CONTINUUM MECHANICS (STATE OF STRESS)

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

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

Stress matrix n1 n3 n2 p2 p3 p1 σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33 σ11 , σ12 , σ13 σ21 , σ22 , σ23 σ31 , σ32 , σ33 Tσ Normal stresses Shear stresses Positive and negative stresses x2 x1 x3 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.

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

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

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

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

x2 x3 n1 n3 n2 x1 σ11=2 σ13=1 σ21=0 σ22= -1 σ23=-3 σ31=1 σ32=-3 σ33=2 σ12=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.

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

i=1 i=2 i=3 j=1 j=2 j=3 Principal stresses in the explicit form: 1 i=1 i=2 i=3 j=1 j=2 j=3 The above is set of 3 linear equations with respect to 3 unknowns ni 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 where invariants I1 , I2 , I3 are following determinants of σij matrix Solution of this algebraic equation of the 3rd order yields 3 roots being real numbers due to symmetry of σij matrix These roots being eigenvalues of matrix σij are called principal stresses

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

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.

Principal stresses 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 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. 2 1 3 x2 x1 x3 n1 n3 n2 p2 p3 p1 σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33

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

1 3 2 1 3 2 1 3 2 max σ max 

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

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