PSI - Issue 35

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David J. Unger et al. / Procedia Structural Integrity 35 (2022) 2–9 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Table 1. Cubic equations and their relationships to yield criteria through deviatoric stress invariants. Yield Condition X Y k

Cubic Equation 2 3 3 2 Y X X = − +

Tresca

J k J k J k J k J k

0 0.500 σ

3 3 2

J k

2 2 3 −

3 3

Drucker

3 2 J k J k

J k

0 0.540 σ

2 3 1 Y X = −

3 3

2 2

E1

0 0.542 σ

2 3 Y X X = −

2

−

3 3

2 2

E2

0 0.573 σ

2 3 6 Y X X = − +

2 2 3 −

3 3

von Mises

-

0 0.577 σ

1 X =

2 2

In Table 1, one finds four different yield conditions representable in the form of (4), plus the von Mises yield condition, Chakrabarty (1987), for comparison. The yield criteria tabulated as E1 and E2 are elliptic curves whose properties are investigated by the author. Their priority remains unknown to the author, but because of their simplicity, the author assumes none. In Table 1, the symbol 0 σ represents the yield strength in tension. In Fig. 1 (a), the various yield conditions presented in Table 1 are plotted in the XY plane. In Fig. 1 (b), the

a

b

Drucker

Y

4

3

σ /σ

E

1.0

von Mises (outer curve) Tresca Drucker (inner curve)

2

Tresca

0.5

1

E

X

-2

-1

0

2

3

0.0

σ /σ

X=1 Y=0 von Mises

-1

E (outer curve)

-0.5

E (inner curve)

-2

-3

-1.0

X=1

-4

-0.5

0.5

1.0

0.0

-1.0

Fig. 1. (a) various yield conditions in the XY plane; (b) yield conditions in the normalized principal stress plane.