Objective: Presentation of the theory of Cauchy fluxes in the - PowerPoint PPT Presentation

✬ ✩ 1 C AUCHY ’ S F LUX T HEOREM IN L IGHT OF G EOMETRIC I NTEGRATION T HEORY Guy Rodnay and Reuven Segev Department of Mechanical Engineering, Ben-Gurion University, Beer-Sheva, Israel ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 2 Objective: Presentation of the theory of Cauchy fluxes in the framework of geometric integration theory as formulated by H. Whitney and extended recently by J. Harrison. ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 3 Traditional Approach: In terms of scalar extensive property in space, one assumes: T (∂ A ) + S ( A ) = 0 Balance : � � Regularity : S ( A ) = A b A d v , and T (∂ A ) = ∂ A t A da Locality (pointwise) : b A ( p ) = b ( p ) , and t A ( p ) = t ( p , n ) Continuity : t ( · , n ) is continuous. Cauchy’s theorem asserts that t ( p , n ) depends linearly on n . There is a vector field τ such that t = τ · n . Considering smooth regions such that Gauss-Green Theorem may be applied, the balance may be written in the form of a differential equation as div τ + b = 0 . ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 4 Contributions in Continuum Mechanics - I Noll (1957): t ( n ) implied by local dependence on open sets of the boundary. Gurtin & Williams (1967): Interaction I ( A , B ) on a universe of bodies bi-additive: I ( A � B , C ) = I ( A , C ) + I ( B , C ) , bounded: | I ( A , B ) | ≤ l area (∂ A ∩ ∂ B ) + k volume ( A ) , Pairwise balanced: I ( A , B ) = − I ( B , A ) , Continuity: t ( · , n ) is continuous (omitted in later works). Continued later by Noll (1973,1986), Gurtin, Williams & Ziemer (1986), Noll & Virga (1988), etc. ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 5 Contributions in Continuum Mechanics - II Gurtin & Martins (1975): Relaxing the continuity of t ( p , n ) in p , proved linearity in n almost everywhere. ˇ Silhav´ y (1985,1991): Admissible bodies are sets of finite perimeter in E n , and the assumptions and results are assumed to hold for “almost every subbody”, in a way which allows singularities. The resulting flux vector t has an L p weak divergence. Degiovanni & Marzocchi & Musesti (1999) generalize ˇ Silhav´ y by considering fluxes which are only locally integrable. The field b = − div τ is meaningful only in the weak sense. Fosdick & Virga (1989) prove Cauchy’s theorem directly from an integral balance equation using a variational approach. Geometric measure theory is used for specifying the class of bodies, ✫ ✪ generalized definitions of n , generalized Gauss Theorem. Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 6 Previous work: Segev 1986, 1991 Stress theory for manifolds without a metric using a weak formulation. Stresses may be as irregular as measures. Works for continuum mechanics of any order. Segev 2000, Segev & Rodnay 1999: Classical Cauchy approach on general manifolds using differential forms. ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 7 The Proposed Formulation • Uses Geometric Integration Theory by Whitney (1947, 1957), Wolf (1948), and later Harrison (1993,1998), rather than Geometric Measure Theory (e.g., Federer, Fleming, de Giorgi). 1. Building blocks: r -dimensional oriented cells in E n . 2. Formal vector space of r -cells: polyhedral r -chains. 3. Complete w.r.t a norm: Banach space of r -chains. 4. Elements of the dual space: r -cochains. • Relevance to Continuum Mechaincs: – The total flux operator on regions is modelled mathematically by a cochain. – Cauchy’s flux theorem is implied by a representation theorem for cochains by forms. ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 8 Features of the Proposed Formulation • It offers a common point of view for the analysis of the following aspects: class of domains, integration, Stokes’ theorem, and fluxes. • Irregular domains and flux fields. Smoother fluxes allow less regular domains and vise versa. Examples: 1. Domains as irregular as Dirac measure and its derivatives—differentiable flux fields. 2. L 1 regions—bounded and measurable flux fields • Boundedness of flux operator is optimally associated with continuity: 1. Largest class of domains s.t. bounded fluxes are continuous. 2. Largest class of fluxes s.t. continuity implies boundedness. • Codimension not limited to 1. Allows membranes, strings, etc. Not only the boundary is irregular, but so is the domain itself. ✫ ✪ • Compatible with a formulation on general manifolds. Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 9 The Structure of the Presentation • Cells and polyhedral chains • Algebraic cochains • Norms and the complete spaces of chains (flat, sharp, natural) • The representation of cochains by forms: – Multivectors and forms – Integration – Representation – Coboundaries and balance equations ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 10 Cells and Polyhedral Chains ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 11 Oriented Cells • A cell , σ , is a non empty bounded subset of E n expressed as an intersection of a finite collection of half spaces. • The plane of σ is the smallest affine subspace containing σ . • The dimension of σ is the dimension of its plane. • An oriented r -cell is an r -cell with a choice of one of the two orientations of the vector space associated with its plane. • The orientation of σ ′ ∈ ∂σ is determined by the orientation of σ : 1. Choose independent (v 2 , . . . , v r ) in σ ′ . 2. Order them such that given v 1 in σ which points out at σ ′ , (v 1 , . . . , v r ) are positively oriented relative to σ . ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 12 Oriented Cells (Illustration) The plane of the cell An oriented 2-cell e 1 v 1 e 2 v 2 − σ + -oriented σ ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 13 Polyhedral Chains • A polyhedral r-chain in E n is an element of the vector space spanned by formal linear combinations of r -cells, together with: 1. The polyhedral chain 1 σ is identified with the cell σ . 2. We associate multiplication of a cell by − 1 with the operation of inversion of orientation, i.e., − 1 σ = − σ . 3. If σ is cut into σ 1 , . . . , σ m , then σ and σ 1 + . . . + σ m are identified. • The space of polyhedral r -chains in E n is now an infinite-dimensional vector space denoted by A r ( E n ) . • The boundary of a polyhedral r-chain A = � a i σ i is ∂ A = � a i ∂σ i . Note that ∂ is a linear operator A r ( E n ) − → A r − 1 ( E n ) . ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 14 Polyhedral Chains (Illustration - I) A ∂ A ∂ A = ∂ : A r → A r − 1 ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 15 A Polyhedral Chain as a Function A = � a i σ i ∂ A = � a i ∂σ i a · · · · · · σ 1 σ 2 ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 16 Total Fluxes as Cochains A cochain: Linear T : A r → R . Algebraic implications: • additivity, • interaction antisymmetry. σ 2 σ T · σ σ 1 T · ( − σ) σ 1 + σ 2 T · ( − σ) = − T · σ, T · (σ 1 + σ 2 ) = T · σ 1 + T · σ 2 ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 17 Norms and the Complete Spaces of Chains ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 18 The Norm Induced by Boundedness Boundedness: | T ∂ A | � N 2 | ∂ A | , | T ∂ A | � N 1 | A | . As a cochain: | T · A | � N 2 | A | , | T · ∂ D | � N 1 | D | , A ∈ A r , D ∈ A r + 1 . Thus, | T · A | = | T · A − T · ∂ D + T · ∂ D | � | T · A − T · ∂ D | + | T · ∂ D | � N 1 | A − ∂ D | + N 2 | D | � C T ( | A − ∂ D | + | D | ) , Continuity: Regard the flux as a continuous mapping of chains w.r.t. a norm | T · A | � C T � A � . Set: The flat norm (smallest) � A � = | A | ♭ = inf ✫ ✪ D {| A − ∂ D | + | D |} . Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 19 Flat Chains • The mass of a polyhedral r -chain A = � a i σ i is | A | = � | a i || σ i | . • The flat norm , | A | ♭ , of a polyhedral r -chain: | A | ♭ = inf {| A − ∂ D | + | D |} , using all polyhedral ( r + 1 ) -chains D . • Completing A r ( E n ) w.r.t the flat norm gives a Banach space denoted by A ♭ r ( E n ) , whose elements are flat r -chains in E n . • Flat chains may be used to represent continuous and smooth submanifolds of E n and even irregular surfaces. • The boundary of a flat ( r + 1 ) -chain A = lim ♭ A i , is the a flat r -chain ∂ A = lim ∂ A i . ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 20 Flat Chains, an Example (Illustration - I): A i A i L 1 i L 1 i D D d i d i L 2 i L 2 i L d i | A i | = 2 L , | A i | = 2 d i , | A i | ♭ � ( L + 2 ) d i → 0. | A i | ♭ � 2 d i → 0. ✫ ✪ Truesdell Memorial Symposium, June 2002 Rodnay & Segev

✬ ✩ 21 The Staircase: B 0 A 1 A 2 The dashed lines are for reference only. A 3 B 3 ✫ ✪ | A i | ♭ � 2 i − 1 2 − 2 i = 2 − i / 2 Truesdell Memorial Symposium, June 2002 Rodnay & Segev