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A version of the relations for the damage tensor of an elastoplastic medium
Journal of Applied Mathematics and Mechanics 75 (2011) 5–9
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
A version of the relations for the damage tensor of an elastoplastic medium夽 R.A. Vasin, P.A. Mossakovskii Moscow, Russia
a r t i c l e
i n f o
Article history: Received 22 September 2010
a b s t r a c t An approach to the description of damage accumulation in the deformation of an elastoplasticmaterial, based on Il’yushin’s ideas in the theory of elastoplastic processes, is proposed. A version of the relations for the damage tensor is formulated, from which the well-known Kolmogorov model follows as a special case. As an example, a class of deformation processes is considered for which the trajectories of the deformations are broken lines with extended links. © 2011 Elsevier Ltd. All rights reserved.
1. Models of damage accumulation There is an enormous literature devoted to fracture criteria and models of damage accumulation in deformable solids. While not aiming to review this literature, we mention two characteristic tendencies in investigations on this theme namely, the development of an approach based on describing the damage of a material within the limits of the mechanics of a damaged continuous medium and the use of tensor measures of damage. Relations, similar in structure to the constitutioe equations in Il’yushin’s1,2 theory of elastoplastic processes, are proposed in this paper. The constitiutive equtions of a damaged continuous medium are some or other connstitutive equations of an inelastic medium (a schleronomous elastoplastic medium, a rheonomic elastoviscoplastic medium) in which a parameter (or several parameters) appears reflecting the extent to which the material is damaged; an additional kinetic equation is introduced for this parameter. It is important that the constitutiove equtions of a damaged continuous medium are “coupled”, that is, the damage characteristics and the stress-strain state cannot be found independently. Several models of damage accumulation in creep theory (for example, see Ref 3 or Korotkikh’s4 model of thermoviscoplasticity) can serve as examples of such constitutive equations. The account of the damage described differs from the more traditional account in which the accumulated damage and (or) a check on the fracture condition are estimated using the independently determined stress-strain state of the material. An example is Bondar’s model of inelasticity 5 which is constructed, like Korotkikh’s model, on the basis of the flow theory with isotropic kinematie hardening. Another example is the fracture condition that is widely used in calculations of metal forming processes and proposed by Kolmogorov 6 for elastoplastic processes with small curvature deformation trajectories
(1.1)
Here, s is the arc length of the plastic deformation trajectory, p is a function of the temperature T, the essestive plastic strain rate q and the stress state characteristics, that is, the Lode parameter and the indicator of the form of the stress state k, which is equal to the ratio of the hydrostatic pressure to the essestive stress. In the case of simple loading, when and k are constants, the quantity p is equal to the arc length s at the instant of fracture (for given T(t) and u (t), where t is the time).
夽 Prikl. Mat. Mekh. Vol. 75, No. 1, pp. 8–14, 2011. E-mail address: [email protected] (R.A. Vasin). 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.04.002
6
R.A. Vasin, P.A. Mossakovskii / Journal of Applied Mathematics and Mechanics 75 (2011) 5–9
It is obvious that the law of essentially linear summation (1.1) is of little use in the case of complex loading. In Kolmogorov’s approach for intrinsically complex loading, represented by a stress trajectory in the form of a multi-link broken line, condition (1.1) is replaced by the condition
(1.2) in which n is the number of steps (links in the multi-link broken line) and the values of and k are constant for each step. Conditions (1.1) and (1.2) explicitly take account of the loading history of the material. It should be noted, however, that condition (1.2) does not continuously change into (1.1) (except in the obvious cases of n = 1 or ai = 1) and, in this sense, it is mathematically incorrect. One of the first detailed treatments of damage as a tensor object and of the damage measures constructed on its basis was proposed by Il’yushin7 , In his approach, the tensor characterizing the damage accumulation is assumed to be a function of the state of a macroparticle and, according to the principle of macroscopic determinability,2 it is uniquely lesinded by the loading process, that is, by the stress tensor with the components ij (t) (and, generally speaking, by the moments of different orders) and T(t). Specific versions of the damage accumulation theory were proposed by Il’yushin using a symmetric second rank damage tensor and different damage measures for the one and the same material. One version of Il’yushin’s approach was specified by Kiiko8 in terms of the deformation space in the following form. The damage is described by a second rank deviatorie tensor to which a vector P with components
(1.3) Corresponds in deformation space. Here, j () are the components of the strain rate vector v() ; = |v| and Aij are experimentally determined functions (the corresponding functionals can appear in relation (1.3) instead of the functions Aij ); pu = |P| is taken as one of the measures of accumulated plastic strain, and the condition
then corresponds to the exhaustion of plasticity resources. For the particular form of relations (1.3) when
p
−1
the material functions Ai = (i (T, k, )) (i = 1, 3) are found from tensile experiments when 1 = 10 = const and torsional experiments p p when 3 = 30 = const; 1 and 3 are the ultimate plasticity functions for stretching and torsion respectively. Other approaches to the description of the tensor nature of the damage of materials are also known. In particular, the damage tensor is specified (as in this paper) by some kinetic equation. Thus, the equation for the damage tensor ˝ij was taken9 in the form
(1.4) where ␣ij is the deviatorie residual microstress (“additional” stress) tensor, W is the fracture energy, nij is the inelastic strain and is a material function. An approach to the construction of a fracture criterion is proposed below in which allowance for the loading history is based on Il’yushin’s ideas in the theory of elastoplastic processes1,2 and from which a relation of the type of (1.1) follows as a special case. 2. Construction of the relations for the damage tensor For simplicity, these relations are constructed for a schleronomous, initially isotropic material in a geometrically linear formulation. The general procedure for obtaining the relations for a damage tensor with the components ij consists of successive repetition of a scheme according to which the constitutive relations are constructed in the theory of elastoplastic processes1 with the replacement of p the strain space E5 by the plastic deformations space E5 and the replacement of the stress tensor by the damage tensor. The vector , which is defined in the space ˝5 and is assumed to be one of the physical vectors of the loading process, corresponds to the symmetric p second-rank deviatorie tensor ij . At each point of the plastic deformation trajectory in the space E5 , the vector can be represented in a
Frenet reference frame Pi in the form
Here, i and = |ω| are functionals of the process. By analogy with the coplanarity hypothesis (see Ref. 10), the evolutionary equation for the vector can be chosen in the form
where de is an increment in the plastic strain vector, ds = de, and A and B are functionals of the process.
(2.1)
R.A. Vasin, P.A. Mossakovskii / Journal of Applied Mathematics and Mechanics 75 (2011) 5–9
7
The condition for attaining a critical value of the damage trajectory arc length (of the hodograph of the vector in the space 5 ) can be taken as the criterion of the fracture of the material.
(2.2) or the more traditional condition (2.3) Versions of relations of the type (2.1), that take account of the tensor character of damage accumulation, are known in the literature (for example, see relation (1.4) presented above). However, in these relations, the dependence of the functionals A and B on the most characteristic “vector” parameter, that is, the convergence angle ; (2.4) has not been investigated to any great extent At the same time, the “vector” properties (the misalignment of the vectors 0 and p1 ) are of undoubted interest. We will represent relations (2.1) in the form of separate equations for , the characteristics of the “scalar” properties, and for , the characteristics of the vector properties. Multiplying both sides of relation (2.1) by 0 and p1 , after some reduction we obtain (a prime denotes differentiation with respect to s)
(2.5) Henceforth, the discussion is confined to the class of deformation trajectories in the form of multi-link broken lines. The deformation trajectory curvature is then equal to zero for each link and, consequently, the second equation of (2.5) is simplified:
(2.6) Assuming that F and A in relations (2.5) and (2.6) (or A and B in Eq. (2.1)) are certain functions of s, of the process parameters ␣i and the structural parameters of the material j (and, correspondingly, supplementing relations (2.5) and (2.6) by the equations for determining these parameters), specific special versions of the relations for calculating of can be formulated. In choosing the function F, account can be taken of the fact that the writing of the kinetic equation for is often based on the use of the energy approach when the condition of proportionality of d and dW is assumed:
(the latter expression was used in the Novoshilov school; for ␣ij , see relation (1.4)). We will now consider the simple case when the lengths of the deformation trajectory links are fairly large such that the “retardation trace” for the vector is exhausted and, consequently, the change in the angle at each step can be considered to depend on the previous history. We will assume that the first of equations (2.5) has the form (2.7) p
Here r is of the material constant, is a function that has the same meaning as in formula (1.1), exept that the angles defining rays in p the E5 space are annded to its arguments, * is a piecewise-constant function of the process.
(2.8) 0k is the deflection angle of the k-th link of the deformation trajectory, f is a function which is subject to experimental determination (from a series of experiments on two-link trajectories with different angles of deflection 0 , for example) and which satisfies the conditions f (0) = 0, f and (1) = 1. Assuming that p is constant in the k-th link, integration of Eq. (2.7) gives
(2.9) for the k-th link. On the basis of this relation using expression (2.8) and condition (2.3), the constant r can be determined from an experiment with torsion in one direction up to the value s = s1 and then in the opposite direction until fracture when s = s2:
where is the deformation corresponding to the ultimate of the static torsional strength.
8
R.A. Vasin, P.A. Mossakovskii / Journal of Applied Mathematics and Mechanics 75 (2011) 5–9
Within the limits of the proposed model, with certain simplifications (r = 2, the ultimate deformations are identical under tension, and torsion and infinite under compression,), a comparison of the lengths of the deformation trajectories at the instant of fracture as a function of the magnitude of a side of the square was carried out for deformation trajectories in the form of squares with their centre at the coordinate origin (see Ref. 11). As is also observed in experiments, it was found that the deformation trajectory arc at the instant of fracture will have a greater length in the case of “walking” along a trajectory in the form of a square with a smaller side. The determination of the “retardation trace” for the vector (together with the establishment of the form of the functions A and B in relation (2.1)) should be designated as one of the urgent experimental problems in the investigation of damage accumulation. Formula (2.9), in combination with the fracture criterion (2.3) or (2.2), allows of direct experimental verification and enables one to determine the plasticity margin at each instant of the process up to fracture. A fracture condition of the form (1.2) follows from it as a special case but only condition (2.9), unlike (1.2), is mathematically correct and does not lead to a formal contradiction, for example, when a straight link AB of the deformation trajectory is replaced by two links AC and CB. 3. Formulation of the approach (2.1) - (2.5) in a geometrically non-linear statement Equations. (2.1) - (2.5), written in tensor form, are interpreted as the governing equations with reference to a certain mobile “quasima(m) terial” reference frame ki (that is, a reference frame with an orientation relative to the oblique Lagrange reference frame i that is solely determined by the prehistory of the metric gij (), gij = i i . During the deformation of a particle of a continuous medium, the reference (m)
frame ki
(m) according to the equation rotates with respect to its initial position ei with an angular velocity ˝
(3.1)
(m) is the skew-symmetric tensor associated with the axial angular velocity vector. The components of the strain tensor εij (with where ˝ (m)
respect to the reference frame ki
) are determined by the equation (3.2)
is the strain rate tensor. where The usual relations (3.3) are taken for the elastic strain deviator with the components eije , where G is the shear modulus. Then, (3.4) We now present the covariant tensor form of relations (2.1) - (2.5) for the two most common versions of setting the reference frame ki
(1) = ω (2) = ˝ R, ˝ (R) is the angular velocity ot the relative rotation of the right and left principal Here ˝ , ω is the Jaumann spin and ˝ strain trihedra. In tensor form, Eqs. (2.1) - (2.5) are as follows;
where
(x)
A corotational derivative of a tensor, taken with respect to the trihedron ki asterisk.
(x) is denoted by an rotating with an angular velocity ˝
References 1. 2. 3. 4. 5. 6. 7. 8.
Il’yushin AA. On the relation between stresses and small strains in the continuam. Prikl Mat Mekh 1954;18(6):641–66. Il’yushin AA. Plasticity. Foundations of the General Mathematical Theory. Moscow: Izd Akad Nauk SSSR; 1963. Shesterikov SA. Some problems of long-term durability and creep. In Shesterikov SA. Selected Papers Moscow:Izd MGU 2007:134–48. Volkov IA, Korotkikh YuG. Equations of State of Viscoelastoplastic Media with Defects. Moscow:Fizmatgiz;2008. Bondar’ VS. Inelasticity. Versions of the Theory. Moscow:Fizmatgiz;2004. Kolmogorov VL. Mechanics of the Metal Forming Processes. Eekaterinburg:Izd. Ural’sk Gos. Tekhn Univ. (UPI);2001. Il’yushin AA. On one theory of long-term strength. Inzh Zh MTT 1967;3:21–35. Zavoichinskaya EB, Kiiko IA. Introduction to the Theory of Fracture of Solid. Moscow: Izd MGU; 2004.
R.A. Vasin, P.A. Mossakovskii / Journal of Applied Mathematics and Mechanics 75 (2011) 5–9
9
9. Bondar’ VS. Mathematical model of the inelastic behaviour as a material and damage accumulation. In: Applied Problems of Strength and Plasticity. Methods of Solution. Gorkii:Gork.Univ.;1987:24-8. 10. Vasin RA, Il’yushin AA. A representation of the elasticity and plasticity laws in plane problems. Izd Akad Nauk SSSR MTT 1983;4:114–8. 11. Ohashi Y, Tanaka E, Ooka M. Plastic deformation behaviour of type 316 stainless steel sbuject to out-of-phase strain cycles//. Trans Jap Soc Mech Eng 1985;51(461):72–9.
Translated by E.L.S.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
A version of the relations for the damage tensor of an elastoplastic medium夽 R.A. Vasin, P.A. Mossakovskii Moscow, Russia
a r t i c l e
i n f o
Article history: Received 22 September 2010
a b s t r a c t An approach to the description of damage accumulation in the deformation of an elastoplasticmaterial, based on Il’yushin’s ideas in the theory of elastoplastic processes, is proposed. A version of the relations for the damage tensor is formulated, from which the well-known Kolmogorov model follows as a special case. As an example, a class of deformation processes is considered for which the trajectories of the deformations are broken lines with extended links. © 2011 Elsevier Ltd. All rights reserved.
1. Models of damage accumulation There is an enormous literature devoted to fracture criteria and models of damage accumulation in deformable solids. While not aiming to review this literature, we mention two characteristic tendencies in investigations on this theme namely, the development of an approach based on describing the damage of a material within the limits of the mechanics of a damaged continuous medium and the use of tensor measures of damage. Relations, similar in structure to the constitutioe equations in Il’yushin’s1,2 theory of elastoplastic processes, are proposed in this paper. The constitiutive equtions of a damaged continuous medium are some or other connstitutive equations of an inelastic medium (a schleronomous elastoplastic medium, a rheonomic elastoviscoplastic medium) in which a parameter (or several parameters) appears reflecting the extent to which the material is damaged; an additional kinetic equation is introduced for this parameter. It is important that the constitutiove equtions of a damaged continuous medium are “coupled”, that is, the damage characteristics and the stress-strain state cannot be found independently. Several models of damage accumulation in creep theory (for example, see Ref 3 or Korotkikh’s4 model of thermoviscoplasticity) can serve as examples of such constitutive equations. The account of the damage described differs from the more traditional account in which the accumulated damage and (or) a check on the fracture condition are estimated using the independently determined stress-strain state of the material. An example is Bondar’s model of inelasticity 5 which is constructed, like Korotkikh’s model, on the basis of the flow theory with isotropic kinematie hardening. Another example is the fracture condition that is widely used in calculations of metal forming processes and proposed by Kolmogorov 6 for elastoplastic processes with small curvature deformation trajectories
(1.1)
Here, s is the arc length of the plastic deformation trajectory, p is a function of the temperature T, the essestive plastic strain rate q and the stress state characteristics, that is, the Lode parameter and the indicator of the form of the stress state k, which is equal to the ratio of the hydrostatic pressure to the essestive stress. In the case of simple loading, when and k are constants, the quantity p is equal to the arc length s at the instant of fracture (for given T(t) and u (t), where t is the time).
夽 Prikl. Mat. Mekh. Vol. 75, No. 1, pp. 8–14, 2011. E-mail address: [email protected] (R.A. Vasin). 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.04.002
6
R.A. Vasin, P.A. Mossakovskii / Journal of Applied Mathematics and Mechanics 75 (2011) 5–9
It is obvious that the law of essentially linear summation (1.1) is of little use in the case of complex loading. In Kolmogorov’s approach for intrinsically complex loading, represented by a stress trajectory in the form of a multi-link broken line, condition (1.1) is replaced by the condition
(1.2) in which n is the number of steps (links in the multi-link broken line) and the values of and k are constant for each step. Conditions (1.1) and (1.2) explicitly take account of the loading history of the material. It should be noted, however, that condition (1.2) does not continuously change into (1.1) (except in the obvious cases of n = 1 or ai = 1) and, in this sense, it is mathematically incorrect. One of the first detailed treatments of damage as a tensor object and of the damage measures constructed on its basis was proposed by Il’yushin7 , In his approach, the tensor characterizing the damage accumulation is assumed to be a function of the state of a macroparticle and, according to the principle of macroscopic determinability,2 it is uniquely lesinded by the loading process, that is, by the stress tensor with the components ij (t) (and, generally speaking, by the moments of different orders) and T(t). Specific versions of the damage accumulation theory were proposed by Il’yushin using a symmetric second rank damage tensor and different damage measures for the one and the same material. One version of Il’yushin’s approach was specified by Kiiko8 in terms of the deformation space in the following form. The damage is described by a second rank deviatorie tensor to which a vector P with components
(1.3) Corresponds in deformation space. Here, j () are the components of the strain rate vector v() ; = |v| and Aij are experimentally determined functions (the corresponding functionals can appear in relation (1.3) instead of the functions Aij ); pu = |P| is taken as one of the measures of accumulated plastic strain, and the condition
then corresponds to the exhaustion of plasticity resources. For the particular form of relations (1.3) when
p
−1
the material functions Ai = (i (T, k, )) (i = 1, 3) are found from tensile experiments when 1 = 10 = const and torsional experiments p p when 3 = 30 = const; 1 and 3 are the ultimate plasticity functions for stretching and torsion respectively. Other approaches to the description of the tensor nature of the damage of materials are also known. In particular, the damage tensor is specified (as in this paper) by some kinetic equation. Thus, the equation for the damage tensor ˝ij was taken9 in the form
(1.4) where ␣ij is the deviatorie residual microstress (“additional” stress) tensor, W is the fracture energy, nij is the inelastic strain and is a material function. An approach to the construction of a fracture criterion is proposed below in which allowance for the loading history is based on Il’yushin’s ideas in the theory of elastoplastic processes1,2 and from which a relation of the type of (1.1) follows as a special case. 2. Construction of the relations for the damage tensor For simplicity, these relations are constructed for a schleronomous, initially isotropic material in a geometrically linear formulation. The general procedure for obtaining the relations for a damage tensor with the components ij consists of successive repetition of a scheme according to which the constitutive relations are constructed in the theory of elastoplastic processes1 with the replacement of p the strain space E5 by the plastic deformations space E5 and the replacement of the stress tensor by the damage tensor. The vector , which is defined in the space ˝5 and is assumed to be one of the physical vectors of the loading process, corresponds to the symmetric p second-rank deviatorie tensor ij . At each point of the plastic deformation trajectory in the space E5 , the vector can be represented in a
Frenet reference frame Pi in the form
Here, i and = |ω| are functionals of the process. By analogy with the coplanarity hypothesis (see Ref. 10), the evolutionary equation for the vector can be chosen in the form
where de is an increment in the plastic strain vector, ds = de, and A and B are functionals of the process.
(2.1)
R.A. Vasin, P.A. Mossakovskii / Journal of Applied Mathematics and Mechanics 75 (2011) 5–9
7
The condition for attaining a critical value of the damage trajectory arc length (of the hodograph of the vector in the space 5 ) can be taken as the criterion of the fracture of the material.
(2.2) or the more traditional condition (2.3) Versions of relations of the type (2.1), that take account of the tensor character of damage accumulation, are known in the literature (for example, see relation (1.4) presented above). However, in these relations, the dependence of the functionals A and B on the most characteristic “vector” parameter, that is, the convergence angle ; (2.4) has not been investigated to any great extent At the same time, the “vector” properties (the misalignment of the vectors 0 and p1 ) are of undoubted interest. We will represent relations (2.1) in the form of separate equations for , the characteristics of the “scalar” properties, and for , the characteristics of the vector properties. Multiplying both sides of relation (2.1) by 0 and p1 , after some reduction we obtain (a prime denotes differentiation with respect to s)
(2.5) Henceforth, the discussion is confined to the class of deformation trajectories in the form of multi-link broken lines. The deformation trajectory curvature is then equal to zero for each link and, consequently, the second equation of (2.5) is simplified:
(2.6) Assuming that F and A in relations (2.5) and (2.6) (or A and B in Eq. (2.1)) are certain functions of s, of the process parameters ␣i and the structural parameters of the material j (and, correspondingly, supplementing relations (2.5) and (2.6) by the equations for determining these parameters), specific special versions of the relations for calculating of can be formulated. In choosing the function F, account can be taken of the fact that the writing of the kinetic equation for is often based on the use of the energy approach when the condition of proportionality of d and dW is assumed:
(the latter expression was used in the Novoshilov school; for ␣ij , see relation (1.4)). We will now consider the simple case when the lengths of the deformation trajectory links are fairly large such that the “retardation trace” for the vector is exhausted and, consequently, the change in the angle at each step can be considered to depend on the previous history. We will assume that the first of equations (2.5) has the form (2.7) p
Here r is of the material constant, is a function that has the same meaning as in formula (1.1), exept that the angles defining rays in p the E5 space are annded to its arguments, * is a piecewise-constant function of the process.
(2.8) 0k is the deflection angle of the k-th link of the deformation trajectory, f is a function which is subject to experimental determination (from a series of experiments on two-link trajectories with different angles of deflection 0 , for example) and which satisfies the conditions f (0) = 0, f and (1) = 1. Assuming that p is constant in the k-th link, integration of Eq. (2.7) gives
(2.9) for the k-th link. On the basis of this relation using expression (2.8) and condition (2.3), the constant r can be determined from an experiment with torsion in one direction up to the value s = s1 and then in the opposite direction until fracture when s = s2:
where is the deformation corresponding to the ultimate of the static torsional strength.
8
R.A. Vasin, P.A. Mossakovskii / Journal of Applied Mathematics and Mechanics 75 (2011) 5–9
Within the limits of the proposed model, with certain simplifications (r = 2, the ultimate deformations are identical under tension, and torsion and infinite under compression,), a comparison of the lengths of the deformation trajectories at the instant of fracture as a function of the magnitude of a side of the square was carried out for deformation trajectories in the form of squares with their centre at the coordinate origin (see Ref. 11). As is also observed in experiments, it was found that the deformation trajectory arc at the instant of fracture will have a greater length in the case of “walking” along a trajectory in the form of a square with a smaller side. The determination of the “retardation trace” for the vector (together with the establishment of the form of the functions A and B in relation (2.1)) should be designated as one of the urgent experimental problems in the investigation of damage accumulation. Formula (2.9), in combination with the fracture criterion (2.3) or (2.2), allows of direct experimental verification and enables one to determine the plasticity margin at each instant of the process up to fracture. A fracture condition of the form (1.2) follows from it as a special case but only condition (2.9), unlike (1.2), is mathematically correct and does not lead to a formal contradiction, for example, when a straight link AB of the deformation trajectory is replaced by two links AC and CB. 3. Formulation of the approach (2.1) - (2.5) in a geometrically non-linear statement Equations. (2.1) - (2.5), written in tensor form, are interpreted as the governing equations with reference to a certain mobile “quasima(m) terial” reference frame ki (that is, a reference frame with an orientation relative to the oblique Lagrange reference frame i that is solely determined by the prehistory of the metric gij (), gij = i i . During the deformation of a particle of a continuous medium, the reference (m)
frame ki
(m) according to the equation rotates with respect to its initial position ei with an angular velocity ˝
(3.1)
(m) is the skew-symmetric tensor associated with the axial angular velocity vector. The components of the strain tensor εij (with where ˝ (m)
respect to the reference frame ki
) are determined by the equation (3.2)
is the strain rate tensor. where The usual relations (3.3) are taken for the elastic strain deviator with the components eije , where G is the shear modulus. Then, (3.4) We now present the covariant tensor form of relations (2.1) - (2.5) for the two most common versions of setting the reference frame ki
(1) = ω (2) = ˝ R, ˝ (R) is the angular velocity ot the relative rotation of the right and left principal Here ˝ , ω is the Jaumann spin and ˝ strain trihedra. In tensor form, Eqs. (2.1) - (2.5) are as follows;
where
(x)
A corotational derivative of a tensor, taken with respect to the trihedron ki asterisk.
(x) is denoted by an rotating with an angular velocity ˝
References 1. 2. 3. 4. 5. 6. 7. 8.
Il’yushin AA. On the relation between stresses and small strains in the continuam. Prikl Mat Mekh 1954;18(6):641–66. Il’yushin AA. Plasticity. Foundations of the General Mathematical Theory. Moscow: Izd Akad Nauk SSSR; 1963. Shesterikov SA. Some problems of long-term durability and creep. In Shesterikov SA. Selected Papers Moscow:Izd MGU 2007:134–48. Volkov IA, Korotkikh YuG. Equations of State of Viscoelastoplastic Media with Defects. Moscow:Fizmatgiz;2008. Bondar’ VS. Inelasticity. Versions of the Theory. Moscow:Fizmatgiz;2004. Kolmogorov VL. Mechanics of the Metal Forming Processes. Eekaterinburg:Izd. Ural’sk Gos. Tekhn Univ. (UPI);2001. Il’yushin AA. On one theory of long-term strength. Inzh Zh MTT 1967;3:21–35. Zavoichinskaya EB, Kiiko IA. Introduction to the Theory of Fracture of Solid. Moscow: Izd MGU; 2004.
R.A. Vasin, P.A. Mossakovskii / Journal of Applied Mathematics and Mechanics 75 (2011) 5–9
9
9. Bondar’ VS. Mathematical model of the inelastic behaviour as a material and damage accumulation. In: Applied Problems of Strength and Plasticity. Methods of Solution. Gorkii:Gork.Univ.;1987:24-8. 10. Vasin RA, Il’yushin AA. A representation of the elasticity and plasticity laws in plane problems. Izd Akad Nauk SSSR MTT 1983;4:114–8. 11. Ohashi Y, Tanaka E, Ooka M. Plastic deformation behaviour of type 316 stainless steel sbuject to out-of-phase strain cycles//. Trans Jap Soc Mech Eng 1985;51(461):72–9.
Translated by E.L.S.