Deviatoric Stress

Similar to strain, we can split the stress into its mean and deviatoric parts. The mean stress is defined by

We shall also refer to $p$ as a pressure. The deviatoric stress is then defined as

In Figure 2.6, the components of stress remaining after subtracting the hydrostatic stress have a special significance. They are called the deviatoric, or reduced stresses and are defined by

In plane stress, this may also be written in terms of the stress ratio, i.e.

An average stress is defined as:

By decomposing an existing stress state as given in Equation 1.3, we may define the total stress as the sum of the average stress and another stress term, which is known as deviatoric stress as given below:

It can easily be seen that the deviatoric stress actually reflects the shear stress level. It is therefore important to also determine the principal deviatoric stresses. The method used is identical to that of Equation 3.2, except σ_x is replaced by σ_x-σ_m and so on. The deviatoric invariants can therefore be obtained by substituting the normal stress components in the invariants of Equation 3.4. The result is:

In many cases, it is useful to decompose the stress tensor in a volumetric (i.e. spherical) part and a deviatoric (i.e. distortional) part. The above can be mathematically expressed as

where $p$ is a scalar quantity called mean stress and $s_{\mathit{ij}}$ is a tensor characterised by zero trace called deviatoric stress tensor (or stress deviator). The mean stress can be written in rectangular Cartesian coordinates as

The deviatoric stress tensor in rectangular Cartesian coordinates reads

where $s_{\mathit{ii}}={\sigma }_{\mathit{ii}}-p$.

Eq. (4.13) can therefore be rewritten in matrix form as

In order to consider the general principal stress case it has been shown necessary, in § 14.6, to add to the mean stress $\overline{\sigma }$ in the three perpendicular directions, certain so-called deviatoric stress values to return the stress system to values of σ₁, σ₂ and σ₃. These deviatoric stresses are then associated directly with change of shape, i.e. distortion, without change in volume and the strain energy associated with this mechanism is shown to be given by

Using the concepts of deviatoric stress (σ_d) and mean principal stress (σ_m) in continuum mechanics, we have:

By substituting Eqs. (128) and (129) into Eq. (127), a simpler relation can be expressed as:

The above Eq. (130) is the mathematical relation expressed by the deviatoric stress for porous materials under triaxial loading failure, which is derived from the octahedral model theory. This is the relationship between the nominal deviatoric stress of failure and the porosity for these materials under triaxial loading. For the present modeling theory, the second item on the left side of Eq. (130) has a relatively small weight on the failure of the porous body, and the failure is mainly the result of the first item of deviatoric stress. Therefore, this equation can be further approximately expressed as follows:

The main problem in the application of a “standard” displacement formulation to incompressible or nearly incompressible problems lies in the determination of the mean stress or pressure which is related to the volumetric part of the strain (for isotropic materials). For this reason it is convenient to separate this from the total stress field and treat it as an independent variable. Using the Voigt notation of stress, recall from Section 2.2.8.2 that the mean stress or pressure is given by

where $\mathbf{m}$ for the general three-dimensional state of stress is given by

For isotropic behavior the “pressure” is related to the volumetric strain, ${\epsilon }_v$, by the bulk modulus of the material, $K$. Thus,

For an incompressible material $K=\infty$$(\nu \equiv 0.5)$ and the volumetric strain is simply zero.

The deviatoric strain ${\mathbf{∊}}^d$ is defined by

where ${\mathbf{I}}_d$ is a deviatoric projection matrix which also proves useful in problems with more general constitutive relations [2]. In isotropic elasticity the deviatoric strain is related to the deviatoric stress by the shear modulus $G$ as

where the diagonal matrix [viz. Eq. (2.58a)]

is introduced because of the Voigt notation. A deviatoric form for the elastic moduli of an isotropic material is written as

for convenience in writing subsequent equations.

The above relationships are but an alternate way of determining the stress strain relations shown in Chapters 2 and 7Chapter 2Chapter 7 , with the material parameters related through

and indeed Eqs. (10.4) and (10.2) can be used to define the standard $\mathbf{D}$ matrix in an alternative manner.

As for the deviatoric stresses the deviatoric strains are also defined with reference to some selected "false zero" or datum value,

Thus, referred to the new datum, the principal strain values become

and these are the so-called deviatoric strains. It may now be observed that the following relationship applies:

It can also be shown that the deviatoric strains are related to the principal strains as follows:

The deviatoric stress invariants have a straightforward geometrical interpretation in principal stress space (see Section 2.1.2), as illustrated in Fig. 1.10. Eq. (1.48) is the equation of a circle centred on $\bar{\sigma }$, with the normal pointing along the hydrostatic axis ${\sigma }_1={\sigma }_2={\sigma }_3$. Thus the distance from a point $({\sigma }_1,{\sigma }_2,{\sigma }_3)$ in principal stress space to the hydrostatic axis is

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Figure 1.10. Geometrical interpretation of the deviatoric stress invariants in principal stress space. The dashed lines are the projections of the principal stress axes onto a deviatoric plane (i.e. a plane normal to the hydrostatic axis σ₁ = σ₂ = σ₃, also called the π-plane) passing through the point (σ₁,σ₂,σ₃). The angle ϑ_L is called the Lode angle.

It can further be shown (see e.g. Chen and Han, 1988) that the angle ${\vartheta }_{\mathrm{L}}$, called the Lode angle (in honour of the Lode, 1926 paper), indicated in Fig. 1.10, is given by the invariants as

(Note that since arccos is a multi-valued function, the Lode angle computed from Eq. (1.54) is not unique. If we choose the principal branch of arccos, then the result will be in the range 0^∘ to 60^∘ even if the actual stress state corresponds to another value.)