Punch problem

King, R. B. (1987), Elastic analysis of some punch problems for a layered medium, International Journal of Solids and Structures 23, 1657-1664. 

Hill, J.M. Wu, Y-H. (1996) The punch problem for shear-index granidar material. J. Mechanics Applied Mathematics, 49, (1) 81-105. 

The model describing interaction between two bodies, one of which is a deformed solid and the other is a rigid one, we call a contact problem. After the deformation, the rigid body (called also punch or obstacle) remains invariable, and the solid must not penetrate into the punch. Meanwhile, it is assumed that the contact area (i.e. the set where the boundary of the deformed solid coincides with the obstacle surface) is unknown a priori. This condition is physically acceptable and is called a nonpenetration condition. We intend to give a mathematical description of nonpenetration conditions to diversified models of solids for contact and crack problems. Indeed, as one will see, the nonpenetration of crack surfaces is similar to contact problems. In this subsection, the contact problems for two-dimensional problems characterizing constraints imposed inside a domain are considered. 

Thus, the relations (1.36) or (1.37) describe the interaction between a plate and a punch. To derive the contact model for an elastic plate, one needs to use the constitutive law (1.25). Contact problems for inelastic plates are derived by the utilizing of corresponding inelastic constitutive laws given in Section 1.1.4. 

The equilibrium problem for the plate contacting with the punch z = x, y)... 

Considering the crack, we impose the nonpenetration condition of the inequality type at the crack faces. The nonpenetration condition for the plate-punch system also is the inequality type. It is well known that, in general, solutions of problems having restrictions of inequality type are not smooth. In this section, we establish existence and regularity results related to the problem considered. Namely, the following questions are under consideration ... 

We note that if the crack opening is zero on F,, i.e. [%] = 0, the value of the objective functional Js u) is zero. We also assume that near F, the punch does not interact with the shell. It turns out that in this case the solution X = (IF, w) of problem (2.188) is infinitely differentiable in a neighbourhood of points of the crack. This property is local, so that a zero opening of the crack near the fixed point guarantees infinite differentiability of the solution in some neighbourhood of this point. Here it is undoubtedly necessary to require appropriate regularity of the curvatures % and the external forces u. The aim of the following discussion is to justify this fact. At this point the external force u is taken to be fixed. 

In so doing, the boundary value of on F is assumed to provide nonemptiness of the set K. The equilibrium problem for the plate contacting with the punch and having the crack can be formulated as a variational one ... 

Khludnev A.M. (1983) A contact problem of a linear elastic body and a rigid punch (variational approach). Appls. Maths. Mechs. 47 (6), 999-1005 (in Russian). 

It s not very easy having this problem, because kids at school tease me. I don t want to get pushed around so I just say, You wouldn t have such a big mouth if I punched you That backs them off. People will stop bugging you if you stand up for yourself. Sometimes I stand up for my friends, too. 

In the earhest days of computer systems highly inflated estimates of the power, potential, and accuracy of systems created a strong pathos of proof. The computer says so became the rall3nng cry and defense of billing agents, government clerks, and bureaucrats the world over. As stories of enormous and humorous computer errors flooded popular culture in later years, however, a computer error became as common a punch line as the check is in the mail computer professionals fell from godlike status to a reputation probably far below the reasonable norm of accurate and reliable system function. The result was, and is, an appropriate demand for controls, even as most reviews demonstrate that those controls are preventive rather than corrective of real problems. 

M awe of my favorite Funky Winkerbean cartoons starts out with something like, When Henry was twice as old as Hank was two years after half of Hank s age... And the punch line How many apples is an orange worth Some people think this is hilarious. Others don t find a bit of humor in these very involved, contrived, convoluted problems. 

The earliest type of automatic digital calculator to become generally available, starting at about 1945, was the punched-card calculator. These machines were at first entirely electromechanical, but in their modern form are largely electronic. Their intended use was initially in accounting applications, but their utility in various technical problems was soon discovered. Many of the present users of large-scale calculators were introduced to computing by the punched-card calculator. 

The operations of a punched-card calculator are determined by the wiring of a control panel. The panel, in effect, completes circuits between components in the machine so that desired operations are carried out. The panels may be removed from the machine and saved permanently. Thus several differently wired panels may be kept on hand for different types of problems, making it a simple matter to change the functions of the machine. 

There are many problems, particularly in the field of accounting, for which the small punched-card calculator is suitable. However, for many technical or scientific calculations it is of. only limited utility, primarily because of its limited speed and a basic lack of adaptability to lengthy sequential calculations. The speed limitation comes about principally because the operation of these machines is dependent upon the mechanical movement of cards. Thus, even though electronic calculations may be performed at high speed, this feature cannot be fully exploited. 

Another evident trend is the application of mechanical computers for the rapid calculation of distillation problems. At least two papers have already appeared, one by Weil (57) on the use of an electronic computer, and the other by Rose and Williams (47) on the application of the IBM card punch technique. If a short rigorous method is not forthcoming, machine calculations may be the answer to accurate design of fractionating equipment. 

Recall that considering a simpler reduced model (4.3.1) instead of (4.3.5) is motivated by the smallness of the parameter e and the appropriate results for a nowhere vanishing N x) = 0(1) are expected to be valid for the nonreduced problem, unless very large voltages are applied, causing punch through. ... 

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