Inelastic deformation during sill and laccolith emplacement: Insights from an analytic elastoplastic model

Numerous geological observations evidence that inelastic deformation occurs during sills and laccoliths emplacement. However, most models of sill and laccolith emplacement neglect inelastic processes by assuming purely elastic deformation of the host rock. This assumption has never been tested, so that the role of inelastic deformation on the growth dynamics of magma intrusions remains poorly understood. In this paper, we introduce the first analytical model of shallow sill and laccolith emplacement that accounts for elasto-plastic deformation of the host rock. It considers the intrusion's overburden as a thin elastic bending plate attached to an elastic-perfectly-plastic foundation. We find that, for geologically realistic values of the model parameters, the horizontal extent of the plastic zone lp is much smaller than the radius of the intrusion a. By modeling the quasi-static growth of a sill, we find that the ratio lp/a decreases during propagation, as 1/ \sqrt a 4 $\Delta$P , with $\Delta$P the magma overpressure. The model also shows that the extent of the plastic zone decreases with the intrusion's depth, while it increases if the host rock is weaker. Comparison between our elasto-plastic model and existing purely elastic models shows that plasticity can have a significant effect on intrusion propagation dynamics, with e.g. up to a doubling of the overpressure necessary for the sill to grow. Our results suggest that plasticity effects might be small for large sills, but conversely that they might be substantial for early sill propagation.

Memory in random bouncing ball dynamics

The bouncing of an inelastic ball on a vibrating plate is a popular model used in various fields, from granular gases to nanometer-sized mechanical contacts. For random plate motion, so far, the model has been studied using Poincar{\'e} maps in which the excitation by the plate at successive bounces is assumed to be a discrete Markovian (memoryless) process. Here, we investigate numerically the behaviour of the model for continuous random excitations with tunable correlation time. We show that the system dynamics are controlled by the ratio of the Markovian mean flight time of the ball and the mean time between successive peaks in the motion of the exciting plate. When this ratio, which depends on the bandwidth of the excitation signal, exceeds a certain value, the Markovian approach is appropriate; below, memory of preceding excitations arises, leading to a significant decrease of the jump duration; at the smallest values of the ratio, chattering occurs. Overall, our results open the way for uses of the model in the low excitation regime, which is still poorly understood.Comment: Final published version, 5 pages, 4 figure

Understanding fast macroscale fracture from microcrack post mortem patterns

Dynamic crack propagation drives catastrophic solid failures. In many amorphous brittle materials, sufficiently fast crack growth involves small-scale, high-frequency microcracking damage localized near the crack tip. The ultra-fast dynamics of microcrack nucleation, growth and coalescence is inaccessible experimentally and fast crack propagation was therefore studied only as a macroscale average. Here, we overcome this limitation in polymethylmethacrylate, the archetype of brittle amorphous materials: We reconstruct the complete spatio-temporal microcracking dynamics, with micrometer / nanosecond resolution, through post mortem analysis of the fracture surfaces. We find that all individual microcracks propagate at the same low, load-independent, velocity. Collectively, the main effect of microcracks is not to slow down fracture by increasing the energy required for crack propagation, as commonly believed, but on the contrary to boost the macroscale velocity through an acceleration factor selected on geometric grounds. Our results emphasize the key role of damage-related internal variables in the selection of macroscale fracture dynamics.Comment: 9 pages, 5 figures + supporting information (15 pages

Role of friction-induced torque in stick-slip motion

We present a minimal quasistatic 1D model describing the kinematics of the transition from static friction to stick-slip motion of a linear elastic block on a rigid plane. We show how the kinematics of both the precursors to frictional sliding and the periodic stick-slip motion are controlled by the amount of friction-induced torque at the interface. Our model provides a general framework to understand and relate a series of recent experimental observations, in particular the nucleation location of micro-slip instabilities and the build up of an asymmetric field of real contact area.Comment: 6 pages, 5 figure

Adhesion in soft contacts is minimum beyond a critical shear displacement

The most direct measurement of adhesion is the pull-off force, i.e. the tensile force necessary to separate two solids in contact. For a given interface, it depends on various experimental parameters, including separation speed, contact age and maximum loading force. Here, using smooth contacts between elastomer spheres and rigid plates, we show that the pull-off force also varies if the contact is sheared prior to separation. For shear displacements below a critical valueabout 10% of that necessary to yield gross sliding, the pull-off force steadily decreases as shear increases. For larger shear, the pull-off force remains constant, at a residual value 10%--15% of its initial value. Combining force measurements and in situ imaging, we show how the unloading path leading to contact separation is modified by the initial shear. In particular, we find that the residual pull-off force prevails if the contact reaches full sliding during unloading. Based on those observations, a first modeling attempt of the critical shear displacement is proposed,involving a competition between jump instability and transition to sliding. Overall, those results offer new insights into the interplay between adhesion and friction, provide new constraints on adhesion measurements and challenge existing adhesive models. They will be useful wherever soft contacts undergo both normal and shear stresses, including tire grip, soft robotics, haptics and animal locomotion

Experimental evidence of non-Amontons behaviour at a multicontact interface

We report on normal stress field measurements at the multicontact interface between a rough elastomeric film and a smooth glass sphere under normal load, using an original MEMS-based stress sensing device. These measurements are compared to Finite Elements Method calculations with boundary conditions obeying locally Amontons' rigid-plastic-like friction law with a uniform friction coefficient. In dry contact conditions, significant deviations are observed which decrease with increasing load. In lubricated conditions, the measured profile recovers almost perfectly the predicted profile. These results are interpreted as a consequence of the finite compliance of the multicontact interface, a mechanism which is not taken into account in Amontons' law

The role of fingerprints in the coding of tactile information probed with a biomimetic sensor

In humans, the tactile perception of fine textures (spatial scale <200 micrometers) is mediated by skin vibrations generated as the finger scans the surface. To establish the relationship between texture characteristics and subcutaneous vibrations, a biomimetic tactile sensor has been designed whose dimensions match those of the fingertip. When the sensor surface is patterned with parallel ridges mimicking the fingerprints, the spectrum of vibrations elicited by randomly textured substrates is dominated by one frequency set by the ratio of the scanning speed to the interridge distance. For human touch, this frequency falls within the optimal range of sensitivity of Pacinian afferents, which mediate the coding of fine textures. Thus, fingerprints may perform spectral selection and amplification of tactile information that facilitate its processing by specific mechanoreceptors.Comment: 25 pages, 11 figures, article + supporting materia

Finite deformations govern the anisotropic shear-induced area reduction of soft elastic contacts

Solid contacts involving soft materials are important in mechanical engineering or biomechanics. Experimentally, such contacts have been shown to shrink significantly under shear, an effect which is usually explained using adhesion models. Here we show that quantitative agreement with recent high-load experiments can be obtained, with no adjustable parameter, using a non-adhesive model, provided that finite deformations are taken into account. Analysis of the model uncovers the basic mechanisms underlying shear-induced area reduction, local contact lifting being the dominant one. We confirm experimentally the relevance of all those mechanisms, by tracking the shear-induced evolution of tracers inserted close to the surface of a smooth elastomer sphere in contact with a smooth glass plate. Our results suggest that finite deformations are an alternative to adhesion, when interpreting a variety of sheared contact experiments involving soft materials.Comment: Version accepted at J. Mech. Phys. Solids. It includes Supplementary Informatio

1D model of precursors to frictional stick-slip motion allowing for robust comparison with experiments

We study the dynamic behaviour of 1D spring-block models of friction when the external loading is applied from a side, and not on all blocks like in the classical Burridge-Knopoff-like models. Such a change in the loading yields specific difficulties, both from numerical and physical viewpoints. To address some of these difficulties and clarify the precise role of a series of model parameters, we start with the minimalistic model by Maegawa et al. (Tribol. Lett. 38, 313, 2010) which was proposed to reproduce their experiments about precursors to frictional sliding in the stick-slip regime. By successively adding (i) an internal viscosity, (ii) an interfacial stiffness and (iii) an initial tangential force distribution at the interface, we manage to (i) avoid the model's unphysical stress fluctuations, (ii) avoid its unphysical dependence on the spatial resolution and (iii) improve its agreement with the experimental results, respectively. Based on the behaviour of this improved 1D model, we develop an analytical prediction for the length of precursors as a function of the applied tangential load. We also discuss the relationship between the microscopic and macroscopic friction coefficients in the model.Comment: 13 pages, 14 figures, accepted in Tribology Letter