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Influence of the startup and shutdown phases on the viscoplastic structural analysis of the thrust chamber wall
Aerospace Science and Technology 34 (2014) 84–91
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Aerospace Science and Technology www.elsevier.com/locate/aescte
Influence of the startup and shutdown phases on the viscoplastic structural analysis of the thrust chamber wall Jinhui Yang, Tao Chen, Ping Jin, Guobiao Cai ∗ School of Astronautics, BeiHang University, Beijing, 100191, PR China
a r t i c l e
i n f o
Article history: Received 13 December 2012 Received in revised form 27 August 2013 Accepted 9 September 2013 Available online 18 September 2013 Keywords: Viscoplastic Structural analysis Life prediction Startup phase Shutdown phase
a b s t r a c t Detailed structural analysis with Robinson’s viscoplastic model of the thrust chamber wall has been completed to explain its damage process phase by phase, indicating that under the same level of thermal–structure loadings, the startup and shutdown phases play an important role of the failure. Different startup and shutdown durations were applied to study the phase influence on structural analysis and predicted life. Results reveal that the startup process mainly affects the remaining strain, while the shutdown process contributes more to the remaining stress on the cyclic effect; quick startup and longer shutdown duration conduce to prolong the chamber life and the life can be promoted 22% through reducing start-up duration from 1.5 s to 0.3 s, while 5% through increasing shut-down duration from 0.15 s to 0.9 s. © 2013 Elsevier Masson SAS. All rights reserved.
1. Introduction As one of the critical components of the reusable rocket engine, the thrust chamber is designed to operate in severe conditions of elevated temperature and pressure for improving the engine performance. At these elevated temperatures, in turn, the thrust chamber wall experiences significant inelastic strains. For an accurate estimation of the life of the chambers and to predict their progressive deformation with the number of loading cycles, a realistic stress–strain analysis must be made [8]. Unified viscoplastic analyses provide realistic descriptions of high-temperature inelastic behavior of materials, in which all inelastic strains (e.g. creep, plastic, relaxation, and their interactions) are accounted for as a single, time-dependent quantity. Arya and Arnold applied the Robinson’s model [2] and Freed’s model [1] to assess the inelastic deformation, which qualitatively replicated the doghouse effect, as shown in Fig. 1; Ray and Dai, basing on the Freed’s viscoplastic model, captured the nonlinear effects to represent the inelastic strain ratcheting, progressive bulging out, and thinning in the thrust chamber wall, which contributed lots to the on-line service-life prediction and damage analysis of the thrust chamber [5]; Sung et al. applied conventional and viscoplastic models to predict a small scale experimental annular plug nozzle thruster life, and validated by cyclic test, which revealed that the viscoplastic model showed better agreement in predicting the life cycle failure when both low cycle fatigue and fatigue–creep inter-
*
Corresponding author. Tel.: +86 010 82336222; fax: +86 010 82338798. E-mail address: [email protected] (G. Cai).
1270-9638/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ast.2013.09.004
Fig. 1. Cross section sketch of thrust chamber and analytical model.
action were taken into account [19,18]; Schwarz et al. developed a viscoplastic model including ageing that successfully captured the observed accrued thinning of the hot wall and quantitatively reproduced the doghouse deformation [17]. However, most of the previous researches focus on the final state analysis, while the damage process study phase by phase using the viscoplastic theory is blank. The thrust chamber used in experiments at the NASA Lewis Research Center was analyzed in this paper [11], which used NARloy-Z as inner wall and Ni as the jacket, as shown in Fig. 1. The right figure depicts the finite element model applied in the analysis. It consists of 2800 elements and 2941 nodes. The coupled thermal–mechanical plan strain solid elements were used to model the smallest repeating segment of the cylinder wall. Because the wall was symmetrical, only one-half of a cooling channel was modeled. Experimental results published by Quentmeyer [11,12] and Pavli [10] demonstrate that the crack initiates at point D (Fig. 1)
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
85
Table 1 Cyclic thermal and pressure loadings. Loadings
hf,cool kW/(K m2 )
T cool K
P cool MPa
hf,cool kW/(K m2 )
T hot K
P hot MPa
Precooling/post cooling Hot run
102 48.3
28 50
5.1 6.55
0 20.2
278 3364
0.0965 2.78
˙ in ˙ th (i , j = 1, 2, 3) ε˙ i j = ε˙ el i j + εi j + εi j
(3)
The elastic strain rate for an isotropic material is governed by Hooke’s law:
1+ν
ε˙ el ij =
Fig. 2. Working phases and loading history.
of the inner wall, i.e. the “roof” point of the doghouse. In order to explain the damage process, detailed viscoplastic stress–strain analysis of point D was conducted in this paper. Due to considerable influence over the thrust chamber structural and life analysis exerted by the startup and shutdown phases, different durations of the two phases were studied in the paper. Our results show that quick startup and longer shutdown duration conduce to prolong the chamber life, which gives an improvement method for the reusable rocket engine.
E
˙ in ij
ε =
F > 0 and S i j Σ i j > 0 or F > 0 and S i j Σ i j 0
(5)
Evolutionary law:
a˙ i j =
⎧ h in ⎪ ⎨ G β ε˙ i j − ⎪ ⎩
where
Coupled thermal–mechanical analysis of the thrust chamber was completed under the boundary conditions [2] shown in Table 1 and Fig. 2. Five phases are taken into account as one working cycle: precooling, startup, hot run, shut-down and post cooling. Table 1 shows the cyclic loadings working on the coolant side and the hot gas side, which respectively represented by subscript “cool” and “hot”. The convective film coefficient h f and bulk temperature of the fluid T characterize the cyclic thermal loading, and P means the coolant and hot gas pressure. The general heat equation for dynamic problems without internal heat sources is given as:
F=
εth is decided by the temperature increij
A F nΣij √ J2
0F 0
2.1. Thermal analysis
conductivity. The thermal strain ment:
(4)
E
where E is the Young’s modulus, v is the Poisson’s ratio, and σ i j is the stress. The repeated subscript in Eq. (4) and elsewhere implies summation over their range, and δ i j is the Kronecker delta function. A dot over a symbol denotes its derivative with respect to time t. The nonisothermal multiaxial inelastic constitutive equations for the model are given below: Flow law:
2. Thermal–mechanical analysis models
∂ T (x, t ) ρc = − div λ∇ T (x, t ) (1) ∂t where the ρ is density, c is specific heat, t is time and λ is thermal
ν σ˙ i j − σ˙ kk δ i j
G= J2 =
J2 K2 I2
h β G0
ε˙ in ij −
rG√m−β I2 m−β rG 0 √
I2
ai j
G < G 0 and S i j ai j > 0
ai j
G G 0 or G < G 0 and S i j ai j > 0
−1
(7) (8)
K 02 1
(9)
ΣijΣij
2 1 I 2 = ai j ai j 2 Σ i j = S i j − ai j and
Sij = σ ij − ai j = α i j −
(6)
1 3 1 3
(10) (11)
⎫
⎬ σ kk δ i j ⎪
⎪ αkk δ i j ⎭
(12)
where αi j ( T ) is the thermal expansion coefficient. Equivalent nodal forces are calculated from the thermal strain increment and then added to the nodal force vector for the solution of the problem.
The material parameters for Robinson’s model are taken from Ref. [2], which qualitatively replicated the “doghouse” effect and the thinning of the coolant channel wall. Effects of temperature are incorporated in the model through the temperature dependent parameters: Young’s modulus E, hardening variable K , recovery or softening material parameter R and so on. Detailed viscoplastic stress–strain analysis of thrust chamber wall basing on the Robinson’s model is conducted in the following section.
2.2. Robinson’s viscoplastic model
2.3. Uniformly valid asymptotic integration algorithm
Robinson’s model [16] employs a dissipation potential to derive the flow and evolutionary laws for the inelastic strain and internal state variables. The total strain rate ε˙ i j is decomposed into elastic ε˙ el i j , inelas-
Chulya and Walker present one uniformly valid asymptotic integration algorithm for the viscoplastic theory [4], which is implicit and has high stability and allows large time increments. By incorporating the algorithm into the finite element program MARC through a user subroutine called HYPELA2, a stress–strain analysis of the cylindrical thrust chamber wall was performed. Following
∂ ε th ij ∂t
= αi j ( T )
∂T ∂t
(2)
˙ th tic ε˙ in i j (including plastic, creep, relaxation, etc.), and thermal ε ij strain rate components. Thus
86
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
gives the derivation for the uniformly valid asymptotic expansion form of the Robinson’s model. To smooth the discontinuous boundaries in Robinson’s viscoplastic model for facilitating numerical computations, define a ”spline function” [16,3] P (x) on the interval (−1, 1) as:
⎧ (1+x)2 ⎪ ⎪ ⎨ 2 (1−x)2 P (x) = 1 − 2 ⎪ ⎪ ⎩1
1x<0 0x1 x>1 x < −1
0
(13)
SijΣij
W1
F
(14)
where the weighting function W 1 is to be selected by the user and the angular brackets denote:
x =
(15)
The discontinuities in ai j are removed by replacing the function G by G ∗ :
G1 =
G 2G 0
G, G2 4G 0
+ G 0 , G < 2G 0
S i j ai j ∗ + G0 G = (G 1 − G 0 ) P W2
(16) (17)
so the flow law becomes:
˙ in ij
ε =
A F n ( S i j − ai j )
√
J2 = K
ε˙ in ij =
2μ
2μ
J2
(18)
( S i j − ai j )
(19)
L˙ =
3μ R˙ √ ˙ K ( 23AR )1/n + 1
(26)
The inelastic strain rate tensor is written in the form of the deviatoric strain tensor e˙ i j as:
˙ in ε˙ in ij = e ij −
S˙ i j
(27)
2μ
2με˙ ii j − S˙ i j = L˙ ( S i j − ai j ) Let
y i j = S i j − ai j y˙ i j = S˙ i j − a˙ i j
AF n =√
(20)
J2
(28)
(29)
Then Eq. (28) becomes:
y˙ i j + L˙ y i j = 2μe˙ i j − a˙ i j
(30)
Eq. (30) is in the form of a first order differential equation and can be integrated as:
∂ εi j ∂ ai j 2μ ∂ εkk − δi j − ∂ε 3 ∂ε ∂ε ε =0 × exp − L (t ) − L (ε ) dε
S i j (t ) = ai j (t ) +
as:
2μ
(31)
With the smoothing function, the evolutionary law becomes:
a˙ i j =
The second invariant of the inelastic strain rate tensor is written
h G∗ β
ε˙ in ij −
rG ∗ (m−β)
√
I2
ai j
(32)
Rewrite Eq. (32) into a first order differential equation as:
a˙ i j + Q˙ ai j =
h G∗ β
ε˙ in ij
(33)
where
R˙ =
2 3
1/2 ˙ in ε˙ in ij ε ij
(21)
L˙
R˙ = √
3μ
rG ∗ (m−β)
√
J2
(22)
ai j (t ) =
h
exp − Q (t ) − Q (ε )
ε =0
3μ R˙ L˙
(23)
ε˙ in i j dε
(35)
2μ
δ i j εkk − ai j + 2μ ε i j − 3 1 − exp(− L ) × L
(24)
This is different from the original uniformly valid asymptotic integration algorithm, which is an adaption for the discontinuous problems. What’s more, this adaption contributes lots to the convergence of the algorithm application on Robinson’s viscoplastic
S i j (t + t ) = ai j (t + t ) + exp(− L ) S i j (t ) − ai j (t )
When substituted into Eq. (20) gets:
2 AF n R˙ = √
G∗ β
Furthermore, the uniformly valid asymptotic expansion form of the Robinson’s model is:
√
3
(34)
I2
t
Rearranging Eq. (22) yields:
J2 =
Q˙ =
Eq. (33) can be integrated to give:
Substitution of Eq. (19) into Eq. (21) gives:
(25)
t
where
L˙
2A
+1
Notice that Eq. (24) can be extracted directly from the right side of Eq. (20), and substituting Eq. (25) into Eq. (20) gives:
Eq. (18) can be rewritten as:
L˙
√ ˙ 1/n 3R
Equating Eq. (19) with Eq. (27) forms:
x0 x<0
x, 0,
√
The function F in Eq. (5) is then replaced by a function F defined as:
F=P
model, which can be applied to other discontinuous constitutive equations. J When R˙ = 0, from F = K 22 − 1 we can get:
ai j (t + t )
= exp(− Q )ai j (t ) +
h G
ε in ij ∗β
1 − exp(− Q )
Q
(36)
(37)
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
87
where
√ L = L˙ (t + t ) t =
K
Q = Q (t + t ) t =
3μ R˙ t
√
(
3 R˙ 1/n ) 2A
+1
rG m−β t
√
I2
(38)
t + t
(39)
t + t
2.4. Life prediction model A life prediction approach with cyclic fatigue and quasi static fatigue two failure mechanisms developed by Riccius [14,13] is applied. Damage due to cyclic fatigue is simply the reciprocal of the number of cycles to fatigue failure as expressed below:
uc =
1
(40)
Nc
Fig. 3. Temperature evaluation of points A and D.
where N c is the number of cycles until failure, which can be obtained easily from the material ε − N curve basing on the structural analysis of thrust chamber wall. Usage factors are assumed to add up for each passed cycle and therefore, after N c identical cycles, u c 1.0 is obtained, which indicates failure a usage factor of for cyclic fatigue. The quasi static (or ratcheting) usage factor u qs is defined as:
u qs =
max(0, (εend − εbegine ))
(41)
εu
with the remaining strain εend and beginning strain εbegine after the considered cycle. εu is the ultimate strain of the combustion chamber wall material. After N qs = u1 identical cycles, the ultiqs
mate strain εu obtained, which indicates failure for quasi static fatigue. The total usage factor u t is defined as the sum of the cyclic and quasi static usage factors:
u t = u c + u qs
Fig. 4. Path temperature of different phases.
(42)
Finally, the total number of cycles N F until failure is calculated as the reciprocal value of the total usage factor u t :
NF =
1 ut
(43)
3. Results and discussion 3.1. Thermal analysis Fig. 3 shows the temperature evaluation of points A and D, while Fig. 4 illustrates the temperature along the path D-C-N-MB-A at every phase end. Due to the higher thermal conductivity of NARloy-Z and smaller inner wall thickness, thermal delay effects between the considered points A and D during the whole working phases is demonstrated in Fig. 3. At the precooling and shut-down phases, temperature of point D decreases quickly to 60 K, while main part of the jacket remains at ambient temperature at the end of the two phases. During the startup and hot run phases, the extreme thermal loads together with the regenerative cooling introduce enormous temperature gradients in the combustion chamber wall, as shown in Fig. 4, and the largest temperature difference between the point D and B researches almost 550 K, which causes large heat fluxes and will induce extremely large thermal stresses.
Fig. 5. NARloy-Z stress–strain data at different temperatures.
3.2. Detailed viscoplastic structural analysis Detailed structural analysis of the thrust chamber wall is completed with Robinson’s viscoplastic model in this part. Firstly, the implemented model is validated by Fig. 5 and Fig. 6. The predicted temperature dependent stress–strain responses of NARloy-Z with Robinson’s model in Fig. 5 coincide well with the experiment data [6], especially at high temperature. What’s more, the simulated hysteresis loops of NARloy-Z compare favorably with the experiment data [7] under the cyclic load as shown in Fig. 6,
88
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
Fig. 8. Evolution of the second invariant of the inelastic strain. Fig. 6. Saturated hysteresis loops of NARloy-Z.
Fig. 7. Circumferential stress–mechanical strain response of point D.
which the strain rate ε˙ = 0.004 s−1 and the temperature of 811 K. These excellent agreements obtained indicate that the Robinson’s model with the uniformly valid asymptotic integration algorithm could be used for more complex problems. The circumferential (x-direction) stress–mechanical strain response of point D is shown in Fig. 7. In the precooling state, the inner wall’s contracting trend is constrained by the jacket due to the thermal expansion ratio difference of the copper alloy and nickel, which lead to the tensile strain of point D. For the sudden loading suffered by the inner wall, the relaxation process appears at the later period of precooling. This recovery phenomenon for copper has been investigated by Kolsky through experiments [9]. During the startup and hot run phases, enormous temperature gradient is introduced in the combustion chamber wall. Therefore, large compressive strain of the copper alloy appears for the jacket tending to contract and the inner wall tending to expand. In the shutdown and post cooling states, the compressive strain releases and evolves into tensile strain for the same reason as in the precooling state. As shown in Fig. 7, the circumferential stress and mechanical strain vary widely throughout the startup and shutdown phases. What’s more, the inelastic strain is accumulated mostly in these two phases, as revealed in Fig. 8. So while the two phases are short, an important role is played of the whole thermal–structural analysis for the combustion chamber wall. Stress is determined by the Young’s modulus (Fig. 9) and elastic strain during the elastic phase. As indicated in Figs. 10 and 11, no inelastic strain variation appears from 0.1 to 0.4 s and 1.8 to 1.84 s. For the beginning startup phase, even if the total strain and elastic strain decrease slightly during this period, Fig. 7 shows that the stress reduces straightly until reaching the maximum compressive
Fig. 9. Evolution of the Young’s modulus.
Fig. 10. Evolution of the circumferential strains during the startup phase.
value. This considerable stress reduction is mainly resulted from the large Young’s modulus (> 140 GPa). The same elastic period lasts about 0.04 s during the shutdown phase. For relatively low Young’s modulus (approximate to 90 GPa), the stress just increases 100 MPa. After the elastic phases, as shown in Figs. 10 and 11, the copper alloy yields and inelastic strain accumulates quickly. Though the compression inelastic strain and total strain increase during the remaining startup phase, for sudden rise of temperature, the yield stress reduces nearly 50 MPa. Same to the shutdown phase, the hardening state appears with the tensile inelastic increasing. The stress increases slightly with large strain variation, which results from the temperature decrease. From 2.15 to 2.25 s, the cooper
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
Fig. 11. Evolution of the circumferential strains during the shutdown phase.
Fig. 12. Circumferential stress–mechanical strain response with different startup durations of point D.
alloy appears a softer state, i.e. both the stress and strain recovery are taken. The hot run and post cooling phases could also be seen as the softer states following hardening periods of startup and shutdown respectively, which is revealed in Fig. 7. Recovery plays a major role in the problem of thermal ratcheting, which has been observed in thrust chamber wall of reusable rocket engines. The presence of mechanisms of recovery can allow creep strain rates to increase following periods of hardening and thus cause acceleration of creep buckling [15]. 3.3. Influence on structural analysis of different durations In order to analyze the influence of different durations on structural results, the remaining strain and stress are adopted as “cyclic effect” characters, which contribute directly to the ratcheting damage. What’s more, the accumulation of plastic strain, decrease of the inner wall thickness and “doghouse” failure are the physical phenomenon for “cyclic effect”, which can also be calculated or explained by the remaining strain and stress. Fig. 12 presents the comparison between different startup durations of 1.5 s, 0.9 s, and 0.3 s. The solid part of these curves shows the circumferential stress–mechanical strain response during the startup phase. As illustrated in the figure, no difference appears through the elastic analysis. For the yield stress will increase with higher strain rate under dynamic loadings, so the shorter of the startup duration, the lower of compress stress appears during the hardening period, but after the whole working phase, this distinction is eliminated. The cyclic influence of the startup duration is revealed by the remaining strain, which has close relation to the
89
Fig. 13. Circumferential stress–mechanical strain response with different shutdown durations of point D.
Fig. 14. Evolution of the circumferential strains for different startup durations.
thrust chamber life. Fig. 12 shows the remaining strain increases with longer startup duration. The circumferential stress–mechanical strain response with shutdown durations of 0.9 s, 0.45 s and 0.15 s are shown in Fig. 13. Same as the startup duration analysis, the elastic periods are similar, and the yield stress decreases as the shutdown duration increases. What’s more, the hardening effect is strengthened when the engine shut-down quickly, which leads to larger maximum stress and remaining compressive stress. Comparing Fig. 13 with Fig. 12 on the cyclic effect, it can be found that the startup process mainly affects the remaining strain, while the shutdown process contributes more to the remaining stress. This difference is principally caused by the following recovery process, i.e. the hot run and the post cooling states respectively. For the startup duration analysis, as illustrated in Fig. 14, the inelastic strain is lower with 0.3 s and the compressive total strain recovers less within the limited recovery time for greater hardening effect. As the yield stress is fixed with temperature in the last two phases, so no stress difference appears and the strain variation is enlarged; while with enough recovery duration in the post cooling phase, as shown in Fig. 15, the strains of different shutdown duration recover to the same state, so the hardening effect on stress lasts till the end of cycle. 3.4. Influence on predicted life of different durations Depending on the cyclic structure analysis in Fig. 16 and Fig. 17, the influence of different startup and shutdown durations on life is evaluated by Riccius developed life prediction model. As shown in Fig. 16, the remaining strain difference at the pre-cycle is enlarged
90
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
Table 2 Predicted life for different startup/shutdown durations. Startup/shutdown duration (s) Strain range Increment of remaining strain Life (cycles)
0.3/0.45 0.0148 0.0021 124
0.9/0.45 0.0147 0.0024 105
Fig. 15. Evolution of the circumferential strains for different shutdown durations.
Fig. 16. Cyclic stress–mechanical strain response with different startup durations of points D.
in the next-cycle startup phases, which induces large strain range distinction. As the yield stress is fixed with temperature, the different startup durations have no effect on the stress in shutdown and post cooling phases. With the same reason, for different shutdown durations in Fig. 17, after the startup and hot run phases, the stress difference is eliminated and the tide of curves in next-cycle almost keeps unchanged with the pre-cycle. As shown in Table 2, for the start-up at prescribed shut-down duration of 0.45 s, if the start-up duration is reduced from 1.5 to 0.3 s, the life increases 22%; for the shut-down at prescribed start-up duration of 0.9 s, if the shut-down duration is increased from 0.15 to 0.9 s, the life increases 5%. As the result illustrates, quick startup and longer shutdown duration conduce to prolong the chamber life. Comparing the influence of startup and shutdown durations on life, it can be found that the increment of remaining strain, which represents the ratcheting damage, contributes more to the failure. As a result, the effect of startup durations on life is greater. 4. Conclusion The damage process of thrust chamber wall was explained by detailed viscoplastic structure analysis phase by phase. Analysis re-
1.5/0.45 0.0142 0.0025 102
0.9/0.15 0.0148 0.0025 101
0.9/0.9 0.0141 0.0024 106
Fig. 17. Cyclic stress–mechanical strain response with different shutdown durations of points D.
vealed that under the same level of thermal–structure loadings, the startup and shutdown phases play an important role in the stress and strain evolution. Both the startup and shutdown phases include one elastic period and one hardening period. Resulted from the corporate contribution of the Young’s modulus and elastic strain, considerable stress varies with slight strain variation in the former period and contrary in the other period for sudden variation of temperature. Influence of startup and shutdown durations on structural analysis and predicted life indicated that no difference appears in the elastic period; hardening effect is strengthened with short durations; on the cyclic effect, startup process mainly affects the remaining strain, while the shutdown process contributes more to the remaining stress; quick startup and longer shutdown duration conduce to prolong the chamber life, which gives an improvement method for the reusable rocket engine. Results indicate that the life can be promoted 22% through reducing start-up duration from 1.5 to 0.3 s, while 5% through increasing shut-down duration from 0.15 to 0.9 s. The analysis result can also be extended to explain the stress– strain response under the quickly varying thermal–structural load like the startup and shutdown phases of steam and gas turbines, jet engines, steam boilers, nuclear reactors, etc. References [1] V.K. Arya, Nonlinear structural analysis of cylindrical thrust chambers using viscoplastic models, NASA CR-185253, 1991. [2] V.K. Arya, S.M. Arnold, Viscoplastic analysis of an experimental cylindrical thrust chamber liner, AIAA J. 30 (1992) 781–789. [3] V.K. Arya, A. Kaufman, Finite element implementation of Robinson’s unified viscoplastic model and its application to some uniaxial and multiaxial problems, NASA TM-89891, 1987. [4] A. Chulya, K. Walker, A new uniformly valid asymptotic integration algorithm for elasto-plastic-creep and unified viscoplastic theories including continuum damage, NASA TM-102344, 1989. [5] X. Dai, A.K. Ray, Life prediction of the thrust chamber wall of a reusable, J. Propuls. Power 11 (1995) 1279–1287. [6] J.J. Esposito, R.F. Zabora, Thrust chamber life prediction, vol. I. Mechanical and physical properties of high performance rocket nozzle materials, NASA CR-134806, 1975.
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[7] A.D. Freed, K.P. Walke, M.J. Verrilli, Extending the theory of creep to viscoplasticity, J. Press. Vessel Technol. 116 (1994) 67–75. [8] N.P. Hannum, H.G. Price, Some effects of thermal-cycle-induced deformation in rocket thrust chambers, NASA TP-1834, 1981. [9] H. Kolsky, An investigation of the mechanical properties of materials at very high rates of loading, Proc. Phys. Soc. 62 (1949) 676–700. [10] A.J. Pavli, J.M. Kazaroff, R.S. Jankovsky, Hot fire fatigue testing results for the compliant combustion chamber, NASA TP-3223, 1992. [11] R.J. Quentmeyer, Experimental fatigue life investigation of cylindrical thrust chambers, AIAA 77-893, 1977. [12] R.J. Quentmeyer, Rocket combustion chamber life-enhancing design concepts, AIAA 90-2116, 1990. [13] J.R. Riccius, M.R. Hilsenbeck, O.J. Haidn, Optimization of geometric parameters of cryogenic liquid rocket combustion chambers, AIAA 2001-3408-631, 2001.
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Aerospace Science and Technology www.elsevier.com/locate/aescte
Influence of the startup and shutdown phases on the viscoplastic structural analysis of the thrust chamber wall Jinhui Yang, Tao Chen, Ping Jin, Guobiao Cai ∗ School of Astronautics, BeiHang University, Beijing, 100191, PR China
a r t i c l e
i n f o
Article history: Received 13 December 2012 Received in revised form 27 August 2013 Accepted 9 September 2013 Available online 18 September 2013 Keywords: Viscoplastic Structural analysis Life prediction Startup phase Shutdown phase
a b s t r a c t Detailed structural analysis with Robinson’s viscoplastic model of the thrust chamber wall has been completed to explain its damage process phase by phase, indicating that under the same level of thermal–structure loadings, the startup and shutdown phases play an important role of the failure. Different startup and shutdown durations were applied to study the phase influence on structural analysis and predicted life. Results reveal that the startup process mainly affects the remaining strain, while the shutdown process contributes more to the remaining stress on the cyclic effect; quick startup and longer shutdown duration conduce to prolong the chamber life and the life can be promoted 22% through reducing start-up duration from 1.5 s to 0.3 s, while 5% through increasing shut-down duration from 0.15 s to 0.9 s. © 2013 Elsevier Masson SAS. All rights reserved.
1. Introduction As one of the critical components of the reusable rocket engine, the thrust chamber is designed to operate in severe conditions of elevated temperature and pressure for improving the engine performance. At these elevated temperatures, in turn, the thrust chamber wall experiences significant inelastic strains. For an accurate estimation of the life of the chambers and to predict their progressive deformation with the number of loading cycles, a realistic stress–strain analysis must be made [8]. Unified viscoplastic analyses provide realistic descriptions of high-temperature inelastic behavior of materials, in which all inelastic strains (e.g. creep, plastic, relaxation, and their interactions) are accounted for as a single, time-dependent quantity. Arya and Arnold applied the Robinson’s model [2] and Freed’s model [1] to assess the inelastic deformation, which qualitatively replicated the doghouse effect, as shown in Fig. 1; Ray and Dai, basing on the Freed’s viscoplastic model, captured the nonlinear effects to represent the inelastic strain ratcheting, progressive bulging out, and thinning in the thrust chamber wall, which contributed lots to the on-line service-life prediction and damage analysis of the thrust chamber [5]; Sung et al. applied conventional and viscoplastic models to predict a small scale experimental annular plug nozzle thruster life, and validated by cyclic test, which revealed that the viscoplastic model showed better agreement in predicting the life cycle failure when both low cycle fatigue and fatigue–creep inter-
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1270-9638/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ast.2013.09.004
Fig. 1. Cross section sketch of thrust chamber and analytical model.
action were taken into account [19,18]; Schwarz et al. developed a viscoplastic model including ageing that successfully captured the observed accrued thinning of the hot wall and quantitatively reproduced the doghouse deformation [17]. However, most of the previous researches focus on the final state analysis, while the damage process study phase by phase using the viscoplastic theory is blank. The thrust chamber used in experiments at the NASA Lewis Research Center was analyzed in this paper [11], which used NARloy-Z as inner wall and Ni as the jacket, as shown in Fig. 1. The right figure depicts the finite element model applied in the analysis. It consists of 2800 elements and 2941 nodes. The coupled thermal–mechanical plan strain solid elements were used to model the smallest repeating segment of the cylinder wall. Because the wall was symmetrical, only one-half of a cooling channel was modeled. Experimental results published by Quentmeyer [11,12] and Pavli [10] demonstrate that the crack initiates at point D (Fig. 1)
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
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Table 1 Cyclic thermal and pressure loadings. Loadings
hf,cool kW/(K m2 )
T cool K
P cool MPa
hf,cool kW/(K m2 )
T hot K
P hot MPa
Precooling/post cooling Hot run
102 48.3
28 50
5.1 6.55
0 20.2
278 3364
0.0965 2.78
˙ in ˙ th (i , j = 1, 2, 3) ε˙ i j = ε˙ el i j + εi j + εi j
(3)
The elastic strain rate for an isotropic material is governed by Hooke’s law:
1+ν
ε˙ el ij =
Fig. 2. Working phases and loading history.
of the inner wall, i.e. the “roof” point of the doghouse. In order to explain the damage process, detailed viscoplastic stress–strain analysis of point D was conducted in this paper. Due to considerable influence over the thrust chamber structural and life analysis exerted by the startup and shutdown phases, different durations of the two phases were studied in the paper. Our results show that quick startup and longer shutdown duration conduce to prolong the chamber life, which gives an improvement method for the reusable rocket engine.
E
˙ in ij
ε =
F > 0 and S i j Σ i j > 0 or F > 0 and S i j Σ i j 0
(5)
Evolutionary law:
a˙ i j =
⎧ h in ⎪ ⎨ G β ε˙ i j − ⎪ ⎩
where
Coupled thermal–mechanical analysis of the thrust chamber was completed under the boundary conditions [2] shown in Table 1 and Fig. 2. Five phases are taken into account as one working cycle: precooling, startup, hot run, shut-down and post cooling. Table 1 shows the cyclic loadings working on the coolant side and the hot gas side, which respectively represented by subscript “cool” and “hot”. The convective film coefficient h f and bulk temperature of the fluid T characterize the cyclic thermal loading, and P means the coolant and hot gas pressure. The general heat equation for dynamic problems without internal heat sources is given as:
F=
εth is decided by the temperature increij
A F nΣij √ J2
0F 0
2.1. Thermal analysis
conductivity. The thermal strain ment:
(4)
E
where E is the Young’s modulus, v is the Poisson’s ratio, and σ i j is the stress. The repeated subscript in Eq. (4) and elsewhere implies summation over their range, and δ i j is the Kronecker delta function. A dot over a symbol denotes its derivative with respect to time t. The nonisothermal multiaxial inelastic constitutive equations for the model are given below: Flow law:
2. Thermal–mechanical analysis models
∂ T (x, t ) ρc = − div λ∇ T (x, t ) (1) ∂t where the ρ is density, c is specific heat, t is time and λ is thermal
ν σ˙ i j − σ˙ kk δ i j
G= J2 =
J2 K2 I2
h β G0
ε˙ in ij −
rG√m−β I2 m−β rG 0 √
I2
ai j
G < G 0 and S i j ai j > 0
ai j
G G 0 or G < G 0 and S i j ai j > 0
−1
(7) (8)
K 02 1
(9)
ΣijΣij
2 1 I 2 = ai j ai j 2 Σ i j = S i j − ai j and
Sij = σ ij − ai j = α i j −
(6)
1 3 1 3
(10) (11)
⎫
⎬ σ kk δ i j ⎪
⎪ αkk δ i j ⎭
(12)
where αi j ( T ) is the thermal expansion coefficient. Equivalent nodal forces are calculated from the thermal strain increment and then added to the nodal force vector for the solution of the problem.
The material parameters for Robinson’s model are taken from Ref. [2], which qualitatively replicated the “doghouse” effect and the thinning of the coolant channel wall. Effects of temperature are incorporated in the model through the temperature dependent parameters: Young’s modulus E, hardening variable K , recovery or softening material parameter R and so on. Detailed viscoplastic stress–strain analysis of thrust chamber wall basing on the Robinson’s model is conducted in the following section.
2.2. Robinson’s viscoplastic model
2.3. Uniformly valid asymptotic integration algorithm
Robinson’s model [16] employs a dissipation potential to derive the flow and evolutionary laws for the inelastic strain and internal state variables. The total strain rate ε˙ i j is decomposed into elastic ε˙ el i j , inelas-
Chulya and Walker present one uniformly valid asymptotic integration algorithm for the viscoplastic theory [4], which is implicit and has high stability and allows large time increments. By incorporating the algorithm into the finite element program MARC through a user subroutine called HYPELA2, a stress–strain analysis of the cylindrical thrust chamber wall was performed. Following
∂ ε th ij ∂t
= αi j ( T )
∂T ∂t
(2)
˙ th tic ε˙ in i j (including plastic, creep, relaxation, etc.), and thermal ε ij strain rate components. Thus
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J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
gives the derivation for the uniformly valid asymptotic expansion form of the Robinson’s model. To smooth the discontinuous boundaries in Robinson’s viscoplastic model for facilitating numerical computations, define a ”spline function” [16,3] P (x) on the interval (−1, 1) as:
⎧ (1+x)2 ⎪ ⎪ ⎨ 2 (1−x)2 P (x) = 1 − 2 ⎪ ⎪ ⎩1
1x<0 0x1 x>1 x < −1
0
(13)
SijΣij
W1
F
(14)
where the weighting function W 1 is to be selected by the user and the angular brackets denote:
x =
(15)
The discontinuities in ai j are removed by replacing the function G by G ∗ :
G1 =
G 2G 0
G, G2 4G 0
+ G 0 , G < 2G 0
S i j ai j ∗ + G0 G = (G 1 − G 0 ) P W2
(16) (17)
so the flow law becomes:
˙ in ij
ε =
A F n ( S i j − ai j )
√
J2 = K
ε˙ in ij =
2μ
2μ
J2
(18)
( S i j − ai j )
(19)
L˙ =
3μ R˙ √ ˙ K ( 23AR )1/n + 1
(26)
The inelastic strain rate tensor is written in the form of the deviatoric strain tensor e˙ i j as:
˙ in ε˙ in ij = e ij −
S˙ i j
(27)
2μ
2με˙ ii j − S˙ i j = L˙ ( S i j − ai j ) Let
y i j = S i j − ai j y˙ i j = S˙ i j − a˙ i j
AF n =√
(20)
J2
(28)
(29)
Then Eq. (28) becomes:
y˙ i j + L˙ y i j = 2μe˙ i j − a˙ i j
(30)
Eq. (30) is in the form of a first order differential equation and can be integrated as:
∂ εi j ∂ ai j 2μ ∂ εkk − δi j − ∂ε 3 ∂ε ∂ε ε =0 × exp − L (t ) − L (ε ) dε
S i j (t ) = ai j (t ) +
as:
2μ
(31)
With the smoothing function, the evolutionary law becomes:
a˙ i j =
The second invariant of the inelastic strain rate tensor is written
h G∗ β
ε˙ in ij −
rG ∗ (m−β)
√
I2
ai j
(32)
Rewrite Eq. (32) into a first order differential equation as:
a˙ i j + Q˙ ai j =
h G∗ β
ε˙ in ij
(33)
where
R˙ =
2 3
1/2 ˙ in ε˙ in ij ε ij
(21)
L˙
R˙ = √
3μ
rG ∗ (m−β)
√
J2
(22)
ai j (t ) =
h
exp − Q (t ) − Q (ε )
ε =0
3μ R˙ L˙
(23)
ε˙ in i j dε
(35)
2μ
δ i j εkk − ai j + 2μ ε i j − 3 1 − exp(− L ) × L
(24)
This is different from the original uniformly valid asymptotic integration algorithm, which is an adaption for the discontinuous problems. What’s more, this adaption contributes lots to the convergence of the algorithm application on Robinson’s viscoplastic
S i j (t + t ) = ai j (t + t ) + exp(− L ) S i j (t ) − ai j (t )
When substituted into Eq. (20) gets:
2 AF n R˙ = √
G∗ β
Furthermore, the uniformly valid asymptotic expansion form of the Robinson’s model is:
√
3
(34)
I2
t
Rearranging Eq. (22) yields:
J2 =
Q˙ =
Eq. (33) can be integrated to give:
Substitution of Eq. (19) into Eq. (21) gives:
(25)
t
where
L˙
2A
+1
Notice that Eq. (24) can be extracted directly from the right side of Eq. (20), and substituting Eq. (25) into Eq. (20) gives:
Eq. (18) can be rewritten as:
L˙
√ ˙ 1/n 3R
Equating Eq. (19) with Eq. (27) forms:
x0 x<0
x, 0,
√
The function F in Eq. (5) is then replaced by a function F defined as:
F=P
model, which can be applied to other discontinuous constitutive equations. J When R˙ = 0, from F = K 22 − 1 we can get:
ai j (t + t )
= exp(− Q )ai j (t ) +
h G
ε in ij ∗β
1 − exp(− Q )
Q
(36)
(37)
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
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where
√ L = L˙ (t + t ) t =
K
Q = Q (t + t ) t =
3μ R˙ t
√
(
3 R˙ 1/n ) 2A
+1
rG m−β t
√
I2
(38)
t + t
(39)
t + t
2.4. Life prediction model A life prediction approach with cyclic fatigue and quasi static fatigue two failure mechanisms developed by Riccius [14,13] is applied. Damage due to cyclic fatigue is simply the reciprocal of the number of cycles to fatigue failure as expressed below:
uc =
1
(40)
Nc
Fig. 3. Temperature evaluation of points A and D.
where N c is the number of cycles until failure, which can be obtained easily from the material ε − N curve basing on the structural analysis of thrust chamber wall. Usage factors are assumed to add up for each passed cycle and therefore, after N c identical cycles, u c 1.0 is obtained, which indicates failure a usage factor of for cyclic fatigue. The quasi static (or ratcheting) usage factor u qs is defined as:
u qs =
max(0, (εend − εbegine ))
(41)
εu
with the remaining strain εend and beginning strain εbegine after the considered cycle. εu is the ultimate strain of the combustion chamber wall material. After N qs = u1 identical cycles, the ultiqs
mate strain εu obtained, which indicates failure for quasi static fatigue. The total usage factor u t is defined as the sum of the cyclic and quasi static usage factors:
u t = u c + u qs
Fig. 4. Path temperature of different phases.
(42)
Finally, the total number of cycles N F until failure is calculated as the reciprocal value of the total usage factor u t :
NF =
1 ut
(43)
3. Results and discussion 3.1. Thermal analysis Fig. 3 shows the temperature evaluation of points A and D, while Fig. 4 illustrates the temperature along the path D-C-N-MB-A at every phase end. Due to the higher thermal conductivity of NARloy-Z and smaller inner wall thickness, thermal delay effects between the considered points A and D during the whole working phases is demonstrated in Fig. 3. At the precooling and shut-down phases, temperature of point D decreases quickly to 60 K, while main part of the jacket remains at ambient temperature at the end of the two phases. During the startup and hot run phases, the extreme thermal loads together with the regenerative cooling introduce enormous temperature gradients in the combustion chamber wall, as shown in Fig. 4, and the largest temperature difference between the point D and B researches almost 550 K, which causes large heat fluxes and will induce extremely large thermal stresses.
Fig. 5. NARloy-Z stress–strain data at different temperatures.
3.2. Detailed viscoplastic structural analysis Detailed structural analysis of the thrust chamber wall is completed with Robinson’s viscoplastic model in this part. Firstly, the implemented model is validated by Fig. 5 and Fig. 6. The predicted temperature dependent stress–strain responses of NARloy-Z with Robinson’s model in Fig. 5 coincide well with the experiment data [6], especially at high temperature. What’s more, the simulated hysteresis loops of NARloy-Z compare favorably with the experiment data [7] under the cyclic load as shown in Fig. 6,
88
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
Fig. 8. Evolution of the second invariant of the inelastic strain. Fig. 6. Saturated hysteresis loops of NARloy-Z.
Fig. 7. Circumferential stress–mechanical strain response of point D.
which the strain rate ε˙ = 0.004 s−1 and the temperature of 811 K. These excellent agreements obtained indicate that the Robinson’s model with the uniformly valid asymptotic integration algorithm could be used for more complex problems. The circumferential (x-direction) stress–mechanical strain response of point D is shown in Fig. 7. In the precooling state, the inner wall’s contracting trend is constrained by the jacket due to the thermal expansion ratio difference of the copper alloy and nickel, which lead to the tensile strain of point D. For the sudden loading suffered by the inner wall, the relaxation process appears at the later period of precooling. This recovery phenomenon for copper has been investigated by Kolsky through experiments [9]. During the startup and hot run phases, enormous temperature gradient is introduced in the combustion chamber wall. Therefore, large compressive strain of the copper alloy appears for the jacket tending to contract and the inner wall tending to expand. In the shutdown and post cooling states, the compressive strain releases and evolves into tensile strain for the same reason as in the precooling state. As shown in Fig. 7, the circumferential stress and mechanical strain vary widely throughout the startup and shutdown phases. What’s more, the inelastic strain is accumulated mostly in these two phases, as revealed in Fig. 8. So while the two phases are short, an important role is played of the whole thermal–structural analysis for the combustion chamber wall. Stress is determined by the Young’s modulus (Fig. 9) and elastic strain during the elastic phase. As indicated in Figs. 10 and 11, no inelastic strain variation appears from 0.1 to 0.4 s and 1.8 to 1.84 s. For the beginning startup phase, even if the total strain and elastic strain decrease slightly during this period, Fig. 7 shows that the stress reduces straightly until reaching the maximum compressive
Fig. 9. Evolution of the Young’s modulus.
Fig. 10. Evolution of the circumferential strains during the startup phase.
value. This considerable stress reduction is mainly resulted from the large Young’s modulus (> 140 GPa). The same elastic period lasts about 0.04 s during the shutdown phase. For relatively low Young’s modulus (approximate to 90 GPa), the stress just increases 100 MPa. After the elastic phases, as shown in Figs. 10 and 11, the copper alloy yields and inelastic strain accumulates quickly. Though the compression inelastic strain and total strain increase during the remaining startup phase, for sudden rise of temperature, the yield stress reduces nearly 50 MPa. Same to the shutdown phase, the hardening state appears with the tensile inelastic increasing. The stress increases slightly with large strain variation, which results from the temperature decrease. From 2.15 to 2.25 s, the cooper
J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
Fig. 11. Evolution of the circumferential strains during the shutdown phase.
Fig. 12. Circumferential stress–mechanical strain response with different startup durations of point D.
alloy appears a softer state, i.e. both the stress and strain recovery are taken. The hot run and post cooling phases could also be seen as the softer states following hardening periods of startup and shutdown respectively, which is revealed in Fig. 7. Recovery plays a major role in the problem of thermal ratcheting, which has been observed in thrust chamber wall of reusable rocket engines. The presence of mechanisms of recovery can allow creep strain rates to increase following periods of hardening and thus cause acceleration of creep buckling [15]. 3.3. Influence on structural analysis of different durations In order to analyze the influence of different durations on structural results, the remaining strain and stress are adopted as “cyclic effect” characters, which contribute directly to the ratcheting damage. What’s more, the accumulation of plastic strain, decrease of the inner wall thickness and “doghouse” failure are the physical phenomenon for “cyclic effect”, which can also be calculated or explained by the remaining strain and stress. Fig. 12 presents the comparison between different startup durations of 1.5 s, 0.9 s, and 0.3 s. The solid part of these curves shows the circumferential stress–mechanical strain response during the startup phase. As illustrated in the figure, no difference appears through the elastic analysis. For the yield stress will increase with higher strain rate under dynamic loadings, so the shorter of the startup duration, the lower of compress stress appears during the hardening period, but after the whole working phase, this distinction is eliminated. The cyclic influence of the startup duration is revealed by the remaining strain, which has close relation to the
89
Fig. 13. Circumferential stress–mechanical strain response with different shutdown durations of point D.
Fig. 14. Evolution of the circumferential strains for different startup durations.
thrust chamber life. Fig. 12 shows the remaining strain increases with longer startup duration. The circumferential stress–mechanical strain response with shutdown durations of 0.9 s, 0.45 s and 0.15 s are shown in Fig. 13. Same as the startup duration analysis, the elastic periods are similar, and the yield stress decreases as the shutdown duration increases. What’s more, the hardening effect is strengthened when the engine shut-down quickly, which leads to larger maximum stress and remaining compressive stress. Comparing Fig. 13 with Fig. 12 on the cyclic effect, it can be found that the startup process mainly affects the remaining strain, while the shutdown process contributes more to the remaining stress. This difference is principally caused by the following recovery process, i.e. the hot run and the post cooling states respectively. For the startup duration analysis, as illustrated in Fig. 14, the inelastic strain is lower with 0.3 s and the compressive total strain recovers less within the limited recovery time for greater hardening effect. As the yield stress is fixed with temperature in the last two phases, so no stress difference appears and the strain variation is enlarged; while with enough recovery duration in the post cooling phase, as shown in Fig. 15, the strains of different shutdown duration recover to the same state, so the hardening effect on stress lasts till the end of cycle. 3.4. Influence on predicted life of different durations Depending on the cyclic structure analysis in Fig. 16 and Fig. 17, the influence of different startup and shutdown durations on life is evaluated by Riccius developed life prediction model. As shown in Fig. 16, the remaining strain difference at the pre-cycle is enlarged
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J. Yang et al. / Aerospace Science and Technology 34 (2014) 84–91
Table 2 Predicted life for different startup/shutdown durations. Startup/shutdown duration (s) Strain range Increment of remaining strain Life (cycles)
0.3/0.45 0.0148 0.0021 124
0.9/0.45 0.0147 0.0024 105
Fig. 15. Evolution of the circumferential strains for different shutdown durations.
Fig. 16. Cyclic stress–mechanical strain response with different startup durations of points D.
in the next-cycle startup phases, which induces large strain range distinction. As the yield stress is fixed with temperature, the different startup durations have no effect on the stress in shutdown and post cooling phases. With the same reason, for different shutdown durations in Fig. 17, after the startup and hot run phases, the stress difference is eliminated and the tide of curves in next-cycle almost keeps unchanged with the pre-cycle. As shown in Table 2, for the start-up at prescribed shut-down duration of 0.45 s, if the start-up duration is reduced from 1.5 to 0.3 s, the life increases 22%; for the shut-down at prescribed start-up duration of 0.9 s, if the shut-down duration is increased from 0.15 to 0.9 s, the life increases 5%. As the result illustrates, quick startup and longer shutdown duration conduce to prolong the chamber life. Comparing the influence of startup and shutdown durations on life, it can be found that the increment of remaining strain, which represents the ratcheting damage, contributes more to the failure. As a result, the effect of startup durations on life is greater. 4. Conclusion The damage process of thrust chamber wall was explained by detailed viscoplastic structure analysis phase by phase. Analysis re-
1.5/0.45 0.0142 0.0025 102
0.9/0.15 0.0148 0.0025 101
0.9/0.9 0.0141 0.0024 106
Fig. 17. Cyclic stress–mechanical strain response with different shutdown durations of points D.
vealed that under the same level of thermal–structure loadings, the startup and shutdown phases play an important role in the stress and strain evolution. Both the startup and shutdown phases include one elastic period and one hardening period. Resulted from the corporate contribution of the Young’s modulus and elastic strain, considerable stress varies with slight strain variation in the former period and contrary in the other period for sudden variation of temperature. Influence of startup and shutdown durations on structural analysis and predicted life indicated that no difference appears in the elastic period; hardening effect is strengthened with short durations; on the cyclic effect, startup process mainly affects the remaining strain, while the shutdown process contributes more to the remaining stress; quick startup and longer shutdown duration conduce to prolong the chamber life, which gives an improvement method for the reusable rocket engine. Results indicate that the life can be promoted 22% through reducing start-up duration from 1.5 to 0.3 s, while 5% through increasing shut-down duration from 0.15 to 0.9 s. The analysis result can also be extended to explain the stress– strain response under the quickly varying thermal–structural load like the startup and shutdown phases of steam and gas turbines, jet engines, steam boilers, nuclear reactors, etc. References [1] V.K. Arya, Nonlinear structural analysis of cylindrical thrust chambers using viscoplastic models, NASA CR-185253, 1991. [2] V.K. Arya, S.M. Arnold, Viscoplastic analysis of an experimental cylindrical thrust chamber liner, AIAA J. 30 (1992) 781–789. [3] V.K. Arya, A. Kaufman, Finite element implementation of Robinson’s unified viscoplastic model and its application to some uniaxial and multiaxial problems, NASA TM-89891, 1987. [4] A. Chulya, K. Walker, A new uniformly valid asymptotic integration algorithm for elasto-plastic-creep and unified viscoplastic theories including continuum damage, NASA TM-102344, 1989. [5] X. Dai, A.K. Ray, Life prediction of the thrust chamber wall of a reusable, J. Propuls. Power 11 (1995) 1279–1287. [6] J.J. Esposito, R.F. Zabora, Thrust chamber life prediction, vol. I. Mechanical and physical properties of high performance rocket nozzle materials, NASA CR-134806, 1975.
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