ENDOCHRONIC THEORY OF PLASTICITY AT FINITE DEFORMATIONS
Yulij Kadashevich, Sergei Pomytkin
Technological University of Plant Polymers Saint-Petersburg, Russia
ABSTRACT. Report is devoted to analysis of constitutive equations of endochronic type at finite deformations. Some new parametric endochronic variants of theory are presented. Methods of extension of endochronic theory of plasticity for finite deformations region are proposed. The various incremental theories of plasticity are initial. The variant-companion of endochronic theory is presented for each type of incremental one. Against the classical endochronic theory, the equations in parametric tensor form ( differential type ) are considered. The notions of reduced stresses, reduced strains and its rates are introduced into constitutive relations for generalization of endochronic theory on the finite deformations domain. Additionally the original strain and stress measures adequate to the approach are proposed. The new strain measure is compared with the wellknown ones on the simple shear test. The effective method of tensor function calculations is presented. The method is derived from formulae by Novozhilov and the common properties of matrix functions. The possibilities of presented theory for applications are demonstrated for simple and complex loading paths including the cyclic ones. Numerical simulations and theoretical predictions are compared with experimental data.
ENDOCHRONIC THEORY FOR SMALL STRAINS
Against endochronic theory presented by Valanis (1, 2) the constitutive equations of endochronic theory of plasticity in differential form can be written as (3)
(1.1)
or
,
, .
Here - deviators of stress tensor, strain tensor and parametric one, - endochronic parameter (), - shear modulus, - analog of strain yield limit (), - analog of hardening coefficient ().
Obviously that if then , and relations have the view
.
It would be pointed that if then , ( - deviator of plastic strain tensor, - Odquist’s parameter ) and above mentioned endochronic theory is incremental theory with linear Prager’s kinematical hardening.
At the same time another new endochronic equations of differential type can be used in plasticity
or
.
, , .
For all variants of equations the curve “stress-strain” under uniaxial active loading tend to an asymtote
.
INTERCOUPLING INCREMENTAL AND ENDOCHRONIC TEORIES
Analysing various linear equations between stress and plastic strain tensors within the framework of incremental theory with yield surface it was proved (3) that the highest order of derivate of plastic strain tensor must be greater by unit than the highest derivate of stress tensor. For example ( is deviator of backstress tensor, is tensor of active stress, , , , are constants),
;
or
, ;
or
, etc.
If the orders of derivates in left and right parts of are equal then the notion of yield surface is vanished. Therefore adding the items with augmented order of stress derivate to the left part of equation we can find the relation of endochronic type (without yield surface and without loading-unloading condition). So any variant of classical incremental theory of plasticity with isotropic and kinematical hardening has its own variant-companion of endochronic theory. For example,
- incremental theory
- endochronic theory-companion
.
More detail explanations can be found in (4).
ENDOCHRONIC THEORY AT FINITE DEFORMATIONS
In the process of extension of endochronic constitutive equations for finite (large) deformations domain next statements will be foundation of approach:
a) at small strains the theory is obliged to transform to ordinary theory of endochronic type;
b) oscillations of stress and deformations have to be absent in basic monotonic loading;
c) stress and deformation tensors are indifferent;
d) constitutive relations of the theory are to satisfy to objectiivty principle.
Suppose that the motion of material point with material coordinates () into a position with the spatial coordinates () is described by the vector function . (Further we omitt the lower indexes for tensors but the upper sign “T” is operation of matrix transposition). The deformation gradient is defined as and can be polar decomposed into . Here the symmetric positive tensors and are the right and left stretch tensors, and the proper orthogonal tensor is the rotation tensor. Further, for the velocity gradient the following additive decomposition formulas can be used
; ; .
Stretching tensor is the symmetric part of and vorticiy tensor is its skew-symmetric part. ( In the general case it would not confuse with the time rate of and with the derivative of strain measure tensor with respect to time). Additionally, the obvious relations can be indicated , .
Using these notations, reduced stresses, reduced strains and its rates are defined
.
Then these definitions are introduced into relations (1.1). We yield the endochronic constitutive equations for finite deformations ():
,
,
Proposed equations satisfy the principles a)-d). In particular, it is well known that any tensor transformed according to formula is called indifferent or independent of the observer.
MORE USUAL VIEW OF CONSTITUTIVE EQUATIONS
In order to apply the endochronic constitutive equations at finite deformations in more comfortable and usual manner we realize inverse convolution transformation by using operator . As result we yield
, (4.1)
where for any tensor we suppose that ( is named spin tensor)
, .
It can be seen from these equations that if then the relations have the simplest form
.
If deformation rates are set then the equations are solved very simply. If stress rates are assigned then the system are integrated a little more complex.
ON STRAIN MEASURE
According Hill’s proposals (5) a general class of Eulerian strain measures is used in mechanics usually
,
where are the distinct eigenvalues of symmetric positive definitive right stretch tensor . In particular, definition
,
is applied in practice. In this case defines the Hencky’s strain measure, assigns linear deformation and determines Green’s measure.
In fact the reduced strains and its rates
define a strain measure
. (5.1)
In equivalent form relations (3) are a system of ordinary differential equations
, .
For simple shear when deformation gradient and stretch rate are
and (5.2)
components of strain tensor, according (3), are defined by relations as
.
Strains calculated on basis of Green’s measure and Henky’ one are evaluated as
and
,
,
respectively. Some numerical values of strains are presented in table 1. Here it was accounted for that
and .
Table 1
Parameter / Measure (3) / Hencky's measuret / 11 / 12 / 11 / 12
0,5 / 0,208 / 0,435 / 0,215 / 0,431
1,0 / 0,571 / 0,693 / 0,623 / 0,623
5,0 / 2,140 / 3,333 / 2,267 / 0,454
10,0 / 3,126 / 7,830 / 2,983 / 0,300
SOME EXAMPLES
By tradition now consider the behaviour of stresses in simple shear of unit cube. Using the constitutive equations (4.1) with , , under deformation gradient (5.2) we yield the stress response presented in Fig.1. (Simulation was made in undimentional form).
Fig. 1
If the stress rate tensor
is given then coordinates of motion vector can be looked like relations
.
For this case deformation gradient and stretch rate are defined by tensors
and ,
rotation tensor and spin one are determinated as
and , .
The results of calculations of variables , and are displayed in Fig.2. The material constants were taken to be , , , , . The data of this numerical experiment has the good agreement with relults published in (6).
Fig. 2
Additionally it would be noted that under the cyclic deformation in simple shear conditions the Baushinger effect is described. Its response and magnitude don’t depend on strain rate and endochronic parameter.
Analysis of some another spin tensors and stress behaviour in the frameworks of incremental theory can be found, for example, in (7).
ON MATRIX FUNCTIONS
In the processes analytical and numerical simulation of finite deformations there is a need to calculate matrix functions very often. The basic method of calcucation for matrix its matrix function is well-known
.
Here is unit tensor, , , are eigen-values of tensor which are found from cubic equation usually. In two-dimensional case (when ) eigen-value is equal to always and residuary quadric equation is solved easy. But in the general case the solving of cubic equation can be laborious. Using the Novozhilov’s formulae (8)
, , ,
, , ,
the problem can be simplified.
ACKNOWLEDGMENTS
The research described in this publication was made possible in part by Grant No. 03-01-00770 from Russian Foundation for Basic Researches and Grant No. E-02-4.0-158 from Ministry of Education of Russian Federation.
REFERENCES
1. Valanis K.C.: A theory of viscoplasticity without a yield surface. Archiwum Mechaniki Stosowanej 1971 23 (4) 517–551.
2. Valanis K.C.: Fundamental consequences of a new intrinsic time measure: Plasticity as limit of the endochronic theory: Archives of Mechanics 1980 32 (2) 171-191.
3. Kadashevich Yu.I.: On various variants of tensoral linear relations in theory of plasticity. Researches in elasticity and plasticity 1967 6 39-45 (in russian).
4. Kadashevich Yu.I., Mikhailov A.N.: On the theory of plasticity without the yield of surface. Doklady Akademii Nauk SSSR 1980 254 (3) 574-576 (in russian).
5. Hill R.: Aspects of invariance in solid mechanics. Advances in Applied Mechanics 1978 18 1-75.
6. Xia Z., Ellyin F.: A finite elastoplastic constitutive formulation with new co-rotational stress-rate and strain-hardening rule. Journal of Applied Mechanics 1995 62 (4) 733-739.
7. Chen L.S., Zhao X.H., Fu M.F.: The simple shear oscillation and the restrictions to elastic-plastic constitutive relations. Applied Mathematics and Mechanics (English Edition) 1999 20 (6) 593-603.
8. Novozhilov V.V.: Elasticity theory. Leningrad, Sudpromgiz, 1958 (in russian).
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