Introduction to linear and nonlinear programming

The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

Literature survey of contact dynamics modelling

Vibration of simple beams due to trains moving at high speeds

The objective of this study is to develop an element that is both accurate and efficient for modeling the vehicle-bridge interaction (VBI) in analysis of railway bridges carrying high-speed trains, which may consist of a number of cars in connection. In this study, a train is modeled as a series of sprung masses lumped at the bogie positions and a bridge with track irregularities by beam elements. Two sets of equations of motion that are coupled can be written, one for the bridge and the other for each of the sprung masses. To resolve the problem of coupling, the sprung mass equation is first discretized using Newmark's finite difference formulas and then condensed to that of the bridge element in contact. The element derived is referred to as the vehicle-bridge interaction element, which has the same number of degrees of freedom (DOF) as the parent element, while possessing the properties of symmetry and bandedness in element matrices. For this reason, conventional assembly procedures can be employed to ...

Vehicle-Bridge Interaction Element for Dynamic Analysis

An analytical-numerical method is presented that can be used to determine the dynamic behavior of beams, with different boundary conditions, carrying a moving mass. This article demonstrates the transformation of a familiar governing equation into a new, solvable series of ordinary differential equations. The correctness of the results has been ascertained by a comparison, using finite element models, and very good agreement has been obtained. Furthermore, the article shows that the response of structures due to moving mass, which has often been neglected in the past, must be properly taken into account because it often differs significantly from the moving force model.

Numerical solution for response of beams with moving mass

Closure of "Numerical Solution for Response of Beams With Moving Mass"

Experimental study on cyclic behavior of trapezoidally corrugated steel shear walls

Investigation of critical influential speed for moving mass problems on beams

Modeling the water flow in cohesive fracture is a fundamental issue in the crack growth simulation of cracked concrete gravity dams and hydraulic fracture problems. In this paper, a mathematical model is presented for the analysis of fracture propagation in the semi-saturated porous media. The solid behavior incorporates a discrete cohesive fracture model, coupled with the flow in porous media through the fracture network. The double-nodded zero-thickness cohesive interface element is employed for the mixed mode fracture behavior in tension and contact behavior in compression. The modified crack permeability is applied in fracture propagation based on the data obtained from experimental results to implement the roughness of fracture walls.

Massood Mofid

Papers

Numerical solution for response of beams with moving mass

Closure of "Numerical Solution for Response of Beams With Moving Mass"

Experimental study on cyclic behavior of trapezoidally corrugated steel shear walls

Investigation of critical influential speed for moving mass problems on beams

Modeling of cohesive crack growth in partially saturated porous media; a study on the permeability of cohesive fracture