Continuum mechanics usually considers the theoretical and computational description of standard single-phasic materials in the framework of either solid mechanics, fluid mechanics or gas dynamics. ...However, growing complexity in material modelling combined with the request of users leads to a growing interest in porous-media mechanics, where porous solid materials with fluid or gaseous pore content are investigated on a macroscopic scale. In this regard, the present article reviews the theoretical and numerical framework for the description of geomechanical and biomechanical problems including elastic, elasto-plastic and visco-elastic solid behaviour partly combined with electro-active properties. For this purpose, the Theory of Porous Media is applied for an elegant consideration of the coupling phenomena of porous solids with pore fluids, no matter if the fluids have to be treated as inert fluids or as fluid mixtures. In the sense of a review article, different computational examples are presented to illuminate the possibilities and challenges of porous-media mechanics.
Human brain tissue is complex and multi-component in nature. It consists of an anisotropic hyperelastic solid material composed of tissue cells and blood vessel walls. Brain tissue is permeated by ...two viscous pore liquids, the interstitial fluid and the blood. Both liquids are mobile within the tissue and exhibit a significant anisotropic perfusion behaviour. To model this complex aggregate, the well-founded Theory of Porous Media, a continuum-mechanical approach for the description of multi-component aggregates, is used. To include microscopic information, the model is enhanced by tissue characteristics obtained from medical imaging techniques. Moreover, the model is applied to invasive drug-delivery strategies, i.e. the direct extra-vascular infusion of therapeutic agents. For this purpose, the overall interstitial fluid is treated as a real two-component mixture of a liquid solvent and a dissolved therapeutic solute. Finally, the continuum-mechanical model results in a set of strongly coupled partial differential equations which are spatially discretised using mixed finite elements and solved in a monolithic manner with an implicit Euler time-integration scheme. Numerical examples demonstrate the applicability of the presented model.
The well-known phase-field approach applied to fracturing solids has recently been embedded in the Theory of Porous Media for the description of dynamic hydraulic fracturing scenarios based on fully ...saturated porous media. This method has further been enhanced by the introduction of a crack-opening indicator to distinguish between open and closed cracks accompanied by a switch between Darcy-type and Navier–Stokes-type flow situations in the unbroken porous domain and in fully broken areas. Based on these achievements, the present article extends the challenging matter of fully saturated media by the introduction of partially saturated scenarios, where the pore space contains both a liquid, such as water or oil, and a pore gas, such as air or natural gas. Proceeding from the Theory of Porous Media, the setup of the model is based on first principles of continuum mechanics, while the numerical study proceeds from the Finite-Element Method, where coupled problems of fracturing multi-component and multi-phasic media are treated by a monolithic solution strategy provided by a solver for coupled problems. By use of this procedure, it can be shown by comparison of fully and partially saturated porous media that the existence of pore gas slows down the fracture evolution. It is furthermore pointed out that the existence of closed precracks combined with external loads does not only lead to opening and evolving fractures, but also to fractures that do not open as a result of compressive boundary conditions.
The medical relevance of brain tumours is characterised by its locally invasive and destructive growth. With a high mortality rate combined with a short remaining life expectancy, brain tumours are ...identified as highly malignant. A continuum‐mechanical model for the description of the governing processes of growth and regression is derived in the framework of the Theory of Porous Media (TPM). The model is based on medical multi‐modal magnetic resonance imaging (MRI) scans, which represent the gold standard in diagnosis. The multi‐phase model is described mathematically via strongly coupled partial differential equations. This set of governing equations is transformed into their weak formulation and is solved with the software package FEniCS. A proof‐of‐concept simulation based on one patient geometry and tumour pathology shows the relevant processes of tumour growth and the results are discussed.
The outcome of vertebroplasty is hard to predict due to its dependence on complex factors like bone cement and marrow rheologies. Cement leakage could occur if the procedure is done incorrectly, ...potentially causing adverse complications. A reliable simulation could predict the patient-specific outcome preoperatively and avoid the risk of cement leakage. Therefore, the aim of this work was to introduce a computationally feasible and experimentally validated model for simulating vertebroplasty. The developed model is a multiphase continuum-mechanical macro-scale model based on the Theory of Porous Media. The related governing equations were discretized using a combined finite element–finite volume approach by the so-called Box discretization. Three different rheological upscaling methods were used to compare and determine the most suitable approach for this application. For validation, a benchmark experiment was set up and simulated using the model. The influence of bone marrow and parameters like permeability, porosity, etc., was investigated to study the effect of varying conditions on vertebroplasty. The presented model could realistically simulate the injection of bone cement in porous materials when used with the correct rheological upscaling models, of which the semi-analytical averaging of the viscosity gave the best results. The marrow viscosity is identified as the crucial reference to categorize bone cements as ‘high- ’or ‘low-’ viscosity in the context of vertebroplasty. It is confirmed that a cement with higher viscosity than the marrow ensures stable development of the injection and a proper cement interdigitation inside the vertebra.
Hydraulically induced fracturing is widely used in practice for several exploitation techniques. The chosen macroscopic model combines a phase‐field approach to fractures with the Theory of Porous ...Media (TPM) to describe dynamic hydraulic fracturing processes in fully‐saturated porous materials. In this regard, the solid's state of damage shows a diffuse transition zone between the broken and unbroken domain. Rocks or soils in grown nature are generally inhomogeneous with material imperfections on the microscale, such that modelling homogeneous porous material may oversimplify the behaviour of the solid and fluid phases in the fracturing process. Therefore, material imperfections and inhomogeneities in the porous structure are considered through the definition of location‐dependent material parameters. In this contribution, a deterministic approach to account for predefined imperfection areas as well as statistical fields of geomechanical properties is proposed. Representative numerical simulations show the impact of solid skeleton heterogeneities in porous media on the fracturing characteristics, e. g. the crack path.
This contribution aims to study the influence of the saturation degree on the fracturing process. Therefore, a continuum‐mechanical model for fluid‐driven fracturing in partially saturated porous ...material is presented on the basis of the Theory of Porous Media and the phase‐field approach to fracture. The material is described on the macroscopic scale as an immiscible mixture of three phases, i.e. a solid phase, representing the solid skeleton, and two fluid phases percolating the pore space. Hereby, capillarity effects between the fluid phases are taken into account. The fracturing process is modelled with a phase‐field approach, characterising a diffuse crack pattern. The crack propagation in the solid skeleton is driven by the pressure field of the injected fluid. Finally, a numerical example showing the coupled process of crack propagation in partially saturated porous materials is discussed.
The formation of extracellular ice within plant tissues is regarded as one of their crucial factors to withstand subzero temperatures without any (biologically irreversible) damage. In this regard, ...extracellular ice implies two important consequences, which are the dehydration of the tissue cells to prevent intracellular ice formation, which would be fatal for the plant, and the attraction of water towards the freezing site. However, the pattern of ice formation may vary significantly among various types of plants. There might be rather dispersed ice formation distributed in large parts of the plant or localised ice formation at internal surfaces. Within this contribution, the latter‐mentioned case is addressed with a macroscopic modelling approach based on the Theory of Porous Media. The appearing water management is discussed at a representative numerical example.
Abstract Under in-situ conditions, natural hydraulic fractures (NHF) can occur in permeable rock structures as a result of a rapid decrease of pore water accompanied by a local pressure regression. ...Obviously, these phenomena are of great interest for the geo-engineering community, as for instance in the framework of mining technologies. Compared to induced hydraulic fractures, NHF do not evolve under an increasing pore pressure resulting from pressing a fracking fluid in the underground but occur and evolve under local pore-pressure reductions resulting in tensile stresses in the rock material. The present contribution concerns the question under what quantitative circumstances NHF emerge and evolve. By this means, the novelty of this article results from the combination of numerical investigations based on the Theory of Porous Media with a tailored experimental protocol applied to saturated porous sandstone cylinders. The numerical investigations include both pre-existing and evolving fractures described by use of an embedded phase-field fracture model. Based on this procedure, representative mechanical and hydraulic loading scenarios are simulated that are in line with experimental investigations on low-permeable sandstone cylinders accomplished in the Porous Media Lab of the University of Stuttgart. The values of two parameters, the hydraulic conductivity of the sandstone and the critical energy release rate of the fracture model, have turned out essential for the occurrence of tensile fractures in the sandstone cores, where the latter is quantitatively estimated by a comparison of experimental and numerical results. This parameter can be taken as reference for further studies of in-situ NHF phenomena and experimental results.
Cancer is one of the most serious diseases for human beings, especially when metastases come into play. In the present article, the example of lung-cancer metastases in the brain is used to discuss ...the basic problem of cancer growth and atrophy as a result of both nutrients and medication. As the brain itself is a soft tissue that is saturated by blood and interstitial fluid, the biomechanical description of the problem is based on the Theory of Porous Media enhanced by the results of medication tests carried out in in-vitro experiments on cancer-cell cultures. Based on theoretical and experimental results, the consideration of proliferation, necrosis and apoptosis of metastatic cancer cells is included in the description by so-called mass-production terms added to the mass balances of the brain skeleton and the interstitial fluid. Furthermore, the mass interaction of nutrients and medical drugs between the solid and the interstitial fluid and its influence on proliferation, necrosis and apoptosis of cancer cells are considered. As a result, the overall model is appropriate for the description of brain tumour treatment combined with stress and deformation induced by cancer growth in the skull.
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