Strengthening behavior of crystalline solids containing uniform dispersion of fine particle

Ansell, G. S.

Hirschhorn, J. S.

English

NASA Technical Reports Server

c .* 
. 
C r  
Dispersion Strengthening Nodels 
3. S. Hirschhorn 
G. S. Ansell 
Materials Engineering Department 
Rensselaer Polytechnic Institute 
Troy, Hew York 
GPO PRICE S 
CSFTl PRICE(S) S 
Hard copy (HC) ,/. 0 
Microfiche (MF) ,523 
ff 653 July 65 
J . .  . 
https://ntrs.nasa.gov/search.jsp?R=19650024105 2020-03-17T01:25:10+00:00Z
i . 
Dispe$,sF Strengthening M o d e l $  
{ 1-3; In recent years a number of theoretical models 
have been proposed to explain the observed strengthening 
behavior of crystalline solids containing a uniform dispersion 
of fine particles, A cursory examination of the structure- 
property relationships derived from these models shows them 
apparently to be quite dissimilar. 
however, reveals a basic similarity of approach utilized in 
each model. It is therefore appropriate to consider why 
these basically similar models predict such differing structure- 
strength relationships, 
A more detailed examination, 
The dislocation model in each case ascribes yielding to 
occur when the applied stress on the bulk alloy permits 
dislocations to freely move through the lattice by cutting 
through the distributed second phase. In calculating the yield 
strength Ansell (l) assumed that the stress required to cut  
the distributed phase arose from the magnification of the 
applied stress due to a piled-up array of curved dislocations, 
9 on 
Kelly and Hicholson (2 )  and Marcinkowski and Wriedt ( 3 )  
the other hand, related the work done dJring deformation, 
due to the applied stress, to the energy required to fracture 
the particles. 
to pile-up was originally derived on the basis of such a work 
consideration (4) . 
The magnification of the applied stress due 
Therefore all three models use essentially 
- 1 -  
. the same dislocation-pWtitle interact ion model and similar 
c r i t e r i a  for  yielding. Vhere these models widely d i f f e r ,  
however, l ies i n  the authors' treatments of the description 
of these interact ions i n  re la t ion t o  the  d is t r ibu t ion  of the 
p a r t i c l e s  within the so l id .  
Each of the  models considers the  interact ion of dis locat ions 
with the pa r t i c l e s  contaiaed within a s l i p  plane, Kelly and 
Nicholson and Marcinkowski and klriedt assume a par t icular  
d i s t r ibu t ion  function for  the arrangement of the pa r t i c l e s  i n  
the slip plaae and t h e n  extrapolate t h i s  d i s t r ibu t ion  function 
t o  the  d i s t r ibu t ion  of par t ic les  contained within the solid.  
Ansell (l), however, first considers the nature of the dis t r ibu-  
t i o n  i n  three  dimensioils and from t h i s  determines the  second 
phase d i s t r ibu t ion  within a s l i p  palne, On the  surfaces these 
two approaches would appear t o  be the same; however, t he  authors' 
use of these approaches ( i .e . ,  planar and voluinetric 
d i s t r ibu t ion)  are not compatible and lead t o  t h e i r  divergent 
results. 
The planar approach invokes the  assumption t h a t  the  
d.ispersed p a r t i c l e s  may be imagined t o  be arranged on a 
square-mesh i n  the  s l i p  plane. I n  r e a l i t y  there  a re  a 
mul t ip l ic i ty  of such s l i p  planes, especial ly  i n  cubic 
s t ructures .  Thus, according t o  t h i s  approach the  p a r t i c l e s  
would not be uniformly dis t r ibuted within the  matrix l a t t i c e .  
- 2 -  
e -  
* Instead, one must imagine a rather peculiar s p a t i a l  
- d i s t r ibu t ion  such t h a t  many random planes in te rsec t  the 
p a r t i c l e s  i n  such a manner t h a t  on these planes the p a r t i c l e s  
are arranged on a square-mesh. I n  addition, one Itlust consider 
an average e f fec t ive  diameter of the par t i c l e s  i n  such 
planes. Indeed, there can be no such three dimensional 
d i s t r ibu t ion  of spheres i n  a solid which would yield these 
re su l t s ,  
The volumetric approach s t a r t s  with the assumption that  
the pa r t i c l e s  a re  uniformly d i s t r ibu ted  i n  space: and t h a t  
t h i s  d i s t r ibu t ion  may be represented by a cubic  arrangement 
of the par t ic les .  Thus any random plane within the valume 
of the  a l loy  possesses an equally probable planar d i s t r ibu t ion  
of pa r t i c l e s  . 
It is then possible t o  derive two equations for  each 
of the proposed models; t h a t  is, one representing a planar 
d i scr ip t ion  and the other representing a volumetric description 
of the dispersion, To do th is  w e  must m a k e  note of some 
basic equations appropriate for each approach. I n  the 
planar method the  center t o  center  spacing between p a r t i c l e s  
is given by (2 )  
- 3 -  
where d is be true dime-& of the spherical  pa r t i c l e s  and 
F is the volume fract ion of dispersed particles., Tfie 
ef fec t ive  intercept  diameter (ai) of the pa r t i c l e s  on 
planes is given by 
d i  = (2/3) ' d 
or 
-3 
d i  = (&d/2) 6 
In the  volumetric method L is given by (1) 
and the t r u e  diameter of t h e  pa r t i c l e s  is  considered, 
The first model proposed was t h a t  of AnselI (1) * 
the  
(2) 
(3 )  
(4) 
Using 
the volumetric approach he derives 
1/3 F 
where 2, is  the yield stress of the alloy, r m  is the strength I 
* of "&e matrix, ,AA is the  shear modulus of the  dispersed , 
phase, and C is  a constant equal t o  about 30, W i t h  t h e  
I use  of eqns. (1) and ( 2 )  we obtain from this m o d e l ,  for 
I 
the  planar case I ,
L -  \" /  
The model of Kelly and Nicholson i n  which the planar - f  
approach was used gave 
- 4 -  
b 
where $is the t=rl-v-rrvr -- -' 3 1  =E the additiotiai matrix-particle 
in te r face  produced by the  shear and the energy of the dis- 
locations produced a t  the interface and b is the  Burgers 
vector of the  dislocation cutt ing the particle. 
use of eqn, (4) w e  obtain from t h e i r  model, for the  
With the  
volumetric case 
Marcinkowski and Wriedt  a l so  used the planar approach 
and found 
2' = pm i- 2 ( T . ,  - 2- ) m (9) 
where z- 
we obtain from t h e i r  m o d e l ,  for the volumetric case 
is t he  strength of the  par t ic le .  Using eqn. (4) P 
The m o d e l  of Dew-Hughes may also be s imilar ly  treated; 
and this would yield equations s imilar  t o  eqns. 
This model involves the cut t ing of dis locat ion loops by other 
dislocations;  and did not use e i the r  the planar or volumetric 
(5) and ( 6 ) .  
type of description, but  left t he  f i n a l  equation i n  terms 
of general parameters. 
- 5 -  
- -  
* We now can conclude tha t  if we use a similar m e t h o d  of 
describing the geometrical d i s t r ibu t ion  of the dispersion, 
then a l l  three m o d e l s  y i e ld  the same basic relationship; 
t h a t  is, i f  the planar description is used, then the yield 
strength is a function of F 35 , and i f  the volumetric 
description is used, then the yield strength is  a function 
1/3 of F . 
The c ruc ia l  question t o  now consider is whether the  
planar o r  volumetric type of description is more r e a l i s t i c ,  
We have already noted tha t  it is  impossible t o  extrapolate a 
th ree  dimensional d i s t r ibu t ion  which is compatible with 
the planar description employed, we conclude t h a t  the 
volumetric type of description represents more f a i t h f u l l y  
the actual  physical dis t r ibut ion of dispersed phase pa r t i c l e s ,  
Thus, regardless of the  model one believes correct ly  treats 
the dislocation-particle interaction, w e  believe t h a t  the  
y ie ld  strength is a function of F 1/3 . Mathematically one 
f inds  t h a t  equally good agreement may be obtained i f  one 
!2 or ? vs, F ; t hus ,  the data  i n  the p lo t s  2 VS. F 1/3 
found t o  be i n  equally good agreement with F 1/3 . 
?z 
l i t e r a t u r e  found t o  agree w i t h  the  F dependence may be 
- 6 -  
3 4  
, *  ' 
The work reported in th i s  & 
National Aeronautics and Space 
paper was supported by the 
Interdisciplinary Materials Research Center, Rensselaer 
Polytechnic Institute. 
J. S. Hirschhorn 
G. S. Ansell 
Materials Engineering Department 
Rensselaer Polytechnic Institute 
Troy, New York 
- 7 -  
1, 
2. 
3, 
4, 
5. 
References 
# 
G. S, A n s e l l ,  gcta Met,, 9, 518 ( 1 9 6 1 ) -  4 
A, Kelly and R, B. Nicholson, Proqr, Materials Science, 
10, 148 (1963)- 
M, J, Marcinkowski and D. F. Wriedt, J. Electrochemical Soc,, 
111, 92 (1954). 
A, H, Cottrell, Dislocations and Plastic Plow in 
Crystals, Oxford, London (1953) , 
D, Dew-Hughes, Acta  Met., 8, 816 (1960). 
- 8 -  


Dispersion strengthening models

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server