PSI - Issue 13

Milan Micunovic et al. / Procedia Structural Integrity 13 (2018) 2158–2163 Micunovic, Kudrjavceva / Structural Integrity Procedia 00 (2018) 000 – 000

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2.2. Hooke's Law By Homogenization Approach For treating elastic meso and micro-strains the effective medium variant of the self-consistent method [5, 11] is applied with an assumption that for monocrystals residual microstrains are negligible. Moreover, for considered fcc polycrystals slight disorder is assumed such that all the grains are almost aligned giving rise to anisotropy. Then the effective Hook e’s tensor has the approximate simplified form:     1 , eff M        D D D S D SD 1 (2) where D is stiffness 4-tensor (subscript M stands for the imagined matrix – average of grains and  for a typical grain) and S is the Eshelby 4-tensor. Angular parentheses denote a spatial averaging like in [4, 10]. It is worth noting that the simplest linear approximation for the effective grain stiffnesses is worthless since it does not include Eshelby 4 tensor at all. 2.3. Evolution equation – micro to meso transition A special attention is paid to the associativity of flow rule based on the loading function  and derived by Rice [2]. The experimental evidence (cf. [6]) has shown that real time in the evolution equation for plastic stretching has to be replaced by a thermodynamic Vakulenko’s time  (cf. [1, 9]). If such a time is the same for all the grains, then normality of the plastic stretching onto a yield (or a loading) surface could hold. In such a case, the meso evolution equation for RVE reads ( T is the stress, e P – plastic strain and M  – Spencer- Boehler’s structural tensor describing anisotropy type,  - Heaviside function, Y – dynamic yield stress):



Y 

  

  

(3)





eq

D

, , P T e M

/ , D Dt 

D

exp( ) . M 

with

1





   

 



P

T

t eq



When thermodynamic time is a nonlinear function of plastic power (i.e. non-steady aging happens) the above equation covers non-steady aging as well. A possible generalization of the function  to include not only plastic power, but also fracture (following [13, 14]) seems worthwhile. A special attention deserves the assumed normality in (3). If there exists a scalar function of resolved shear stresses       T A : i.e.         such that from









D  

,

 

. P D





then it follows that

(4)











T















Therefore, the mentioned normality should hold. Such a function is called the loading function (cf. [2]) . Even if the mentioned simplifications are valid this normality could fail. Indeed, for a monocrystal let us introduce two types of additional resolved shear stresses , : 1 1    T A  2 2 :    T A  and suppose that the loading function has the form

2 1

h

A

B

.  

(5)

 

 



 



1  

2  

  

(summation over slip systems is assumed). In the sequel a special type of tensor representation described in [9] is used. In order to explore whether a simple micro-evolution equation (4) leads to tensor representation it is integrated for some characteristic strain histories in isotropic polycrystals and calibration of an assumed meso-evolution equation is performed in the paper [11]. Then meso-micro transition was used for computer simulation in two cases of a RVE composed of N=1000 as well as N=125 fcc-grains with slightly disordered crystal orientations and 12 potential slip systems. Residual stresses and number of active slip systems were found. The RVE was loaded by slow and fast stresses with three typical stress states: uniaxial tension, uniaxial shear (with one principal stress vanishing and the other two being opposite) and equibiaxial shear. In all cases smooth increase of active slip systems has been remarked during stress growth. A comparison with J 2 -theory (which covers only slow stresses) was given.