Understanding Stress in Continuum Mechanics: The State of Internal Forces

1. CONTINUUM MECHANICS(STATE OF STRESS)

3. Conversation with a StoneI knock at the stone's front door."It's only me, let me come in.I want to enter your insides,have a look round,breathe my fill of you.""Go away," says the stone."I'm shut tight.Even if you break me to pieces,we'll all still be closed.You can grind us to sand,we still won't let you in." W.Szymborska Rozmowa z kamieniemPukam do drzwi kamienia.- To ja, wpuść mnie.Chcę wejść do twego wnętrza,rozejrzeć się dokoła,nabrać ciebie jak tchu.- Odejdź - mówi kamień. -Jestem szczelnie zamknięty.Nawet rozbite na częscibędziemy szczelnie zamknięte.Nawet starte na piaseknie wpuścimy nikogo.

4. Internal forces - stress n {wI} A n {ZI} P1 A I Pn The sum of all internal forces acting on ΔA Δw ΔA – area ofpoint A neighbourhood Neighbourhood of point A Stress vector at A

5. Stress matrix n3 n2 n1 x3 x2 x1 Stress vector is a measure of internal forces intensity and depends on the chosen point and cross section Stress vectors: p1[σ11 , σ12 , σ13 ] σ33 p3 p2[σ21 , σ22 , σ23 ] σ32 p3[σ31 , σ32 , σ33 ] σ31 σ23 p2 σ13 σ11 , σ12 , σ13 p1 σ22 σ21 , σ22 , σ23 Tσ σ11 σ31 , σ32 , σ33 σ31 , σ32 , σ33 σ21 σ12 Stress matrix Tσ(σij) Point A image Components ijof matrix T are called stresses. Stress measure is [N/m2] i.e. [Pa] i,j = 1,2,3

6. Stress matrix σ11, σ12 , σ13 σ21, σ22 , σ23 Tσ n3 σ33 σ31, σ32 , σ33 p3 σ32 n2 σ31 σ23 σ13 p2 p1 n1 σ22 σ11 x3 σ21 σ12 x2 x1 Normal stresses Shear stresses Positive and negative stresses Stress is defined aspositive when the direction of stress vectorcomponentandthe direction of the outward normal to the plane of cross-section are both in the positivesenseorboth in the negative sensein relation to the co-ordinate axes. If this double conjunction of stress component and normal vector does not occur – the stress component is negative one.

7. Stress transformation n3 σ33 p3 σ32 n2 σ31 σ23 σ13 p2 p1 σ’33 n1 σ22 σ11 x3 σ21 x’3 σ12 σ’31 σ’32 σ’21 σ’22 σ’13 σ’23 x’2 σ’11 x2 σ’12 x1 x’1 Tσ[σij] {xi} n’2 [ij] n’3 p’3 T’σ[σ’ij] {x’i} p’1 n’1 p’2 Here Einstein’s summation convention has been applied

8. Stresses on inclined plane x3 x2 x1 X1= 0 ν(νi ) …, … ΔAν n1 ΔA1 Symmetry of stress tensor ΔA2 n2 ΔA3 If we assume: then: n3

9. x3 ν(νi ) ν(νi ) ν(νi ) Diminish area of front side of tetrahedron keeping constant versor ν(νi ) ΔAν0 x2 ν(νi ) x1 Procedure of LIMES transition ΔAν i.e. its direction and length = 1 Lack of tilde over sigma denotes the stress vector just in a given point

10. x3 ν(νi ) ν(νi ) ν(νi ) Diminish area of front side of tetrahedron keeping constant versor ν(νi ) ΔAν0 x2 ν(νi ) x1 Procedure of LIMES transition ΔAν i.e. its direction and length = 1 Lack of tilde over sigma denotes the stress vector just in a given point

11. Stresses on inclined plane i=1 i=2 x3 i=3 j=1 j=2 j=3 x2 x1 i=1 j=1 j=2 j=3

12. x3 Stresses on inclined plane n2 n1 n3 x2 x1 i=3 σ33=2 σ31=1 σ32=-3 i=1 i=2 σ23=-3 σ13=1 σ22= -1 i=3 σ12=0 j=2 j=1 j=3 σ21=0 σ11=2 i=1 i=2

13. x3 n2 σ33=2 σ31=1 σ32=-3 n1 σ23=-3 n3 σ13=1 x2 σ22= -1 σ12=0 σ21=0 σ11=2 x1 On this plane none of the vector components are perpendicular nor parallel to the plane. We will look for such a plane to which vector will be perpendicular, thus having no shear components.

14. 1 if i=j dij= 0 if ij Seeking are :3 components of normal vector : and vector size Principal stresses or Kronecker’s delta We will use Kronecker’s delta to renumber normal vector components i etc… Three equations 4 unknowns

15. Principal stresses in the explicit form: 0 0 1 i=1 i=2 i=3 j=2 j=1 j=3 The above is set of 3 linear equations with respect to 3 unknowns niwith zero-valued constants . The necessary condition for non-zero solution is vanishing of matrix main determinant composed of the coefficients of the unknowns.

16. Principal stresses where invariants I1 , I2 , I3are following determinants of σijmatrix Solution of this algebraic equation of the 3rd order yields 3 roots being real numbers due to symmetry of σijmatrix These roots being eigenvalues of matrix σijare called principal stresses

17. Principal stresses In the special case of plane stress state and:

18. Principal stresses Now, from the set of equations: one can find out components of 3 eigenvectors, corresponding to each principal stress These vectors are normal to three perpendicular planes. The stress vectors on these planes are also perpendicular to them and no shear components of stress vector exist, whereas normal stresses are equal principal stresses.

19. It can be proved that are extreme values of normal stresses (stresses on a main diagonal of stress matrix) . Customary, these values are ordered as follows Principal stresses Surface of an ellipsoid with semi-axis (equatorial radii) equal to the values of principal stresses represents all possible stress vectors in the chosen point and under given loading.

20. Principal stresses n3 σ33 p3 3 σ32 n2 x3 σ31 σ23 σ13 p2 p1 n1 σ22 2 σ11 σ21 σ12 x2 1 x1 With given stress matrix in the chosen point and given loading one can find 3 perpendicular planes such that stress vectors (principal stresses) have only normal components (no shear components).The coordinate system defined by the directions of principal stresses is called system of principal axis.

21. 3 3 3 3 2 2 2 1 1 1 2 1 Stresses on characteristic planes /4

22. 3 3 3 2 2 1 1 2 1 maxσ max 

23. Mohr circles – represent 3D state of stress in a given point – on the plane of normal and shear stresses 0 max 

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