Doctoral Thesis Defence by Daniel Kienle

Daniel Kienle, former doctoral researcher at the Institute of Applied Mechanics (CE), and member within the framework of SFB 1313 and the Integrated Research Training Group IRTG-IMPM, will defend his dissertation: 

Title: "Computational Methods for the Efficient Analysis of Multi-Field Problems with Applications to Hydraulic-Elastic-Plastic Fracturing"
Date: 29 August 2022
Time: 10:30 am CET
Venue: Aula 0.009, Pfaffenwaldring 5a

Abstract

Over the last decades the initiation and growth of fracture in materials was subject of intensive research in the field of continuum mechanics. Thereby the focus was to understand the ongoing processes and predict the initiation as well as the growth of fracture. To do so mathematical formulations in form of partial differential equations have been proposed. The research started with the investigation of fracture in linear elastic materials. Here the material behaviour before the fracture initiation is well known and does not need any special treatment when it comes to modelling. One of the reason for the difficulty in modelling of fracture stems from the non—smooth nature of the cracked areas, since the mathematical description of a continuum is based on smooth functions. In literature several techniques are applied to overcome this issue, such as extended finite element method (XFEM), discrete fracture model or phase-field approach. The know-how gained from the modelling of fracture in elastic materials was soon applied to multi-field problems such as in elastic-plastic, electo-mechanical, magneto-mechanical, chemo-mechanical or hydro-mechanical solids. In these materials a coupling between the deformation and the additional introduced physical aspect takes place. This effects the initiation and propagation of fracture. The multi-field problems demand an efficient numerical analysis due to the increase of unknowns which have to be determined.

In order to propose and design new model formulations which incorporate the desired phenomena the basic concepts of continuum mechanics and thermodynamics are used as a methodical basis. These concepts incorporate the mathematical description of motion and deformation of a body, as well as the definition of mechanical stress and thermal flux and the derivation of physical balance laws. Although this general framework is sketch in this work for pure thermo-mechanical problems it can be extended towards additional physical effects in a straightforward manner. The behaviour of a specific material type is then specified by constitutive functions which are constructed in such a way that they obey certain modelling principles. In a second step the designed mathematical description of the problem can be transferred into a variational formulation which then can be used for the derivation of a numerical treatment. Thereby the geometry of the desired domain together with the fields of the unknowns (displacement, etc.) are discretized by the means of the finite-element method. This finally yields to a range of system of linear equations which then can be solved by an appropriate solution scheme.

In the presented work the above mentioned process of model conception and numerical treatment is used for ductile fracture in porous metals, fracturing in frictional ductile materials and frictional ductile materials at hydraulic fracture. The obtained research results are given in terms of three attached scientific articles. In the first article the phase-field modelling of fracture in isotropic porous solids is formulated in a variational framework. The porous solid, such as metals, can thereby undergo large elastic-plastic deformations. The crack is described by the phase-field approach to fracture which regularizes sharp crack surfaces in a pure continuum setting. It originates both from gradient damage modelling and fracture mechanics. The plastic deformations are characterized by a model for porous plasticity which incorporates the evolution of the void fraction by means of a simple growth law. It is linked to a gradient plasticity formulation. The fracture phase field is driven by the local elastic-plastic work density on which the failure criterion is based on.

With this formulation it is possible to model classical ductile failure problems such as cup-cone failure surfaces. Therefore two material parameters are sufficient to describe the failure behaviour. These parameters are the critical work density and the shape parameter. While the first one specifies the onset of damage the second one controls the growth of the postcritical damage until the final rupture. In order to model damage zones to be inside of plastic zones or vice versa, two length scales are introduced. One controls the regularization of the plastic response and the other the regularization of the damage zone.

The second article presents a model for ductile fracture in frictional materials. It is based on the already mentioned phase-field approach to fracture. Combining a {\it non-associative} DruckerPrager-type elastic-plastic constitutive formulation with the evolution equation for the crack phasefield yields a failure criterion in terms of an elastic-plastic energy density. Due to the non—associative formulation of the Drucker-Prager-type yield function for frictional materials, the model is not formulated within the variational framework. However it still follows similar concepts. The model accounts for large elastic-plastic deformations of the material. The hardening behaviour of the frictional material such as soil is characterized by a isotropic hardening mechanism. It is capable to capture both friction and cohesion hardening. A modified enhanced element formulation is used for the numerical treatment. Hereby it guarantees a locking- and hourglass-free response.

In the last scientific article a model for hydraulically induced fracturing of elastic-plastic solids is proposed. It is formulated within a variational framework yielding a global minimization structure. Again the phase-field approach to fracture is deployed here. It is combined with an associative Drucker--Prager-type yield-criterion function. Thereby this yield-criterion function characterizes the plastic deformations of the full fluid-solid mixture. The elastic fluid storage and the fluid transport within the porous medium are governed by a Darcy--Biot-type material description. The flow within the fractures is characterized by an increase of the permeability in crack direction yielding a Poiseuille--type flow. Similarly to the mechanical strain the fluid storage decomposes into an elastic and plastic part. The elasto-plastic deformations are limited to the infinitesimal strain regime. Due to the global minimization structure a H(div)-conforming finite-element formulation is chosen. Locking phenomena originating from the plastic evolution are eliminated by using an enhanced assumed-strain formulation additionally.

Chair: Prof. Dr.-Ing. Felix Fritzen
First Supervisor: Prof. Dr.-Ing. Marc-André Keip
Secondary Supervisor: Prof. Laura de Lorenzis, ETH Zürich
Secondary Supervisor: Prof. Oliver Sander, TU Dresden