some extensions and analysis of flux and stress theory
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Some Extensions and Analysis of Flux and Stress Theory Reuven Segev Department of Mechanical Engineering Ben-Gurion University Structures of the Mechanics of Complex Bodies October 2007 Centro di Ricerca Matematica, Ennio De Giorgi Scuola


  1. Some Extensions and Analysis of Flux and Stress Theory Reuven Segev Department of Mechanical Engineering Ben-Gurion University Structures of the Mechanics of Complex Bodies October 2007 Centro di Ricerca Matematica, Ennio De Giorgi Scuola Normale Superiore R. Segev ( Ben-Gurion Univ. ) 1 / 20 Flux and Stress Theories Pisa, Oct. 2007

  2. The Global Point of View C n -Functionals R. Segev ( Ben-Gurion Univ. ) 2 / 20 Flux and Stress Theories Pisa, Oct. 2007

  3. Review of Basic Kinematics and Statics on Manifolds is con- T κ Q The mechanical system is man- characterized by its κ configuration space—a manifold Q . ec- Velocities are tangent vectors to the manifold—elements of T Q . a- A Force at the configuration κ is a linear mapping mapping Q F : T κ Q → R . Can we apply this framework to Continuum Mechanics? R. Segev ( Ben-Gurion Univ. ) 3 / 20 Flux and Stress Theories Pisa, Oct. 2007

  4. Problems Associated with the Configuration Space in Continuum Mechanics What is a configuration? Does the configuration space have a structure of a manifold? The configuration space for continuum mechanics is infinite dimensional. R. Segev ( Ben-Gurion Univ. ) 4 / 20 Flux and Stress Theories Pisa, Oct. 2007

  5. Configurations of Bodies in Space A mapping of the body into space; material impenetrability—one-to-one; not “crash” volumes—invertible derivative. continuous deformation gradient (derivative); do not “crash” volumes—invertible derivative. κ κ ( B ) A body B Space ✪ U R. Segev ( Ben-Gurion Univ. ) 5 / 20 Flux and Stress Theories Pisa, Oct. 2007

  6. Manifold Structure for Euclidean Geometry If the body is a subset of R 3 and space is modeled by R 3 , the collection of differentiable mappings C 1 ( B , R 3 ) is a vector space However, the subset of “good” configurations is not a vector space, e.g., κ − κ = 0 —not one-to-one. We want to make sure that the subset of configurations Q is an open subset of C 1 ( B , R 3 ) , so it is a trivial manifold. configurations configurations C 1 ( B , R 3 ) C 1 ( B , R 3 ) all di c erentiable mappings all di c erentiable mappings R. Segev ( Ben-Gurion Univ. ) 6 / 20 Flux and Stress Theories Pisa, Oct. 2007

  7. The C 0 -Distance Between Functions The C 0 -distance between functions measures the maximum difference between functions. A configuration is arbitrarily close to a “bad” mapping. Space a configuration solid “bad mapping” dotted Body R. Segev ( Ben-Gurion Univ. ) 7 / 20 Flux and Stress Theories Pisa, Oct. 2007

  8. The C 1 -Distance Between Functions The C 1 distance between functions measures the maximum difference between functions and their derivative | u − v | C 1 = sup {| u ( x ) − v ( x ) | , | Du ( x ) − Dv ( x ) |} . A configuration is always a finite distance away from a “bad” mapping. Space a configuration solid “bad mapping” dotted Body R. Segev ( Ben-Gurion Univ. ) 8 / 20 Flux and Stress Theories Pisa, Oct. 2007

  9. Conclusions for R 3 If we use the C 1 -norm, the configuration space of a continuous body in space is an open subset of C 1 ( B , R 3 ) -the vector space of all u ( κ ( x )) = d κ ( x ) differentiable mapping. dt Q is a trivial infinite dimensional κ { B } manifold and its tangent space at any point may be identified with C 1 ( B , R 3 ) . A tangent vector is a velocity field. R. Segev ( Ben-Gurion Univ. ) 9 / 20 Flux and Stress Theories Pisa, Oct. 2007

  10. For Manifolds Both the body B and space U are differentiable manifolds. The configuration space is the collection Q = Emb ( B , U ) of the embeddings of the body in space. This is an open submanifold of the infinite dimensional manifold C 1 ( B , U ) . The tangent space T κ Q may be characterized as ∗ → | ◦ } T κ Q = C 1 ( κ ∗ T U ) . T κ Q = { w : B → T Q | τ ◦ w = κ } , or alternatively, κ T M κ ∗ ( T M ) T x M projection τ w x B a body κ M space manifold R. Segev ( Ben-Gurion Univ. ) 10 / 20 Flux and Stress Theories Pisa, Oct. 2007

  11. Representation of C 0 -Functionals by Integrals Assume you measure the size of a function using the C 0 -distance, � w � = sup {| w ( x ) |} . A linear functional F : w �→ F ( w ) is continuous with respect to this norm if F ( w ) → 0 when max | w ( x ) | → 0 . Riesz representation theorem: A continuous linear functional F with respect to the C 0 -norm may be represented by a unique measure µ in the form � F ( w ) = w d µ . B F ( w ) = � F ( w ) = � w φ dx w φ dx B B δ δ F isn’t sensitive to the derivative force density force density φ φ Velocity Velocity w w Body B Body B R. Segev ( Ben-Gurion Univ. ) 11 / 20 Flux and Stress Theories Pisa, Oct. 2007

  12. Representation of C 1 -Functionals by Integrals Now, you measure the size of a function using the C 1 -distance, � w � = sup {| w ( x ) | , | Dw ( x ) |} . A linear functional F : w �→ F ( w ) is continuous with respect to this norm if F ( w ) → 0 when both max | w ( x ) | → 0 and max | Dw ( x ) | → 0 . Representation theorem: A continuous linear functional F with respect to the C 1 -norm may be represented by measures σ 0 , σ 1 in the form � � F ( w ) = w d σ 0 + Dw d σ 1 . B B F is sensitive to the derivative F ( w ) = � φ 0 wdx + � φ 1 Dwdx B B δ “self” force density φ 0 stress density φ 1 Velocity w Body B ✪ Velocity gradient Dw R. Segev ( Ben-Gurion Univ. ) 12 / 20 Flux and Stress Theories Pisa, Oct. 2007

  13. Non-Uniqueness of C 1 -Representation by Integrals We had an expression in the form � � w ′ d σ 1 . F ( w ) = w d σ 0 + B B If we were allowed to vary w and w ′ independently, we could determine σ 0 and σ 1 uniquely. This cannot be done because of the condition w ′ = Dw . • F ( w ) = � φ 0 wdx + � φ 1 w ′ dx B B δ “self” force density φ 0 stress density φ 1 Velocity w Body B w ′ ✪ R. Segev ( Ben-Gurion Univ. ) 13 / 20 Flux and Stress Theories Pisa, Oct. 2007

  14. Unique Representation of a Force System Assume we have a force system, i.e., a force F P for every subbody P of B . We can approximate pairs of non-compatible functions w and w ′ , i.e., w ′ � = Dw , by piecewise compatible functions. w ′ � Dw approximation of � w B wd σ 0 Calculate � B wd σ 0 Body B P 1 P 2 . . . . . . B w ′ d σ 1 approximation of � w ′ B w ′ d σ 1 � Calculate P 1 P 2 Body B . . . . . . This way the two measures are determined uniquely. One needs consistency conditions for the force system. R. Segev ( Ben-Gurion Univ. ) 14 / 20 Flux and Stress Theories Pisa, Oct. 2007

  15. Generalized Cauchy Consistency Conditions P 1 • Additivity: P 2 F P 1 ∪ P 2 ( w | P 1 ∪ P 2 ) = F P 1 ( w | P 1 ) + F P 2 ( w | P 2 ) . • Continuity: If P i → A , then F P i ( w | P 1 ) P i converges and the limit depends on A only. A • Uniform Boundedness: There is a K > 0 such that for every subbody P and every w , | F P ( w | P ) ≤ K � w P � . Main Tool in Proof: Approximation of measurable sets by bodies with smooth boundaries. R. Segev ( Ben-Gurion Univ. ) 15 / 20 Flux and Stress Theories Pisa, Oct. 2007

  16. Generalizations All the above may be formulated and proved for differentiable manifolds. This formulation applies to continuum mechanics of order k > 1 (stress tensors of order k ). One should simply use the C k -norm instead of the C 1 -norm. The generalized Cauchy conditions also apply to continuum mechanics of order k > 1 . This is the only formulation of Cauchy conditions for higher order continuum mechanics. R. Segev ( Ben-Gurion Univ. ) 16 / 20 Flux and Stress Theories Pisa, Oct. 2007

  17. Locality and Continuity in Constitutive Theory R. Segev ( Ben-Gurion Univ. ) 17 / 20 Flux and Stress Theories Pisa, Oct. 2007

  18. Global Constitutive Relations (Elasticity for Simplicity) Q , the configuration space of a body B . C 0 � � R 3 , R 3 �� B , L , the collection of all stress fields over the body. Ψ : Q → C 0 � � R 3 , R 3 �� B , L , a global constitutive relation. � � �� space stress Ψ configuration stress field κ σ = Ψ ( κ ) Global constitutive relation. Body B Body B R. Segev ( Ben-Gurion Univ. ) 18 / 20 Flux and Stress Theories Pisa, Oct. 2007

  19. Locality and Materials of Grade- n Germ Locality: If two configurations κ 1 and κ 2 are equal on a subbody containing X , then the resulting stress fields are equal at X . space stress Ψ Ψ ( κ 1 ) κ 1 Ψ ( κ 2 ) κ 2 X X P P Body B Body B Material of Grade-n or n-Jet Locality: If the first n derivatives n n n of κ 1 and κ 2 are equal at X , then, Ψ ( κ 1 )( X ) = Ψ ( κ 2 )( X ) . (Elastic = grade 1.) X X X 1 2 space stress Ψ Ψ ( κ 1 ) κ 1 Ψ ( κ 2 ) κ 2 X X P P Body B Body B R. Segev ( Ben-Gurion Univ. ) 19 / 20 Flux and Stress Theories Pisa, Oct. 2007

  20. n -Jet Locality and Continuity Basic Theorem: If a constitutive relation Ψ : Q → C 0 � � R 3 , R 3 �� B , L is local and continuous with respect to the C n -norm, then, it is n -jet local. In particular, if Ψ is continuous with respect to the C 1 -topology, the material is elastic. space space space Whitney’s restriction extension κ ′ κ 1 | P 1 κ 1 κ ′ κ 2 | P 2 κ 2 X X X P P Body B Body B Body B Ψ Ψ stress stress Ψ ( κ ′ Ψ ( κ 1 ) 1 ) Ψ ( κ ′ Ψ ( κ 2 ) 2 ) X X P P Body B Body B R. Segev ( Ben-Gurion Univ. ) 20 / 20 Flux and Stress Theories Pisa, Oct. 2007

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