Dynamics of continental collision zones


*   Rheology of the continental lithosphere

*   The India Asia collision 

*   Indentation

*   Upper bound solution for Indentation

*   Slip line field solution for Indentation

*   Recommended Reading


Rheology of the continental lithosphere


Continents are softer than oceans…..one obtains this impression when looking at continental collision zones where at least one colliding continental plate has given up the paradigm of plate tectonic rigidity and forms mountain belts of sometimes large proportions. The following topographic map shows an image of the India-Asia collision. The scale of deformation is several thousand kilometers.


Figure 1 Topography of the India-Asia collision featuring large-amplitude vertical deformation of the Asian plate. India does not appear to experience significant deformation.


Such a large amount of deformation is not seen in oceanic plates. Three important differences play a role:

1)    Continental lithosphere has a much thicker and highly differentiated buoyant crust.

2)    The continental lithosphere has differentiated a significant amount of radiogenic element in its uppermost crust providing significant heat input into its uppermost layer. This upper crustal heat source is missing in oceanic lithosphere.

3)    The rheology of the differentiated crust is a lot weaker than that of the more primitive oceanic crust.


The most abundant mineral in the continental crust is quartz, which controls its rheology. The lower part of the crust is governed by feldspar rheology, which is stronger than quartz but not so strong as the olivine of the mantle. We have already discussed the mantle rheology for oceanic plates the same rheology applies for the mantle portion of continental plates. The crust differs, however and contains soft quartz and feldspar layers. The following figure shows the two soft layers. These diagrams are normally shown with depth as the vertical axis. The varying soft and strong layers give the impression of a Christmas tree when plotting the strength for tectonic compression (positive) and extension (negative). 


Figure 2 Classical (Christmas tree, here in resting position) strength profile of continental lithosphere.  The two lines show the differences between "wet" and "dry" rheologies. Only one strain rate contour is shown.


The competent mantle section of the continental Christmas tree is similar to the strength profile of the oceanic lithosphere. The behavior of this strong layer cannot be observed directly. However the soft crustal top can be seen. It adds buoyancy to continental plates thereby preventing them to subduct back into the mantle. This property and the requirement of continuity calls for a new style of tectonics that does not exist in oceans. The India Asia collision is a good example





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The India Asia collision


The following sketches show how the India Asia collision may have evolved with time. The position of India relative to Eurasia has been derived from rigid plate reconstruction as discussed in the chapters on kinematics and mechanics.




Figure 3 Subduction of the Tethys ocean was completed  about 40 Ma ago. Subsequent deformation must involve some vertical as well as horizontal  mass transfer.


While the Tethys ocean existed oceanic slab pull forces were still active. After continent-continent collision the convergence velocity reduced to half its earlier value (now around 5 cm/yr) but the penetration still goes on at present.


Figure 4 GPS measurements show that about 2 cm/yr convergence goes into underthrusting at the frontal Himalayan arc.


While GPS measurements in Pamir and India have shown that a significant amount of deformation goes into underthrusting at the frontal Himalayan arc the deformation at depth rests obscure. One may also ask: where are the remaining 3 cm/yr inferred from kinematic reconstructions at plate tectonic scale. The focal mechanism map gives some indicative answers.

Figure 5 showing focal mechanism around the Indian plate



The frontal Himalayan arc clearly witnesses prominent underthrusting, however in Asia a lot of deformation far away from the thrust plane is recorded. There appears to be a mixture of thrust (Tien Shan) strike slip and dip slip (Tibet). Deformation clearly does not stop in the immediate vicinity of the colliding Indian block but reaches up to the Lake Baikal as shown in the epicenter map.


Figure 6 Epicentre map shows far field deformation up to Lake Baikal.


Deformation at such a large scale with thrusting, strike slip and extension can be reproduced by an elasto-plastic rheology. We will for simplicity only discuss the horizontal (strike slip) solution.



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The Indian Plate has been penetrating the Eurasian Plate with considerable vertical deformation forming the high Himalayas and the Tibetan plateau.



Figure 7 showing a sketch map of the Himalayas and the Tibetan Plateau as well as arcuate deformation pattern which appears to be related to the collision



Vertical deformation is manifest in the main thrust systems of the Himalayas with some ingestion of the Indian plate.



Figure 8  Ingestion of the Indian Plate as shown by the reference point (yellow). The net volume displaced by the Indian Indenter is mostly accommodated by deformation within Asia.

However, the volume displaced by the Indian plate at depth can no longer be pushed down vertically to form deep continental subduction for reasons of buoyancy.  The Indian plate must push some Eurasian material away sideways in the way an icebreaker pushes the ice away to the sides. This mechanism is called indentation and the Indian Plate is called the “indenter” for reasons of its apparent rigidity. Indentation involves vertical and horizontal deformation. We will deal here with the horizontal solution only. In such a “plane strain” solution it is assumed that the deformation field is planar but the stress is allowed to vary in a 3-D fashion.


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Upper bound solution for Indentation


In the chapter on rheology we have concluded that the oceanic lithosphere can be treated in a first order approach as an elasto-plastic body. Viscous creep can be neglected for reasons of very large Maxwell times. By analogy the same holds true for the strong mantle section of the continental plates. For deformation at plate tectonic scale we also only need to consider the mantle section of the lithosphere because it is the strongest layer. In other words we are not interested in the small-scale perturbations in the weak crust. Such a crust might – like the icing on a solid cake – be disturbed locally but for the plates to give way and loose its rigidity it is still necessary to form shear zones in the strong mantle part.


The mathematics that underlies such deformation has also been discussed briefly in the chapter on rheology. In order to understand the method of characteristics, so called “slip lines”, we will first develop an indentation solution without knowledge of the solution to the hyperbolic differential equations of plane strain deformation. A simple approximate solution can be derived with an upper bound approach.


Such an approach relies on assuming faults where the elastic stress just reaches the yield stress k and becomes deformable. One then has to show that these faults describe the deformation in an approximate manner not violating mass conservation. As an (often neglected) corollary the positive work requirement must be shown to hold on the assumed faults, i.e. a left lateral shear stress must be associated with a left lateral shear displacement. The solution does not attempt to find an elasto-plastic stress equilibrium and is entirely constructed by kinematics. Obviously the solution cannot predict the true indentation pressure that is necessary to cause deformation on the faults. From a basic theorem in plasticity it can be shown that kinematic solutions overestimate the contact pressure while solutions that are derived from elasto-plastic stress equilibrium only underestimate the contact pressure. The former is called an “upper bound solution” and the latter is called a “lower bound” solution. A complete solution is found if lower and upper bound coincide. The upper bound solution can be derived using the hodograph concept introduced in the chapter on kinematics.


Assume a six plate Eurasian Plate model indented by a single rigid Indian block.


Figure 9 showing an upper bound approach for indentation.  It is assumed that the deformation is only accommodated within the strike slip faults separating the triangles.


The rate of external work done by the indenter is the contact pressure p on the surface a times the unit thickness and the indentation velocity v




The rate of internal work expended on a unit thickness of the strike slip faults in the right half of the field is



where O corresponds to the Asian plate. By symmetry the left half should expend the same rate of internal work. Plasticity requires that the strike slip faults should reach the shear yield stress k, the velocity can be derived from the hodograph and the work dissipated is



The work dissipated on the strike slip fault CO is twice as large as on the other faults because the dashed line moves into the Asian plate with velocity v resulting in an escape velocity which is twice as large as the shear displacement on the other faults. The upper bound theorem states that



Assuming that we are close to equality we obtain.



and the upper bound contact pressure for indentation is


The energy that is dissipated internally is released as heat on the “assumed” shear zones. We will now compare this kinematic solution with a more complete approach.


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Slip Line Field solution for Indentation

The slip line field solution for indentation of India and Asia is discussed in Hochstein and Regenauer-Lieb (1998) which is available as a pdf-file. The paper analyses anomalous heat release during collision. As a working hypothesis shear heating is suggested and the quantity that can be predicted from a slip line field solution is compared to the quantity of anomalous heat released by convective geothermal systems. The geothermal systems are aligned along major faults and form about 40km wide linear features (heat lines).



Figure 10 Hot springs in Asia


There are three different slip line field solutions that are in mechanical equilibrium. One is a solution that causes near field strike slip deformation (escape tectonics), the second is a solution that allows splitting of Asia and a third is a solution for thickening of Asia. The slip line field solution complete solution (upper and lower bounds coincide) of the near field strike slip deformation and splitting (cutting) mechanics is shown here:


Figure 11 Complete solution of the indentation problem. The enlarged inset shows minor slip lines (continuous shearing) and major slip lines (strike slip faults) and streamlines of the escape flow.

A problem in plasticity is that an infinite number of velocity fields is compatible with one and the same stress field. A possible velocity field is shown in the inset. This near field escape solution is similar in style to the upper bound solution we have discussed earlier. However, there now exists in addition to the strike slip faults (major slip lines or velocity discontinuities) of the upper bound solution an array of minor slip lines with homogeneous shearing.


The third solution for thickening deformation predict semicircular thickening trajectories with a decaying topography from the innermost semicircle to the outermost one.


Figure 12 Thickening trajectories of the indented plate.


We now proceed to superpose the three solutions onto a topographic map of Asia.


Figure 13 Slip line field solutions for thickening, extension (cutting) and strike slip (escape, indentation) deformation.


In Hochstein and Regenauer-Lieb (1998) the near field indentation solution (escape trajectories) was selected to model the anomalous heat release in Asia. There exist other suggestions in the literature.



The solution predicts for both internal and external work that:



The external work has already been specified above to be p a v  on a unit thickness, hence using equality of internal and external work one obtains:




We can see that the complete solution requires about 10% less work than the upper bound approach. The style of deformation is, however, rather similar. Using the simple upper bound approach we can understand the essence of plastic deformation. For those who are interested in a more detailed account of deformation modeling the textbook by Backofen is recommended as an introduction.


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Recommended Reading

*   Backofen, W.A. Deformation Processing, Addison Wesley Publishing, 326,  New York, 1979



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Last revised: Date  4 December 2000