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ECF23 - European Conference on Fracture

Volume 42 • 2022

ISSN 2452-3216

ELSEYlER

Stliuctu�al

2 3 E uro p ean Conference on Fracture

Guest Editors: P edro Moreira , L uis Filipe Galrao dos R eis

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Procedia Structural Integrity 42 (2022) 1–2

23 European Conference on Fracture - ECF23 Editorial Pedro Moreira a, *, Luís Reis b a INEGI Institute of Science and Innovation in Mechanical and Industrial Engineerin, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal b Instituto Superior Técnico, University of Lisbon, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract The International European Conference on Fracture (ECF23) was organized from 27 June to 1 July 2022. ECF started out in 1973 in France, as the 1st European Colloquium on Fracture, and rapidly become a reference event focused on Fracture Mechanics and Structural Integrity related topics. It has been traveling around Europe, and the previous event was held in Belgrade in 2018. It has the mission to promote research focused on topics related to all fracture phenomena, structural integrity and monitoring, contributing for the improvement of the reliability of systems, through focused R&D in applied mechanics. This conference was the twentieth edition of a globally recognized event in the field of structural mechanics. The importance of the event required us to apply for its organization in 2014 in its edition in Trondheim, Norway. Since then, it has passed through Catania and Belgrade, now having Funchal as its destination in 2022, after the postponement due to the pandemic situation in 2020. A Summer School was also included in this organization from the 25th to the 26th of June, held at the premises of the Rectory of the University of Madeira. This first event was mainly dedicated to young students and researchers, who can enjoy the experience of renowned speakers and exchange experiences with each other. Despite all difficulties related to the pandemic, the scientific community response was outstanding: a total of 505 delegates from 40 countries attended the event. A total of 616 abstracts were submitted, and 457 were accepted for oral presentations, divided in 96 parallel sessions, and 71 for Poster presentations. Apart from the publication of the proceedings in Procedia Structural Integrity, special issues in Theoretical and Applied Fracture Mechanics, Engineering Fracture Mechanics, International Journal of Fatigue and

* Corresponding author. Tel.: +351 938758186 E-mail address: pmoreira@inegi.up.pt

2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23

Pedro Moreira et al. / Procedia Structural Integrity 42 (2022) 1–2 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

Engineering Failure Analysis were also offered. In the two days prior to the ECF23 conference, a Summer School on Fracture Mechanics and Structural Integrity took place, and four well known Lecturers attracted the presence of 90 delegates. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of 23 European Conference on Fracture - ECF23 Keywords: Type your keywords here, separated by semicolons ;

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Procedia Structural Integrity 42 (2022) 1291–1298 Structural Integrity Procedia 00 (2019) 000–000 Structural Integrity Procedia 00 (2019) 000–000

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© 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the 23 European Conference on Fracture – ECF23 Abstract Crack growth simulations by way of the traditional Finite Element Method claim progressive remeshing to fit the geometry of the fracture, severely increasing the computational e ff ort. Methods such as the eXtended Finite Element Method (XFEM) allow to overcome this limitation by means of nodal shape functions multiplied by Heaviside step function to enrich finite element nodes. Through the medium of a discontinuous field, the entire geometry of the discontinuity can be modelled regardless of the mesh, avoiding remeshing. In this paper two shell-type XFEM elements (a three-node triangular element and a four-node quadrangular element) to evaluate crack propagation in brittle materials are presented. These elements have been implemented into the widespread opensource framework OpenSees to evaluate crack propagation into a plane shell subjected to monotonically increasing loads. Moreover, in the perspective of fracture propagation simulations, the problem of managing multiple cracks without remeshing or operating subdivisions on the integration domain has been investigated and a four-node quadrangular finite element for the computational analysis of double crossed discontinuities by the means of equivalent polynomials is presented in this paper. Equivalent polynomials allow to overcome inaccuracies on the results when performing standard numerical integration (e.g. Gauss Legendre quadrature rule) over the entire domain of XFEM elements, without the need of defining integration subdomains. The presented work and the computational strategy behind it may be extremely useful not only in the field of fracture mechanics, but also to solve complex geometry problems or material discontinuities. 2020 The Authors. Published by Elsevier B.V. is is an open access article under the CC BY-NC-ND license (http: // creativec mmons.org / licenses / by-nc-nd / 4.0 / ) er-review under responsibility of 23 European Conference on F acture – ECF23 . Keywords: Extended Finite Element Method ; Discontinuities ; Equivalent Polynomials 23 European Conference on Fracture – ECF23 2D finite elements for the computational analysis of crack propagation in brittle materials and the handling of double discontinuities Sebastiano Fichera a, ∗ , Bruno Biondi b , Giulio Ventura a a Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy b S.T.S. srl – Software Tecnico Scientifico, Via Tre Torri 11, Sant’Agata Li Battiati, 95030, Catania, Italy Abstract Crack growth simulations by way of the traditional Finite Element Method claim progressive remeshing to fit the geometry of the fracture, severely increasing the computational e ff ort. Methods such as the eXtended Finite Element Method (XFEM) allow to overcome this limitation by means of nodal shape functions multiplied by Heaviside step function to enrich finite element nodes. Through the medium of a discontinuous field, the entire geometry of the discontinuity can be modelled regardless of the mesh, avoiding remeshing. In this paper two shell-type XFEM elements (a three-node triangular element and a four-node quadrangular element) to evaluate crack propagation in brittle materials are presented. These elements have been implemented into the widespread opensource framework OpenSees to evaluate crack propagation into a plane shell subjected to monotonically increasing loads. Moreover, in the perspective of fracture propagation simulations, the problem of managing multiple cracks without remeshing or operating subdivisions on the integration domain has been investigated and a four-node quadrangular finite element for the computational analysis of double crossed discontinuities by the means of equivalent polynomials is presented in this paper. Equivalent polynomials allow to overcome inaccuracies on the results when performing standard numerical integration (e.g. Gauss Legendre quadrature rule) over the entire domain of XFEM elements, without the need of defining integration subdomains. The presented work and the computational strategy behind it may be extremely useful not only in the field of fracture mechanics, but also to solve complex geometry problems or material discontinuities. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of 23 European Conference on Fracture – ECF23 . Keywords: Extended Finite Element Method ; Discontinuities ; Equivalent Polynomials 23 European Conference on Fracture – ECF23 2D finite elements for the computational analysis of crack propagation in brittle materials and the handling of double discontinuities Sebastiano Fichera a, ∗ , Bruno Biondi b , Giulio Ventura a a Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy b S.T.S. srl – Software Tecnico Scientifico, Via Tre Torri 11, Sant’Agata Li Battiati, 95030, Catania, Italy

∗ Corresponding author. E-mail address: sebastiano.fichera@polito.it ∗ Corresponding author. E-mail address: sebastiano.fichera@polito.it

2452-3216 © 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the 23 European Conference on Fracture – ECF23 10.1016/j.prostr.2022.12.164 2210-7843 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of 23 European Conference on Fracture – ECF23 . 2210-7843 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of 23 European Conference on Fracture – ECF23 .

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1. Introduction

The eXtended Finite Element Method (XFEM) is a versatile approach for the analysis of problems characterised by discontinuities and singularities such as localised deformations, material discontinuities or cracks. It was first proposed by Belytschko et al. (1999), and later improved by Moe¨s et al. (1999). In XFEM formulation the discontinuous displacement field is modelled along the crack surface through additional nodal degrees of freedom and enrichment shape functions. XFEM allows to define the finite elements mesh independently to the discontinuity position and does not require any mesh refinement close to the discontinuities: that is a major advantage with respect to the standard Finite Element Method. Moreover, when using XFEM in crack propagation problems, remeshing during the analysis to track the evolution of the crack is not needed, dramatically decreasing the computational e ff ort. Since enrichment shape functions are discontinuous and non-di ff erentiable, numerical problems arise if a quadra ture rule (e.g. Gauss-Legendre) is used to evaluate the sti ff ness matrix of elements containing discontinuities. This problem can be over-come by partitioning these elements into sub-elements, so that the integrands are continuous and di ff erentiable into each subdomain. Alternatively, a solution by means of equivalent polynomials that does not require partitioning of the integration domain has been proposed by Ventura (2006) and by Ventura et al. (2015). In this paper, the implementation into OpenSees of a three-node triangular and a four-node quadrangular shell XFEM elements is presented. These elements are an enhancement of the finite elements with drilling degrees of freedom recently presented by the authors (Fichera et al. (2019)). The proposed elements are able to model crack propagation in brittle materials and have been used to perform static incremental analysis on plane shells.

2. XFEM formulation overview

The Extended Finite Element Method (XFEM) is a numerical method, based on the Finite Element Method (FEM), that is especially designed for handling discontinuities (Belytschko et al. (1999), Moe¨s et al. (1999)). In standard FEM, the displacement field of a single element of a domain Ω can be expressed as:

n i = 1

T ( x ) u

N i ( x ) u i = N

u ( x ) =

(1)

where n is the number of nodes of the element, N i ( x ) are the element shape functions and u i are the nodal displacement components. Eq. (1) cannot describe the behaviour of the displacement field when discontinuities or singularities exist within the element. To overcome this limit, one can enrich the interpolation on Eq. (1) by means of an enrichment function Ψ ( x ) and a certain number of additional degrees of freedom a i :

n i = 1

u ( x ) =

N i ( x ) u i +

N i ( x ) Ψ ( x ) a i

(2)

The nature of the enrichment function Ψ ( x ) depends on the nature of the discontinuity that has to be described. In the case of a strong discontinuity in the displacement field (discontinuities in the solution variable of a problem, e.g. a crack), the most appropriate enrichment function is the Heaviside step function:

Ψ ( x ) = H ( φ ( x ))

(3)

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H ( φ ( x )) = −

1 φ ( x ) < 0 1 φ ( x ) > 0

(4)

where φ ( x ) is the signed distance of the discontinuity from the evaluation point. XFEM formulation allows to adequately represent discontinuities or singularities in a suitable way and with strong performance in case of a pronounced non-polynomial behaviour of the solution. It has to be noted that if the element is crossed by a discontinuity, the standard Gauss quadrature rule cannot be used, due to the non-polynomial nature of the enrichment function. This problem can be overcome by splitting the integration domain Ω in two parts Ω − and Ω + along the discontinuity, so that the enrichment function is continuous and di ff erentiable into each integration subdomain: Ω Ψ ( x ) P n ( x ) d Ω = ⇒ Ω − Ψ ( x ) P n ( x ) d Ω + Ω + Ψ ( x ) P n ( x ) d Ω (5) ( P n ( x ) is a generic n-degree polynomial function, e.g. a term of the element sti ff ness matrix). Partitioning into sub domains introduces a sort of ‘mesh’ condition in the elegance of the XFEM formulation. A technique to eliminate the requirement of sub-cells generation without introducing any approximation in the quadrature by means of equivalent polynomials has been proposed by Ventura (2006) and Ventura et al. (2015). It has been demonstrated that an equiv alent polynomial function exists such that its integral gives the exact values of the discontinuous / non-di ff erentiable function integrated on sub-cells. The polynomial is defined in the entire element domain, so that it can be easily integrated by Gauss quadrature, and no quadrature sub-domains have to be defined. Eq. (5) thus becomes: Ω Ψ ( x ) P n ( x ) d Ω = ⇒ Ω − Ψ ( x ) P n ( x ) d Ω + Ω + Ψ ( x ) P n ( x ) d Ω = Ω ˜ Ψ ( x ) P n ( x ) d Ω (6) where ˜ Ψ ( x ) is the equivalent polynomial function. In the case of strong discontinuities, Heaviside function is usually used as enrichment function. Eq. (6) thus be comes: Ω − H ( x ) P n ( x ) d Ω + Ω + H ( x ) P n ( x ) d Ω = Ω ˜ H ( x ) P n ( x ) d Ω (7) where ˜ H ( x ) is the equivalent polynomial function for this particular case. Equivalent polynomials avoid the quadrature domain splitting at the cost of doubling the polynomial degree of the integrand function.

3. Handling multiple discontinuities using equivalent polynomials

Analysing a body containing multiple fractures is not an uncommon problem in fracture mechanics. Such problems can be still addressed by means of XFEM formulation, but quadrature domain splitting becomes more burdensome. Moreover, the equivalent polynomials law defined in Eq. (7) is able to take into account a single discontinuity for each integration domain Ω . To overcome this problem, a new equivalent polynomials formulation to manage double discontinuities has been recently proposed by the authors. Let us examine a body Ω and let us assume it is split in four parts by the discontinuity lines q and r , as shown in Fig. 1. Let us define Ω A the partition obtained when the normal of both discontinuities have a positive value of their b

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component, therefore both normal vectors point inwards Ω A . Starting from Ω A , the remaining partitions ( Ω B , Ω C and Ω D ) are defined counterclockwise by convention.

Fig. 1: 2D domain Ω crossed by two discontinuity lines: q and r .

Let us assume a n-degree polynomial P n ( x ) to be integrated over the subdomains Ω A , Ω B , Ω C or Ω D obtained partitioning a parallelogram with two lines q and r , so that: I i = Ω i P n ( x ) d Ω i (8) where i = { A , B , C , D } To solve the integral in Eq. (8) for each value of i it is required to define each domain of integration Ω i . This is not always an easy task because integration domains may result in rather not trivial polygonal shapes. Said problem has been solved by the authors finding an equivalent polynomial ˜ H i that allows, for each partition Ω i , to perform the integration over the entire domain Ω of the element with a traditional quadrature rule without partitioning Ω into subdomains. So, Eq. (8) becomes: I i = Ω i P n ( x ) d Ω i = Ω ˜ H i ( x ) P n ( x ) d Ω (9)

The equivalent polynomial ˜ H i ( x ) depends on the equations of the two discontinuity lines q and r and has the same degree of P n ( x ).

4. XFEM shell-type elements implementation into OpenSees

In this paper, two XFEM shell-type elements to solve fracture mechanics problems using OpenSees have been presented. The versatility of OpenSees framework made the implementation process fairly straightforward. First of all, a new ‘node’ Class (alternative to the original one in the code) had to be defined to dynamically change the number of degrees of freedom of each node during the analysis. This is a crucial part in order to comply the solution for the displacement field in Eq. (2). The introduction of a new Class of nodes made possible the subsequent implementation of two new XFEM plane shell-type elements: a three-node triangular element and a four-node quadrangular element. In this first implemen-

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tation, the in-plane behaviour of the proposed elements is indefinite linear elastic in the case of compression and elastic-fragile in the case of tensile stress. The out-of-plane behaviour is always linear elastic. A pre-existing fracture can be assigned to the elements. It can be defined by means of a crossing point and the normal to the discontinuity itself. The elements can be also defined as initially undamaged; in this case they may crack during the incremental loading process. When the principal tensile stress overcomes the material tensile resistance, a fracture will arise in the element. The fracture position is defined by means of the coordinates of the element point where this limit is exceeded. The proposed elements can also be used to study the formation and the consequent propagation of fractures in brittle materials. Elements are modelled so that the displacement field is described by the interpolation law in Eq. (1) if they are undamaged and by the one in Eq. (2) if cracking arises. Thus, the number of degrees of freedom of the nodes will increase as the analysis progresses, since the enriched degrees of freedom a i will be added to the standard degrees of freedom u i . At the current state of the implementation, each element of the mesh can handle just one discontinuity. Elements able to handle multiple discontinuity has been analysed and will be introduced in the upcoming developments. Also, a non-linear compressive behaviour for the elements will be included.

5. Numerical applications

The proposed XFEM elements have been used for the analysis of plane shells containing discontinuities.

5.1. Damaged cantilever beam

In the first example the cantilever beam shown in Fig. (2) is analysed. The beam contains a pre-existing fracture and it is subject to two constant forces applied to its free end: axial force F x and cross-sectional force F y . Geometrical

Fig. 2: Damaged cantilever beam subject to constant forces F x and F y . Fracture in the structural element is highlighted in red.

and mechanical properties of the beam are shown in Tab. (1, 2).

Table 1: Geometrical properties of the cantilever beam.

Geometrical properties

L = 50 [ cm ]

b = 2 . 5 [ cm ]

h = 20 [ cm ]

The beam has been modelled both using standard FEM shell-type elements and the proposed XFEM shell-type quadrangular elements. The discretisation mesh for the structural element is shown in Fig. (3). In the case of standard FEM (Fig. (3b)), the mesh had to be refined near the fracture to follow the geometry of the discontinuity, so triangular finite elements and distorted quadrangular finite elements had to be used.

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Fichera et al. / Structural Integrity Procedia 00 (2019) 000–000 Table 2: Mechanical properties of the cantilever beam.

Mechanical properties

E = 2796 [ kN / cm 2 ]

ν = 0 . 2

x = 20 [ kN ]

F y = 20 [ kN ]

(a) XFEM discretisation

(b) Standard FEM discretisation.

Fig. 3: Mesh discretisation for the cantilever beam.

The deformed configurations obtained from the analysis are shown in Fig. (4). It is clear that both the standard FEM model and the XFEM one yield the exact same results in terms of displacement. In particular, the deflection of point A is the same in both models, as shown in Tab. (3).

(a) XFEM discretisation

(b) Standard FEM discretisation.

Fig. 4: Results of the analysis in terms of deformed configurations.

Table 3: Deflection of point A.

Standard FEM Model

Proposed XFEM Model

u x = 0 . 00037 [ cm ] u y = 0 . 00177 [ cm ]

These results validate the proposed elements.

5.2. Undamaged cantilever beam – progressive cracking

In the second example an undamaged cantilever beam has been analysed. The behaviour of the beam is assumed to be linear elastic in case of tensile stress and elastic-fragile in the case of compression. The material tensile resistance is defined as f t = 1 . 5 [ kN / cm 2 ]. In this case, an analysis with a monotonically increasing cross-sectional load F y has been performed. The results of this analysis, until the collapse is reached, are shown in Fig. (5). It has to be noted that the XFEM model does not have the aim of following the exact crack propagation path, but to determine the displacement field of a structural element that is subject to progressive cracking. Thus, the fracture scheme in Fig. (5) is just indicative.

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Fig. 5: Cantilever beam subject to a monotonically increasing cross-sectional load. Progressive cracking arises as the load increases with the time-step until failure is reached.

5.3. Undamaged hinged beam – progressive cracking

Finally, the undamaged hinged beam shown in Fig. (6) has been studied. As in the previous example, the beam is

Fig. 6: Undamaged hinged beam subject to a monotonically increasing cross-sectional load F y .

subject to a monotonically increasing force applied in the centreline. The results of the analysis are shown in Fig. (7), with the evolution of the XFEM elements under progressive cracking.

Fig. 7: Progressive cracking arises as the load increases with the time-step until failure is reached.

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6. Conclusions

In this paper a method to integrate over an entire integration domain containing two discontinuities without splitting it into multiple subdomains has been presented. This method is an enhancement of the solution by means of equivalent polynomials proposed by Ventura (2006) and Ventura et al. (2015) and it can be a powerful tool in the context of XFEM, when addressing multiple fracture problems. Also two shell-type XFEM elements have been presented: a three-node triangular element and a four-node quad rangular element. These elements have been implemented into OpenSees in order to evaluate crack propagation in brittle materials. The proposed XFEM elements are an enhancement of the finite elements with drilling degrees of freedom recently presented by the authors Fichera et al. (2019). The proposed XFEM elements have been used to evaluate crack propagation into a plane shell subject to monotonically increasing loads. The results presented in the numerical applications validate the proposed formulation and enable the use of OpenSees framework to address frac ture mechanics problems. Future developments for the proposed elements will include the handling of multiple discontinuities by a single XFEM element, as well as a non-linear compressive behaviour.

Acknowledgements

Reaserch support by S.T.S. srl - Software Tecnico Scientifico, Italy is gratefully acknowledged. Research support by Politecnico di Torino, Italy is gratefully acknowledged.

References

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1. Introduction New advanced alloys and technologies are developing as the response to the required improvement of components performances (Gorsse et al., 2018; Mitrica et al., 2021; Borsato et al., 2016; Ferro et al., 2020). When facing the new

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challenges linked to sustainability, such as the improvement of engines thermal efficiency or mitigation of the supply risk and economical importance of critical raw materials (CRMs) identified by the European Commission, Nickel based alloys and their joining techniques are certainly worthy of investigations. In fact, in recent works (Ferro and Bonollo 2019; Ferro et al., 2020) dealing with design for recycling and substitution in a critical raw materials perspective, it was shown how nickel-based alloys are excellent candidates for application undergoing severe environmental conditions. Generally speaking, nickel superalloys, such as Inconel 625 (IN625), present at high temperature an exceptional combination of high mechanical strength and excellent corrosion resistance (Reed, 2006). A recent review of the status of technology in design and manufacture of wrought polycrystalline Ni-base superalloys for critical engineering applications can be found in (Hardy et al., 2020). Ni-base superalloys currently represent more than 50% of the weight of advanced aeronautical turbines and are also important for applications in energy production, naval propulsion, extraction of oil and gas, spacecraft, nuclear reactors, heat exchangers (Pollock and Sammy, 2006) and hydrogen technology. Even though Ni-based superalloys cost from 3 to 5 times the iron-based ones, their use is expanding especially in gas turbine components to produce energy because higher temperatures of the thermal cycle guarantee greater efficiency and reduction of polluting emissions. The demand of Ni-based superalloys is expected to expand also for the energy production through conventional steam turbine plants for achieving super critical conditions with a predicted increase of efficiency to 60% and reduction of CO 2 to about 0.7 ton/kWatth while actual sub-critical power plants have an efficiency of 35% and produce 1.2 ton/kWatth of CO 2 . However, higher operating temperatures involve more severe degradation of the mechanical components due to several factors: i) microstructure evolution including formation of undesired phases, coalescence of  ’ precipitates, degeneration of car bides due to fatigue and creep exposure; ii) the formation of cracks. In the first case, heat treatments or Hot Isostatic Pressing (HIP) are used to restore the original microstructure as much as possible; in the second case, joining techniques are used, which could give rise to defects or imperfections, such as discontinuities, local segregations, inadequate penetration, resulting in poor joint resistance (David et al., 1997). Furthermore, cracks may form also in the production process of components made of Ni-based superalloys for energy applications such as flanges, valve bodies, and headers (Viswanathan et al., 2005; deBarbadillo et al., 2016). According to Andersson (2018), Ni-alloy welding issues can be broken down into two broad classifications: geometrical and metallurgical. The geometrical issues include the shape of the weld pool (tear drop tending to be more crack prone), location of the weld (concave tending to be more crack prone), as well as residual stress (RS) and weld defects [13]. The metallurgical issues mainly include strain age cracking (SAC) (Berry and Hughes, 1969; Franklin and Savage, 1974; Thompson et al., 1968) and hot cracking (DuPont et al., 2009; Cieslak et al., 1988; Cieslak et al., 1990). Welding of Ni-base superalloys has gained increasing importance in the repair of mechanical components for the first stages of turbines. The microstructure in the welded metal (WM) of the joint forming during solidification depends on two phenomena: the formation of dendrites and the partition of the solute with consequent possible formation of carbides, borides and different intermetallic phases. Some of these low-melting compounds can trigger micro-cracks during PWHT (Ojo et al., 2004; Henderson et al., 2013). Furthermore, during the solidification localized stresses can develop in the welded area causing mechanical failures (Jensen, 2002). Prediction of such localized stresses is not an easy task but extremely important to correctly design a welding process. In this regard, finite element method (FEM) is certainly a powerful tool provided that the main phenomena inducing residual stresses are considered and correctly modelled. However, for design reasons, the models should not be time consuming. When residual stresses (RSs) are the goal of the simulation, computational welding mechanics (CWM) could be a reliable numerical strategy (Ferro et al., 2005). As a matter of fact, the fusion zone (FZ) is modelled by using proper power density distribution functions reproducing the volume and shape of the molten pool without solving the complex equations of fluid dynamics (Ferro et al., 2010). Moreover, solid-state phase transformations effects on RS can be easily considered using alloy metallurgical constitutive equations (Ferro, 2012; Ferro and Petrone, 2009). Unfortunately, to the best of the authors knowledge, very few works dealing with welding simulation of IN165 can be found in literature. Thejasree et al. (2021) proposed a thermal model of the laser beam welding of IN625 basing on CWM and Sysweld® numerical code. Even if thermal results were found in good agreement with experiments, no mechanical model was implemented for RS prediction. A numerical model of IN625 TIG welding was also developed by Siwek et al. (2013), who implemented an adaptive mesh approach to speed up the simulation process. However, again authors gave up assessing RSs. Finally,

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Solomon et al. (2018) carried out a numerical model of IN625 welding process proposing regression equations for the temperature and stress; however, it was not specified how the model was validated. It must point out that the above mentioned papers didn’t focus on RSs. To fill this gap, gas tungsten arc welding (GTAW) simulation of IN625 was caried out with the main objective to predict residual stresses. The model, validated through residual stresses X-Ray diffraction (XRD) measurements, is thought to be a powerful tool in welding process design of such kind of superalloy, even used for repair operations. 2. Materials, geometry, and method 2.1. Experimental investigation The composition of the parent (IN625) and filler metal (ERNiCrMo-3) used in the experiments are summarized in table 1.

Table 1. Chemical composition of the parent and filler metals (wt%). Ni C Mn S Cu

Others

PM: Inconel 625 FM: ErNiCrMo-3

Min 58 Min 64

Max 0.1 Max 0.1

Max 0.5 Max 0.5

Max 0.015 Max 0.015

Max 0.5 Max 0.50

20-23

Max 0.015 Max 0.015

Fe 5, Al 0.40, Mo 8-10, Ti 0.1 Fe 1.0, Al 0.40, Nb 3.6 4.5, Mo 0.015, Ti 0.40

22.0 23.0

Plates 100 mm x 60 mm x 5 mm (Fig. 1) were butt welded using a multi-pass arc welding process (GTAW) whose parameters are collected in table 2.

Fig. 1. Cross section of the butt-welded joint with dimensions (mm) and V-groove shape.

Table 2. Welding process parameter. Pass No Current (I) (Amps)

Voltage (V) (Volts)

Root gap (mm)

Argon Gas Flow rate (LPM)

Time for each pass (Sec)

Pass-1 Pass-2 Pass-3

145 145 145

15 15 15

36 31 27

An inter-pass temperature of about 200 °C was maintained between the passes to avoid hot cracking. Residual stresses were measured by means of XRD technique. Fig. 2 shows a schematic of the goniometer used for detecting the Bragg’s angle used in the Bragg’ s law (Eq. (1)): n λ = 2dsin θ (1) where n refers to the order of reflection beam, d is the interplanar lattice spacing,  is the wavelength of the incident wave and  is the scattering angle. The residual stress was measured on the top surface of the welded joint in the transverse direction. More in detail, their calculation was carried out via Eq. (2): (2) σ φ = m d 0 E 1 + ν æ èç ö ÷ø

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where E is the Young’s modulus,  is the Poisson’s coefficient, d 0 (1.0771016 Å) is the is the stress-free lattice spacing, m is the gradient of the d vs. sin2ψ curve and ψ is the angle between the normal of the sample and the normal of the diffracting plane (bisecting the incident and diffracted beams).

Fig. 2. Goniometer head with a detector (a) and Bruker D8-Discover TM diffractometer used for the measurements (tube, Mn K  )

Table 3 summarizes the main parameters used for XRD measurements.

Table 3. Parameters used for XRD measurements. Parameter Value Type of Tube Mn-K-Alpha Voltage [KV] 20 Current in tube [mA] 4 Bragg’s Angle (°) 155 Wavelength [Å] 2.103

2.2. Numerical Model A thermo-mechanical numerical model was developed using Sysweld® numerical code with the aim at estimating RSs. The temperature history at each node is obtained by solving the heat flow balance equation:

æ è ç

ö ø ÷ + q(x, y, z, t ) = ρ c

¶ R y ¶ y

¶ R x ¶ x

¶ R z ¶ z

¶ T(x, y, z, t ) ¶ t

−

(3)

where the rate of heat flow per unit area is denoted by R x , R y and R z ; the current temperature is T(x,y,z,t), the rate of internal heat generation is q(x,y,z,t), c is the specific heat,  is the alloy density, and t is the time. Eq. (3) can be revised by bringing Fourier‘s heat flow as

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263

ì í ï ï ï î ï ï ï

¶ T ¶ x ¶ T ¶ y ¶ T ¶ z

R x = − k x

R x = − k y

(4)

R x = − k z

where k x , k y and k y are the alloy thermal conductivity in the three directions x, y and z, respectively. Due to temperature-dependent material properties (k and c), the associated material behavior was nonlinear. Now, by substituting Eqs. (4) into Eq. (3), the differential governing heat conduction equation can be rewritten as:

é ë ê

ù û ú +

¶ T ¶ t

¶ ¶ x

¶ T ¶ x

¶ ¶ y

¶ T ¶ y

¶ ¶ z

¶ T ¶ z

é ë ê

ù û ú +

é ë ê

ù û ú + q

ρ c

k x

k y

k z

(5)

The solution of Eq. (5) requires the definition of initial conditions:

T(x, y, z,0) = T 0 (x, y, z)

(6)

as well as boundaries conditions:

æ èç

ö ø÷ + q s + h c (T − T a ) + h r (T − T r ) = 0

¶ T ¶ x

¶ T ¶ y

¶ T ¶ z

N x + k y

N y + k z

k x

N z

(7)

In Eq. (6), T 0 (x,y,z) is taken equal to the ambient temperature (T a = 20 °C) or eventually the pre-heating temperature. In Eq. (7) N x , N y and N z are the direction cosine of the outward projected normal to the boundary; h c (25 W/m 2 K (Solomon et al., 2018)) and h r are the heat transfer coefficients of convection and radiation, respectively; T is the surface temperature of the model and T r is the temperature of the heat source instigating radiation. The boundary heat flux is designated by q s . The heat transfer coefficient of radiation can be written as:

2 − T r

2 )(T + T

h r = σε F(T

r )

(8)

where  is the Stefan Boltzmann’s constant,  is the emissivity (0.7) and F is the configuration factor. The heat generation due to the heat source moving over the weld line and quantified by the term q in Eq. (5) is described by the power distribution function proposed by Goldak et al. (1984) for arc welding processes:

(9)

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In previous expressions (9), q F and q R represent the frontal and rear power density, respectively; Q W is the welding heat input estimated from the input current (I) and voltage (V) parameters (QW=  VI, with  the thermal efficiency found to be equal to 0.33); f f (= 0.6) and f r (= 1.4) denotes the fractions of heat present in the front and rear parts of the heat source, while a, b, c f and c r are Gaussian parameters of the Goldak’s heat source (1984), as described in Fig. 3 that were chosen in a way that it produces a proper molten weld pool (Chen et al., 2018). All Goldak’s heat source parameters adopted in the FE analyses have been summarized in Table 4.

c f

c r

adiabatic conditions

Fig. 3. Schematic of Goldak’s heat source shape and parameters

Table 4. Goldak’s heat source parameters used in the simulations . Welding pass Q (=VI) [W] a [mm] b [mm]

c f [mm]

c r [mm]

v [mm/sec]

1 2 3

3.5

2.2 1.8 2.5

4.5 4.0 3.0

9.0 9.0 3.0

1.66 1.93 2.22

7 8

2175

In the frame of CWM, the temperature history obtained at each node of the model is used as input load for the mechanical computation (uncoupled thermo-mechanical analysis). In this simplification, the heat generation due to plastic deformation is neglected, being much lower than the heat induced by the arc heat source. The equilibrium equations to be solved are:

ì í ï îï

σ ij + ρ b i = 0 σ ij = σ ji

(10)

where  ij and b i are the stress tensor and body forces, respectively. Moreover, the problem solving requires the formulation of the constitutive equations for the thermal elastic-plastic material as:

ì í ï îï

d σ éë ùû = D ep éë ùû d ε éë ùû − D th éë ùû dt D ep éë ùû = D e éë ùû + D p éë ùû

(11)

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where elastic, plastic and thermal stiffness matrix are denoted by [D e ], [D p ] and [D th ], respectively, while stress, strain and temperature increment are represented by d  , d  and dT, respectively. Among the components of total strain, elastic, plastic, and thermal strains are governed by the isotropic Hooke’s law, Von Misses criterion, and temperature dependent coefficients of thermal expansion, respectively. The differential form of total strain in terms of the above three components are: (12) Finally, the kinematic strain hardening model was chosen while clamping condition was imposed as isostatic and thermo-mechanical properties as a function of temperature were taken from Special Metals® data base (www.specialmetals.com) Fig. 4 shows the mesh used in the numerical model with a detail of the elements group, B1, B2 and B3 that were progressively activated according to the simulated welding pass. It is observed that for symmetry reasons only one half of the joint was modelled using 25340 8-node brick elements. Moreover, the mesh was refined near the weld bead to correctly capture the thermal gradients induced by the heat source. d ε ij = d ε ij e + d ε ij p + d ε ij th

Fig. 4. Mesh of the numerical model with a detail of elements groups (B1, B2 and B3) used to simulate the multi-pass welding process.

3. Results and discussion Metallographic and X-ray radiography analyses of the welded joints revealed sound welds without defects and a fully penetrated bead. This allowed confirming the good welding parameters used in the experiments. The starting point of any CWM-based simulation is the calibration of the heat source parameters (Tab. 4) via experimental analysis. In this work, a cross section of the welded joint was used to compare the FZ dimensions and shape with those obtained by the numerical model. The results of the calibration are shown in Fig. 5 while the corresponding source parameters values were already reported in Table 4. The thermal calibration resulted quite good and sufficient to obtain a good residual stress-strain field prediction. Figure 5 shows the 3D shape of isotherms, as well, according to the analyzed welding pass.

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