Finite Element Analysis of Influence of Non-homogenous Temperature Field on Designed Lifetime of Spatial Structural Elements under Creep Conditions

Structural elements of responsible objects function often under long-term static or cyclic force loading. The process of creep or fatigue, accompanied by the gradual accumulation of scattered damage, the formation and growth of macroscopic defects (fracture zones) are occurs under such a loading conditions. This problem, similarly as well as other aspects of reliability analysis [1, 2, 5], is very important for a reliable determination of long-term strength and lifetime.

A description of above mentioned processes, which took the name “continual fracture”, may be fulfilled efficiently using phenomenological scalar damage parameter, proposed in the works of V. Bolotin, L. Kachanov and Yu. Rabotnov [7, 13]. This approach is developed and implemented for different loading conditions in the publications of Ukrainian scientists M. Bobyr, V. Golub, G. L’vov, Yu. Shevchenko [5, 6, 9, 14] and foreign ones (Chen G., Hayhurst D., Lemaitre J., Murakami S., Otevrel I., etc.), particularly in [10–12, 15, 16]. It is shown that the damage accumulation process should be taking into account for correct final definition of part’s operating time. However, finite element solution of this problem requires creation of effective finite element base and special algorithms for regarding of temperature deformation and damage accumulation process.

The purposes of this paper are:

To highlight the effective finite elements for decision of two-dimensional and spatial stress-strained state problem and the developed technique for modeling of continual fracture process under creep loading condition regarding non-homogenous temperature field.

A problem of stress-strain state parameters determination under linear and nonlinear deformation process performed by the algorithm based on the use of the implicit integration over the time scheme with help of Newton-Kantorovich iterative procedure:

{ΔUl}nm={ΔUl}n−1m+β[Kll]−1({Ql}m−{Rl}nm),$$\begin{array}{} \displaystyle \{ \Delta {\mkern 1mu} {U_l}\} _n^m = \{ \Delta {\mkern 1mu} {U_l}\} _{n - 1}^m + \beta {\left[ {{K_{ll}}} \right]^{ - 1}}\left( {{{\{ {Q_l}\} }^m} - \{ {R_l}\} _n^m} \right), \end{array}$$(8)

where 1 ≤ β < 2 is the relaxation parameter, {Q}^m is the vector of full loads in nodes in step m, [K_ll] is the FE stiffness matrix, and {Rl}nm$\begin{array}{} \displaystyle \left\{ {{R_l}} \right\}_n^m \end{array}$ is the vector of nodes reactions in the iteration n of step m.

For isotropic material, temperature components of strain tensor components are obtained by Eq. 3:

εijT=αTΔTgij,$$\begin{array}{} \displaystyle \varepsilon _{ij}^T = {\alpha _T}\Delta T{g_{ij}}, \end{array}$$(9)

where α_T = α_T(z^k′, T) is the linear expansion coefficient of the material, ΔT = T − T₀ is an increase of temperature in a covered point of the body relative to the initial state T = T₀.

During step-by-step representation of deformation process expansion coefficient of stresses increments in the step m are determined by values of increments of total Δε_ij, and temperature (Δε_ij)^T strains:

(Δσij)m=Cijkl({Δεkl}m−{(Δεkl)T}m).$$\begin{array}{} \displaystyle [{\left( {\Delta {\sigma _{ij}}} \right)_m} = {C_{ijkl}}\left( {{{\left\{ {\Delta {\mkern 1mu} {\varepsilon _{kl}}} \right\}}_m} - {{\left\{ {{{\left( {\Delta {\mkern 1mu} {\varepsilon _{kl}}} \right)}^T}} \right\}}_m}} \right). \end{array}$$(10)

Creep problem solution considering damage accumulation is being executed by means of step-by-step algorithm on the parameter of time. When starting each iteration n of a step m, stress values σ_ij are calculated considering creep deformation process by the formula:

(σij)nm=13δij(σij)nm+(sij)nm.$$\begin{array}{} \displaystyle \left( {{\sigma _{ij}}} \right)_n^m = \frac{1}{3}{\delta ^{ij}}\left( {{\sigma _{ij}}} \right)_n^m + \left( {{s^{ij}}} \right)_n^m. \end{array}$$(11)

Components of stress tensor σ_ij are defined in compliance with the Hook’s law considering an increment of total and temperature strains:

(σij)nm=(σij)n−1m+(Δσij)nm,$$\begin{array}{} \displaystyle \left( {{\sigma _{ij}}} \right)_n^m = \left( {{\sigma _{ij}}} \right)_{n - 1}^m + \left( {\Delta {\mkern 1mu} {\sigma _{ij}}} \right)_n^m, \end{array}$$(12)

while the components of stress deviator (sij)nm$\begin{array}{} \displaystyle \left( {{s^{ij}}} \right)_n^m \end{array}$ relate to an increment in creep deformation Δεijc$\begin{array}{} \displaystyle \Delta {\mkern 1mu} \varepsilon _{ij}^c \end{array}$:

(Δεijc)nm=(ξijc)nmΔtm,(sij)nm=(sij)nm−G1(Δεijc)nm,$$\begin{array}{} \displaystyle \left( {\Delta {\mkern 1mu} \varepsilon _{ij}^c} \right)_n^m = \left( {\xi _{ij}^c} \right)_n^m{\mkern 1mu} \Delta {t_m},\qquad \left( {{s^{ij}}} \right)_n^m = \left( {{s^{ij}}} \right)_n^m - {G_1}\left( {\Delta \varepsilon _{ij}^c} \right)_n^m, \end{array}$$(13)

where (ξijc)nm=32[ξic]mn(sij)nm(σi)mn$\begin{array}{} \displaystyle \left( {\xi _{ij}^c} \right)_n^m = \frac{3}{2}\left[ {\xi _i^c} \right]_m^n\frac{{\left( {{s_{ij}}} \right)_n^m}}{{\left( {{\sigma _i}} \right)_m^n}} \end{array}$ components of creep deformation rate tensor, G₁ = E/(1 − 2μ) are elastic constants, and Δt_m is the time interval value.

The DP values addition (Δω)_m and accumulated DP values ω_m in a time interval m calculated with next relation:

ωm=ωm−1+(Δω)m=ωm−1+(dωdt)mΔtm.$$\begin{array}{} \displaystyle {\omega _m} = {\omega _{m - 1}} + {\left( {\Delta {\mkern 1mu} \omega } \right)_m} = {\omega _{m - 1}} + {\left( {\frac{{d{\mkern 1mu} \omega }}{{d{\mkern 1mu} t}}} \right)_m}{\mkern 1mu} \Delta {\mkern 1mu} {t_m}. \end{array}$$(14)

The criterion of local loss of the material bearing capacity is ω(t*) > ω*, where ω* ≈ 1 is the critical DP value. Its fulfillment in some point of studied object K with coordinates (z^i′)_K = z^i′* = {z^1′*, z^2′*, z^3′*}, indicates the transition of the scattered damages accumulation process, which accounted integrally using DP, to occurrence of macroscopic defects – initial areas of continual fracture. These points in time determining the value of the designed lifetime of studied object.

The allowed approaches have been applied to solving of practical problems regarding consideration of influence of non-homogenous temperature field on designed lifetime value of responsible structure element – the blade of a gas turbine under creep. Analysis has been carried out separately for root and blade; this is because of specifics of semianalytic finite element model-building.

Based on results of elastic deformation modeling of blade, based on three-dimensional FEM, a dangerous cross section R₀ was chosen. This section is characterized with combination of average stress σ₀ and average temperature T₀, which leads to the most intensive creep process and damage accumulation. Listed values (σ₀ and T₀) are used further to describe the design scheme and the results of the problem solving.

Fig. 3

The description of creep and damage accumulation processes is carried out with the following equation:

dεcdt=Bσn(1−ω)r,dωdt=C(σ1−ω)m1(1−ω)q,$$\begin{array}{} \displaystyle \frac{{d{\varepsilon _c}}}{{d{\mkern 1mu} t}} = \frac{{B{\sigma ^n}}}{{{{(1 - \omega )}^r}}},\qquad \frac{{d{\mkern 1mu} \omega }}{{d{\mkern 1mu} t}} = C{\left( {\frac{\sigma }{{1 - \omega }}} \right)^m}{\mkern 1mu} \frac{1}{{{{(1 - \omega )}^q}}}, \end{array}$$(15)

where B = B(T ),C = C(T),m = m(T),n = n(T),r = r(T),q = q(σ,T) are the material constants, and T is the temperature.

The root of gas turbine blade (Fig. 4, left) is a prismatic body that carries heat and force loading. Interaction between blade an root has been modeled by nonuniformly distributed load q. Teeth of the root bear on corresponding slots in wheel rim, which are being deformed. Thus, boundary conditions in the form of elastic supports employed alongside of surfaces of connection between root’s teeth and wheel rim.

Fig. 4

Finite element model is obtained for one-half of the detail due to symmetry relative to vertical axis. Boundary conditions are obtained from symmetry conditions and constraint in radial (z¹) direction. Analysis of convergence of obtained results depending on number of unknowns in finite element model has been carried out using comparison of distribution of dimensionless stress intensity [10] for meshes with 1074, 3344, and 9596 unknowns. Inaccuracy of stresses determination under elastic deformation reaches around 2%. Thus, in the future analysis the mesh with 3344 unknowns (Fig. 4, right) was used further for creep process modeling.

In terms of analysis, evaluation of influence of non-homogenous temperature field on evolution of stress-strain state parameters and designed lifetime of root has been performed. Linear law is describing temperature change across the height of root’s cross section. Maximum and minimum values of temperature lie in ±0.5% range from temperature in section that located at a distance of 0.9345R₀ from rotation axis of wheel rim.

During analysis, the provision has been made that due to small size of the root throughout its height the influence of temperature strains on stress-strain state is insignificant. At the same time, dependence of constants in Eq. (15) on temperature leads to significant differences in the flow of creep process with consideration of non-homogenous distribution of temperature, which leads to change of designed lifetime.

For analysis of influence of temperature field, an examination of differences in distribution of dimensionless stress intensity (Fig. 5, left) and damage (Fig. 5, right) throughout the whole running time has to be performed. These values have been taken at the point, where maximum stress occurs with (T ≠ const) and without (T = const) consideration of non-homogenous temperature field.

Fig. 5

For such temperature distribution the value of designed lifetime is t*(T) = 0.91t*, where t* is a lifetime value obtained under constant temperature in whole root volume. Thus, regard of non-homogenous temperature field allows adjusting of designed lifetime of blade root on 10% approximately.

Stationary gas turbine blade is a spatial body of complicated shape. The blade is swirled at the vertical axis, has a variable at height cross-sectional area. It is influenced by centrifugal forces in a heterogeneous, both in height and in cross-section, temperature field (Fig. 6, left). Creep deformation process modeling is made for a blade fragment with size 0.94R₀ < R < 1.06R₀.

Fig. 6

Fragment is loaded with its own centrifugal load p. The simulation of the upper part of blade in section R = 1.06R₀ implemented with unevenly distributed load q = q(z^1′,z^2′) that meets stress values, been applied in this cross-section.

To account for the impact of blade swirling at the vertical axis has been applied a change of the original material density on a cross-section plane. For a central part of cross section (in the vicinity of the middle channel), for which swirling leads to increased stress, the output material density has been increased, while for peripheral parts of the section it was reduced. The averaged material density in whole cross section remained unchanged.

The distribution of dimensionless stress intensity along the height of blade fragment obtained based on this approach is identical with the one results obtained from the three-dimensional FEM solution under elastic deformation.

Temperature distribution in blade fragment and finite element model of gas turbine blade are shown at Fig. 6, right.

Solving of given problem using SFEM showed that consideration of non-homogenous distribution of temperature across cross section does not have a significant influence on stress-strain state and value of lifetime. Under non-homogenous distribution of temperature along the height of the blade, the maximum difference in stresses intensity values (around 10%) is observed at vicinity of point 2, where damage value is insignificant. Changing of stresses across the height of the blade is illustrated in Fig. 7, left. But obtained time-dependent damage parameter values at point 1 show (Fig. 7, right), that even such small reduction of stresses leads to adjustment of main designed lifetime of the blade up to 9% (t*(T) = 1.09t*).

Fig. 7

Developed in this paper methods for modeling of continual fracture process allows to determine of designed lifetime values for responsible structural elements that work under long-term static loading condition. Consideration of non-homogenous temperature distribution in studied object allows to adjust the value of designed lifetime in the range of 10% approximately.