Abstract

This work shows the variability of the friction angle based on the geological history, over the bearing capacity of shallow foundations. This study was developed through the generation of random fields of friction angle using the matrix decomposition technique for a self-correlated lognormal distribution. In addition, to recreate the geological history, the anisotropic random fields were created using the rotation matrix with angles of 0, 45 and 90 degrees. Subsequently, these random fields were implemented in a finite element model of a continuous footing. A linear elastoplastic constitutive model was selected to represent stress-strain soil behavior together with the Mohr-Coulomb failure criteria. This last work was done in order to understand the influence of geological history and the variability of the friction angle in the bearing capacity of shallow foundations. Results include the weight of the variability of the friction angle in the bearing capacity and the influence of the dip over the friction angle and the bearing capacity.

Abstract

A state of the art on simulation methods in stochastic structural analysis is presented. The purpose of the
paper is to review some of the di erent methods available for analysing the effects of randomness of models
and data in structural analysis. While most of these techniques can be grouped under the general name of
Monte Carlo methods, the several published algorithms are more suitable to some objectives of analysis than
to others in each case. These objectives have been classiffed into the following cathegories: (1), The Statistical
Description of the structural scattering, a primary analysis in which the uncertain parameters are treated as
random variables; (2) The consideration of the spatial variability of the random parameters, that must then
be modelled as Random Fields (Stochastic Finite Elements); (3) The advanced Monte Carlo methods for
calculating the usually very low failure probabilities (Reliability Analysis) and, (4), a deterministic technique
that depart from the random nature of the above methods, but which can be linked with them in some cases,
known as the Response Surface Method. All of these techniques are critically examined and discussed. The
concluding remarks point out some research needs in the field from the authors' point of view.

Abstract

Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic fracturing process by considering various system parameters to be random. Our analysis is based on the Perkins-Kern-Nordgren (PKN) model for hydraulic fracturing. This baseline model enables computation of high fidelity solutions, which avoids pollution of our stochastic results by inaccuracies in the deterministic solution procedure. In order to obtain the desired degree of accuracy of the computed solution, we supplement the employed time-dependent moving-mesh finite element method with two new enhancements: (i) global conservation of volume is enforced through a Lagrange multiplier; (ii) the weakly singular behavior of the solution at the fracture tip is resolved by supplementing the solution space with a tip enrichment function. This tip enrichment function enables the computation of the tip speed directly from its associated solution coefficient. A novel incremental-iterative solution procedure based on a backward-Euler time-integrator with sub-iterations is employed to solve the PKN model. Direct Monte-Carlo sampling is performed based on random variable and random field input parameters. The presented stochastic results quantify the dependence of the fracture evolution process—in particular the fracture length and fracture opening—on variations in the elastic properties and leak-off coefficient of the formation, and the height of the fracture.

Abstract

Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in material properties are frequently modeled through random fields, which typically induces the need to solve finite element problems for a large number of realizations. In this context, we make use of reduced order modeling to solve these problems at an affordable computational cost. This paper proposes a reduced order modeling framework to predict crack propagation in brittle materials with random heterogeneities. The framework is based on a combination of the Proper Generalized Decomposition (PGD) method with Griffith’s global energy criterion. The PGD framework provides an explicit parametric solution for the physical response of the system. We illustrate that a non-intrusive sampling-based technique can be applied as a postprocessing operation on the explicit solution provided by PGD.We first validate the framework using a global energy approach on a deterministic two-dimensional linear elastic fracture mechanics benchmark. Subsequently, we apply the reduced order modeling approach to a stochastic fracture propagation problem.

Abstract