丁 軍，黃 霞，李文中，王國超重慶理工大學(xué)機械工程學(xué)院，重慶 400054

1．Introduction

The application of thermal barrier coatings(TBCs)continue to increase as the sustainable growth in the requirement for greater engine efficiency in aircraft and gas-turbine industries has steadily increased．However，the mechanical performance of TGO significantly influences the durability and reliability in TBCs．The coating system shielding the underlying metal substrate against high temperature environment consists typically of four layers:the top ceramic coating，the thermally grown oxide(TGO)，commonly alumina，the bond coat(BC)between the substrate and ceramic coating，as well as the metal substrate［1-3］．When the TBC system works in high temperature environment and experiences thermal cycling，TGO forms at the dwell time in a thermal cycle as a result of a chemical interaction of the outward diffusion of metal cations and inward diffusion of oxygen anions in air［4］．And the formation of TGO has everlastingly been identified as one of the most important factors affecting the durability of TBC system since it performs as a protective oxide scale in oxidizing and corrosive environments at the elevated temperature．Consequently，the in-depth investigation into the formation using FEM spreads across the present research on TBC system．

The review for the literatures shows that considerable researches have so far been conducted on the numerical simulation for TGO formation at the high temperature． However， most of the simulation schemes behave in a limitation of TGO thickness．The most typical research was conducted by Prof．Karlsson［5］．They developed a tractable numerical scheme to make up for the deficiency by imposing a TGO thickening strain in order to consider the TGO thickness change in oxidation，and come up with a good result in comparison with experimental observation．Even so，it cannot simulate TGOformation for a large TGO thickness．

In this work，our purpose is to develop a new numerical scheme based on UMAT(a user subroutine)in ABAQUSsoftware to simulate TGOformation at the high temperatures which can be considered as an effective means to break up the limitation of TGO thickness．In order to verify the accuracy and validity of the numerical simulation scheme，a comparison between the results from this work and from the authoritative work(by Karlsson)for small TGO thickness has been made indicating that a good agreement was arrived as a proof of the validity to the work．

2．FEM model

Since the purpose of the work is to compare that of Karlsson，it needs to firstly clarify the details such as FEM model and simulation scheme used in Karlsson’s work．For simplicity，the description of the FEM model can be as follows．Fig．1 shows the geometrical model，F(xiàn)EM model and TGO formation simulation scheme．were constrained．The emphasis was placed on the simulation of TGO growth at the high temperature．They considered TGO formation caused by two strains，lateral growth strain and thickening strain．The thermal expansion was implemented into the FEM calculation with the aid of the subroutine UEXPAN in ABAQUS．With regard to modeling for TGO thickening strain，an initial TGOlayer with the thickness of 1μm is in advance supposed to simulate the pre-oxidation of the metal substrate．This layer was then meshed into six sub-layers of elements．The TGO thickening strain is implemented by making the layer which is the closest to the bond layer to have an expansion in the TGO thickness direction．

3．Numerical simulation procedure

Fig．1 The FEM model

The groove shape was employed to model the pre-existence of the fault or imperfection on the interface，and three layers，including TGO layer，bond coat layer and Fecralloy substrate layer are arranged in terms of top to down in constructing FEM model．Considering the geometric symmetry，the symmetric boundary condition was engaged at the central axis of the specimen configuration．The periodic boundary condition was taken to model the random existence of the imperfection in TBC system．In order to keep the same displacement in x direction for the nodes located at the right side，the command equation in ABAQUS software was employed to impose x displacement in horizontal axis，and the y displacement sat the nodes at the bottom side of the metal substrate

3．1．The merits of UMAT

UMAT is the abbreviation for User Materials in ABAQUSsoftware，which means user of software can define custom material behavior by virtue of the command．It provides for the FEM user with a very strong interface to develop user routine to realize its special functionality in terms of FORTRAN，C or C++source code［6］．In this paper，we employed the same FEM model as that in Karlsson’s work，including geometrical configuration of the model，material property for TGO，BCand metal substrate，as well as the similar boundary conditions constrained at the symmetry and the periodic boundary conditions．

The biggest difference in comparison with［5］is the numerical simulation scheme modeling TGO formation．In previous work，the realization for TGO formation along lateral and normal direction was accomplished with the aid of user subroutine uexpan in ABAQUS software，which is much simpler than UMAT in programming to simulate TGO formation at high temperature，and it can only realize rather single function．Fig．2 shows interface comparison between UEXPAN and UMAT in ABAQUS software．It is known that coding UMAT looks much more difficult than UEXPAN since it needs to define some fundamental parameters such as Jocobi matrix using UMAT while define only thermal expansion using UEXPAN．

Fig．2 The comparison between interfaces of UMAT and UEXPAN

3．2．The simulation scheme

The main idea developing the simulation scheme is to realize the formation of TGO;that is，TGO is created at high temperature changing from metal substrate material into another material property，TGO，in essence;the material property transferred in simulation．Material properties for TGO，BC and substrate are taken perfectly plastic material behavior．The whole FEM model can be divided into initial TGO，TGO thickening，and substrate layer in terms of thickness of TGO．TGO thickening layer，which is in essence different from initial TGO，represents that alumina which are produced from Fecralloy substrate through chemical interaction，while the initial TGO has already existed prior to the onset of thermal oxidation．TGOthickening layer was meshed into twenty four sub-layers with each layer representing a thickness of TGO formed in one respective thermal cycle．Prior to the analysis，the TGO thickening layer was assigned to have substrate property．The reason for meshing 24 sub-layers，not six sub-layers in Karlsson work are that TGO formation is supposed only to happen in the layer most next to the metal substrate，while in this work it is supposed that TGO formation occurs in all of TGO layers，realizing by material property transformation．

In programming with UMAT，a solution dependent variable(SDV)should be firstly defined in ABAQUSinput file．The statement for SDV invokes the parameter state(1)in UMAT together with user subroutine，usdfld，in ABAQUS．The SDV(solution dependent variable)in ABAQUS appearing in the subroutine code was used to control over the material property change from substrate to TGO．Namely，in the first thermal cycle，the uppermost sub-layer of TGO thickening layer changes its material property to TGO by controlling the value of SDV to indicate the formation of new alumina．Afterwards，this layer keeps having TGO property．The same procedure was repeated for the next cycle．So TGO formation has been modeled after twenty-four thermal cycles．

4．Results and discussions

Employed the same FEM model except for the numerical simulation scheme for TGO formation，the results from［5］and this work is compared in Fig．3 for investigating the magnitude of total displacement and x displacement，respectively．It can be seen from Fig．3(a)and(b)that the values for total displacement of TBCs are almost identical．The magnitude of total displacement for this work looks a little smaller than Ref．［5］，and the peripheral area for both cases develops the maximum displacement value，while for this work，the area next to the bottom of the TGO also happens bigger displacement．For the displacement in x direction，the contour map for two cases looks much closer and the values for them are also closer，especially for the maximum displacement(red area in contour map)．It shows the accuracy and validity for the numerical simulation scheme in this work．

Fig．3 The comparison of displacement of TBCs between the results from［5］and from this work

5．Conclusions

A new numerical simulation scheme has been programmed to simulate TGO formation at the elevated temperature based on UMAT in ABAQUS software．Compared to the other simulation scheme，the procedure can model the rather real physical process of TGO formation at high temperature which can break up the limitation of TGOthickness，providing a good way to predict the mechanical performance of TBCs．

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