Homogenisation approach in analysis of creep behaviour in multipass weld

Introduction

The long term monitoring of equipment working under high pressure at high temperatures shows that it is important to take into account influence of creep and damage in weldments. ¹ Quite often, welded constructions are found damaged before the predicted lifetime of components. One of the reasons for that is a mismatch in creep deformation properties between the weld and the parent metal in combination with non-favourable weld shapes, stress concentrations developed within the weldment. It should be noted that the design rules of weldments under pressure are based only on the long term fracture properties of weldable material at uniaxial tension. However, in reality, such constructions operate at multiaxial stress state conditions. In addition, mechanical properties in the zone of the weld material depend on the direction. These problems are especially important for modern turbine industry among others. The high temperature of steam and high pressure in supercritical steam turbines require application of steels with improved properties with long term durability. For implementation of these requirements, a family of new heat resistant steels with 9–12Cr is designed. The 9–12Cr steels are used in both boilers and in steam turbines for many components including pipes, headers, rotors and casings with a maximum operating temperature of 620°C. ² In general, these alloys have lower coefficients of thermal expansion and higher thermal conductivities than austenitic steels and should therefore be more resistant to thermal cycling. A typical weld in a component consists of parent material, heat affected zone and weld metal. The two parent materials joined by the weld may be made from the same material or different. The weld and parent material can have the same or different composition. However, even for welds joined by the weld metals with the same composition as the parent materials, the creep properties in parent, heat affected zone and weld materials will be different. ³ Thus, the weldments are highly complex heterogeneous structures. Moreover, in the case of multipass welds, the weld material is also inhomogeneous. It consists of overlapping weld beads that will create specific heat affected zones within the weld metal because of cooling and heating from the next pass. A single weld bead generally consists of a columnar solidification structure. However, in multipass weld, when the further bead is laid over the previous one, part of it will be recrystallised, and this will create coarse and fine grained structure. ⁴ All of these factors make it important to take anisotropy into account in modelling creep behaviour of the multipass weld. ⁵

Constitutive equations of anisotropic creep

The strain rate–stress relations for creep for anisotropic materials are based on the assumption of the existence of the creep potential. The creep potential hypothesis is widely used for continuum mechanics modelling of isotropic and anisotropic creep. ⁶ During the secondary creep stage, strain rate is defined by the scalar valued potential W(σ) and the flow rule (1) For simple determination of the creep potential on the basis of uniaxial creep tests, the equivalent stress σ_eq is introduced as intermediate scalar argument W[σ_eq(σ)]. The Norton–Bailey's power law is assumed here for the approximation of the strain rate–stress relations (2) where the material parameters K and n depend on the temperature.

The equivalent stress is invariant with respect to the coordinate system. The form of equivalent stress depends on the symmetry type of material. Materials of columnar, coarse and fine grained zones are assumed to be isotropic. In case of isotropic creep, equivalent stress is suggested as von Mises type (3) where S is the stress deviator (4) and . For the Norton–Bailey type potential the flow rule results in (5) To model the microstructure of the multipass weld as an equivalent continuum on the macroscopic scale, the representative volume element (RVE) is considered. The macroscopic creep properties of equivalent continuum are anisotropic in general case. The symmetry type of the equivalent media is determined by the geometric structure of multipass weld and the creep properties of the microstructure components. For modelling anisotropic creep behaviour, the equivalent stress is assumed in the general quadratic form (6) where ⁽⁴⁾ B is a symmetric positively definite fourth rank tensor. The number of independent tensor components depends on the symmetry class of the equivalent continuum. For most multipass weld geometric structure, RVE allows to consider equivalent continuum as an orthotropic solid. In this case, the tensor ⁽⁴⁾ B can be presented in the base of orthonormal vectors n₁, n₂, n₃, which are perpendicular to symmetry planes of the orthotropic solid (7) In this case, tensor ⁽⁴⁾ B includes nine non-zero independent components b_ijkl (including K in Norton–Bailey's law). With the equivalent stress (equation (6)), the secondary creep equation for the orthotropic solid can be written as follows (8) Creep law for all microstructure zones of the weld reflects the incompressibility of materials. Therefore, it is assumed that the creep deformation does not produce a change in volume of the equivalent continuum. The spherical part of the creep rate tensor is set to be zero (9) The volume constancy assumption reduces number of independent components of tensor B to 6. Identification of independent material constants in the equivalent stress and the creep potential for homogeneous solid is carried out experimentally. Only for secondary creep stage, six independent stress states should be implemented in order to identify six constants of equivalent stress. In addition, for identification of the parameter n in the power law (equation (8)), a pair of uniaxial creep tests for different constant stress values should be executed. The parameter identification problem is complicated by the high scatter of test results. For homogeneous materials, creep rates of specimens cut-out from the same rod differ >10. In this case, it is difficult to recognise whether the difference of creep rates for specimens’ cut from the same material in orthogonal directions is a result of the anisotropy or scatter. More clear creep anisotropy shows up in the heterogeneous materials. Thus, polymer composites reinforced by fibres show quite different creep rates in directions of fibres and orthogonal direction. Essential creep anisotropy appears in weld metal ⁷ for specimens cut in welding directions and transverse direction. For periodic heterogeneous structures, the theoretical averaging method allows to determine effective creep properties on base of known creep properties of components. Theoretical determination of effective properties of multiphase periodical materials has many advantages with comparison to experimental investigation. The analytical or numerical homogenisation technique for periodical media enables to find effective creep properties for many variants of multiphase structures. Here, the paper FEM is used for the determination of averaged parameters of the creep law of the multipass weld.

Numerical analysis of anisotropic creep in multipass welds

To model the anisotropic creep of multipass welding, the RVE is created as prismatic body with the transversal section shown on the Fig. 1. This section is considered as the repeated unit cell periodically distributed in the plane OXY. The axis Z is directed along the weld seam. Material properties of weld metal grain size zones are assumed isotropic. To describe the creep behaviour of weld metal zones, Norton creep law is used (equation (5)).

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1

Finite element model of RVE

Material parameters used for equation (5) are taken from Ref. 8 and are presented in Table 1. It should be noted that it is impossible to make the specimens directly from fine and coarse grained zones independently, that is why for this heat affected zones, material properties in Ref. 8 are assumed equal.

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Table 1

Parameters of Norton law for weld metal zones

Zone type	K/	n
Columnar	2·74×10−21	7·86
Coarse grained	1·37×10−20	7·86
Fine grained	1·37×10−20	7·86

The creep law for the homogenised continuum is presented by the averaged components in the volume V of the unit cell (10) (11) where and 〈σ_kl〉 are the averaged creep strain rates and stresses, which correspond to uniform macroscopic strain rate and stress (12) For identification of six material constants in equation (10), six independent numerical tests should be executed. For these purposes, two- and three-dimensional models of RVE are created in the finite element code ABAQUS. For these purposes, a model of RVE is developed in the finite element code ABAQUS. Since the structure of the weld material with a sufficiently large number of passes can be assumed periodical, three types of representative volume are created: a reference one (eight passes), the realisation with twice more passes and realisation with half of the number of passes. Owing to the fact that in different zones on the microscale level creep behaviour is described by the different relations, numerical experiments should be carried out to define the possibility to how equation (10) approximates the creep behaviour on the macroscopic level.

Let us consider a numerical experiment performed on an RVE of uniaxial tension in the 11 direction. Average creep strain rate in case of uniaxial tension is related with average stress through the following equation (if b₁₁₁₁ = 1) (13) Averaging by volume of stresses and strains can be reduced to averaging by area of RVE's cross-section. To lower the computational costs in calculation of average strains, one can use the Green equation (14) where s is the surface area of the RVE, L is the perimeter, x and y are the coordinates in coordinate system that coincide with 11 and 22 directions, a and b are the dimensions of RVE in 0XY coordinate system. Similar simplifications can be made to obtain stresses (15) At first, it is necessary to determine the conditions of approaching steady state stage for the strain rate. Series of creep analysis of RVE are performed under the constant uniform stress for different time levels. From the results of numerical experiments, one can extract the set of strain rate values for the different moments of time t_i (i = 1, 2, …, N).

Strain rate values on the macroscopic level can be obtained by the least squares method (16) To derive the exponent in creep law, a set of numerical experiments under different macrostress levels are made. The result of this series of M experiments is the set of for different stress values (i = 1, 2, …, M).

To process the results of numerical experiments, relation (13) is presented in logarithmic coordinates (17) Material constants for creep law of homogenised material are defined by processing results of numerical experiments using the least squares method (18) (19) The other components of the tensor ⁽⁴⁾B can be defined by processing the results of numerical experiments of uniaxial macroscopic stress state of other types. For example, in uniaxial tension analysis along axis OY with the stress , creep law of homogenised material will be as follows (20) Processing the results of numerical experiments by the least squares method will make it possible to obtain parameter b₂₂₂₂ (21) In similar way, all other components of tensor ⁽⁴⁾B can be defined. Parameters of the considered material are presented in Table 2.

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Table 2

Parameters of tensor B multiplied by K/MPa⁻ⁿ s⁻¹

	8 pass	16 pass	4 pass
b 1111	1·64×10−5·25	1·61×10−5·25	1·66×10−5·25
b 2222	1·51×10−5·25	1·58×10−5·25	1·49×10−5·25
b 3333	1·16×10−5·25	1·18×10−5·25	1·2×10−5·25
b 1212	1·33×10−5·25	1·36×10−5·25	1·35×10−5·25
b 1313	1·26×10−5·25	1·27×10−5·25	1·27×10−5·25

Distributions of the components σ₁₁ and σ₂₂ on steady state creep stage during numerical experiments on tension are shown in Fig. 2 respectively.

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2

Equivalent creep stress redistribution after 1000 h of creep a σ₁₁ and b σ₂₂ (MPa)

Conclusions

As a result of finite element analysis on one-component loadings of a representative volume equivalent, total strains are obtained, and by processing them using averaging procedure, the material constants for constitutive equation of the equivalent material are found. To take into account scaling effects and also the welding operational features, three types of RVEs with different number of passes were considered.