Elasticity: part 1 – stress analysis

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Modeling stress state in a loaded body and use it to verify the failure and stability criteria are the main objectives of mechanics of solids. In structural mechanics, stress are usually represented in integral forms such as internal forces and moments. In mechanics of materials, failure criteria are directly verified on the stress tensor and its invariants. This option provides some basic analysis related to the stress tensor including the calculation of the principle stresses, the stress invariants, the deviatoric stress tensor, Lode's angle, stress on an inclined surface, Mohr's circles, etc (Fung, 1977; Fjaer et al., 2008; Salençon, 2012; Love, 2013).


References
Love, A. E. H. (2013). A treatise on the mathematical theory of elasticity. Cambridge university press.
Fjaer, E., Holt, R. M., Raaen, A. M., Risnes, R., & Horsrud, P. (2008). Petroleum related rock mechanics (Vol. 53). Elsevier.
Salençon, J. (2012). Handbook of continuum mechanics: general concepts thermoelasticity. Springer Science & Business Media.
Fung, Y. C. (1977). A first course in continuum mechanics. Englewood Cliffs, NJ, Prentice-Hall, Inc., 1977. 351 p.

Input the components of a stress tensor in the orthogonal coordinates Oxyz:

σx (MPa)

σy (MPa)

σz (MPa)

τyz (MPa)

τxz (MPa)

τxy (MPa)

(Note: the first index of the shear stress stands for the face's normal, the second one is for the stress' direction)



(Click Compute button to run the calculation then continue the reading)

The stress invariants those are independent of the coordinate axis are:

I1 = tr(σ) = 0.00 (MPa),

I2 = (σij * σji - σii * σjj)/2 = 0.00 (MPa2), (Einstein's summation is considered)

I3 = det(σ) = 0.00 (MPa3).

The deviatoric stress tensor is defined by σd = σ - I1/3 * 1 where 1 is the second order idendity tensor. The deviatoric stress invariants are:

J1 = tr(σd) = 0,

J2 = 0.00 (MPa2),

J3 = det(σd) = 0.00 (MPa3).

Others useful stress invariants:

von Mises stress (equivalent stress q): 0.00 (MPa)
Stress invariant r: 0.00 (MPa) (see e.g. Fjaer et al., 2008)
Lode's angle θ: 0.00 (°), note that cos(3θ) = (r/q)3

The three principle stresses are calculated by solving equation det(f(σ)) = 0 for σ with f(σ) = σ - σ 1. This equation can be explicitly expressed in terms of the stress invariants as: -σ3 + I1 * σ2 + I2 * σ + I3 = 0. Solutions to this problem can be found by a graphical way as shown on the figure below. On this figure, the principle stresses correspond to the intersections between the curve and the σ-axis.

Once the principle stresses are determined, the three Mohr's circles can be ploted. Note that Mohr's circles are important for failure analysis, especially for brittle materials.

Input the three principle stresses obtained from the chart above in the text fields below then click Plot button to plot or update the Mohr's circles (assume that σ1 ⋝ σ2 ⋝ σ3).

σ1 (MPa)

σ2 (MPa)

σ3 (MPa)

For the particular case of a 2D stress or a 2D strain problem (all the shear stresses on one face vanish) closed-form solutions exist for the two principle stresses on the 2D plane. For example, if τxz = τyz = 0, the principle stresses on the X-Y plane are:

σ1 = 0.00 (MPa), σ2 = 0.00 (MPa).

Analytical solutions also exist for the normal and shear stresses on any inclined surface. Let's consider for example a surface of which the normal makes an angle

θ = (°) to the X-axis.



(Note: If the default value of θ is modified, click the Update button above to update the results).

The normal and shear stresses on the considered inclined surface are:

σ = 0.00 (MPa), τ = 0.00 (MPa).