~=,/
D=o versus the homogeneity parameter of the material m.
versus form of values of m R = 1 for m-.
MPG-6 with an average homogeneity parameter m a v = 6.2 and for medium-grained graphite VPP ( m a v = i0), calculated from the experimental fracture curves of the indicated kinds of graphite shown in [6]. LITERATURE CITED i. 2. 3. 4. 5. 6.
P . M . Vitvitskii, Fiz.-Khim. Mekh. Mater., No. 5, 52-58 (1970). P . M . Vitvitskii, Probl. Prochn., No. 4, 13-17 (1971). P . M . Vitvitskii and S. Yu. Popina, Fiz.-Khim. Mekh. Mater., No. 2, 43-48 (1975). V . V . Bolotin, Statistical Methods in Structural Mechanics [in Russian], Stroiizdat (1961). V. I. Mossakovskii and M. T. Rybka, Prikl. Mat. Mekh., No. 2, 291-296 (1965). A . M . Fridman, Yu. P. Anufriev, and V. N. Barabanov, Probl. Prochn., No. i, 52-55 (1973).
TESTING THE STRENGTH OF MATERIALS UDC 539.4
D. Yu. Mochernyuk
It is well known [1-5] that determination of the strength of materials is accomplished experimentally most simply in the uniaxial stress state. For a complex stress state the danger of material fracturing depends on both the magnitude and the relations of components of the stress tensor. The complexity of conducting experiments and the unlimited number of combination of relations among the principal stresses, taking their signs into account, compel us to reject experiment and to seek theoretical means of evaluating the strength of material in a complex stress state [2-5]. It is generally assumed that we compare quantitatively the complex stress state with the uniaxial on the basis of test results and the strength derived from some particular theory. In this procedure it is tacitly assumed that the level of critical stresses in the material in a uniaxial stress state models the critical state in a complex stress field if the theory of strength employed furnishes a basis in this approach for solving the problem. In order to show that experimental determination of the strength of material for any type of stress state is feasible, we turn first to the widely known [2-4] mathematical treatment of the investigation of a stress state at a point with a given stress tensor. We write the characteristic equation for the stress state in the vicinity of some point of the investigated solid [2-4]: - I,o +
- 4 = 0.
(1)
L'vov Polytechnic Institute. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 13, No. 2, pp. 51-55, March-April, 1977. Original article submitted December 24, 1975.
154
0038-5565/77/1302-0154507.50
9 1978 Plenum Publishing Corporation
where o n is normal stress in the plane passing through this point; 11, I=, and 13 are invariants of the stress tensor (2)
Ii "=%, + oy + a, = o x + 0= + %; xy
f3
=
" O-rOY~z "~ ~''.ryY'YZ~'Zx ~
"yz O'~T'yz ~-
zar -- r
~ a2 -~- o2 ~ -P ~ al;
(3)
O,~'~-rY= ~1 02 03;
(4)
r ZJ:~
ax, ay, and a z are the normal, Txy , Tyz, and ~zx the tangential, cipal stresses in three mutually perpendicular planes.
The total octahedral
stress
a2, and ~3 the prin-
(2) and (3), after simplification of
Joint consideration of the right sides of Eqs. their transforms, yields the equation
/,
and ~ ,
= 1~ - 2/, = +; + o~ + ~,+.
(5)
[2-4] is
toot = 3
2
3,
-
(6)
3
The normal octahedral stress in this same plane is __ V / , (l + 2A) ~OCI =
--'"
(7)
3
where A is a dimensionless parameter characterizing point in the solid
the type of stress state at the given
A =-12 = :2 4 ]~ 2I~ +
(8)
For the tangential stress on the octahedral plane, there is the well-known relation
r After substituting Eqs.
0"2tOot -- "~gCt'
(9)
(6) and (7) in Eq. (9), we obtain
89
Ai.
Solution of the problem is based on an analysis of change in components of the total octahedral stress and also in components of the balance of potential energy in elastic strain of a continuous medium with different invariants of the stress tensor given in dimensionless form. For this purpose Eqs. (7), (9), and (I0) may be represented as dimensionless linear functions:
~2
:o~t 1 a-L' = ~(I+2A);
(Ii)
:oo~ ~ (~ -- A);
(12)
~~oct+
3-77-,=3 4
3
,r
t
~=
1.
(13)
From these equations we see that the extreme numerical values of the dimensionless parameter (8) may be found within the limits--I/2 < A < +I~ A graphical representation of Eqs. (11)-(13) is shown in Fig. i. We shall investigate the processes of redistribution of components in the balance of specific potential strain energy c consumed, respectively, in change of volume ~v and form 9f of the material~ depending on the kind of stress state.
155
~0
~o=~ v.,f
'>\
Fig. i. Dimensionless functional relations characterizing the changes in normal and tangentialoctahedral stresses and also the distribution of balance of the potential elastic strain energy: i, 2, 3) dimensionless relations (ii), (12), and (13); 4, 5) dimensionless relations
a's \,x~ 3o
(18).
~.sY The balance equation of potential strain energy has the form ~v+3f=3,
(14)
where ~ is the total energy of specific elastic deformation of the material, determined by the Beltrami equation [2-4]
3 =~(i--2vA).
(15)
The part of the specific elastic strain energy spent on change in volume [2-4] is
`gv= 1~-2~_z~=(1-294(t +2A) 6E
0s
=
30-2~)o~ 2E
(16)
" oct"
The remaining specific elastic strain energy is expended on change in form:
3f = (] +~)4 ( 1 - - A ) = 3(I ~-------) + ~-gc, ' 3E
(17)
2F
where E is Young's modulus and v is Poisson's ratio. Curves illustrating the relations A ~ `9v/`9 and A - 9f/~ are shown in Fig. i. After substituting Eqs. (15)-(17) in Eq. (14) and after dividing both sides by the value of (15), we obtain components of the balance of specific elastic potential strain energy in the dimensionless form
3_v+ -~_f= (1 -2~)(1 + 2A) + 2(1 +~)(I --A) = I, 3
3
3(1 - 2~A)
(18)
3(1 - 2~A)
where`gv/3 and 3 f $ ~ are the components of balance of specific elastic potential strain energy expended on changes in volume and form of the solid. On the basis of Eq. (18) we may conclude that the redistribution of energy going into changes in volume and form depends only on the dimensionless value A, i.e., on the ratio of the invariants 12/14 characterizing the stress state. Using Eqs. (7), (i0), and (18) and assuming v = 0.3, let us examine the particular examples whose analysis permits us to arrive at definite conclusions. i.
The uniaxial stress state (~: = ~; ~2 = us = 0):
]',=~2; .
o
V~
`gv
aoct==~ ~ "=oct= --~ a; ~-=0.133;
~f --=0.867.,9
(19)
Isotropic tension or compression (02 = u2 = us = +-P);
A=I;
156
.4=0;
I2=[4=3W'; % c t = - - P ;
%ct=0;
3--~ 3
'
3f__=0. ,9
(20)
~5
Fig.
3.
The functional relation (24).
The case of pure shear (ox = ~; ~2 = 0; g3 =--~):
A=--~; 4.
2.
[~=2~2; aoct=0; " = 0 c t = ~ ;
The plane stress state 1
A=~;
I4=2~;
~V ~- -----0; "~f
0-=1.o.
(21)
9f = -ff 0.62.
(22)
(al = ~2 = ~; os = 0); %ct=2a;
3v ~; -ff=0.38;
%ct=
Let us examine the triaxial stress state obtained by the application of hydrostatic pressure on a uniaxial stress state in accord with the scheme
(23) The dimensionless value (8) for t h i s case i s s_2_ ~ A 1~ P p+
Here 1 2 = ~
3--2
;14=P ~ 3--2~-}-
~
Assigning different values to the ratio o/P, we may obtain values for the dimensionless value A within the limits --I/2 < A < +i. Analysis of the graphical relation in dimensionless coordinates (Fig. 2) permits us to draw a very important conclusion from the practical point of view, namely, that By the application of hydrostatic pressure on a uniaxial stress state it is possible to reproduce the same values of the parameter A that hold for any other stress state. In substituting each time in Eq. (24) the value of A obtained for different stress states (19)-(22) and solving the quadratic equation, we find the ratio o/P corresponding to each case. Equation (18) gives identical values of the components of the potential energy balance when A = const. It remains for us now to select the components of the stress state in accord with the scheme of (23) in such a fashion that we satisfy the equation of invariants for "nature" 14n and the model l~m and thereby obtain equal values of normal (7) and tangential (i0) octahedral stresses for equivalent cases. In this way it is possible to use fracture processes of materials in a complex state in our investigations. LITERATURE CITED
le 2. 3. 4~ 5.
P. W. Bridgman, Studies in Large Plastic Flow and Fracture with Emphasis on the Effects of Hydrostatic Pressure, McGraw-Hill, New York (1952). J. C. Jaeger, Elasticity, Strength, and Creep [in Russian], Mashgiz (1961). A. I. Ii'yushin, Plasticity: Principles of General Mathematical Theory [in Russian], Izd. Akad. Nauk SSSR (1963). S. P. Timoshenko, A Course in Elasticity Theory (edited by E. I. Grigolyuk) [in Russian], Naukova Dumka (1972). S. P. Timoshenko, Strength and Vibration of Constructional Elements [in Russian], Nauka (1975).
157