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Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 13 Go Back Full Screen Close Quit

Von Mises Failure Criterion in Mechanics of Materials: How to Efficiently Use it Under Interval and Fuzzy Uncertainty

Gang Xiang1, Andrzej Pownuk2, Olga Kosheleva3, and Scott A. Starks3

Departments of 1Computer Science, 2Mathematics,

3Electrical and Computer Engineering

University of Texas at El Paso El Paso, Texas 79968, USA gxiang@utep.edu, sstarks@utep.edu

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 13 Go Back Full Screen Close Quit

1. Outline

• One of the main objective of mechanics of materials

is to predict when the material experiences fracture (fails), and to prevent this failure.

• With this objective in mind, it is desirable to use duc-

tile materials, i.e., materials which can sustain large deformations without failure.

• Von Mises criterion enables us to predict the failure of

such ductile materials.

• To apply this criterion, we need to know the exact

stresses applied at different directions.

• In practice, we only know these stresses with interval
• r fuzzy uncertainty.
• In this paper, we describe how we can apply this cri-

terion under such uncertainty, and how to make this application computationally efficient.

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 13 Go Back Full Screen Close Quit

2. Basics of Mechanics of Materials: the Notion of Stress and Case of Small Stress

• When a force is applied to a material, this material

deforms and at some point breaks down.

• We can gauge the effect of the force by the stress, the

force per unit area.

• The larger the stress, the larger the deformation.
• At some point, larger stress leads to a breakdown.
• When the stress is small, no irreversible damage occurs,

all deformations are reversible.

• Thus, under small stress, the material returns to its
• riginal shape once the force is no longer applied.
• Such reversible deformation (which returns to the orig-

inal shape) is called elastic.

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 13 Go Back Full Screen Close Quit

3. Case of Larger Stress

• An increased level of stress causes irreversible damage.
• In this case, after the force is no longer applied, the

material does not return to its original shape.

• Irreversible deformation is called plastic or yielding.
• As the stress increases, the material experiences frac-

tures.

• Often, the fractured material can no longer fulfil its

duties, it fails.

• Material failure can have catastrophic consequences.
• As a result, it is extremely important to predict when

a material fails.

• It is also important to know when the yielding starts,

because the irreversible damage can lead to a failure in the long run.

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 13 Go Back Full Screen Close Quit

4. Ductile Materials and their Practical Importance

• It is desirable to use ductile materials which can sustain

large deformations without failure.

• When the force is only applied in one direction, the

yielding and the failure start when the stress becomes large enough: – there is a threshold σy after which yielding starts; – there is a threshold σf > σy after which the mate- rial fails.

• In real life, we often have a combination of stresses

coming from different directions.

• It is desirable:

– given the three stresses σ1, σ2, and σ3 applied at three orthogonal directions, – to predict when a ductile material fails under these stresses.

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 13 Go Back Full Screen Close Quit

5. Maxwell’s Mathematical Solution to the Problem

• Main idea: combine σi into a numerical criterion f(σ1, σ2, σ3)

so that the material fails if f ≥ f0.

• A function f can be expanded into Taylor series:

f(σ1, σ2, σ3) = a0 +

3

• i=1

ai · σi +

3

• i=1

3

• j=1

aij · σi · σj + . . .

• Due to symmetry, ai = a1, aii = a11, and aij = a12.
• Isotropic stress σ1 = σ2 = σ3 leads to isotropic defor-

mations – no fractures (we cannot get f > f0).

• Thus, a1 = 0, a11 = −2a12, and f = a0 − a12 · V , where

V (σ1, σ2, σ3)

def

= (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2.

• Since f linearly depends on V , the condition f ≥ f0 is

equivalent to V ≥ V0 for some V0.

• For 1-D stress, V = 2σ2

1, hence V0 = 2σ2 f.

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 13 Go Back Full Screen Close Quit

6. Von Mises Criterion and Its Practical Use

• In 1913, von Mises experimentally confirmed Maxwell’s

formula: failure when V ≥ 2σ2

f.

• Problem: we only know the σi with uncertainty.
• Case of interval uncertainty: we only know the bounds

σi and σi on σi.

• Criterion: no-failure is guaranteed if V < 2σ2

f, where

V is the largest possible value of V .

• Case of fuzzy uncertainty: we have fuzzy numbers cor-

responding to σi.

• Fact: with certainty α, the material does not fail if the

α-cut of V is below 2σ2

f.

• Reduction to interval case: the α-cut for V can be com-

puted based on interval α-cuts for σi.

• Conclusion: we must be able to compute V .

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 13 Go Back Full Screen Close Quit

7. Computing V under Interval Uncertainty

• Observation: V = (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2

is proportional to the sample variance of σi.

• Known fact: variance is a convex function on the box

[σ1, σ1] × [σ2, σ2] × [σ3, σ3].

• Conclusion: V is attained at one of the vertices.
• Computation: to compute V , it is sufficient to compute

V for 23 = 8 combinations of σi and σi.

• Each computation of V requires:
• 3 subtractions (to compute σi − σj),
• 3 multiplications (to compute the squares), and
• 2 additions (to compute V ),
• Total: 3 · 8 = 24 multiplications, (2 + 3) · 8 = 40 ±s.
• Problem: can we compute V faster?

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 13 Go Back Full Screen Close Quit

8. New Faster Algorithm for Computing V

• Idea: maximum is never attained on all σi or on all σi.
• Algorithm with 12 mult. and 24 ±s: first, compute

(σ1 − σ2)2, (σ1 − σ2)2, (σ1 − σ2)2, (σ1 − σ2)2; (σ2 − σ3)2, (σ2 − σ3)2, (σ2 − σ3)2, (σ2 − σ3)2; (σ3 − σ1)2, (σ3 − σ1)2, (σ3 − σ1)2, (σ3 − σ1)2.

• Compute V as the largest of the 6 sums

(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2; (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2; (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2; (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2; (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2; (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2.

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 13 Go Back Full Screen Close Quit

9. Practically Important Situation: Special Shapes

• General case: stresses can be of the same size.
• In practice: we often have a linear or a planar shape.
• In such cases: stresses in the direction of the shape are

usually much larger than in the other directions.

• Planar shape: we have σ3 ≪ σ1 and σ3 ≪ σ2.
• Reformulation in precise terms: we have

σ3 < σ1 and σ3 < σ2.

• Linear shape: we have σ2 ≪ σ1 and σ3 ≪ σ1.
• Reformulation in precise terms: we have

σ2 < 1 2 · σ1 and σ3 < 1 2 · σ1.

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 13 Go Back Full Screen Close Quit

10. Faster Algorithm for the Planar Case

• Can be proven: maximum of V is attained when

σ3 = σ3.

• Algorithm: Compute

(σ1 − σ2)2, (σ1 − σ2)2, (σ1 − σ2)2; (σ1 − σ3)2, (σ1 − σ3)2; (σ2 − σ3)2, (σ2 − σ3)2.

• Compute V as the largest of the 3 sums

(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2; (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2; (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2.

• Overall, we need 7 multiplications and 13 additions and

substractions.

Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 13 Go Back Full Screen Close Quit

11. Faster Algorithm for the Linear Case

• Problem (reminder): compute the range of

V = (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2.

• Linear case (reminder): σ2 ≪ σ1 and σ3 ≪ σ1, i.e.,

σ2 < 1 2 · σ1 and σ3 < 1 2 · σ1.

• Can be proven: maximum is attained when σ1 = σ1,

σ2 = σ2, and σ3 = σ3.

• Algorithm: Compute

V = (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2.

• Computation time: we need 3 multiplications and 5

additions and substractions.

Outline Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Criterion . . . Computing V under . . . New Faster Algorithm . . . Practically Important . . . Faster Algorithm for . . . Faster Algorithm for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 13 of 13 Go Back Full Screen Close Quit

12. Acknowledgments This work was supported in part:

• by the University Research Institute grant from the

University of Texas at El Paso,

• a Teacher Quality grant from the University of Texas at

Austin Charles E. Dana Center and Texas Education Agency, and

• the Texas Department of Transportation grant No. 0-

5453. The authors are thankful to the anonymous referees for important suggestions.