Von Mises Failure Criterion in Mechanics of Materials- How to Eff

8/13/2019 Von Mises Failure Criterion in Mechanics of Materials- How to Eff

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University of Texas at El Paso

DigitalCommons@UTEP

Departmental Technical Reports (CS) Department of Computer Science

4-1-2007

Von Mises Failure Criterion in Mechanics ofMaterials: How to Eciently Use It Under Interval

and Fuzzy UncertaintyGang XiangAndrzej PownukUniversity of Texas at El Paso, [email protected]

Olga KoshelevaUniversity of Texas at El Paso, [email protected]

Sco A. StarksUniversity of Texas at El Paso, [email protected]

Follow this and additional works at: hp://digitalcommons.utep.edu/cs_techrep

Part of the Computer Engineering CommonsComments:Technical Report: UTEP-CS-07-24Published in: Marek Reformat and Michael R. Berthold (eds.),Proceedings of the 26th InternationalConference of the North American Fuzzy Information Processing Society NAFIPS'2007, San Diego,California, June 24-27, 2007, pp. 570-575.

is Article is brought to you for free and open access by the Department of Computer Science at DigitalCommons@UTEP. It has been accepted for

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[email protected].

Recommended CitationXiang, Gang; Pownuk, Andrzej; Kosheleva, Olga; and Starks, Sco A., "Von Mises Failure Criterion in Mechanics of Materials: How toEciently Use It Under Interval and Fuzzy Uncertainty" (2007).Departmental Technical Reports (CS). Paper 145.hp://digitalcommons.utep.edu/cs_techrep/145
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y f> y


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1 2 3

i

f

1 2 3

i

1 2 3 f(1, 2, 3)

f0

f(1, 2, 3)< f0

f(1, 2, 3) f0

f(1, 2, 3)

(1, 2, 3)

R3

S

S R3

f :R3 R f0

f(x) f0 x S f(x)< f0 x S

f

S

f(x) = 1

x S

f(x) = 0

x S

f0= 1

S

f(x)

f(1, 2, 3) i

f(1, 2, 3) = a0+3

i=1

ai i+3

i=1

3j=1

aij i j+ . . .

a0 ai aij

i

f(1, 2, 3)

f(1, 2, 3) = a0+3

i=1

ai i.


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ai a1 =a2= a3

f(1, 2, 3) = a0+a1 3

i=1 i,

f(1, 2, 3) = a0+ 3a1

1

3

3i=1

i

.

f f f0

1 = 2 = 3

1 = 2 = 3 f=a0+ 3a1 1

a1= 0 a1= a2= a3= 0

i

f ai = 0

f

f

f

f(1, 2, 3) = a0+3

i=1

3j=1

aij i j.

1 2 1 3

aii

a11 aij i =j

a12

f(1, 2, 2) = a0+a11 3

i=1

2i +a12 i=j

i j .

1= 2= 3 =

f = a0+ (3a11+ 6a12) 2

a11= 2a12 f=a0 a12 V

V(1, 2, 3)def= 221+2

22+2

23212222231=

(1 2)2 + (2 3)

2 + (3 1)2.

f

V

f f0 V V0

V0

V0

1= 0 2= 3= 0

1 f V = 221 V V =2f V0= 2

2f

V 22f

V def

= (1 2)2 + (1 3)

3 + (2 3)2.

1 2 3

V 22f V def

=(1 2)

2 + (1 3)3 + (2 3)

2.

V 22y


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i

i i

[i, i]

i

i [i, i]

V

V i V

[V , V]

V


6/7

{, +} x+idef= xi x

i

def= xi V

2n V(x11 , . . . , xnn )

[x1, x1] [xn, xn]

V 2n 2

V(x11 , . . . , xnn ) (1, . . . , n) (1, . . . , n) =

(+, . . . , +) (1, . . . , n) = (, . . . , )

(1, . . . , n) (1, . . . , n) = (+, . . . , +) (1, . . . , n) = (, . . . , )

n [x1, x1] [xn, xn]

V

(1, . . . , n)

V (x1, . . . , xn) [xi, xi] xi= xi

(x1, . . . , xi1, xi, xi+1, . . . , xn).

xi < xi i

i0 xi0 n xi xi0 (x1, . . . , xn) xi0

V

V

(x1, . . . , xn)

xi

V = 0

xi0 < xi0 xi

V >0 xi

xi0 E xi0 =xi0 xi0 < E

V = 1

n

ni=1

x2i E2.

V

xi0=

1

n (2xi0 2E) =

2

n (xi0 E).

xi0 < E xi0 < xi0 V V

V

(x1, . . . , xn) [xi, xi] xi = xi

(x1, . . . , xi1, xi, xi+1, . . . , xn).

xi < xi i

i0 xi0 n

xi xi

V = 0

xi0 > xi0 xi

V >0

xi

xi0 E

xi0 =xi0 E < xi0

V

xi0=

2

n (xi0 E)> 0 xi0 > E

xi0 > xi0 V

(1, . . . , n) (+, . . . , +) (, . . . , )

> 0

[xi, xi] = [1, 1+] i=

[xi, xi] = [1 , 1] i= +

xi 1

xi 1 + E=

x1+. . .+xnn

E (n 1) 1 + (1 +)

n =

n 2 +

n = 1

2

n .

2

n > 2 > n 2> (n + 1)

1 + xi

E > xi

V

xi=

2

n (xi E) xi

V

xi = xi xi

V

xi = xi

V

V 36 = 18 (2 + 3) 6 = 30

V


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V

(i j)2

i = i i = i

j

i j

3 4 = 12 3 4 = 12

612 = 12

(1 2)

2 (1 2)

2

(1 2)2

(1 2)2

(2 3)2

(2 3)2

(2 3)2

(2 3)2 (31)

2 (31)

2 (31)

2

(3 1)2

(1 2)2 + (2 3)

2 + (3 1)2;

(1 2)2 + (2 3)

2 + (3 1)2;

(1 2)2 + (2 3)

2 + (3 1)2;

(1 2)2 + (2 3)

2 + (3 1)2;

(1 2)2 + (2 3)

2 + (3 1)2;

(1 2)2 + (2 3)

2 + (3 1)2.

V

3 1 3 2 3< 1 3< 2

2 1 3 1

2 3 2 > 3 1 +2 +3 > 33 E > 3 3 = 1

V

1 2

V

(12)2

xi (3 1)2

(31)2

3 3 = 6 V

2< 1 3< 1

E < 1

1> 22

1+2+3>32 E > 2 E > 2 1 2 3

V = (1 2)2 + (2 3)

3 + (3 1)2,

Von Mises Failure Criterion in Mechanics of Materials- How to Eff

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