VARIATIONAL PRINCIPLES IN THE MATHEMATICAL THEORY OF PLASTICITY

1956 ◽  
Author(s):  
D. C. DRUCKER
Keyword(s):  
Mathematical Theory ◽  
Variational Principles ◽  
Theory Of Plasticity

The Stress-Strain Laws of the Mathematical Theory of Plasticity—a Survey of Recent Progress

1948 ◽  
Vol 15 (3) ◽  
pp. 226-233
Author(s):  
William Prager
Keyword(s):  
Test Data ◽  
Mathematical Theory ◽  
Recent Progress ◽  
Variational Principles ◽  
Stress Strain ◽  
Theory Of Plasticity

Abstract Typical stress-strain laws of flow and deformation types are discussed with particular reference to the conditions of continuity and uniqueness which these laws must fulfill if they are to make sense physically. Alternative forms of some of these laws are presented, and conditions are discussed under which different laws yield identical results. It is shown how these laws may be generalized so as to fit test data more readily. Methods of integration are indicated and the use of variational principles is stressed.

Strain Hardening of a Porous Limestone

1967 ◽  
Vol 7 (03) ◽  
pp. 229-234 ◽  
Author(s):  
J.B. Cheatham
Keyword(s):  
Yield Strength ◽  
Mechanical Behavior ◽  
Strain Hardening ◽  
Rock Mechanics ◽  
Mathematical Theory ◽  
Confining Pressure ◽  
Plasticity Theory ◽  
Single Type ◽  
Microscopic Structure ◽  
Theory Of Plasticity

Abstract Applications of the mathematical theory of plasticity promises to lead to the solution of many drilling and rock mechanics problems. Because of mathematical considerations, the inelastic behavior of rock has frequently been represented by a perfectly plastic model in conjunction with a yield criteria of the Coulomb or Mohr type. The totality of all stress states for which a solid ceases to behave elastically can be represented as a limit surface in stress space. Probing of such limit surfaces indicates details of strain hardening which are not provided by the standard triaxial testing procedure. Probing tests of the limit surfaces have been performed on Cordova Cream limestone to provide data for extending plasticity theory to cover situations in which consolidation and strain hardening are present. Test results indicate that this highly porous limestone undergoes a permanent volume decrease when it is subjected to hydrostatic pressures in excess of 3,500 psi. A virgin sample tested under a confining pressure of 1,500 psi has a yield strength of 1,700 psi; however, if the sample is subjected to a consolidation pressure of 5,000 psi, before testing at 1,500 psi, the yield strength is raised to 2,300 psi. Thus, both consolidation and strain hardening are important considerations in describing the mechanical behavior of this limestone. Tests conducted with the axis of the core having different orientations indicate that this rock is also anisotropic. Portions of the initial and subsequent limit surfaces are determined for samples loaded either perpendicular or parallel to the bedding planes. INTRODUCTION Previous experimental work in rock mechanics indicates that no mathematically tractable constitutive theory is inclusive enough to describe the mechanical behavior exhibited by all types of rocks under all conditions of stress and temperature. Indeed, the type of deformation encountered in a single type of rock is known to depend upon the stress and temperature conditions in the rock during deformation.1-5 Certain rocks and minerals, notably those minerals composed of ionic salts, have been shown to exhibit plastic deformation when tested under conditions of high confining pressure.2 Since the mathematical theory of plasticity provides simplifications over the theory of elasticity in certain types of problems, such as those in which limit analysis can be applied, it is of interest to know under what conditions plasticity theory may be applied to rock mechanics problems. The following factors determine the nature of the deformation a particular specimen will undergo:the microscopic structure of the rock, i.e., the structure visible under an optical microscope, including number of phases, porosity, distribution of phases,the mineralogical structure of the solid phases,the conditions of stress and the rate of change of stress andthe temperature. Extensive experiments on Yule marble, Carthage marble, Solenhofen limestone and other calcerous rocks indicate that these relatively nonporous rocks deform plastically under certain conditions of loading, and creep under other conditions of loading.1-3 This study is concerned with the behavior of Cordova Cream limestone (Austin chalk) which is also composed almost entirely of calcite and thus has the same mineralogical composition but, because of a rather large porosity, it possesses a different microscopic structure. This investigation was undertaken to learn if Cordova Cream limestone deforms plastically despite the embrittling effect of the pore spaces, and to provide data which can be used to determine whether the mathematical theory of plasticity can describe the mechanical behavior of Cordova Cream limestone.

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