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Development of analogy method for thermal-fatigue crack propagation in pressurized cylinder by using permeation diffusion-fracture model
Journal Pre-proofs Development of analogy method for thermal-fatigue crack propagation in pressurized cylinder by using permeation diffusion-fracture model Ziming Yan, Minjin Tang, Gang Chen, Tao Wang, Xiang Li, Zhuo Zhuang PII: DOI: Reference:
S0013-7944(19)30688-5 https://doi.org/10.1016/j.engfracmech.2019.106710 EFM 106710
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
3 June 2019 28 August 2019 4 October 2019
Please cite this article as: Yan, Z., Tang, M., Chen, G., Wang, T., Li, X., Zhuang, Z., Development of analogy method for thermal-fatigue crack propagation in pressurized cylinder by using permeation diffusion-fracture model, Engineering Fracture Mechanics (2019), doi: https://doi.org/10.1016/j.engfracmech.2019.106710
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© 2019 Published by Elsevier Ltd.
Development of analogy method for thermal-fatigue crack propagation in pressurized cylinder by using permeation diffusionfracture model Ziming Yan1, Minjin Tang2, Gang Chen2, Tao Wang1, Xiang Li1, Zhuo Zhuang1 1Applied Mechanics Lab., Dept. of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084 China 2
Shanghai Electric Power Generation Equipment Co., Ltd. Turbine Plant No. 333, Jiangchuan Road, Minhang District, Shanghai 200240 China
Abstract Natural crack propagation analysis in thermal-fatigue fracture problem is a challenging problem. Reliable predictions for the crack path and residual live of structure provide essential information for the structural integrity assessment and repairment. Various conventional predictive fatigue models have been proposed to evaluate the remaining life, but most of them can’t build an explicit relation between damage degree and fatigue life. In recent years, the advanced techniques on crack propagation analyses were proposed, which were used to simulate the fatigue crack growth, but no coupling thermal-fracture function existed in the commercial finite element codes. Fortunately, the extended finite element method (XFEM) in ABAQUS has the coupling permeation diffusion-fracture model. In this paper, an analogy method from coupling permeation diffusion-fracture to coupling thermal-fatigue is proposed to compute fatigue fracture, which is applied for predicting the fatigue life in a specific steam turbine by directly using ABAQUS. Typical fatigue experiments by compact specimens of cast iron are carried out for the fatigue parameters measurement. By using analogy method and Paris’ Law, an entire analysis process for the turbine units is built to estimate the thermalfatigue life. The simulation results predicting remaining life and analyzing fatigue crack growth for the pressurized cylinder are agreed well with the measured engineering data. Keywords: steam turbine, analogy model, thermal-fatigue life, permeation diffusionfracture, XFEM
Corresponding author. Tel.: +86- 10-62783014; Fax: +86- 10-62783014. Email: [email protected] (Z Zhuang)
Nomenclature
qw
fluid flow
a
current crack length
qT
heat flux
ac
critical crack length
t
current time
B, W
thickness and width of specimen
T f , Ts
the fluid and solid temperature at the boundary layer
c
volumetric specific heat
Biot’s coefficient
C, m
parameters of Paris’ Law
T
thermal expansion coefficient
E ,
elasticity modulus and poisson ratio
w
fluid’s proportion
a g W
shape parameter of stress intensity factor
c
crack surface
G
shear modulus
u , t
h
coefficient of heat transfer
T , q
displacement and force boundary temperature and heat flux boundary
kT
thermal conductivity
ij
Kronecker delta
kp
permeability coefficient
ij
strain
K
bulk modulus
v
volumetric strain
Ks , K f
grain’s and fluid’s bulk modulus threshold of stable fatigue fracture
lame constant
material’s density
K
range of stress intensity factor
w , s
liquid’s and solid’s density
K IC
fracture toughness
v
volumetric strain
ms
mass of solid grain
ij
stress
M
elastic constant defined in Biot’s theory
ijp , ijT
stress in heat diffusion and in flow diffusion
N
cycle times
void ratio
pf
pore pressure
IP
intermediate pressure
ps
pore pressure of solid at the boundary
SIF
stress intensity factor
P
force in fatigue experiment
K th
1. Introduction Combined cycle steam turbine is an important part of combined-cycle power plant (CCPP) that drives generator set through gas and steam turbines[1]. When the turboset begins to work, the gas turbine starts and generates residual high-temperature steam. Then the waste heat from gas turbine is routed to the nearby steam turbine, which generates extra power. In long-term work, it turns out that the ullage of turboset fatigue life depends on thick-walled steel structures and movable parts under frequent starts and stops, of which intermediate pressure (IP) inner cylinder (Fig. 1) is a typical component. For the inner cylinder, it is a typical thick-walled pressure vessel, which produces high thermal stress in flanges with a significant temperature gradient, especially those located in horizontal and vertical middle planes, respectively. Fatigue crack initiates in the internal surface at the flange corner and extends in a direction to the bolt hole, as shown in Fig. 2. When surface cracks develop into through cracks, the inner cylinder cannot be repaired. Fatigue crack always appears in these areas along the circle under long-term cycle thermal-mechanical load. In actual engineering, it is difficult to identify the degree of fatigue crack propagation in real-time observation. However, if a feasible method can be used to estimate the fatigue crack propagation of working turboset, it will be helpful to determine the overhaul downtime to avoid the permanent damage.
Fig. 1 IP inner cylinder
Fig. 2 Surface crack appeared on the flange
Previous work has made extensive research and developed advance techniques on crack propagation analysis, such as finite element method (FEM) based nodal force release technique[2, 3], FEM based local or global remeshing[4-9], extended finite element method (XFEM)[10, 11] and generalized finite element method (GFEM)[12, 13], finite element alternating method[14, 15], phase field method[16-18] and so on. However, there is no commercial FEM software yet that could compute the fatigue crack propagation directly under coupling thermal-mechanical load. Fortunately, the natural essence of XFEM in ABAQUS has the coupling permeation diffusion-fracture capability, which supports a calculation module to solve hydraulic fracturing problem based on Biot’s model[19]. In this paper, an analogy method from coupling permeation diffusion-fracture to coupling thermal-fatigue is proposed to compute fatigue fracture, which is applied for predicting the fatigue life in a specific steam turbine by directly using ABAQUS. The dual relation between heat diffusion equation and seepage diffusion equation is obtained. Therefore, the fatigue crack propagation under thermalmechanical load can be conveniently computed by using hydraulic fracturing calculation module. In order to estimate the remaining service life of turboset reasonably, an integral analysis process is built to predict fatigue life on the components of the steam turbine. Typical fatigue experiments by compact specimens of cast iron are also carried out for the fatigue parameters measurement. By using analogy method and Paris’ Law, an entire analysis process for the turbine units is built to estimate the thermal-fatigue life. Consequently, the simulation results predicting remaining life and analyzing fatigue crack growth for the pressurized cylinder are agreed well with the measured engineering data.
2. Analogy method to solve coupling thermal-mechanical problem For crack propagation under thermal-mechanical coupling load, its solution domain 0 in three-dimension space is a discontinuous displacement field, as shown in Fig. 3, where c is crack surface, u is displacement boundary, t is force boundary, T is temperature boundary and q is heat flux boundary.
Fig. 3 Solution domain in three-dimension space
Duflot[20] considered crack propagation calculated by XFEM based on thermoelastic medium and established new XFEM scheme including discontinuous temperature field. Pathak[21, 22] considered thermal fatigue crack problem based on a three-dimensional thermoelastic medium and analyzed different types of cracks such as plane crack, non-plane crack under the thermal shock load, adiabatic load and isothermal load. Habib[23] changed previous sequential coupling methods and established XFEM scheme based on fully coupled thermal-mechanical model to solve the quasi-static problem. Although these works have considered fracture under coupling thermal-mechanical load, some unsolved problems are left for the fatigue fracture of actual structure under cycling thermal-mechanical load and complex operating conditions. Nowadays, the reliable analyses for crack initiation and propagation are given by AREVA during the Pressurized Thermal Shock (PTS) event for thick-walled pressure vessel under thermal-mechanical coupling load [24] and by Alstom for crack propagation of turboset under high pressure and temperature [25]. Both of them fully decoupled heat diffusion field and discontinuous displacement field through computing temperature field without fracture. Neither of them considers the coupling heat diffusion and fracture. Thus, these computing methods can’t solve the situation that newly formed crack surface becomes a new heat boundary, which influences on the temperature field. As mentioned in Section 1, the commercial code ABAQUS supports calculation to solve hydraulic-fracture problem by using XFEM, which includes a flow diffusion equation in porous medium. In contrast, we need to solve a thermal-fracture problem with a heat diffusion equation in thermal medium, as shown in Fig. 4. Based on relation of mathematical equations between two physical processes, the thermal-fracture problem can be solved through hydraulic-fracture solver. To establish the integrated
dual relation, both governing equations and boundary conditions must be considered.
Fig.4. heat diffusion in thermal medium and flow diffusion in porous medium
For thermo-elastic medium, the constitutive equation is
ij v ij 2G ij 3T K T T0 ij
(1)
where is lame constant, G is shear modulus, T is thermal expansion coefficient and K is bulk modulus. T and T0 are current temperature and initial temperature. For permeation diffusion problem, the constitutive theory of poroelasticity [26-28] is used to describe permeation diffusion where pore pressure p f is used to describe fluid flow in porous medium. This theory simplifies primitive complex coupled fluidmechanical problem and its constitutive equation for the porous elastic medium is ij = v ij 2G ij p f ij
(2)
where is the material constant defined in Biot’s theory and p f is pore pressure which is a field variable as the fluid influence. In comparation between Eq. (1) and (2), if temperature variation T T0 is equal to pore pressure p f numerically and corresponding relations of the other parameters are found, the thermal-fracture problem can be analogically solved from hydraulicfracture problem. The deformation and stress fields from hydraulic-fracture can be transformed to the thermal-fracture problem. For the numerical simulation of mechanical problem with diffusion field, the problem can be divided into static and transient analytical models. The former does not consider the diffusion process with time but the latter does. Firstly, the dual relation of mathematical equations is established between two physical processes based on static analytical models. T ij v ij 2G ij 3T K T T0 ij ijp = v ij 2G ij p f ij
(3)
Generally, pore pressure p f is non-positive, so the constitutive equation should be expressed as follows if p f is equal to temperature variation T T0 numerically
ijp = v ij 2G ij p f ij Consequently,
to
make
ijT ijp
and
(2)
p f T T0
numerically,
the
corresponding relation is
3T K
E T 1 2v
(5)
Besides, grain’s bulk modulus Ks is always used in numerical simulation instead of Biot’s constant in static analysis. The relation between Ks and is
1
K Ks
(6)
The corresponding relation between Ks and thermal parameter is given by Ks
K K 1 1 3 T K
(7)
Thus, the dual relation in static analysis is given in Table 1. Table 1 Corresponding relation between two physical process in static analysis
Permeation-diffusion problem
Thermal-mechanical problem
pf
T T0
3T K
1 K s
K
For a transient problem, the variation of diffusion field with time must be considered. More parameters are required whether in a thermal-mechanical problem or a permeation-diffusion problem. Table 2 shows the required parameters in static and transient analyses, where,
is density, c is volumetric specific heat, qT is heat
flux, kT is conductivity, k p is permeability, K f is fluid’s bulk modulus, is
void ratio,
w is fluid’s proportion and qw is fluid flow.
Table 2 Required parameters of two physical processes in numerical simulation
Permeation-diffusion problem
Thermal-mechanical problem
Parameters in static analysis
E , , T , kT
E,v,k p ,K s
Parameters in transient analysis
, c, qT
K f , , w , qw
In thermal analysis, Fourier’s Law is used to define the heat flux qT
qT kT T
(8)
In seepage analysis, seepage equation is used to define the fluid flow qw
qw
kp
w
p
(9)
If pore pressure p f is non-positive considered and qT qw , the corresponding relation between kT , k p and w is given below
kT
kp
(10)
w
To identify other parameters’ corresponding relation, we need to make the contrast of two diffusion equations. For the heat diffusion, the diffusion equation is T kT 2 T 0 t c
(11)
For the seepage diffusion, the diffusion equation in Biot’s theory is p f t
k p M 2 p f M
v t
(12)
where M is an elastic constant defined in Biot’s theory. Apparently, this constant situated in both sides of Eq. (12) and can’t be directly found the relation between M and the other heat parameters. This diffusion equation should be modified until the relation can be figured out. For porous medium, suppose mass conservation for inflowed fluid, we can get
w wqw t
(13)
Considering the seepage diffusion described in Eq. (9), the form of Eq. (13) can be changed by
wk p2 p f
w d w t dt
(14)
According to the definition of fluid’s bulk modulus K f , the density of fluid w is a function of time, pore pressure and fluid’s bulk modulus. Thus, the change rate of
w is expressed as w w p f p f w K f t t p f t
(15)
For solid grain, supposing the mass conservation for this portion, we can get
ms 1 s
(16)
D ms D 1 s d 0 Dt D t
(17)
1 d s v d 1 dt t s dt
(18)
where ms is mass of solid grain and is domain of porous model. For a fatigue problem refer to cast iron under long-term cycling loading, the change rate of solid grain’s density is negligible. That is
d s 0 dt
(19)
d 1 v dt t
(20)
So that, Eq. (10) can be written as
Considering Eq. (15) and Eq. (20), then Eq. (14) may be expressed as follows,
p f t
kpK f
2 p f 1
v t
(21)
Consequently, the modified seepage diffusion equation is obtained, which can be made comparison with Eq. (11). Obviously, let 1 , the corresponding relation of
the other heat parameters is
wK f 1 c
(22)
With the relation given above, the dual relation in transient analysis is given in Table 3. Table 3 Corresponding relation between two physical process in transient analysis
Permeation-diffusion problem
Thermal-mechanical problem
qw
qT
3T K
1 K s
K
kp
w wK f
kT 1 c
Based on the corresponding relation shown in Table.3, the primitive coupled thermal-mechanical problem could be solved by using hydraulic fracturing solver in commercial FEM codes. In addition, the temperature of inner cylinder is 20 C at the initial state and the highest temperature of whole cylinder is up to 600 C in normal working state. More narrowly, the material’s parameters in thermal-mechanical problems dependent on the temperature should be considered, especially for the materials of turbine which have significant change of temperature during each cycle. In analogy method, the influence of temperature on material is considered. These parameters should be also dependent on the pore pressure through the corresponding relation with material thermo-physical parameters. Consequently, all analysis of computing result in the paper has considered the effect of temperature. To ensure the integrity of whole analogy, the corresponding relation between boundary conditions is required to be confirmed. The most common thermal boundary conditions [29] are temperature boundary, heat flux boundary and boundary layer condition (Fig. 5) for turbine units.
Fig. 5 Boundary layer condition
Since both of two physical models are diffusion problem, the form of boundary conditions is also of analogy. For temperature boundary condition, its effect is the same as the pore pressure boundary condition, T p f . For heat flux boundary condition, its effect is the same as a seepage boundary condition, qT qw . Here, the value of seepage boundary becomes negative because the pore pressure is non-positive by default. For boundary layer condition, it is expressed as qT h T f Ts , where h is coefficient of heat transfer, T f is the fluid temperature at the boundary layer and Ts is the solid temperature at the boundary layer. So we can define a new formation of fluid flow as qw h p f ps , where p f is pore pressure at the boundary and ps is the pore pressure of solid at the boundary. In ABAQUS, the subroutine DFLOW can be used to build the new formation of fluid flow. The analogy method redefines the value of equation parameters, so the physical meaning of the analogous model is no longer a permeation-diffusion problem but a coupled thermal-mechanical problem. Using the proposed corresponding relations on governing equations and boundary conditions, the equivalence is achieved between coupled thermal-mechanical fracture and permeation-diffusion fracture problem, which makes subsequent computation of thermal-fracture propagation. The program is easily implemented into the commercial FEM codes. Based on the simple plane model, as shown in Fig. 6, the diffusion fields are computed as shown in Fig. 7 in order to verify the above relations. The material is cast iron, CrMoV[30], and parameters changing with temperature. The bottom of model is
fixed. The heat flux into the right side is qT 10mJ/ mm 2 s
and the initial
temperature is 0 C . The loading time is 300s. The size of element is 2mm 2mm and the type of element is CPE4RT (Fig. 7(a)), which is the reduced 4-node plane strain element with the freedom of temperature.
Fig. 6 Plane model
(a)
(b)
Fig. 7 (a) show the temperature at 300s and (b) show the pore pressure at 300s. Both results are computed under dual relations
Then the diffusion field is computed through the use of transferred material parameters based on relationship shown in Table 3. The fluid flow into the right side is
qw 10mD/ mm 2 s and the initial pore pressure is 0.0MPa . The load time is 300s. The size of element is 2mm 2mm and the type of element is CPE4RP (Fig. 7(b)), which is the reduced 4-node plane strain element with the freedom of pore pressure. Extracting the variation of temperature and pore pressure at point A in Fig. 6, we can see the value of diffusion field at the value point is identical (Fig. 8). The results prove the correctness of corresponding relations and the fitness of permeation-diffusion solver, which is applied to solve the coupled thermal-mechanical problem through transferred parameters. It illustrates that we have realized the accurate simulation for a coupled thermal-mechanical problem through another simulating module, which can be utilized to calculate crack propagation.
Fig. 8 The contrast between temperature and pore pressure at point A
3. The analysis process of fatigue crack propagation based on submodels When simulating the fatigue crack propagation for actual engineering structures, like IP inner cylinder, the computational efficiency is supposed to be taken into account. Considering that the macrocracks are mainly formed in the surface of volute flanges, the computing time will be rather long if using the full-size model to compute fatigue propagation. Hence, adopting submodels is an efficient way to reduce computational expense. For IP inner cylinder, the submodels should contain the parts where main fatigue cracks always appear. The submodels are partitioned from the cylinder model shown in Fig. 15. Here, the classic formula of Paris’ Law is applied to describe the relationship between crack growth rate and stress intensity factor da m C K (23) dN where
da is a crack growth rate, K is the stress intensity factor (SIF) range, C dN
and m are material constants defined by experiments. The fatigue total-life-cycle[31-33] is generally divided into three stages: fatigue crack initiation, crack steady propagation and rapid crack propagation. At the stage of fatigue crack initiation, the complex reasons and initial microcracks are not always the leading cracks which develop perforated cracks. At the stage of rapid crack propagation, the time of whole process is short. When fatigue macrocracks extend at the stage, the time can be neglected as the structure would be almost failure or destroyed rapidly. In this paper, the simulation aims to compute the time of fatigue crack growth at the steady
stage. As simulating fatigue crack propagation by using XFEM, the criterion of crack growth should be established. Furthermore, it is difficult to guarantee the fatigue crack stay in steady propagation stage, so the criterion contains two modes: In steady propagation, Paris’ Law is applied to describe the fatigue crack growth length[34], and the direction of maximum principal stress decides the propagation direction and fracture initiation time. In rapid crack propagation, the maximum principal stress criterion[35] could be utilized to decide the time of crack initiation and propagation direction. Besides, the stress intensity factor range K is used to judge which mode of the criterion is applied:
When
ΔK th K ΔK IC , fatigue crack is in steady
propagation; when K ΔK th , the crack arrest; when K K IC , the crack becomes rapid propagation stage. The phantom nodes technology[36-38] is applied in XFEM scheme of ABAQUS, in which it is negative for the parts of displacement field at the crack tip. Under this circumstance, the crack front must penetrate the entire element if the criterion is satisfied. Meanwhile, considering that the global model simulation with fatigue crack propagation requires very high cost of computation in each cycle, an average method is utilized to simplify the whole analytical process. In each cycle time, the crack increment a is a crack growth length through a whole element. Suppose that the stress intensity factor range K keeps constant, the average growth rate in this cycle time is
a a m C K . So, the actual cycle times is equal to N m . In N C K
every simulating cycle, the cycle times N equals to the minimum of all N computed on those elements along the crack front. Besides, the numbers of elements to be penetrated would also be recalculated according to the minimum N . If the cycle n
time is equal to n in simulation, the total crack growth length is equal to a ai i 1
and corresponding stress intensity factor range is equal to Ki . The total actual cycle time is equal to n
n
i 1
i 1
N N i
ai
C K i
m
(24)
The process implementation to realize the average method is: In every simulating cycle, the fatigue crack propagation is limited to elements along the front of crack (Fig. 9). If the crack extends through one element under the criterion of crack steady
propagation, it cannot extend again unless the stress intensity factor satisfies K K IC . In this method, the stress intensity factor K will be calculated in each
simulating cycle[20-22]. At the same time, the corresponding crack increment a can be computed. When the total crack growth length up to the required value, the actual cycle times can be computed by Eq. (24).
Fig. 9 In every cycle, we assume that the fatigue crack propagation only appears in the elements along the front of crack
Combining the submodel technology and above method to compute actual cycle times, the fatigue crack analysis process is obtained, as shown in Fig. 10.
Fig. 10 The left flow diagram of fatigue crack analysis process based on submodel. The right flow diagram of fatigue crack analysis process based on steam turbine cylinder
The whole analytical process can be divided into two part. The first is analogy of two physical models and the second is to obtain the value of freedoms of global model to drive submodels. Consequently, the entire fatigue crack analysis process includes the following four steps: a) transforming the parameters through analogy method; b) computing the full model without crack through one cycle to get the value of freedoms to drive submodel, including three displacement freedoms and temperature freedom
(the temperature freedom is replaced with pore pressure freedom); c) computing the submodel with crack through several cycle times; d) evaluating the residual service life based on Paris’ Law in the end.
4. Fatigue experiment by using CT specimen For the fatigue simulation, it is necessary to describe fatigue crack propagation based on proper fatigue model. The purpose of fatigue experiment is to gain material parameters that influence fatigue crack growth rate[39, 40]. In this study, Paris’ Law [41] is applied to establish the relationship between fatigue crack growth and stress intensity factor (SIF) in the crack growth stage. The compact tension (CT) specimen made by cast iron, CrMoV[30, 42], is applied to fatigue experiment, as shown in Fig. 11. The effective width of CT specimen is 66mm and thickness is 12.7mm. The cycle loading is sine wave load which makes the specimen stay in tension, of which load frequency is 15Hz. During the entire process of loading, the minimum tensile force is 2.7kN and the maximum is 27kN in each cycle with a constant ratio R 0.1 of them.
Fig. 11 Compact tension specimen in fatigue experiment
With the cycle loading, the crack moves nearly in a straight line and the crack growth rate increases during it propagates. According to the Eq. (25), which shows the computing SIF formula towards CT specimen in fatigue experiment, SIF becomes more significant with crack length increase. When cycle times up to about 45,000, the mode of crack propagation becomes unstable and the specimen is destroyed rapidly, as shown in Fig. 12.
K
P a g 1/2 BW W
(25)
where P is the force that tensile testing machine exerts, B is the thickness of
a specimen, W is the width of specimen and g is the shape parameter of stress W intensity factor[43].
(a)
(b)
Fig.12. The fatigue experiment process. (a) shows the crack length a 1.3545mm when cycle times
N 5， 600 ; (b) shows the crack length a 6.9106mm when cycle times N 16， 500 ; And the crack propagation entered the phase of instability when cycle times N was approximately equal to 45,000
To take the logarithm of both sides of Eq. (23), the equation is written as da ln ln C m ln K which is applied to plot the curve shown in Fig. 13 and dN confirm the Paris’ Law’s parameters with a linear square fit. The slope of fitted curve is m and the intercept of fitted curve is ln C .
Fig. 13 The relation between fatigue crack growth rate
da and stress intensity factor range K of dN
cast iron, CrMoV
This experiment is carried out to determine two material parameters of Paris’ Law
C=2.28 1010 , m 2.3432 , respectively. Generally, K th corresponds to the K when
da is equal to 107 mm/cycle . According to the tested formula of Paris’ Law, dN
K th is equal to 0.7035MPa m . KIC is equal to material’s fracture toughness, K IC , as the region of crack tip under compression during cycle loading.
5. The fatigue analysis on steam turbine cylinder The finite element model of IP inner cylinder is shown in Fig. 14. For cast iron used in the cylinder, the fracture toughness KIC 134.8MPa m . The fatigue crack growth rate is represented by Paris’ Law,
da 2.3432 . 2.28 1010 K dN
According to the operating feature of hot start, the single working state is divided into three typical stages: the time of heating stage is 7200s by transient analysis; the steady stage is computed by steady analysis; and time of cooling stage is 28,800s by transient analysis. In the normal working state, the steam enters into the intake chamber through the volute and turns the turbine blades, as shown in Fig. 14. The inner walls of volute and intake chamber endure high-temperature steam with high pressure, while the outer walls of cylinder and other surface endure steam with relatively low-temperature steam with relative low pressure.
Fig. 14 The left is IP inner cylinder model including upper and lower cylinders which are connected by bolted joints. The right shows the flow of inlet and outlet steam in inner cylinder. Both inside and outside surface endure different temperature and pressure
The partitioned submodels are given in Fig. 15, and regions of flange included. The preexisting crack in submodels is a quarter-elliptic surface crack. The semimajor axis of the preexisting crack is 15mm and the semi-minor axis of preexisting crack is 5mm. The degree of angle between the crack surface and horizontal plane is 45 degrees.
Fig. 15 (a) The figure shows the connection between global models and submodels: both submodels are partitioned through red cutting plane. As connecting planes between global model and submodel, the results calculated by global model, including displacement and temperature, should be applied on these red planes as submodels’ boundary conditions. The other loadings and boundary conditions on original surfaces should keep the same with global models. (b) The figure shows the temperature field of inner cylinder (heating stage, 7200s) under the above load and boundary conditions. (c) The figure shows the connected submodels and the location of preexisting cracks. (d) The figure shows the temperature field of inner cylinder (heating stage, 7200s) under submodel boundary conditions and original loadings and boundary conditions.
Through calculation under cycle loading, the results illustrate the variation of stress in submodel and the behavior of fatigue crack growth, as shown in Fig. 16. At the heating stage, the pressure and temperature increase as the high-temperature steam enters into the cylinder. The corner is still under pressure and the crack keeps in closure. With the increase of temperature gradient, the pressure becomes higher. When entering the cooling stage, the cooling process leads to a large temperature gradient and the pressure at the corner is reduced rapidly. With the process of cooling, the stress states at the corner are changed from compressive stress to tensile stress. As a result, the crack opens gradually and crack propagation appears at this stage.
Fig. 16 The field variable named UVARM4 represents the maximum principal stress. (a) (1800s, heating stage) shows the crack kept in closure; (b) (6500s, heating stage) shows the crack kept in closure; (c) (4000s, cooling stage) shows the crack opened; and (d) (8600s, cooling stage) shows the crack further opened.
Through computation circularly, the surface crack at the middle plane is extended from 5mm to 65mm (Fig.17). The crack path is near along the direction of preexisting crack. With the fatigue crack propagating, the growth rate increases gradually. Fig. 18 illustrates that the K at crack tip on the surface of middle plane is also increased as the fatigue crack extended. Fig. 19 illustrates the relationship between the crack length and actual cycle times through Paris’ Law. When the crack length is up to 65mm, the actual cycle is equal to 3,257 times. In the normal working state, the times of startup and shutdown is about 200. Consequently, the corresponding service is 16.3 years. This value is quite close to the estimating years from overall detection, about 20 years, but it is more conservative.
Fig. 17 The crack propagation on the surface of the middle plane
Fig. 18 The relation between K and cycle times(simulation)
Fig. 19 The relation between the length of crack growth and actual cycle times
Actually, the fatigue crack in flange does not extend to the range what we have computed. The turboset must be shut down and maintained during furnace operation and periodic repair before the depth of surface crack is up to several millimeters. However, an accurate estimation on the states of crack with experience is a hard work. In the practical use, the method can be utilized to support a proper downtime before the crack extends to a critical length or depth a c . Based on this method, we do not need to shutdown the turbine only by experience. We can help engineers to estimate whether the fatigue cracks extend up to the critical depth or length, support reliable downtime to maintain the turboset, and remove the formed crack to prolong the use-life to the designed-life as far as possible.
6. Conclusion In this paper, the dual analogous relationship of equation parameters has been established between the thermal-diffusion and permeation-diffusion problem. A thermal-fracture problem for thermal medium can be transformed into a hydraulic
fracture problem for the porous medium by the method. The method supports a smart way to solve the fracture problem under coupled thermal-mechanical loads, which has comparatively strong practicality in engineering. In addition, the general process of fatigue crack analysis on turbine units is built under high temperature and pressure cycle loadings. Under above process, the relation between use-life and fatigue crack propagation is established explicitly. It is convenient to obtain the cycle times corresponding to a certain crack length specified. Relying on the actual model of steam turbine cylinder, this work analyzed the characteristic of fatigue crack growth under coupled thermal-mechanical cycle loadings and summarized the relationship between fatigue crack growth rate and cycle times in hot start. The calculating result is close to the one by overall detection. Meanwhile, some problems remain to be solved on the service life prediction. The element size has an effect on estimation of cycle times through an average method. A proper mesh will result in a more precise estimation. Besides, this work studies the fatigue crack growth when the structure has preexisting macrocracks. In fact, the fatigue crack initiation period is a main stage of the total fatigue life. To build a numerical simulation method to predict the total fatigue life is meaningful for structure design and optimization.
7. Acknowledgement This work was supported by the National Science Foundation of China. (Grant No.11532008)
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Highlights
Dual relationship between heat diffusion equation and seepage diffusion equation An analogy method to compute fatigue fracture in steam turbine by using ABAQUS An analysis process for turboset to estimate the remaining thermal-fatigue life To supply a structural assessment and lifetime prediction for pressure vessels
S0013-7944(19)30688-5 https://doi.org/10.1016/j.engfracmech.2019.106710 EFM 106710
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
3 June 2019 28 August 2019 4 October 2019
Please cite this article as: Yan, Z., Tang, M., Chen, G., Wang, T., Li, X., Zhuang, Z., Development of analogy method for thermal-fatigue crack propagation in pressurized cylinder by using permeation diffusion-fracture model, Engineering Fracture Mechanics (2019), doi: https://doi.org/10.1016/j.engfracmech.2019.106710
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© 2019 Published by Elsevier Ltd.
Development of analogy method for thermal-fatigue crack propagation in pressurized cylinder by using permeation diffusionfracture model Ziming Yan1, Minjin Tang2, Gang Chen2, Tao Wang1, Xiang Li1, Zhuo Zhuang1 1Applied Mechanics Lab., Dept. of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084 China 2
Shanghai Electric Power Generation Equipment Co., Ltd. Turbine Plant No. 333, Jiangchuan Road, Minhang District, Shanghai 200240 China
Abstract Natural crack propagation analysis in thermal-fatigue fracture problem is a challenging problem. Reliable predictions for the crack path and residual live of structure provide essential information for the structural integrity assessment and repairment. Various conventional predictive fatigue models have been proposed to evaluate the remaining life, but most of them can’t build an explicit relation between damage degree and fatigue life. In recent years, the advanced techniques on crack propagation analyses were proposed, which were used to simulate the fatigue crack growth, but no coupling thermal-fracture function existed in the commercial finite element codes. Fortunately, the extended finite element method (XFEM) in ABAQUS has the coupling permeation diffusion-fracture model. In this paper, an analogy method from coupling permeation diffusion-fracture to coupling thermal-fatigue is proposed to compute fatigue fracture, which is applied for predicting the fatigue life in a specific steam turbine by directly using ABAQUS. Typical fatigue experiments by compact specimens of cast iron are carried out for the fatigue parameters measurement. By using analogy method and Paris’ Law, an entire analysis process for the turbine units is built to estimate the thermalfatigue life. The simulation results predicting remaining life and analyzing fatigue crack growth for the pressurized cylinder are agreed well with the measured engineering data. Keywords: steam turbine, analogy model, thermal-fatigue life, permeation diffusionfracture, XFEM
Corresponding author. Tel.: +86- 10-62783014; Fax: +86- 10-62783014. Email: [email protected] (Z Zhuang)
Nomenclature
qw
fluid flow
a
current crack length
qT
heat flux
ac
critical crack length
t
current time
B, W
thickness and width of specimen
T f , Ts
the fluid and solid temperature at the boundary layer
c
volumetric specific heat
Biot’s coefficient
C, m
parameters of Paris’ Law
T
thermal expansion coefficient
E ,
elasticity modulus and poisson ratio
w
fluid’s proportion
a g W
shape parameter of stress intensity factor
c
crack surface
G
shear modulus
u , t
h
coefficient of heat transfer
T , q
displacement and force boundary temperature and heat flux boundary
kT
thermal conductivity
ij
Kronecker delta
kp
permeability coefficient
ij
strain
K
bulk modulus
v
volumetric strain
Ks , K f
grain’s and fluid’s bulk modulus threshold of stable fatigue fracture
lame constant
material’s density
K
range of stress intensity factor
w , s
liquid’s and solid’s density
K IC
fracture toughness
v
volumetric strain
ms
mass of solid grain
ij
stress
M
elastic constant defined in Biot’s theory
ijp , ijT
stress in heat diffusion and in flow diffusion
N
cycle times
void ratio
pf
pore pressure
IP
intermediate pressure
ps
pore pressure of solid at the boundary
SIF
stress intensity factor
P
force in fatigue experiment
K th
1. Introduction Combined cycle steam turbine is an important part of combined-cycle power plant (CCPP) that drives generator set through gas and steam turbines[1]. When the turboset begins to work, the gas turbine starts and generates residual high-temperature steam. Then the waste heat from gas turbine is routed to the nearby steam turbine, which generates extra power. In long-term work, it turns out that the ullage of turboset fatigue life depends on thick-walled steel structures and movable parts under frequent starts and stops, of which intermediate pressure (IP) inner cylinder (Fig. 1) is a typical component. For the inner cylinder, it is a typical thick-walled pressure vessel, which produces high thermal stress in flanges with a significant temperature gradient, especially those located in horizontal and vertical middle planes, respectively. Fatigue crack initiates in the internal surface at the flange corner and extends in a direction to the bolt hole, as shown in Fig. 2. When surface cracks develop into through cracks, the inner cylinder cannot be repaired. Fatigue crack always appears in these areas along the circle under long-term cycle thermal-mechanical load. In actual engineering, it is difficult to identify the degree of fatigue crack propagation in real-time observation. However, if a feasible method can be used to estimate the fatigue crack propagation of working turboset, it will be helpful to determine the overhaul downtime to avoid the permanent damage.
Fig. 1 IP inner cylinder
Fig. 2 Surface crack appeared on the flange
Previous work has made extensive research and developed advance techniques on crack propagation analysis, such as finite element method (FEM) based nodal force release technique[2, 3], FEM based local or global remeshing[4-9], extended finite element method (XFEM)[10, 11] and generalized finite element method (GFEM)[12, 13], finite element alternating method[14, 15], phase field method[16-18] and so on. However, there is no commercial FEM software yet that could compute the fatigue crack propagation directly under coupling thermal-mechanical load. Fortunately, the natural essence of XFEM in ABAQUS has the coupling permeation diffusion-fracture capability, which supports a calculation module to solve hydraulic fracturing problem based on Biot’s model[19]. In this paper, an analogy method from coupling permeation diffusion-fracture to coupling thermal-fatigue is proposed to compute fatigue fracture, which is applied for predicting the fatigue life in a specific steam turbine by directly using ABAQUS. The dual relation between heat diffusion equation and seepage diffusion equation is obtained. Therefore, the fatigue crack propagation under thermalmechanical load can be conveniently computed by using hydraulic fracturing calculation module. In order to estimate the remaining service life of turboset reasonably, an integral analysis process is built to predict fatigue life on the components of the steam turbine. Typical fatigue experiments by compact specimens of cast iron are also carried out for the fatigue parameters measurement. By using analogy method and Paris’ Law, an entire analysis process for the turbine units is built to estimate the thermal-fatigue life. Consequently, the simulation results predicting remaining life and analyzing fatigue crack growth for the pressurized cylinder are agreed well with the measured engineering data.
2. Analogy method to solve coupling thermal-mechanical problem For crack propagation under thermal-mechanical coupling load, its solution domain 0 in three-dimension space is a discontinuous displacement field, as shown in Fig. 3, where c is crack surface, u is displacement boundary, t is force boundary, T is temperature boundary and q is heat flux boundary.
Fig. 3 Solution domain in three-dimension space
Duflot[20] considered crack propagation calculated by XFEM based on thermoelastic medium and established new XFEM scheme including discontinuous temperature field. Pathak[21, 22] considered thermal fatigue crack problem based on a three-dimensional thermoelastic medium and analyzed different types of cracks such as plane crack, non-plane crack under the thermal shock load, adiabatic load and isothermal load. Habib[23] changed previous sequential coupling methods and established XFEM scheme based on fully coupled thermal-mechanical model to solve the quasi-static problem. Although these works have considered fracture under coupling thermal-mechanical load, some unsolved problems are left for the fatigue fracture of actual structure under cycling thermal-mechanical load and complex operating conditions. Nowadays, the reliable analyses for crack initiation and propagation are given by AREVA during the Pressurized Thermal Shock (PTS) event for thick-walled pressure vessel under thermal-mechanical coupling load [24] and by Alstom for crack propagation of turboset under high pressure and temperature [25]. Both of them fully decoupled heat diffusion field and discontinuous displacement field through computing temperature field without fracture. Neither of them considers the coupling heat diffusion and fracture. Thus, these computing methods can’t solve the situation that newly formed crack surface becomes a new heat boundary, which influences on the temperature field. As mentioned in Section 1, the commercial code ABAQUS supports calculation to solve hydraulic-fracture problem by using XFEM, which includes a flow diffusion equation in porous medium. In contrast, we need to solve a thermal-fracture problem with a heat diffusion equation in thermal medium, as shown in Fig. 4. Based on relation of mathematical equations between two physical processes, the thermal-fracture problem can be solved through hydraulic-fracture solver. To establish the integrated
dual relation, both governing equations and boundary conditions must be considered.
Fig.4. heat diffusion in thermal medium and flow diffusion in porous medium
For thermo-elastic medium, the constitutive equation is
ij v ij 2G ij 3T K T T0 ij
(1)
where is lame constant, G is shear modulus, T is thermal expansion coefficient and K is bulk modulus. T and T0 are current temperature and initial temperature. For permeation diffusion problem, the constitutive theory of poroelasticity [26-28] is used to describe permeation diffusion where pore pressure p f is used to describe fluid flow in porous medium. This theory simplifies primitive complex coupled fluidmechanical problem and its constitutive equation for the porous elastic medium is ij = v ij 2G ij p f ij
(2)
where is the material constant defined in Biot’s theory and p f is pore pressure which is a field variable as the fluid influence. In comparation between Eq. (1) and (2), if temperature variation T T0 is equal to pore pressure p f numerically and corresponding relations of the other parameters are found, the thermal-fracture problem can be analogically solved from hydraulicfracture problem. The deformation and stress fields from hydraulic-fracture can be transformed to the thermal-fracture problem. For the numerical simulation of mechanical problem with diffusion field, the problem can be divided into static and transient analytical models. The former does not consider the diffusion process with time but the latter does. Firstly, the dual relation of mathematical equations is established between two physical processes based on static analytical models. T ij v ij 2G ij 3T K T T0 ij ijp = v ij 2G ij p f ij
(3)
Generally, pore pressure p f is non-positive, so the constitutive equation should be expressed as follows if p f is equal to temperature variation T T0 numerically
ijp = v ij 2G ij p f ij Consequently,
to
make
ijT ijp
and
(2)
p f T T0
numerically,
the
corresponding relation is
3T K
E T 1 2v
(5)
Besides, grain’s bulk modulus Ks is always used in numerical simulation instead of Biot’s constant in static analysis. The relation between Ks and is
1
K Ks
(6)
The corresponding relation between Ks and thermal parameter is given by Ks
K K 1 1 3 T K
(7)
Thus, the dual relation in static analysis is given in Table 1. Table 1 Corresponding relation between two physical process in static analysis
Permeation-diffusion problem
Thermal-mechanical problem
pf
T T0
3T K
1 K s
K
For a transient problem, the variation of diffusion field with time must be considered. More parameters are required whether in a thermal-mechanical problem or a permeation-diffusion problem. Table 2 shows the required parameters in static and transient analyses, where,
is density, c is volumetric specific heat, qT is heat
flux, kT is conductivity, k p is permeability, K f is fluid’s bulk modulus, is
void ratio,
w is fluid’s proportion and qw is fluid flow.
Table 2 Required parameters of two physical processes in numerical simulation
Permeation-diffusion problem
Thermal-mechanical problem
Parameters in static analysis
E , , T , kT
E,v,k p ,K s
Parameters in transient analysis
, c, qT
K f , , w , qw
In thermal analysis, Fourier’s Law is used to define the heat flux qT
qT kT T
(8)
In seepage analysis, seepage equation is used to define the fluid flow qw
qw
kp
w
p
(9)
If pore pressure p f is non-positive considered and qT qw , the corresponding relation between kT , k p and w is given below
kT
kp
(10)
w
To identify other parameters’ corresponding relation, we need to make the contrast of two diffusion equations. For the heat diffusion, the diffusion equation is T kT 2 T 0 t c
(11)
For the seepage diffusion, the diffusion equation in Biot’s theory is p f t
k p M 2 p f M
v t
(12)
where M is an elastic constant defined in Biot’s theory. Apparently, this constant situated in both sides of Eq. (12) and can’t be directly found the relation between M and the other heat parameters. This diffusion equation should be modified until the relation can be figured out. For porous medium, suppose mass conservation for inflowed fluid, we can get
w wqw t
(13)
Considering the seepage diffusion described in Eq. (9), the form of Eq. (13) can be changed by
wk p2 p f
w d w t dt
(14)
According to the definition of fluid’s bulk modulus K f , the density of fluid w is a function of time, pore pressure and fluid’s bulk modulus. Thus, the change rate of
w is expressed as w w p f p f w K f t t p f t
(15)
For solid grain, supposing the mass conservation for this portion, we can get
ms 1 s
(16)
D ms D 1 s d 0 Dt D t
(17)
1 d s v d 1 dt t s dt
(18)
where ms is mass of solid grain and is domain of porous model. For a fatigue problem refer to cast iron under long-term cycling loading, the change rate of solid grain’s density is negligible. That is
d s 0 dt
(19)
d 1 v dt t
(20)
So that, Eq. (10) can be written as
Considering Eq. (15) and Eq. (20), then Eq. (14) may be expressed as follows,
p f t
kpK f
2 p f 1
v t
(21)
Consequently, the modified seepage diffusion equation is obtained, which can be made comparison with Eq. (11). Obviously, let 1 , the corresponding relation of
the other heat parameters is
wK f 1 c
(22)
With the relation given above, the dual relation in transient analysis is given in Table 3. Table 3 Corresponding relation between two physical process in transient analysis
Permeation-diffusion problem
Thermal-mechanical problem
qw
qT
3T K
1 K s
K
kp
w wK f
kT 1 c
Based on the corresponding relation shown in Table.3, the primitive coupled thermal-mechanical problem could be solved by using hydraulic fracturing solver in commercial FEM codes. In addition, the temperature of inner cylinder is 20 C at the initial state and the highest temperature of whole cylinder is up to 600 C in normal working state. More narrowly, the material’s parameters in thermal-mechanical problems dependent on the temperature should be considered, especially for the materials of turbine which have significant change of temperature during each cycle. In analogy method, the influence of temperature on material is considered. These parameters should be also dependent on the pore pressure through the corresponding relation with material thermo-physical parameters. Consequently, all analysis of computing result in the paper has considered the effect of temperature. To ensure the integrity of whole analogy, the corresponding relation between boundary conditions is required to be confirmed. The most common thermal boundary conditions [29] are temperature boundary, heat flux boundary and boundary layer condition (Fig. 5) for turbine units.
Fig. 5 Boundary layer condition
Since both of two physical models are diffusion problem, the form of boundary conditions is also of analogy. For temperature boundary condition, its effect is the same as the pore pressure boundary condition, T p f . For heat flux boundary condition, its effect is the same as a seepage boundary condition, qT qw . Here, the value of seepage boundary becomes negative because the pore pressure is non-positive by default. For boundary layer condition, it is expressed as qT h T f Ts , where h is coefficient of heat transfer, T f is the fluid temperature at the boundary layer and Ts is the solid temperature at the boundary layer. So we can define a new formation of fluid flow as qw h p f ps , where p f is pore pressure at the boundary and ps is the pore pressure of solid at the boundary. In ABAQUS, the subroutine DFLOW can be used to build the new formation of fluid flow. The analogy method redefines the value of equation parameters, so the physical meaning of the analogous model is no longer a permeation-diffusion problem but a coupled thermal-mechanical problem. Using the proposed corresponding relations on governing equations and boundary conditions, the equivalence is achieved between coupled thermal-mechanical fracture and permeation-diffusion fracture problem, which makes subsequent computation of thermal-fracture propagation. The program is easily implemented into the commercial FEM codes. Based on the simple plane model, as shown in Fig. 6, the diffusion fields are computed as shown in Fig. 7 in order to verify the above relations. The material is cast iron, CrMoV[30], and parameters changing with temperature. The bottom of model is
fixed. The heat flux into the right side is qT 10mJ/ mm 2 s
and the initial
temperature is 0 C . The loading time is 300s. The size of element is 2mm 2mm and the type of element is CPE4RT (Fig. 7(a)), which is the reduced 4-node plane strain element with the freedom of temperature.
Fig. 6 Plane model
(a)
(b)
Fig. 7 (a) show the temperature at 300s and (b) show the pore pressure at 300s. Both results are computed under dual relations
Then the diffusion field is computed through the use of transferred material parameters based on relationship shown in Table 3. The fluid flow into the right side is
qw 10mD/ mm 2 s and the initial pore pressure is 0.0MPa . The load time is 300s. The size of element is 2mm 2mm and the type of element is CPE4RP (Fig. 7(b)), which is the reduced 4-node plane strain element with the freedom of pore pressure. Extracting the variation of temperature and pore pressure at point A in Fig. 6, we can see the value of diffusion field at the value point is identical (Fig. 8). The results prove the correctness of corresponding relations and the fitness of permeation-diffusion solver, which is applied to solve the coupled thermal-mechanical problem through transferred parameters. It illustrates that we have realized the accurate simulation for a coupled thermal-mechanical problem through another simulating module, which can be utilized to calculate crack propagation.
Fig. 8 The contrast between temperature and pore pressure at point A
3. The analysis process of fatigue crack propagation based on submodels When simulating the fatigue crack propagation for actual engineering structures, like IP inner cylinder, the computational efficiency is supposed to be taken into account. Considering that the macrocracks are mainly formed in the surface of volute flanges, the computing time will be rather long if using the full-size model to compute fatigue propagation. Hence, adopting submodels is an efficient way to reduce computational expense. For IP inner cylinder, the submodels should contain the parts where main fatigue cracks always appear. The submodels are partitioned from the cylinder model shown in Fig. 15. Here, the classic formula of Paris’ Law is applied to describe the relationship between crack growth rate and stress intensity factor da m C K (23) dN where
da is a crack growth rate, K is the stress intensity factor (SIF) range, C dN
and m are material constants defined by experiments. The fatigue total-life-cycle[31-33] is generally divided into three stages: fatigue crack initiation, crack steady propagation and rapid crack propagation. At the stage of fatigue crack initiation, the complex reasons and initial microcracks are not always the leading cracks which develop perforated cracks. At the stage of rapid crack propagation, the time of whole process is short. When fatigue macrocracks extend at the stage, the time can be neglected as the structure would be almost failure or destroyed rapidly. In this paper, the simulation aims to compute the time of fatigue crack growth at the steady
stage. As simulating fatigue crack propagation by using XFEM, the criterion of crack growth should be established. Furthermore, it is difficult to guarantee the fatigue crack stay in steady propagation stage, so the criterion contains two modes: In steady propagation, Paris’ Law is applied to describe the fatigue crack growth length[34], and the direction of maximum principal stress decides the propagation direction and fracture initiation time. In rapid crack propagation, the maximum principal stress criterion[35] could be utilized to decide the time of crack initiation and propagation direction. Besides, the stress intensity factor range K is used to judge which mode of the criterion is applied:
When
ΔK th K ΔK IC , fatigue crack is in steady
propagation; when K ΔK th , the crack arrest; when K K IC , the crack becomes rapid propagation stage. The phantom nodes technology[36-38] is applied in XFEM scheme of ABAQUS, in which it is negative for the parts of displacement field at the crack tip. Under this circumstance, the crack front must penetrate the entire element if the criterion is satisfied. Meanwhile, considering that the global model simulation with fatigue crack propagation requires very high cost of computation in each cycle, an average method is utilized to simplify the whole analytical process. In each cycle time, the crack increment a is a crack growth length through a whole element. Suppose that the stress intensity factor range K keeps constant, the average growth rate in this cycle time is
a a m C K . So, the actual cycle times is equal to N m . In N C K
every simulating cycle, the cycle times N equals to the minimum of all N computed on those elements along the crack front. Besides, the numbers of elements to be penetrated would also be recalculated according to the minimum N . If the cycle n
time is equal to n in simulation, the total crack growth length is equal to a ai i 1
and corresponding stress intensity factor range is equal to Ki . The total actual cycle time is equal to n
n
i 1
i 1
N N i
ai
C K i
m
(24)
The process implementation to realize the average method is: In every simulating cycle, the fatigue crack propagation is limited to elements along the front of crack (Fig. 9). If the crack extends through one element under the criterion of crack steady
propagation, it cannot extend again unless the stress intensity factor satisfies K K IC . In this method, the stress intensity factor K will be calculated in each
simulating cycle[20-22]. At the same time, the corresponding crack increment a can be computed. When the total crack growth length up to the required value, the actual cycle times can be computed by Eq. (24).
Fig. 9 In every cycle, we assume that the fatigue crack propagation only appears in the elements along the front of crack
Combining the submodel technology and above method to compute actual cycle times, the fatigue crack analysis process is obtained, as shown in Fig. 10.
Fig. 10 The left flow diagram of fatigue crack analysis process based on submodel. The right flow diagram of fatigue crack analysis process based on steam turbine cylinder
The whole analytical process can be divided into two part. The first is analogy of two physical models and the second is to obtain the value of freedoms of global model to drive submodels. Consequently, the entire fatigue crack analysis process includes the following four steps: a) transforming the parameters through analogy method; b) computing the full model without crack through one cycle to get the value of freedoms to drive submodel, including three displacement freedoms and temperature freedom
(the temperature freedom is replaced with pore pressure freedom); c) computing the submodel with crack through several cycle times; d) evaluating the residual service life based on Paris’ Law in the end.
4. Fatigue experiment by using CT specimen For the fatigue simulation, it is necessary to describe fatigue crack propagation based on proper fatigue model. The purpose of fatigue experiment is to gain material parameters that influence fatigue crack growth rate[39, 40]. In this study, Paris’ Law [41] is applied to establish the relationship between fatigue crack growth and stress intensity factor (SIF) in the crack growth stage. The compact tension (CT) specimen made by cast iron, CrMoV[30, 42], is applied to fatigue experiment, as shown in Fig. 11. The effective width of CT specimen is 66mm and thickness is 12.7mm. The cycle loading is sine wave load which makes the specimen stay in tension, of which load frequency is 15Hz. During the entire process of loading, the minimum tensile force is 2.7kN and the maximum is 27kN in each cycle with a constant ratio R 0.1 of them.
Fig. 11 Compact tension specimen in fatigue experiment
With the cycle loading, the crack moves nearly in a straight line and the crack growth rate increases during it propagates. According to the Eq. (25), which shows the computing SIF formula towards CT specimen in fatigue experiment, SIF becomes more significant with crack length increase. When cycle times up to about 45,000, the mode of crack propagation becomes unstable and the specimen is destroyed rapidly, as shown in Fig. 12.
K
P a g 1/2 BW W
(25)
where P is the force that tensile testing machine exerts, B is the thickness of
a specimen, W is the width of specimen and g is the shape parameter of stress W intensity factor[43].
(a)
(b)
Fig.12. The fatigue experiment process. (a) shows the crack length a 1.3545mm when cycle times
N 5， 600 ; (b) shows the crack length a 6.9106mm when cycle times N 16， 500 ; And the crack propagation entered the phase of instability when cycle times N was approximately equal to 45,000
To take the logarithm of both sides of Eq. (23), the equation is written as da ln ln C m ln K which is applied to plot the curve shown in Fig. 13 and dN confirm the Paris’ Law’s parameters with a linear square fit. The slope of fitted curve is m and the intercept of fitted curve is ln C .
Fig. 13 The relation between fatigue crack growth rate
da and stress intensity factor range K of dN
cast iron, CrMoV
This experiment is carried out to determine two material parameters of Paris’ Law
C=2.28 1010 , m 2.3432 , respectively. Generally, K th corresponds to the K when
da is equal to 107 mm/cycle . According to the tested formula of Paris’ Law, dN
K th is equal to 0.7035MPa m . KIC is equal to material’s fracture toughness, K IC , as the region of crack tip under compression during cycle loading.
5. The fatigue analysis on steam turbine cylinder The finite element model of IP inner cylinder is shown in Fig. 14. For cast iron used in the cylinder, the fracture toughness KIC 134.8MPa m . The fatigue crack growth rate is represented by Paris’ Law,
da 2.3432 . 2.28 1010 K dN
According to the operating feature of hot start, the single working state is divided into three typical stages: the time of heating stage is 7200s by transient analysis; the steady stage is computed by steady analysis; and time of cooling stage is 28,800s by transient analysis. In the normal working state, the steam enters into the intake chamber through the volute and turns the turbine blades, as shown in Fig. 14. The inner walls of volute and intake chamber endure high-temperature steam with high pressure, while the outer walls of cylinder and other surface endure steam with relatively low-temperature steam with relative low pressure.
Fig. 14 The left is IP inner cylinder model including upper and lower cylinders which are connected by bolted joints. The right shows the flow of inlet and outlet steam in inner cylinder. Both inside and outside surface endure different temperature and pressure
The partitioned submodels are given in Fig. 15, and regions of flange included. The preexisting crack in submodels is a quarter-elliptic surface crack. The semimajor axis of the preexisting crack is 15mm and the semi-minor axis of preexisting crack is 5mm. The degree of angle between the crack surface and horizontal plane is 45 degrees.
Fig. 15 (a) The figure shows the connection between global models and submodels: both submodels are partitioned through red cutting plane. As connecting planes between global model and submodel, the results calculated by global model, including displacement and temperature, should be applied on these red planes as submodels’ boundary conditions. The other loadings and boundary conditions on original surfaces should keep the same with global models. (b) The figure shows the temperature field of inner cylinder (heating stage, 7200s) under the above load and boundary conditions. (c) The figure shows the connected submodels and the location of preexisting cracks. (d) The figure shows the temperature field of inner cylinder (heating stage, 7200s) under submodel boundary conditions and original loadings and boundary conditions.
Through calculation under cycle loading, the results illustrate the variation of stress in submodel and the behavior of fatigue crack growth, as shown in Fig. 16. At the heating stage, the pressure and temperature increase as the high-temperature steam enters into the cylinder. The corner is still under pressure and the crack keeps in closure. With the increase of temperature gradient, the pressure becomes higher. When entering the cooling stage, the cooling process leads to a large temperature gradient and the pressure at the corner is reduced rapidly. With the process of cooling, the stress states at the corner are changed from compressive stress to tensile stress. As a result, the crack opens gradually and crack propagation appears at this stage.
Fig. 16 The field variable named UVARM4 represents the maximum principal stress. (a) (1800s, heating stage) shows the crack kept in closure; (b) (6500s, heating stage) shows the crack kept in closure; (c) (4000s, cooling stage) shows the crack opened; and (d) (8600s, cooling stage) shows the crack further opened.
Through computation circularly, the surface crack at the middle plane is extended from 5mm to 65mm (Fig.17). The crack path is near along the direction of preexisting crack. With the fatigue crack propagating, the growth rate increases gradually. Fig. 18 illustrates that the K at crack tip on the surface of middle plane is also increased as the fatigue crack extended. Fig. 19 illustrates the relationship between the crack length and actual cycle times through Paris’ Law. When the crack length is up to 65mm, the actual cycle is equal to 3,257 times. In the normal working state, the times of startup and shutdown is about 200. Consequently, the corresponding service is 16.3 years. This value is quite close to the estimating years from overall detection, about 20 years, but it is more conservative.
Fig. 17 The crack propagation on the surface of the middle plane
Fig. 18 The relation between K and cycle times(simulation)
Fig. 19 The relation between the length of crack growth and actual cycle times
Actually, the fatigue crack in flange does not extend to the range what we have computed. The turboset must be shut down and maintained during furnace operation and periodic repair before the depth of surface crack is up to several millimeters. However, an accurate estimation on the states of crack with experience is a hard work. In the practical use, the method can be utilized to support a proper downtime before the crack extends to a critical length or depth a c . Based on this method, we do not need to shutdown the turbine only by experience. We can help engineers to estimate whether the fatigue cracks extend up to the critical depth or length, support reliable downtime to maintain the turboset, and remove the formed crack to prolong the use-life to the designed-life as far as possible.
6. Conclusion In this paper, the dual analogous relationship of equation parameters has been established between the thermal-diffusion and permeation-diffusion problem. A thermal-fracture problem for thermal medium can be transformed into a hydraulic
fracture problem for the porous medium by the method. The method supports a smart way to solve the fracture problem under coupled thermal-mechanical loads, which has comparatively strong practicality in engineering. In addition, the general process of fatigue crack analysis on turbine units is built under high temperature and pressure cycle loadings. Under above process, the relation between use-life and fatigue crack propagation is established explicitly. It is convenient to obtain the cycle times corresponding to a certain crack length specified. Relying on the actual model of steam turbine cylinder, this work analyzed the characteristic of fatigue crack growth under coupled thermal-mechanical cycle loadings and summarized the relationship between fatigue crack growth rate and cycle times in hot start. The calculating result is close to the one by overall detection. Meanwhile, some problems remain to be solved on the service life prediction. The element size has an effect on estimation of cycle times through an average method. A proper mesh will result in a more precise estimation. Besides, this work studies the fatigue crack growth when the structure has preexisting macrocracks. In fact, the fatigue crack initiation period is a main stage of the total fatigue life. To build a numerical simulation method to predict the total fatigue life is meaningful for structure design and optimization.
7. Acknowledgement This work was supported by the National Science Foundation of China. (Grant No.11532008)
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Highlights
Dual relationship between heat diffusion equation and seepage diffusion equation An analogy method to compute fatigue fracture in steam turbine by using ABAQUS An analysis process for turboset to estimate the remaining thermal-fatigue life To supply a structural assessment and lifetime prediction for pressure vessels