Finite deformation model of simple shear of fault with microrotations: apparent strain localisation and en-echelon fracture pattern

Strain localisation is a widespread phenomenon often observed in shear and compressive loading of geomaterials, for example, the fault gouge. It is believed that the main mechanisms of strain localisation are strain softening and mismatch between dilatancy and pressure sensitivity. Observations show that gouge deformation is accompanied by considerable rotations of grains. In our previous work as a model for gouge material, we proposed a continuum description for an assembly of particles of equal radius in which the particle rotation is treated as an independent degree of freedom. We showed that there exist critical values of the model parameters for which the displacement gradient exhibits a pronounced localisation at the mid-surface layers of the fault, even in the absence of inelasticity. Here, we generalise the model to the case of finite deformations characteristic for the gouge deformation. We derive objective constitutive relationships relating the Jaumann rates of stress and moment stress to the relative strain and curvature rates, respectively. The model suggests that the pattern of localisation remains the same as in the linear case. However, the presence of the Jaumann terms leads to the emergence of non-zero normal stresses acting along and perpendicular to the shear layer (with zero hydrostatic pressure), and localised along the mid-line of the gouge; these stress components are absent in the linear model of simple shear. These additional normal stresses, albeit small, cause a change in the direction in which the maximal normal stresses act and in which en-echelon fracturing is formed.

Caputo Derivatives in Viscoelasticity: A Non-Linear Finite-Deformation Theory for Tissue

Mathematics Subject Classification: 26A33, 74B20, 74D10, 74L15

InSAR deformation measurements of the earthquake cycle in transcurrent tectonic domains, analytical and analog modeling

I applied the SBAS-DInSAR method to the Mattinata Fault (MF) (Southern Italy) and to the Doruneh Fault System (DFS) (Central Iran). In the first case, I observed limited internal deformation and determined the right lateral kinematic pattern with a compressional pattern in the northern sector of the fault. Using the Okada model I inverted the observed velocities defining a right lateral strike slip solution for the MF. Even if it fits the data within the uncertainties, the modeled slip rate of 13-15 mm yr-1 seems too high with respect to the geological record. Concerning the Western termination of DFS, SAR data confirms the main left lateral transcurrent kinematics of this fault segment, but reveal a compressional component. My analytical model fits successfully the observed data and quantifies the slip in ~4 mm yr-1 and ~2.5 mm yr-1 of pure horizontal and vertical displacement respectively. The horizontal velocity is compatible with geological record. I applied classic SAR interferometry to the October–December 2008 Balochistan (Central Pakistan) seismic swarm; I discerned the different contributions of the three Mw > 5.7 earthquakes determining fault positions, lengths, widths, depths and slip distributions, constraining the other source parameters using different Global CMT solutions. A well constrained solution has been obtained for the 09/12/2008 aftershock, whereas I tested two possible fault solutions for the 28-29/10/08 mainshocks. It is not possible to favor one of the solutions without independent constraints derived from geological data. Finally I approached the study of the earthquake-cycle in transcurrent tectonic domains using analog modeling, with alimentary gelatins like crust analog material. I successfully joined the study of finite deformation with the earthquake cycle study and sudden dislocation. A lot of seismic cycles were reproduced in which a characteristic earthquake is recognizable in terms of displacement, coseismic velocity and recurrence time.

Elastoplastic consolidation at finite strain. Part 2: finite element implementation and numerical examples

A mathematical model for finite strain elastoplastic consolidation of fully saturated soil media is implemented into a finite element program. The algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the Kirchhoff pore water pressure. A two-field mixed finite element formulation is employed in which the nodal solid displacements and the nodal pore water pressures are coupled via the linear momentum and mass balance equations. The constitutive model for the solid phase is represented by modified Cam—Clay theory formulated in the Kirchhoff principal stress space, and return mapping is carried out in the strain space defined by the invariants of the elastic logarithmic principal stretches. The constitutive model for fluid flow is represented by a generalized Darcy's law formulated with respect to the current configuration. The finite element model is fully amenable to exact linearization. Numerical examples with and without finite deformation effects are presented to demonstrate the impact of geometric nonlinearity on the predicted responses. The paper concludes with an assessment of the performance of the finite element consolidation model with respect to accuracy and numerical stability.

A unified friction description and its application to the simulation of frictional instability using the finite element method

This article first summarizes some available experimental results on the frictional behaviour of contact interfaces, and briefly recalls typical frictional experiments and relationships, which are applicable for rock mechanics, and then a unified description is obtained to describe the entire frictional behaviour. It is formulated based on the experimental results and applied with a stick and slip decomposition algorithm to describe the stick-slip instability phenomena, which can describe the effects observed in rock experiments without using the so-called state variable, thus avoiding related numerical difficulties. This has been implemented to our finite element code, which uses the node-to-point contact element strategy proposed by the authors to handle the frictional contact between multiple finite-deformation bodies with stick and finite frictional slip, and applied here to simulate the frictional behaviour of rocks to show its usefulness and efficiency.

Finite element analysis of particle impact problems

Particle impacts are of fundamental importance in many areas and there has been a renewed interest in research on particle impact problems. A comprehensive investigation of the particle impact problems, using finite element (FE) methods, is presented in this thesis. The capability of FE procedures for modelling particle impacts is demonstrated by excellent agreements between FE analysis results and previous theoretical, experimental and numerical results. For normal impacts of elastic particles, it is found that the energy loss due to stress wave propagation is negligible if it can reflect more than three times during the impact, for which Hertz theory provides a good prediction of impact behaviour provided that the contact deformation is sufficiently small. For normal impact of plastic particles, the energy loss due to stress wave propagation is also generally negligible so that the energy loss is mainly due to plastic deformation. Finite-deformation plastic impact is addressed in this thesis so that plastic impacts can be categorised into elastic-plastic impact and finite-deformation plastic impact. Criteria for the onset of finite-deformation plastic impacts are proposed in terms of impact velocity and material properties. It is found that the coefficient of restitution depends mainly upon the ratio of impact velocity to yield Vni/Vy0 for elastic-plastic impacts, but it is proportional to [(Vni/Vy0)*(Y/E*)]-1/2, where Y /E* is the representative yield strain for finite-deformation plastic impacts. A theoretical model for elastic-plastic impacts is also developed and compares favourably with FEA and previous experimental results. The effect of work hardening is also investigated.

Shell curvature as an initial deformation: A geometrically exact finite element approach

An alternative approach for the analysis of arbitrarily curved shells is developed in this paper based on the idea of initial deformations. By `alternative` we mean that neither differential geometry nor the concept of degeneration is invoked here to describe the shell surface. We begin with a flat reference configuration for the shell mid-surface, after which the initial (curved) geometry is mapped as a stress-free deformation from the plane position. The actual motion of the shell takes place only after this initial mapping. In contrast to classical works in the literature, this strategy enables the use of only orthogonal frames within the theory and therefore objects such as Christoffel symbols, the second fundamental form or three-dimensional degenerated solids do not enter the formulation. Furthermore, the issue of physical components of tensors does not appear. Another important aspect (but not exclusive of our scheme) is the possibility to describe exactly the initial geometry. The model is kinematically exact, encompasses finite strains in a totally consistent manner and is here discretized under the light of the finite element method (although implementation via mesh-free techniques is also possible). Assessment is made by means of several numerical simulations. Copyright (C) 2009 John Wiley & Sons, Ltd.

Finite element models of coseismic deformation due to the 2009 L'Aquila (Italy) and 2008 Wenchuan(China) earthquakes

The topic of my Ph.D. thesis is the finite element modeling of coseismic deformation imaged by DInSAR and GPS data. I developed a method to calculate synthetic Green functions with finite element models (FEMs) and then use linear inversion methods to determine the slip distribution on the fault plane. The method is applied to the 2009 L’Aquila Earthquake (Italy) and to the 2008 Wenchuan earthquake (China). I focus on the influence of rheological features of the earth's crust by implementing seismic tomographic data and the influence of topography by implementing Digital Elevation Models (DEM) layers on the FEMs. Results for the L’Aquila earthquake highlight the non-negligible influence of the medium structure: homogeneous and heterogeneous models show discrepancies up to 20% in the fault slip distribution values. Furthermore, in the heterogeneous models a new area of slip appears above the hypocenter. Regarding the 2008 Wenchuan earthquake, the very steep topographic relief of Longmen Shan Range is implemented in my FE model. A large number of DEM layers corresponding to East China is used to achieve the complete coverage of the FE model. My objective was to explore the influence of the topography on the retrieved coseismic slip distribution. The inversion results reveals significant differences between the flat and topographic model. Thus, the flat models frequently adopted are inappropriate to represent the earth surface topographic features and especially in the case of the 2008 Wenchuan earthquake.

Finite element modeling of flow/compression-induced deformation of alginate scaffolds for bone tissue engineering

Trauma or degenerative diseases such as osteonecrosis may determine bone loss whose recover is promised by a "tissue engineering“ approach. This strategy involves the use of stem cells, grown onboard of adequate biocompatible/bioreabsorbable hosting templates (usually defined as scaffolds) and cultured in specific dynamic environments afforded by differentiation-inducing actuators (usually defined as bioreactors) to produce implantable tissue constructs. The purpose of this thesis is to evaluate, by finite element modeling of flow/compression-induced deformation, alginate scaffolds intended for bone tissue engineering. This work was conducted at the Biomechanics Laboratory of the Institute of Biomedical and Neural Engineering of the Reykjavik University of Iceland. In this respect, Comsol Multiphysics 5.1 simulations were carried out to approximate the loads over alginate 3D matrices under perfusion, compression and perfusion+compression, when varyingalginate pore size and flow/compression regimen. The results of the simulations show that the shear forces in the matrix of the scaffold increase coherently with the increase in flow and load, and decrease with the increase of the pore size. Flow and load rates suggested for proper osteogenic cell differentiation are reported.

Finite element analysis predicts experimental failure patterns in vertebral bodies loaded via intervertebral discs up to large deformation

Vertebral compression fracture is a common medical problem in osteoporotic individuals. The quantitative computed tomography (QCT)-based finite element (FE) method may be used to predict vertebral strength in vivo, but needs to be validated with experimental tests. The aim of this study was to validate a nonlinear anatomy specific QCT-based FE model by using a novel testing setup. Thirty-seven human thoracolumbar vertebral bone slices were prepared by removing cortical endplates and posterior elements. The slices were scanned with QCT and the volumetric bone mineral density (vBMD) was computed with the standard clinical approach. A novel experimental setup was designed to induce a realistic failure in the vertebral slices in vitro. Rotation of the loading plate was allowed by means of a ball joint. To minimize device compliance, the specimen deformation was measured directly on the loading plate with three sensors. A nonlinear FE model was generated from the calibrated QCT images and computed vertebral stiffness and strength were compared to those measured during the experiments. In agreement with clinical observations, most of the vertebrae underwent an anterior wedge-shape fracture. As expected, the FE method predicted both stiffness and strength better than vBMD (R2 improved from 0.27 to 0.49 and from 0.34 to 0.79, respectively). Despite the lack of fitting parameters, the linear regression of the FE prediction for strength was close to the 1:1 relation (slope and intercept close to one (0.86 kN) and to zero (0.72 kN), respectively). In conclusion, a nonlinear FE model was successfully validated through a novel experimental technique for generating wedge-shape fractures in human thoracolumbar vertebrae.

Simplified Finite Element Assessment of Beams With and Without Shear Deformation

This paper presents a simplified finite element (FE) methodology for solving accurately beam models with (Timoshenko) and without (Bernoulli-Euler) shear deformation. Special emphasis is made on showing how it is possible to obtain the exact solution on the nodes and a good accuracy inside the element. The proposed simplifying concept, denominated as the equivalent distributed load (EDL) of any order, is based on the use of Legendre orthogonal polynomials to approximate the original or acting load for computing the results between the nodes. The 1-span beam examples show that this is a promising procedure that allows the aim of using either one FE and an EDL of slightly higher order or by using an slightly larger number of FEs leaving the EDL in the lowest possible order assumed by definition to be equal to 4 independently of how irregular the beam is loaded.

Finite-difference schemes compared for wave-deformation characteristics in mathematical modeling of two-dimensional long-wave propagation /

"October 1970."