Continuum plasticity theory
- Related Topics:
- mechanics
- Tresca criterion
The macroscopic theory of plastic flow has a history nearly as old as that of elasticity. While in the microscopic theory of materials, the word “plasticity” is usually interpreted as denoting deformation by dislocation processes, in macroscopic continuum mechanics it is taken to denote any type of permanent deformation of materials, especially those of a type for which time or rate of deformation effects are not the most dominant feature of the phenomenon (the terms viscoplasticity, creep, or viscoelasticity are usually used in such cases). Coulomb’s work of 1773 on the frictional yielding of soils under shear and normal stress has been mentioned; yielding denotes the occurrence of large shear deformations without significant increase in applied stress. His results were used to explain the pressure of soils against retaining walls and footings in the work of the French mathematician and engineer Jean Victor Poncelet in 1840 and the Scottish engineer and physicist William John Macquorn Rankine in 1853. The inelastic deformation of soils and rocks often takes place in situations for which the deforming mass is infiltrated by groundwater, and Austrian-American civil engineer Karl Terzaghi in the 1920s developed the concept of effective stress, whereby the stresses that enter a criterion of yielding or failure are not the total stresses applied to the saturated soil or rock mass but rather the effective stresses, which are the difference between the total stresses and those of a purely hydrostatic stress state with pressure equal to that in the pore fluid. Terzaghi also introduced the concept of consolidation, in which the compression of a fluid-saturated soil can take place only as the fluid slowly flows through the pore space under pressure gradients, according to Darcy’s law; this effect accounts for the time-dependent settlement of constructions over clay soils.
Apart from the earlier observation of plastic flow at large stresses in the tensile testing of bars, the theory of continuum plasticity for metallic materials begins with Henri Edouard Tresca in 1864. His experiments on the compression and indentation of metals led him to propose that this type of plasticity, in contrast to that in soils, was essentially independent of the average normal stress in the material and dependent only on shear stresses, a feature later rationalized by the dislocation mechanism. Tresca proposed a yield criterion for macroscopically isotropic metal polycrystals based on the maximum shear stress in the material, and that was used by Saint-Venant to solve an early elastic-plastic problem, that of the partly plastic cylinder in torsion, and also to solve for the stresses in a completely plastic tube under pressure.
The German applied mechanician Ludwig Prandtl developed the rudiments of the theory of plane plastic flow in 1920 and 1921, with an analysis of indentation of a ductile solid by a flat-ended rigid indenter, and the resulting theory of plastic slip lines was completed by H. Hencky in 1923 and Hilda Geiringer in 1930. Additional developments include the methods of plastic limit analysis, which allowed engineers to directly calculate upper and lower bounds to the plastic collapse loads of structures or to forces required in metal forming. Those methods developed gradually over the early 1900s on a largely intuitive basis, first for simple beam structures and later for plates, and were put on a rigorous basis within the rapidly developing mathematical theory of plasticity about 1950 by Daniel C. Drucker and William Prager in the United States and Rodney Hill in Great Britain.
The Austrian-American applied mathematician Richard von Mises proposed in 1913 that a mathematically simpler theory of plasticity than that based on the Tresca yield criterion could be based on the second tensor invariant of the deviatoric stresses (i.e., of the total stresses minus those of a hydrostatic state in which pressure is equal to the average normal stress over all planes). An equivalent yield criterion had been proposed independently by the Polish engineer Maksymilian Tytus Huber. The Mises theory incorporates a proposal by M. Levy in 1871 that components of the plastic strain increment tensor are in proportion to one another just as are the components of deviatoric stress. This criterion was generally found to provide slightly better agreement with experiment than did that of Tresca, and most work on the application of plasticity theory uses this form. Following a suggestion of Prandtl, E. Reuss completed the theory in 1930 by adding an elastic component of strain increments, related to stress increments in the same way as for linear elastic response. This formulation was soon generalized to include strain hardening, whereby the value of the second invariant for continued yielding increases with ongoing plastic deformation, and was extended to high-temperature creep response in metals or other hot solids by assuming that the second invariant of the plastic (now generally called “creep”) strain rate is a function of that same invariant of the deviatoric stress, typically a power law type with Arrhenius temperature dependence.
This formulation of plastic and viscoplastic, or creep, response has been applied to all manner of problems in materials and structural technology and in flow of geologic masses. Representative problems addressed include the growth and subsequent coalescence of microscopic voids in the ductile fracture of metals, the theory of the indentation hardness test, the extrusion of metal rods and rolling of metal sheets, design against collapse of ductile steel structures, estimation of the thickness of the Greenland Ice Sheet, and modeling the geologic evolution of the Plateau of Tibet. Other types of elastic-plastic theories intended for analysis of ductile single crystals originate from the work of G.I. Taylor and Hill and base the criterion for yielding on E. Schmid’s concept from the 1920s of a critical resolved shear stress along a crystal slip plane, in the direction of an allowed slip on that plane; this sort of yield condition has approximate support from the dislocation theory of plasticity.
Viscoelasticity
The German physicist Wilhelm Weber noticed in 1835 that a load applied to a silk thread produced not only an immediate extension but also a continuing elongation of the thread with time. This type of viscoelastic response is especially notable in polymeric solids but is present to some extent in all types of solids and often does not have a clear separation from what could be called viscoplastic, or creep, response. In general, if all of the strain is ultimately recovered when a load is removed from a body, the response is termed viscoelastic, but the term is also used in cases for which sustained loading leads to strains that are not fully recovered. The Austrian physicist Ludwig Boltzmann developed in 1874 the theory of linear viscoelastic stress-strain relations. In their most general form, these involve the notion that a step loading (a suddenly imposed stress that is subsequently maintained constant) causes an immediate strain followed by a time-dependent strain which, for different materials, either may have a finite limit at long time or may increase indefinitely with time. Within the assumption of linearity, the strain at time t in response to a general time-dependent stress history σ(t) can then be written as the sum (or integral) of terms that involve the step-loading strain response due to a step loading dt′dσ(t′)/dt′ at time t′. The theory of viscoelasticity is important for consideration of the attenuation of stress waves and the damping of vibrations.
A new class of problems arose with the mechanics of very-long-molecule polymers, which do not have significant cross-linking and exist either in solution or as a melt. These are fluids in the sense that they cannot long support shear stress, but at the same time they have remarkable properties like those of finitely deformed elastic solids. A famous demonstration is to pour one of these fluids slowly from a beaker and to cut the flowing stream suddenly with scissors; if the cut is not too far below the place of exit from the beaker, the stream of falling fluid immediately contracts elastically and returns to the beaker. The molecules are elongated during flow but tend to return to their thermodynamically preferred coiled configuration when forces are removed.
The theory of such materials came under intense development in the 1950s after the British applied mathematician James Gardner Oldroyd showed in 1950 how viscoelastic stress-strain relations of a memory type could be generalized to a flowing fluid. This requires that the constitutive relation, or rheological relation, between the stress history and the deformation history at a material “point” be properly invariant to a superposed history of rigid rotation, which should not affect the local physics determining that relation (the resulting Coriolis and centrifugal effects are quite negligible at the scale of molecular interactions). Important contributions on this issue were made by the applied mathematicians Stanisław Zaremba and Gustav Andreas Johannes Jaumann in the first decade of the 1900s; they showed how to make tensorial definitions of stress rate that were invariant to superposed spin and thus were suitable for use in constitutive relations. But it was only during the 1950s that these concepts found their way into the theory of constitutive relations for general viscoelastic materials; independently, a few years later, properly invariant stress rates were adopted in continuum formulations of elastic-plastic response.
Computational mechanics
The digital computer revolutionized the practice of many areas of engineering and science, and solid mechanics was among the first fields to benefit from its impact. Many computational techniques have been used in this field, but the one that emerged by the end of 1970s as, by far, the most widely adopted is the finite-element method. This method was outlined by the mathematician Richard Courant in 1943 and was developed independently, and put to practical use on computers, in the mid-1950s by the aeronautical structures engineers M.J. Turner, Ray W. Clough, Harold Clifford Martin, and LeRoy J. Topp in the United States and J.H. Argyris and Sydney Kelsey in Britain. Their work grew out of earlier attempts at systematic structural analysis for complex frameworks of beam elements. The method was soon recast in a variational framework and related to earlier efforts at deriving approximate solutions of problems described by variational principles. The new technique involved substituting trial functions of unknown amplitude into the variational functional, which is then rendered stationary as an algebraic function of the amplitude coefficients. In the most common version of the finite-element method, the domain to be analyzed is divided into cells, or elements, and the displacement field within each element is interpolated in terms of displacements at a few points around the element boundary (and sometimes within it) called nodes. The interpolation is done so that the displacement field is continuous across element boundaries for any choice of the nodal displacements. The strain at every point can thus be expressed in terms of nodal displacements, and it is then required that the stresses associated with these strains, through the stress-strain relations of the material, satisfy the principle of virtual work for arbitrary variation of the nodal displacements. This generates as many simultaneous equations as there are degrees of freedom in the finite element model, and numerical techniques for solving such systems of equations are programmed for computer solution.
