REGLAMENTO DE LA LEY GENERAL PARA LA PREVENCIÓN Y GESTION INTEGRAL DE LOS RESIDUOS (SUELOS)

A Lithospheric Strength Envelope (LSE) shows the relation-ship of lithospheric strength with depth derived from elasti-city, failure criteria, frictional sliding laws and flow laws (Fig. 9.2). The term yield strength envelope is commonly used (e.g. Goetze and Evans 1979), but since the upper part of the envelope is governed by cataclastic failure, LSE is a better term. LSEs are a powerful way of describing the

be-haviour of the lithosphere and have important applications to large-scale tectonic problems (e.g. Burov and Diament 1995, 1996).

The upper part of the LSE is usually constructed using Byerlee’s Law for frictional sliding to represent cataclastic behaviour. Coulomb or Griffith failure criteria can also be used to model the behaviour of intact lithosphere (cf. Sib-son 1983, Kusznir and Park 1987). A power law for disloca-tion creep is usually assumed for the lower, plastic part of the LSE, justified by observations that suggest that lithospheric stress/temperature/grain size conditions are appropriate for this deformation mechanism. Dislocation glide can also be included (e.g. Burov and Diament 1995).

The LSE is based on a model of the compositional structure of the earth. Gross simplifications are necessary due to the limited availability of rock mechanics data, and the difficulty of applying them to the earth. The upper crust is commonly modelled by wet quartzite, the mid to lower crust by feldspar or diorite, and the mantle as olivine. A temperature-depth profile is chosen for the crustal model. Appropriate steady state geotherms can be calculated from surface heat flux (e.g.

Kusznir and Park 1984, 1987), but Burov and Diament (1995) have modelled an evolving geotherm due to cooling after oro-geny.

The LSE is delineated by the lesser of the cataclastic or plastic strengths. The stresses can either be calculated for a given strain rate, or by applying a constant stress at the boundaries of a model that analyses the stress and strain with time. The latter approach shows that stress applied uniformly at the boundaries of the lithosphere is redistributed into the more competent layers which deform elastically, a process referred to as stress amplification (Kusznir 1982). Eventu-ally the elastic strength of all layers is exceeded, leading to whole lithosphere failure, and deformation continues by fric-tional sliding and intracrystalline plasticity (Kusznir and Bott 1977, Kusznir and Park 1982, 1984, 1987). Hence steady state LSEs have no part that is controlled by elastic strength.

The shape of the simplest LSE (for a lithosphere of homo-geneous composition) consists of a rapid linear increase in strength with depth from the surface of the earth governed by the high normal-stress dependence of Byerlee’s Law or a fail-ure criterion (e.g. the top part of Fig. 9.2). The cataclastic part of the LSE depends on the orientation of the failure sur-face: differential stresses for faulting or frictional sliding at any level of the crust increase in the order normal - strike-slip - thrust fault. The strength of the lithosphere reaches a maximum where the cataclastic strength is equal to the flow stress for intracrystalline plasticity. Below this point the flow stress is less than the cataclastic strength, and continues to decrease with depth as temperature increases. LSEs for layered models of the earth generally show the above pattern repeated in each layer (Fig. 9.2). There are dramatic strength contrasts across layers, especially between the crust and the mantle. This suggests that detachment could occur at these layer boundaries, which are sometimes misleadingly called

“brittle-ductile transitions”, but are more accurately referred to as brittle-plastic transitions (Section 1.3, 1.5).

However, saw-tooth LSEs such as those in Fig. 9. 2 can be criticized because the validity of Byerlee’s law has not

98 CHAPTER 9. FROM MICROSTRUCTURES TO MOUNTAINS

been established at mid to lower crustal pressures, and the high stresses predicted by extrapolation of the law to these conditions are unrealistic (Ord and Hobbs 1989). Moreover, the semibrittle deformation regime is not recognised, prob-ably because of the paucity of published flow laws. More realisitic possibilities for the shape of the LSE at mid crustal depths are either a vertical line of constant stress with a value equal to the stress at the base of the seismogenic zone (Hobbs et al. 1986), or non-linear, pressure-dependent functions de-rived from Mohr-Coulomb constitutive laws for permanent deformation (Ord 1991). The experimental basis for the lat-ter laws is very limited, and they have the disadvantage of predicting strain-dependent values of stress.

9.10 Palaeopiezometry

9.10.1 Methods and calibration

Palaeopiezometry is the determination of past stress fields, which can provide major constraints on tectonic models. Mi-crostructural methods have concentrated mainly on measur-ing differential stress (Sections 9.10.2-9.10.6), and the

ori-entation of the stress tensor (Sections 9.10.7-9.10.8). Pa-laeopiezometers are calibrated by experiments and theory, but there are large variations in calibrations of the same pa-laeopiezometer. Furthermore, different palaeopiezometers yield considerably different values when applied to the same rocks. The latter variations can potentially lead to further tec-tonic insights (Section 9.10.9). Determination of the mean stress is considered in the separate Section on geothermoba-rometry (Section 9.11).

9.10.2 Recrystallized grain size

can be related to differential stress ( MPa) by an expression of the form:

The size of grains recrystallized during deformation (d, mm)

where m and are constants, which can be determined ex-perimentally or derived theoretically (e.g. White 1979). Val-ues of m and are given in Table 9.5, and the calibrations for quartz are plotted in Fig. 9.3. The differential stresses for quartz predicted by different calibrations differ by up to four orders of magnitude, for at least seven possible reasons,

100 CHAPTER 9. FROM MICROSTRUCTURES TO MOUNTAINS

which are worth examining in detail because they highlight the problems of recrystallized grain size palaeopiezometry.

1.

Recrystallized grain size appears to depend on the re-crystallization mechanism: subgrain rotation (regime 2) gives smaller grains than grain boundary migration (re-gime 3) (e.g. Drury et al. 1985, Drury and Urai 1990, Hirth and Tullis 1992).

The importance of the effect of water is obvious from comparing the dry and wet calibrations of Ord and Christie (1984). However, it does not appear to be im-portant in olivine (Van der Wal 1993).

The possibility that the recrystallized grain size is sens-itive to temperature is suggested by some data on metals and olivine (e.g. Mercier et al. 1977, Ross et al. 1980), and predicted by theory (Mercier 1980a).

The possibility that the recrystallized grain size is de-pendent on quartz phase is suggested by the flow law dependence on phase (9.4.2.2).

Kinetic effects may severely hamper determination of the calibration constants if equilibrium grain size is not attained during experiments (Twiss 1977). The lack of such equilibrium for grain sizes on the order of mm is strongly suggested by the experiments of Kronenberg and Tullis (1984); hence recent experiments have been on much finer grain sizes (e.g. Post and Tullis 1999).

Equilibrium grain size may not be achieved if grain boundaries are pinned by impurities (e.g. Evans and White 1984).

Recrystallized grain size has been measured by various techniques and may be specified by different parameters:

a standard practice is to measure the mean linear inter-cept grain size and multiply by 1.5 to allow for trunca-tion and sampling effects (e.g. Christie and Ord 1980).

However, this will not be correct for grains with a fabric.

9.10.3 Subgrain size

Subgrain size can be related to stress by an expression of the form:

where and are calibration constants, is the shear mod-ulus, and is the most common Burgers vector. Values of 1 and 2 for have been proposed on theoretical grounds (Twiss 1977). Another possible relationship proposed by Twiss (1986) is:

where C and are constants, implying that there is a stable subgrain size at zero stress. However, the second rela-tionship does not fit data for olivine and quartz better than the first. This may be due to the lack of accurate, low stress meas-urements (Twiss 1986). Until these are available, empirically derived constants for the first relationship are probably the most satisfactory calibration of the subgrain size palaeopiezo-meter, as given in Table 9.6. Problems with the application of this palaeopiezometer include a possible temperature de-pendence, the sensitivity of subgrain size to water known for olivine (Twiss 1986, Van der Wal 1993) and the problem of subgrain size measurement. Subgrain sizes may be measured under the optical microscope in reflected light after etching (see Ord and Christie 1984 for the method) or by electron mi-croscopy. Optically determined subgrain sizes are commonly an order of magnitude larger than subgrain sizes measured by electron microscopy (Ord and Christie 1984). Available data at present may not allow reliable extrapolation outside the range of subgrain sizes used to calibrate the palaeopiezo-meter, since alternative calibrations fit the experimental data almost equally well, yet give enormous differences when ex-trapolated (Twiss 1986).

9.10.4 Dislocation density

The dislocation density relationship can be

written as:

where D and are constants, implying a steady state free dislocation density. However, data for olivine and quartz are not fitted any better by the second relationship. Table 9.7 presents calibration constants for quartz and olivine using the first expression. Problems with this palaeopiezometer are possible temperature dependence and the difficulty of meas-uring dislocation density accurately. Invisibility of tions under the TEM may cause an underestimate in disloca-tion density of 30% (Ord and Christie 1984).

9.10.5 Twinning - differential stress

Three different approaches have been proposed to relate twin-ning in carbonates to differential stress. Two of these (Jam-ison and Spang 1976 and Laurent et al. 1981, 1990) take no account of the fact that grain size is known to have an effect on the twinning, and are therefore less suitable than where k and u are constants, and and are defined as above.

A theoretical value of u is 0.5 (Weathers et al. 1979), but measured values have a range from 0.45 to 3.33 (Twiss 1986).

An alternative form of the relationship sometimes used is:

the approach of Rowe and Rutter (1990), who have calib-rated three twinning palaeopeizometers from laboratory ex-periments, which all appear to be independent of temperature, strain rate and strain.

Twinning incidence,

is the proportion of grains with optically visible twins in any grain size class. It can be related to differential stress

In practice, stresses are calculated from for each grain size class and plotted against grain size: they should lie on lines of constant slope for different values of The standard error in the experimental calibration was 31 MPa.

Twin density,

is the number of twins per mm. This is calculated by the slope of the relation between number of twins in a grain size interval and grain size. appears to be independent of grain size, and is related to with a standard error of 43 MPa, by:

Maximum twin volume,

is the maximum volume proportion (%) of twinned ma-terial in a grain size class. In the experiments, the area frac-tion of twins was taken as equal to the volume fracfrac-tion, and was measured from a two-dimensional thin section. is related to with a standard error of 41 MPa, by :

As with the previous palaeopeizometers, there are a number of potential problems. Clearly if twinning is the only deform-ation mechanism, the amount of twinning measured by any of the three parameters will be a function of strain. The method thus assumes that after twinning, equilibrium is reached with

a non-twinning deformation mechanism. In principle, grain size distribution could have an effect on this palaeopiezo-meter because stress distributions in grains are strongly in-fluenced by the size of neighbouring grains (Newman 1994), but this has not yet been evaluated or demonstrated. This palaeopiezometer has been calibrated only for calcite (other carbonates twin at different stress levels), and since the cal-ibration experiments were carried out at 400°C and above, it is probably not appropriate for low temperature deformation (Burkhard 1993). The best results will be obtained in samples that have experienced a single, coaxial strain event (the con-ditions of the experimental calibration), otherwise the method will overestimate stress (Rowe and Rutter 1990).

There is some experimental evidence that twinning in pyroxene could be used as a palaeopiezometer: Tullis (1980) suggests that the minimum stress needed for twinning in pyroxenes is 100 MPa.

9.10.6 Deformation lamellae

The appearance of deformation lamellae (Section 4.7) in ex-periments is characteristic of flow laws with an exponential relationship between stress and strain rate, which occur at relatively high stresses. This observation leads to the sug-gestion that there may be a critical differential stress for the formation of deformation lamellae, dependent on the type of bonding and crystal structure of the material (Blenkinsop and Drury 1988). Values of 100-200 MPa have been suggested for quartz. However, it is possible that the critical stress may be inversely temperature-dependent, so that at present deformation lamellae in quartz should only be taken as qual-itative indicators of relatively high stress levels (Drury 1993).

The relation derived by Koch and Christie is ( MPa):

(MPa) and grain size (d, mm) by the following equation:

Spacing of deformation lamellae (s, mm) may constitute a palaeopiezometer (Koch and Christie 1981, McLaren 1991).

102 CHAPTER 9. FROM MICROSTRUCTURES TO MOUNTAINS

9.10.7 Principal stress orientations from de-formation lamellae

Techniques of using deformation lamellae in quartz to de-duce principal stress orientations were developed by Carter and Friedman (1965) and Carter and Rayleigh (1969). Of the three methods proposed by Carter and coworkers, the “ar-row” method is the easiest and yields consistent results. The orientations of the c-axis and deformation lamellae are first measured from individual quartz grains. The plane containing the pole to a lamella and the c-axis is constructed (Fig. 9.4).

and are located within this plane at 45° to the lamella plane. is distinguished from by the fact that the c-axis lies closest to An arrow from the c-axis to the lamella pole will point from to (Fig. 9.4). Unique and orientations can be derived by eigenvector analysis from a number of such measurements (e.g. Spang and Van der Lee 1975). This construction implies that the deformation lamel-lae form in a plane of maximum resolved shear stress with a maximum shear stress direction parallel to the projection of the c-axis onto the lamellae plane. The evidence presented in Section 4.7 shows that recovery is an integral part of lamellae development, so that a simple analogy with a slip system for the formation of lamellae may not be entirely valid. However, the strength of the method is based on its empirical success (Pavlis and Bruhn 1988, Drury 1993), even if its theoretical basis remains unclear. The principal stress axes from deform-ation lamellae are in good agreement with those calculated from the carbonate twin method from the same samples (e.g.

Spang and Van der Lee 1975). It is possible to consider other angular relationships between and the lamella plane, but some results suggest that the stress axes do not change ap-preciably from the 45° constraint (Spang and Van der Lee 1975). It is also possible to analyze for the principal stresses without using a fixed angular relationship between and the lamella plane, by using the right dihedra technique (cf. An-gelier 1984).

9.10.8 Principal stress orientations and strains from twins

Twins have been used to deduce principal stress orientations by assuming that the twin plane is a shear plane and the twin direction is parallel to the maximum resolved shear stress

(e.g. Turner 1953, Weiss 1954). Twinning in calcite occurs on the e planes in the direction of the plane contain-ing the pole to the twin and the c-axis (Fig. 9.4). and are therefore at 45° to the twin plane in the plane of the c-axis and the pole to the twin - a similar geometry to that used for the quartz deformation lamellae method. However, in con-trast to that method, is in the opposite quadrant from the c-axis (Fig. 9.4). A similar construction can be applied to f twins in dolomite, but here is in the same quadrant as the c-axis (Fig. 9.4).

A development of this technique is to use the right dihedra method to determine the and orientations that are com-patible with the largest numbers of twins. The value of the maximum number of compatible twins (the MAX number) gives a measure of the degree to which the data can be fitted by a single stress orientation (Pfiffner and Burkhard 1987).

This method has the advantage that it does not assume a fixed angle between and the twin plane. A further development is the determination of principal stress orientations and ab-solute magnitudes by assuming homogeneous stress distribu-tion and a critical resolved shear stress for twinning (Laurent et al. 1981, 1990).

It is possible to calculate a finite strain tensor by incorpor-ating the shear strain necessary for twinning into the orient-ation analysis (e.g. Groshong 1972, 1974, 1984). A com-puter program to carry out the analyses is described by Evans and Groshong (1994). The measurements necessary for the determination include the c-axis orientation, twin set orienta-tion, average thickness and number of twins, and grain width.

25 grains in two perpendicular thin sections should be meas-ured.

Twins which do not fit the calculated strain or stress fields are a problem in all these methods. Two basic philosophies to deal with this problem have been adopted. The first is to recognize that some twins will inevitably develop in incom-patible orientations due to stress inhomogeneity. The propor-tion of such twins (negative expected values, NEV, Groshong 1972) can be used to test the homogeneity of the sample, and the incompatible grains can be removed from the data set.

Alternatively, the incompatible twins are presumed to reflect different superimposed stress fields, and they can be used to separate out a number of different stress fields (e.g. Lacombe et al. 1990). This method has been criticized by Burkhard (1993) because it has yielded unrealistic results, and the

as-104 CHAPTER 9. FROM MICROSTRUCTURES TO MOUNTAINS

sumption of stress homogeneity is clearly false in a polysized grain aggregate.

9.10.9 General problems with palaeopiezomet-ers

A general problem with all palaeopiezometers is the interpret-ation of the results. The significance of the stress level recor-ded can only be evaluated in the context of an evolving stress field, analogous to the problem of interpreting results from geothermobarometry within a P-T path. Interpretations be-come especially problematic when considering results from the same sample that differ according to the method used. For example, stresses measured by dislocation density, subgrain size and recrystallized grain size may differ because they have variable dependence on strain magnitude. Dislocation dens-ities may be reset after less than 1% strain, compared to 5%

for subgrain sizes and perhaps 30% for recrystallized grain sizes; dislocation densities may therefore record later tectonic events with low strains such as stress during uplift (e.g. White 1979a). This problem can potentially be overcome by plot-ting stress levels derived from one palaeopiezometer against another. If the stresses are recorded by both palaeopiezomet-ers from the same part of the stress-time path, they will lie on a line with a unit slope, while deviations from this line indicate records from different parts of the stress path.

9.11 Geothermobarometry

9.11.1 Methods and calibration

The determination of past temperature and mean stress (geo-thermobarometry) is traditionally the domain of thermody-namics and mineral chemistry. However, there are a num-ber of promising new developments in the application of mi-crostructures to geothermobarometry, which are particularly robust because they are calibrated from naturally deformed samples.

In addition to these new, quantitative approaches, experi-ments and thermodynamic results allow some generalizations about the minimum temperatures for plasticity. Plasticity in the form of twinning can occur even at room temperature in calcite, but appears to require a temperature of 300°C in

In addition to these new, quantitative approaches, experi-ments and thermodynamic results allow some generalizations about the minimum temperatures for plasticity. Plasticity in the form of twinning can occur even at room temperature in calcite, but appears to require a temperature of 300°C in