GENERALIZED STRESS CONCENTRATION FACTORS Reuven Segev Department of Mechanical Engineering Ben-Gurion University, Beer-Sheva, Israel ICTAM2004, Warsaw
✬ ✩ 2 Stress Concentration for Engineers ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 3 Generalized Stress Stress Concentration Factors: • Assume a body Ω is given (open, regular with smooth boundary). • Assume a force F is given in terms of a body force b and a surface force t and let σ be a stress field that is in equilibrium with F . • The stress concentration factor associated with the pair F , σ is ess sup x {| σ( x ) |} K F ,σ = x ∈ Ω, y ∈ ∂Ω. ess sup x , y {| b ( x ) | , | t ( y ) |} , • Denote by � F the collection of all possible stress fields that are in equilibrium with F . (There are many such stress fields because material properties are not specified.) ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 4 • The optimal stress concentration factor for the force F is defined by � � K F = inf K F ,σ . σ ∈ � F • The generalized stress concentration factor K —a purely geometric property of Ω —is defined by � � ess sup x {| σ( x ) |} K = sup { K F } = sup inf . ess sup x , y {| b ( x ) | , | t ( y ) |} σ ∈ � F F F Result: K = � δ � , where, δ is a mapping that extends vector fields from the interior of the body to its closure and is defined on Sobolev spaces or the related L D -spaces. ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 5 First case: General mechanics – simpler mathematics • Forces may have non-vanishing resultants and total torques. • The stress object contains a tensor field σ i j and a self force field σ 0 i . • The principle of virtual work is of the form � � � � b i w i d V + t i w i d A = σ 0 i w i d V + σ i j w i , j d V . Ω ∂Ω Ω Ω • The stress tensor σ i j need not be symmetric. • With the self force field � � � � � | σ 0 i ( x ) | , �� ess sup i , j , k , x � σ jk ( x ) K = sup inf . ess sup i , x {| b i ( x ) | , | t i ( x ) |} σ ∈ � F F ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 6 Forces and Stresses as Linear Functionals A Force: A linear functional (power functional) on virtual velocity fields, � � F (w) = b i w i d V + t i w i d A . Ω ∂Ω A Stress: A linear functional (power functional) on the space of tensor fields (gradients of virtual velocity fields) � σ(χ) = σ i j χ i j d V . Ω We will generalize stresses to include self-forces so � � σ(χ) = σ 0 i χ i d V + σ i j χ i j d V . Ω Ω Equilibrium: F (w) = σ(χ) if χ i = w i and χ i j = w i , j . ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 7 The L 1 and L ∞ -Norms and Their Duality Objective: The maximal absolute value of a stress component will be the magnitude or norm of the stress � σ � L ∞ = ess sup � � � | σ 0 i ( x ) | , �� � σ jk ( x ) . i , j , k , x Duality: If we use the L 1 -norm on the space of “local deformations” { χ = (χ i , χ jk ) } , � χ � L 1 = � � � � � d V , � � | χ i | d V + � χ i j i i , j Ω Ω then every stress with finite L ∞ -norm is continuous and | σ(χ) | � σ � L ∞ = � σ � L 1 ∗ = sup � χ � L 1 = | σ(χ) | . sup χ � χ � L 1 = 1 ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 8 The Measure µ and the Corresponding Norms Objective: Set � � � F � = max {| b i ( x ) |} , ess sup {| t j ( y ) |} ess sup . i , x j , y This will be the dual norm of a force if we use the norm � w � L 1 ,µ = � � | w i | d A = � w � L 1 + � w | ∂Ω � L 1 . � � | w i | d V + i i Ω ∂Ω This is the L 1 -norm relative to the measure µ , denoted L 1 ,µ , such that µ( D ) = V ( D ∩ Ω) + A ( D ∩ ∂Ω) . Conclusion: We want to find a relation between the L ∞ -norm of the stress field and the L ∞ ,µ -norm � F � of the force. ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 9 The Relation Between L 1 ,µ (Ω, R 3 ) and L 1 (Ω, R 12 ) j δ L 1 ,µ (Ω, R 3 ) → L 1 (Ω, R 12 ) ← − − − − − − − − ∗ j ∗ δ ∗ L 1 ,µ (Ω, R 3 ) ∗ − L 1 (Ω, R 12 ) ∗ − − − − → ← − − − � � � � � � L ∞ ,µ (Ω, R 3 ) L ∞ (Ω, R 12 ) ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 10 Sobolev mappings, traces and extensions • We consider the Sobolev space W 1 1 (Ω, R 3 ) of L 1 -mappings whose distributional derivatives are also L 1 -mappings. The Sobolev space is a Banach space under the norm � φ i � L 1 + � L 1 � φ � W 1 � � 1 = � � � φ j , k . i j , k • There is a continuous linear mapping, the trace mapping γ : W 1 → L 1 (∂Ω, R 3 ), 1 (Ω, R 3 ) − ∂Ω )( y ) = u ( y ), y ∈ ∂Ω, u ∈ C (Ω, R 3 ). γ ( u � � Thus, there is a K ∂ > 0 such that � γ (w) � L 1 � K ∂ � w � W 1 1 . ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 11 Implications to the Present Situation • Clearly, we have a continuous inclusion mapping ι 0 : W 1 → L 1 (Ω, R 3 ), 1 (Ω, R 3 ) − satisfying � ι 0 (w) � L 1 � � w � W 1 1 . • Hence, there is a linear injection—the extension to the boundary— δ : W 1 → L 1 ,µ (Ω, R 3 ), 1 (Ω, R 3 ) − δ(w)( x ) = w( x ), x ∈ Ω, δ(w)( y ) = γ (w)( y ), y ∈ ∂Ω. • The extension to the boundary is continuous and its norm is � ˆ � d A � � � � | w | d V + w � δ(w) � L 1 ,µ Ω ∂Ω � δ � = sup = sup |∇ w | d V . � � � w � W 1 | w | d V + w w ∈ W 1 1 1 (Ω, R 3 ) Ω Ω ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 12 ∗ -Norms of Forces The Relation Between the L ∞ ,µ and W 1 1 • As δ is a linear continuous injection, the dual mapping δ ∗ : L 1 ,µ (Ω, R 3 ) ∗ = L ∞ ,µ (Ω, R 3 ) − 1 (Ω, R 3 ) ∗ , → W 1 δ ∗ ( F )(w) = F (δ(w)), for all w ∈ W 1 1 (Ω, R 3 ) , is continuous. • A basic implication of the Hahn-Banach theorem: � δ ∗ � = � δ � . Thus, ∗ � δ ∗ ( F ) � W 1 1 F ∈ L ∞ ,µ (Ω, R 3 ). = � δ � , sup � F � L ∞ ,µ F ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 13 1 -Forces by Stresses in L ∞ (Ω, R 12 ) The Representation of W 1 • Consider the injection j : W 1 → L 1 (Ω, R 12 ), 1 (Ω, R 3 ) − j (φ) = (φ i , φ l , m ). • We note that � φ � W 1 1 = � j (φ) � L 1 . • It follows that every W 1 1 -force S may be represented (non-uniquely) by some stress σ in L ∞ (Ω, R 12 ) in the form S = j ∗ (σ). • In addition, S = j ∗ (σ) � σ � L ∞ . ∗ � S � W 1 = inf 1 ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 14 Result The situation so far j δ L 1 ,µ (Ω, R 3 ) − W 1 → L 1 (Ω, R 12 ) 1 (Ω, R 3 ) ← − − − − − − − j ∗ δ ∗ L 1 ,µ (Ω, R 3 ) ∗ → W 1 1 (Ω, R 3 ) ∗ − L 1 (Ω, R 12 ) ∗ − − − − ← − − − � � � � � � L ∞ ,µ (Ω, R 3 ) L ∞ (Ω, R 12 ) ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 15 ∗ � δ ∗ ( F ) � W 1 1 � δ � = � δ ∗ � = sup � F � L ∞ ,µ F ∈ L ∞ ,µ (Ω, R 3 ) � � � � | σ 0 i ( x ) | , ��� inf σ, δ ∗ ( F ) = j ∗ (σ) ess sup i , j , k , x � σ jk ( x ) = sup ess sup i , x {| b i ( x ) | , | t i ( x ) |} F ∈ L ∞ ,µ (Ω, R 3 ) We recall that δ ∗ ( F ) = j ∗ (σ) means δ ∗ ( F )(w) = j ∗ (σ)(w). It follows that σ ∈ � F because � � � � b i w i d V + t i w i d A = σ 0 i w i d V + σ i j w i , j d V . Ω ∂Ω Ω Ω Hence, � δ � = K ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 16 Adaptation to Equilibrated Forces and Stresses Basic idea: consider forces in the various dual spaces that do not perform power on rigid velocity fields (in- finitesimal displacement fields). ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 17 Stretchings and Rigid Velocity Fields • For a velocity field w ∈ W , the associated stretching (strain, deformation) ε(w) is the tensor field ε(w) im = 1 2 (w i , m + w m , i ). • A rigid velocity (or displacement) field is of the form w( x ) = a + ω × x , x ∈ Ω. • R – the collection of rigid velocity fields—a 6-dimensional subspace of the spaces of velocity fields. • Considering the kernel of the stretching mapping ε : w �→ ε(w), a theorem whose classical version is due to Liouville states that Kernel ε = R . In particular, ε(w + r ) = ε(w) . ✫ ✪ R. Segev ICTAM2004, Warsaw
✬ ✩ 18 Distortions and Approximations by Rigid Velocities • For a space W of velocity fields, the associated space of distortions is R ( [ w ] ) is well defined. We have W R . On W R the stretching map ε/ / / the natural projection π : W → W R . / R by �[ w ]� = inf r ∈ R � w − r � – the error • A norm is induced on W / of the best approximation by rigid motion. • For the L 2 -norm, the best approximation is the rigid motion that gives the same linear and angular momentum as w . This gives a projection π R : W → R . – Setting W 0 = Kernel π R ⊂ W , we have an isomorphism R ∼ W / = W 0 and ι R π 0 − − − − → R − − − − → W − − − − → W R − − − − → 0 / π R ι 0 ← − − − − R ← − − − − W ← − − − − ← − − − − 0 . W 0 – Thus, W ∼ ✫ ✪ = W 0 ⊕ R . R. Segev ICTAM2004, Warsaw
Recommend
More recommend
