The purpose of the present paper is development of a Non-singular Method of Fundamental Solutions (NMFS) for two-dimensional isotropic linear elasticity problems. The NMFS is based on the classical Method of Fundamental Solutions (MFS) with regularization of the singularities. This is achieved by replacement of the concentrated point sources by distributed sources over circular discs around the singularity, as originally suggested by [Liu (2010)] for potential problems. The Kelvin’s fundamental solution is employed in collocation of the governing plane strain force balance equations. In case of the displacement boundary conditions, the values of distributed sources are calculated directly and analytically. In case of traction boundary conditions, the respective desingularized values of the derivatives of the fundamental solution in the coordinate directions, as required in the calculations, are calculated indirectly from the considerations of two reference solutions of the linearly varying simple displacement fields. The developments represent a first use of NMFS for solid mechanics problems. With this, the main drawback of MFS for these types of problems is removed, since the artificial boundary is not present. In order to demonstrate the feasibility and accuracy of the newly developed method, is the NMFS solution compared to the MFS solution and analytical solutions for a spectra of plane strain elasticity problems, including bi-material problems. NMFS turns out to give similar results than the MFS in all spectra of performed tests. The lack of artificial boundary is particularly advantageous for using NMFS in multi-body problems.

COBISS.SI-ID: 2750203

The vertical organization of karst conduit networks has been in the focus of speleogenetic studies for more than a century. The four state model of Ford and Ewers (1978), which still is considered as most general, relates the geometry of caves to the frequency of permeable fissures. According to this model, one would expect that water table caves would be common in areas with high fissure frequency. However, in Alpine karst systems, water table caves are more an exception than a rule. Alpine speleogenesis is influenced by high uplift and valley incision rates and irregular recharge. To study the potential role of these processes for speleogenesis in the dimensions of length and depth, we apply a simple mathematical model based on coupling of flow, dissolution and transport. We assume a master conduit draining the water to the spring at the base level. Incision of the valley triggers evolution of deeper flow pathways, which are initially in a proto-conduit state. The master conduit evolves into a canyon following the valley incision, while the deep pathways evolve towards maturity and tend to capture the water from the master conduits. Two outcomes are possible: a) deep pathways evolve fast enough to capture all the recharge, leaving the master conduit dry; or b) the canyon reaches the level of deep pathways before these evolve to maturity. We introduce the Loop-to-Canyon ratio (LCR), which predicts which of the two outcomes is more likely to occur in certain settings. Our model is extended to account for transient flow conditions. In the case of an undulating master conduit, floodwater is stored troughs after the flood retreat. This water seeps through sub-vertical fractures (proto-soutirages) connecting the master conduit with the deep pathways. Therefore, the loops evolve also during the dry season, and the LCR is considerably increased. Although the model is very simple, it leads to some important conclusions for vertical organization of karst conduit networks and stresses the importance of base level changes and transient recharge conditions. It therefore gives an explanation of speleogenesis that relies much more on the dynamic nature of water flow than on the static one of fracture density.

COBISS.SI-ID: 36592685

We use measurements of dissolution rates of limestone tablets placed along a cave stream to estimate rates of modern incision. Dissolution rates within the stream display a systematic decrease with downstream distance. We discuss a variety of mechanisms that could be responsible for the longitudinal decrease in dissolution rates and develop simple mathematical models for each. The dissolutional length scales that arise from each model allow a first order estimate of the plausibility of each mechanism and motivate further field studies to test each possibility. Water chemistry and other field data suggest that a decrease in the concentration of CO2 along the cave stream is responsible for the observed decrease in dissolution rates. We propose two potential mechanisms that could trigger this reduction in dissolved CO2 and discuss the plausibility of each mechanism in light of the field data collected. Either of these mechanisms introduces a feedback loop whereby the stream profile of a channel in soluble bedrock indirectly influences CO2 concentrations in the water, via either microbial or hydraulic processes.

COBISS.SI-ID: 35231533