An experimental and numerical investigation of ...

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An experimental and numerical investigation of fracture resistance behaviour of a dissimilar metal welded joint M K Samal1∗ , K Balani2 , M Seidenfuss3 , and E Roos3 1 Health Safety and Environment Group, Bhabha Atomic Research Centre, Mumbai, India 2 Department of Materials and Metallurgical Engineering, Indian Institute of Technology Kanpur, Kanpur, India 3 Institut für Material Prüfung Werkstoffkunde und Festigkeitslehre, Universität Stuttgart, Germany The manuscript was received on 30 October 2008 and was accepted after revision for publication on 23 December 2008. DOI: 10.1243/09544062JMES1416

Abstract: Dissimilar welds impose a challenge to the engineers concerned with the structural integrity assessment of these joints. This is because of the highly inhomogeneous nature of these joints in terms of their microstructure, mechanical, thermal, and fracture properties. Fracture mechanics-based concepts cannot be directly used because of the problems associated with the definition of a suitable crack-tip loading parameter such as J -integral crack tip opening displacement (CTOD), etc. Again, depending upon the location of initial crack (i.e. base, weld, buttering, different interfaces, etc.), further crack propagation can occur in any material. The objective of the current work is to use micro-mechanical models of ductile fracture for initiation and propagation of cracks in the bimetallic welds. The authors have developed a finite element formulation that incorporates the porous plasticity yield function due to Gurson– Tvergaard–Needleman and utilized it here for the analysis. Experiments have been conducted at MPA Stuttgart using single edge-notched bend (SEB) specimens with cracks at different locations of the joint. The micro-mechanical (Gurson) parameters of four different materials (i.e. ferrite, austenite, buttering, and weld) have been determined individually by simulation of fracture resistance behaviour of SEB specimens and comparing the simulated results with those of the experiment. In order to demonstrate the effectiveness of the damage model in predicting the crack growth in the actual bimetallic-welded specimen, simulation of two SEB specimens (with initial crack at ferrite–buttering and buttering–weld interface) has been carried out. The simulated fracture resistance behaviour compares well with those of the experiment. Keywords: dissimilar metal weld joint, ferrite–buttering interface, buttering–weld interface, fracture resistance, Gurson–Tvergaard–Needleman’s model, micro-mechanical parameters

1

INTRODUCTION

email: [email protected]

alloy and austenitic stainless-steel components. The nickel-base alloy dissimilar metal welds are typically made of alloy 182 and alloy 82. Recently, alloy 52 has been used both in new constructions as well as in repair welding in pressurized water reactors, while in boiling water reactors alloy 82 is still considered to be the best choice [1, 2]. The interest in alloys with higher content of chromium is driven by the cracking observed in alloy 182 and, recently, also in alloy 82. The operating experience of major nuclear power plant pressure boundary components has recently shown that dissimilar metal weld joints can markedly affect the plant availability and safety because of increased incidences of environment-assisted cracking of alloy 600 and corresponding weld metals (alloys 182/82). Alloy 690 and associated weld metals (alloys 152/52)

JMES1416 © IMechE 2009

Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science

In general, dissimilar metal welds are used in nuclear power plants and in oil refineries at locations where two different types of materials (e.g. carbon steel and austenitic stainless steel or nickel-base alloy) are joined together. Nickel-base weld metals are used throughout the light water reactors to join the low alloy steel pressure vessel, the pressurizer, and the steam generator nozzles to wrought nickel-base ∗ Corresponding

author: Reactor Safety Division, Bhabha Atomic

Research Centre, Hall-7, Trombay, Mumbai, Maharashtra 400085, India.

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M K Samal, K Balani, M Seidenfuss, and E Roos

are widely used for repair and replacement of the affected thick-section components [1–9]. Dissimilar metal welds impose a challenge to the engineers concerned with the structural integrity assessment of these joints. This is because of the highly inhomogeneous nature of these joints in terms of their microstructure, mechanical, thermal, and fracture properties. Fracture mechanics-based concepts cannot be directly used because of the problems associated with the definition of a suitable crack-tip loading parameter such as J -integral, crack tip opening displacement (CTOD), etc. Again, depending upon the location of initial crack (i.e. base, weld, buttering, different interfaces, etc.), further crack propagation can occur in any material [5]. Moreover, for interface cracks, there can be a change in crack path from one material to the other. The complex conditions and properties of the welded joint, as resulting from the elaborate interaction of different microstructures with gradients in material properties, have limited the ability of currently existing methods to construct the assessment on the basis of actual failure mechanisms of bimetallic welds. At present, the knowledge of homogeneous-welded joints is utilized for design, and no specific methods exist for appraising approvable assessment of structural integrity. Simplified engineering treatment models have been developed in literature for resolving the problem of dissimilar welds utilizing the concept of equivalent properties so that the integrity assessment methods for single materials can be used for them [10–15]. However, it may be argued that the interaction between under and overmatching local microstructures is one of the key elements prohibiting the optimized estimation of integrity of bimetallic components. The objective of the current work is to use micromechanical models of ductile fracture for initiation and propagation of cracks in the dissimilar metal welds. The approach described in the article is appropriate for ductile crack propagation failure mechanisms and this would certainly be typical of properly fabricated joints. It may be noted that this model is not intended to describe dissimilar joints where the failure is caused by a weak or brittle bonding interface between materials. It is well known that, the ductile fracture process involves the typical stages of nucleation, growth, and coalescence of voids in the micro-scale. In order to take into account the effects of these voids on the stress carrying capability of a mechanical continuum during simulation, damage mechanics models, such as those of Rousselier and Gurson–Tvergaard–Needleman (GTN), are widely used in the literature [16–20]. These models have been highly successful in simulating the fracture resistance behaviour of different specimens and components made of a wide spectrum of engineering steels [21–42]. The application of these damage models to predict fracture resistance behaviour of dissimilar welded

joints is not systematically investigated in research literature. The approach involves determination of micro-mechanical parameters of different material regions of the joint, prediction of fracture resistance behaviour of the joint with initial crack located at different regions and validation by experiment. In this work, the authors have developed a finite element (FE) formulation that incorporates the porous plasticity yield function due to GTN and utilized it here for the analysis. Experiments have been conducted at MPA Stuttgart using single edge-notched bend (SEB) specimens with cracks at different locations of the joint. The micro-mechanical (Gurson) parameters of four different materials (i.e. ferrite, austenite, buttering, and weld) have been determined individually by simulation of fracture resistance behaviour of SEB specimens and comparing the simulated results with those of experiment. In order to demonstrate the effectiveness of the damage model in predicting the crack growth in the actual bimetallic-welded specimen, simulation of two SEB specimens (with initial crack at ferrite–buttering and buttering–weld interface) has been carried out. This article is organized in seven sections. A brief introduction to the subject and relevant literature has already been provided in this section. The details of the weld joint, material properties, and the specimens are discussed in section 2. Section 3 describes the FE formulation developed by the authors for analysis of the crack initiation and propagation in ductile materials. The details of determination of micromechanical parameters of the four different materials (namely austenite, ferrite, weld, and buttering) are discussed in section 4. Sections 5 and 6 describe the comparison of results from experiment and analysis of the two SEB specimens with initial cracks at the ferrite–buttering and the buttering–weld interface respectively, followed by conclusive remarks in section 7.



2

MATERIAL AND EXPERIMENT

A dissimilar weld joint was prepared between two pieces of ferritic and austenitic pipes (of 332 mm outer diameter and 29 mm thickness). The ferritic steel is DIN 20MnMoNi55 (similar to A508 Cl.3) and the austenitic steel is DIN 1.4550. The ferritic steel is an optimized reactor pressure vessel steel material. It has a bainitic structure at room temperature. From the metallographic examination of cross-sections of the deformed tensile specimens, it was observed that voids usually form from detachment of manganese sulphide (MnS) inclusions at the particle–matrix interface [21, 43]. The MnS particles inside the voids have almost a spherical shape. After nucleation of the voids from the MnS inclusions, there is no more adhesion between the particles and the matrix material.

Fracture resistance behaviour of a dissimilar metal welded joint

Therefore the initial void volume fraction of the material can be equated to the volume fraction of MnS inclusions in the ferritic steel. The austenitic material DIN 1.4550 (similar to SS304) is a piping material and has a face-centred cubic lattice crystal structure as opposed to the bodycentred cubic lattice structure of the ferritic material. A metallographic cut through a deformed tensile specimen also exhibits voids in the cross-section. However, the voids usually develop at the niobium carbo-nitride inclusions, which normally have elongated shapes. The voids initiate mainly by the fracture of the slender carbo-nitride particles. It has also been observed that most particles are in an already broken state as the piping material is usually subjected to plastic deformation during the manufacturing of the pipes. However, the adhesion between the particles and the matrix is much stronger than that of the ferritic material. The particles do not lie loose in the voids. Hence, the initial void volume fraction for the austenitic material does not correspond to the particle volume (but much lower) because the newly developed voids have only the same volume of the actual internally present cracks of the carbo-nitride particles. After edge preparations on both the pipe ends, a buttering layer (of ≈4 mm thickness) was deposited through a nickel-rich electrode on the ferritic pipe face. The buttering layer is of over-alloyed nickelenriched E309L (24 per cent Cr and 12 per cent Ni). After this, welding is carried out between the buttering layer and the austenitic face of the pipe. The thickness of the weld is ≈25 mm. The weld is deposited with matching E308L (18 per cent Cr and 8 per cent Ni) material. The use of a nickel-enriched buttering layer is a common approach for power-plant applications to reduce toughness degradation as a consequence of carbon precipitation faced with matching austenitic filler materials. The schematic weld configuration is shown in Fig. 1. The regions critical for the performance of the weld are the coarse-grained heat-affected zone, the

Fig. 1

Schematic configuration of the dissimilar weld joint


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fusion line, its immediate vicinity, and the buttering layer. Degradation of fusion zone toughness has been ascribed to the formation of upper-bainitic coarsegrained heat-affected zone microstructure, as well as a narrow martensitic layer (as a result of carbon diffusion from the ferritic steel towards the austenitic material) [2, 3]. The chemical compositions of the austenitic, ferritic piping, and the weld material are given in Tables 1, 2, and 3, respectively [5]. Various locations of the weld joint are etched to clearly locate the interfaces. An ammonium chloride mixture was used to etch the weld and the buttering regions. Electrolytic oxalic acid etching is also carried out for 2–3 min with 1.5–3.0 V electric potential with a cathodic area of 25 mm2 . From the weld joint of the pipe, single edge-notched bend (SEB) specimens were machined. The sketch of the specimen with various dimensions is shown in Fig. 2. The proportion of dimensions of the specimen is in accordance with the ASTM E1820-99 standard. The pipe is welded in the circumferential direction. The specimens are machined such that the length of the specimen is in the longitudinal direction of the pipe, thickness of the specimen is in the thickness direction of the pipe, and the width of the specimen is in the circumferential direction of the pipe. As the outer diameter of the pipe is 332 mm, machining of a specimen of width 15 mm is easy. The specimen is a single-edge cracked and fatigue precracked beam loaded in three-point bending and supported over a span S. The specimens were microetched in order to locate the metallurgically different micro-structural zones and to machine initial cracks at the ferrite–buttering and the buttering–weld interface of the welded region of the specimen. The different interfaces and the initial cracks are schematically shown in Fig. 3. The notch was initially prepared with saw-cutting and the finish cutting was done by electrodischarge machining. The initial crack size is 15 mm. Later, fatigue precracking is done in steps of 1 mm up to a crack size of 18 mm. The specimens were 20 per cent side-grooved in order to obtain a plane strain condition for crack growth during subsequent loading. The crack mouth opening displacement (CMOD) was recorded through a COD gauge, which is attached to the data acquisition module of the universal testing machine, while the load was recorded directly from the load cells of the machine. The crack growth during the test was measured through the standard unloading (i.e. 20 per cent unloading was done) compliance method. The J -integral for the specimens were calculated from the load–displacement data (i.e. using area under the load–displacement curve and suitable η and γ factors) following the standard ASTM method as described below. The J -integral can be defined as the quantity that represents the rate of energy release with respect to Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science

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Table 1

Chemical composition of the austenitic steel (DIN 1.4550)

Element

C

Si

Mn

P

S

Cr

Ni

Mo

Nb

Fe

Wt%

0.045

0.3

1.89

0.028

0.004

17.6

10.6

0.28

0.45

Rest

Table 2

Chemical composition of the ferritic steel (DIN 20MnMoNo55)

Element

C

Si

Mn

P

S

Cr

Ni

Mo

V

Al

Cu

Fe

Wt%

0.21

0.21

1.3

0.009

0.001

0.05

0.068

0.494

0.01

0.029

0.02

Rest

Table 3

Chemical composition of the weld material

Element

C

Si

Mn

P

S

Cr

Ni

Mo

Nb

Ti

Fe

Wt%

0.025

0.38

4.81

0.01

0.005

18.2

Rest

1.3

2.17

0.06

3.48

Fig. 2

Schematic figure of the SEB specimen (a0 = 18 mm, L = 135 mm, S = 120 mm, W = 29 mm, and B = 15 mm)

the growing crack surface. According to Sumpter and Turner [44], the J -integral can be divided into an elastic and a plastic part, Jel and Jpl , whose expressions are written as Upl a K 2 (1 − ν 2 ) f + J = Jel + Jpl = E Anet w Upl K 2 (1 − ν 2 ) = +η (1) E Anet where K is the elastic stress intensity factor, ν is the Poison’s ratio, E is the Young’s modulus of elasticity,

Anet is the net cross-sectional area of the specimen that carries the load (i.e. product of effective thickness Bnet considering side-grooving in the specimen and the remaining ligament length (W − a), where W and a are the width and remaining ligament, respectively), Upl is the plastic part of the area of the load–displacement curve, and η is the geometric factor that depends upon the dimensions (i.e. initial crack length to the width ratio) of the specimen. As the elastic part of the area of the load–displacement curve Uel is very small compared to the plastic part of the area, the J -integral can be evaluated in terms of the total area (i.e. Utot = Uel + Upl ) and a correction factor (in terms of a geometric factor γ ) is introduced to take into account of the increased crack length during crack growth. Hence, the expression of J -integral can be expressed as [45] γ a J = J0 1 − (W − a0 ) η Utot γ a = 1− Bnet (W − a0 ) (W − a0 )

Fig. 3

Schematic representation of a crack at two different interfaces


(2)



where a0 is the initial crack length and a is the increment in crack length during loading (it increases as one traverses the load–displacement curve). The geometric factors η and γ depend upon the specimen geometry and crack length-to-width ratio (a/W ). For a standard compact tension (CT) specimen, η = 2 + 0.522(1 − a/W ) and γ = 0.75η − 1. For a standard SEB specimen, η = 2 and γ = 1. In this way, the loadCMOD and the J -integral versus crack growth data were obtained from the tests and these are discussed later while comparing the results with those of analysis. Before discussing the comparison of data from experiment and analysis, a brief description of the FE formulation using the GTN’s porous plasticity yield function is provided here. 3

DEVELOPMENT OF AN FE FORMULATION FOR THE GTN DAMAGE MODEL

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as [18–20] p f˙growth = (1 − f )˙εkk

(5)

The voids quantified by f are either initially present or nucleated by the deformation process. In the latter case, some void nucleation law has to be specified. Chu and Needleman [46] proposed the following law for calculating increment of void volume fraction due to nucleation, which tells that the increase in void volume fraction f is due to the void growth and void nucleation processes. Hence, the increment of void volume fraction includes the growth law (i.e. equation (5)) and a strain-controlled void nucleation mechanism (also a stress-controlled mechanism if it is favourable in the material), which is described as p p ˙ + (1 − f )˙εkk f˙ = f˙nucleation + f˙growth = Aε˙ eq + B(q˙ − p)

(6) 3.1

Material yield function

The yield potential of the GTN’s damage model is written as [11–13] φ=

q2 p 2 ∗ − 1 − q3 f ∗ = 0 + 2 q f cosh −1.5 q 1 2 σm2 σm (3)

where q is the von-Mises equivalent stress, f ∗ is the modified ductile void volume fraction, p is the hydrostatic pressure, and σm is the true stress in the matrix material, which is a function of equivap lent plastic strain εeq in the matrix material (i.e. the material stress–strain curve). q1 , q2 , and q3 are the constants introduced by Tvergaard and Needleman [18, 19] in order to simulate the observed experimental fracture behaviour in many different materials more accurately. The function f ∗ was also introduced by Tvergaard and Needleman [18, 19] and the modified void volume fraction f ∗ is related to the actual void volume fraction f through the relationship ⎧ if f fc ⎨f f∗ = (4) fu∗ − fc ⎩fc + (f − fc ) if f > fc ff − f c where fc is the critical void volume fraction (signifying void coalescence), ff is the actual void volume fraction at final fracture, and fu∗ is the modified void volume fraction at fracture (usually fu∗ = 1/q1 ).

where the constants A and B for strain-controlled nucleation are written as 2

p fNε 1 εeq − εn A=√ , exp − 2 snε 2πsnε B=0

Growth of the void volume fraction in a ductile material

(7)

and for stress-controlled nucleation

fnσ 1 (q − p) − σn 2 A = 0, B = √ exp − 2 snσ 2πsnσ ˙ >0 for (q˙ − p)

(8)

where fnε and fnσ are the values of void volume fractions of void nucleating particles at mean nucleation strain and stresses εn and σn , respectively. snε and snσ are the standard deviations of equivalent plastic strain and sum of equivalent and hydrostatic stress, respectively, which are responsible for void nucleation (assuming a Gaussian distribution for the void nucleation process). The superscripts ε and σ denote the strain- and stress-controlled nucleation, respectively. Again, the increment in equivalent plastic strain of the matrix material ε˙ eq can be written (from the principle of plastic work equivalence between the voided and the unvoided matrix material) as p

p ε˙ eq

3.3 3.2

p for ε˙ eq >0

=

σij ε˙ ij

(1 − f )q

i = 1, . . . , 3

j = 1, . . . , 3

(9)

Consistent material tangent stiffness matrix for the elasto-plastically deforming material (with damage)

The void growth rate is obtained using the plastic incompressibility condition of the matrix material

In order to derive the material constitutive matrix for the elasto-plastically deforming material, the additive



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decomposition of total strain increment ε˙ into elastic ε˙ e and plastic ε˙ p parts is assumed, i.e. p

ε˙ ij = ε˙ ije + ε˙ ij

(10)

The porous plastic yield function (equation (3)) can be written in terms of mean hydrostatic and deviatoric parts of stress tensor and other field variables as φ(p, q, H α ) = 0

(11)

where p and q are the hydrostatic pressure and von Mises equivalent stress, respectively, and are defined as 1 p = − σkk , 3 3 sij sij q= 2

k = 1, . . . , 3

and shear modulus of the material, respectively. By solving the above equations simultaneously, the corrections in the hydrostatic and equivalent parts of the incremental plastic strain (i.e. ∂ ε˙ p and ∂ ε˙ q ) in the form of two simultaneous equations are expressed as A11 ∂ ε˙ p + A12 ∂ ε˙ q = b1 A21 ∂ ε˙ p + A22 ∂ ε˙ q = b2 (19) where the expressions for the constants A11 , A12 , A21 , A22 , b1 , and b2 are given in reference [47]. After solving equation (19), the values of ε˙ p and ε˙ q can be updated as ε˙ p ≈ ε˙ p + ∂ ε˙ p ε˙ q ≈ ε˙ q + ∂ ε˙ q (20)

(12) H α ’s are the thermodynamic internal state variables such as hardening parameter, void volume fraction f , etc., I is the Kronecker-delta or second-order identity tensor, and s is the deviatoric part of stress tensor σ . The generalized flow rule for the increment of plastic strain tensor from time t to t + t can be written as [47] ε˙ p =

1 ε˙ p I + ε˙ q nt+t 3

(13)

where ε˙ p and ε˙ q are the hydrostatic and equivalent parts of plastic strain increment (the superscript ‘p’ and the tensorial subscripts ‘ij’ have been omitted for simplicity). The direction vector of plastic flow at time t + t (i.e. nt+t ), is defined in terms of the deviatoric part of the elastic predictor stress s tr and von Mises equivalent stress q tr (corresponding to elastic predictor stress σ tr ) as nt+t =

3s tr 2q tr

(14)

where the superscript ‘tr’ refers to the trial state of the quantities. The terms ε˙ p and ε˙ q are related through the equation ε˙ p

∂g ∂g + ε˙ q =0 ∂q ∂p

(15)

where g is the plastic potential and for associated plasticity, g = φ. In order to calculate the increment in stress tensor, the following sets of equations are to be solved simultaneously, i.e. equation (11) and p = ptr + K ε˙ p c

(16)

q = q − 3G ε˙ q tr

(17)

H = h (˙εp , ε˙ q , p, q, H ) α

α

β

(18)

Using these updated values of ε˙ p and ε˙ q , the stress tensor at end of time step t + t is evaluated as σt+t = σ tr − K ε˙ p I − 2G ε˙ q nt+t

(21)

where nt+t is defined in equation (14). Taking variation of equation (15), the variation of ε˙ p and ε˙ q in terms of increment of total strain is expressed as ∂ ε˙ p = (mpl I + mpn n) : ∂ε ∂ ε˙ q = (mql I + mqn n) : ∂ε (22) where the coefficients mpl , mpn , mql , and mqn are defined in reference [47]. Now, taking variation of equation (21) and substituting equation (22) for ∂ ε˙ p and ∂ ε˙ q , the increment of stress tensor is obtained as ∂σ = Cep : ∂ε

(23)

where the elasto-plastic material tangent stiffness matrix can be defined as q J + K (1 − mpl )II q tr 4 3 q + G 1 − tr − mqn nn 3 q 2

Cep = 2G

− 2Gmql nI − Kmpn In

(24)

In the above expression, J is the fourth-order unit tensor and J = J − (II )/(3). 3.4 Variational statement of the governing differential equation

where the quantities with superscript ‘tr’ refers to ‘trial’ or the elastic predictor part, and K and G refer to bulk

In order to derive the FE matrices for the elasto-plastic continuum, one must know the governing partial




differential equation and the associated boundary conditions, which are written as [48] ∇ · σ + fb = 0

(25)

σ · n| f = τs

(26)

u| u = u0

(27)

where fb is the body force per unit volume, σ and τs are the Cauchy’s stress tensor and traction vector, respectively, equation (26) is Cauchy’s traction law, equation (27) is the geometric or essential displacement boundary condition, and n is the unit vector normal to the boundary surface f where the force or Neumann boundary condition is specified. u is the boundary surface at which displacement or geometric boundary condition is specified (i.e. u = u0 ). The weighted residual statement of the governing mechanical equilibrium equation (26) can be defined for the mechanical continuum as δu(∇σ + fb ) d = 0 (28)

is the non-linear strain–displacement transformation ˆ T , the terms matrix [48]. For arbitrary value of δ(u) inside the bracket of equation (30) can be put to zero and hence this equation reduces to the set of FE algebraic equations, which can be written in convenient (matrix) form as int ˆ = {R ext [K L + K NL ]{u} m − Rm }

(31)

where the matrices and the force vectors can be expressed as KL =

BTu Cep Bu · d

K NL =

BTNL t σ BNL · d T R ext = N f d + NTu τs d m u b int Tt R m = Bu σ · d

where δu is the weight function (which is taken as variation of displacement field u here) and the quantity within the brackets of equation (28) is the residual as we will substitute the displacement field u with its approximation (which is a interpolation of the nodal displacement values uˆ at the material point) as discussed in equation (29). 3.5

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FE discretization

In order to derive the FE equations from equation (28), we write the generalized displacement u and strain vectors t e at any material point inside the FE in terms of the generalized nodal variables uˆ (i.e. FE discretization) as ˆ u = Nu u,

t

e = Bu uˆ

(29)

where Nu is the shape function matrix for the displacement variable uˆ and Bu is the matrix containing the derivatives of the shape function Nu [48]. Expanding equation (28) and using the consistent material tangent stiffness matrix from equation (23), and substituting the expressions for u and t e from equation (29), one obtains ˆ T δ(u) BTu Cep Bu · d uˆ + BTNL t σBNL · d × uˆ + BTu t σBNL · d − NTu fb d

− NTu τs d

(32) where the left superscript ‘t’ refers to the quantities at the end of previous iteration step of the incremental non-linear FE analysis. The element level equations derived so can be assembled and solved for the global degrees of freedom when the required boundary and int loading conditions are specified. R ext m and R m are the externally applied mechanical force vector and internal mechanical force vector (due to internal stresses), respectively. 4

4.1

DETERMINATION OF MICRO-MECHANICAL PARAMETERS FOR DIFFERENT MATERIALS Evolution of the void volume fraction due to plastic deformation and subsequent void coalescence

where t σ is the matrix and t σ is the vector containing Cauchy’s stress components at time t, and BNL

The void volume fraction (which is a measure of damage evolution in ductile materials) evolves from the initial void volume fraction f0 (volume fraction of relevant inclusions or void nucleation sites) in the material and with straining; it increases to the critical void volume fraction for coalescence fc . After fc , the increase in void volume fraction gets accelerated and final fracture of the material point occurs at final void volume fraction ff . In the continuum damage mechanics models, these parameters are determined from combined numerical simulation and metallurgical observations of microscopic voids as described in references [21], [27], [31], [37] to [39], [42], and [43]. For the materials under consideration here, the details of the selection of the parameters will be discussed in the next section.



=0

(30)

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Usually, a fracture mechanics specimen such as a compact tension specimen is numerically simulated and by comparing the experimentally obtained J -integral value at crack initiation (JIC ) with the numerically simulated value, the value of fc is fixed. Many times, the notched tensile specimens are also used [23, 25, 39]. The critical void volume fraction for coalescence fc is determined by comparing the true fracture strain from experiment with that of numerical simulation. It was observed from numerical simulation of notched tensile specimens that the growth of voids is usually slow and the maximum growth takes place in the centre of the specimens (in case of low constraint notches; however, the notch tip can see a higher void growth compared to the centre in case of specimens with sharp notches). The rate of void growth becomes higher only after necking takes place. However, just before the final fracture (by formation of cup and cone mechanism), the void growth gets accelerated by coalescence of the voids at the centre of the notched tensile specimens and most of the elements at the ligament attain the final fracture condition for void volume fraction [18, 19, 39]. This phenomenon of acceleration in the void growth is characterized by the void coalescence parameter fc and hence Tvergaard and Needleman [19] had earlier introduced the modification through equation (4) in the Gurson model [17]. Hence, this point of drastic load drop (before final separation) in the notched tensile specimens is typically used to fit the parameter fc . It may be noted that the void volume fractions, such as f0 and fc , are material properties and hence they can be used for simulation in all types of specimens and components of the same material. 4.2

Determination of the initial void fraction f0 and void fraction fn at saturated condition of nucleation

As discussed in section 2, the initial void volume fraction f0 to be used in the micro-mechanical model not only depends upon the distribution of the inclusions and second-phase particles, but also upon the state of damage of these particles (by either decohesion

Fig. 4

of the interface between the particle and the matrix or by cracking of the particle). For the ferritic steel (DIN 20MnMoNi55), the initial volume fraction of the MnS particles was found to be 0.005, whereas for the austenitic steel (DIN 1.4550) the carbo-nitrides are tightly bonded to the matrix and hence the value of f0 is very small (of the order 1e-6). In order to confirm these parameters and obtain the void volume fraction at saturated condition of nucleation fn (assuming void nucleation to be following a Gaussian statistical distribution and is strain-controlled as proposed by Chu and Needleman [46]), the analysis of the SEB specimen is performed as discussed in section 2 using individual material properties of the four different materials for the whole specimen. The two-dimensional plane strain eight-noded isoparametric elements were used in the FE analysis. The FE mesh is shown Fig. 4(a) with an enlarged view near the crack tip in Fig. 4(b). The mesh size is of the order of 0.2 mm near the crack tip and in the region of crack propagation. This mesh size is a requirement of FE analysis for correct prediction of crack initiation and propagation using porous plasticity models and this refers approximately to the mean distance between relevant inclusions in the material. For further discussions, the readers can refer to references [21], [23] to [25], [28], and [31] to [33]. The material stress–strain curve (true stress versus true plastic strain) at room temperature for the four different materials under consideration (i.e. austenitic, ferritic, weld, and buttering) used in the FE analysis are shown in Fig. 5. The specimen is loaded at the centre with an applied displacement from the machine loading attachment. For the statistical void nucleation model, the value of mean strain for void nucleation εn was taken to be 0.3 for all the materials. Similarly, the standard deviation of void nucleation strain sn was taken as 0.1. The void volume fraction for coalescence fc is taken as 0.05 and the value of final void volume fraction ff is taken as 0.3. The values of Tvergaard and Needleman’s parameters (i.e. q1 , q2 , and q3 ) were taken from literature as 1.5, 1.0, and 2.25, respectively [21, 27, 31, 37–39, 42, 43]. We are left with two parameters, i.e. f0 and fn . Figure 6

(a) FE mesh of the cracked SEB specimen and (b) enlarged view near the crack tip




Fig. 5 True stress–plastic strain curves of the materials used in the analysis

Fig. 6

J -resistance curve of the SEB specimen of the austenitic material (experiment versus analysis)

shows the J -resistance (J -integral versus crack growth) curve of the SEB specimen made of austenitic steel. Analysis was carried out with several values of fn and it can be observed that the results from analysis compare very well with that of experiment when the parameters f0 and fn are chosen as 1e-6 and 0.0055, respectively. Similar procedure was followed for other materials. The results of parametric study of analysis (along with the experimental results) are shown in Figs 7, 8, and 9 for the ferritic, weld, and buttering material, respectively. The final values of the parameters f0 and fn , where the results of FE analysis matches satisfactorily with those of experiment, are enumerated in Table 4 for the different materials. Once the micro-mechanical parameters are determined, these are used for analysis of the SEB specimens made from the dissimilar-welded joint with initial crack located


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Fig. 7

J -resistance curve of the SEB specimen of the ferritic material (experiment versus analysis)

Fig. 8

J -resistance curve of the SEB specimen of the weld material (experiment versus analysis)

Fig. 9

J -resistance curve of the SEB specimen of the buttering material (experiment versus analysis)


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Table 4

Parameters for different materials of the dissimilar weld joint

UTS (MPa)

Initial void fraction f0

Void fraction at saturated nucleation condition fn

245 536.4 448.1 417.83

588 675 587 631

0.000 001 0.005 0.001 0.004

0.0055 0.008 0.008 0

Age elongation (%)

Age reduction in area (%)

Vickers hardness

68.1 25.1 14.2 14.8

73.9 74.1 17.9 18.5

153 160 234 228

Material

Young’s modulus (MPa)

Yield stress (MPa)

Austenitic steel Ferritic steel Weld material Buttering material

1.94E5 2.271E5 1.82E5 2.0455E5

Austenitic steel Ferritic steel Weld material Buttering material

at different interfaces (i.e. ferrite–buttering and the buttering–weld interface) and the results are described in the next section. 5

ANALYSIS OF SEB SPECIMEN WITH CRACK AT THE FERRITE–BUTTERING INTERFACE

The standard dimensions of SEB specimen are the same as discussed in section 2 and shown in Fig. 2. The FE mesh used in the analysis is the same as discussed in section 4.2 and shown in Fig. 4. The full specimen is meshed as we have different materials at both sides of the crack and hence the symmetric conditions cannot be used. The schematic representation of the specimen with crack location and materials at both sides is shown in Fig. 10. The actual FE mesh with thickness of each layer of material is shown in Fig. 11, where one can see that the crack is at the ferrite–buttering interface. It may be noted that, the interface will not be perfectly straight in reality. However, for simplicity of analysis, a straight interface is considered. In order to have a better insight, the different results of analysis of the specimen are presented here, i.e. progress of plastically deformed zone, crack growth, and its path in the specimen, load versus CMOD and

Fig. 11

FE mesh of the SEB specimen with different material properties at different regions and initial crack at the ferrite–buttering interface

Schematic representation of the SEB specimen with initial crack at the ferrite–buttering interface

J -resistance behaviour, etc. Figures 12 and 13 show the progress of plastically deformed zone and ductile crack growth respectively near the central portion of the SEB specimen through the contour plots of equivalent plastic strain and damage from the analysis (the snapshots s1–s8 are represented by ‘star’ symbols in the load-CMOD and J versus crack growth curves of the specimen in Figs 14 and 15, respectively). The contour representing the region of maximum plastic deformation is near to the crack tip (as expected). As the crack grows, the plastic deformation also spreads along the crack path as shown in Fig. 12 (snap shots are shown at different loading levels, i.e. at different values of applied displacement). One can also observe that the deformed shape is not symmetric and, similarly, the plastically deformed regions (which looks similar to the slip lines) on both sides are non-symmetric. This is because there exists a material with low yield stress (austenitic steel) on the right-hand side and the material with a high yield stress (ferritic steel) on the left-hand side of the crack. The slip lines are more prominent on the austenitic side of the specimen. The initiation and progress of crack from the initial, crack tip along with its path is shown in Fig. 13 in a series of snap shots taken at different intervals during the loading by displacement controlled condition. As can be seen from Fig. 13, the crack initiates on the buttering side (right-hand side of the initial crack tip)



Fig. 10


Fig. 12

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Progress of plastic deformation near the crack tip of the SEB specimen for the initial crack at the ferrite–buttering interface (contour plot of equivalent plastic strain). s1–s8 are the snapshots taken at different instances of loading as shown in Figs 14 and 15, respectively



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Fig. 13

Growth of crack through the buttering region of the SEB specimen with initial crack at the ferrite–buttering interface (contour plot of damage). s1–s8 are the snapshots taken at different instances of loading as shown in Figs 14 and 15, respectively

6

ANALYSIS OF SEB SPECIMEN WITH CRACK AT THE BUTTERING–WELD INTERFACE

and also grows in this material zone. This may be because of the reason that the buttering material is less resistant to crack initiation and propagation compared to the ferritic steel. Similar conclusions are also derived from experimental observations [5]. The load versus CMOD and J -resistance behaviour of this specimen is obtained from the analysis and compared with those of experiment in Figs 14 and 15, respectively. As can be seen from these figures, the results of analysis are in excellent agreement with those of experiment and, hence, it is demonstrated that the FE formulation with the porous plastic yield function of the GTN model is an excellent tool for analysis of crack growth and path in dissimilar weld joints. The parameters of individual materials could be successfully used in the weld joint fracture mechanics specimens and hence these are transferable. This method can be used to design a dissimilar weld joint where the parameters such as width of buttering and weld layer, material properties of electrodes, etc. can be optimized to obtain a weld with desired mechanical strength and fracture properties.

The standard dimensions of SEB specimen are the same as discussed in section 2 and shown in Fig. 2. The FE mesh used in the analysis is the same as discussed in section 4.2 and shown in Fig. 4. The schematic representation of the specimen with crack location and materials at both sides is shown in Fig. 16. The actual FE mesh with thickness of each layer of material is shown in Fig. 17, where one can see that the crack is at the buttering–weld interface. Figures 18 and 19 show the progress of plastically deformed zone and ductile crack growth respectively near the central portion of the SEB specimen through the contour plots of equivalent plastic strain and damage from the analysis (the snapshots s1–s8 are represented by ‘star’ symbols in the load-CMOD and J versus crack growth curves of the specimen in Figs 20 and 21, respectively). The contour representing the region of maximum plastic deformation is near the




Fig. 14

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Fig. 16

Schematic representation of the SEB specimen with initial crack at the buttering–weld interface

Fig. 17

FE mesh of the SEB specimen with different material properties at different regions and initial crack at the buttering–weld interface

Load–CMOD response of the SEB specimen with initial crack at the ferrite–buttering interface (experiment versus analysis). s1–s8 are the points (star symbols) at which snapshots were taken at different instances of loading as shown in Figs 12 and 13, respectively

crack tip (as expected). As the crack grows, the plastic deformation also spreads along the crack path as shown in Fig. 18. It can be observed that the plastic deformation is initially slightly symmetric as the material properties of the buttering and weld region are nearly similar. However, with further loading, the plastic deformation band spreads to the right-hand side of the initial crack. This is because of the presence of austenitic

steel in the right-hand side and hence the deformation becomes asymmetric. However, the path of crack propagation is different. Initially, crack initiates on both sides of the initial crack (i.e. propagates exactly at the interface between the two materials) as shown in Fig. 19. As the buttering region has less fracture toughness compared to the weld region, crack deviates towards the buttering region during subsequent loading as can be seen from the different snap shots of Fig. 19. Because of crack propagation in the buttering region (left-hand side of the initial crack), the maximum plastic deformation region also shifts towards the buttering region as seen from the snap shots of Fig. 18. However, the slip line is still dominant in the righthand side, i.e. in the side of the austenitic material. This is an important observation from the analysis and has also been verified from the experiment [5]. The load versus CMOD and J -resistance behaviour of this specimen is obtained from the analysis and compared with those of the experiment in Figs 20 and 21, respectively. As can be seen from these figures, the results of analysis are in good agreement with those of the experiment and hence it is demonstrated that the parameters of individual materials could be successfully used to simulate the fracture resistance behaviour of this weld joint fracture mechanics specimens with an initial crack at the buttering–weld interface.



Fig. 15

J -resistance behaviour of the SEB specimen with initial crack at the ferrite–buttering interface (experiment versus analysis). s1–s8 are the points (star symbols) at which snapshots were taken at different instances of loading as shown in Figs 12 and 13, respectively

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Fig. 18

Progress of plastic deformation near the crack tip of the SEB specimen for the initial crack at the buttering–weld interface (contour plot of equivalent plastic strain). s1–s8 are the snapshots taken at different instances of loading as shown in Figs 20 and 21, respectively




Fig. 19

Fig. 20

7

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Growth of crack through the buttering region of the SEB specimen with initial crack at the buttering–weld interface (contour plot of damage). s1–s8 are the snapshots taken at different instances of loading as shown in Figs 20 and 21, respectively

Load–CMOD response of the SEB specimen with initial crack at the buttering–weld interface (experiment versus analysis). s1–s8 are the points (star symbols) at which snapshots were taken at different instances of loading as shown in Figs 18 and 19, respectively

CONCLUSIONS

Fig. 21

J -resistance behaviour of the SEB specimen with initial crack at the buttering–weld interface (experiment versus analysis). s1–s8 are the points (star symbols) at which snapshots were taken at different instances of loading as shown in Figs 18 and 19, respectively

Design and safety analysis of dissimilar weld joints are an important issue for ascertaining the structural integrity of those joints that are unavoidable at several locations of thermal and nuclear power plants. In order to optimize the dimensions and properties, one must have a tool to predict the performance of these joints during actual and accidental loading conditions. To prove the safety of these joints,

initial cracks of different sizes are usually postulated at different locations, especially the interfaces. The fracture mechanics method cannot be directly used because of difficulties in defining the crack tip loading parameters. The use of a porous plasticity yield function for initiation and propagation of cracks in ductile materials has several advantages, e.g. the material stiffness degrades with progress of damage and simulates a crack (material with negligible stress carrying capability) at the material



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point with maximum plastic deformation and stress triaxiality. An FE formulation of the GTN model has been used here to predict the crack path and fracture resistance behaviour of an SEB specimen made from the dissimilar weld joint with initial crack at two different interfaces. The micro-mechanical parameters have been determined for the individual material and it was demonstrated that they can be safely used to predict the crack growth in the weld joint specimen. However, there are several other issues to be investigated further. It may be noted that the material properties (strength, hardness, and fracture properties) and micro-structures vary continuously from one material to the other at the interface. This is because of mixing two different materials during the welding process. However, in the present analysis, it is assumed that material properties drastically change at the interface. Of course, this is not a limitation of the FE method (to accommodate gradually varying material properties), detailed experimental investigations are required for this purpose and it needs to be seen whether use of these fine details makes any substantial difference to the outcome of the final results. There are other phenomena such as consideration of carbon diffusion across the interface, presence of residual stresses, etc., that need further research. The use of the exact shape of the interface of different regions can also be considered and it may provide better results. For a more refined and sophisticated design and safety analysis, FE simulation of the fusion welding process (heat transfer, thermal stresses, evolution of microstructure, and material properties, etc.) is being considered by many researchers and inputs from the same may be considered for analysis of crack initiation and propagation in these joints.

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REFERENCES 16 1 Hänninen, H., Brederholm, A., Saukkonen, T., Gripenberg, H.,Toivonen A., Ehrnstén, U., and Aaltonen, P. Hot cracking and environment assisted cracking susceptibility of dissimilar metal welds. VTT Research Notes 2399, VTT Technical Research, Finland, 2007. 2 Hänninen, H., Aaltonen, P., Brederholm, A., Ehrnstén, U., Gripenberg, H., Toivonen A., Pitkänen, J., and Virkkunen, I. Dissimilar metal weld joints and their performance in nuclear power plant and oil refinery conditions. VTT Research Notes 2347, VTT Technical Research, Finland, 2006. 3 Nevasmaa, P., Laukkanen, A., and Ehrnsten, U. Structural integrity of bimetallic components (BIMET). TG2 – material characterization report. VTT Report VALC517, 1999. 4 Nevasmaa, P., Laukkanen, A., and Ehrnstén, U. The role of metallurgy and micromechanisms of fracture in assessing structural integrity of bi-metallic ferrite–austenite pipeline welds. In Proceedings of Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science

17

18

19

20 21

the 3rd International Pipeline Technology Conference, Brügge, 21–24 May 2000, vol. I, pp. 253–268. Balani, K. Experimental and numerical studies on fracture behaviour of dissimilar welded joints. Masters Thesis, MPA Universität Stuttgart, Germany, 2001. Schwalbe, K. H., Cornec, A., and Lidbury, D. Fracture mechanics analysis of the BIMET welded pipe tests. Int. J. Press. Vessels Pip., 2004, 81, 251–277. Wäppling, D., Gunnars, J., and Stähle, P. Crack growth across a strength mismatched bimaterial interface. Int. J. Fract., 1998, 89, 223–243. Chhibber, R., Arora, N., Gupta, S. R., and Dutta, B. K. Use of bimetallic welds in nuclear reactors: associated problems and structural integrity assessment issues. Proc. IMechE, Part C: J. Mechanical Engineering Science, 2006, 220(C8), 1121–1133. DOI: 10.1243/09544062JMES135. Laukkanen, A., Nevasmaa, P., Ehrnstén, U., and Rintamaa, R. Characteristics relevant to ductile failure of bimetallic welds and evaluation of transferability of fracture properties. Nucl. Eng. Des., 2007, 237, 1–15. Kim,Y.-J., Schwalbe, K.-H., and Ainsworth, R. A. Simplified J-estimations based on the engineering treatment model for homogeneous and mismatched structures. Eng. Fract. Mech., 2001, 68(1), 9–27. Schwalbe, K.-H. and Zerbst, U. The engineering treatment model. Int. J. Press. Vessels Pip., 2000, 77(14–15), 905–918. Kim, Y.-J. and Schwalbe, K.-H. Numerical analyses of strength mis-match effect on local stresses for ideally plastic materials. Eng. Fract. Mech., 2004, 71(7–8), 1177–1199. Zerbst, U., Primas, R., Schindler, H.-J., Heerens, J., and Schwalbe, K.-H. The fracture behaviour of a welded tubular joint-an ESIS TC1-3 round robin on failure assessment methods. Part 4: application of the ETM 97/1. Eng. Fract. Mech., 2002, 69(10), 1129–1148. Kim, Y.-J. and Schwalbe, K.-H. Compendium of yield load solutions for strength mis-matched DE(T), SE(B) and C(T) specimens. Eng. Fract. Mech., 2001, 68(9), 1137–1151. Kim, Y.-J. and Schwalbe, K.-H. Mismatch effect on plastic yield loads in idealised weldments: I. Weld centre cracks. Eng. Fract. Mech., 2001, 68(2), 163–182. Rousselier, G. Ductile fracture models and their potential in local approach of fracture. Nucl. Eng. Des., 1987, 105, 97–111. Gurson, A. L. Continuum theory of ductile rupture by void nucleation and growth. Part-1: yield criteria and flow rules for porous ductile media. Trans. ASME, J. Eng. Mater. Technol., 1977, 99, 2–15. Needleman, A. and Tvergaard, V. An analysis of ductile rupture in notched bars. J. Mech. Phys. Solids, 1984, 32, 461–490. Tvergaard, V. and Needleman, A. Analysis of cup-cone fracture in a round tensile bar. Acta Metall., 1984, 32(1), 157–169. Tvergaard, V. Material failure by void growth to coalescence. Adv. Appl. Mech., 1990, 27, 83–151. Seidenfuss, M. Untersuchungen zur Beschreibung des Versagensverhaltens mit Hilfe von Schädigungsmodellen am Beispiel des Werkstoffes 20MnMoNi5-5. Doctoral Dissertation, Fakultät Energietechnik der Universität Stuttgart, Germany, 1992. JMES1416 © IMechE 2009


1523

22 Roos, E., Seidenfuss, M., Krämer, D., Krolop, S., Eisele, U., and Hindenlang, U. Application and evaluation of different numerical methods for determining crack resistance curves. Nucl. Eng. Des., 1991, 130, 297–308. 23 Brocks, W., Klingbeit, D., Kunecke, G., and Sun, D. Z. Application of the Gurson model to ductile tearing resistance. In Constraint effects in fracture: theory and applications: second volume, ASTM STP 1244 (Eds M. Kirk and A. Bakker), 1995, pp. 232–252 (American Society for Testing and Materials, West Conshohoken). 24 Brocks, W., Sun, D.-Z., and Hönig, A. Verification of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials. Int. J. Plast., 1995, 11(8), 971–989. 25 Xia, L. and Shih, C. F. Ductile crack growth. I: a numerical study using computational cells with microstructurally based length scales. J. Mech. Phys. Solids, 1995, 43, 223–259. 26 Ruggieri, C., Panontin, T. L., and Dodds, R. H. Numerical modeling of ductile crack growth in 3-D using computational cell elements. Int. J. Fract., 1996, 82, 67–95. 27 Eisele, U., Seidenfuss, M., and Pitard-Bouet, J.-M. Comparison between fracture mechanics and local approach models for the analysis of shallow cracks. J. Phys. IV, 1996, 6, 75–89. 28 Brocks, W., Sun, D.-Z., and Hönig, A. Verification of micromechanical models for ductile fracture by cell model calculations. Comput. Mater. Sci., 1996, 7(1–2), 235–241. 29 Steglich, D. and Brocks, W. Micromechanical modelling of the behaviour of ductile materials including particles. Comput. Mater. Sci., 1997, 9(1–2), 7–17. 30 Steglich, D., Siegmund, T., and Brocks, W. Micromechanical modeling of damage due to particle cracking in reinforced metals. Comput. Mater. Sci., 1999, 16(1–4), 404–413. 31 Pitard-Bouet, J.-M., Seidenfuss, M., Bethmont, M., and Kussmaul, K. Experimental investigations on the ‘shallow crack effect’ on the 10 MnMoNi 5 5 steel and computational analysis in the upper shelf by means of the global and local approaches. Nucl. Eng. Des., 1999, 190, 171–190. 32 Gullerud, A. S., Gao, X., Dodds Jr, R. H., and Haj-Ali, R. Simulation of ductile crack growth using computational cells: numerical aspects. Eng. Fract. Mech., 2000, 66, 65–92. 33 Besson, J., Steglich, D., and Brocks W. Modeling of crack growth in round bars and plane strain specimens. Int. J. Solids Struct., 2001, 38(46–47), 8259–8284. 34 Nègre, P., Steglich, D., Brocks,W., and Koçak, M. Numerical simulation of crack extension in aluminium welds. Comput. Mater. Sci., 2003, 28(3–4), 723–731.

35 Besson, J., Steglich, D., and Brocks W. Modeling of plane strain ductile rupture. Int. J. Plast., 2003, 19(10), 1517– 1541. 36 Chabanet, O., Steglich, D., Besson, J., Heitmann, V., Hellmann, D., and Brocks W. Predicting crack growth resistance of aluminium sheets. Comput. Mater. Sci., 2003, 26, 1–12. 37 Poussard, C., Sainte Catherine, C., Forget, P., and Marini, B. On the identification of critical damage mechanisms parameters to predict the behavior of charpy specimens on the upper shelf. J. ASTM Int., 2004, 1, 1–18. 38 Nègre, P., Steglich, D., and Brocks, W. Crack extension in aluminium welds: a numerical approach using the Gurson–Tvergaard–Needleman model. Eng. Fract. Mech., 2004, 71(16–17), 2365–2383. 39 Pavankumar, T. V., Samal, M. K., Chattopadhyay, J, Dutta, B. K., Kushwaha, H. S., Roos, E., and Seidenfuss, M. Transferability of fracture parameters from specimens to component level. Int. J. Press. Vessels Pip., 2005, 82(5), 386–399. 40 Steglich, D., Pirondi, A., Bonora, N., and Brocks, W. Micromechanical modelling of cyclic plasticity incorporating damage. Int. J. Solids Struct., 2005, 42(2), 337–351. 41 Pirondi, A., Bonora, N., Steglich, D., Brocks, W., and Hellmann, D. Simulation of failure under cyclic plastic loading by damage models. Int. J. Plast., 2006, 22(11), 2146–2170. 42 Seebich, H. P. Mikromechanisch basierte schädigungsmodelle zur Beschreibung des Versagensablaufs ferritischer Bauteile. Doctoral Dissertation, Institut für Materialprüfung Werkstoffkunde und Festigkeitslehre, Universität Stuttgart, Germany, 2007. 43 Kussmaul, K., Eisele, U., and Seidenfuss, M. On the applicability of local approach models for the determination of the failure behaviour of steels with different toughness. In ASME PVP. Fatigue and fracture mechanics in pressure vessel and piping (Eds Mehta et al.), vol. 304, book no. H00967, 1995. 44 Sumpter, J. D. G. and Turner, C. E. Method for laboratory determination of JIC . ASTM STP 601, Philadelphia, PA, 1976. 45 Anderson, T. L. Fracture mechanics: fundamentals and applications, 2nd edition, 1995 (CRC Press, Boca Raton, Florida). 46 Chu, C. C. and Needleman, A. Void nucleation effects in biaxially streched sheets. Trans. ASME J. Eng. Mater. Technol., 1980, 102, 249–256. 47 Aravas, N. On the numerical integration of a class of pressure-dependent plasticity models. Int. J. Numer. Methods Eng., 1987, 24, 395–1416. 48 Bathe, K. J. Finite element procedures, 1995 (PrenticeHall, Englewood Cliffs, NJ).

