PARTE EXPERIMENTAL

P er v a siv e fracturing o f rock at h igh tem p erature is a d o m in a n t featu re o f v o lc a n ic sy stem s: w h eth e r this is fracturing o f the co u n tr y ro ck su rroun ding m a gm a c h a m b ers (in ferred from s e is m ic a ctiv ity ; N e w h a ll and P u n o n b a y a n , 1996); or fracturing o f lava d o m e s (F in k and G riffith s, 1 9 9 8 ); fracturing o f th e crust o f m o b ile lava flo w s ; or th eir s o lid ify in g fronts (K ilb u rn , 1 9 9 3 , 2 0 0 0 ). T w o d iv e rse e x a m p le s are the c o n d itio n s for m a g m a a sc e n t and th e e m p la ce m en t o f lava flo w s. T h u s, m a g m a a sc e n t is a ffec te d by m a n y in d ep en d en t fa cto rs su ch as the te cto n ic se ttin g and pressu re c h a n g e s w ith in th e m a gm a. H o w e v e r , it is the failu re o f the h o s t rock b e lo w the v o lc a n ic e d ific e , in v o lv in g fracture at e le v a te d tem p eratu res, that lea d s to th e o p e n in g o f n e w p a th w a y s th rou gh w h ic h fresh m a g m a is ab le to reach the su rface (G u d m u n d sso n , 1 9 9 8 , 19 9 5 ; R yan , 1994; S h a w , 1 9 8 0 ). T h e s e c o n d e x a m p le , lava flo w s , w h ic h is our m ain su b ject o f stu d y , in v o lv e s the prim ary hazard from e f fu s iv e v o lc a n o e s , and m o d e llin g rates o f flo w a d v a n ce and m a x im u m p o ten tia l len gth are im portant g o a ls for hazard m itig a tio n (K ilb u m , 1 9 9 6 , 2 0 0 0 ). M a n y fa cto rs a ffe c t the w a y a flo w e v o lv e s , in c lu d in g e ffu s io n rate, c o o lin g p r o c e s s e s , ch a n n el p ressu res and the to p o g ra p h y o f th e ground o v e r w h ic h it travels. It has lo n g b e en r e c o g n ise d that m aterial p ro p erties, su ch as r h e o lo g y are im portant co n tro ls on lava b eh a v io u r ( N ic h o ls , 1 9 3 9 ), but o n ly recen tly h as the im p o rta n ce o f lava strength a lso b een a d d ressed (K ilb u m , 1 9 9 6 ). In d eed , fracture in general p la y s a cru cial role in lim itin g m aterial stren gth (G riffith , 1 9 2 0 ). T o u n d erstan d h o w rock stren gth at high

C h i i p t c r J - T lu 'o r y o f l i r i t f l c I 'n i c t u r c (uu l D u c t i l e D c j o n i n i t i o i i

temperature varies with extrinsic and intrinsic conditions it is essential to understand how these affect both flow and fracture, and how flow ‘strength’ is limited by fracture. This can be clearly seen in principal stress maps.

Models and data for rock failure are also commonly plotted on two other representations: the Mohr construction and the deformation mechanism map (Jaeger and Cook, 1976; Frost and Ashby, 1982; Murrell, 1990). The Mohr construction plots shear stress against normal stress, but is poor at representing strain-rate dependent processes (Section 3.4.1). The deformation mechanism map plots shear stress against temperature, but is poor at representing the effect of normal stress (or pressure). The principal stress failure map can be employed to demonstrate both the effects of pressure and strain-rate-dependent processes. Therefore it is the best means of representing ductile flow, which is strongly dependent on strain-rate, and the role of fracture in limiting strength, as fracture is strongly dependent on normal stress or pressure.

3.4.1.

Empirical Shear Fracture Criteria- Mohr and Coulomb Theories

Mohr proposed that when a shear failure takes place across a plane, the normal stress Gn and the shear stress T across the plane are related by relation characteristic of the material,

M = / K ) [3.9]

This relation is equivalent to the envelope enclosing the Mohr circles at fracture and can be derived experimentally. This relation is curvilinear, concave towards the normal stress axis. However, the linear form of the Mohr criterion can be written as the Coulomb criterion,

\^p\

where |i is the coefficient of internal friction, Tq is the cohesive strength of the material. The Mohr circle and the Coulomb fracture criterion are shown in Figure 3.6.

C h u p t i ’r J - I h c o r y o f B r i / f lc I 'r a c tu r c o i t d D i u filc D e f o r m a t i o n

Figure 3.6- Coulomb criterion for shear fracture (from Jaeger and Cook, 1976).

In the axisymmetric case, the shear and normal stresses may be expressed as.

T^p = - - (o-, -

-

) sin 2 6 ,

CT„ = J (ct, + cr ) + j (cr, -

-

) cos 2 6 ,

[3.11]

[3.12]

where 0 is the fracture angle as shown in Figure 3.6. Substituting equation 3.11 and 3.12 into equation 3.10, fracture under principal stresses may be expressed as.

(<T, - crj sin

29 = tan ^

“ crj “

(<^i “ <^

) cos

26

where ±pi = tancj). Following the analysis of Jaeger and Cook (1976), by differentiating equation with respect 0, and since sin20 = (p,^ +1)'’^^ and cos 20 = -pi(pi^ +1)"'^^, the criterion of fracture may be rewritten as.

C l u i p r c r .î - T h e o r y o f 'B r iir ic I ' r o c f u r r m u ! D iic r ile D e f o r m a t i o n

O', - or. {/J.^ + 1) 2 = 2Zr [3.14]

This is a straight line in the Ci, G; plane. The Coulomb fracture criterion has a number of limitations: it does not reproduce the concave nature of experimental observations; the criterion is valid only in compression; and the coefficient of internal friction cannot be identified with any real frictional parameter, since the fracture has not yet formed. This is reconciled in the modified Griffith criterion where p; is the friction acting of the faces of microcracks (see Section 3.4.3).

3.4.2. The Griffith Energy Balance Concept

The Griffith failure criteria (1920, 1924) and linear elastic fracture mechanics are based on the energy balance concepts derived below. For a static crack in an elastic-brittle solid the total energy U, is given

by,

U — (-Up + Ug) + Us — Um + Us [3.15]

Where Um is the total mechanical energy in the system made up by the applied load (-Up) and the elastic strain energy stored in the solid (Ug), and Us is the surface energy. The mechanical energy term will cause the crack to extend, while the surface energy component will oppose the crack propagation, since the system needs energy to overcome the cohesive forces in order to create a new crack surface area. By differentiating equation 3.15 with respect to the surface area A, the equilibrium condition for each incremental change in crack surface area is obtained (cf. Meredith, 1990),

d U

dA dA

+

dA

= - G ç + G = 0 , [3.16]

where Gc is the energy release rate (Irwin, 1958) and expresses the rate at which mechanical energy decreases as the crack extends, and Gr is the energy rate resistance and characterises the resisting forces necessary for the creation of new crack surfaces. In an ideal brittle solid the energy rate resistance is twice the free surface energy y, since two new surfaces are created with every crack extension.

3.4.3. The Griffith and the Extended Griffith Crack Theory

With one of the most significant advances in the theory o f fracture, Griffith (1920) showed that fracture initiation occurs in brittle solids once the local tensile stress at the tip of the most favourable oriented pre-

C h u p l i ’v J I ’h c o r y o f B r i t t l e I ' n u t u r c a n d D a c t i l c D c f o n n a t i o n

existing crack exceeds the uncracked m ateriars tensile strength, oy. Griffith’s theory states that fracture is caused by stress concentrations at the tips of the so called Griffith cracks which pervade the material, and that fracture is initiated when the maximum stress near the tip of the most favourable oriented crack reaches a value characteristic of the material. Griffith (1924) analysed the biaxial stress field around an elliptical crack and found that the critical orientations that yielded the greatest tensile stress concentrations can be written as,

(cr, - <J^Y = 8(7y,(cr, + c r j f o r cr, + 3cr^ > 0

[3.17]

(T3 = - ( j j f o r cr, + 3(T3 < 0

where CTt is the uniaxial tensile strength and ai and C] are the principal and least compressive stresses. These equations produce a parabolic Mohr envelope (derived by Murrell (1958)).

= AgjCJ - 4cry^ = 0 [3.18]

For tensile fracture, the most crucially oriented crack is parallel to a i . For shear fracture, it is inclined at an angle 0 away from the a 1 axis given by,

COS 20 = - (cr, - <3-3) /(cr, + 0-3) [3.19]

A consequence of Griffith’s theory is that the presence of internal cracks can give rise to tensile stresses large enough for fracture even when the components of applied stress are both compressive, as long as they are not equal. Griffith’s criterion, based on microscopic fracture mechanisms, combines tensile and shear fracture into one single criterion. It predicts that the uniaxial compressive strength will be eight times the uniaxial tensile strength, which is smaller than the ratio commonly measured for rocks (McLintock and Walsh, 1962).

The plane Griffith criterion neglects the fact that cracks may close under a high compressive stress. Digby and Murrell (1976) established that cracks close under a compressive stress ten times the uniaxial tensile strength or the rock. Frictional stresses acting on the walls of the closed cracks tend to decrease the stress concentration at the crack tips resulting in an increase in the strength of the rock compared to one which crack walls were stress free. Hence the effect of crack closure in the presence of frictional stresses is to increase the strength a brittle body. The degree of strengthening depends on the coefficient of friction between the crack walls (Murrell and Digby, 1970). This criterion is know as the modified Griffith criterion and is expressed as, (Jaeger and Cook, 1976)

Chujitc r 3 - T h eo ry of Britrlc F rocriirc a n d D uctile D e f o r m a tio n + > ) 1 2 - yi _{- 0 - 3} + 1 ' 1 2 + / i = 4 7 ; + 2/i(cT3 - c r j [3.20]

if CTc, which is the normal stress required to close a penny shaped crack, can be neglected, equation 3.20 reduces to,

I

1

( 2

1V7

O']

\jd + \y - fd - 0-3

[jU

+ I f + //

which is the same as the Coulomb criterion, with t = 2Tq and p is now physically identifiable with the friction acting across crack walls. From equation 3.20 it follows that if all cracks are closed there is a linear relation between O; and 03; if all cracks are open the Griffith theory of equation 3.17 applies.