A boundary value problem in the theory of plastic wave propagation
Author:
E. H. Lee
Journal:
Quart. Appl. Math. 10 (1953), 335-346
MSC:
Primary 73.2X
DOI:
https://doi.org/10.1090/qam/52301
MathSciNet review:
52301
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Abstract: In the analysis of boundary value problems in the theory of plasticity, the general situation arises that different partial differential equations are to be satisfied depending on whether the material is in the plastic or elastic state. The criterion determining the state at any material point depends on the dependent variables and their derivatives with respect to time. Thus the regions of application of the different differential equations must be determined from the boundary and initial conditions as the solution is developed. The theory of the propagation of plastic waves in one dimension is a case in which the solution, including the determination of the unknown plastic-elastic boundaries, can be treated. An example is presented in this paper which illustrates the many types of boundary determination conditions which must be used. The method is based on the numerical integration along the characteristics of the hyperbolic equations arising, one linear and one quasi-linear. The development is possible since forward integration along characteristics enables the unknown boundaries to be determined independently of the subsequent solution. This situation is contrasted with other problems in the theory of plasticity. The complexity of the procedure indicates the difficulty to be anticipated with analytical treatment of such problems, and with the numerical treatment of problems involving more extensive plastic flow.
Th. von Kármán, H. F. Bohnenblust and D. H. Hyers, The propagation of plastic waves in tension specimens of finite length, NDRC Report No. A-103 (OSRD No. 946), 1943.
G. I. Taylor, Propagation of earth waves from an explosion, British Official Report, R.C. 70, 1940, The plastic wave in a wire extended by an impact load, R.C. 329, 1942.
Th. von Kármán, On the propagation of plastic deformation in solids, NDRC ReDort No. A-29, OSRD No. 365. 1942.
- Theodore von Kármán and Pol Duwez, The propagation of plastic deformation in solids, J. Appl. Phys. 21 (1950), 987–994. MR 39531
H. F. Bohnenblust, Comments on White and Griffis’ theory of the permanent strain in a uniform bar due to longitudinal impact, NDRC Memo. A-47M, OSRD No. 781, 1942.
M. P. White and L. Griffis, Wave propagation in a uniform bar whose stress-strain curve is concave upward, NDRC Report No. 152, OSRD 1302, 1943, J. Appl. Mech. 70, 256 (1948).
E. H. Lee, Plastic waves in compression, British Official Report A.P.P., Co-ord. Sub-Committee No. 57, 1943.
Th. von Kármán, H. F. Bohnenblust and D. H. Hyers, The propagation of plastic waves in tension specimens of finite length, NDRC Report No. A-103 (OSRD No. 946), 1943.
G. I. Taylor, Propagation of earth waves from an explosion, British Official Report, R.C. 70, 1940, The plastic wave in a wire extended by an impact load, R.C. 329, 1942.
Th. von Kármán, On the propagation of plastic deformation in solids, NDRC ReDort No. A-29, OSRD No. 365. 1942.
Th. von Kármán and Pol Duwez, The propagation of plastic deformation in solids, J. Appl. Phys. 21, 987-994 (1950).
H. F. Bohnenblust, Comments on White and Griffis’ theory of the permanent strain in a uniform bar due to longitudinal impact, NDRC Memo. A-47M, OSRD No. 781, 1942.
M. P. White and L. Griffis, Wave propagation in a uniform bar whose stress-strain curve is concave upward, NDRC Report No. 152, OSRD 1302, 1943, J. Appl. Mech. 70, 256 (1948).
E. H. Lee, Plastic waves in compression, British Official Report A.P.P., Co-ord. Sub-Committee No. 57, 1943.
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Article copyright:
© Copyright 1953
American Mathematical Society