The modified generalized Tresca yield condition with application to mode I perfectly plastic crack problems

Basic properties of a newly defined yield criterion are explored and then applied to plane stress, mode I, perfectly plastic crack problems. This yield condition began as a perturbation of the traditional Tresca yield condition. In a previous study, a relationship was found for a yield criterion that spanned the Tresca to the von Mises yield criteria in the principal stress plane. In this study, a distinct but related yield criterion is used for those cases where the yield criterion lies on or outside the von Mises yield condition in the principal stress plane. For these cases, the associated mathematics becomes more complicated than those cases where the yield condition lies within or on the von Mises yield condition in the principal stress plane. The perfectly plastic mode I crack problem is solved for this modified version of the generalized Tresca yield condition. It is found that the maximum normal stress for this yield condition is higher than that for the equivalent mode I crack problem under the von Mises yield condition. Material parameters associated with the generalized Tresca yield condition are found for both the BCC and FCC crystal structures based on models proposed previously in the literature.


Introduction
In [1], a yield condition was defined that was based on a perturbation of the traditional Tresca yield condition. As such, it was referred to as the generalized Tresca yield condition. This yield condition was found by perturbing a relationship that defines the Tresca yield condition in terms of its deviatoric stress invariants (J 2 , J 3 ) . The particular representation of the Tresca yield condition used for this perturbation is referred to as Weierstrass form. In general, Weierstrass form is a cubic algebraic relationship that is associated with elliptic curves [2]. Elliptic curves are known for their special group properties within mathematical circles [3].
The Weierstrass form in Cartesian coordinates (X, Y ) is commonly defined by the upper sign in the following expression: where c 1 and c 2 are constants. The upper sign in (1) was used in the derivation of the generalized Tresca yield condition [1,4]. In that case, a yield criterion was generated that lies between the Tresca and von Mises yield conditions in the principal stress plane, with the exception of six points, which are common to both yield criteria. In [4], it was noted that replacing the positive sign in (1) with the negative sign generates a yield condition that lies on or outside the von Mises yield condition. Because this relationship can be obtained by replacing Y by iY in (1) under the positive sign, it was named the modified generalized Tresca yield condition. This naming convention was chosen because it is commonly adopted for mathematical functions that are generated by replacing a real argument by an imaginary argument.
For both cases of generalized Tresca yield criteria, the relationships between the Cartesian coordinates in (1) and the deviatoric stress invariants are where J 2 and J 3 are the second and third invariants of the deviatoric stress tensor [5], respectively. In (2), the symbols (k, ε) are parameters that are specific to each individual yield criteria. In terms of the first σ 1 and second σ 2 principal stresses, the deviatoric stress invariants in (2) assume the following forms for plane stress loading conditions In [1,4] it was determined that the generalized Tresca yield condition has the form by adopting the positive sign in (1). Similarly, by assigning a negative value to the ambiguous sign in (1), one finds which defines the modified generalized Tresca yield condition [4]. The lower limit on ε in (5) is a consequence of a material stability requirement [4]. As they stand, both (4) and (5) apply to plane stress problems as the third principal stress has been set equal to zero in the derivations. Note that the simple change of sign in (1) alters the form of the yield condition considerably. For the generalized Tresca yield condition, the parameters appearing in (4) assume the following relationships with the yield strength is tension σ 0 and the yield strength in pure shear τ 0 [4] Similarly, for the modified generalized Tresca yield condition, the corresponding relationships are [4] A plot of representative loci of the three types of yield criteria in the normalized principal stress plane is shown in Fig. 1. The generalized Tresca yield condition (4) is marked by the symbol ε = 1 and is shown in blue. The modified generalized Tresca yield condition (5) is labeled as ε = −1 and is plotted in black. The von Mises yield condition is designated under that name and is drawn in red. The maximum possible normal stress σ max for the generalized Tresca yield condition was derived in [4] as

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The modified generalized Tresca yield condition Page 3 of 11 1 By a similar process of setting σ 1 = σ max and σ 2 = σ max /2 in (5), the maximum possible normal stress for the modified generalized Tresca yield condition is By taking limits as ε → 16, both the modified and unmodified versions of the generalized Tresca yield condition reduce to von Mises yield condition relationships. For ε = 0, relationships for the generalized Tresca yield condition [4] assume the standard forms of the traditional Tresca yield condition.

Analytical and numerical solutions for plane stress loading conditions
In [4], a general solution for the generalized Tresca yield condition was developed analytically. Here, the analogous solution for the modified generalized Tresca yield condition will be presented. As in [4], a stress function φ is introduced here that is analogous to an Airy stress function [6] in its relationship to derived stresses in polar coordinates (r, θ) . For mode I perfectly plastic crack problem [7], the stresses are independent of the radial coordinate r, i.e., where f (θ) is an arbitrary function of the transverse angle θ, σ θ and σ r are normal stresses, and τ rθ is the shear stress. The primes on f (θ) denote the first and second derivatives of this function with respect to the angle θ. Now the principal stresses are related to those in polar form by A notation, introduced in [1,4,8-10], will also be employed here With these substitutions (11)-(13), the modified generalized Tresca yield condition (5) assumes the following form: where Ordinary differential Eq. (14) belongs to the Clairaut classification of nonlinear equations [11,12]. A technique used to solve analytically a similar equation for the generalized Tresca yield condition is given in [4]. Using an identical technique here gives the solution as The constant q c of (16) and (17) is a particular value of the variable q of (14) for a specific solution. The angle α in (16)  Each individual value of q c of the general solution produces an ellipse in the phase plane as shown in Fig. 2. The limits on this parameter are −2 ≤ q c /σ 0 ≤ 2 in order to avoid violating the yield condition. The choice of phase angle α is arbitrary here, and it affects neither the size nor shape of the curves shown in Fig. 2.
The envelope of the ellipses shown in Fig. 2 constitute the locus of the singular solution [11,12] of the differential equation. This solution cannot be obtained directly from the general solution by selecting particular values of the constants q c and α. Instead, a technique used previously to solve the singular solution of the generalized Tresca yield condition in [4] will be employed.
This method relies on the principal of duality [11]. It is also known as a contact transformation. The particular transformation chosen is tabulated as (41.4) of [13] where the three contact variables are x, y, and dy/dx. By substituting (18) into (14), the variable A, defined in (15), reduces to just -y. This characteristic transforms the differential equation into the algebraic This attribute is a consequence of (14) being of Clairaut form. Now to introduce the locus of the envelope of ellipses shown in Fig. 2 into the formulation, one differentiates (19) with respect to x under the assumption y = y (x) to obtain Upon introduction of the following inverse contact transformation in (19) and (20) a system of differential algebraic equations is generated that governs the singular solution of (14), so easily accomplished here. The highest order of q found in (22) is six and the highest order of q found in (23) is five. In addition, the subroutine Eliminate in the symbolic computer program Mathematica was unable to eliminate q from this system of equations within a reasonable amount of computational time.
In order to circumvent this, q was found from (23), using the root finding program, FindRoot, provided in Mathematica . Pairs of data points (f, p) were generated from the yield condition (22) in this manner and then plotted using the subroutine ContourPlot of Mathematica . Two distinct roots of the five generated solving (23) were necessary to obtain one quarter of the locus plotted in Fig. 3 for ε = −1, where k = σ 0 / √ 3 by the first equality in (7). The remainder of the phase plane was then completed using symmetry. Numerical data were then extracted from this graph to find an interpolating function representing p (f ) . Using this relationship, a first-order differential equation was then defined to obtain the primitive An initial condition, θ (0) = 0, was used to solve (24) numerically using the subroutine NDSolve of Mathematica . An interpolating function for the relationship θ (f ) was generated by the software. By plotting this interpolating function, the period of the function could be determined numerically for a specific value of ε. Data points from this interpolating function were then used to generate an interpolating function for the inverse function. Using this procedure an interpolating function of the form f (θ) was determined for the singular solution. One quarter of the period of f (θ) for ε = −1 was found to be approximately 1.65.
In the phase plane, Fig. 3, three singular solution loci are depicted. The von Mises yield criterion is labeled by that name and is represented by the red curve [4,9]. The generalized Tresca yield condition is labeled by ε = 1 and is shown as the blue curve [4]. The modified generalized Tresca yield condition   [4] is labeled by ε = −1 and is shown as the black curve. Note that the curve labeled as ε = −1 in Fig. 3 matches the size and shape of the locus of the envelope of ellipses shown in Fig. 2 for the same value of ε. This correlation supports the validity of the technique used for obtaining the singular solution.

Plane stress mode I perfectly plastic crack problem
The coordinate system used in this analysis for the mode I crack problem is shown in Fig. 4. Only the upper half plane needs to be considered due to symmetry. The original plane stress, mode I, perfectly plastic solution was given in [14] under the von Mises yield condition. Three distinct regions in the upper half plane were indicated. They are divided by angles labeled as θ 12 and θ 23 in Fig. 4. The solution of [4,14] is plotted in Fig. 5, where the specific angles dividing sectors are listed in Table 1 under the name von Mises. In the original analysis [14], the crack was oriented to the left along line OD instead of the right OA as shown in Fig. 5. The singular solution governs region 3 in Fig. 4, which is located ahead of the crack tip. The two other mode I solutions shown in Fig. 5 are for the generalized Tresca yield condition, ε = 1, and for the modified generalized Tresca yield condition, ε = −1. Details of the solution for ε = 1 and also for the von Mises solution are provided in [4]. Values of the angles separating the three sectors are given in Table 1 under the entrees for von Mises and ε = 1.
For the modified generalized Tresca yield condition, the solution in region 1 is uniaxial compression which is also true for the other two yield criteria as well. Note that solution (25) is a special case of the solution defined by (14) and (16). This solution is shown as the heavy blue curve on the phase plane in Fig. 2.
The solution for the second region for the modified generalized Tresca yield condition is obtained from (16) by using relationships (11). This solution is shown as the heavy green curve on the phase plane in Fig. 2. The third region requires use of the interpolating function of the singular solution described in Sect. 2 of this article. The maximum value of this function occurs ahead of the crack tip where θ = π and can be determined as where σ max is determined for a particular value of ε and σ 0 from (9). An adjustment of the angle θ as determined in Sect. 2 is necessary so that the maximum value of the interpolating function coincides with the angle θ = π and with the corresponding value of f (θ) given in (26). This solution is shown as the heavy red curve on the phase plane in Fig. 2. The angles θ 12 and θ 23 must be determined numerically so that the functions σ θ and τ rθ agree at the boundaries between adjacent sectors. The subroutine FindRoot of Mathematica was used for this purpose. These angles are given in Table 1 for the entry ε = −1.

Closing remarks
For a yield criterion that lies outside the von Mises locus, except for six common points, a representative value of the modified version of the generalized Tresca yield condition was chosen as ε = −1. The greatest qualitative difference among the various mode I crack solutions involving generalized Tresca yield criteria is the stress fields for σ r . This can be observed in Fig. 5 just before the angle θ = π.
One of the main features of generalized Tresca yield criteria is that analytical general solutions can be obtained for mode I perfectly plastic solutions under plane stress loading conditions. In [4], a spectrum of solutions was plotted between the Tresca and von Mises yield criteria spanning ε = 0 to ε = 16. In the case of the generalized Tresca yield condition, a plot of the yield criterion is shown in the normalized principal plane in Fig. 6 for the case ε = 2. This locus is compared to the data points provided in [15] for the BCC crystal structure. A good correlation exists among the data points and the yield condition (Fig. 6).  For comparison, a very similar correlation was made using the same data points [15] to those of a different yield criterion given in [16] as where n was determined as 1.6 for the best fit to the data. Although excellent agreement also exists among the data points and this prescribed yield criterion [16], it is doubtful that (27) can be solved analytically to obtain a general solution for a mode I problem similar to [4] due to the fractional powers of the principal stresses alone. A correlation between the generalized Tresca yield condition and FCC materials [17,18] is shown in Fig. 7 for ε = 0.6367. This curve represents an arbitrary FCC material [18] in the Π-plane, see Fig.  6.3.5 in [19] for an expanded view of this plane in the Haigh-Westergaard principal stress space. It is virtually indistinguishable from the plot provided in [18] based on derived theory. Note that in [18] it was mentioned that τ 0 /σ 0 = 0.5396, which was used here to determine the appropriate value of ε. It is curious to note that this value also agrees to four decimal places with the ratio τ 0 /σ 0 as determined under the Drucker yield condition [9].
In summary, this article completes the study of the behavior exhibited by mode I crack solutions under plane stress loading conditions for the two distinct cases of generalized Tresca yield criteria for perfectly plastic materials.

Author contributions
The author is the sole contributor to this article including the preparation of figures and table.
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