Basics of Mechanics of . . . Case of Larger Stress Ductile Materials and . . . Maxwell’s . . . Von Mises Failure Criterion Von Mises Criterion . . . in Mechanics of Materials: Computing V under . . . New Faster Algorithm . . . How to Efficiently Use it Practically Important . . . Faster Algorithm for . . . Under Interval and Fuzzy Faster Algorithm for . . . Uncertainty Acknowledgments Title Page Gang Xiang 1 , Andrzej Pownuk 2 , ◭◭ ◮◮ Olga Kosheleva 3 , and Scott A. Starks 3 ◭ ◮ Departments of 1 Computer Science, 2 Mathematics, Page 1 of 13 3 Electrical and Computer Engineering University of Texas at El Paso Go Back El Paso, Texas 79968, USA Full Screen gxiang@utep.edu, sstarks@utep.edu Close Quit
Basics of Mechanics of . . . Case of Larger Stress 1. Outline Ductile Materials and . . . • One of the main objective of mechanics of materials Maxwell’s . . . is to predict when the material experiences fracture Von Mises Criterion . . . (fails), and to prevent this failure. Computing V under . . . • With this objective in mind, it is desirable to use duc- New Faster Algorithm . . . tile materials, i.e., materials which can sustain large Practically Important . . . deformations without failure. Faster Algorithm for . . . Faster Algorithm for . . . • Von Mises criterion enables us to predict the failure of Acknowledgments such ductile materials. Title Page • To apply this criterion, we need to know the exact ◭◭ ◮◮ stresses applied at different directions. ◭ ◮ • In practice, we only know these stresses with interval or fuzzy uncertainty. Page 2 of 13 Go Back • In this paper, we describe how we can apply this cri- terion under such uncertainty, and how to make this Full Screen application computationally efficient. Close Quit
Basics of Mechanics of . . . Case of Larger Stress 2. Basics of Mechanics of Materials: the Notion of Stress and Case of Small Stress Ductile Materials and . . . Maxwell’s . . . • When a force is applied to a material, this material Von Mises Criterion . . . deforms and at some point breaks down. Computing V under . . . New Faster Algorithm . . . • We can gauge the effect of the force by the stress , the force per unit area. Practically Important . . . Faster Algorithm for . . . • The larger the stress, the larger the deformation. Faster Algorithm for . . . • At some point, larger stress leads to a breakdown. Acknowledgments Title Page • When the stress is small, no irreversible damage occurs, all deformations are reversible . ◭◭ ◮◮ ◭ ◮ • Thus, under small stress, the material returns to its original shape once the force is no longer applied. Page 3 of 13 • Such reversible deformation (which returns to the orig- Go Back inal shape) is called elastic . Full Screen Close Quit
Basics of Mechanics of . . . Case of Larger Stress 3. Case of Larger Stress Ductile Materials and . . . • An increased level of stress causes irreversible damage. Maxwell’s . . . Von Mises Criterion . . . • In this case, after the force is no longer applied, the material does not return to its original shape. Computing V under . . . New Faster Algorithm . . . • Irreversible deformation is called plastic or yielding . Practically Important . . . • As the stress increases, the material experiences frac- Faster Algorithm for . . . tures. Faster Algorithm for . . . • Often, the fractured material can no longer fulfil its Acknowledgments duties, it fails . Title Page • Material failure can have catastrophic consequences. ◭◭ ◮◮ • As a result, it is extremely important to predict when ◭ ◮ a material fails. Page 4 of 13 • It is also important to know when the yielding starts, Go Back because the irreversible damage can lead to a failure in Full Screen the long run. Close Quit
Basics of Mechanics of . . . Case of Larger Stress 4. Ductile Materials and their Practical Importance Ductile Materials and . . . • It is desirable to use ductile materials which can sustain Maxwell’s . . . large deformations without failure. Von Mises Criterion . . . • When the force is only applied in one direction, the Computing V under . . . yielding and the failure start when the stress becomes New Faster Algorithm . . . large enough: Practically Important . . . Faster Algorithm for . . . – there is a threshold σ y after which yielding starts; Faster Algorithm for . . . – there is a threshold σ f > σ y after which the mate- Acknowledgments rial fails. Title Page • In real life, we often have a combination of stresses ◭◭ ◮◮ coming from different directions. ◭ ◮ • It is desirable: Page 5 of 13 – given the three stresses σ 1 , σ 2 , and σ 3 applied at three orthogonal directions, Go Back – to predict when a ductile material fails under these Full Screen stresses. Close Quit
Basics of Mechanics of . . . Case of Larger Stress 5. Maxwell’s Mathematical Solution to the Problem Ductile Materials and . . . • Main idea: combine σ i into a numerical criterion f ( σ 1 , σ 2 , σ 3 ) Maxwell’s . . . so that the material fails if f ≥ f 0 . Von Mises Criterion . . . • A function f can be expanded into Taylor series: Computing V under . . . New Faster Algorithm . . . 3 3 3 � � � f ( σ 1 , σ 2 , σ 3 ) = a 0 + a i · σ i + a ij · σ i · σ j + . . . Practically Important . . . i =1 i =1 j =1 Faster Algorithm for . . . • Due to symmetry, a i = a 1 , a ii = a 11 , and a ij = a 12 . Faster Algorithm for . . . Acknowledgments • Isotropic stress σ 1 = σ 2 = σ 3 leads to isotropic defor- Title Page mations – no fractures (we cannot get f > f 0 ). ◭◭ ◮◮ • Thus, a 1 = 0, a 11 = − 2 a 12 , and f = a 0 − a 12 · V , where ◭ ◮ = ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 . def V ( σ 1 , σ 2 , σ 3 ) Page 6 of 13 • Since f linearly depends on V , the condition f ≥ f 0 is Go Back equivalent to V ≥ V 0 for some V 0 . • For 1-D stress, V = 2 σ 2 1 , hence V 0 = 2 σ 2 Full Screen f . Close Quit
Basics of Mechanics of . . . Case of Larger Stress 6. Von Mises Criterion and Its Practical Use Ductile Materials and . . . • In 1913, von Mises experimentally confirmed Maxwell’s Maxwell’s . . . formula: failure when V ≥ 2 σ 2 f . Von Mises Criterion . . . • Problem: we only know the σ i with uncertainty. Computing V under . . . New Faster Algorithm . . . • Case of interval uncertainty: we only know the bounds Practically Important . . . σ i and σ i on σ i . Faster Algorithm for . . . • Criterion: no-failure is guaranteed if V < 2 σ 2 f , where Faster Algorithm for . . . V is the largest possible value of V . Acknowledgments • Case of fuzzy uncertainty: we have fuzzy numbers cor- Title Page responding to σ i . ◭◭ ◮◮ • Fact: with certainty α , the material does not fail if the ◭ ◮ α -cut of V is below 2 σ 2 f . Page 7 of 13 • Reduction to interval case: the α -cut for V can be com- Go Back puted based on interval α -cuts for σ i . Full Screen • Conclusion: we must be able to compute V . Close Quit
Basics of Mechanics of . . . Case of Larger Stress 7. Computing V under Interval Uncertainty Ductile Materials and . . . • Observation: V = ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 Maxwell’s . . . is proportional to the sample variance of σ i . Von Mises Criterion . . . Computing V under . . . • Known fact: variance is a convex function on the box [ σ 1 , σ 1 ] × [ σ 2 , σ 2 ] × [ σ 3 , σ 3 ] . New Faster Algorithm . . . Practically Important . . . • Conclusion: V is attained at one of the vertices. Faster Algorithm for . . . • Computation: to compute V , it is sufficient to compute Faster Algorithm for . . . V for 2 3 = 8 combinations of σ i and σ i . Acknowledgments Title Page • Each computation of V requires: ◭◭ ◮◮ • 3 subtractions (to compute σ i − σ j ), ◭ ◮ • 3 multiplications (to compute the squares), and • 2 additions (to compute V ), Page 8 of 13 • Total: 3 · 8 = 24 multiplications, (2 + 3) · 8 = 40 ± s. Go Back Full Screen • Problem: can we compute V faster? Close Quit
Basics of Mechanics of . . . Case of Larger Stress 8. New Faster Algorithm for Computing V Ductile Materials and . . . • Idea: maximum is never attained on all σ i or on all σ i . Maxwell’s . . . Von Mises Criterion . . . • Algorithm with 12 mult. and 24 ± s: first, compute Computing V under . . . ( σ 1 − σ 2 ) 2 , ( σ 1 − σ 2 ) 2 , ( σ 1 − σ 2 ) 2 , ( σ 1 − σ 2 ) 2 ; New Faster Algorithm . . . ( σ 2 − σ 3 ) 2 , ( σ 2 − σ 3 ) 2 , ( σ 2 − σ 3 ) 2 , ( σ 2 − σ 3 ) 2 ; Practically Important . . . Faster Algorithm for . . . ( σ 3 − σ 1 ) 2 , ( σ 3 − σ 1 ) 2 , ( σ 3 − σ 1 ) 2 , ( σ 3 − σ 1 ) 2 . Faster Algorithm for . . . • Compute V as the largest of the 6 sums Acknowledgments ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 ; Title Page ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 ; ◭◭ ◮◮ ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 ; ◭ ◮ ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 ; Page 9 of 13 ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 ; Go Back ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 . Full Screen Close Quit
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