PSI - Issue 6

N.S. Selyutina et al. / Procedia Structural Integrity 6 (2017) 77–82 Author name / StructuralIntegrity Procedia 00 (2017) 000 – 000

80

4

2. Relaxation model of plasticity

Let us consider the behaviour of the yield strength at the initial instant of plastic deformation within the structural-temporal approach based on the incubation time concept (Gruzdkov and Petrov (1999), Gruzdkov et al. (2002), Gruzdkov et al. (2009)):



    

   

s ( )

t

t Int

ds

 ( ) 1

.

( ) 1,  t Int p where

(4)



p





t





y

Here, Σ(t) is a function describing the time dependence of stress,  is the incubation time, σ y is the static yield stress, α is a coefficient of amplitude sensitivity of the material. Note that the onset of macroscopic yield t * is determined from the condition of equality (4). The introduced time parameter  , independent of the specific features of deformation and sample geometry, makes it possible to predict the behaviour of the yield strength of material under static and dynamic loads (Gruzdkov and Petrov (1999)). It was shown by Selyutina et al. (2016) that the incubation time can be related to different physical mechanism of plastic deformation. Let us assume the linear elastic deformation law ( ) ( ) t E tH t     , where E is the Young’s modulus and   is the constant strain rate under load, H(t) is the Heaviside function. Having written the left-hand side of (4) under the condition that the yield starts at time t * , one can express the dynamic yield stress ( ) ( ) * t d      in terms of the strain rate of material:





      

1 /





    1 / 1 

1







    y



y



E

1

,

;















1 E



  ( )  

(5)





1/









   

   

d

1

y

E

  1

,

.



  

 





y

E

1 /





1





Thus, the set of parameters ( , , )    y describes the behaviour of the material and it is independent on the strain rate and temporal characteristics of impact. One can use an elastic approximation of stress ( ) 2 ( ) t G t    and consider the case of equality in the criterion (4) in order to define the macroscopic time of the plastic flow beginning t * ; here G is the shear modulus. We propose a primary version of the relaxation model in the present paper for the case of a linear increase of strain ( ) ( ) t tH t     together with time starting from the zero time moment 0  t . Let us introduce a dimensionless relaxation function 0 ( ) 1   t  , defined as follows

          



    

     ds

( ) s

t

1,

1

1,



0 y





t







    

     ds

( ) s

1

t

( ) t

(6)





, 1

1,



1

0 y





t





   

   





    

   

( ) s

1

t

ds



0 y





t





Equality ( ) 1  t  in (6) relates with the elastic deformation accumulation before the starting time t * of the macroscopic plastic flow. Gradual decrease of the relaxation function in the range 0 ( ) 1   t  corresponds to the material transition into the plastic stage of deformation. During the plastic stage of deformation, * t t  , the relaxation function satisfies the condition