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Use a rock strength envelope: at each depth plot the smaller differential stress needed for frictional failure and for sustained ductile material creep at a specified strain rate. Shallow rock generally strengthens with depth because effective confining pressure increases; sufficiently deep hot rock weakens because thermally activated material creep becomes efficient.

At shallow depth, Byerlee's law or a Mohr-Coulomb failure criterion gives a relation of the formwhere $C$ is cohesion, $\mu$ a coefficient of friction, ${\sigma }_n$ the compressive normal stress and $P_f$ the pore pressure. Increasing overburden generally increases effective stress, hence the frictional strength. Conversion from fault shear stress to differential stress depends on the stress regime and fault orientation; a sketch should not label a universal numerical slope. High pore pressure or weak fault gouge can lower the frictional strength substantially.

A simplified dislocation creep relation isHere $Q$ is activation energy, $V^{\ast }$ activation volume, $R$ the molar gas constant and $T$ absolute temperature; $A$ and $n$ depend on composition and deformation mechanism. At fixed strain rate, increasing temperature usually decreases the required stress strongly. The competing effects of pressure, water weakening of rocks, grain size and composition also matter. The intersection with frictional failure gives a brittle-ductile transition, not a change from solid to liquid.

For the Indian continental interior, a useful first sketch has an upper-crustal strength peak, a weaker lower continental crust, and renewed strength in lithospheric mantle beneath the Moho. Quartz-bearing rocks can undergo ductile material creep at conditions where colder, drier olivine-rich mantle retains appreciable strength. This creates a possible two-peak, or “jelly sandwich”, rock strength envelope. With further depth the mantle also weakens. A cold craton can retain strong lithospheric mantle to much greater depths than a tectonically heated continental region.

For the central Pacific, show thin mafic oceanic crust above a relatively thick layer of strong lithospheric mantle. There need not be a conspicuous weak lower-crustal layer. Much of the integrated strength can reside below the Moho, with a broad strength peak followed by ductile weakening towards the asthenosphere. Cooling and age control the thickness: young hot oceanic lithosphere would give a much shallower and weaker profile than an old central-Pacific plate.

The plotted Moho depths of about 38 km and 7 km, and all plotted stresses, are illustrative assumptions. “India” alone does not specify a unique geothermal gradient, water content or strain rate, and the Pacific includes plates of different ages. The robust comparison is the possible crust–mantle layering in a continent versus the thin oceanic crust and mantle-dominated strength of an old oceanic plate. The Moho is a compositional boundary; neither it nor the deepest small plotted strength is automatically the seismic lithosphere-asthenosphere boundary.

Support for the left curve comes from high laboratory friction and substantial in situ differential stress; support for the right curve comes from low inferred fault traction, weak fault materials or high pore pressure. No single observation directly measures the whole plotted envelope.

For the left, stronger shallow-crust model:

For the right, weak shallow-layer model:

To discriminate, measure the strength-controlling quantities through depth and across both faults and their surroundings. Obtain borehole stress, pore pressure and temperature profiles where feasible; use borehole breakouts and hydraulic tests to constrain effective stress. Recover fault and host rocks and test their friction and material creep at relevant fluid conditions. The right curve specifically predicts an end to the shallow weakening near its sharp rise: investigate changes in permeability of a porous medium, alteration and fault gouge with depth, rather than testing only a surface sample.

Combine those measurements with heat flux transects, magnetotelluric sounding for conductive fluids, accurate earthquake depths and focal mechanisms, and GNSS or InSAR observations of deformation. The latter can help constrain the depth distribution of locking and postseismic relaxation, but neither locking nor seismicity alone gives absolute strength. Earthquake stress drop is a change in stress, not the pre-event stress. Low heat flux can also reflect heat transport or transient weakening; high electrical conductivity can have causes other than free water. A joint explanation of all these data is needed to choose between the models locally.

Effective stress and frictional strength generally increase with depth through the shallow crust undergoing brittle fracture. At $1\mathrm{km}$, small overburden and often weak, fractured near-surface materials limit the shear stress and elastic energy that can accumulate. Some shallow materials also undergo stable sliding rather than unstable earthquake nucleation. Near $15\mathrm{km}$, a locked fault prone to brittle fracture can sustain a much larger differential stress before frictional instability releases stored elastic energy.

This depth is plausibly near the lower, strong part of the seismogenic layer: below it, increasing temperature eventually promotes material creep across the brittle-ductile transition. Once nucleated at depth, a dynamic finite-source rupture can propagate into shallow regions that could not independently nucleate the same event. Depth alone does not determine the nucleation point: local pore pressure, frictional stability and stress heterogeneity also matter.

Pore fluid pressure $P$ reduces the compressive load carried across grain contacts or an accessible fault. In the usual elementary approximation,At fixed total stress, increasing $P$ translates the entire Mohr circle left by $P$ without changing its radius or the differential stress. The circle therefore approaches the frictional failure envelope even though the applied shear has not increased. Equivalently, in total-stress coordinates the strength on an existing plane is $c+{\mu }_f({\sigma }_n-P)$, reduced by ${\mu }_{f}P$. For a known orientation and constant frictional parameters, the ideal critical pressure isprovided the plane is in contact and sliding failure is reached before tensile opening.

For the diagram's illustrative ${\sigma }_1=80$, ${\sigma }_3=20$, $c=10\mathrm{MPa}$ and ${\mu }_f=0.6$, the dry circle is below failure. The freely oriented tangency threshold is about $P=8.4\mathrm{MPa}$; the $15\mathrm{MPa}$ circle is already beyond that threshold and shows the loading tendency, not a stable post-failure state. If $P$ grows enough that ${\sigma }_3-P\le -T_0$, fluid pressure can instead open a tensile fracture. A porous-continuum effective stress law may use ${\sigma }_i^{\prime }={\sigma }_i-\alpha P$ with a coefficient $\alpha$ different from one. Also, changing fluid pressure can change total stress through constrained deformation; pure horizontal translation assumes those total stresses remain fixed.^[ref]

Draw a failure envelope in the $({\sigma }_n^{\prime },\tau )$ plane. A useful brittle shear-failure approximation is the Mohr-Coulomb failure criterionwhere $c$ is cohesion, $\phi$ the internal friction angle, and ${\sigma }_n^{\prime }$ the effective compressive normal stress. A circle wholly inside the envelope is below this ideal failure threshold. Increasing the differential stress expands it until it first touches the envelope: the tangent points identify the failing plane orientations. For a linear envelope, tangency impliesWith the angle convention above, the upper tangent point is at $2\theta =90^{\circ }+\phi$. Its plane therefore makes $45^{\circ }-\phi /2$ with the maximum-compression direction; there is a conjugate plane on the other side. For $\phi =30^{\circ }$, this is $30^{\circ }$, rather than the $45^{\circ }$ plane of maximum shear stress. An existing weak fault can reactivate first, with its own lower cohesion and friction, and need not have this optimum new-fracture orientation.

For tensile brittle fracture, add a tensile strength cutoff: failure occurs when ${\sigma }_3^{\prime }\le -T_0$, where $T_0>0$. The opening fracture is approximately perpendicular to the minimum-compression direction. Real rock envelopes curve, especially at small confinement; a straight shear envelope must not be extrapolated as an exact tensile law. High confinement can instead favor distributed or ductile deformation, so merely drawing a larger circle does not establish that brittle failure is the active mechanism.^[ref]

A depth-dependent estimate of the differential stress a rock column can sustain at a specified strain rate, composition, thermal structure and fluid state. In a simple competing-mechanism model, failure occurs by whichever of frictional sliding and ductile material creep requires the lower stress, giving the displayed lower envelope.

Effective confining pressure generally increases frictional strength with depth. In contrast, a power-law creep model $\dot{ϵ}=A{\sigma }^n\exp [-(Q+P{V}^{\ast })/(RT)]$ gives ${\sigma }_{\mathrm{creep}}=(\dot{ϵ}/A{)}^{1/n}\exp [(Q+P{V}^{\ast })/(nRT)]$. At fixed strain rate, the rise of temperature along a geothermal gradient generally lowers the required creep stress. Their crossing is a brittle-ductile transition, while composition changes at the Moho can produce additional peaks. A rock strength envelope is not uniquely determined by depth or by the labels “continental” and “oceanic”.