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Creep fracture of plates with variable bending moments in the presence of an aggressive medium
Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Creep fracture of plates with variable bending moments in the presence of an aggressive medium夽 A.M. Lokoshchenko, L.V. Fomin The Scientific Research Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 30 March 2015 Available online 11 July 2016
a b s t r a c t The dissipated fracture of a rectangular plate with bending under conditions of a time-varying plane stress state taking account of the effect of an aggressive medium is investigated. A method for determining the time up to the fracture of such a plate when it is successively bent in different planes is proposed. Using Rabotnov’s kinetic theory, the times up to fracture are compared in the case of scalar and vector damage parameters using a power creep model. The deviations of the sum of the partial times from unity are investigated in the case of piece-wise constant relations between the level of the bending moment and time. An analogy with the results of experiments on the creep strength of rods with a piecewise-constant tensile stress is shown. © 2016 Elsevier Ltd. All rights reserved.
1. Statement of the problem The creep fracture of a recangular plate of thickness 2H in an aggressive surrounding medium that is under the action of linear bending moments M1 and M2 distributed along its edges (Fig. 1) is considered. The effect of the aggressive medium is determined by the diffusive penetration of its elements into the plate material, leading to a reduction in the duration of the operating capacity of the plate. The problem of determining the time up to the fracture of the plate for different combinations of distributed bending moments along its edges is formulated on the basis of Rabotnov’s kinetic theory1 of creep and creep strength taking account of scalar and vector damage parameters. Here, the natural assumption is used that the thickness 2H is considerably smaller than the other two dimensions of the plate and the diffusion process along the plate edges is therefore not taken into account. 2. Approximate method of solving the diffusion equation Exact solutions of the diffusion equation for bodies with constant boundaries are usually represented in the form of trigonometric series or series consisting of special functions that do not always allow obtaining a representation of the sought after characteristics in comprehensible form. These solution are very cumbersome, and, to obtain an acceptable accuracy in the calculations, a large number of terms have to be retained in the series. For analysing the process of the diffusion of an aggressive surrounding medium into the plate, an approximate method of solving the diffusion equation is used based on the introduction of a diffusion front that propagates from the plate surface.2,3 This approach enables us to separate all the cross section of the plate into a perturbed region (in which the medium has already penetrated into the material) and an unperturbed region (into which there is not yet any penetration of the medium), and then to track the motion of the boundary between these regions in time, . The coefficient of diffusion of the aggressive medium into the plate material is denoted by D and, as the dimensionless variables, we take
夽 Prikl. Mat. Mekh., Vol. 80, No. 2, pp. 276–284, 2016. E-mail address: [email protected] (A.M. Lokoshchenko). http://dx.doi.org/10.1016/j.jappmathmech.2016.06.006 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
199
Fig. 1.
where Z is the coordinate along the thickness of the plate (Z = 0 on the middle line of the cross section of the plate), C is the concentration of the aggressive medium in the plate and C0 = const is the concentration level on the boundary between the plate material and the external medium. From the condition for the symmetry of the diffusion process, a half of the cross section of the plate with respect to its thickness is considered, In these variables, the one-dimensional diffusion equation takes the form
(2.1) The initial and boundary conditions
(2.2) are satisfied exactly and the diffusion equation is satisfied integrally in the whole cross section of the plate. The relation between the concentration c and the coordinate z is taken in the form of a quadratic polynomial that satisfies the initial and boundary conditions. Two stages in the diffusion process are considered here: the stage of the front penetration and the saturation stage which are separated by the instant t0 :2,3
(2.3) where l(t) is the coordinate of the diffusion front, t0 is the time of the change between the stages of the diffusion process (l(t0 ) = 0) and B(t) is the concentration at the centre of the plate cross section when t ≥ t0 (on the middle line z = 0). The unknown relations l(t) and B(t) are determined from the condition of the integral satisfaction of diffusion equation (2.1) by the function c(z, t) (2.3):
(2.4) There is no modulus sign in the integrand since it follows from the results obtained earlier4 that, in this case, the difference between the approximate solution of the diffusion equation obtained and the exact solution is just a few percent. Substituting expression (2.3) into (2.1) taking account of relations (2.2) and (2.4), we obtain the relations between the coordinate of the diffusion front l(t) and the concentration B(t) and time (2.5) The time t0 = 1 is determined on the basis of the boundary condition l(t0 ) = 0.
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A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
Using relation (2.5), from expression (2.3) we obtain
(2.6) The integrally mean concentration in the volume of the plate
(2.7) (2.8) will later be used to analyse the effect of the aggressive medium on the time up to the fracture of the plate. 3. Determination of the components of the stress-strain state The effect of the aggressive medium on the time up to when the plate fractures is taken into account by introducing a function of the integrally mean concentration of the aggressive medium in the plate material into the constitutive power relation: (3.1) where p˙ u is the intensity of the creep strain rates, the dot over pu denotes a derivative with respect to t, u is the stress intensity, and A1 and n are constants. In addition, we assume that a plane stress state (3 = 0) is realized in the plate, the deformations are small, the condition of the incompressibility of the plate material is satisfied and that elastoplastic deformations are not taken into account. Neglecting the change in the length of the middle line in a section, we obtain (3.2) ˙ i are the rates of change of the curvature (from now on, the subscript i takes the values 1 and 2 everywhere). In this case,
(3.3) From constitutive relation (3.1), taking account of equality (3.3) we express the stress intensity:
(3.4) We use the hypothesis of the proportionality of the stress deviators and the creep strain rates p˙ ij for a plane stress state
(3.5) On the basis of relations (3.4) and (3.5) and the plane-sections hypothesis (3.2), we write
(3.6) From equality (3.6), taking account of relation (3.2) we obtain the expressions for the stresses
(3.7) The equilibrium equations have the form
(3.8) The equality to zero of the axial forces is satisfied automatically according to equality (3.7). Substituting expressions (3.7) into equilibrium equations (3.8), we obtain
(3.9)
A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
201
Fig. 2.
We now introduce the quantity Mu and, using relations (3.3) and (3.9), we obtain the relation between Mu and ˙ u :
(3.10) We connect the rates of change of the curvatures and the bending moments by means of the following equations (in analogy with the connection between the stress deviators and the creep strain rates):
(3.11) Substituting the rates of change of the curvatures (3.11) in the expressions for the stresses (3.7), taking account of relation (3.10) we obtain (3.12) Hence, the stresses (unlike creep strain rates) are independent of the presence of the aggressive medium. The relation between the dimensionless stress 1 /d, where
and the dimensionless transverse coordinate of the plate z = Z/H is shown in Fig. 2 for 2H = 1 mm, M1 = 1 N and n = 3. 4. Creep rupture of a plate for a piecewise-constant relation between the bending moments and the time We will consider the following programme for loading the plate. Initially (when t = 0) a bending moment M1 > 0 is instantaneously applied to a pair of oppositely located edges of the plate. This moment acts during a time 0 < t ≤ t1∗ /2 (where t1∗ is the time up to when the plate fractures under the action of this moment M1 ) and, here, M2 = 0 (Fig. 1). The conditions M1 = 0, M2 > 0 are then satisfied when t1∗ /2 < t < t ∗ (t* is the time up to when plate fractures). To determine the time t*, we use Rabotnov = s kinetic theory of creep and creep rupture and, with this aim, we introduce a function of the integrally mean concentration of the aggressive surrounding medium in the plate material into the kinetic equation. Taking account of the scalar defectiveness parameter, we consider the kinetic equation in the form
(4.1) where 0 is a constant quantity of dimension MPa. Since, according to the loading mode adopted, only one of the two bending moments acts at each stage (the other moment is equal to zero), the time up to fracture can be determined taking the following expressions into account: (4.2) It follows from relations (3.12), (4.1) and (4.2) that the equality of the damage parameter to unity begins for the first time on the outer side of the surface Z = H and, at the same time, a fracture front arises on it. For real values of the exponent 3 ≤ k ≤ 9 the fracture front moves
202
A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
quite rapidly in view of the fact that we take the time at which = 1 when Z = H as the time up to the fracture of the plate,. It then follows from relations (3.12), (4.1) and (4.2) that
(4.3) When the damage parameter vector is taken into account, the kinetic equation is taken in the
form5
(4.4) where i is the projection of the damage parameter onto the i-th axis of the system of coordinates. The damage magnitude (the modulus 21 + 22 . In accordance with the kinetic theory, the condition (t*) = 1, where t* is the time up of the damage vector) is defined as = to the fracture of the plate, is taken as the plate fracture criterion. We now introduce the constant dimensionless quantity
where M0 is an arbitrary quantity with the dimension N (Newton). Equations (4.4) then take the form
(4.5) where Mi is the magnitude of the bending moment in the i-th stage of the loading. According to the loading mode considered, the direction of the stress i due to the moment Mi in the i-th loading stage coincides with the i-th axis of the system of coordinates. The one and the same subscript is therefore used in relations (4.5) to denote the projection of the damage vector and the loading stage. Since an arbitrary quantity of dimension N is considered as M0 in the expression for G, we choose it such that the equality G = 1 is satisfied and specify that M1 = M0 . Then,
and, on the first and second loading stage, relations (4.5) take the form
(4.6)
(4.7) The times up to the fracture of the plate for different values of the constant b are determined on the basis of (4.6) and (4.7). Using the fracture criterion = 1, the time t1∗ is determined from the first equation of relation (4.6) taking account of the linear form of the function f (cm (t)) (in the last equality of (3.1), we take a = 9.56,7 ) and relation (2.7). We will only consider the first stage of the diffusion process:
Since t1∗ < 1, it is actually necessary to use expression (2.7) that corresponds to the first stage. During the course of the whole time 0 ≤ t ≤ t1 , the damage 2 (t) ≡ 0 (due to the zero value of the moment M2 = 0 throughout the whole of the first stage). The components i of the damage vector for the first stage of the loading therefore have the form
The damage vector components in the second stage of loading have the form
A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
203
Fig. 3.
The time up to fracture t* is determined using the following fracture criterion:
5. Results of calculations The relations (t) and (t) for the values b = 0.5, 1, 2, obtained using the scalar (the solid curves) and the vector (the dashed curves) ∗ and damage parameters, are shown in Fig. 3. A table is presented in the lower part of Fig. 3 where the values of the times up to fracture t(1) ∗ t , obtained using the scalar and vector damage parameters respectively, are given as well as the sum of the partial times
(t2∗ is the time up to the fracture of the plate under the action of the moment M2 ) and the values of S and S are determined using the scalar or vector approach respectively. It follows from the table that the vector approach always leads to a greater value of the time up to fracture and the following inequalities are satisfied in all the loading modes considered: S > 1 when b < 1 and 0 < S < 1 when b = 1. This result is in agreement with the results of experiments on the creep strength of metals for a piecewise-constant relation between the tensile stress and time.8–10 In these experiments, a deviation in the value of S from unity in the direction of an increase was also observed on lowering the stress (1 > 2 ), and the opposite effect (0 < S < 1) was observed when the stress was increased (1 < 2 ). For example, the corresponding values of S took the values 3.15 and 0.778 , 1.26 and 0.79 and 1.04 and 0.84.10 Acknowledgement This research was financed by the Russian Foundation for Basic Research(No. 14-08-00570a). References 1. 2. 3. 4.
Rabotnov YuN. The Creep of Structural Elements. Moscow: Nauka; 1966. Lokoshchenko AM. Creep and the Creep Strength of Metals in Aggressive Media. Moscow: Izd MGU; 2000. Lokoshchenko AM. The Modelling of the Creep Process and Creep Strength of Metals. Moscow: Mosk Gas Industr Univ; 2007. Kulagin DA, Lokoshchenko AM. Analysis of the influence of the surrounding medium on the long-term strength by using a probabilistic approach. Mech solids 2001;36(1):102–10. 5. Namestnikova IV, Shesterikov SA. Vector representation of the damage parameter. In: The Deformation and Fracture of Solid Bodies. Collection of Papers of Institute of Mechanics the M. V. Lomonosov Moscow State University. Moscow: Izd MGU; 1985. p. 43–52. 6. Oding IA, Fridman ZG. The role of surface layers in the long-term fracture of metals under creep conditions. Zavod Lab 1959;25(3):329–32.
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A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
7. Fomin LV. Description of the creep strength of stretched rods of rectangular and circular cross section in a high temperature air medium. Vestnik Samar Gos Tekhn Inst. Ser Fiz-Mat Nauki 2013;3(32):87–97. 8. Gulyayev VN, Kolesnichenko MG. On estimating durability in the creep process for a stepwise change in the load. Zavod Lab 1963;29:748–52. 9. Marriott DI, Penny RK. Strain Accumulation and rupture during creep under variable uniaxial ten-silt loading. J. Strain Analysis 1973;8(3):151–9. 10. Osasyuk VV, Olisov AN. On the question of the hypotheses concerning the summation of relative durabilities. Probl Prochnosti 1979;11:31–3.
Translated by E. L. S.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Creep fracture of plates with variable bending moments in the presence of an aggressive medium夽 A.M. Lokoshchenko, L.V. Fomin The Scientific Research Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 30 March 2015 Available online 11 July 2016
a b s t r a c t The dissipated fracture of a rectangular plate with bending under conditions of a time-varying plane stress state taking account of the effect of an aggressive medium is investigated. A method for determining the time up to the fracture of such a plate when it is successively bent in different planes is proposed. Using Rabotnov’s kinetic theory, the times up to fracture are compared in the case of scalar and vector damage parameters using a power creep model. The deviations of the sum of the partial times from unity are investigated in the case of piece-wise constant relations between the level of the bending moment and time. An analogy with the results of experiments on the creep strength of rods with a piecewise-constant tensile stress is shown. © 2016 Elsevier Ltd. All rights reserved.
1. Statement of the problem The creep fracture of a recangular plate of thickness 2H in an aggressive surrounding medium that is under the action of linear bending moments M1 and M2 distributed along its edges (Fig. 1) is considered. The effect of the aggressive medium is determined by the diffusive penetration of its elements into the plate material, leading to a reduction in the duration of the operating capacity of the plate. The problem of determining the time up to the fracture of the plate for different combinations of distributed bending moments along its edges is formulated on the basis of Rabotnov’s kinetic theory1 of creep and creep strength taking account of scalar and vector damage parameters. Here, the natural assumption is used that the thickness 2H is considerably smaller than the other two dimensions of the plate and the diffusion process along the plate edges is therefore not taken into account. 2. Approximate method of solving the diffusion equation Exact solutions of the diffusion equation for bodies with constant boundaries are usually represented in the form of trigonometric series or series consisting of special functions that do not always allow obtaining a representation of the sought after characteristics in comprehensible form. These solution are very cumbersome, and, to obtain an acceptable accuracy in the calculations, a large number of terms have to be retained in the series. For analysing the process of the diffusion of an aggressive surrounding medium into the plate, an approximate method of solving the diffusion equation is used based on the introduction of a diffusion front that propagates from the plate surface.2,3 This approach enables us to separate all the cross section of the plate into a perturbed region (in which the medium has already penetrated into the material) and an unperturbed region (into which there is not yet any penetration of the medium), and then to track the motion of the boundary between these regions in time, . The coefficient of diffusion of the aggressive medium into the plate material is denoted by D and, as the dimensionless variables, we take
夽 Prikl. Mat. Mekh., Vol. 80, No. 2, pp. 276–284, 2016. E-mail address: [email protected] (A.M. Lokoshchenko). http://dx.doi.org/10.1016/j.jappmathmech.2016.06.006 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
199
Fig. 1.
where Z is the coordinate along the thickness of the plate (Z = 0 on the middle line of the cross section of the plate), C is the concentration of the aggressive medium in the plate and C0 = const is the concentration level on the boundary between the plate material and the external medium. From the condition for the symmetry of the diffusion process, a half of the cross section of the plate with respect to its thickness is considered, In these variables, the one-dimensional diffusion equation takes the form
(2.1) The initial and boundary conditions
(2.2) are satisfied exactly and the diffusion equation is satisfied integrally in the whole cross section of the plate. The relation between the concentration c and the coordinate z is taken in the form of a quadratic polynomial that satisfies the initial and boundary conditions. Two stages in the diffusion process are considered here: the stage of the front penetration and the saturation stage which are separated by the instant t0 :2,3
(2.3) where l(t) is the coordinate of the diffusion front, t0 is the time of the change between the stages of the diffusion process (l(t0 ) = 0) and B(t) is the concentration at the centre of the plate cross section when t ≥ t0 (on the middle line z = 0). The unknown relations l(t) and B(t) are determined from the condition of the integral satisfaction of diffusion equation (2.1) by the function c(z, t) (2.3):
(2.4) There is no modulus sign in the integrand since it follows from the results obtained earlier4 that, in this case, the difference between the approximate solution of the diffusion equation obtained and the exact solution is just a few percent. Substituting expression (2.3) into (2.1) taking account of relations (2.2) and (2.4), we obtain the relations between the coordinate of the diffusion front l(t) and the concentration B(t) and time (2.5) The time t0 = 1 is determined on the basis of the boundary condition l(t0 ) = 0.
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A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
Using relation (2.5), from expression (2.3) we obtain
(2.6) The integrally mean concentration in the volume of the plate
(2.7) (2.8) will later be used to analyse the effect of the aggressive medium on the time up to the fracture of the plate. 3. Determination of the components of the stress-strain state The effect of the aggressive medium on the time up to when the plate fractures is taken into account by introducing a function of the integrally mean concentration of the aggressive medium in the plate material into the constitutive power relation: (3.1) where p˙ u is the intensity of the creep strain rates, the dot over pu denotes a derivative with respect to t, u is the stress intensity, and A1 and n are constants. In addition, we assume that a plane stress state (3 = 0) is realized in the plate, the deformations are small, the condition of the incompressibility of the plate material is satisfied and that elastoplastic deformations are not taken into account. Neglecting the change in the length of the middle line in a section, we obtain (3.2) ˙ i are the rates of change of the curvature (from now on, the subscript i takes the values 1 and 2 everywhere). In this case,
(3.3) From constitutive relation (3.1), taking account of equality (3.3) we express the stress intensity:
(3.4) We use the hypothesis of the proportionality of the stress deviators and the creep strain rates p˙ ij for a plane stress state
(3.5) On the basis of relations (3.4) and (3.5) and the plane-sections hypothesis (3.2), we write
(3.6) From equality (3.6), taking account of relation (3.2) we obtain the expressions for the stresses
(3.7) The equilibrium equations have the form
(3.8) The equality to zero of the axial forces is satisfied automatically according to equality (3.7). Substituting expressions (3.7) into equilibrium equations (3.8), we obtain
(3.9)
A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
201
Fig. 2.
We now introduce the quantity Mu and, using relations (3.3) and (3.9), we obtain the relation between Mu and ˙ u :
(3.10) We connect the rates of change of the curvatures and the bending moments by means of the following equations (in analogy with the connection between the stress deviators and the creep strain rates):
(3.11) Substituting the rates of change of the curvatures (3.11) in the expressions for the stresses (3.7), taking account of relation (3.10) we obtain (3.12) Hence, the stresses (unlike creep strain rates) are independent of the presence of the aggressive medium. The relation between the dimensionless stress 1 /d, where
and the dimensionless transverse coordinate of the plate z = Z/H is shown in Fig. 2 for 2H = 1 mm, M1 = 1 N and n = 3. 4. Creep rupture of a plate for a piecewise-constant relation between the bending moments and the time We will consider the following programme for loading the plate. Initially (when t = 0) a bending moment M1 > 0 is instantaneously applied to a pair of oppositely located edges of the plate. This moment acts during a time 0 < t ≤ t1∗ /2 (where t1∗ is the time up to when the plate fractures under the action of this moment M1 ) and, here, M2 = 0 (Fig. 1). The conditions M1 = 0, M2 > 0 are then satisfied when t1∗ /2 < t < t ∗ (t* is the time up to when plate fractures). To determine the time t*, we use Rabotnov = s kinetic theory of creep and creep rupture and, with this aim, we introduce a function of the integrally mean concentration of the aggressive surrounding medium in the plate material into the kinetic equation. Taking account of the scalar defectiveness parameter, we consider the kinetic equation in the form
(4.1) where 0 is a constant quantity of dimension MPa. Since, according to the loading mode adopted, only one of the two bending moments acts at each stage (the other moment is equal to zero), the time up to fracture can be determined taking the following expressions into account: (4.2) It follows from relations (3.12), (4.1) and (4.2) that the equality of the damage parameter to unity begins for the first time on the outer side of the surface Z = H and, at the same time, a fracture front arises on it. For real values of the exponent 3 ≤ k ≤ 9 the fracture front moves
202
A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
quite rapidly in view of the fact that we take the time at which = 1 when Z = H as the time up to the fracture of the plate,. It then follows from relations (3.12), (4.1) and (4.2) that
(4.3) When the damage parameter vector is taken into account, the kinetic equation is taken in the
form5
(4.4) where i is the projection of the damage parameter onto the i-th axis of the system of coordinates. The damage magnitude (the modulus 21 + 22 . In accordance with the kinetic theory, the condition (t*) = 1, where t* is the time up of the damage vector) is defined as = to the fracture of the plate, is taken as the plate fracture criterion. We now introduce the constant dimensionless quantity
where M0 is an arbitrary quantity with the dimension N (Newton). Equations (4.4) then take the form
(4.5) where Mi is the magnitude of the bending moment in the i-th stage of the loading. According to the loading mode considered, the direction of the stress i due to the moment Mi in the i-th loading stage coincides with the i-th axis of the system of coordinates. The one and the same subscript is therefore used in relations (4.5) to denote the projection of the damage vector and the loading stage. Since an arbitrary quantity of dimension N is considered as M0 in the expression for G, we choose it such that the equality G = 1 is satisfied and specify that M1 = M0 . Then,
and, on the first and second loading stage, relations (4.5) take the form
(4.6)
(4.7) The times up to the fracture of the plate for different values of the constant b are determined on the basis of (4.6) and (4.7). Using the fracture criterion = 1, the time t1∗ is determined from the first equation of relation (4.6) taking account of the linear form of the function f (cm (t)) (in the last equality of (3.1), we take a = 9.56,7 ) and relation (2.7). We will only consider the first stage of the diffusion process:
Since t1∗ < 1, it is actually necessary to use expression (2.7) that corresponds to the first stage. During the course of the whole time 0 ≤ t ≤ t1 , the damage 2 (t) ≡ 0 (due to the zero value of the moment M2 = 0 throughout the whole of the first stage). The components i of the damage vector for the first stage of the loading therefore have the form
The damage vector components in the second stage of loading have the form
A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
203
Fig. 3.
The time up to fracture t* is determined using the following fracture criterion:
5. Results of calculations The relations (t) and (t) for the values b = 0.5, 1, 2, obtained using the scalar (the solid curves) and the vector (the dashed curves) ∗ and damage parameters, are shown in Fig. 3. A table is presented in the lower part of Fig. 3 where the values of the times up to fracture t(1) ∗ t , obtained using the scalar and vector damage parameters respectively, are given as well as the sum of the partial times
(t2∗ is the time up to the fracture of the plate under the action of the moment M2 ) and the values of S and S are determined using the scalar or vector approach respectively. It follows from the table that the vector approach always leads to a greater value of the time up to fracture and the following inequalities are satisfied in all the loading modes considered: S > 1 when b < 1 and 0 < S < 1 when b = 1. This result is in agreement with the results of experiments on the creep strength of metals for a piecewise-constant relation between the tensile stress and time.8–10 In these experiments, a deviation in the value of S from unity in the direction of an increase was also observed on lowering the stress (1 > 2 ), and the opposite effect (0 < S < 1) was observed when the stress was increased (1 < 2 ). For example, the corresponding values of S took the values 3.15 and 0.778 , 1.26 and 0.79 and 1.04 and 0.84.10 Acknowledgement This research was financed by the Russian Foundation for Basic Research(No. 14-08-00570a). References 1. 2. 3. 4.
Rabotnov YuN. The Creep of Structural Elements. Moscow: Nauka; 1966. Lokoshchenko AM. Creep and the Creep Strength of Metals in Aggressive Media. Moscow: Izd MGU; 2000. Lokoshchenko AM. The Modelling of the Creep Process and Creep Strength of Metals. Moscow: Mosk Gas Industr Univ; 2007. Kulagin DA, Lokoshchenko AM. Analysis of the influence of the surrounding medium on the long-term strength by using a probabilistic approach. Mech solids 2001;36(1):102–10. 5. Namestnikova IV, Shesterikov SA. Vector representation of the damage parameter. In: The Deformation and Fracture of Solid Bodies. Collection of Papers of Institute of Mechanics the M. V. Lomonosov Moscow State University. Moscow: Izd MGU; 1985. p. 43–52. 6. Oding IA, Fridman ZG. The role of surface layers in the long-term fracture of metals under creep conditions. Zavod Lab 1959;25(3):329–32.
204
A.M. Lokoshchenko, L.V. Fomin / Journal of Applied Mathematics and Mechanics 80 (2016) 198–204
7. Fomin LV. Description of the creep strength of stretched rods of rectangular and circular cross section in a high temperature air medium. Vestnik Samar Gos Tekhn Inst. Ser Fiz-Mat Nauki 2013;3(32):87–97. 8. Gulyayev VN, Kolesnichenko MG. On estimating durability in the creep process for a stepwise change in the load. Zavod Lab 1963;29:748–52. 9. Marriott DI, Penny RK. Strain Accumulation and rupture during creep under variable uniaxial ten-silt loading. J. Strain Analysis 1973;8(3):151–9. 10. Osasyuk VV, Olisov AN. On the question of the hypotheses concerning the summation of relative durabilities. Probl Prochnosti 1979;11:31–3.
Translated by E. L. S.