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A method of continuous-discontinuous cellular automaton for modeling the growth and coalescence of multiple cracks in brittle material is presented. The method uses the level set to track arbitrary discontinuities, and calculation grids are independent of the discontinuities and no remeshing are required with the crack growing. Based on Griffith fracture theory and Mohr-Coulumb criterion, a mixed fracture criterion for multiple cracks growth in brittle material is proposed. The method treats the junction and coalescence of multiple cracks, and junction criterion and coalescence criterion for brittle material are given, too. Besides, in order to overcome the tracking error in the level set approximation for crack junction and coalescence, a dichotomy searching algorithm is proposed. Introduced the above theories into continuous-discontinuous cellular automaton, the present method can be applied to solving multiple crack growth in brittle material, and only cell stiffness is needed and no assembled global stiffness is needed. Some numerical examples are given to shown that the present method is efficient and accurate for crack junction, coalescence and percolation problems.
The Anti-Drug Abuse Act provided mandatory minimum sentences of imprisonment forpossessionwith intent to distribute powder and crack cocaine. In this statute Congress established a quantitative100-to-1 sentence ratio between the two ( i.e ., it takes 100 times as much powder cocaineas crackcocaine to trigger the same sentence). Under this distinction, a person convicted of possession withintent to distribute a pound of powder cocaine (453.6 grams) would serve considerably less time ina federal prison than one convicted of possession with intent to distribute 5 grams of crack. TheUnited States Sentencing Commission incorporated the ratio into its generally binding sentencingguidelines.Since enactment, it has become apparent that the incidence of this sentencing differential fallsdisproportionately on African-American defendants. The disparate impact has been attacked withoutgreat success on several judicial fronts. Equal protection and due process arguments have flounderedon the finding that the distinction was not motivated by racial animus or discriminatory intent, butrather was related to the legitimate government purpose of protecting the public against the greaterdangers of crack cocaine. Thus far, defendants have encountered similar difficulties proving therequisite corrupt motivation to establish selective prosecution or sentencing entrapment defenses.Further, the federal appellate courts have found that the stiff minimum sentences for offensesinvolving crack cocaine are rational and not disproportionate to the seriousness of those offenses.Consequently, they do not offend the cruel and unusual punishment clause of the Eighth Amendment.And the courts have been no more receptive to pleas to mitigate the disparate impact by departingfrom the severity of the sentencing guidelines.Instructed to study the situation, the Sentencing Commission promulgated amendments thatwould equate crack and powder cocaine for sentencing purposes and recommended that Congressdrop the 100-to-1 ratio from its own mandatory penalties. Congress rejected both the amendmentsand the suggestion for equation, but directed the Commission to re-examine the issue and report backrecommendations reflecting more moderate adjustments. The Commission subsequentlyrecommended that the penalties be adjusted to a ratio somewhere between 1-to-1 2/3 and 1-to 15.The Commission has made no further recommendations.Legislative efforts to reduce or eliminate the disparity have thus far come to impasse over twoissues: (1) the appropriate ratio and (2) whether and to what extent crack penalties should be reducedor powder penalties enhanced to achieve the proper balance. 2b1af7f3a8