical value of
€2
was(1.04 f 0.10) x giving
fC = (0.47
+
0.05) pm according toequation (7). It should be mentioned that
the calculated value of iC was not affected by the arbitrary choice of grain size. This choice was made because of the
extensive grain size determinations carried out during strength tests (Sinha 1981b) on ice produced, essentially, by the method used by Gold (1972).
If tfc is the time required to develop
€5
and hence the formation of the firstcracks, then the second term in equation (2) gives
Substitution of ci from (7) in (8) gives
which is independent of grain size. Dependence of tfc on stress at -1O0C, calculated on the basis of equation (91,
F I R S T C R A C K S O N L Y . 66 TESTS 0 F I R S T C R A C K D SECOND C R A C K a
8
l a T H I R D C R A C K Fig. 1. Dependence of time to formation of first cracks on stress in S-2 ice, T = 263 K(-10°C). Experimental results are from Gold (1967), 1972).
-
@ F I R S T C R A C K (66 TESTS) 0 F I R S T C R A C K a SECOND C R A C K 32 T E S I S a T H I R D C R A C K-
3
?
5 -Q
-
Fig. 2. Computed 4 -IY delayed elastic strain
a for the formation of
0 1
I
I
first cracks in S-20.5 1.0 1.5 2.0 2 . 5 ice of average grain
diameter of 4.5 m
UPLOADED I N < 1 second I THEORY, ce 2
-
I I 0 -.
I I I TIME, t. sFig. 3. Dependence on time of experimentally observed stress, strain, and theoretical strain and its three components for 5-2 ice of average
grain size of 4.5 mm at -lO°C, constant cross-head rate of
7.5 x 1 0 - ~ m m s - l or i = 3 x s-l. n
GC
= 0.47 vm, and the constants in Table 1, is shown in Fig. 1 by the solid line. It'min =
N'
(10)shows that tfc increases rapidly as stress decreases. The minimum stress, o,h,
-2
required to produce any cracking will be which gives amin = 0.50
+
0.05 MN-m for the level of stress for which tfc =-.
x -C = (0.47+
0.05) pm and the values of Equation ( 9 ) gives, for this condition, other constants in Table 1. This explainsTable 1. Creep parameters for ice It is realized that a systematic study of independently obtained from early creep the formation of cracks and their influence experiments (Sinha 1978a, 1979) on rheological response is required for
further improvement of the theory. In E = 9.5 G N - ~ - ~ ; G = 3.8 G N . ~ - ~ spite of this recognized limitation,
Q = 67 kJ/mol (16 k calfmol) theoretical total strain compares well with
cl = 9 experimental observations and supports the
dl = l m m method of calculation. Figure 3 also shows
.
s = 1 the experimental and theoretical recovery
n = 3 curve after unloading. This part of the
b =0.34 calculations will be reported elsewhere,
a~ (T = 263 K) = 2.5 x s-l but the example shows the possibility of
-
Ev = 1.76 x s-l; 01 = 1 ~ ~ - r n - ~ , predicting the recovery path. It must be1 T = 263 K mentioned, however, that the measurement of
K = 1 (assumed) strain recovery in ice often shows large
scatter (as in Fig. 3) and is therefore less reliable if cracking activity occurs during the loading period. The presence of why Gold (1967, 1972) did not observe cracks at or near the contact areas under cracking activity in ice within the the knife edges of the strain gauges is experimental time for stresses less than thought to influence the response of the about 0.6 MN-m-'. The above numerical strain detectors. As the number of cracks value for omin agrees exactly with the in a specimen increases with time, the minimum stress for cracking of 0.5 M N . ~ - ~ uncertainty in the strain measurement also
estimated by Zaretsky et a1 (1979) from increases with the progress of the test. studies of acoustic emission in columnar This could partly explain the large
grained ice. It also agrees with the discrepancy in the output of the two strain minimum stress of about 0.5 MN.m-2 for the gauges during
onset of acoustic emissions noted by the It is significant that although the author (sinha 1982b) during strength tests. specimen strain rate near the peak stress The most convincing experimental evidence in Fig. 3 is not far from the nominal for the nondependence of omin on grain strain rate tn, estimated from the imposed
size, as predicted (lo), constant cross-head displac-ent rate, the
however, from Currier (1981). He observed initial strain rate is totally different. no grain size dependence of level The theory upholds these observations well.
(0.44
+
0.17 MN-m-') for onset of acoustic Calculations give, also, insight into the emission in isotropic granular ice with material response. The initial strain rate average grain size 7.3 mm is governed by the combined elastic-delayedat -10°C. elastic effect, whereas the peak strain
rate is controlled mainly by the viscous strain rate. Note the observed stress, 6 VARIABLE STRESS CRACKING ofc, strain eft, and timing, tfc, associated
with the first three cracks, as well as the Cracking activity in optically transparent calculated strain components, particularly S-2 ice samples (50 x 100 x 250 mm) was the values of the delayed elastic strains investigated at -lO°C during compressive corresponding to the times for the strength testing under nominally constant formation of these cracks. ~h~ average strain rates (Sinha 1981b). An example of time (458 s) for the first three cracks the results is shown in Fig. 3. The gives cd equal to 9.6 x (or It = 0.43 um dependence of stress and strain on time from equation (7)). This compares well during a test is illustrated, along with with
E~ estimated from constant stress times of formation of the first three experiments (Fig. 2) using the same grain d visible cracks. This experiment was size. Similar observations have been made carried out at a constant cross-head rate, during other tests (Sinha 1981b) in a wide i, of 7.5 x mm s-l, or at a nominal strain rate range of 10-7 .-I to strain rate,
t
= i/L, of 3 x 10'~ s'l, 1 x s-l. Details of thesewhere is thenspecimen length' Figure observations will be presented elsewhere. also shows theoretical strain and its
components as functions of time.
Calculations were made on the basis of 7 DISCUSSION equation (3), the information in Table 1,
and a measured average grain diameter of What, then is the significance of these 4.5 mm. The stress path was divided into observations? The beginning of internal about 200 steps for the calculations. damage due to microcracking does not
necessarily indicate failure. This is certainly true at low rates of loading. At moderate to high rates of loading, however, the stress at which the cracks form first may be sufficient to cause propagation under tensile loading conditions and dictate the ultimate
, strength. If the mode of loading is
compressive and the loading rate is high, then the loading cycle might continue, after the formation of the first cracks,
-
until the lateral extensional strain couldgenerate sufficient stress concentrations at the newly formed crack tips to help them propagate, leading to splitting type of failure, often referred to as brittle failure. Such premature compression failure will depend upon many factors such as grain size (which will determine the initial size of the cracks formed), temperature, position of cracks with respect to specimen end surfaces, quality of the end surfaces, type of platens, etc. It may be seen, therefore, that these compressive splitting failures would depend to a considerable extent on sample preparation and other test conditions
(Sinha 1981b).
Formation of the first cracks in tension, however, could signify the tensile strength under moderate rates of loading, even if the crack were in the bulk of the material, provided the stress satisfied the above condition. Such a possibility has been
examined (Sinha 1982b). It has been shown
that the strain rate and stress rate dependence of the stress level required for the onset of cracking activity in ice correlates well with the few available results on rate sensitivity of tensile strength. This correlation makes the proposed concept of crack nucleation more attractive than mere academic interest.
Such a discussion would not be complete without mention of another significant aspect of the proposed criterion of crack initiation. This has to do with the predictability of delayed fracture given by
equation (9). Usually the relation of
cracking time to applied stress is given by empirical equations. Creep failures for many materials can be described as
(Zhurkov 1965; Bartenev and Zuyev, 1968) 9,-ao
tf = t exp
-
kT - - A ( T ) exp (-Bao) (11)line if log tf is plotted against a; this conforms well, in a general sense, with the observed creep rupture or fracture
phenomena in a wide range of materials. Gold (1967, 1972) used equation (11) for describing the results in Fig. 1, but noted the deviation of the results from a straight line, particularly at lower stresses and temperatures. The obvious physical absurdity in this relation is the prediction that failure would occur even
for extremely small stress (o =
o+).
Thiserroneous characteristic is absent in the present formulation of equation (9), which does, however, predict a minimum stress for cracking and nonlinearity in the dependence of log tfc on o.
8 CONCLUSION
Previously developed constitutive equations in conjunction with experimental observa- tions have been used to show that cracks are initiated after a critical delayed elastic
strain,
€2,
has accumulated during constantstress creep tests as well as during constant strain rate strength tests. It
has been shown that
€2
is independent ofstress during creep tests in the stress
range of a/E = 6 x lov5 to 2 x This
critical value of
€8
has been correlated toa critical grain boundary displacement,
sc.
A-constitutive equation for uniaxial creep has been used to formulate an explicit relation between cracking time, tfc, and o. It represents the experimental observations better than a Zhurkov type relation. Moreover, it predicts that cracking
activity occurs only when a critical stress is exceeded and that tf, is independent of grain size.
9 ACKNOWLEDGEMENT
The author is indebted to L.W. Gold for making available all his original
experimental data on constant stress tests on ice. This paper is a contribution from the Division of Building Research, National Research Council of Canada, and is published with the approval of the Director of the Division.
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where tf is failure time and to, A, B and a Bartenev, G.M., and Y.S. Zuyev. 1968,
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and Qo is implied as the apparent activation materials, Pergamon Press, N.Y.
energy for failure at zero stress. Currier, J.H. 1981, lhe brittle to ductile
tension, M.Sc. Thesis, Dartmouth College, Hanover, N.H., U.S.A.
Duval, P. and H. LeGac. 1980, Does the permanent creep-rate of polycrystalline ice increase with crystal size?, J. Glaciol. 25:151-157.
Gifkins, R.C. 1959, Mechanisms of
intergranular fracture at elevated
temperature. Fracture (ed, by
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,
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first cracks in ice.* 9 Physics of Snow
and Ice, (ed. by H. Oura), Inst. Low Temp. Sci., Hokkaido University, Japan, 359-370.
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polycrystalline ice, J. Glaciol. 21:457-473.
Sinha, N.K. 1979. Grain boundary sliding in polycrystalline materials, Phil. Mag. 40~825-842.
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of ice-like materials in engineering applications. &Mechanics of Structured Media, Proc. Int. Symp. on Mechanical Behaviour of Structured Media, Ottawa.
Sinha, N.K. 1981b, Rate sensitivity of compressive strength of columnar-grained ice, Exper. Mechan. 21:209-218.
Sinha, N.K. 1982a, Constant strain- and stress-rate compressive strength of columnar-grained ice. J. Materials Sci. 17:785-802.
Sinha, N.K. 1982b, Acoustic emission and microcracking in ice, Proc. 1982 SESAIJAP. Soc. Mech. Eng., Honolulu/Maui, Hawaii.
Zaretsky, Yu. K., B.D. Chumichev and
V.I. Solomatin. 1979, Ice behaviour under load, Eng. Geol. 13:299-309. Zener, C. 1948, The micro-mechanism of
fracture. In Fracturing of Metals, Am. Soc
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,
~ e t a l c Cleveland, Ohio, 3-31. Zhurkov, S.N. 1965, Kinetic concept ofthe strength of solids, Inter. J. Fract. Mech. 1:311-323.
D e f o r m a t i o n a n d f a i l u r e o f g r a n u l a r m a t e r i a l s
International Union o f T/ieoretical and Applicd
Mechalzics symposium o n deformation and faihtrc
of granular materials, Delft,
31
Aug.-
3
Sept.
1982
1982, 25 x 18 cm, 671 pp., 72 contributions.
Cloth, Hfl.130
/
$50.00/
£29.00 9 0 6 1 9 1 2 2 4 5 Because of computer facilities there has recently been a tre~nendous progress in the field of constitutive modelling (rheology). Constitutive modelling has become a specialisn~ within soil mechanics, and new items are being considered, such as the role of fabric, the localization of deformation in shear bands and numerical techniques for solving partic- ular problems. The proceedings contain contributions on current research activities by active scientists within the field of mechanics of granular materials.CONTENTS
1. Role and description of fabric. Orientational distributions of contact forces as memory parameters in a granular material (B. Cambou, Ecole Centr., Lyon); Configuration hardening or soft- ening of blocky niaterials (R.J.lzbicki, Techn.Univ., Wroclaw, Poland); On the niechanics of granular material (C.Thornton & D.J.Barncs, Univ. Aston, UK); etc.
2. General aspects of constitutive modelling. Granulates with interfacial layers (J.Cl~ristoffersen. Techn. Univ., Lyngby); Bounding surface elastoplasticity-viscoplasticity for particulate cohesive media (Y.F.Dafalias, Univ. Calif., Davis); Kinematic elasto-plasticity for soil mechanics (F.Molenkamp, Delft Soil Lab, Nethl.); An endochronic geomechanical model for soils (K.C. Valanis, Univ. Cincinnati, Ohio); Some rcmarks on the principle of effective stress (A.Verruijt, Delft Univ. Technol., Nethl.); etc.
3. Constitutive models. A family of tensorial constitutive cqua- tions of a rate type for soils (R.Chambon, Inst.Mdcaniquc, Gre-
'
noble); Computer implcnientation of a doublc-hardening modelfor sand (D.V.Griffits & I.M.Smith, Univ. Manchester & F.Molen-
1
kamp, Delft Soil Lab., Nethl.); A viscoplastic constitutive model for normally consolidated clay (R.Nova, Techn.Univ., Milan); De- layed elastic strain criterion for f i s t cracks in ice (N.K.Sinha, Natl. Res. Council, Ottawa); etc.4. Experimental studies with reference to constitutive models. Plloto-elastic stress analysis and strains in simple shear (H.G.B. Allcrsma, Delft Univ.Technol., Netlll.); Numerical experiments on granular asscmblics: Mcasuremcnts and observations (P.A. Cundall, A.Droscl~cr & O.D.L.Strack, Univ. Minnesota); Expcri- lncntal and theoretical studies of powdcr conipaction (M.Hchen- bcrgcr, P.Saniuelson, O.Alm, L.Nilsson & T.Olofsson, llniv.Lulea, Sweden); Planc straining of clay ( K.Kuntschc, Univ.K;lrlsruhc); Correlating drained and undraincd 3D tests on loose sand (P.1. Lewin, Y.Yamada & K.lshihara, Building Res.Stn., Watford, UK & Univ. Tokyo); etc.
5. Shear band formation. Rupture layers in granular media (J.K.F.
Arthur & T.Dunstan, Univ.Coll., London); Localization effects in triaxial test on sand (P.V.Ladc, Univ. Calif., Los Angeles); Experi-
1
mental observations of shear band patterns in direct shear tests (G. Scarpelli & D.M.Wood, Univ. Rome & Univ. Cambridge, UK); A simple shear-band analysis using compliances (P.A.Vern~cer); etc.6. Limit analysis and granular flow. Calculation of collapsc loads using Iligl~er order elclnents (R.de Borst, Delft Univ.Technol.); Computer simulation of chute flows of granular niatcrials (S.C. Campbell & C.F..Brcnnen, Calif.lnst.Tcclino1., Pasadena); Appli- cation 01- the kinematical clcment ~nctliod to collapse-problems of earth structures (P.Gussmann, Univ.Stuttgart); Rectilincar flow of granular materials (A.J.M.Spcnccr & T.C.O'Mahony, University Nottingham); etc.
7. Model tests and analytical solutions. In-situ tests for mcasure- mcnts of soil properties in bouldcr deposit (S.Prakash & G.Ranjan, Univ. Roorkcc, India); Tlie inllucncc of finite strains on the expen- sion of a cylindrical cavity in a comprcssible or dilatant soil (A.P.S. Selvadurai, Carleton Univ., Ottawa); ctc.
Dungar,
R.,
G.N. Pande & J.A. Studer (editors) Numerical models in geomechanics - Internationalsymposium, Zurich, 13.1 7 September 1982
1982,25 cm, 831 pp., Hfl.125
/
$48.00/
£28.00 Successfully solving problems by the finite element method depends o n the choice of appropriate numer- ical models and their associated parameters. Emphasis is given to the verification and evaluation of models for practical application such as embankment dams, offshore structures, etc. Monotonic, cyclic and random loading are discussed. Theoretical aspects of models for soils and rocks; experimental behavior and evaluation of models; transient, thermal and limit equilibrium problems; application to solution of practical problems. Eisenstein, Z. (editor) 9 0 6 1 9 1 2 4 6 6 Numerical methods in geomechanics 1982 - 4th inter-national conference, Edmonton, 31 May
-
4 June 19821982,29 cm, 1300 pp., 3 vols., Hfl.365/$145/£79.50 Numerical techniques and programming; Numerical treat- ment of constitutive laws; Flow and consolidation; Soil and rock dynamics; Modelling of underground reservoirs; Tunnels and underground space; Earth structures and slopes; Shallow foundations; Piles; Soil-structure inter- action; Marine geotechnology; Frozen soil problems. Wittke, W. (editor) 9 0 6 1 9 1 0 4 0 4 Numerical methods in geomechanics - Proceedings o f the 3rd international conference, Aachen, 2-6 April 1979
1979,25 cm, 15 12 pp. in 4vols.. Hfl.395
/
$187/
£86 130 papers from 3 0 countries contain information on recent research projects and on successful applications of numerical methods in soil mechanics, foundation engineering, rock mechanics, geological engineering and related fields of geomechanics. Theoretical develop- ments; flow & consolidation: constitutive laws; rock behaviour; underground openings; soil-structure-inter- action; embankments & slopes; dynamics. Editor: Prof. Univ. Aachen.Verruijt, A. & F.B.J.Barends (editors)
Flow and transport in porous media - Proceedings o f
Euromech 143, Delft, 2-4 September 1981
1981,25 cm, 240 pp., Hfl.95
/
$35.00/
£20.70 The present book gives an up-to-date review of the state of the art in the field of flow and transport in porous media. This branch of science is rapidly growing, especiall? because of the importance of problems of pollution of groundwater, of heat transport and storage o f solar energy in the ground, of hydraulics and mechanics in large granu- lar structures, and of the simultaneous flow of different fluids in porous media. All these subjects are covered in this volume, together with a number of papers o n the fun- damental aspects of flow through porous media.A.A.Balkc~rna Ntblisl~rrs,
/'.O
Box 16 7.5,Lbr USA & Canada only: A.A.Ba/ken
NI, 3000 BK Rottt,rdanr. Nrflrc,rlui~ds la, 99 Main Street, Salrrn, NH 03079
This publication i b being d i s t r i b u t e d by the Division of Building R e s e a r c h of tile National R e s e a r c h Council of Canada. I t should not be r e p r o d u c e d In whole o r in p a r t without p e r m i s s i o n of t h e o r i g i n a l 1~11blisher. T h e Di- v i s i o n would be glad to l,c uf assistance in obtaining s u c h p e r m i s s i o n .
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