1 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress CONTINUUM MECHANICS (STATE OF STRESS)

2 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress

3 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress  Conversation with a Stone I 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." Rozmowa z kamieniem Pukam 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 częsci będziemy szczelnie zamknięte. Nawet starte na piasek nie wpuścimy nikogo. W.Szymborska

4 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress n A P1P1 PnPn I {wI}{wI} {ZI}{ZI} Internal forces - stress Neighbourhood of point A The sum of all internal forces acting on Δ A Δ A – area of point A neighbourhood ΔwΔw Stress vector at A n A

5 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress Stress vector is a measure of internal forces intensity and depends on the chosen point and cross section x2x2 x1x1 x3x3 n1n1 n3n3 n2n2 p2p2 p3p3 p1p1 σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 p 1 [σ 11, σ 12, σ 13 ] p 2 [σ 21, σ 22, σ 23 ] p 3 [σ 31, σ 32, σ 33 ] Point A image σ 11, σ 12, σ 13 σ 21, σ 22, σ 23 σ 31, σ 32, σ 33 TσTσ T σ (σ ij ) Stress vectors: Stress matrix i,j = 1,2,3 Components  ij of matrix T are called stresses. Stress measure is [N/m 2 ] i.e. [Pa] Stress matrix

6 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress x2x2 x1x1 x3x3 n1n1 n3n3 n2n2 p2p2 p3p3 p1p1 σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 σ 11, σ 12, σ 13 σ 21, σ 22, σ 23 σ 31, σ 32, σ 33 TσTσ Normal stresses Shear stresses Positive and negative stresses Stress matrix Stress is defined as positive when the direction of stress vector component and the direction of the outward normal to the plane of cross-section are both in the positive sense or both in the negative sense in 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 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress x’ 2 x’ 1 x’ 3 n1n1 n3n3 n2n2 p2p2 p3p3 p1p1 σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 x2x2 x1x1 x3x3 n’ 1 n’ 3 n’ 2 p’ 2 p’ 3 p’ 1 σ’ 12 σ’ 11 σ’ 13 σ’ 21 σ’ 22 σ’ 23 σ’ 31 σ’ 32 σ’ 33 {xi}{xi} {x’ i } T σ [ σ ij ] T’ σ [ σ’ ij ] [  ij ] Stress transformation Here Einstein’s summation convention has been applied

8 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress x2x2 x1x1 x3x3 n2n2 n1n1 n3n3 ΔA1ΔA1 ΔA3ΔA3 ΔA2ΔA2 ν(  νi ) ΔAνΔAν  X 1 = 0 …, … If we assume: Stresses on inclined plane then: Symmetry of stress tensor

9 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress x2x2 x1x1 x3x3 ν(  νi ) Procedure of LIMES transition Diminish area of front side of tetrahedron keeping constant versor ν(  νi ) i.e. its direction and length = 1 Lack of tilde over sigma denotes the stress vector just in a given point ΔAνΔAν ΔA ν 0

10 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress x2x2 x1x1 x3x3 ν(  νi ) Procedure of LIMES transition Diminish area of front side of tetrahedron keeping constant versor ν(  νi ) i.e. its direction and length = 1 Lack of tilde over sigma denotes the stress vector just in a given point ΔAνΔAν ΔA ν 0

11 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress x1x1 x3x3 x2x2 i=1 j=1j=2 j=3 i=1 i=2 i=3 j=1j=2 j=3 Stresses on inclined plane

12 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress x2x2 x3x3 n1n1 n3n3 n2n2 x1x1 σ 11 =2 σ 13 =1 σ 21 =0 σ 22 = -1 σ 23 =-3 σ 31 =1 σ 32 =-3 σ 33 =2 σ 12 =0 i=1 j=1 j=2 j=3 i=1 i=2 i=3 i=2 i=3 Stresses on inclined plane

13 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress x2x2 x3x3 n1n1 n3n3 n2n2 x1x1 σ 11 =2 σ 13 =1 σ 21 =0 σ 22 = -1 σ 23 =-3 σ 31 =1 σ 32 =-3 σ 33 =2 σ 12 =0 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 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress or  ij = 1 if i = j 0 if i  j and vector size Seeking are : 3 components of normal vector : 4 unknowns Principal stresses We will use Kronecker’s delta to renumber normal vector components  i Kronecker’s delta etc… Three equations

15 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress in the explicit form: i=1 i=2 i=3 j=1 j=2 j= The above is set of 3 linear equations with respect to 3 unknowns  i with zero-valued constants. The necessary condition for non-zero solution is vanishing of matrix main determinant composed of the coefficients of the unknowns. Principal stresses

16 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress where invariants I 1, I 2, I 3 are following determinants of σ ij matrix Solution of this algebraic equation of the 3rd order yields 3 roots being real numbers due to symmetry of σ ij matrix These roots being eigenvalues of matrix σ ij are called principal stresses Principal stresses

17 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress In the special case of plane stress state and: Principal stresses

18 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress 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. Principal stresses

19 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress 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 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. Principal stresses

20 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress x2x2 x1x1 x3x3 n1n1 n3n3 n2n2 p2p2 p3p3 p1p1 σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 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 Principal stresses

21 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress  /4 Stresses on characteristic planes

22 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress max σ max 

23 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress max  Mohr circles – represent 3D state of stress in a given point – on the plane of normal and shear stresses 0

24 /19 M.Chrzanowski: Strength of Materials SM1-07: Continuum Mechanics: State of stress  stop