A Method of Calculating Principal Stress Trajectories in Powder and Porous Materials Obeying a Piecewise Linear Yield Criterion

The present paper deals with the system of equations comprising the pyramid yield criterion together with the stress equilibrium equations under plane strain conditions. The stress equilibrium equations are written relative to a coordinate system in which the coordinate curves coincide with the trajectories of the principal stress directions. The general solution of the system is found giving a relation connecting the two scale factors for the coordinate curves. This relation is used for developing a method for finding the mapping between the principal lines and Cartesian coordinates with the use of a solution of a hyperbolic system of equations. In particular, the mapping between the principal lines and Cartesian coordinates is given in parametric form with the characteristic coordinates as parameters.


Introduction
In the case of rigid perfectly plastic solids, several efficient methods that utilize this or that property of special coordinate systems are used for solving plane strain boundary value problems.Examples are characteristic coordinates and Mikhlin's coordinates [1,2].An important relation between the scale factors of a principal line coordinate system has been derived in [3].Using this property it is possible to develop an efficient method of calculating principal stress trajectories.This has been demonstrated in [4] where the Mohr-Coulomb yield criterion has been adopted.The material model used in [3] is obtained as a special case.All of the aforementioned methods for smooth yield criteria (or for a smooth portion of piece-wise smooth yield criteria).However, in the case of the pyramid yield criterion used for powder and porous materials [5] plane strain deformation occurs at an edge of the yield surface.Therefore, the aforementioned methods are not applicable.The method of Mikhlin's coordinates has been generalized on the pyramid yield criterion in [6].In the present paper, a method for finding principal stress trajectories is proposed.It is known that the use of principal lines coordinate systems proves to be advantageous [7,8].In particular, a principal line theory of axially symmetric plastic deformation has been developed in [9] for the face regime of the Tresca yield criterion and its associated flow rule.

Geometry of principal stress trajectories
The pyramid yield criterion proposed in [5] as a generalization of Tresca's yield criterion on powder and porous materials reads Here 1 V , 2 V and 3 V are the principal stresses, V is the hydrostatic stress, s W is the shear yield stress and s p is the yield stress in hydrostatic compression.In general, both s W and s p depend on the relative density.However, by assumption, the material is homogeneous.Therefore, s W and s p are constant.Since the yield criterion is singular, several edge and face regimes are possible.It has been shown in [5] that plane stain deformation occurs at one of edge regimes.For definiteness, it is assumed that 0 With no loss of generality, it is possible to assume that the principal axis corresponding to the principal stress 3 V is orthogonal to planes of flow and that 1 2

V V
! .Then, the plane strain yield criterion following from (1) is It follows from this equation that The system of equations comprising the equilibrium equations and the yield criterion (4) can be investigated without using velocity equations.If a boundary value problem is statically determinate then this system allows the stress field to be found.Let us introduce a curvilinear Here h D and h E denote the scale factors for the D and E curves, respectively.Equation ( 4) can be rewritten as Eliminating E V in the first equation in ( 5) and D V in the second equation in ( 5) by means of (6) yields Each of these equations can be immediately integrated using (2) to give Here Equations ( 9) and ( 10) combine to give Thus the problem of finding the field of stress has been reduced to the problem of finding an orthogonal coordinate system whose scale factors satisfy (11).Once such a system of coordinate has been found, the principal stresses are determined from ( 8) and (10).

Method of determining principal stress trajectories
Introduce a Cartesian coordinate system , x y .Let \ be the angle between the D lines and the x axis, measured anticlockwise positive from the x axis (Fig. 1).These equations and the compatibility equations in the form  Using a standard technique it is possible to show that this system of equations is hyperbolic.The characteristic curves are determined from The characteristic relations can be immediately integrated to give Here 3 C K is independent of [ and 4 C [ is independent of K .Solving the equations in (18) for \ and ln h D results in If both [ and K lines are curved, then their parameterization can be chosen such that 3 C q Equation ( 16) can be rewritten as Eliminating h D in this equation by means of (20) yields Introduce the new quantities H and Q by Substituting ( 23) into ( 22) gives Then, equation ( 24) becomes The equations in (25 . In this case it is possible to find from (25) that

. 1 m q m
Moreover, equation (26) transforms to These equations can be rewritten as where Z H Q and T H Q .It is evident that (29) is equivalent to the equations of telegraphy: This equation is integrated by the method of Riemann.In particular, the Green's function is the Bessel function of zero order.The methods for finding solutions of the equation of telegraphy in conjunction with boundary conditions typical in plasticity theory have been well documented [1,2,11].Assuming that x and y are functions of D and E and using the chain rule it is possible to find that , , , .
Here q should be eliminated by menas of (27).Moreover, using (27) it is possible to transform (20) to

Conclusions
On the assumption of plane strain conditions the system of equations comprising the piramyd yield criterion (1) and the equilibrium equations has been solved in a curvilinear orthogonal coordinate system in which the coordinate curves coincide with trajectories of the principal stress directions.It has been shown that the scale factors of the coordinate curves should satisfy equation (11).The principal stresses are expressed in terms of the scale factors using equations ( 8) and (10).
Mapping between the principal line and Cartesian coordinates is derived in parametric form with characteristic variables as parameters.In particualar, equations (34) can be integrated along any path in the ,

{
, respectively, the equilibrium equations in the , D E coordinate system may be written in the following form[10]

Figure 1 .
Figure 1.Principal line and Cartesian coordinate systems.It follows from the geometry of this figure that Eliminating h D and \ in (34) by means of(35) the coefficients of the derivatives D [ w w and D K w w in (34) are represented as functions of [ and K .The derivatives D [ w w and D K w w are readely determined from (23) if the functions , H [ K and, Q [ K are known.The latter are found from a solution of (30) and the equations 2 can be integrated numerically along any convenient path in the , [ K space to determine the principal stress trajectories.
are determined from (23) and any solution of the equations in (30).It is worthy of note here that 2 2 C E and 2C D merely change the scale of the E and D curves, respectively.