L8. Basic concepts of stress and equilibrium
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1 L8. Basic concepts of stress and equilibrium Duggafrågor 1) Show that the stress (considered as a second order tensor) can be represented in terms of the eigenbases m i n i n i. Make the geometrical representation of the stress in the principal stress base and consider the stress subdivided in terms of the mean stress and the stress deviator. Additional issue 6) Formulate the equivalence between the weak and strong representations of equilibrium for a solid subjected to quasistatic forces. Basic concept of stress Stress considered as second order tensor Consider the stress representation: ij e i e conv. j ij e i e j j 1 sum. Note the introduction of the dyadic product, e.g. e i e j. It is defined by A a b (where A is a second order tensor) and the contraction (or projection) is defined as A c a b c Α a where the scalar Α is defined by the scalar Α b c. Consequences: Traction w.r.t arbitrary orientation obtained via contraction! t n t i ij n j Holds for Cartesian basis; "Normally" taken as e i,2, e x, e y, e z t t i e i n ij e i e j n n n k e k ij n k e i e j e k ij n j e i t i ij n j jk Follows from "equilibrium" considerations of stress state. Consider 2D-stress state in e x e y -system:
2 2 L8.nb Considerations: Equilibrium A x t x A y t y A t Geometry n x 1 A x A ; n y 1 A y A A x A n x, A y A n y t n x t x n y t y Representation on matrixform t x n t x x y Τ xy t x n y y t y x y stress matrix n x n y t i ij n j stress matrix for a 2D plane stress state Principal stress state Consider the traction vector w.r.t. a given surface (defined by the orientation vector n):
3 L8.nb t Τ n n Τ t n n From the Cauchy stress theorem: t n Τ t n n n n n n 1 n 1= second order identity tensor Define the principal stresses: Consider "shear stress free" orientations: Τ! Τ n 1 n Egenvalueproblem det n 1 A Coeff. matrix (1) deta x Τ xz y Τ yz Τ xz Τ yz z n n n... n I 1 2 n I 2 n I Secular ekvation (2) The invariants: I 1 x y z I 2 x y x z y z Τ 2 xy Τ 2 2 xz Τ yz I Det x y z z Τ 2 xy y Τ 2 2 xz 2 Τ xz Τ yz x Τ yz (2) principal stresses: i,2, (ordered as 1 2 ) (1) principal orientations: n i,2, from i 1 n i with i given! Principal orientations: n i,2, orthogonal! Orthogonality of principal stress orientations Reconsider eigenvalue problem: i 1 n i,2, Orientations have "unit length" n i n i : 1 Consider: 1) n j i 1 n i
4 4 L8.nb 2) n i j 1 n j Assume distinct eigenvalues: i j when i j i n j n i n j n i j n i n j n i n j i n j n i j n i n j i j n i n j Orthogonality properties of principal orientations if i j : i j, n i n j 1 if i j : i j, n i n j om i j om i j Representation of stress state in eigenbases Properties of Cartesian basis: Consider 2nd order unit tensor 1 in Cartesian basis m ij e i e j : 1 ij m ij m ii m i Note! m i n i n i,2, are the eigenbases of the stress state 1 ij! Consider stress representation in Cartesian basis and in the eigenbases as ij m ij i m i where it was used that the shear stress components take on zero values in the principal coordinates. Hence, it suffices to consider the summation of stress components with respect to the eigenbases m i (as done in the last equality). Please note that the principal stresses i as well as the eigenbases m i are functions of the stress state itself, i.e. i,2, principal stresses Geometric interpretation of stress state - eigenbases 1 How about the scalar product, e.g ! How to formulate? Consider geometry! 2 : i j m i : m j j 1 Scalar product formulated in terms of double contraction m i : m j
5 L8.nb 5 m i : m j n i n i : n j n j def. orto. 1 n j n i n i n j om i j om i j 2 i 2 OK Consider the generalization to the Cartesian basis m ij : 2 ij ij This follows directly from j 1 2 sum conv. : ij m ij : kl m kl ij kl m ij : m kl k 1 l 1 ij kl e i e j : e k e l ij kl e i e k e j e l ij kl ik jl ij ij Mean stress m Def. mean stress: "the projection of the stress onto the mean stress axis 1" m 1 1 : 1 i 1 : m i 1 i n i 1 n i 1 i 1 I 1 1 Tr Geometric interpretation of unit tensor 1 m i mean stress axis! Consider the generalization to the Cartesian basis m ij : 1 : ij 1 : m ij ij e i 1 e j ij ij ii j 1 Consider also single contraction: j 1 1 ij kl m ij m kl il m il
6 6 L8.nb Stress deviator dev Consider the stress subdivided, cf. Fig. above, We have Consider dev m 1 stress deviator dev m 1 dev m 1 dev dev,i m i dev : dev dev dev,1 dev,2 dev, 1 : dev i m OK 1 and dev are orthogonal! Momentum balance - Strong and weak formats Consider equilibrated solid: t f t t n t f strong form of equilibrium t f x
7 L8.nb 7 Momentum balance in terms of virtual work W W u t u f u t u f W virtual work produced by virtual displacement field u. u t t : u u t virtual work W t : u u t u f along with equilibrium satisfied in "weak" sense, i.e. u t f Link between strong and weak forms of equilibrium W considered w.r.t. "all virtual displacements fields" u t f f t if taken u U Strong form of the equilibrium formulated as balance of internal - external virtual work W t : u u t u f u U irtual work W = work rate W if u u W t : u Displacement rate gradient u named the "spatial velocity gradient" l d u l dx with l u x u Compact notation, introduce the notation l l u W t : l, W t : l
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