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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) ij d ij drij rij (1.1) 2G 2G dr dr g or d ij drij 1 g ij rij , dr dr g ij rij ij (1 ) , dr drij drij . 2G Here ij , ij , rij - deviators of stress tensor, strain tensor and parametric one, - endochronic parameter ( 0 1), G - shear modulus, - analog of strain yield limit ( 0 ), g - analog of hardening coefficient ( g 0 ). Obviously that if 1 then rij ij , dr d and relations have the view ij d ij d ij ij . 2G 2G d d g 1 It would be pointed that if 0 then rij ij , dr d ( ij - deviator of plastic p p 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 3 – 105 d ij drij (1 g ) ij rij dr dr (1 g ) g or ij d ij drij rij . 2G 2G dr dr (1 g ) g ij g rij (1 ) ij , dr drij drij , . 2G 1 g For all variants of equations the curve “stress-strain” under uniaxial active loading tend to an asymtote (1 g ) g , g 0; 0 . 2G 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 ( ij is deviator of backstress tensor, is tensor of active stress, , a , b , k are constants), ij ij ij ~ ij ij i i ~ d ij p ij k ij ; p d or ~ d ij p d ij d ij p ij ij , ij a1 b1 ij b2 p ; d d d or d ij d ij p d 2 ij p ij a2 b4 b3 ij p b5 , etc. d d d2 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 ~ d ij k p p ij d ij - endochronic theory-companion d ij ~ drij ~ ij k rij . 2G dr dr More detail explanations can be found in (4). 3 – 106 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 X ( 1, 2, 3 ) into a position with the spatial coordinates x ( 1, 2, 3 ) is described by the vector function x x( X , t ) . (Further we omitt the lower indexes for tensors but the upper sign “T” is operation of matrix transposition). The deformation gradient x is defined as F and can be polar decomposed into F R u v R . Here the X symmetric positive tensors u and v are the right and left stretch tensors, and the proper orthogonal tensor R is the rotation tensor. Further, for the velocity gradient L F F 1 the following additive decomposition formulas can be used 1 1 L D W ; D ( L LT ) ; W ( L LT ) . 2 2 Stretching tensor D is the symmetric part of L and vorticiy tensor W is its skew- symmetric part. ( In the general case it would not confuse W with the time rate of R and D with the derivative of strain measure tensor with respect to time). Additionally, the obvious relations can be indicated u 2 F T F , R F u 1 . Using these notations, reduced stresses, reduced strains and its rates are defined E R R ; E R DR ; T R R ; T R R ; N R r R ; N R r R . T T T T T T Then these definitions are introduced into relations (1.1). We yield the endochronic constitutive equations for finite deformations ( ): T 1 r T N r N, 2G 2G g T r D (1 ) , N E (1 ) 2G 2G Proposed equations satisfy the principles a)-d). In particular, it is well known that any tensor A transformed according to formula A R T A R 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 R R T . As result we yield 3 – 107 r r r r 2G 2G g D , (4.1) r D (1 ) 2G where for any tensor A we suppose that ( is named spin tensor) A A A A , R RT . It can be seen from these equations that if 1 then the relations have the simplest form D D D 2G 2G 1 g . D 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 3 3 u k I f (u ) f (i ) , i 1 k 1 i k k i where i are the distinct eigenvalues of symmetric positive definitive right stretch tensor u . In particular, definition un I , n 0, 1, 2 n is applied in practice. In this case n 0 defines the Hencky’s strain measure, n 1 assigns linear deformation and n 2 determines Green’s measure. In fact the reduced strains and its rates E RT R E RT D R define a strain measure t R ( R T D R) dt R T . (5.1) 0 In equivalent form relations (3) are a system of ordinary differential equations D , R RT . For simple shear when deformation gradient F and stretch rate D are 3 – 108 1 2t 0 0 1 0 F 0 1 0 and D 1 0 0 (5.2) 0 0 1 0 0 0 components of strain tensor, according (3), are defined by relations as 11 22 cos 2 (2 ln cos ) sin 2 (2 tg ) 12 cos 2 (2 tg ) sin 2 (2 ln cos ) . t tg Strains calculated on basis of Green’s measure and Henky’ one are evaluated as 11 0 , 22 2t 2 , 12 t and t ln 2 ln 2 11 , 12 1 t2 1 t 2 1 1, 2 1 t 2 t , 3 1 t 2 t , respectively. Some numerical values of strains are presented in table 1. Here it was accounted for that cos sin 0 0 1 0 1 R sin cos 0 and 1 0 0 . 0 1 t2 0 0 0 0 1 Table 1 Parameter Measure (3) Hencky's measure t 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 1 , 2G 1 , 1 /( g ) 0,1 under deformation gradient (5.2) we yield the stress response presented in Fig.1. (Simulation was made in undimentional form). 3 – 109 Fig. 1 If the stress rate tensor 0 0 ij 0 0 0 0 0 is given then coordinates of motion vector can be looked like relations x1 a ( X 1 e X 2 ) x 2 bX 2 . x cX 3 3 For this case deformation gradient and stretch rate are defined by tensors a ae a 0 2b a a e 0 a e b F 0 b 0 and D 0 , 0 0 c 2b b c 0 0 c rotation tensor and spin one are determinated as cos sin 0 0 1 0 ae R sin cos 0 and 1 0 0 , tg . 0 0 0 0 ab 0 1 The results of calculations of variables tg , b and a are displayed in Fig.2. The material constants were taken to be E 10 5 MPa , 0.25 , 0 207 MPa , 1 , 3 – 110 1 g 0.5 . 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 a its matrix function f (a) is well-known 3 3 (a k I ) f (a) f (i ) . i 1 k 1 (i k ) k i Here I is unit tensor, 1 , 2 , 3 are eigen-values of tensor a which are found from cubic equation det (a I ) 0 usually. In two-dimensional case (when a13 a 23 a31 a32 0 ) eigen-value 1 is equal to a33 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) 2 J 2 2 J 1 J 2 sin 1 , 2 J 2 sin 1 , 3 3 3 3 3 2 4 J 3 J 2 sin 1 , 3 3 3 6 J3 J 1 aii , J 2 aij a ji , J 3 aij a jk a ki , sin 3 3 J 22 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. 3 – 111 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. 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