User:Axelwang1/sandbox

As for any material, the strength of Earth's lithosphere is defined as the minimum force per unit area, i.e. stress, required to deform the Earth's lithosphere. Strength therefore has the same units as stress. Because Earth's lithosphere encompasses materials of varying composition, experiencing different pressure, temperature and water content, and because stresses are applied on different timescales, there is no single mode that can describe all deformations for the whole lithosphere. Instead, at a given set of conditions, one particular mode will allow the minimum stress to induce a strain, and the lithosphere under this set of conditions is said to be in the regime of deformation corresponding to that mode. For example, in the upper 10–15 km of the lithosphere, deformation is in the elastic-brittle regime, and is described by Mohr-Coulomb criterion (for opening of new fractures) and Byerlee's law (for frictional sliding along existing fractures). As depth (as a proxy for temperature and pressure) increases, deformation transitions to the plastic-ductile regime, and is described by Dorn's law (for smaller differential stress) and Goetze's criterion, both are forms of power-law creep (slow flow). As such, deformations in the lithosphere exhibit characteristics of both solid and fluid. Rheology, a collective term referring to different subjects in continuum mechanics including plasticity and fluid mechanics, deals with this type of combined deformation. All of the equations mentioned above are determined by fitting experimental data. Plotting these relations, usually as the log of the difference between the largest and smallest principal stresses, under some given conditions (e.g. layers of different mineralogy, geotherms, water content etc.), against lithospheric depth, one can find the part of the curves having the smallest values of differential stress resemble the shape of a Christmas tree. These "Christmas- tree diagrams" are theoretical predications of vertical variations of lithospheric strength, for the conditions under which they are produced.

Strength controls how the lithosphere responds to both short-term (seismic) and long-term (tectonic) forces. Lithospheric response to long- term tectonic forces, such as vertical load and subduction, is characterized by its equivalent elastic thickness, which combined with the specific state of stress, is related to the maximum depth possible for generation of earthquakes. These two depths values have been essential in studying rheological models for a particular area.

Rheological laws and basic variables
In rheology, all equations of deformation follow the basic form of mechanical laws, i.e. a relation between a dynamic (force) variable and a kinematic (motion) variable, with some intrinsic properties of the object as constants of proportionality. As a simple example, for Newton's second law, $$F=ma$$, the dynamic variable is the force $$F$$, kinematic variable the acceleration $$a$$, and the intrinsic property the mass $$m$$. All laws of rheology therefore take the form, following the notation of Ranalli (1995), $$R(\epsilon,\dot{\epsilon},\sigma,\dot{\sigma},...,\{M\})=0$$where $$\epsilon$$,$$\sigma$$ denote strain and stress, respectively. An overdot indicates a time- derivative, and $$\{M\}$$ is a collection of intrinsic material properties, such as compressibility, viscosity, flexural rigidity, creep activation energy. It must be noted that although the intrinsic properties are material- dependent, they are also dependent on extrinsic properties such as temperature, pressure and water content. Hence, even for a given material, its intrinsic properties can be different under different sets of extrinsic conditions. The effect of extrinsic properties on the rheological laws is on the specific form of the rheological function $$R$$.

For rheological laws, the dynamic variable is the stress, and the kinematic variables can be both the strain and the strain rate. For deformations in the elastic-brittle regime, the deformation is instantaneous, i.e. the material fails without going through much strain, hence it is natural in this setting to use strain as the kinematic variable. For the plastic-ductile deformation regime, the degree of deformation, or strain, changes in response to different levels of stress, hence it is natural in this setting to use the strain rate $$\dot{\epsilon}$$as the kinematic variable. Note that strain rate is calculated from strain by taking a total derivative with respect to time.

Brittle fracture and frictional sliding
In the upper crust, deformation occurs as brittle fracture or frictional sliding. Strength mainly derives from compressional overburden pressures, so it increases as depth increases. Once the applied stress exceeds the overburden pressure, brittle fracture happens. The presence of pore fluid can effectively reduce the amount of stress needed to induce brittle fracture. This is captured by the Mohr-Coulomb criterion:

$$\tau = S_0 +\mu \sigma_n(1- \lambda)$$

where $$\tau$$ is the shear stress; $$\sigma_n$$ is the normal (overburden) stress; $$S_0$$ and $$\mu$$ are coefficients of cohesion and friction, respectively; $$\lambda$$ is the pore fluid factor, which is defined by a ratio of pore fluid pressure over the lithostatic (overburden) pressure $$\lambda = p_f / \rho g z = p_f / \sigma_n$$, where $$\rho$$ is overburden density, $$g $$ is the gravitational acceleration and $$z$$ is depth. By adjusting the values of $$S_0$$ and $$\mu$$, the Mohr-Coulomb criterion gives the required shear stress to overcome the overburden pressure to open a new fracture. Byerlee's law gives the shear stress required for frictional sliding along a pre- existing fracture:

$$\tau=0.85 \sigma_n, \quad \sigma_n<200 \ MPa $$

$$\tau=0.5+0.6 \sigma_n, \quad 200 \ MPa <\sigma_n<1700 \ MPa$$

It is determined from experimental data and based on the Mohr -Coulomb criterion. It is only accurate for temperatures below about 400 $^oC$. Given a specific fault geometry (normal with respect to the maximum principal stress direction), the shear and normal stresses above can be represented by the minimum and maximum principal stresses.

Transition zones and deformations within
Transition zones are bounded by the brittle-ductile transition (BDT) in the upper limit and the brittle-plastic transition (BPT) in the lower limit. Brittle-ductile transition refers to a change in the distribution of induced strains, from localized to distributed. Pure brittle fractures is localized on the pre-existing plane of weakness, while ductile deformations do not have such localizations. Brittle-plastic transition corresponds to changes in the dominant deformation mechanism, hence it happens somewhere away (further down in depth) from the last depth of brittle frictional sliding, well into the depths where rheological laws of plastic flow dictates the stress-strain relation. Transition zones correspond to the corners of the Christmas-trees (strength profiles) and represent the strongest parts of the lithosphere. There will be more than one transition zone in the lithosphere if there are multiple transitions from brittle to ductile deformation regimes. This will be dependent on lithologic layering and on the specific tectonic settings.

Deformation in the transition zone is the least well-understood. There is no constitutive relation describing them. They are often viewed as a mix of brittle frictional sliding and plastic flow, thereby exhibiting complex stress-strain relations.

Non- Newtonian flow
Flow in the lower lithosphere is modeled by power- law creep, which is a form of non-Newtonian fluid where strain rate is proportional to an integer power of stress.

Goetze's criterion states

$$\dot{\epsilon}=A (\sigma_1- \sigma_3)^3 \exp{(-H/RT)}, \quad (\sigma_1- \sigma_3)< 200 \ MPa$$

and for differential stress above 200 mega- pascals, Dorn's law predicts

$$\dot{\epsilon}=B \left[1-\left(\frac{\sigma_1-\sigma_3}{A_d}\right)^2\right] \exp{(-H/RT)}, \quad (\sigma_1-\sigma_3) > 200 \ MPa $$

In the equations, $$A$$ and $$A_d$$ are stress constants; $$B$$ is strain rate constant; $$H$$ is the creep activation energy; $$R$$ is the gas constant; $$T$$ is absolute temperature. It should be noted that these stress-strain rate relations decays exponentially as temperature increases, which is why strength of the lithosphere sharply decreases below the transition zone. These laws form the curved branch of the Christmas tree.

Strength profiles
Strength profiles plot the maximum differential stress lithospheric materials can bear before deformation as a function of depth. They always contain linear part(s) corresponding to brittle deformation and non- linear part(s) corresponding to power-law creep. The number of these parts and their respective depths will be dependent on the specific extrinsic and intrinsic conditions. Goetze and Evans (1979) were the first to introduce them, in a study for the oceanic lithosphere.

Estimating a strength profile
Lithospheric strength profiles can only be estimated, based on results from rock-physics experiments. The estimated profiles can then be cross-checked with observations of earthquake depth distributions, because only brittle and transitional regimes can produce earthquakes.

To estimate a lithospheric strength profile one must first determine which lithospheric model to use. As an example, the following are parameters of the lithospheric model that Ranalli and Murphy (1987) used to estimate lithospheric strength in a variety of tectonic settings: lithospheric thickness, crustal thickness, composition of the crust, type of faulting, and geothermal gradient. In their study, the composition of lithospheric mantle is fixed to be ultrabasic, whereas the composition of the crust is either one layer that is quartz-granitic or two-layer that is quartz-granitic over intermediate-basic. Types of faulting comes in to play when converting the Mohr-Coulomb criterion to be in terms of the differential stress. Following the modified Anderson theory, one can write the brittle deformations collectively in terms of the differential stress as

$$\sigma_d=\alpha \rho g z (1- \lambda)$$

where the value of $$\alpha$$ depends on the type of faulting and $$\sigma_d = \sigma_1- \sigma_3$$ is the differential stress. For the plastic regime, it is straightforward to write either Goetze's criterion or Dorn's law with the differential stress isolated. For example, Goetze's criterion can be written as

$$\sigma_d = \left(\frac{\dot{\epsilon}}{A_D} \right)^{(1/3)} \exp{(H/3RT)}$$. Notice that differential stress in case of ductile flow is strain-rate-dependent, meaning each strength profile is drawn for a specific strain rate. Background mean strain rate is often known within one-order-of-magnitude accuracy, and this degree of precision is sufficient because it produces strength differences of at most 10%. Now, given a lithospheric model, the above two equations will predict differential stress as a function of depth (using geothermal gradient to convert depth to temperature), and the smaller of the two at a given depth will form part of a strength profile.

Caveats
Uncertainties of a theoretical strength profile can come from the following:


 * Composition. Seismic investigation can only yield blurry images for average subsurface composition, and even if the composition is perfectly correct, rheological parameters of different rocks under different extrinsic conditions carry experimental uncertainties themselves.
 * Geotherm. Geotherm can have an uncertainty as large as 100 $$K$$ in the lower crust, due to large variations of surface heat flow, from which geotherms are typically estimated.
 * Distribution of stress. In the above analysis, it is assumed that the distribution of stress is homogenous and that the strain rate is constant. This is not true in reality. Rutter and Brodie (1992) showed large inhomogeneity in the distribution of shear stress in the lower crust.
 * Extrapolation of the Mohr- Coulomb criterion. Due to the large overburden pressure at depth, extending the Mohr-Coulomb criterion to depth can lead to very large differential stress that could be unrealistic. Presence of water (i.e. pore fluid pressure) can help mitigate this issue.

Seismogenic thickness ($$T_s$$) and equivalent elastic thickness ($$T_e$$)
Theoretically speaking, given a lithospheric strength profile, any depth above the BPT (brittle-plastic transition; i.e. above the transition zone) can generate earthquakes. This depth corresponds to the maximum possible seismogenic thickness, $$T_s(max)$$. However, the true seismogenic thickness is dependent on the specific state of stress of the lithosphere. Tectonic forces have to "load" the lithosphere to "close" to the critical point (i.e. touching the strength profile), so that further small stress perturbations can cause faults to slide, making an earthquake possible. Here, earthquakes refer to intraplate earthquakes and do not include the ones associated with subduction, whose depth range is clearly not related with the seismogenic zone. Flexure of the lithosphere results from tectonic forces such as vertical loads including mountains on land and in ocean, as well as during subduction. Flexure of a purely elastic plate due to vertical load or is undergoing subduction is illustrated to the right. Black arrows in the center of the plot are stresses experienced by a "fibre" in the plate. As an example of how stress levels are related to the curvature, for the case of a vertical load, let $$x_3$$ be the vertical direction and $$x_1$$ be the horizontal direction parallel to the page, then the horizontal stresses along $$x_1$$ are

$$\sigma_{11}= -\frac{E}{1-\nu^2}x_3 \frac{d^2 \xi}{d x_1^2}$$

where $$E$$ is Young's modulus; $$\nu$$ is Poisson's ratio; $$x_3$$ is depth and $$\xi$$ is deflection from the horizontal. Since the vertical stresses are small compared to the horizontal stresses, this formula is in fact representative of the differential principal stress. Clearly, it is dependent on material properties, depth, and curvature of the flexure.

The elastic picture gives the maximum equivalent elastic thickness $$T_e(max)$$. It equals to the entire lithospheric thickness, if the entire lithosphere is made of purely elastic materials. Realistically, the lithosphere deforms according to different rheology, but an equivalent elastic thickness can still be established. $$T_e(min)$$ consists mainly of the depth region where stresses induced by vertical loads (or subduction) are "far" from the strength profile (the region between the blue bars in the figure with oceanic and continental strength profiles), and therefore these areas, referred to as elastic cores, will either not easily fail or they will only deform by creep. Brittle and ductile layers above and below elastic cores also contribute to elastic strength, hence the minimum equivalent elastic thickness, $$T_e(min)$$,is the thickness of the elastic core plus small distances above and below. This equivalent elastic layer is the layer that supports the flexure. Most methods to estimate $$T_e$$ fit observed wavelength of flexures from gravity anomalies to theoretical calculations assuming a completely elastic lithosphere. Realistic $$T_s(min)$$ for a region is defined by the maximum earthquake focal depth that has been recorded for that region. $$T_s(min)$$ is indicative of the current stress levels in the lithosphere, whereas $$T_e$$ represents its integrated strength, which mainly comes from a competent, load- supporting layer whose strength is mostly derived from an aseismic zone (elastic core). For more details, see, for example, Burov (2011) and Watts and Burov (2003).

The seismogenic thickness and the equivalent elastic thickness are not equivalent. However, similarity in the numerical values of $$T_e(min)$$ and $$T_s(min)$$  for oceanic lithosphere has led to confusion between the two; their dissimilarity for continental lithosphere has informed the debates about general rheological model for the lithosphere.

Rheological models
Using the concept of strength profiles, as well as the observation and estimation of $$T_s(min)$$ and $$T_e(min)$$, rheological models for the lithosphere can be constructed. Early successes in constructing a general rheological model for the oceanic lithosphere has led to similar idea being applied to the continent. "Jelly- Sandwich" model, characterized by strong upper crust and mantle (the sandwich) and a weak lower crust (the jelly), used to be viewed as a general rheological model for the continental lithosphere (e.g. Chen and Molnar, 1983 ). This view has recently been challenged by the "Crème- brûlée" model, which features a single strong crust and a weak mantle (e.g. Maggi et al 2000 a&b ). More recently, it has been realized that there shouldn't be one single rheological model for the entire continental lithosphere. Depending on parameters like $$T_e(min)$$, Moho depth and age of the lithosphere, rheological models can have sharply different characteristics and thus imply different values of $$T_s(min)$$ (e.g. Burov 2011 ). In fact, even for the oceanic lithosphere, the general model does not apply to areas with vertical loads, such as around a seamount or volcanic island.

Oceanic lithosphere
The strength profiles for oceanic lithosphere is represented by Byerlee's law for the brittle part and power- law creep for the ductile part, which is highly temperature dependent. A single rheological model can work for a vast area of oceanic lithosphere, thus making the model somewhat "general", because there exists simple relations between the geotherm in the ocean and the bathymetry. In areas with no thermal disturbances, such as from a hot spot, oceanic geotherm can usually be modeled by a half- space cooling model and its cooling time can be easily obtained from depth of the ocean (e.g. Parsons and Sclater, 1977 ). Oceanic $$T_e(min)$$ can be estimated from gravity measurements, from which Moho or basement topography can be calculated and compared with local isostatic models. Any deviations can then be attributed to the plate strength. Both forward (from different values of $$T_e(min)$$ find the one that produces best- fitting Moho or basement topography. e.g. Watts, 2001 ) and inverse (spectral method which uses both $$T_e(min)$$ and topography to invert for flexural wavelength. e.g.  Forsyth, 1985 ) models exist and show that $$T_e(min)$$ varies from 2 – 40 km from near the ridge to older areas. Since mean oceanic crust depth is only about 7 km, oceanic upper mantle undoubtedly contributes most of the strength. Initially, the discovery of increasing plate strength as age increases presented a conundrum for initiation of subduction: as the plate reaches its maximum negative buoyancy, so does it reach the maximum strength that makes it very hard to bend. Plastic hinge zones, proposed by e.g. McAdoo et al, 1985, can create a local 20- 30% reduction in $$T_e(min)$$ and thus making subduction possible.

There is little doubt that many earthquakes happen in the oceanic upper mantle. In fact, compilation of $$T_e(min)$$ and $$T_s(min)$$ measurements as a function of age show that they correlate and are nearly equal to each other (e.g. Watts. 2001 ). However, when compressional and extensional events are separated, compressional events are systematically found to occur lower than extensional events. This is a strong suggestion that oceanic lithosphere only consists of one strong layer, as if there were two, the deeper earthquakes should be extensional as well.

Jelly- sandwich model
Following the widespread acceptance of a strong oceanic lithospheric mantle, and results from rock experiments suggesting behaviors of oceanic and continental materials cannot be significantly different from each other, researchers have deduced that there must also be a strong continental upper mantle. Early observations of seismicity depth show, in many areas such as Tibet, Tien Shan, Iran, and Aegean, earthquake depth have a bimodal distribution where the upper crust and mantle are seismically active and the lower crust is dormant. This discovery has led to the "Jelly- sandwich" rheological model for continental lithosphere.

One of the criticisms the jelly- sandwich model receives is the scarcity of recorded upper mantle earthquake. Some researchers have also doubted their very existence, based on arguments of better resolved earthquake and Moho depths. While most of the early examples have now been refuted, it is unequivocal that some earthquakes in Tibet do occur at depths as great as 80–90 km and thus are very likely to be sub- Moho. Examples also show upper- mantle earthquakes around the Aegean sea, but all of them occur in areas where crustal thickness is smaller than 20– 30 km. As for the general scarcity of upper- mantle earthquakes, it has been shown that typical intraplate tectonic forces is only about $$10^{13} \ N$$ per meter while at 50 km depth, brittle rock strength is about 2 GPa and one would need a force that is one or two orders of magnitude higher to initiate brittle failure, assuming a 100 km thick lithosphere. Further, at large depth, pre- existing cracks that is necessary for applying Byerlee's law might be healed due to high temperature and pressure, making brittle failure even less probable.

Jelly- sandwich model's strong upper mantle makes it favorable for supporting areas with old and thick lithosphere, called craton s. This is a major advantage of Jelly- sandwich model against Crème- brûlée model.

Crème- brûlée model
Re- examinations of earthquake depth distribution revealed not only are there not many continental upper mantle earthquakes, but also, partly due to increased amount of data, in some areas, such as east Africa, the Ganges basin, and Tien Shan, earthquakes occur throughout the crustal depth, making an aseismic lower crust absent, while in some other areas, like Tibet (with some of the 80–90 km deep earthquakes excluded) and Iran, earthquakes are only located in the upper crust. These new discoveries suggest rigidity in the upper or even the whole crust, while also pointing out a weak upper mantle. This has led to the birth of the "crème- brûlée" rheological model for continental lithosphere.

One criticism the crème- brûlée model receives is the dilemma that earthquakes in the oceanic lithosphere can occur down to temperatures of 700-750 $^oC$, while continental mantle is as cold as 300- 500 $^oC$ , so why is the former seismogenic and the latter is not? This has been largely explained by an re- investigation on both the oceanic and continental geotherms. Previous geotherm values were obtained with out consideration of the large radiogenic heat generated in thick curst, and that modeling parameters were not temperature dependent. After these corrections, it is found that now oceanic earthquakes occur only up to about 600 $^oC$ and that continental Moho temperature in shields could be larger than this, validating the claim that the mantle should be aseismic. Another doubt faced by crème- brûlée model, as mentioned above, is its insufficiency to support the overlying crust, especially for thick cratons. It has been argued that melt extraction of mantle due to downward heating from crust might stablize the mantle, and effect of this downward heating can be seen from surface wave tomography.

One advantage of crème- brûlée model over jelly- sandwich model is its ability to explain the depth pattern of focal mechanisms: normal (associated with extension) events all happen shallower than thrust (associated with compression) events. While this distinction is clear in the oceanic lithosphere, it is more ambiguous on land, partly because the scarcity of continental upper mantle earthquakes.