# Plasticity, mathematical theory of

The theory of deformable plastic solids in which one studies problems of determining the fields of the displacement vector or the velocity vector and of the strain tensor , or the deformation rate and of the stress tensor , that arise in a body occupying a region with boundary under body forces and surface forces subject to given initial and boundary conditions, as well as the determination of the loads and processes for which the equilibrium (motion) of the body is unstable. Features of the mathematical theory of plasticity are:

a) the relations are non-linear and non-reversible, and, in general, are described by functionals (yield conditions), so the problems in the theory are essentially non-linear;

b) the configurations of the unknown quasi-static fields are determined by the changes in the given functions over an interval , and not by the instantaneous values at a time ;

c) when and vary, one gets regions of elastic strain, active plastic strain (loading) and passive strain (unloading), in which the relations differ. These regions should be determined in solving the problem;

d) the equations for the boundary value problem are in general of different types (elliptic or hyperbolic) in different parts of the body.

For an arbitrary complicated process only very general features are known for the plasticity functionals, and their explicit analytic structure has not been clarified. Concrete relations that do not contain functionals have been established and have experimentally been justified for several typical deformation processes. Sometimes one considers also artificial "models" for which only partly reflect the elastic-plastic properties of materials.

## Static boundary value problems.

In the theory of elastic-plastic processes [1], the isotropy postulate due to A.A. Il'yushin implies that the relations can be represented by

 (1)

(summation with respect to from 1 to 6), where is a basis constructed from the tensor of small deformation and the are functionals in the invariants of the deformation tensor, the pressure , the temperature , and possibly other invariant quantities of a non-mechanical nature. The unknown functions , , satisfy the following equations at equilibrium:

 (2)
 (3)
 (4)
 (5)
 (6)

with given functions , , and regions , and . Here the are the direction cosines of the outward normal to the boundary surface , is the parameter of the process (for example, the true or nominal time) and is the density of the material. Equation (2) represents the differential equations of equilibrium, (3) are the kinematic relations between the small deformations and displacements, and (5) and (6) are the boundary conditions in terms of the stresses and displacements, correspondingly. The equations (2)–(6), strictly speaking, do not give a formulation of a boundary value problem, since one cannot determine the existence of a solution if the structure of the plasticity functionals is unknown.

Then in order to solve (2)–(6), taking account of the indeterminacy of these functionals, one assumes that the solution exists for an arbitrary complex loading. Here the following method for computer solution has been proposed [1]. In what follows, the system composed of (2), (3), (5), and (6) will be called the incomplete system (B). In connection with the use of a finite-difference procedure, is split up into cells (elements), in each of which the unknown functions have constant (mean) values depending on the parameter (the relevant time interval is split up into steps) and equations (4) are replaced in cell by the approximate relations

 (7)

where are functions of such that the solution of (B), (7) exists. Suppose that in a first approximation with specifically given functions (as far as possible specified in the simplest way, e.g. in accordance with a generalized Hooke law) one has

 (7prm)

where the solution of (B), (7prm) is , , . The set of invariants including the functions and determines the set of programs for testing specimens in a uniform state of stress. The tests by means of the combine-loading matrix provide the true relationships in the processes , , . These determine the refined approximating relations

 (7prmprm)

which are used in the numerical solution of (B), (7prmprm) in the second approximation. The subsequent approximations are constructed similarly. The convergence is evaluated in the norms of the differences between two successive approximations.

There is a difference between the variants of theoretical-experimental methods for solving boundary value problems with known (4) and that used here, in that one uses tests on standard specimens in uniform states of stress by a standard method, and not tests on a natural object or on its model in a complex state of stress.

An a posteriori existence criterion has been proposed [2]: If the iteration process converges, then the solution to (2)–(6) with an unknown structure for the functionals exists and is determined with a given degree of accuracy in approximation .

If (4) is known but complicated, this method can be used with (4) instead of a combine-loading testing machine.

The complexity in formulating and solving the boundary value problem in plasticity theory in the general case is substantially reduced by considering particular classes of processes.

The complexity of the deformation process at a point in a body is determined by comparing the curvatures of the strain trajectories (these paths represent the changes in the deformation deviator in the five-dimensional Euclidean space) with a typical delay trace for each material, which is determined by experiment. In this way one distinguishes in particular classes of processes for which (4) may be specified in detail and does not contain functionals explicitly. For such a class of processes, the boundary value problem (2)–(6) is completely defined and allows one to prove existence and uniqueness theorems and to construct general solution methods. Then, however, there is the problem of the physical reliability of the solution, since with given , and the processes determined by the solution need not correspond to the reliability region for the particular form of (4) used in the formulation. This can be stated as a problem in determining the class of given functions , and , and, possibly, of constraints on (4) of a particular form, such that (2)–(6) is compatible when supplemented by equations defining the corresponding class of processes.

In the theory of small elastic-plastic deformations [3], which relates to simple deformation processes (zero curvature), (4) takes the form

 (8)

where is the deformation intensity, is the experimentally-determined hardening function and is a constant (the bulk modulus of elasticity), .

Under the conditions

 (9)

( is a number), applicable to constructional materials, it has been found that the boundary value problem (B), (8) is elliptic; existence and uniqueness theorems for the solution have been proved, minimality principles have been established and corresponding variational formulations have been given. A theorem on simple loading has been proved, defining the class of functions , and (a one-parameter loading) for which the solution is physically reliable. In principle one can use Ritz' method and the Bubnov–Galerkin method to solve (B), (8), but these are ineffective because of the non-linearity of the problem. The elastic-solution method is widely used [3]; it converges under the conditions (9): In each successive approximation one solves a simpler boundary value problem in the linear theory of elasticity [3]. In the course of solving one determines the regions of elastic deformation in which the generalized form of Hooke's law holds. The method of the variable elasticity parameter has also been used [4].

The formulation (B), (8) together with these solution methods have been used also in thermoplasticity. The temperature field is defined by the solution to the thermal-conductance problem, and in (8) one puts , with the term replaced by the expression . In view of the functional nature of , the use of is restricted to a certain class of thermal processes. Special interest attaches to cyclic loading [5] accompanied by the periodic occurrence of regions of unloading and loading with opposite sign.

In the theory of elastic-plastic processes of small curvature (the maximum curvature much less than ) with

 (10)

where

is the arc length on the strain trajectory, and with the kinematic equations

 (11)

where are the coordinates of the velocity vector for a particle in the medium, one may pose the boundary value problem (2), (10), (11), (5), (6), for which existence and uniqueness theorems have been proved, variational principles have been formulated and a method of successive approximation has been proposed. Test of the physical reliability of the solution have not been clarified. This problem has been posed in particular for the calculation of steady-state plastic flow in a hardening metal in a technological metal working process (pressing, rolling, etc.).

Similarly, the stress-strain relations have been established and the boundary value problem has been formulated and analyzed for the strain process of moderate curvature and the polygonal (bilinear) process. In both cases no a priori criterion of physical reliability has been proved.

For an arbitrary complex loading the stress-strain relations of the local theory of elastic-plastic processes,

 (12)

have been experimentally corroborated [12], [13]. The corresponding boundary value problem has been analyzed and algorithms for numerical solution have been given [14]. Here the functions of the variables could be determined experimentally for both loading and unloading processes, and the dot means "time" derivative. The realiability of the relations (12) ensures the physical reliability of the corresponding mathematical model, and provides an acceptable solution of the problem.

In the modern theory of flow, relations of the following form have been derived for the plastic part of the deformation tensor from plasticity postulates of energy type:

 (13)

while the generalized form of Hooke's law is used for the elastic part:

Here and are the Lamé constants, the loading function is a functional of the loading process , and the hardening function is also dependent of the increments . It is often assumed that is independent of . Linearization of (13) with respect to the increments enables one to rigorously pose a boundary value problem of the type (B), (13) and to prove various general theorems and minimality principles [6]; procedures have been given for a numerical-analytic solution. The advantage of the simplicity provided by linearization is seriously weakened by the fact that the region in which the linearized relations (13) are physically reliable has not yet been established.

In the mathematical theory of plasticity one frequently used a formulation of the boundary value problem on the basis of the Prandtl–Reuss plasticity theory, which is described by the relation

where are elasticity constants and is a function of . The reliability region of these equations is bounded (and has not been determined exactly).

A theory of ideally plastic bodies has been developed [7], [8] based on the Saint-Venant–Levy–von Mises physical relations. These are formally derived from equations (10) of small curvature if one puts ( is the yield stress of a material) and one adopts the incompressibility condition:

 (14)

The concept of ideal plasticity also implies that a yield condition of the following form is satisfied in the regions of active plastic deformation:

 (15)

for example, the Hencky–von Mises conditions , , or the Tresca–Saint-Venant conditions , where is the largest tangential stress and and are constants of the material.

A boundary value problem (2), (5), (6), (11), (14), (15) is usually of hyperbolic type (sometimes of elliptic type). The complete task of the theory of ideal plasticity is to solve (2), (11), (14), (15) in the part in which there is plastic deformation, and (2), (3) together with the generalized Hooke law in the region of elastic deformation, with corresponding boundary conditions and kinematic and dynamic conditions for linking the solutions at the unknown boundary of and (the elastic-plastic problem).

Sometimes, results of practical use are given by means of a rigid-plastic analysis, in which the elastic deformations are ignored, and the parts of the body outside the plastic regions are considered undeformable. Usually stationary surfaces of discontinuity in the particle velocities arise, which is incompatible with classical concepts in the mechanics of continuous media. In cases where the boundary conditions are given in terms of stresses, the equilibrium equations together with conditions of the type of (15) can be considered as constituting an autonomous system of hyperbolic type. It is then possible that the regions of plastic deformation and stresses in these regions are not uniquely determined in this region, and the conflict is frequently resolved by means of some heuristic arguments. This ambiguity does not arise in solving the elastic-plastic problem.

Among the problems handled by methods from the theory of ideal plasticity one may mention the problem of steady-state plastic flow in a non-hardening material in "channels" , which involves the determination of velocities and stresses, including contact ones, and which has applications to technological metal working problems (forging, pressing or stamping).

The theory of plastic flow in thin layers of metal, which is based on various rational hypotheses of kinematic and physical character, is of special importance in applications to engineering. One determines conditions necessary for maintaining plastic flow (for example, the pressure force) and determines the velocity distribution, enabling one to predict the shape of the manifactured object.

Applied problems in the mathematical theory of plasticity (e.g. on shell equilibrium) lead to boundary value problems containing non-linear partial differential equations of a high order (for example, the fourth).

Regarding equilibrium stability under plastic deformation, the typical process is the change of the type of deformation at the instant of buckling, with the formation of corners on the strain trajectory. For example, under simple loading complex stresses arise at the instant of bifurcation, and to solve the stability problem it is necessary to use relations describing the infinitesimal process after a corner [1].

## Dynamic problems in plasticity theory.

A general formulation of the dynamic problems is obtained if (2) is replaced by the equation of motion

and if one adds initial conditions to (2)–(6), e.g. of the form

Here, difficulties of two types arise:

1) since different types of waves with different velocities of propagation arise, depending on the absolute value of the deformation, the deformation at various points in the body may differ in complexity even with simple types of loads applied (and the same applies to a given point at different times), and one cannot establish a priori the possibility of using dynamic relations of a particular form;

2) it is necessary to use dynamic relations allowing for the time dependence of the deformation.

The dynamic plasticity functional has not been fully examined even for a one-dimensional process, i.e. for simple deformation. Only very scattered information is available on the dynamic characteristics of materials in combined states of stress in complicated deformation processes, and the data are not sufficient to establish even the qualitative features of the dynamic plasticity functionals. Consequently, dynamic problems in plasticity theory are posed using static relations, which has provided methods of solution and has elucidated mechanical effects specific to dynamic cases, while giving solutions that are useful as estimates for practical purposes. This approach as a stage in research is justified by the extreme importance of studies on dynamic processes in structures and constructions.

It has been found that combined discontinuities of various modes typical only for non-linear problems arise in the propagation of active non-linear deformations in complicated states of stress. As far as can be concluded from the available results, the most complicated processes occur in regions adjoining the surfaces of discontinuity, while the complexity is restricted in the main regions of motion as regards the curvatures of strain trajectories and the development over time. It is therefore possible to solve dynamic problems in plasticity theory for complicated cases by the combined use of relationships of a particular form (e.g. for moderate-curvature processes) in the main region while in the vicinity of discontinuity surfaces the theory for polygonal (bilinear) trajectories can be used. A non-trivial and specific aspect of the dynamic theory is that of finding the surfaces of unloading, separating regions of active and passive deformation.

Particularly detailed studies have been made on the propagation of one-dimensional elastic-plastic waves [9]. The propagation of unloading waves in a rod subjected to tension and compression have been examined [9]. There are also studies on the propagation and interaction of elastic-plastic longitudinal waves in a rod, where allowance has been made for the different types of relationship between the mechanical load and the deformation rate. The solutions, in turn, have been used in interpreting experiments (basic or control results) in research on the dynamic properties of materials [10].

## Mathematical problems in the theory of the equations of state.

The phenomenon of plasticity is so complicated that it is impossible to construct equations of state ( relations) by a direct generalization from experimental data. There is therefore a predominant tendency to set up a general theory, which in particular determines the theory applicable to experiments. In this way mathematical studies have been conducted on the admissible forms of equations of state that do not conflict with the laws of mechanics and thermodynamics. One has studied curves and surfaces in -dimensional space in relation to studies on processes and limiting configurations and one has formulated definitive principles for certain classes of materials (in some way idealized) that indicate the independent state parameters and determine the structure of the relationships (in fairly general form), which enables one to examine the features of plasticity functionals and to elucidate the classes of processes allowing a simpler mathematical description within the framework of the general equations [1], [11]. The development of the theory of the representation of functionals is very important. The results of this theory enable one, in particular, to establish the physical reliability of equations of state having complicated functional structures from the analysis of phenomena that are relatively easy to reproduce in nature.

#### References

 [1] A.A. Il'yushin, "Plasticity: principles of the general mathematical theory" , Moscow (1963) (In Russian) [2] A.A. Il'yushin, V.S. Lenskii, "A model and algorithm" Prikladn. Probl. Prochnost. i Plastichnost. , 1 (1975) (In Russian) [3] A.A. Il'yushin, "Plasticity" , Elastoplastic strains , 1 , Moscow-Leningrad (1948) (In Russian) [4] I.A. Birger, "Some general solution methods for problems in plasticity theory" Prikl. Mat. Mekh. , 15 : 6 (1951) pp. 765–770 (In Russian) [5] V.V. Moskvitin, "Plasticity with variable loads" , Moscow (1965) (In Russian) [6] W.T. Koiter, "General theorems for elastic-plastic media" I.N. Sneddon (ed.) R. Hill (ed.) , Progress in Solid Mechanics , 1 , North-Holland & Interscience (1960) pp. 167–224 [7] V.V. Sokolovskii, "Theorie der Plastizität" , Deutsch. Verlag Wissenschaft. (1955) (Translated from Russian) [8] R. Hill, "The mathematical theory of plasticity" , Oxford Univ. Press (1950) [9] Kh.A. Rakhmatulin, Yu.A. Dem'yanov, "Strength under intense short loading" , Moscow (1961) (In Russian) [10] R.A. Vasin, V.S. Lenskii, E.V. Lenskii, "Dynamic relations between stress and deformation" , Problems in the dynamics of elastoplastic media , Moscow (1975) (In Russian) [11] C. Truesdell, "A first course in rational continuum mechanics" , Acad. Press (1977) [12] V.S. Lenskii, "Experimental corroboration of basic postulates of a general theory of elastic-plastic strains" , Problems of the Theory of Plasticity , Moscow (1961) pp. 58–82 (In Russian) [13] V.S. Lenskii, E.V. Lenskii, "Trinomial relation in the general theory of plasticity" Mekh. Tverd. Tela , 4 (1985) pp. 111–115 (In Russian) [14] Dao Zuy Bic, "On the uniqueness theorem in the theory of plasticity based on the hypothesis of local determinability" Mekh. Tverd. Tela , 1 (1982) pp. 119–124 (In Russian)