Multi-exponential model to describe pressure-dependent P-and S-wave velocities and its use to estimate the crack aspect ratio

We present new quantitative model describing the pressure dependence of acoustic P-and S-wave velocities.Assuming that a variety of individual mechanisms or defects(such as cracks,pore collapse and grain crushing)can contribute to the pressure-dependent change of the wave velocity,we order a characteristic pressure to all of them and allow a series of exponential terms in the description of the(Pand S-waves)velocity-pressure function.We estimate the parameters of the multi-exponential rock physical model in inversion procedures using laboratory measured P-and S-wave velocity data.As is known,the conventional damped least squares method gives acceptable results only when one or two individual mechanisms are assumed.Increasing the number of exponential terms leads to highly nonlinear ill-posed inverse problem.Due to this reason,we develop the spectral inversion method(SIM)in which the velocity amplitudes(the spectral lines in the characteristic pressure spectrum)are only considered as unknowns.The characteristic pressures(belonging to the velocity amplitudes)are excluded from the set of inversion unknowns,instead,they are defined in a set of fixed positions equidistantly distributed in the actual interval of the independent variable(pressure).Through this novel linear inversion method,we estimate the parameters of the multi-exponential rock physical model using laboratory measured P-and S-wave velocity data.The characteristic pressures are related to the closing pressures of cracks which are described by well-known rock mechanical relationships depending on the aspect ratio of elliptical cracks.This gives the possibility to estimate the aspect ratios in terms of the characteristic pressures.

Efficient Degradation 4-Nitrophenol by Photocatalysis from Modified TiO_2 Doped Copper

In the present study,a film consisting of TiO_2 doped with copper was prepared for efficiently decomposing 4-nitrophenol(4-NP) by photocatalysis.The preparing process of TiO_2 doped with copper includes two procedures:preparing Ti(OH)_4 doped with copper and synthesizing anatase and rutile TiO_2 doped with copper.Ti(OH)_4 doped with copper could be achieved by hydrolyzing TiCl_4in the mixed solution containing deionized water and copper oxalate.The Ti(OH)_4 doped with copper can be gained successfully by the following procedures:rinsing,drying and vacuum drying.The Ti(OH)_4 doped with copper could be converted into anatase TiO_2 doped with copper and rutile TiO_2 doped with copper by incineration for 4.5 h at 723 and 1 073 K,respectively.Characterizations of anatase TiO_2 doped with copper and rutile TiO_2 doped with copper were determined by X-ray diffraction(XRD) and energy dispersion of X-ray(EDX).Anatase and rutile TiO_2 doped with copper were dissolved in a mixed solution containing isopropanol and diethylamine.Stainless electrode was submerged into with the solutions,the film of TiO_2 was formed by drying the thin layer at a ramp rate of 3℃/min until 373 K,and this temperature was held for 1 h.The temperature of the oven was subsequently increased to a final temperature of 823 K at a ramp rate of 3℃/min,and was held at this value for 1 h.The stainless steel covered with modified TiO_2 film was utilized as the anode.The stainless steel mesh was used as the cathode.The cathode and anode were connected with the source and immersed into the solution with 100 mg/L 4-NP.The whole reaction on photocatalysis was perfectly carried out after ultraviolet radiation and aerator were run.The experimental results showed that:cracking ratio of 4-NP ring,the removal ratio of chemical oxygen demand(COD) and total organic carbon(TOC) were respectively more than 90%,80% and 80% within 2 h.Degradation of 4-NP implied its potential application in associated wastewater.

Crack elongation and its width of large depth reinforced concrete beams

In order to meet the requirement of structural inspection,the crack spacing and crack width at various heights in the tensile zone of six large depth reinforced concrete beams were measured under several loading levels of serviceability state.The effects of the depth of normal section beams on the crack spacing and crack width were analyzed,and the modified model is proposed for calculating the average crack spacing by thinking about the depth of normal section,the reinforcement arrangement and the effective reinforcement ratio.The relationships of crack widths at any position in the tensile zone and at the reinforcement level on the side surface of beam were studied.By theoretical and statistical analysis,a method is proposed to calculate the ratios of crack widths between any position and the reinforcement level on the side surface of large depth reinforced concrete beams.

Interaction between collinear periodic cracks in an infinite piezoelectric body

The problem of collinear periodic cracks in an infinite piezoelectric body is studied. Effect of saturation strips at the crack-tips is taken into account. By means of the Stroh formalism and the conformal mapping technique, the general periodic solutions for collinear cracks are obtained. The stress intensity factors and the size of saturation strips are derived analytically, and their dependencies on the ratio of the periodicity on the half-length of the crack are analyzed in detail. Numerical results show the following two facts. （1） When h/l 〉 4.0, the stress intensity factors become almost identical to those of a single crack in an infinite piezoelectric body. This indicates that the interaction between cracks can be ignored in establishing the criterion for the crack initiation in this case. （2） The speed of the saturation strip size of periodic cracks approaching that of a single crack depends on the electric load applied at infinity. In general, a large electric load at infinity is associated with a slow approaching speed.