33.5: Principle of Normality
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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
σ21+σ22−σ1σ2=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):
δε1σ1−12(σ2+σ3)=δε2σ2−12(σ3+σ1)=δε3σ3−12(σ1+σ2)
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:
2σ1dσ1+2σ2dσ2−σ2dσ1−σ1dσ2=0
Rearranging this,
dσ2dσ1=2σ1−σ22σ2−σ1
Now, examining equation (2) for the situation where σ3 = 0, it follows that
δε2δε1=2σ2−σ12σ1−σ2
so that
δε2δε1⋅dσ2dσ1=−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
dσijdεPij≥0
where dσij are stress increments created by an external agency which produce very small plastic strain increments in strain, dεijP (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, σa, minus the radial compressive stress, σr, and where p' is the mean effective compressive stress generates the condition
dp′dv+dqdε≥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.