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STRAIN CHARACTERISTICS OF MATERIALS FRACTURE
UNDER LOW-CYCLE PLASTIC DEFORMATION
A. Khromov, D. Fedorchenko1, E. Kocherov2, A. Bukhanko
Samara State Aerospace University named after S.P. Korolyov
34, Moskovskoe shosse, Samara, SU-443086, Russia;
[email protected], [email protected]
1
Kuznetsov Samara Research und Engineering complex
2
Samara Machine – Building Design Bureau
ABSTRACT
Plastic fracture of materials under quasistatic deformation processes is characterized by deformation and energy fracture
criteria. At the same time such criteria are tightly bound among themselves as any process of plastic deformation is
connected with energy dissipation. The basic problem will consist in separation of the deformation and energy parameters
independent from each other which characterizes tendency of the material to fracture. Principles of the theory
development of rigid-plastic body excluding nonuniqueness of plastic flow developed by authors are incorporated in the
basis of the suggested approach. These principles are based on sequential application of nonequilibrium thermodynamics
principles.
1. Introduction
One of features of the theory of ideal rigid-plastic body is the possibility of the description of finite strain tensor fields in
neighborhood of zones of discontinuity of body shape (angular point of notch, crack tip, etc.), [1]. These regions are strain
concentrators and, as a rule, fracture sources. Therefore the formulation of strain criteria of fracture is actual problem. The
present approach is formulated in papers [2-8]. Below generalization of this approach is offered with the formulation of
fracture criterion including both the strain state of material particles, and the specific energy dissipation accomplished by
the particle. Below generalization of this approach is offered with the formulation of fracture criterion including both the
strain state of material particles, and the specific energy dissipation accomplished by the particle. Necessity of such
criteria arises in the description of fracture processes on conditions of composite deformation, in particular, under cyclic
change of plastic strains. The basic problem of the formulation of fracture criterion (including deformation states and
energy dissipation on plastic deformation) is their communication which depends on the formulation of plasticity condition
and a particle deformation path.
2. Determination of the strain fields
As a measure of strains we take finite strain tensors of Cauchy Cij and Almansi E ij :
C ij = H ki H kj , Eij =
where H ij =
∂X i
∂x j
(
)
1
δ ij − Cij , i , j = 1,2,3
2
(1)
, X i and x i are Lagrangian and Eulerian coordinates of a material particle, respectively.
Associated law of plastic flow is
ε ij = λ
∂f
,
∂σ ij
(
λ > 0,
i , j = 1,2,3
(2)
)
1
ε ij = Vi , j + V j ,i ,
2
(
where f σ ij , E ij
) − loading function, σ ij
− stress tensor, ε ij − velocity strain tensor, Vi − the vector of displacement
velocity. We consider the rigid-plastic body on plasticity condition, to satisfying incompressibility condition, Bykovcev et al
[1]. The relationship between tensors E ij and ε ij is
DE ij
Dt
=
∂E ij
∂t
+
∂E ij
∂x k
Vk + E ik
∂Vk
∂Vk
+ E jk
= ε ij .
∂x j
∂x i
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(3)
3. Deformation states of incompressibility rigid-plastic body
The incompressibility condition
ε 1 + ε 2 + ε 3 = 0;
or
C1C 2C 3 = 1, C1 > 0, C 2 > 0, C 3 > 0;
(4)
or
(1 − 2E1 )(1 − 2E 2 )(1 − 2E 3 ) = 1,
determines in space E i a hyperbolic surface of the third order , fig. 1. The point O represents the initial strainless state.
In figure the projection of the surface
onto deviator plane with a normal line n is submitted (fig. 2). Here projections of
intersection lines of the surface
coordinate origin:
with the plane which is parallel to deviator plane located on distance h
h = (E1 + E 2 + E 3 ), (1 − 2E1 )(1 − 2E 2 )(1 − 2E 3 ) = 1 .
Figure 1. Deformation states surface of incompressibility rigid-plastic body
Figure 2. Level line of deformation states surface
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3 prior to the
(5)
4. Simple deformation processes
We consider velocities field of kind
V1 = x1ε 1 (t ), V2 = x 2 ε 21 (t ), V3 = x 3 ε 3 (t ),
(6)
where ε i (t ) − principal values of velocity strain tensor, which are functions of time t . In athermic plasticity theories the
time scale is not determined and can change during deformation. In particular, from three functions ε i (t ) it is possible to
set one function at will, for example, ε 1(t ) ≡ 1 . In this case from (4) follows, that
ε 1(t ) = 1,
ε 2 (t ) = −1 − ε 3 (t ) .
(7)
I.e. only one function ε 3 (t ) defines simple deformation process in space E i . If deformation process begins from
strainless state (point O, fig.1,2) then the system of the equations (4) under condition (6) and initial conditions
E ij
t =0
= 0 has the solution:
Ei =
(
)
1
1 − eτ i ( t ) , i = 1,2,3
2
(8)
t
∫
E12 =E 13 = E 23 ≡ 0, τ i (t ) = −2 ε i (t )dt .
0
5. Orthogonal deformation processes
We shall represent simple deformation processes by curves l on deviator plane (fig. 2). As process parameter (time) we
shall choose value h = E1 + E 2 + E 3 , E i = E i (h ) . If in the equation (8) to replace t with h , then
(
)
h
∫
1
h = 3 − eτ 1( h ) − eτ 2 ( h ) − eτ 3 ( h ) , τ i = −2 ε i (h )dh .
2
(9)
0
For simple deformation processes principal directions of velocity strain tensor are orthogonal to projections of lines (5) on
deviator plane which are level lines for function h = E1 + E 2 + E 3 . For other processes the values
εi
are connected by
relationships:
ε 1 + ε 2 + ε 3 = 0,
ε 1(1 − 2E1 ) + ε 2 (1 − 2E 2 ) + ε 3 (1 − 2E 3 ) = 1,
(10)
ε 1(1 − 2E1 )(E 2 − E 3 ) + ε 2 (1 − 2E 2 )(E 3 − E1 ) + ε 3 (1 − 2E 3 )(E1 − E 2 ) = 0.
Here the first equation defines incompressibility, the second equation follows from (9) after its differentiation by time h, the
third equation follows from a orthogonality condition of deformation process to projections of lines (5) on deviator plane. If
we shall introduce new process parameter t then carrying out change of variable h = h(t ) within (9) and differentiating by
t, we shall have the equation
ε1
(1 − 2E1 )h′ + ε 2 (1 − 2E 2 )h′ + ε 3 (1 − 2E 3 )h′ = 1.
h′
h′
h′
(11)
If strain and stress are connected to function h(t ) relationships σ i = (1 − 2E i )h ′ , ε i* =
εi
, then the equations (10) will
h′
become
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ε 1* + ε 2* + ε 3* = 0,
ε 1*σ 1 + ε 2*σ 2 + ε 3*σ 3 = 1,
(12)
ε 1*σ 1(σ 2 − σ 3 ) + ε 2*σ 2 (σ 3 − σ 1 ) + ε 3*σ 3 (σ 1 − σ 2 ) = 0.
The equations (12) define a cylindrical loading surface with directional line on deviator plane conterminous to projections
of lines (5) and generatrix parallel to n with hardening parameter. The first and third equations follow from the associated
flow law (2) for all deformation processes of incompressible rigid-plastic bodies. The second equation (12) will be
determined for all deformation processes if for process parameter to accept energy dissipation. The loading surface has
the following properties: the particle performs same specific energy dissipation D0 =
∫ ε σ dl
i
i
under a material
l
deformation by any simple deformation process of a datum point of strainless state up to strain level h = E1 + E 2 + E 3 ;
under the deformation, distinct from simple deformation process, it is required greater energy dissipation. For any
structural material the loading surface can be determined from standard experiment for uniaxial tension throughout
dependence of yield point from parameter h ( σ T = σ T (h) ).
6. Loading surface related with level lines of strain surface
We shall determine the equations (5) in components Cauchy’s tensor
H=
1
3
(C1 + C2 + C3 ),
C1C 2C 3 = 1, C i = 1 − 2E i ,
(13)
and we shall insert new coordinate system which is connected with deviator plane and a normal line to it by transformation
H=
x=−
1
2
C 1+
1
2
1
3
C1 +
1
3
y =−
C2 ,
C2 +
1
6
On deviator planes the equations (13) define level lines for function
1
3
C1 −
C3 ,
1
6
C2 +
2
6
C 3.
H:
2 x 3 − 6 xy 2 − 3 2Hx 2 − 3 2Hy 2 + 2 2H 3 − 6 6 = 0
(14)
or in polar coordinates x = ρ cos ϕ, y = ρ sin ϕ :
2 ρ 3 cos 3ϕ − 3 2Hρ 2 + 2 2H 3 − 6 6 = 0 .
The equations (14) and (15) determine the cylindrical loading surface as f (σ 1 − σ 2 , σ 2 − σ 3 , H ) = 0 , where
x=
−1
2h′
(σ 1 − σ 2 ),
H=
−2
3
y =−
1
3
x−
2
6h′
(σ 2 − σ 3 ),
(E1 + E 2 + E3 ) + 3 .
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(15)
Figure 3. Loading curve related with level lines
On fig. 3 curves (14) and (15) are submitted for distinctive values Н which show, that at small values Н they do not differ
almost from circumference. The curve 1 – for H =
3 + 10 −5 , the curve 2 – for H = 3 + 0.5 , the curve 3 – for
H = 3 + 1.5 . Yield point for tension and compression become essentially the distinctive at increase Н. In fig. 4 the
by a plane passing through an axis C1 and normal line n , which characterizes relationships of yield
section of surface
points for tension and compression ( σ TP , σ TC ) is submitted. It is possible to show, that lim
σ TP
H →∞ σ C
T
= 2 , lim
H→ 3
σ TP
σ TC
= 1.
Figure 4. Section of deformation states surface by plane of the symmetry passing
through an axis C1 and normal line n
7. Fracture conditions
Standard experimental researches (for tension of plane, cylindrical samples) show, that materials fracture occur under the
certain deformations E i* , and permit to define the minimal system of points on a surface , approximated by some
critical line. It is known, what even under small cyclically changeable plastic deformations there is the fracture of all
materials practically that is connected first of all with energy dissipation D =
∫ ε σ dt . Therefore the equation of critical
ij
ij
curve can be as
Ф (E1, E 2 , E 3 , D ) = 0,

(1 − 2E1 )(1 − 2E 2 )(1 − 2E 3 ) = 1.
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(16)
It is postulated, that when a curve corresponding to deformation process, crosses a critical line, there is a fracture of a
material.
Position of critical curve (5) can be determined for each structural material experimentally. If to assume, that stress-strain
properties of material correspond to loading surface of item 6, both the second and the third invariant of Almansi's strain
tensor a little influence for fracture of material then the equations of critical line (16) can be as
E1 + E 2 + E 3 = h(D ),

(1 − 2E1 )(1 − 2E 2 )(1 − 2E 3 ) = 1.
(17)
The critical line (17) coincides with line of the greatest possible hardening of the material, the determined value
hmax = h(D0max ) , where D0 is the energy dissipation under orthogonal deformation process. In this connection it is
supposed, that additional energy dissipation, producible by the material volume element under nonorthogonal deformation
reduces the material ability to ultimate hardening. The equations (16), (17) imply that the critical level line determining the
fracture point of each particle, verge towards strainless state during plastic deformation according to energy dissipation.
Function h(D ) should be determined experimentally.
References
[1] Bykovcev, G.I., & Ivlev, D.D., Plasticity Theory, Vladivostok, Russia, 1998.
[2] Khromov, A.I., Localization of plastic strains and fracture of ideal rigid-plastic bodies. Doklady Physics, 43(9), pp.
202–205, 1998.
[3] Kozlova, O.V. & Khromov, A.I., Fracture constants for ideal rigid-plastic bodies. Doklady Physics, 47(7), pp. 548–551,
2002.
[4] Khromov, A.I., Bukhanko, A.A. & Stepanov, A.L., Strain Raisers. Doklady Physics, 51(4), pp. 223–226, 2006.
[5] Khromov, A.I. Fracture of rigid-plastic bodies, fracture constants. Izv. RAS Mech. Solid, 3, pp. 137–152, 2005.
[6] Khromov, A.I., Deformation and fracture of rigid-plastic bar pulled tension, Izv. RAS Mech. Solid, 1, pp. 136–142,
2002.
[7] Khromov, A.I., Bukhanko, A.A., Kozlova, O.V. & Stepanov, A.L., Plastic Constants of Fracture. J. Appl. Mech. and
Techn. Phys. 47(2), pp. 274-281, 2006.
[8] Khromov, A.I., Kozlova, O.V., Fracture of Rigid-plastic Bodies. Fracture Constants, Vladivostok, Russia, 2005.
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