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Mechanics of snow slab failure from a geotechnical perspective
McClung, D. M.
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Mechanics of Snow Slab Failure
From a Geotechnical
by D.M. McClung
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Avalanche Formation, Movement and Effects
(Proceedings
of the Davos Symposium, September 1986)
IAHS Publication No. 162, 1987, p. 475-508
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September 1986). IAHS Publ. no. 162,1987.
Mechanics of snow
slab failure from a geotechnical
perspective
D. M. McCLUNG
National Research Council of Canada, 3650 Wesbrook Mall, Vancouver, BC, Canada V6S 2L.2
ABSTRACT F i e l d o b s e r v a t i o n s and measurements show t h a t a1 1 snow s l a b f a i l u r e s have some common features. The p h y s i c a l mechani sms g o v e r n i ng t h e r e 1 ease can d i f f e r g r e a t l y depending on t h e c h a r a c t e r o f t h e deformation i n t h e weak l a y e r o r i n t e r f a c e beneath t h e s l a b where t h e r e l e a s e process i n i t i a t e s . Here two p r o t o t y p e s f o r s l a b avalanche i n i t i a t i o n a r e
c o n s i d e r e d : I. r e l e a s e o f d r y s l a b avalanches where weak l a y e r shear f a i l u r e beneath t h e s l a b i s r e q u i r e d and, 11. r e l e a s e o f f u l l depth avalanches caused by r a p i d g l i d i n g . For avalanches caused by g l i d e , i t i s proposed t h a t t h e snowpack p a r t i a1 l y separates from t h e g l i d e i n t e r f a c e . T h i s reduces d r a g and i n c r e a s e s g l i d e speed. The d e f o r m a t i o n processes f o r b o t h t y p e s of s l a b f a i l u r e a r e reviewed and suggestions f o r f u t u r e r e s e a r c h a r e given.
MEcanique de l a r u p t u r e des d a l l e s de n e i g e dans une p e r s p e c t i v e ggotechnique
RESUME Les o b s e r v a t i o n s e t mesures s u r l e t e r r a i n rnontrent que t o u t e s l e s r u p t u r e s de d a l l e s de n e i g e o n t des t r a i t s communs. Les m6cani sines physiques r E g i ssant 1 e dEcl enchement peuvent v a r i e r grandement s e l o n 1e c a r a c t g r e de l a d E f o r m a t i o n de l a couche f a i b l e ou de 1 ' i n t e r f a c e se t r o u v a n t sous l a d a l l e , oC s'amorce l e processus de dEclenchement. L ' a u t e u r E t u d i e i c i deux hypothgses expl i quant l e dEcl enchement d ' a v a l anches en d a l l e s : I. l e dgclenchement d'avalanches ssches en d a l l e s n 6 c e s s i t a n t l a r u p t u r e de c i s a i l l e m e n t de l a couche f a i b l e se t r o u v a n t sous l a d a l l e , e t 11. l e dEclenchement d'avalanches p l e i n e hauteur causE p a r l e g l i s s e m e n t r a p i d e . Dans l e cas des avalanches dues au g l i ssement, on avance que 1 'a g g l omEration de n e i g e se s6pare p a r t i e l 1 ement de 1
'
i n t e r f a c e de g l i ssement.
Cela r 6 d u i t l e f r o t t e m e n t e t a c c r o 7 t l a v i t e s s e de g l i ssement. L ' a u t e u r examine 1 es processus ded 6 f o r m a t i o n pour l e s deux t y p e s de r u p t u r e de d a l l e s e t il propose des v o i e s de recherche pour 1 'a v e n i r .
SHEAR STRENGTH PROPERTIES OF ALPINE SNOW
F r a c t u r e l i n e s t r a t i g r a p h y i m p l i e s t h a t t h e f i r s t f a i l u r e w i l l o c c u r i n o r a t t h e b o u n d a r i e s of t h e weak l a y e r . M a t e r i a l i n a t h i n weak l a y e r s h e a r e d between two t h i c k e r zones w i l l be s u b j e c t t o p l a n e - s t r a i n s i m p l e s h e a r deformation. Slow p l a n e - s t r a i n s i m p l e s h e a r t e s t s w i t h homogeneous samples w i l l s i m u l a t e t h e s e
deformation c o n d i t i o n s u n l e s s i m p e r f e c t i o n s a r e p r e s e n t .
Macroscopic i m p e r f e c t i o n s a r e e x p e c t e d i n t h e f i e l d and t h e y w i l l c a u s e s t r e s s c o n c e n t r a t i o n s . T h e r e f o r e , deformation r a t e s c a n be h i g h e r , i n s t a b i l i t y can o c c u r e a r l i e r , and a p p l i e d s t r e s s e s can be lower t h a n i n l a b o r a t o r y experiments.
Slow d i r e c t s i m p l e s h e a r e x p e r i m e n t s a r e emphasized because t h e d e f o r m a t i o n p r o p e r t i e s of a l p i n e snow vary s i g n i f i c a n t l y w i t h s t r e s s and s t r a i n s t a t e ; t e n s i o n o r compression t e s t s would be of very l i m i t e d u s e t o s i m u l a t e t h i n weak l a y e r f a i l u r e s e x c e p t t o confirm t r e n d s s e e n i n s h e a r t e s t i n g d a t a . Slow t e s t s a r e i m p o r t a n t t o d u p l i c a t e c o n d i t i o n s b e f o r e n a t u r a l s l a b r e l e a s e . For s l a b f a i l u r e s t h e d e f o r m a t i o n s h o u l d change from c r e e p t o f a i l u r e t o f r a c t u r e . T h i s i s t h e same sequence s e e n i n t h e t e s t s . Slow c o n s t a n t r a t e s h e a r t e s t s (McClung, 1 9 7 7 ) r e v e a l a l p i n e snow t o be a p r e s s u r e s e n s i t i v e , d i l a t a n t s t r a i n - s o f t e n i n g m a t e r i a l 476
D.
M.
McClungI. DRY SLAB AVALANCHE RELEASE INTRODUCTION
S t u d i e s a t t h e f r a c t u r e l i n e s of f a l l e n d r y snow s l a b s i n d i c a t e t h a t a r e l a t i v e l y t h i n weak l a y e r o r s t r a t u m i s always sandwiched between two s t i f f e r r e g i o n s : t h e s l a b i t s e l f and t h e m a t e r i a l
underneath. T h i s s i t u a t i o n i m p l i e s c o n c e n t r a t i o n of d e f o r m a t i o n i n t h e weak l a y e r under a p p r o x i m a t e l y p l a n e - s t r a i n s i m p l e s h e a r c o n d i t i o n s p r i o r t o t h e a p p e a r a n c e of i m p e r f e c t i o n s . F i e l d and l a b o r a t o r y measurements i n d i c a t e t h a t c o n c e n t r a t i o n of d e f o r m a t i o n i n t h e weak l a y e r o r f a i l u r e p l a n e i s r e q u i r e d f o r a s e l f - c o n s i s t e n t p i c t u r e of d r y s l a b a v a l a n c h e r e l e a s e . The primary deformation f e a t u r e s of a l p i n e snow from slow p l a n e - s t r a i n , s i m p l e s h e a r experiments a r e : weak s t r a i n - s o f t e n i n g , f r i c t i o n a n g l e s i n e x c e s s of s l o p e f a i l u r e a n g l e s and prominent r a t e e f f e c t s .
A c o u s t i c e m i s s i o n measurements i l l u s t r a t e a c r i t i c a l d e f o r m a t i o n r a t e f o r f r a c t u r e i n i t i a t i o n which i s g r e a t e r t h a n c r e e p r a t e s measured i n t h e f i e l d .
When measured s h e a r s t r e n g t h p r o p e r t i e s of a l p i n e snow a r e i n c o r p o r a t e d i n t o c u r r e n t g e o t e c h n i c a l f a i l u r e models, t h e
p r e f e r r e d mechanism f o r s l a b a v a l a n c h e r e l e a s e emerges. A n a l y s i s shows t h a t dynamic p r o p a g a t i o n of s h e a r f a i l u r e s i n t h e weak l a y e r i s t h e most l i k e l y cause. The f a i l u r e models and t h e mechanical p r o p e r t i e s of a l p i n e snow f o r c e t h i s c o n c l u s i o n independent of t h e p r e c i s e d e t a i l s of weak l a y e r deformation. It i s shown t h a t t h e time s c a l e f o r a v a l a n c h e r e l e a s e i s s e n s i t i v e t o t h e d e t a i l s of weak l a y e r d e f o r m a t i o n b u t n o t t h e r e l e a s e mechanism.
Mechanics of snow slab failure 477 w i t h s i g n i f i c a n t r a t e dependent p r o p e r t i e s . The s h e a r d a t a
p r e s e n t e d i n t h i s paper have been o b t a i n e d by me u s i n g a m o d i f i e d Norwegian G e o t e c h n i c a l I n s t i t u t e (NGI) d i r e c t s i m p l e s h e a r
a p p a r a t u s [Bjerrum and Landva (1966) and McClung (1977)
1.
The t e s t s employ a c y l i n d r i c a l sample 15-20 mm h i g h and 115 mm i n d i a m e t e r s h e a r e d under a c o n s t a n t normal load. These t e s t s have some d e f i c i e n c i e s , a s do a l l l a b o r a t o r y t e s t s . T h e r e f o r e , c a u t i o n i s a d v i s e d i n u s i n g t h e d a t a f o r a n y t h i n g more t h a n g e n e r a l t r e n d s . It was found, f o r example, t h a t p l a n e s t r a i n s i m p l e s h e a rc o n d i t i o n s a r e n o t m a i n t a i n e d when t h e s h e a r i n g d i s p l a c e m e n t s exceed 5 mm f o r t h e a p p a r a t u s used.
The measurements show c l e a r t r e n d s f o r p r e s s u r e s e n s i t i v i t y i n common w i t h d a t a from s o i l s and sand: i n i t i a l sample modulus ( s t i f f n e s s ) , peak s t r e n g t h and r e s i d u a l s t r e n g t h a l l i n c r e a s e w i t h normal l o a d f o r s i m i l a r samples s h e a r e d a t t h e same r a t e and
temperature. The i n i t i a l t a n g e n t modulus ( s t i f f n e s s ) i n c r e a s e s w i t h normal l o a d i n a n e a r l y l i n e a r manner. It i s d i f f i c u l t t o q u a n t i f y p r e s s u r e s e n s i t i v i t y of peak s t r e s s f o r t h e a v a l a n c h e problem. Due t o d a t a s c a t t e r , v a r i a t i o n s i n s t r e n g t h f o r d i f f e r e n t samples from t h e same l a y e r a r e comparable t o i n c r e a s e s i n s t r e n g t h due t o i n c r e a s e d normal l o a d s ( f o r normal l o a d s i n t h e range of i n t e r e s t ) . Approximate v a l u e s of f r i c t i o n a n g l e s ( p r e s s u r e dependence of peak s t r e s s ) a p p e a r t o be i n t h e range 40'-70°.
These a r e somewhat h i g h e r t h a n t h e range of a v a l a n c h e s l o p e f a i l u r e a n g l e s ( P e r l a , 1976).
The r a t i o of peak s h e a r s t r e s s t o normal s t r e s s f o r i n d i v i d u a l t e s t s i s of g r e a t e r i n t e r e s t h e r e t h a n f o r s o i l s l i d e s because snow s l a b s a r e of c o n s t a n t t h i c k n e s s . The r e s u l t i n g a n g l e s , d e f i n e d from t h e a r c t a n g e n t of t h e r a t i o , a r e i n t h e range 50' t o 80'. These a r e d e f i n i t e l y h i g h e r t h a n s l o p e f a i l u r e a n g l e s . P e r l a (1976) summarizes 194 a v a l a n c h e c a s e s : t h e range i s 25'-55O, and t h e mean i s n e a r 38O. The lower v a l u e s i n t h e f i e l d show a p o t e n t i a l i n f l u e n c e from i m p e r f e c t i o n s p r e s e n t t h e r e .
The t e s t s a l s o show t h a t snow d i s p l a y s q u i t e weak s t r a i n - s o f t e n i n g . T h i s i s probably due t h e h i g h p o r o s i t y of a l p i n e snow. The maximum r a t i o of peak t o r e s i d u a l s t r e n g t h , zp / z
,
i s n e a r 2 b u t t h i s v a l u e would be c h a r a c t e r i s t i c of f i n e g r a i n e z e q u i l i b r i u m forms. Coarse g r a i n e d k i n e t i c forms, which a r e more l i k e l y t o be found i n s l a b f a i l u r e l a y e r s , a p p e a r t o d i s p l a y l e s s s o f t e n i n g . Data from 51 samples s h e a r e d a t a r a t e of 0.15 mm/min (normal s t r e s s range 0.5 t o 5.7 kPa) show t h a t t h e mean v a l u e of z / T ~i s1.24 w i t h a range 1.01-1.69 and a s t a n d a r d d e v i a t i o n 0.19.' For t h e s e d a t a , d e n s i t i e s a r e from 150-375 kg/m3 and t e s t t e m p e r a t u r e s a r e -5' t o -lO°C.
F i g u r e 1 compares t e s t f a i l u r e v a l u e s (peak s h e a r s t r e s s ) w i t h s l a b a v a l a n c h e f a i l u r e l e v e l s a s a f u n c t i o n of f a i l u r e d e n s i t y . The a v a l a n c h e f a i l u r e v a l u e s a r e d e f i n e d ( P e r l a , 1976) by t h e s h e a r s t r e s s a t t h e bed s u r f a c e (CgH s i n +) where
p
i s d e p t h averaged d e n s i t y , H i s s l a b t h i c k n e s s , g i s a c c e l e r a t i o n due t o g r a v i t y and+
i s s l o p e a n g l e . The t e s t v a l u e s a r e h i g h e r because t h e yr e p r e s e n t t h e s t r e n g t h of s m a l l homogeneous samples. The a v a l a n c h e f a i l u r e v a l u e s d e f i n e t h e l o w e s t v a l u e of s t r e n g t h f o r a l a y e r of
I E - 2 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 D E N S I T Y , k g l m 3 F I G U R E 1 P E A K S H E A R S T R E S S ( F A I L U R E ) L E V E L S F O R S L O W S H E A R T E S T S O N D R Y S N O W ( A ) C O M P A R E D W I T H E S T I M A T E D S H E A R S T R E S S F A I L U R E L E V E L S F R O M A V A L A N C H E F R A C T U R E L I N E D A T A ( o l .
much l a r g e r s i z e . The a v a l a n c h e v a l u e s may i n c l u d e t h e weakening e f f e c t of macroscopic i m p e r f e c t i o n s which a r e l e s s l i k e l y t o be found i n s m a l l t e s t samples.
Rate e f f e c t s found i n t h e t e s t s a r e i m p o r t a n t c l u e s f o r
u n d e r s t a n d i n g t h e s l a b f a i l u r e mechanism. A c o u s t i c e m i s s i o n s were measured d u r i n g s h e a r i n g t e s t s w i t h a B r u e l and K j a e r a c c e l e r o m e t e r (#4375) coupled t o t h e sample w i t h a s i l i c o n e l u b r i c a n t . The
i n s t r u m e n t h a s f l a t r e s p o n s e f o r f r e q u e n c i e s i n t h e r a n g e of s e v e r a l Hz t o s e v e r a l kHz. F i g u r e 2 i l l u s t r a t e s d a t a f o r two d i f f e r e n t samples of c o a r s e g r a i n e d f a c e t e d snow s h e a r e d a t r a t e s of 0.15 mmlmin and 0.012 mmlmin. A t t h e f a s t e r r a t e , t h e number of e m i s s i o n s i n c r e a s e s w i t h s h e a r s t r e s s u n t i l j u s t p r i o r t o r e a c h i n g a peak on t h e s t r e s s - d i s p l a c e m e n t curve. T h e r e a f t e r , t h e r a t e of e m i s s i o n s d e c r e a s e s d u r i n g s t r a i n - s o f t e n i n g and l e v e l s o f f when t h e r e s i d u a l s t r e n g t h i s a t t a i n e d . For t h e s l o w e r t e s t , a c o u s t i c e m i s s i o n s a r e v i r t u a l l y a b s e n t and, even though t h e sample d i l a t e d , a peak s h e a r s t r e s s was n o t observed d u r i n g 4 mm of s h e a r i n g .
The p a t t e r n of a c o u s t i c e m i s s i o n s observed d u r i n g s h e a r i n g i s s u g g e s t i v e ( b u t n o t p r o o f ) of s l i p s u r f a c e formation. A l l
g e o t e c h n i c a l and g r a n u l a r m a t e r i a l s which d i l a t e and s t r a i n - s o f t e n a r e known t o form s l i p s u r f a c e s . The e m i s s i o n p a t t e r n ( F i g . 2 ) shows t h a t sample damage b e g i n s p r i o r t o r e a c h i n g peak s t r e s s . The r e l a t i v e l y c o n s t a n t r a t e of e m i s s i o n s a t t h e r e s i d u a l s t r e s s l e v e l may i n d i c a t e d e f o r m a t i o n on a s l i p s u r f a c e . I f s l i p s u r f a c e f o r m a t i o n t a k e s p l a c e , a s e x p e c t e d , i m p o r t a n t consequences a r e
Mechanics of snow slab failure 479 0 1 2 3 d 5 D I S P L A C E M E N T , m m F I G U R E 2 1 a l S H E A R S T R E S S - D I S P L A C E M E N T C U R V E - F O R A S A M P L E S H E A R E D A T A R A T E O F 0 . 1 5 m m l m i n . A C O U S T I C E M I S S I O N S ( C O U N T S / ~ ~ ~ . 1 A R E D E N O T E D B Y 0-0 D I S P L A C E M E N T . r n m F I G U R E 2 ( b l C U R V E S F O R A S A M P L E S I M I L A R T O T H A T I N 2 ( a ) S H E A R E D A T A R A T E O F 0 . 0 1 2 m m l r n i n . T E S T T E M P E R A T U R E - 8 " G F O R B O T H S A M P L E S A N D D E N S I T Y 3 2 0 k g / m
.
expected; t h e y w i l l a c t a s s t r e s s c o n c e n t r a t o r s even i f n a t u r a l i m p e r f e c t i o n s a r e a b s e n t .
The r a t e e f f e c t s (Fig. 2) a r e s i m i l a r t o t h o s e shown by S t . Lawrence (1977) f o r samples deformed i n t e n s i o n and
compression. He a t t r i b u t e s t h e l a c k of e m i s s i o n s a t slow r a t e s a s an i n d i c a t i o n t h a t t h e sample i s deforming almost e n t i r e l y by p l a s t i c flow. From Fig. 2 t h e c r i t i c a l r a t e of s h e a r i n g f o r s i g n i f i c a n t numbers of a c o u s t i c e m i s s i o n s i s on t h e o r d e r of 0.1 mmlmin. T h i s i s e q u i v a l e n t t o a s t r a i n - r a t e on t h e o r d e r of 1 0 ~ s e c . St. Lawrence showed t h a t r a t e s n e a r 1 mm/min a r e f a s t enough t o c a u s e f r a c t u r i n g i n t e n s i o n t e s t s f o r 200 mm l o n g samples. Salm (1971) n o t e d s t r a i n - s o f t e n i n g i n t e n s i o n t e s t s a t r a t e s n e a r 0.1 mm/min (150 mm l o n g samples). Salm a l s o n o t e d a
c r i t i c a l r a t e of d e f o r m a t i o n n e a r 1 mm/min a t which b r i t t l e e f f e c t s 1 began t o dominate i n compression t e s t s . K i n o s i t a (1967) found a
c r i t i c a l deformation speed f o r compression t e s t s n e a r 1 mm/min. T h i s d e f i n e d t h e t r a n s i t i o n from p l a s t i c t o b r i t t l e behaviour i n h i s t e s t s . K i n o s i t a a l s o showed t h a t d e f o r m a t i o n speed i s more fundamental t h a n s t r a i n - r a t e i n d e t e r m i n i n g t h e t r a n s i t i o n . I f one u s e s s t r a i n - r a t e i n s t e a d , t h e sample h e i g h t must be s p e c i f i e d .
Here I wish t o a d o p t K i n o s i t a ' s (1967) p r e s c r i p t i o n f o r
d e f o r m a t i o n speed t o compare w i t h f i e l d d a t a . The s h e a r i n g t e s t s show t h a t s i g n i f i c a n t numbers of a c o u s t i c e m i s s i o n s a p p e a r f o r r a t e s n e a r 0.1 mm/min. For a 1 m t h i c k s l a b , d e f o r m a t i o n a t 0.01 mm/min a t t h e t o p of a s l a b i m p l i e s a s h e a r s t r a i n - r a t e of 1 0 ~ / s e c . This i s comparable t o t h e f a s t e s t observed c r e e p r a t e s (McClung, 1975) i n low d e n s i t y snow. C o n c e n t r a t i o n of d e f o r m a t i o n i n t h e weak l a y e r i s r e q u i r e d t o b o o s t r a t e s above t h o s e e x p e c t e d i n t h e s l a b . It seems 1-2 o r d e r s of magnitude h i g h e r r a t e s a r e needed t o a c h i e v e t h e h i g h number of a c o u s t i c e m i s s i o n s and s t r a i n - s o f t e n i n g behaviour which p r e c e d e a f r a c t u r e c o n d i t i o n .
The c r i t i c a l r a t e s f o r f a i l u r e can be a c h i e v e d i n s e v e r a l ways. Yamamoto (1978) s t u d i e d a problem s i m i l a r t o t h e p h y s i c a l s i t u a t i o n h e r e , b u t f o r t e n s i l e s t a t e s of s t r e s s . Assuming a weak l a y e r c o n s i s t i n g of m a t e r i a l w i t h s l i g h t l y l a r g e r v o i d volume f r a c t i o n t h a n t h e remainder of t h e specimen h e showed t h a t t h e r a t e of d e f o r m a t i o n c a n i n c r e a s e c o n t i n u o u s l y w i t h i n t h e band compared t o t h e s l a b . He a l s o found a g r e a t l y reduced s t r a i n a t which
i n s t a b i l i t y o c c u r s i n t h e weak band once d e f o r m a t i o n c o n c e n t r a t e s t h e r e . A second p o s s i b i l i t y f o r i n c r e a s i n g t h e l o c a l r a t e s of deformation under n a t u r a l c o n d i t i o n s i s t h e p r e s e n c e of n a t u r a l i m p e r f e c t i o n s . By analogy t o f r a c t u r e mechanics, t h e d e f o r m a t i o n r a t e i n s i d e o r n e a r t h e boundary of a f l a w may e a s i l y be s e v e r a l o r d e r s of magnitude g r e a t e r t h a n one found i n e x p e r i m e n t s w i t h homogeneous samples. Without f u r t h e r i n f o r m a t i o n , I assume b o t h of t h e s e e f f e c t s can a c t t o g e t h e r t o i n i t i a t e t h e f a i l u r e p r o c e s s f o r n a t u r a l s l a b r e l e a s e .
Another d r a m a t i c r a t e e f f e c t i s s e e n i n t h e t e s t d a t a f o r low d e n s i t y snow. I f a low d e n s i t y snow sample i s s h e a r e d v e r y s l o w l y , i t i s p o s s i b l e t h a t s e t t l e m e n t e f f e c t s (due t o t h e normal l o a d ) can, exceed t h e s h e a r i n g deformation. F i g . 3 shows t e s t d a t a from two s i m i l a r samples s h e a r e d a t d i f f e r e n t r a t e s under t h e same normal
Mechanics of snow slab failure 1 . 0 I I I I
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- 1 . 5 I I H O R I Z O N T A L D I S P L A C E M E N T , m m F I G U R E 3 S H E A R S T R E S S - D I S P L A C E M E N T A N D S E T T L E M E N T - D I L A T I O N V E R S U S H O R I Z O N T A L D I S P L A C E M E N T F O R T W O L O W D E N S I T Y S A M P L E S S H E A R E D A T D I F F E R E N T R A T E S U N D E R T H E S A M E N O R M A L L O A D ( 1 . 8 9 k P a ) . D E N S I T Y 1 8 0 k g l m 2 F O R B O T H S A M P L E S .load. The sample in the slow test contracts and strain-hardens because settlement dominates over shear deformation. Thus, this test becomes effectively a compression test. The other sample is sheared an order of magnitude faster and it exhibits the usual I dilatant, strain- softening behaviour. These observations
illustrate the rate effects observed by Kinosita (1967) in a dramatic way. Also, the effect may help to explain why slab avalanches do not usually initiate on slopes below 2 5 ' . Fgr low density snow (McClung, 1979) settlement effects should be
1
comparable to shearing effects for such low slope angles. This I should make dilatant, strain-softening behaviour (and instability)more difficult to achieve.
Comparison of the tests and field data show that weak layer failures are of a different character than those in the tests;
s l o p e a n g l e s and s t r e n g t h l e v e l s a r e lower i n t h e f i e l d . I f d e f o r m a t i o n c o n c e n t r a t e s i n t h e weak l a y e r due t o s l i p s u r f a c e f o r m a t i o n o r t h e p r e s e n c e of n a t u r a l i m p e r f e c t i o n s , t h e s e
d i s c r e p a n c i e s s h o u l d d i s a p p e a r . Also, t h e s e s t r e s s c o n c e n t r a t o r s s h o u l d i n c r e a s e t h e r a t e s i n t h e weak l a y e r s o t h a t correspondence between t h e t e s t s and t h e f i e l d i s provided. These r e s u l t s show t h e importance of t h e l a b test d a t a . The t e s t s show t h e
c h a r a c t e r i s t i c s of slow s h e a r f a i l u r e and t h e y a s s i s t i n d e f i n i n g t h e c h a r a c t e r of f a i l u r e s i n t h e f i e l d .
The narrow range of r a t e s f o r which s t r a i n - s o f t e n i n g i s o b s e r v e d i s c r u c i a l t o u n d e r s t a n d i n g t h e mechanism f o r d r y s l a b a v a l a n c h e r e l e a s e . For r a t e s much s l o w e r t h a n 0.1 mrnlmin, a c o u s t i c e m i s s i o n s and b r i t t l e e f f e c t s d i s a p p e a r and t h e m a t e r i a l deforms by p l a s t i c flow. For r a t e s an o r d e r of magnitude f a s t e r , l a r g e s c a l e f a i l u r e p l a n e s form and f r a c t u r e o c c u r s . Deformation r a t e s i n a weak l a y e r s h o u l d p a s s through t h i s r a n g e under n a t u r a l c o n d i t i o n s a s
i n s t a b i l i t y i s approached.
MODELS FOR DRY SNOW SLAB FAILURE
Consider t h e l a y e r e d s t r a t i g r a p h y shown by a v a l a n c h e f r a c t u r e l i n e s t u d i e s . The c h a r a c t e r of t h e d e f o r m a t i o n i n t h e weak l a y e r o r a t i t s b o u n d a r i e s i s t h e c r u x of t h e snow s l a b s t a b i l i t y problem. It i s assumed t h a t t h e i n i t i a l f a i l u r e ( a t t a i n m e n t of peak s h e a r s t r e s s ) o c c u r s i n t h e weak l a y e r o r a t i t s b o u n d a r i e s . S i n c e t h e s l a b i s s t i f f e r and s t r o n g e r , i t i s assumed t o deform a s a l i n e a r e l a s t i c m a t e r i a l i n i t i a l l y ; e x t e n s i o n t o t h e more
r e a l i s t i c v i s c o - e l a s t i c c a s e i s d i s c u s s e d a f t e r t h e s i m p l e r e l a s t i c problem i s s o l v e d .
Weak l a y e r d e f o r m a t i o n s have n o t been measured i n t h e f i e l d . T h e r e f o r e , f a i l u r e c h a r a c t e r i s t i c s from l a b o r a t o r y t e s t s and t h e g e n e r a l c o n c e p t s f o r f a i l u r e of g e o t e c h n i c a l m a t e r i a l s c o n s t i t u t e t h e p r e s e n t s o u r c e s of i n f o r m a t i o n . My approach c o n s i d e r s t h e extremes of d e f o r m a t i o n c o n d i t i o n s p r i o r t o c a t a s t r o p h i c f a i l u r e
( f r a c t u r e ) . The amount of d e f o r m a t i o n b e f o r e f r a c t u r e w i l l depend on t h e a p p l i e d l o a d i n g c o n d i t i o n s .
i) Loading Model
F i r s t , c o n s i d e r l o a d s a p p l i e d t o t h e t o p of t h e snowpack a t a c o n s t a n t r a t e t o s i m u l a t e l o a d i n g by new s n o w f a l l (Fig. 4). The weak m a t e r i a l i n a narrow c r a c k - l i k e zone i s assumed t o
s t r a i n - s o f t e n and deform homogeneously; t h i s d e s c r i b e s t h e minimum deformation c o n d i t i o n s p r i o r t o f r a c t u r e . Shear and normal s t r e s s l o a d i n g i n c r e m e n t s a r e denoted by dzs and dos.
Simple i n c r e m e n t a l e l a s t i c - p l a s t i c c o n s t i t u t i v e e q u a t i o n s s u f f i c i e n t l y g e n e r a l t o d e s c r i b e t h e weak l a y e r d e f o r m a t i o n a r e g i v e n by Rudnicki (1977). The e n g i n e e r i n g s h e a r and v e r t i c a l s t r a i n i n c r e m e n t s (Fig. 4) dyi and d ~ ~ c o r r e s p o n d i n g t o s t r e s s i n c r e m e n t s dzi and doi a r e r e l a t e d by
Mechanics of snow slab failure 483
F I G U R E 4
S C H E M A T I C O F A S L A B U N D E R F A R F I E L D L O A D I N G C O N T A I N I N G A W E A K Z O N E W H I C H D E F O R M S A S A S T R A I N - S O F T E N I N G M A T E R I A L .
I
For these equations po is a friction coefficient reflecting pressure dependence of yielding, i3 is the ratio of plastic shear strain increment to plastic normal strain increment. TheI
parameters Ge and Ke are the elastic shear and compression moduli for the slab and h, is a plastic hardening (or softening) modulus which may takepositive
or negative values.The deformation in the slab must be related to that in the weak zone. The relationship between the two depends on the geometry of the failure zone. A good choice of failure zone geometry for the slab avalanche problem is a very thin ellipsoidal zone. Assuming deformation is homogeneous in the weak zone, the loading increments applied to the slab can be easily related to the weak layer
deformation. From a theorem by Eshelby (1957), Rudnicki (1977) developed the incremental constitutive equations to describe this situation for far field loading
The parameter
5
= a/Lb(l-ve)J whezone. For l i n e a r e l a s t i c s l a b d e f o r m a t i o n , t h e r a t i o of s h e a r s t r a i n s f o r t h e weakened zone and t h e s l a b may be c a l c u l a t e d . With i n c r e m e n t a l l o a d i n g dzs/dos = t a n J, = c o n s t . [McClung, 1 9 8 1 ( b ) ] , t h e r a t i o i s
doi
where f* =
-
i s a f u n c t i o n of hi and t h e c o n s t a n t s . dziThe s o f t e n i n g modulus, hi, v a r i e s from p o s i t i v e t o n e g a t i v e v a l u e s w i t h deformation. Thus, a c r i t i c a l v a l u e of hi may be d e f i n e d f o r which t h e s t r a i n increment r a t i o i n q u a t i o n ( 3 )
f
becomes unbounded. The v a l u e i m p l i e s h i + 0 a s-
+ 0. T h i s l i m i t5
i s e q u i v a l e n t t o a f l a t zone. The s o l u t i o n i m p l i e s a dynamic p r o p a g a t i n g i n s t a b i l i t y w i l l o c c u r f o r a narrow c r a c k - l i k e zone when a peak i s reached on t h e s t r e s s - s t r a i n curve. F i g u r e 5 g i v e s
t h e s o l u t i o n i n g r a p h i c a l farm [ a f t e r R i c e , 1 9 7 9 ( b ) ] . I n g e n e r a l , i n c r e a s i n g s h e a r s t r a i n r a t e s i n t h e weak zone a r e p r e d i c t e d a s t h e l o a d i n g i n c r e a s e s . I n s t a b i l i t y o c c u r s when t h e l i n e ( s l o p e
-
Ge/F;) S H E A R S T R E S S1-
-
-
dYi S H E A R S T R A I N F I G U R E 5 S C H E M A T I C O F T H E S O L U T I O N T O T H E L O A D I N G M O D E L [ A F T E R R I C E 1 9 7 9 ( b ) ] . S H E A R S T R A I N - R A T E I N C R E A S E S I N T H E W E A K Z O N E W I T H D E F O R M A T I O N . A N D I N S T A B I L I T Y R E S U L T S W H E N T H E C O N N E C T I N G L I N E O F S L O P E-GI[
B E C O M E S T A N G E N T T O T H E W E A K Z O N E S T R A I N - S O F T E N I N G C U R V E .Mechanics of snow slab failure
485connecting the slab and weak zone deformation curves is tangent to the weak zone deformation curve.
It is not trivial to extend the inclusion analysis to include time dependent loading and visco-elastic effects in the slab. Rice, Rudnicki and Simons (1978) show that the Eshelby relations
( 2 ) must be replaced by integral equations and they provide a solution for a spherical inclusion. From Fig.
5,
approximate extension to the case of visco-elastic deformation (creep) in the slab may be done if loading stops just prior to instability. The elastic unloading stiffness of the surroundings is G,/E. Near to the instability point (peak stress), time dependent reduction of the effective slab modulus by visco-elastic creep will force instability. Time scales should be typically on the order of the appropriate material relaxation time for the loading process in question.This analysis shows how a narrow zone can fail catastrophically near peak stress without significant strain-softening. The result is reinforced by general concepts for failure of granular and geotechnical materials. For these materials, once a peak is reached on the stress-strain curve, slip surfaces form and such is expected for snow. Under the assumed loading conditions, slip surfaces would further enhance the chances of instability near peak stress. The main reason strain-softening can be monitored in a constant rate of deformation test is that the implied incremental rate of horizontal loading on the sample decreases to small values near peak stress. The shear tests on snow reveal that constant rates of 0.15 mm/min sometimes provoke dynamic failure of the sample at or near peak stress. For tests with constant rate of horizontal loading this effect would surely be amplified.
Given the likelihood of slip surface formation once peak stress is achieved or the probability of imperfections occurring
naturally, intense instability is implied when loading causes the applied shear stress to approach peak stress in the layer. This concept fits observations because most snow slabs ( > 90%) occur when snow is falling.
ii) Constant Load Model
Consider the other extreme: the loads on the weak layer are constant and due to gravity forces caused by the overlying slab. The intention is to describe avalanche releases which occur after loading or storm periods are over. Slabs which release without snowfall or externally applied triggering mechanisms are rare. This is expected since the applied loads must be very near
(slightly below) peak shear stress if release is delayed. For constant loads, more deformation may be tolerated in the weak layer prior to a fracture condition than under loading. The assumption is that a slip surface or natural imperfection in the weak layer (or at its boundaries) may extend by shearing. Palmer and Rice (1973) describe when such a slip surface or band of concentrated deformation can become self-propagating. The important concept is the attainment of a critical length for the band using the Griffith
486
D.
M. McClung / G R A V I T Y B A N D - 1 F I G U R E 6 S C H E M A T I C O F A S L A B U N D E R L A I N B Y A S H E A R B A N D W I T H S T R A I N S O F T E N I N G T A K I N G P L A C E A T T H E T I P OF T H E B A N D .criterion in fracture mechanics. With the previous example of a homogeneous zone of fixed size and shape under loading, an exploration of the bounds on weak layer deformation conditions prior to fracture is provided. Neither extreme may be realistic in actual field conditions but they provide reasonable deformation bounds. An intermediate condition seems most appropriate in the majority of cases.
Consider a band of material in a weak layer with inhomogeneous deformation and strain-softening (failure) taking place at the tip of the band. The shear stress drops from peak to a residual value over a characteristic distance w downslope from the band tip. By analogy to small scale yielding in fracture mechanics, the
propagation condition for the shear band (Fig. 6) may be expressed as (Palmer and Rice, 1973).
The integral i[r(b)-rr]db = (rp-rr)6 is performed in the region of strain-softening in which shear stress drops from peak to residual and 6 is slip along the band. The parameter
%
is the mode I1 stress intensity factor from fracture mechanics for in-plane shearing deformation. For an elastic slab and constant residual stress along the band, an expression for the stress intensity factor is [McClung, 1981(b)]where
z
= j z pg sin+
dZ (Fig. 6). Equation (4) may be solved for L, fhe critical length of the band beyond which propagation isMechanics of snow slab failure 487
a s s u r e d i n terms of measured s t r a i n - s o f t e n i n g parameters. From parameters measured i n t h e l a b o r a t o r y , L i s e x p e c t e d t o be many s l a b t h i c k n e s s e s i n t h e f i e l d [McClung, 1 9 8 l ( b )
1.
T e n s i l e s t r e s s e s i n t h e s l a b and i n t e n s e i n s t a b i l i t y a r e b o t h i m p l i e d when t h e p r o p a g a t i o n c o n d i t i o n i s approached. The t e n s i l e s t r e s s e s were s t u d i e d by McClung [ 1 9 8 1 ( b ) ] . I f s e v e r e s t r a i n - s o f t e n i n g i s assumed t o have t a k e n p l a c e o v e r a r e g i o n of l e n g t h L, s i g n i f i c a n t t e n s i l e s t r e s s e s can be g e n e r a t e d . However, t e s t d a t a show t h a t a l p i n e snow i s a weak s o f t e n i n g m a t e r i a l . More
i m p o r t a n t l y , s t r a i n - s o f t e n i n g i s s e e n o n l y f o r a narrow range of d e f o r m a t i o n r a t e s . T h i s r a n g e would be d i f f i c u l t t o m a i n t a i n f o r a f a i l u r e p r o c e s s s p r e a d i n g o v e r a wide a r e a . Based upon measured mechanical p r o p e r t i e s of a l p i n e snow i t seems u n l i k e l y t h a t s e v e r e s t r a i n - s o f t e n i n g c o u l d t a k e p l a c e o v e r l a r g e a r e a s t o g e n e r a t e a slow primary t e n s i l e f r a c t u r e a s i m p l i e d by t h e model of P e r l a and LaChapelle (1970). A p p l i c a t i o n of t h e p r e s e n t model and t h e p r o p e r t i e s of a l p i n e snow i m p l i e s i n i t i a t i o n of a s h e a r band a t a s t r e s s c o n c e n t r a t i o n w i t h growth of t h e band by s h e a r i n g u n t i l a c r i t i c a l l e n g t h i s achieved. U l t i m a t e l y i n c r e a s i n g d e f o r m a t i o n r a t e s and r a p i d p r o p a g a t i o n are p r e d i c t e d .
The s h e a r band p r o p a g a t i o n c o n d i t i o n may be e a s i l y extended t o i n c l u d e time dependent l i n e a r v i s c o - e l a s t i c d e f o r m a t i o n i n t h e s l a b . T h i s r e q u i r e s replacement of t h e modulus l / E 1 i n e q u a t i o n
( 4 ) by t h e a p p r o p r i a t e c r e e p compliance f u n c t i o n C ( t ) where t i s time. The compliance f u n c t i o n i s chosen a c c o r d i n g t o t h e t i m e s c a l e d e f i n e d a s t h e l e n g t h parameter w d i v i d e d by t h e s h e a r band p r o p a g a t i o n speed ( R i c e , 1973). Thus, C(0) = l / E 1 ( 0 ) r e p r e s e n t s t h e h i g h speed modulus and C(m) =1/E1(m) i s t h e l o n g time modulus f o r v e r y slow p r o p a g a t i o n . The l a t t e r d e f i n e s t h e t h r e s h o l d s t r e s s i n t e n s i t y f a c t o r below which p r o p a g a t i o n i s n o t e x p e c t e d
I n t h e h i g h speed c a s e , t h e upper l i m i t s t r e s s i n t e n s i t y f a c t o r i s
For i n t e r m e d i a t e v a l u e s , K';<K~ <K;~, p r o p a g a t i o n o c c u r s f o r t h e v a l u e of C ( t ) a p p r o p r i a t e t o band speed and t h e growth of t h e band i s q u a s i - s t a t i c . The time when
%
t a k e s i n t e r m e d i a t e v a l u e s i s of i n t e r e s t t o compare w i t h t h e time s c a l e s f o r a v a l a n c h e r e l e a s e . Once s t o r m l o a d i n g s t o p s , a v a l a n c h e s sometimes r e l e a s e a day o r two l a t e r . A f t e r e x p l o s i v e c o n t r o l , r e l e a s e can o c c u r a n hour o r s o l a t e r . These examples imply s t a b l e growth p e r i o d s of 10-100 t i m e s t h e a p p r o p r i a t e r e l a x a t i o n t i m e (Shinojima, 1967). The p e r i o d s of development a r e comparable t o s t a b l e growth c a l c u l a t e d f o r d e l a y of f r a c t u r e by v i s c o - e l a s t i c e f f e c t s (Wnuk, 1971). Under c o n s t a n t l o a d , t h e time i n c r e a s e s s i g n i f i c a n t l y f o r a l i n e a r v i s c o - e l a s t i c m a t e r i a l i f t h e i n i t i a l v a l u e of I$r i s much lower t h a n K;. I have488
D.
M.
McClung d i s c u s s e d a p o s s i b l e example of t h i s e f f e c t f o r a v a l a n c h e r e l e a s e a f t e r e x p l o s i v e c o n t r o l [McClung, 1 9 8 1 ( b ) ] . This a n a l y s i s t o g e t h e r w i t h t h a t from t h e i n c l u s i o n model i l l u s t r a t e s t h e i m p l i c a t i o n s f o r t i m i n g of r e l e a s e s . Long d e l a y s f o l l o w i n g l o a d i n g a r e e a s i e r t o e x p l a i n by t h e t i m i n g e f f e c t s p r e d i c t e d f o r t h e s h e a r band model; d e l a y s up t o 100 t i m e s t h e a p p r o p r i a t e r e l a x a t i o n t i m e a r ep o s s i b l e . Delays s h o u l d a l s o depend on time dependent e f f e c t s i n t h e weak l a y e r and i t i s n o t p o s s i b l e , a t p r e s e n t , t o s e p a r a t e e f f e c t s i n t h e l a y e r from t h o s e i n t h e s l a b .
Based on e i t h e r t h e s h e a r band model under c o n s t a n t l o a d o r t h e l o a d i n g model, t h e s p e c t a c u l a r t e n s i l e f r a c t u r e a s s o c i a t e d w i t h t h e d r y s l a b a v a l a n c h e s h o u l d be g e n e r a t e d by a r a p i d p r o p a g a t i n g s h e a r i n s t a b i l i t y . S i n c e t h e s e two p i c t u r e s r e p r e s e n t r e a s o n a b l e
deformation extremes f o r t h e weak l a y e r p r i o r t o p r o p a g a t i o n i t i s l i k e i y t h a t t h e more a p p r o p r i a t e i n t e r m e d i a t e c a s e w i l l r e q u i r e t h e same l o g i c . FRACTURE GEOMETRY A p e r s i s t e n t f i e l d o b s e r v a t i o n i s t h a t t h e t e n s i l e f r a c t u r e l i n e i s n e a r l y p e r p e n d i c u l a r t o t h e s h e a r f a i l u r e p l a n e . For t h e g e o t e c h n i c a l m a t e r i a l s , t e n s i l e f r a c t u r e i s e x p e c t e d p e r p e n d i c u l a r t o t h e maximum p r i n c i p a l t e n s i o n s t r e s s . For a p u r e mode I1 c r a c k , t h e d i r e c t i o n of t h e maximum t e n s i o n s t r e s s i s d e f i n e d by S i h and L i e b o w i t z , 1968 from a s t a t i c a n a l y s i s
I n ( 8 ) , 8 i s t h e a n g l e between t h e bed s u r f a c e p l a n e and t h e l i n e p e r p e n d i c u l a r t o t h e d i r e c t i o n of t h e maximum p r i n c i p a l t e n s i o n n e a r t h e band t i p . The s o l u t i o n of ( 8 ) g i v e s 0 = 70.5'. This a n a l y s i s does n o t i n c l u d e dynamic e f f e c t s and i t i s f o r a n i s o t r o p i c m a t e r i s l .
Once p r o p a g a t i o n b e g i n s e i t h e r from l o a d i n g o r s h e a r band e x t e n s i o n t h e r e w i l l be a r e o r i e n t a t i o n of s t r e s s e s a t t h e t i p of t h e p r o p a g a t i n g band [ R i c e , 1 9 7 9 ( a ) ] . T e n s i l e f r a c t u r e w i l l
i n i t i a t e a t t h e b a s e of t h e s l a b n e a r t h e t i p of t h e p r o p a g a t i n g band. Achenbach and Bazant (1975) and Achenbach e t . a l .
(1976 a , b ) show t h a t l a y e r e d s t r a t i g r a p h y ( r e q u i r e d by f r a c t u r e l i n e s t u d i e s ) and dynamic e f f e c t s t o g e t h e r w i l l f o r c e t h e a n g l e f o r t e n s i o n f r a c t u r e a t t h e b a s e of t h e s l a b i n t o t h e r a n g e , 90'
+
l o 0 , r e p o r t e d f o r snow s l a b s ( P e r l a , 1975). The mechanical p r o p e r t i e s of snow make i t l i k e l y t h a t t h e a n i s o t r o p y and dynamic e f f e c t s a c t t o g e t h e r t o produce a t e n s i l e f r a c t u r e l i n e p e r p e n d i c u l a r t o t h e bed s u r f a c e .DISCUSSION
I have combined t h e mechanical p r o p e r t i e s of a l p i n e snow from s h e a r t e s t s w i t h s i m p l e d e f o r m a t i o n models and t h e g e n e r a l f a i l u r e
Mechanics of snow slab failure 489
c h a r a c t e r i s t i c s of g e o t e c h n i c a l m a t e r i a l s . When t h e s e e l e m e n t s a r e a p p l i e d t o t h e l a y e r e d s t r u c t u r e i m p l i e d by a v a l a n c h e f r a c t u r e l i n e s t u d i e s , one i s f o r c e d t o t h e c o n c l u s i o n t h a t t h e primary c a u s e of d r y s l a b a v a l a n c h e s i s t h e p r o p a g a t i o n of s h e a r i n s t a b i l i t i e s i n t h e weak l a y e r . The g e n e r a l model which e v o l v e s h a s t h e c a p a b i l i t y t o e x p l a i n e v e r y major observed f e a t u r e of d r y s l a b a v a l a n c h e r e l e a s e and i t i s c o n s i s t e n t w i t h known snow f a i l u r e p r o p e r t i e s .
A t p r e s e n t , t h e o n l y e x p e r i m e n t a l d a t a on slow s h e a r f a i l u r e s f o r a l p i n e snow a r e from s m a l l , homogeneous samples. Large s c a l e , slow i n - s i t u s h e a r t e s t s have a chance t o e x p l a i n t h e d i s c r e p a n c i e s between r a t e e f f e c t s and s t r e n g t h v a l u e s i n t h e l a b o r a t o r y and t h e f i e l d . Such t e s t i n g s h o u l d i n c l u d e n a t u r a l i m p e r f e c t i o n s and f a i l u r e s w i t h i n and a t t h e b o u n d a r i e s of weak l a y e r s i n o r d e r t o i n v e s t i g a t e s i z e e f f e c t s .
I have n o t d e a l t w i t h t e m p e r a t u r e e f f e c t s h e r e b u t t h e y have been mentioned i n p r e v i o u s work [McClung 1979, 1 9 8 1 ( b ) ] . A p r o p e r a n a l y s i s of t e m p e r a t u r e e f f e c t s , even f o r t h e s i m p l e models
d e s c r i b e d , h a s n o t y e t been provided. I n t h e s h o r t t e r m ,
i n c r e a s i n g s l a b t e m p e r a t u r e s s h o u l d promote i n s t a b i l i t y f o r e i t h e r model by d e c r e a s i n g s l a b s t i f f n e s s . However, prolonged c o l d s p e l l s
( l a r g e t e m p e r a t u r e g r a d i e n t s ) c a n produce weakened f a c e t e d c r y s t a l s i n t h e f a i l u r e l a y e r by metamorphism a l s o promoting e a r l i e r
i n s t a b i l i t y f o r any d e f o r m a t i o n c o n d i t i o n s t h e r e .
S i n c e weak l a y e r d e f o r m a t i o n s would be very d i f f i c u l t t o measure i n t h e f i e l d , measurement of a c o u s t i c e m i s s i o n s (e.g. Sommerfeld and Gubler, 1983) may y e t h o l d promise f o r fundamental s t u d y of t h e snow s l a b s t a b i l i t y problem. U l t i m a t e l y , v e r i f i c a t i o n of a v a l a n c h e r e l e a s e mechanisms must come mainly from f i e l d measurements.
11. FULL DEPTH AVALANCHES CAUSED BY GLIDING
INTRODUCTION
S l a b a v a l a n c h e s caused by g l i d e ( s l i p of t h e e n t i r e snowpack o v e r t h e ground) r e q u i r e d i f f e r e n t methods f o r p r e d i c t i o n and c o n t r o l t h a n d r y snow f a i l u r e s . F u l l d e p t h a v a l a n c h e s a r e
d i f f i c u l t t o t r i g g e r u s i n g e x p l o s i v e s and t h e y a r e r a r e l y caused by new s n o w f a l l ( L a c k i n g e r , 1986). The r e l e a s e i s u s u a l l y a s s o c i a t e d w i t h t h e a d d i t i o n of f r e e w a t e r i n t o t h e snowpack. The
c o n v e n t i o n a l e x p l a n a t i o n i s t h a t somehow f r e e w a t e r l u b r i c a t e s t h e i n t e r f a c e o v e r which t h e snowpack g l i d e s t o produce a n u n s t a b l e c o n d i t i o n .
It i s w e l l known t h a t t e n s i l e c r a c k s ( g l i d e c r a c k s ) p r e c e d e f u l l depth a v a l a n c h e f o r m a t i o n . F i e l d measurements show t h a t r a p i d g l i d i n g r e s u l t s when f r e e w a t e r i s added t o t h e snowpack i n s u f f i c i e n t q u a n t i t y . However, r a p i d g l i d i n g a l o n e cannot produce h i g h t e n s i l e s t r e s s e s i n a snowcover; a d e c r e a s e i n b a s a l f r i c t i o n downslope from t h e s i t e of c r a c k f o r m a t i o n i s r e q u i r e d . A
d e s c r i p t i o n of b a s a l f r i c t i o n c o n d i t i o n s when r a p i d g l i d e o c c u r s i s t h e c r u x of t h e s t a b i l i t y problem.
490
D.
M.
McClungRapid g l i d e caused by f r e e w a t e r must be due t o changes i n f r i c t i o n c o n d i t i o n s a t o r n e a r t h e snow-earth i n t e r f a c e . A
d e s c r i p t i o n of t h e s e changes r e q u i r e s a n e x t e n s i o n of snow g l i d i n g t h e o r y . The c o n v e n t i o n a l model [McClung, 1 9 8 1 ( a ) ] p r e d i c t s a r e l a t i o n s h i p between snowpack d r a g and g l i d e v e l o c i t y which depends only on b a s a l geometry and snow v i s c o s i t y . A p r i n c i p a l assumption i s t h a t b a s a l w a t e r i s a n i n f i n i t e s i m a l t h i n f i l m . I f a d d i t i o n a l w a t e r r e a c h e s t h e i n t e r f a c e , o r l a y e r s n e a r t h e i n t e r f a c e , t h e
b a s a l geometry o r snow v i s c o s i t y c a n change. I n o r d e r f o r w a t e r t o r e a c h t h e i n t e r f a c e i n s i g n i f i c a n t q u a n t i t y , p a r t i a l s e p a r a t i o n of t h e snowpack from t h e i n t e r f a c e must occur. When t h i s happens, p a r t of t h e s u b s t r a t e may be drowned r e s u l t i n g i n p a r t i a l l o s s of b a s a l f r i c t i o n . Water i n t h e l a y e r a d j a c e n t t o t h e i n t e r f a c e c a n a l s o reduce t h e snow v i s c o s i t y t h e r e t o i n c r e a s e t h e g l i d e v e l o c i t y producing a s i m i l a r e f f e c t . Due t o d r a i n a g e , t h e s e e f f e c t s c a n b o t h produce t h e f l u c t u a t i o n s i n g l i d e v e l o c i t y which a r e a b s e n t from t h e s t e a d y g l i d e t h e o r y b u t p r e s e n t i n f i e l d measurements. F i e l d o b s e r v a t i o n s show t h a t g l i d e c r a c k s c a n form w i t h o u t f u l l d e p t h avalanches. T h i s i m p l i e s p a r t i a l l o s s of b a s a l f r i c t i o n . When a r e g i o n of t h e i n t e r f a c e i s p a r t l y covered w i t h w a t e r , t e n s i l e s t r e s s e s w i l l be g e n e r a t e d i n t h e snowpack. An a n a l y s i s which s i m u l a t e s t h e s e c o n d i t i o n s shows t h a t a s u b s t a n t i a l a r e a of
t h e b a s a l s u r f a c e needs t o be covered w i t h w a t e r t o e x p l a i n known g l i d e c r a c k f e a t u r e s . The e f f e c t of reduced snow v i s c o s i t y by p r e s e n c e of f r e e w a t e r i n t h e l o w e s t l a y e r c a n be e q u a l l y e f f e c t i v e f o r d r a g r e d u c t i o n . These e f f e c t s w i l l probably a c t t o g e t h e r i n f i e l d s i t u a t i o n s .
A review of t h e b a s i c p r o p e r t i e s of g l i d i n g shows t h a t t h e extended t h e o r y e x p l a i n s t h e major f e a t u r e s . The u n c e r t a i n t i e s about t h e p r o c e s s e s i n v o l v e d i n d i c a t e t h a t more r e s e a r c h i s r e q u i r e d t o produce a d e f i n i t i v e p i c t u r e of a v a l a n c h e r e l e a s e . Some d i r e c t i o n i s g i v e n toward t h a t g o a l .
DESCRIPTIVE PROPERTIES OF SNOW GLIDING
The p r o p e r t i e s of snow g l i d i n g o v e r n a t u r a l rock o r v e g e t a t i v e s u r f a c e s a r e now f a i r l y w e l l known and most r e s e a r c h e r s a g r e e on t h e g e n e r a l c h a r a c t e r i s t i c s . I n d e r Gand and Zupancic (1966) l i s t some of t h e p r o p e r t i e s and my own work [McClung [1975, 1 9 8 1 ( a ) ] g i v e s some a d d i t i o n s . A summary of p r o p e r t i e s f o l l o w s .
( 1 ) There i s g e n e r a l agreement t h a t f o r g l i d e t o o c c u r o v e r a s u r f a c e w i t h roughness t y p i c a l of mountain t e r r a i n , t h e snow-earth i n t e r f a c e must be a t O°C. I n t e r f a c e t e m p e r a t u r e s a t O°C g u a r a n t e e t h e p r e s e n c e of some w a t e r t h e r e . The d i s t r i b u t i o n of f r e e w a t e r a t t h e i n t e r f a c e i s n o t w e l l known.
( 2 ) G l i d e c o n s i s t s of a s t e a d y and a f l u c t u a t i n g component. F l u c t u a t i o n s i n c r e a s e when w a t e r from m e l t o r r a i n f a l l r e a c h e s t h e i n t e r f a c e o r when summer h e a t s t o r e d i n t h e ground m e l t s snow a t t h e i n t e r f a c e . Rapid g l i d e from s t o r e d summer h e a t and i n c r e a s e s i n g l i d e r a t e s due t o r a i n f a l l a r e d e p i c t e d i n F i g . 7 (McClung, 1975).
Mechanics of snow slab failure 491 I I I r I I I I I I I
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0 21 I 15 1 15 1 15 1 1 5 1 1 5 1 15 3 0 O C T N O V D E C J A N FE B M A R A PR F I G U R E 7 G L I D I N G M E A S U R E M E N T S [A F T E R M C C L U N G , 19751 S H O W I N G H I G H R A T E S I N E A R L Y S E A S O N D U E T O S U M M E R H E A T S T O R E D I N T H E G R O U N D A N D F L U C T U A T I O N S D U E T O R A I N F A L L . ( 3 ) Steady g l i d e r a t e s seem t o i n c r e a s e a s t h e i n t e r f a c e s h e a r s t r e s s i n c r e a s e s due t o t h e a d d i t i o n of new snow ( i n d e r Gand and Zupancic, 1966). This e f f e c t i s i l l u s t r a t e d i n F i g . 8 (McClung,1975) f o r g l i d i n g on a timbered s l o p e . ( 4 ) G l i d e does n o t u s u a l l y o c c u r on s l o p e s of l e s s t h a n 15' f o r t e r r a i n w i t h roughness normally e n c o u n t e r e d i n a l p i n e a r e a s . ( 5 ) G l i d e does n o t o c c u r on a l l s l o p e s . It i s thought t h a t a r a t h e r smooth i n t e r f a c e s u r f a c e i s r e q u i r e d f o r g l i d i n g t o occur. The f a s t e s t g l i d i n g o c c u r s on smooth r o c k o r g r a s s y s l o p e s . ( 6 ) G l i d e r a t e s on a s u r f a c e covered by v e g e t a t i o n may be s i g n i f i c a n t l y i n f l u e n c e d when t h e v e g e t a t i o n a c t s a s a n anchor. Endo and Akitaya (1978) and Endo (1983, 1985) p r o v i d e a d r a m a t i c
- Ln 0 21 1 15 1 15 1 15 1 15 1 15 1 15 3 0 O C T N O V D EC J A N F E B M A R A P R F I G U R E 8 G L I D I N G M E A S U R E M E N T S [ A F T E R M C C L U N G . 1 9 7 5 1 S H O W I N G H I G H E R R A T E S O F T H E S T E A D Y G L I D E C O M P O N E N T W I T H I N C R E A S E D S N O W D E P T H . T H E M E A S U R M E N T S W E R E T A K E N O N A T I M B E R E D S L O P E A T T H E S A M E L O C A T I O N I N T W O S U C C E S S I V E Y E A R S . i l l u s t r a t i o n of t h i s e f f e c t f o r f u l l d e p t h a v a l a n c h e r e l e a s e on a bamboo covered s l o o e . ( 7 ) On smooth, g r a s s y s l o p e s t h e e n t i r e snowpack c a n s e p a r a t e from t h e g l i d e i n t e r f a c e t o form f o l d s a f t e r g l i d e c r a c k s form.
( 8 ) The s t e a d y component of g l i d i n g slows a s t h e s e a s o n
p r o g r e s s e s . T h i s may be due t o i n c r e a s e s i n snow v i s c o s i t y a t t h e
snow-earth i n t e r f a c e [McClung, 1981(a)
1.
( 9 ) Normal r a t e s f o r s t e a d y g l i d e a r e on t h e o r d e r of a few mmlday b u t r a t e s of s e v e r a l mm/min a r e n o t uncommon w i t h r e s p e c t t o t h e f l u c t u a t i n g component under u n u s u a l c o n d i t i o n s ( i n d e r Gand and Zupancic, 1966; David, e t . a l . , 1974).
( 1 0 ) Rapid g l i d i n g can c a u s e d e s t r u c t i o n of s o i l l a y e r s by plowing a c t i o n and i n j u r y t o young p l a n t s o r t r e e s a s w e l l a s damage t o s t r u c t u r e s ( i n d e r Gand and Zupancic, 1966).
(11) G l i d i n g a p p e a r s t o be more r a p i d on convex s l o p e s t h a n f o r s l o p e s w i t h o u t c u r v a t u r e o r concave s l o p e s .
I n a d d i t i o n , some c h a r a c t e r i s t i c s a r e known a b o u t g l i d i n g b e f o r e
and a f t e r g l i d e c r a c k formation. P r i o r t o c r a c k f o r m a t i o n , f i e l d
measurements i n d i c a t e t h a t g l i d e r a t e s a r e h i g h e r downslope from
t h e f u t u r e s i t e of t h e c r a c k (e.g. Endo, 1983). J u s t b e f o r e o r
d u r i n g c r a c k f o r m a t i o n , a c c e l e r a t i n g motion (which may be m i l d o r
v i g o r o u s ) i s n o t e d downslope from t h e s i t e of c r a c k f o r m a t i o n .
D e c e l e r a t i n g motion i s found u p s l o p e from t h e c r a c k a f t e r i t forms
( s e e Fig. 9 and Endo, 1983). G l i d e r a t e s t e n d t o be g r e a t e s t a t
Mechanics of snow slab failure
493 0 20 3 0 1 0 2 0 3 0 1 0 2 0 28 1 0 2 0 3 0 J A N 1 9 7 4 FE B M A R I I I 1 I I I I I C R A C K-
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D--a- F 1 - I I I F I G U R E 9 G L I D I N G M E A S U R E M E N T S [ A F T E R E N D O . 1 9 8 3 1 I L L U S T R A T I N Gs
I Z E E F F E C T S A N D L O N G I T U D I N A L D I S T R I B U T I O N O F G L I D E D I S P L A C E M E N T P R I O R T O A N D A F T E R G L I D E C R A C K F O R M A T I O N . T H E M E A S U R E M E N T S W E R E M A D E O N A B A M B O O C O V E R E D S L O P E .c r a c k s open f i r s t a t t h e snow-earth i n t e r f a c e (Endo and A k i t a y a ,
1978). Measurements of a c o u s t i c e m i s s i o n s (Gubler, p e r s o n a l
communication) a l s o i n d i c a t e a c c e l e r a t i n g motion p r i o r t o o r d u r i n g c r a c k formation.
G l i d e c r a c k f o r m a t i o n depends upon ground roughness a s w e l l a s
s l o p e angle. I n d e r Gand and Zupancic c i t e 30' a s a n approximate
lower l i m i t on s l o p e a n g l e s f o r g l i d e c r a c k i n i t i a t i o n . T h i s v a l u e
i s f o r g r a s s covered o r smooth rock s u r f a c e s . Rougher s u r f a c e s may
r e q u i r e s l o p e a n g l e s i n e x c e s s of 40'. They a l s o s t a t e t h a t c r a c k s
form more o f t e n on convex and s o u t h f a c i n g s l o p e s i n t h e Swiss Alps.
CONCEPTS FROM SNOW GLIDING THEORY
The fundamental problem of snow g l i d i n g t h e o r y i s t o r e l a t e
snowpack d r a g and t h e g l i d e v e l o c i t y . T h i s complicated problem i s
f a r from s o l v e d , b u t some b a s i c e l e m e n t s have been proposed. F i e l d
measurements show t h a t t h e r a t e of g l i d e depends upon t h e roughness
a t t h e snow-earth i n t e r f a c e and snowpack p r o p e r t i e s . Of c e n t r a l
importance f o r r a p i d g l i d i n g i s t h e o b s e r v a t i o n (e.g. i n d e r Gand
and Zupancic, 1966) t h a t g l i d e can have b o t h a s t e a d y and a
non-steady component. For non-steady g l i d e , v a r i a t i o n s i n g l i d e
v e l o c i t y w i t h o u t new s n o w f a l l on time s c a l e s of a day o r l e s s have
David, e t . a l . , 1974). For t h i s l e n g t h of t i m e , t h e p h y s i c a l c h a r a c t e r i s t i c s of snow a t t h e bottom of t h e snowpack would n o t change s i g n i f i c a n t l y u n l e s s t h e y a r e a f f e c t e d by f r e e w a t e r . I n a d d i t i o n , d a t a and f i e l d o b s e r v a t i o n s show t h a t i m p o r t a n t s o u r c e s of v e l o c i t y i n c r e a s e s a r e p r o c e s s e s which c a n produce a d d i t i o n a l w a t e r a t o r n e a r t h e snow-earth i n t e r f a c e . Examples i n c l u d e : r a i n f a l l , warming t r e n d s c a u s i n g m e l t , geothermal h o t s p o t s o r r a p i d d i s s i p a t i o n of h e a t s t o r e d i n t h e ground a t t h e b e g i n n i n g of t h e snow season. The observed f l u c t u a t i o n s i n g l i d e v e l o c i t y must r e p r e s e n t changes i n t h e b a s a l boundary c o n d i t i o n .
The s t e a d y p a r t of t h e g l i d e p r o c e s s was t h e f i r s t element of t h e problem d e a l t w i t h [McClung, 1 9 8 1 ( a ) ] . A t a g i v e n l o c a t i o n w i t h f i x e d bed geometry and snowpack p r o p e r t i e s , t h e t h e o r y p r e d i c t s a u n i q u e r e l a t i o n s h i p between g l i d e v e l o c i t y and b a s a l s h e a r s t r e s s . It i s assumed t h a t t h e snowpack i s s e p a r a t e d from t h e bed by an i n f i n i t e s i m a l , c o n t i n u o u s w a t e r f i l m . T h i s t h i n w a t e r f i l m a s s u r e s t h a t l o c a l l y t h e c o n t a c t between t h e snowpack and t h e bed does n o t s u p p o r t a s h e a r s t r e s s . Melt-freeze p r o c e s s e s a r e i g n o r e d s o t h e snowpack v e l o c i t y f i e l d i s t a k e n t a n g e n t i a l t o t h e i n t e r f a c e . G l i d e i s c o n s i d e r e d t o o c c u r by c r e e p o v e r t h e i n t e r f a c e roughness o b s t a c l e s . The e f f e c t i v e b a s a l s h e a r s t r e s s o r d r a g i s c a l c u l a t e d a s t h e downslope component of t h e f l u c t u a t i n g normal s t r e s s e s o v e r t h e o b s t a c l e s and t h i s i s r e l a t e d m a t h e m a t i c a l l y t o t h e s l i p v e l o c i t y a t t h e bed.
The s t e a d y t h e o r y t a k e s t h e snow t o deform a s a l i n e a r
c o m p r e s s i b l e v i s c o u s m a t e r i a l ( e q u i v a l e n t t o a Newtonian v i s c o u s f l u i d w i t h n e g l e c t of t h e s t a t i c f l u i d p r e s s u r e t e r m ) (e.g. Salm, 1967). This r e s u l t s i n a l i n e a r e q u a t i o n r e l a t i n g t h e b a s a l s h e a r s t r e s s , T, and t h e g l i d e v e l o c i t y , U , ( F i g . 1 0 )
