A new method of failure analysis

. The present paper develops a new failure analysis method under plane strain conditions considering a generalized linear yield criterion. The yield criterion and the stress equilibrium equations constitute a hyperbolic system of equations. It is shown that two auxiliary variables satisfy the equation of telegraphy. Simple analytical relationships connect these variables and the radii of curvature of the characteristic curves. The calculated radii of curvature allow the corresponding characteristic net to be constructed. Then, the stress field is determined using another set of analytical relationships. Thus, a numerical procedure is only necessary for solving the equation of telegraphy. This equation can be integrated by the method of Riemann. In particular, Green’s function is the Bessel function of zero order. A simple example illustrates the general method.


INTRODUCTION
The application of stress-based yield (or failure) criteria requires a stress analysis of structures.In the case of statically determinate problems, a yield criterion and the stress equilibrium equations can be solved independently of the other constitutive equations.The present paper deals with such statically determinate systems under plane strain conditions, assuming a generalized linear yield criterion.The study is restricted to hyperbolic systems of equations.
Several methods are available for determining stress fields in metal plasticity based on pressure-independent yield criteria.In all cases, the methods aim at determining characteristic nets.The corresponding stress fields are then found using simple relationships.One of these methods derives the equations for the radii of curvature of characteristic lines [1].It has been shown that these radii satisfy separately the equation of telegraphy.This method is usually referred to as the R -S method.Another method was proposed by Mikhlin and is sometimes named the method of moving coordinates [2,3].This method introduces a Cartesian coordinate system whose axes are tangent to characteristic lines at each point of a region.The moving coordinates also satisfy the equation of telegraphy.The third method derives the equations for principal line coordinates.The coordinate lines of principal line coordinate systems coincide with principal stress trajectories.The starting point of this method is the simple relationship between the scale factors of the principal line coordinates derived in [4].As in the previous methods, the equation of telegraphy should be solved for calculating characteristic nets.
Many materials obey pressure-dependent yield criteria, for example, soils, granular materials, porous and powder metals, and concrete.The most widely used yield criterion for such materials is that of Mohr-Coulomb [5 -8].Modified versions of this criterion are also often used for various materials [9 -12].Other piecewise linear yield criteria have been proposed and used in [13 -15], among many others.
The methods above have been extended to various linear yield criteria (i.e., the criteria represented by linear equations in terms of the principal stresses).The R -S method has been extended to the Mohr-Coulomb yield criterion in [16] and the piecewise linear yield criterion proposed in [13] in [17].The method of moving coordinates has been generalized on the Mohr-Coulomb yield criterion in [18] and a generalized linear yield criterion in [19].The method based on the geometric properties of principal line coordinates has been developed in [20] for the Mohr-Coulomb yield criterion and [21] for a generalized linear yield criterion.
It depends on the specific boundary value problem which of the above methods is most convenient.For example, using the moving coordinates method to calculate the stress field near curved traction-free surfaces is advantageous [3,19,22].However, it is worth noting that all the methods apply in a region where both families of characteristics are curved.On the other hand, in many cases, such a region is adjacent to a region where one of the characteristic families is straight.In such cases, the R -S method is most advantageous.The present paper develops the R-S method for a generalized linear yield criterion.Thus, it generalizes one of the most used methods for solving boundary value problems in plane-strain plasticity on any linear yield criterion that results in a system of hyperbolic equations.

SYSTEM OF EQUATIONS
The phrase 'piecewise linear yield criterion' is usually referred to a yield criterion that is represented by linear functions of the principal stresses.Under plane strain conditions, the general piecewise linear criterion is where 1  and 2  are the principal stresses in the planes of flow, 0  is a reference stress, and q is constant.It is assumed without loss of generality that 12 .
  The equations comprising (1) and the stress equilibrium equations are hyperbolic if 0 q  [2].The present paper is restricted to this case.This section briefly describes the characteristic analysis to introduce the equations required for subsequent derivation.
The characteristic directions are inclined at an angle  to the direction of 1  (Figure 1).This angle is determined as arctan . q 172 The two families of characteristics are regarded as the coordinate lines of a curvilinear coordinate system   , .The line of action of 1  falls in the sector between the positive directions of these coordinate lines (Figure 1).Introduce a Cartesian coordinate system   , xy.
Let  be the angle between the x-axis and the direction of 1 These equations can be rewritten as 0 along an line, 0 along a line, where In a region where both families of characteristics are curved, the equations in ( 6) can be represented as Here  is a constant whose value will be specified later, and 0  is a constant of integration.
Solving the equations in (8) for  and  , one gets

METHOD OF CALCULATING CHARACTERISTIC NETS
Equations ( 1), (7), and ( 9) allow the stress field to be found if a characteristic net is determined.A method of calculating characteristic nets is developed in this section.
The radii of curvature of the  and   lines are denoted as R and S, respectively.It is seen from ( 3) that the angle between each characteristic direction and the line of action of the stress 1  is constant.Therefore, the radii of curvature of the characteristic curves can be defined by the following equations: where s   and s   are space derivatives taken along the  and   lines, respectively.It follows from the geometry of Figure 1 that xx ss yy ss Using ( 9) and (10), one can transform (11) to The compatibility equations are Substituting ( 13) into (12) and employing (9) yields These equations can be solved for the derivatives R  and S  .As a result,

R and 0
S by the following relationships: where n and m are constant.Substituting ( 16) into (15) yields It is seen from these equations that 0 R and 0 S separately satisfy the equation of telegraphy.
Methods of solving this equation in conjunction with typical boundary conditions adopted in plasticity theory are well-developed [2].Also, equations ( 9) and ( 16) become and Using (21), one can rewrite (12) as sin 2 cos exp cos 2 , sin 2 sin exp cos 2 , sin 2 sin exp cos 2 .
x R Having (20) and a solution of the equations in (19), one can integrate the equations in (22) along any path in characteristic space.

STRENGTH OF A WELDED JOINT
Welded joints are an important class of structures.A distinguished feature of the highly under matched welded joints is that plastic yielding occurs in the weld, whereas the base material remains elastic at plastic collapse [23 -25].The simplest structure of this type is a welded panel subjected to tension (Figure 2).The thickness of the weld is 2H, and the width of the panel is 2 W. It is required to determine the distribution of the principal stresses along the center line of the weld.It is possible to assume without loss of generality that 1 H  .
The general structure of the characteristic net is shown in Figure 3.The characteristics of both families are straight in Region 1. Therefore, and  are constant.The edge is traction-free.
Therefore, 2 0   in Region 2.Then, it follows from ( 1) and ( 7) that in Region 1.The direction of the principal stress 1  dictates the orientation of the characteristic lines.In particular, the base  and   lines are shown in Figure 3.It is worthy of note that 0   on the base   line and 0   on the base  line.Choosing the Cartesian coordinate system shown in Figure 3   Consider Region 2. The   lines are straight in this region.To construct the characteristic field on the right to the base  and   lines employing the method developed in the previous section, one must find the radii of curvature of these lines.To this end, it is convenient to use a polar coordinate system   , r  .This coordinate system's origin is at point 1 O (Figures 3 and 4).Since the angle between the  and   lines equals 2 , the equation for a generic The general solution of this equation is It follows from ( 25) and ( 26) that the equation of the base  line is The radius of curvature of this line is determined from the following equation:   , (31), and (33) into (35) and integrating, one gets Region 2' (Figure 3) can be treated similarly.As a result, Here B  and B y are the   and y  coordinates of point B (Figure 3), respectively.
Consider Region 3 (Figures 3 and 5).The characteristics of both families are curved in this region.Therefore, equation ( 19) is valid.The equations in (19) can be rewritten as Each of these equations is integrated by the method of Riemann.In particular, the Green's function is where   0 Jz is the Bessel function of zero order.Note that Using ( 33) and (37), one can integrate the equations in (19) along the base characteristics to get where   1 J  is the Bessel function of the first order.Using (54), one can transform (53) to Since 0 ab  , it is convenient to rewrite (55) as     The y-coordinate of point P can be determined from the fourth equation in (22) as where ' A y is the y-coordinate of point ' A and is found from (20) as The value of ' A y is determined from the third equation in (22) at 0   .As a result, with the use of ( 20) and ( 33 Since   at the symmetry axis, the y-coordinate of point P at this axis is found from (57) as  is given in (34).The value of  at this point follows (20).Then, the stress 1  is determined from (7) and the stress 2  from (1).The continuity of the stresses across AOB and (24) require that   0 ln . 1 q q q     (62)


, measured from the axis anticlockwise.Then, the equations of the  and   lines are

Figure 2 .
Figure 2. Schematic diagram of the tensile welded joint.
is convenient.In this case, 0   in Region 1.

Figure 3 .
Figure 3.The general structure of the characteristic net.

Figure 4 .
Figure 4. Characteristic and polar coordinates in Region 2.

where 0 r and 0 
are constant.The base  line passes through point O.It is seen from the geometry of Figure 4 that the coordinates of this point are


28)Here the subscript 'b' emphasizes that it is the radius of curvature of the base  line.Moreover, ' the base  line, the second equation in(20) transforms to sin 2 .It is seen from Figure4that 0   at point A. Then, it follows from (30) and (31is the value of at point A. The y-coordinate of this point is determined from the third equation in(22).In particular,

Figure 5 .
Figure 5. Representation of characteristic curves for Riemann's method of integration.Assume one must find 0 S at point P (Figure5).Consider closed contour B'PA'OB'.The method of Riemann leads to

1 I
 are the modified Bessel functions of zero and first orders, respectively.
and (60), one can evaluate the integral involved in (61) numerically.This result and (59) provide the value of y at the symmetry axis.The parameter a varies in the range 0

Figure 6 . 1 
Figure 6.Distribution of the principal stress 1  along the center line of the welded joint.

Figure 7 . 2 
Figure 7. Distribution of the principal stress 2  along the center line of the welded joint.

Figures 6 and 7
Figures6 and 7illustrate the effect of on the distribution of the principal stresses along the center line.In these figures, s is the distance from the edge (Figure2).It is seen from Figure3that tan .sy 