33.5: Principle of Normality

The principle of normality follows from a detailed consideration of the yield surfaces of work-hardening materials and ideal plastic materials by Daniel Charles Drucker in 1951 in a paper entitled ‘A more fundamental approach to plastic stress strain relations’, Proc. 1st U.S. Nat. Congr. Of Appl. Mech., pp. 487-491. In this paper Drucker established that these yield surfaces must be convex and that the vector sum of the plastic strain increments (or flow increments) at any point on the yield surface is normal to the yield surface. This is the principle of normality.
To illustrate what this means in practice for an ideal plastic metal, in which hydrostatic stress does not cause plastic deformation (see here), we can consider the von Mises yield criterion in plane stress.

In plane stress in principal stress space with principal stresses σ₁ and σ₂ and where σ₃ = 0, the von Mises yield surface is defined by the equation

\[ \sigma^2_1 + \sigma^2_2 - \sigma_1 \sigma_2 = 1 \]

if the uniaxial yield stress, Y, is take to be unity (see here).

At and beyond yield, the flow behaviour of ideal plastic metals is governed by the Lévy-Mises equations (see here):

\[\frac{\delta \varepsilon_1}{\sigma_1 - \frac{1}{2} (\sigma_2 + \sigma_3)} =\frac{\delta \varepsilon_2}{\sigma_2 - \frac{1}{2} (\sigma_3 + \sigma_1)} =\frac{\delta \varepsilon_3}{\sigma_3 - \frac{1}{2} (\sigma_1 + \sigma_2)}\]

for principal plastic strain increments δε₁, δε₂ and δε₃ parallel to the principal stresses σ₁ and σ₂ and σ₃ respectively.

Examining the equation for the von Mises yield surface, the tangent at a point (σ₁, σ₂) on the yield locus can be found be differentiating equation (1) implicitly:

\[ 2 \sigma_1 d \sigma_1 + 2 \sigma_2 d \sigma_2 - \sigma_2 d \sigma_1 - \sigma_1 d \sigma_2 = 0 \]

Rearranging this,

\[ \frac{d \sigma_2}{d \sigma_1} = \frac{2 \sigma_1 - \sigma_2}{2 \sigma_2 - \sigma_1} \]

Now, examining equation (2) for the situation where σ₃ = 0, it follows that

\[ \frac{\delta \varepsilon_2}{\delta \varepsilon_1} = \frac{2 \sigma_2 - \sigma_1}{2 \sigma_1 - \sigma_2} \]

so that

\[ \frac{\delta \varepsilon_2}{\delta \varepsilon_1} \cdot \frac{d \sigma_2}{d \sigma_1} = -1 \]

Hence, in words, the product of the gradient of the tangent at a point (σ₁, σ₂) on the yield surface and the gradient of the vector [δε₁, δε₂] defining the plastic flow increments in the (σ₁, σ₂) plane is minus one, i.e., these two gradients are perpendicular.

Since the third plastic strain increment δε₁ is parallel to the principal stress σ₃, it follows that the vector sum of plastic flow increments [ δε₁, δε₂, δε₃] is normal to the yield surface at (σ₁, σ₂) , i.e., the principle of normality is proved for the von Mises yield criterion.

The animation below shows how in two dimensions in (σ₁, σ₂) space, the vector defining the plastic flow increments (in red) is normal to the von Mises yield surface (in black). The angle can be varied by moving the cursor in the bottom left of the animation.

In the language of soil mechanics, the principle of normality is known as the associated flow rule (see, for example, A. Schofield, Disturbed Soil Properties and Geotechnical Design, Thomas Telford Ltd., London, 2005, p. 91). A further result which follows from the convexity of yield surfaces and this principle is that

\[ d \sigma_{ij} d \varepsilon^P_{ij} \geq 0 \]

where dσ_ij are stress increments created by an external agency which produce very small plastic strain increments in strain, dε_ij^P (as well as elastic strain increments). In words, this is a statement that plastic work done by an external agency is always positive.

Applying equation (7) to a triaxial stress test on a soil where q is the axial compressive stress, σ_a, minus the radial compressive stress, σ_r, and where p' is the mean effective compressive stress generates the condition

\[ dp'dv + dqd \varepsilon \geq 0 \]

where dv is the incremental change in the specific volume of the soil aggregate and where dε is the incremental change in pure distortion. The specific volume is the ratio of the volume occupied by soil to that which it would occupy if the voids were eliminated. In soil mechanics the convention is that dp', dq , dv and dε are all positive.

Equation (8) is relevant to the Original Cam-Clay model of how soils at the critical state and on the ‘wet’ side of the critical state fail.