1. CONTINUUM MECHANICS (STATE OF STRESS)

2. 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

3. 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

4. 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.

5. Stress transformationp2 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

6. 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

7. Stresses on inclined planex3
i=3
j=1
j=2
j=3 x2 x1
i=1
j=1
j=2
j=3

8. Stresses on inclined planex2 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

9. 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.

10. 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

11. 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.

12. 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

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

14. 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.

15. 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.

16. 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

17. Stresses on characteristic planes1 3 2 1 3 2 1 3 2 1 3 2
/4

18. 1 3 2 1 3 2 1 3 2
max σ
max 

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

20. stop