Viscoelasticity at large strains

The Maxwell body of a linear-elastic Hookean spring in series with a Newtonian dashpot is the simplest rheological model for geological deformation. It has been employed to describe many deformation processes.

However current Maxwell models display well-known errors when the associated strains and, importantly, rotations are large. Such conditions are often met in Earth sciences. Large rotations pose a mathematical challenge when elasticity is considered in the rheology of highly transformed materials as one requires an objective formulation of the stress rate (time derivative of stress).

In a new publication, Schrank et al. introduce a new large-strain model for Maxwell viscoelasticity with a logarithmic co-rotational stress rate (the ‘FT model’). An analysis of homogeneous isothermal simple shear with the FT model compared
to a classic small-strain formulation (the ‘SS model’) and a model using the classic Jaumann stress rate (the ‘MJ model’) leads to the following key conclusions:

  • At W  ≤ 0.1, all models yield essentially identical results.
  • At larger W, the models show increasing differences for γ > 0.5. The SS model overestimates shear stresses compared to the FT model while the MJ model exhibits an oscillatory response underestimating the FT model.
  • The MJ model violates the self-consistency condition resulting in stress oscillations and should be disregarded. It does not deliver truly elastic behaviour.
  • In the intermediate-W regime, the shear-stress overestimates of the SS model may constitute acceptable errors if energy consistency is not important. If energy consistency is desirable, the SS model should not be used at W ≥ 0.3.
  • In the high-W regime, stresses in the SS model become unacceptably
    large. The FT model should be used in this domain.

The FT model constitutes a physically consistent Maxwell model for large non-coaxial deformations, even at high Weissenberg numbers (W). It overcomes the conceptual limitations of the SS model, which is limited to small transformations, not objective and not self-consistent. It also solves the problem of the energetically aberrant oscillations of the MJ model.


Controls on sill and dyke-sill hybrid geometry and propagation in the crust: The role of fracture toughness


Analogue experiments using gelatine were carried out to investigate the role of the mechanical properties of rock layers and their bonded interfaces on the formation and propagation of magma-filled fractures in the crust.

Water was injected at controlled flux through the base of a clear-Perspex tank into superposed and variably bonded layers of solidified gelatine. Experimental dykes and sills were formed, as well as dyke-sill hybrid structures where the ascending dyke crosses the interface between layers but also intrudes it to form a sill.

Stress evolution in the gelatine was visualised using polarised light as the intrusions grew, and its evolving strain was measured using digital image correlation (DIC).
During the formation of dyke-sill hybrids there are notable decreases in stress and strain near the dyke as sills form, which is attributed to a pressure decrease within the intrusive network. Additional fluid is extracted from the open dykes to help grow the sills, causing the dyke protrusion in the overlying layer to be almost completely drained.

Scaling laws and the geometry of the propagating sill suggest sill growth into the interface was toughness-dominated rather than viscosity-dominated. We define KIc* as the fracture toughness of the interface between layers relative to the lower gelatine layer (KIcInt / KIcG). Our results show that KIc* influences the type of intrusion formed (dyke, sill or hybrid), and the magnitude of KIcInt impacted the growth rate of the sills. KIcInt was determined during setup of the experiment by controlling the temperature of the upper layer Tm when it was poured into place, with Tm < 24 °C resulting in an interface with relatively low fracture toughness that is favourable for sill or dyke-sill hybrid formation. The experiments help to explain the dominance of dykes and sills in the rock record, compared to intermediate hybrid structures.

Tectonophysics 698 (2017) 109–120


Dyke-sill hybrid formation, with fluorescent particles in the gelatine illuminated by a thin vertical laser sheet. The intrusion is viewed perpendicular to the dyke strike direction


Photo of dyke-sill hybrid formation. The intrusion is viewed with polarised light, approximately perpendicular to the strike direction of the dyke. Interference colours indicate the evolving distribution and intensity of stress within the gelatine host.


Dyke-sill hybrid formation. The intrusion is viewed looking down and from the side, onto the interface between the gelatine layers. The position of the interface against the tank wall is indicated by the dashed line.
A) A penny-shaped dyke has propagated through the lower gelatine layer and slightly protruded into the upper layer, with two small sills intruding the horizontal interface where it is intercepted by the dyke margins.
B) The dyke protrusion in the upper layer quickly became arrested as the sills grew.
C) The sills joined together within the interface, continued to grow and then coalesced with one margin of the dyke to create the final dyke-sill hybrid structure.

Slab breakoff: insights from 3D thermo-mechanical analogue modelling experiments


The detachment or breakoff of subducted lithosphere is investigated using scaled three-dimensional thermo-mechanical analogue experiments in which forces are measured and deformation is monitored using high-speed particle imaging velocimetry (PIV). The experiments demonstrate that the convergence rate in a subduction zone determine if and when slab detachment occurs. Slow subduction experiments (with scaled convergence rates ∼1 cm yr −1) have lower Peclet numbers and are characterised by lower tensile strength subducted lithosphere, causing detachment to occur when the downward pull force exerted by a relatively short subducted slab is relatively low. When continental collision is preceded by slow oceanic subduction, the subducted lithosphere therefore need not be very long or extremely negatively buoyant to cause detachment because the subducted oceanic lithosphere is hot and weak. Under such conditions detachment may occur sooner after the onset of continental subduction than previously predicted. In contrast, if a collision is preceded by rapid subduction (∼10 cm yr −1), breakoff will be delayed and occur only when the convergence rate slowed sufficiently to thermally weaken the slab and cause its eventual failure. The analogue experiments further confirm that slab detachment occurs diachronously as it propagates along the plate boundary. Stereoscopic PIV reveals a characteristic strain pattern that accompanies the detachment. Horizontal contraction and subsidence (with scaled values up to 1200 m) in the trench and forearc area preceeds the passage of the detachment, which is followed by horizontal extension and uplift (up to 900 m). High-frequency monitoring captures rapid propagation of the detachment along the plate boundary at rates of up to 100 cm yr −1. However rate is not constant and interaction between the slab and lower mantle or opening of a backarc basin in the upper plate can reduce or stop slab breakoff propagation altogether.



Successive side views of the models in Experiment 1 and 2. Experiment 1 (a-e), the subducting lithosphere is pushed by the piston at the constant velocity of 2.5 × 10 −4 m s −1 (equivalent to ∼10 cm yr −1 in nature). The slab becomes vertical due to the negative buoyancy but does not break. It folds when hitting the rigid plate that models the impenetrable lower mantle. Experiment 2 (f–j), The model is identical to Experiment 1 but it is the upper plate that is pushed instead of the lower plate. The model evolution is similar to Experiment 1 until the slab touches the lower mantle. The slab angle reduces in the late stages (dashed line in panel j).



Successive side views of the models in Experiment 3. The model is identical to that employed in Experiment 1 (Fig. 4), but the imposed rate is one order of magnitude lower, 2.5 × 10 −5 m s −1 (equivalent to ∼1 cm yr −1 in nature). Very slow subduction leads to multiple slab detachments at 2283, 4266 and 6420 s. We note that the repeated detachment caused extension in the trailing edge of the upper plate, and a slab graveyard sitting on top of the rigid upper mantle.



Sketch of propagating slab detachment with distribution of surface deformation and uplift. Horizontal contraction and surface subsidence is generated ahead of the breakoff tip, while horizontal extension and uplift follow.



Maps of earthquakes hypocenters along the Aleutian subduction zone (a), and Java-Sumatra-Andaman subduction zone (b), with profiles showing the Wadati-Benioff zone. Earthquakes hypocenters are represented by circles with diameter proportional to magnitude, and color indicating depth (see profiles for color scale). Hypocenters are from EHB catalogue (Engdahl et al., 1998). White arrows represent the convergence vectors calculated using the MORVEL global kinematic model (DeMets et al., 2010). V is the convergence rate (in mm yr −1), θ is the obliquity (angle between the normal to the trench and convergence vector), and Vn is the convergence in the direction of the profile (i.e. convergence corrected from obliquity, in mm yr −1). Topography/bathymetry from Smith and Sandwell (1997).




Slow oblique subduction along the northern branch of the Caribbean subduction zone. Map shows the topography of the trench characterized by a deep through between 65 and 67°W. Convergence from MORVEL global kinematic model (DeMets et al., 2010), is only 19 mm/yr at 64°W (white arrow) and the obliquity is approximately 67°, which yields about 7 mm/yr of normal convergence in the subduction west of 64°W. 3 North-South profiles are plotted showing that this area is also characterized by a deep negative free air anomaly (Sandwell et al., 2014), the peak of which is located the forearc. Based on our experimental results we propose that both the topography and gravity anomalies are caused by an excess downward pull in the subducted lithosphere due to ongoing slab detachment. Topography/bathymetry from Smith and Sandwell (1997), gravity from Sandwell et al. (2014).


Modelling of sill inception, propagation and growth

We are pleased to inform about a publication from our research group. The paper is published in Earth And Planetary Science Letters and available online at

A model of magma propagation in the crust is presented using a series of analogue experiments, where dyed water is injected at a constant flux into layers of solidified gelatine. Digital image correlation (DIC) is used to calculate incremental strain and finite strain in the deforming host material as it is intruded. This is mapped in 2D for the developing experimental volcanic plumbing system that comprises a feeder dyke and sill. Since the gelatine deforms elastically, strain measurements correlate with stress. Our results indicate that, for constant magma flux, the moment of sill inception is characterised by a significant magmatic pressure decrease of up to ∼60%. This is evidenced by the rapid contraction of the feeder dyke at the moment the sill forms. Sill propagation is then controlled by the fracture properties of the weak interface, with fluid from the feeder dyke extracted to help grow the sill. Pressure drops during sill inception and growth are likely to be important in volcanic systems, where destabilisation of the magmatic plumbing system could trigger an eruption.


Model of dyke and sill formation

Model of dyke and sill formation



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