flow law
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2021 ◽  
Vol 11 (24) ◽  
pp. 11829
Author(s):  
Ana Alencar ◽  
Ruben Galindo ◽  
Claudio Olalla Marañón

The influence of the non-associative flow law on the bearing capacity of shallow foundations on rock masses is, in general, a subject that is not discussed in the field of rock mechanics. The calculation methods of bearing capacity usually do not define which flow law is adopted and, in some methods, the associative flow rule is assumed without knowing how that hypothesis influences the bearing capacity of the rock mass. In this paper, the study of the influence of the dilatancy angle on the bearing capacity of shallow foundations on rock masses is presented. The variation of the bearing capacity with the associative flow law and the non-associative flow law with zero dilatancy angle is studied using the finite difference method and by considering the influence of the self-weight of rock material. The calculations confirm the great influence of the flow law on the bearing capacity and a correction coefficient is proposed, which makes it possible to estimate the variation of the bearing capacity of the rock mass in terms of the function of the flow law for the hypothesis of weightless rock masses.


2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Sagar Masuti ◽  
Sylvain Barbot

AbstractThe rheology of the upper mantle impacts a variety of geodynamic processes, including postseismic deformation following great earthquakes and post-glacial rebound. The deformation of upper mantle rocks is controlled by the rheology of olivine, the most abundant upper mantle mineral. The mechanical properties of olivine at steady state are well constrained. However, the physical mechanism underlying transient creep, an evolutionary, hardening phase converging to steady state asymptotically, is still poorly understood. Here, we constrain a constitutive framework that captures transient creep and steady state creep consistently using the mechanical data from laboratory experiments on natural dunites containing at least 94% olivine under both hydrous and anhydrous conditions. The constitutive framework represents a Burgers assembly with a thermally activated nonlinear stress-versus-strain-rate relationship for the dashpots. Work hardening is obtained by the evolution of a state variable that represents internal stress. We determine the flow law parameters for dunites using a Markov chain Monte Carlo method. We find the activation energy $$430\pm 20$$ 430 ± 20   and $$250\pm 10$$ 250 ± 10  kJ/mol for dry and wet conditions, respectively, and the stress exponent $$2.0\pm 0.1$$ 2.0 ± 0.1 for both the dry and wet cases for transient creep, consistently lower than those of steady-state creep, suggesting a separate physical mechanism. For wet dunites in the grain-boundary sliding regime, the grain-size dependence is similar for transient creep and steady-state creep. The lower activation energy of transient creep could be due to a higher jog density of the corresponding soft-slip system. More experimental data are required to estimate the activation volume and water content exponent of transient creep. The constitutive relation used and its associated flow law parameters provide useful constraints for geodynamics applications. Graphical Abstract


2021 ◽  
Author(s):  
◽  
Susanne Grigull

<p>A suite of brittle-ductile faults in the central Southern Alps, New Zealand is used as a natural laboratory into the rheology of quartz rocks. The fault array is ~2 km wide and formed in the hanging-wall of the SE-dipping Alpine Fault during the late Cenozoic at >= 25 km depth. It was exhumed in the past few Myr and is now exposed 5-7 km east of the Alpine Fault. The faults are near-vertical, extend laterally and vertically over tens of metres, and strike sub-parallel to the Alpine Fault. They displace quartzofeldspathic Alpine Schist (metagreywacke) in a predominantly brittle way. The faults impinge upon and displace abundant centimetre-thick quartz veins that are discordant to the dominant schist foliation. These quartz veins exhibit a full range of slip from fully brittle to fully ductile. In most quartz veins, a ductile component of slip and a 1-3 cm (n=72) wide ductile shear zone are present. The mean total slip measured in the veins is (7.2 +/- 5.8 ) cm (n=72).  This study first develops a method to determine the true shape and displacement of a geological marker from any outcrop orientation. It then uses a set of geometrical scaling relationships exhibited by the ductilely-to-brittlely sheared quartz veins, and the observed interaction between brittle faults and ductilely deforming quartz veins to develop a series of finite-element models that reproduce the field observations. A flow law of the form de/dt = A*(f_H2O)^m*(sig_d)^n*exp(-Q/(R*T)) is used to model the behaviour of the quartz veins. Flow law parameters for the quartz veins and viscous and frictional strength ratios between quartz and schist are determined from these models. For Q = 135 kJ mol^-1, f_H2O= 200 MPa and m = 1.0, the results show that the scaling relationships in the quartz veins are successfully reproduced for A = 10^(-10 +/- 2) MPa^-n s^-1, and n = 4.  The ratio between ductile-to-total slip (D) were measured for 72 veins throughout the brittle-ductile shear array and are highly variable. In order to understand what has led to this variability, we investigate the following parameters: original vein thickness, deformation temperature, water content, microfracturing, calcite fraction, and total slip. D-ratios appear to scale with original vein thickness, however, significant scattering of the D-values indicates that other factors also control D. The temperature resolution (from Titanium-in-Quartz geothermometry and oxygen isotopy) is not high enough to determine whether temperature influenced the D-values. Fourier Transform Infrared Spectroscopy (FTIR), and optical microscopy reveal that water content, microfracturing, and calcite fraction were very similar from one vein to another and therefore did not control the D-ratios either. Detailed outcrop maps of the brittle-ductile shears and displacement-length profiles along five individual faults indicate that the total slip varied rapidly and on short distances (cm- to m-scale) along the faults. We infer that these varying slip rates led to different flow strain rates in the deforming quartz veins and therefore can explain the variations in D-values.  Optical microscopy reveals abundant fluid inclusions in both the deformed and undeformed parts of the veins. These inclusions indicate that the quartz was ‘wet’ and the veins were weakened with respect to the surrounding schist. We therefore infer that the location of the shear zones was predetermined by the position of the brittle faults propagating through the stronger schist and impinging on the weaker quartz veins.</p>


2021 ◽  
Author(s):  
◽  
Susanne Grigull

<p>A suite of brittle-ductile faults in the central Southern Alps, New Zealand is used as a natural laboratory into the rheology of quartz rocks. The fault array is ~2 km wide and formed in the hanging-wall of the SE-dipping Alpine Fault during the late Cenozoic at >= 25 km depth. It was exhumed in the past few Myr and is now exposed 5-7 km east of the Alpine Fault. The faults are near-vertical, extend laterally and vertically over tens of metres, and strike sub-parallel to the Alpine Fault. They displace quartzofeldspathic Alpine Schist (metagreywacke) in a predominantly brittle way. The faults impinge upon and displace abundant centimetre-thick quartz veins that are discordant to the dominant schist foliation. These quartz veins exhibit a full range of slip from fully brittle to fully ductile. In most quartz veins, a ductile component of slip and a 1-3 cm (n=72) wide ductile shear zone are present. The mean total slip measured in the veins is (7.2 +/- 5.8 ) cm (n=72).  This study first develops a method to determine the true shape and displacement of a geological marker from any outcrop orientation. It then uses a set of geometrical scaling relationships exhibited by the ductilely-to-brittlely sheared quartz veins, and the observed interaction between brittle faults and ductilely deforming quartz veins to develop a series of finite-element models that reproduce the field observations. A flow law of the form de/dt = A*(f_H2O)^m*(sig_d)^n*exp(-Q/(R*T)) is used to model the behaviour of the quartz veins. Flow law parameters for the quartz veins and viscous and frictional strength ratios between quartz and schist are determined from these models. For Q = 135 kJ mol^-1, f_H2O= 200 MPa and m = 1.0, the results show that the scaling relationships in the quartz veins are successfully reproduced for A = 10^(-10 +/- 2) MPa^-n s^-1, and n = 4.  The ratio between ductile-to-total slip (D) were measured for 72 veins throughout the brittle-ductile shear array and are highly variable. In order to understand what has led to this variability, we investigate the following parameters: original vein thickness, deformation temperature, water content, microfracturing, calcite fraction, and total slip. D-ratios appear to scale with original vein thickness, however, significant scattering of the D-values indicates that other factors also control D. The temperature resolution (from Titanium-in-Quartz geothermometry and oxygen isotopy) is not high enough to determine whether temperature influenced the D-values. Fourier Transform Infrared Spectroscopy (FTIR), and optical microscopy reveal that water content, microfracturing, and calcite fraction were very similar from one vein to another and therefore did not control the D-ratios either. Detailed outcrop maps of the brittle-ductile shears and displacement-length profiles along five individual faults indicate that the total slip varied rapidly and on short distances (cm- to m-scale) along the faults. We infer that these varying slip rates led to different flow strain rates in the deforming quartz veins and therefore can explain the variations in D-values.  Optical microscopy reveals abundant fluid inclusions in both the deformed and undeformed parts of the veins. These inclusions indicate that the quartz was ‘wet’ and the veins were weakened with respect to the surrounding schist. We therefore infer that the location of the shear zones was predetermined by the position of the brittle faults propagating through the stronger schist and impinging on the weaker quartz veins.</p>


2021 ◽  
Author(s):  
Casper Pranger ◽  
Patrick Sanan ◽  
David May ◽  
Laetitia Le Pourhiet ◽  
Alice-Agnes Gabriel

2021 ◽  
Vol 15 (9) ◽  
pp. 4589-4605
Author(s):  
Mark D. Behn ◽  
David L. Goldsby ◽  
Greg Hirth

Abstract. Viscous flow in ice is often described by the Glen flow law – a non-Newtonian, power-law relationship between stress and strain rate with a stress exponent n ∼ 3. The Glen law is attributed to grain-size-insensitive dislocation creep; however, laboratory and field studies demonstrate that deformation in ice can be strongly dependent on grain size. This has led to the hypothesis that at sufficiently low stresses, ice flow is controlled by grain boundary sliding, which explicitly incorporates the grain size dependence of ice rheology. Experimental studies find that neither dislocation creep (n ∼ 4) nor grain boundary sliding (n ∼ 1.8) have stress exponents that match the value of n ∼ 3 in the Glen law. Thus, although the Glen law provides an approximate description of ice flow in glaciers and ice sheets, its functional form is not explained by a single deformation mechanism. Here we seek to understand the origin of the n ∼ 3 dependence of the Glen law by using the “wattmeter” to model grain size evolution in ice. The wattmeter posits that grain size is controlled by a balance between the mechanical work required for grain growth and dynamic grain size reduction. Using the wattmeter, we calculate grain size evolution in two end-member cases: (1) a 1-D shear zone and (2) as a function of depth within an ice sheet. Calculated grain sizes match both laboratory data and ice core observations for the interior of ice sheets. Finally, we show that variations in grain size with deformation conditions result in an effective stress exponent intermediate between grain boundary sliding and dislocation creep, which is consistent with a value of n = 3 ± 0.5 over the range of strain rates found in most natural systems.


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