ISSN 10526188, Journal of Machinery Manufacture and Reliability, 2013, Vol. 42, No. 4, pp. 319–324. © Allerton Press, Inc., 2013. Original Russian Text © A.M. Lokoshchenko, K.A. Agakhi, L.V. Fomin, 2013, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2013, No. 4, pp. 70–75.
RELIABILITY, STRENGTH, AND WEAR RESISTANCE OF MACHINES AND STRUCTURES
Bending Creep of Beams in Aggressive Media A. M. Lokoshchenko, K. A. Agakhi, and L. V. Fomin Moscow, Russia Received December 5, 2012
Abstract—The effect of an aggressive medium on hightemperature creep and creeprupture charac teristics of beams in pure bending is considered. Creep modeling is based on the Rabotnov kinetic the ory that involves two structural parameters, i.e., damage and the concentration of the environmental chemical elements in a beam. The problem of beam bending allows for the difference in the material creep processes in tension and compression. The novelty of the problem under consideration is the use of the governing and kinetic equations in the form of singular fractional power dependencies of the creep rate and damage accumulation rate on the stress; instantaneous elastic deformations are neglected because they are low compared to creep deformations. The singularity makes it possible to allow for nonlinear viscosity together with instantaneous fracture characteristics. DOI: 10.3103/S1052618813040079
INTRODUCTION The assessment of the quality and reliability of structures subjected to long hightemperature loading is a topical problem and requires prediction of their durability with allowance for specific factors, includ ing an aggressive working medium. Aggressive media have a pronounced effect on hightemperature creep and creeprupture characteristics of structural materials. The allowance for the effect of an aggressive medium can vary the rupture life of materials and structures by several times. In this work, the process of material creep in an aggressive medium is modeled on the basis of the Rabo tnov kinetic theory [1] that involves two structural parameters, i.e., the damage ω and the concentration c of the environmental chemicals. The equations derived are used to study the pure bending of a beam during creep with allowance for the effect of the aggressive medium. The beam material has different creep characteristics in tension and compression; the creep of the beam in tension is accompanied by gradual damage accumulation. PROBLEM OF BEAM BENDING DURING CREEP WITH ALLOWANCE FOR MEDIUM DIFFUSION AND DIFFERENT MATERIAL RESISTANCE We consider the pure bending of a beam during creep with allowance for the effect of diffusion from the environment; the cross section of the beam is a thin strip with width b and height H (H Ⰶ b). The length of the beam l is substantially greater than its transverse dimensions (H ≤ b ≤ l); therefore, the influ ence of the beam length on specific features of a solution is not considered. The bending moment that acts on the beam in the plane perpendicular to the beam width is designated as M. Let us assume the concen tration c of the aggressive medium in the beam material is equal to zero as the initial condition; the bound ary condition on the beam surface is assumed as c(t) = c0 = const. The beam is made of a material with different tensile and compressive strengths (σb1 > 0 and (σb2 < 0, respectively). Let us assume the system of the governing and kinetic creep relations with allowance for the fractional power function [2, 3] in the following form: n γ1 dp σ ⎞ = A ⎛ 1 + c m ( t )⎠ , ⎝ dt c0 ( σ b1 – σ ) ( σ – σ b2 ) ( 1 – ω )
⎧ σ ⎪ B dω = ⎨ ( σ b1 – σ ) ( σ – σ b2 ) ( 1 – ω ) dt ⎪ ⎩ 0 at σ ≤ 0. 319
m
where
2 ⎛ 1 + γc ⎞ m ( t )⎠ ⎝ c0
ω = 0
at
at
σ > 0,
σ ≤ 0;
(1)
(2)
320
LOKOSHCHENKO et al.
Here p is the creep deformation; σ is the stress; cm(t) is the concentration level integrally averaged over the beam cross section [4, 5]; the coefficients γ1 and γ2 characterize the extent of the effect of the aggressive medium on the creep deformation rate and the damage accumulation rate, respectively; n, m, A, and B are the material constants; and t is the time. We solve the diffusion equation using the approximate method proposed in [4, 5]. In this method, the initial condition and the boundary conditions for the diffusion equation are accurately satisfied, while the equation itself is integral over the beam volume. In this case, the equation for the concentration level inte grally averaged over the beam cross section is written as follows: 2 ⎧ 1 48D H t at 0 t < ≤ , ⎪ 2 48D ⎪3 H cm ( t ) = ⎨ c0 ⎪ 2 48D ⎞ ⎞ ⎛1⎛ t at ⎪ 1 – 3 exp ⎝ 4 ⎝ 1 – 2 ⎠⎠ H ⎩
(3)
2
H , t > 48D
where D = const is the diffusion coefficient. Let us consider the process of the pure bending of a thin strip that experiences the effect of an aggressive medium. The plane section hypothesis takes the following form: p· = χ· ( y – y 0 ),
(4)
where p· is the creep deformation rate; χ· is the rate of beam curvature variation; y is the coordinate that is measured from the beam middle line (–0.5H ≤ y ≤ 0.5H) in the direction of the beam zone under ten sion; y0(t) is the coordinate of the neutral surface on which no stresses arise (σ(y0) ≡ 0). The displacement of the neutral surface of the beam being bent during creep results from different ten sion and compression resistances of the material, as well as from its softening due to damage accumulation during creep. In the kinetic equation, the continuity parameter ψ = 1 – ω is used, where ω is the damage. Let us introduce the following dimensionless variables: σ b2 α = – , σ b1
σ , σ = σ b1
y = 2y , H
t = tA,
4 M, M = 2 bH σ b1
χ = H χ, 2
B = B , A
c c m = m , c0
2 N, N = bHσ b1
(5)
where N is the axial force. With dimensionless variables (5), Eqs. (1) and (2) take the following form: n
dp σ ( 1 + γ c ( t ) ) = 1 m dt ( 1 – σ ) ( α + σ )ψ ⎧ σ ⎪ –B dψ = ⎨ ( 1 – σ ) ( α + σ )ψ dt ⎪ ⎩ 0 at σ ≤ 0.
at
σ > 0,
where
ψ = 1
at
σ ≤ 0;
(6)
m
( 1 + γ2 cm ( t ) )
at
σ > 0,
(7)
With allowance for (5), the concentration cm(t) expressed by formula (3) takes the following form: ⎧1 t ⎪ at t ≤ t 0 , t 0 = ⎪3 t 0 cm ( t ) = ⎨ ⎪ 2 t ⎞⎞ ⎛1⎛ ⎪ 1 – exp ⎝ ⎝ 1 – ⎠ ⎠ at 3 4 t0 ⎩
2
AH , 48D t > t 0.
JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
Vol. 42
No. 4
2013
BENDING CREEP OF BEAMS IN AGGRESSIVE MEDIA
321
The stressstrain state of the bent beam at any moment of time is determined by the axial stresses σ = σ( y ) and the axial creep deformations p = p( y , t ). Instantaneous elastoplastic deformations are neglected. The dimensionless equations of equilibrium for a thin strip are as follows: y0
N =
∫ (σ
y0
1 – ) dy
–1
∫
+ ( σ + ) dy = 0,
M =
1
∫ (σ
– )y dy
–1
y0
∫
+ ( σ + )y dy, y0
where σ – and σ + are the dimensionless stresses in the compressed and stretched zones of the beam, respectively; they depend on c m ( t ). Using Eqs. (4) and (6), we derive the following expressions for σ– and σ+: 2
2
4
2
2
2
– Cψ ( α – 1 ) ± C ψ ( α – 1 ) + 4 ( 1 + Cψ )αCψ , σ +, – = 2 2 ( 1 + Cψ ) y – y0 C ( y, t ) = dχ ⋅ dt 1 + γ 1 c m ( t )
ψ = 1
for
σ – < 0;
2/n
.
Let us transform formula (7) as follows: 2 ⎧ d( ψm + 1 ) σ ⎪ = – B ( m + 1 ) (1 – σ)(σ + α) ⎪ dt ⎨ ⎪ dψ ⎪ = 0 ⎩ dt
m/2
( 1 + γ2 cm ( t ) )
at
σ > 0,
σ ≤ 0.
at
Let us write the total system of equations in the dimensionless variables as follows: y0
1
∫
∫
2
C1 + C ( α – 1 ) C 2 – Cψ ( α – 1 ) dy = 0, N = – dy + 2 2(1 + C) 2 1 Cψ ( + ) –1 y 0
y0
1
∫
∫
2
C1 + C ( α – 1 ) C 2 – Cψ ( α – 1 ) y dy, M = – y dy + 2 2(1 + C) 2 1 Cψ ( + ) –1 y 0
m+1
2
dψ σ = – ( m + 1 )B ( 1 – σ )(σ + α) dt dψ = 0 dt C1 =
2
2
at
σ < 0,
C ( α – 1 ) + 4 ( 1 + C )αC ,
m/2
( 1 + γ2 cm ( t ) )
at
dχ y – y 0 C ( y, t ) = dt 1 + γ 1 c m ( t ) C2 =
2
4
2
σ > 0,
(8)
2/n
, 2
2
C ψ ( α – 1 ) + 4 ( 1 + Cψ )αCψ .
Therefore, the solution of the problem of beam bending is reduced to the solution of system of integro differential equations (8) for the unknown functions y 0 ( t ) , χ ( t ) , and ψ( y , t ) under the initial conditions χ (0) = 0 and ψ( y , 0) = 1. The initial value y 0 (0) coincides with the value obtained when solving a similar problem for steady creep without allowance for damage (see, e.g., [6]). The process of solving system of equations (8) runs until the continuity on the stretched surface layer reaches the zero value ψ( y = 1, t * ) = 0, i.e., the damage is ω( y = 1, t * ) = 1. At this time t = t * , a rupture front appears that propagates into the beam with time. The propagation of the rupture front is characterized by the coordinate ξ( t ). In the stretched zone of the beam, the equations of equilibrium are JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
Vol. 42
No. 4
2013
322
LOKOSHCHENKO et al.
Table 1. Values of functions y 0( t.), χ ( t.), and ω( y = 1, t.) obtained for increasing damage χ ( t.)
y 0( t.)
t.
ω( y = 1, t.)
t0 = 0
–0.061
0
0
t 1 = 0.1
–0.144
0.023
0.312
t 2 = 0.15
–0.215
0.044
0.564
t 3 = 0.17
–0.290
0.060
0.744
t 4 = 0.18
–0.378
0.084
0.889
t 5 = 0.182
–0.425
0.099
0.958
t 6 = t.* = 0.1825
–0.486
0.121
1
Table 2. Values of functions y 0( t.), χ ( t.), and ω( y = ξ, t.) obtained for propagating rupture front ξ
y 0( t.)
t.
χ ( t.)
ω( y = ξ, t.)
1
t 6 = t.* = 0.1825
–0.486
0.121
1
0.95
t 7 = 0.182547
–0.486507
0.152
1
0.93
t 8 = t.** = 0.182548
–0.486508
0.159
1
integrated over this coordinate ( y ≤ ξ( t )). The calculations are carried out until the stresses on the out sides of the stretched and compressed zones reach corresponding values of the ultimate stresses. This value of time t = t ** determines the duration of the disintegration of the beam into two parts, i.e., the duration of its rupture. We carried out, for example, the calculation for a beam with the following values of the parameters: M = 0.5, B = 20, n = m = 3, α = 1.5, γ1 = 0.2, γ2 = 0.8, A = 0.00144 h–1. The calculation results are pre sented in Tables 1 and 2. Figure 1 shows the diagrams of the distribution of the stresses σ over the beam cross section at various values of t ( t 0 – t 6 ). Figures 2 and 3 illustrate the time dependencies of the beam curvature χ ( t ) and the − σ 1.0 0.5 −0.8
−0.4
− t0 t−
− t8
−0.5 −1.0
− t6
0.14
0.8 − y
0.10
1
0.4
− χ 0.18 1
2
0.06 0.02 0
0.05
0.15
0.25 − t
−1.5 Fig. 1.
Fig. 2.
JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
Vol. 42
No. 4
2013
BENDING CREEP OF BEAMS IN AGGRESSIVE MEDIA − − y0(t)
323
− ω(1, t)
−0.1
0.9 1
0.7
2
0.5
−0.3
0.3 1 −0.5
0
0.05
0.15
2
0.1 0.25 − t
0
0.05
Fig. 3.
0.15
0.25 −t
Fig. 4.
coordinate of the neutral axis y 0 ( t ) , respectively, with (curve 1) and without (curve 2) allowance for dif fusion. The time dependence of the damage on the stretched surface layer ω( y = 1, t ) with (curve 1) and without (curve 2) allowance for diffusion (0 < t < t*) is shown in Fig. 4. A comparison of the dimensionless value of the time t ** with the dimensionless value of the corre sponding time obtained without allowance for the effect of an aggressive medium [7] shows it to decrease by 18%. CONCLUSIONS The ultimate creep characteristics of a beam subjected to pure bending in an aggressive medium are determined on the basis of the Rabotnov kinetic theory that involves two structural parameters, i.e., the damage and the concentration of the diffusing environmental chemicals into the beam material. The solution of the problem is obtained using the governing and kinetic equations in the form of sin gular fractional power dependencies. The singularity makes it possible to allow for nonlinear viscosity together with instantaneous fracture characteristics, i.e., the shortterm ultimate strengths of a material at a corresponding elevated temperature. This is an advantage of the solution over solutions based on the use of power governing and kinetic dependencies. The analysis performed has shown that the effect of aggressive media on structural elements during creep manifests itself as a substantial shortening of the rupture life under bending loading. ACKNOWLEDGMENTS This study was supported by the Russian Foundation for Basic Research, projects nos. 110800007a and 110801015a. REFERENCES 1. Rabotnov, Yu.N., Polzuchest’ elementov konstruktsii (Creeping of Structure Elements), Moscow: Nauka, 1966. 2. Shesterikov, S.A. and Yumasheva, M.A., State equation concretization in creeping theory, Izv. Akad. Nauk SSSR, Ser. Mekhan. Tverd. Tela, 1984, no. 1, pp. 86–92. 3. Shesterikov, S.A. and Yumasheva, M.A., The variant of state equation under creeping and its application The variant of state equation under creeping and its application, in Voprosy dolgovremennoi prochnosti energet icheskogo oborudovaniya. Tr. Tsentral’nogo kotloturbinnogo instituta im. I.I. Polzunova (Problems on LongTerm Strength of Power Equipments. Works of Central Boiler and Turbine Institution named after I.I. Polzunov), Leningrad, 1988, issue 246, pp. 74–79. JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
Vol. 42
No. 4
2013
324
LOKOSHCHENKO et al.
4. Lokoshchenko, A.M., Polzuchest’ i dlitel’naya prochnost’ metallov v agressivnykh sredakh (Creeping and Long Term Strength of Metals in Corrosive Mediums), Moscow: Izd. Mosk. Univ., 2000. 5. Lokoshchenko, A.M., Modelirovanie polzuchesti i dlitel’noi prochnosti metallov (Simulation for Creeping and LongTerm Strength of Metals), Moscow: Mosk. gos. industr. univ., 2007. 6. Lokoshchenko, A.M., Agakhi, K.A., and Fomin, L.V., Pure bending of a beam made of differently strength material under creeping, Vestn. Samarsk. Gos. Tekhn. Univ. Ser Fiz.Mat. Nauki, 2012, no. 1(26), pp. 66–73. 7. Lokoshchenko, A.M., Agakhi, K.A., and Fomin, L.V., Beam bending under creeping by considering failure and material’s different resistance, Mashinostr. Inzh. Obraz., 2012, no. 3, pp. 29–35.
Translated by D. Tkachuk
JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
Vol. 42
No. 4
2013
