Numerical Modeling: An Efficient Tool for Analyzing the Behavior of Constructions - Multiscale Geomechanics: From Soil to Engineering Projects - page 267

272 Multiscale Geomechanics

9.3.6. Loading/unloading definition in plasticity

In the case of general plasticity, the calculation of the elastoplastic yield function
is done according to the consistency or compatibility condition expressed by f =0 .
This relationship helps to establish the evolution of the plastic multiplier λ p . The two
following conditions can be distinguished for plastic loading or elastic unloading,
respectively:
λ p > 0;

f =0,

f<0; λ p =0.

These conditions can be assembled:
λ p · f =0 [9.69]
In a general manner, five states of behavior can be distinguished (Figure 9.6)
according to the expression of the yield function in terms of stresses ( f ) or strains( F ):
Conditions Conditions
in stresses in strains
loading (hardening) ∂ σ f :σ>0 ∂ ε F:ε>0
loading (perfect plasticity) ∂ σ f :σ=0 ∂ ε F:ε>0
loading (softening) ∂ σ f :σ<0 ∂ ε F:ε>0

neutral loading ∂ σ f :σ=0 ∂ ε F:ε=0

unloading ∂ σ f :σ<0 ∂ ε F:ε<0

Let us note that the conditions of loading and unloading may not be properly defined
merely by conditions on stresses, whereas conditions on strains are not all ambiguous.
However, as we work with yield functions defined in the stress space, we must find the
relationship between ∂ ε F and ∂ ε F . With a linear elasticity assumption, we can write:
σ = C(ε− ε p ) [9.70]

In this case, the yield function in terms of stresses can be obtained from that
expressed in terms of strains:
F(ε,ε p ,α)=f(σ ,ε p ,α)=f(C(ε−ε p ),α)=0 [9.71]

replacing ∂ ε F by Φ : C where Φ = ∂ σ f . Therefore, in the case of yield functions
written in terms of stresses, the following conditions on the yield function and plastic
multiplier λ p are required to define the various states:

f=0; Φ:C:ε>0 and λ p > 0 then loading

if f =0;

f=0; Φ:C:ε=0 and λ p =0 then neutral loading

if f =0;