Finite element analysis of plastic failure in heat-affected zone of welded aluminium connections

Computers and Structures 88 (2010) 519–528

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Finite element analysis of plastic failure in heat-affected zone of welded aluminium connections Cato Dørum a,b,*, Odd-Geir Lademo a,b,c, Ole Runar Myhr d,b, Torodd Berstad a,b, Odd Sture Hopperstad b,c a

SINTEF Materials and Chemistry, Applied Mechanics and Corrosion, Trondheim, Norway Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation, Norwegian University of Science and Technology, Trondheim, Norway c Department for Structural Engineering, Norwegian University of Science and Technology, Trondheim, Norway d Hydro Aluminium Structures, Raufoss, Norway b

a r t i c l e

i n f o

Article history: Received 2 February 2009 Accepted 7 January 2010 Available online 1 February 2010 Keywords: Aluminium Heat-affected zone Plastic failure Microstructure-based modelling Finite element simulation

a b s t r a c t Finite element analyses of plastic failure in the heat-affected zone of a generic welded aluminium connection are presented. The analyses include process history through multi-scale modelling. The heterogeneous material properties of the heat-affected zone are calculated using welding simulations to obtain the temperature history as input to coupled precipitation, yield strength and work-hardening models. Thermal history-dependent material parameters are mapped as field variables onto the finite element model to account for their spatial variation inside the heat-affected zone. The welded connection is modelled using shell elements, solid elements and cohesive-zone elements. Convergence studies show that very small elements are needed (much less than the plate thickness) to resolve the large strain gradients within the HAZ and to obtain converged solutions. Non-local regularization is vital in shell element analyses to obtain accurate estimates of the ductility of the welded connection. Two methods for estimating the ductility in the welded aluminium connection with coarser meshes are proposed. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction The manufacturing of welded automotive components from age-hardening aluminium alloys involves a series of thermal and mechanical operations. These alloys have a strong memory of the past process steps due to interactions between different types of particles that form at various temperatures [1,2]. In particular, the different heat treatment and welding operations that are used towards the end of the process chain have a large influence on the resulting structural performance. The selection and optimization of alloy composition, temper and process conditions cannot readily be based on experimental testing, since the number of possible combinations is too large. As a result, numerical design tools are desired that take into account the process history and its influence on the mechanical properties of the material in the vicinity of the weld. Realistic prediction of the mechanical performance of welded components and structures calls for the use of a rather sophisticated numerical simulation technique. Recently, Myhr et al. [3] presented a multi-scale modelling approach where precipitate evolution during heat treatment and welding of Al–Mg–Si alloys is predicted through finite element (FE) based welding simulations. * Corresponding author. Address: SINTEF Materials and Chemistry, Applied Mechanics and Corrosion, Trondheim, Norway. E-mail address: [email protected] (C. Dørum). 0045-7949/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2010.01.003

By using a microstructure based constitutive model, this enables realistic predictions of the structural performance of aluminium passing through complex age hardening, welding and Post Weld Heat Treatment (PWHT) processes. For age-hardening aluminium alloys, it is well known that welding leads to heat-affected zones in the vicinity of the weld that, for some alloys and process conditions, may have a significantly reduced strength compared to the base material. The mechanical properties of the heat-affected zones are likely to limit the structural capacity of aluminium connections when the structure is subjected to loads such as in crash situations. The European code for aluminium alloy structures (Eurocode 9) gives design provisions for welded connections [4]. To allow for the influence of the welding process on the material behaviour, the thickness of the material is reduced in the heat-affected zone. These guidelines are based on extensive research in the technical literature on the heat-affected zones in different alloys and tempers. As an example, the major contribution on welded connections in aluminium alloy structures by Soetens [5] is worth mentioning. Even if design codes give provisions for the dimensioning of welded aluminium connections, a robust, efficient and reliable methodology for prediction of the mechanical behaviour of the heat-affected zones in computeraided design of these components is still required. Nègre et al. [6–8] studied ductile tearing of laser welded aluminium sheets both experimentally and numerically. The

C. Dørum et al. / Computers and Structures 88 (2010) 519–528

cohesive-zone solid elements has been used to obtain a first evaluation of the proposed modelling strategy. 2. Problem description and modelling strategy 2.1. Geometry of specimen The geometry of the specimen is shown in Fig. 1. It is assumed that the 2.0 mm thick specimen has been cut out from a larger plate where a weld has been deposited along the centreline. Hence, an HAZ is placed symmetrically transverse to the welding direction at each side of the specimen. The weld metal is ignored in the present study. The analysis assumes an alloy composition corresponding to AA6060, that welding is performed in the initial peak-aged (T6) condition, and that the specimen is room temperature aged for several months after welding before the final mechanical testing. 2.2. Coupling between models Plastic failure in the HAZ has been investigated by numerical simulations of uniaxial tensile specimens using the modelling approach described in Ref. [3]. Fig. 2 shows the coupling between the different models that are in play, i.e. the weld simulation code WELDSIMTM, the microstructure based NaMo model and the explicit FE code LS-DYNA [16]. The thermal module of WELDSIM™ [17–19] is used to predict the temperature field resulting from the welding. As shown in Fig. 2, the calculated thermal history is input to the microstructure model named NaMo (Nano Structure Module) [3]. This contains a precipitation model [1,2,20,21] that calculates the evolution of the Particle Size Distribution (PSD) with time, and can be used to quantify the characteristics of the precipitate structure. The work-hardening behaviour of the different peak-temperature regions of the HAZ is then calculated by employing the work-hardening model in NaMo described in Refs. [22,23], based on inputs from the precipitation model. By combining the results from the yield strength and the work-hardening models, the complete stress–strain curves in any position of the HAZ can be estimated and then transferred to LS-DYNA for simulations of the resulting mechanical response. 3. Mathematical modelling Predictions of the complete stress–strain curve at room temperature require that the flow stress rf can be calculated as a function of the plastic strain ep, where rf is given as follows:

rf ¼ ry þ Drd :

ð1Þ

Here, ry and Drd are the room temperature yield stress and the net contribution from dislocation hardening, respectively. Both variables can be predicted from the particle size distribution (PSD), by

Weld line 200

12.5

40 65

7 2. Ø1

mechanical behaviour of the various zones in vicinity of the weld was characterized by tensile testing using small specimens, while fracture tests were carried out using compact tension specimens with different positions of the initial crack. The crack extension was modelled using the Gurson–Tvergaard–Needleman (GTN) model for ductile damage and a phenomenological cohesive model. Good agreement between experiments and simulations was obtained for the different cases under investigation. Ødegard and Zhang [9] predicted the performance of welded aluminium nodes for car body applications using the finite element method. More recently Zhang et al. [10] integrated a thermal–mechanical microstructure analysis with a load–deformation mechanical analysis to predict the fracture behaviour of aluminium joints. Wang et al. [11–13] studied the strain localization and fracture in various welded aluminium connections using shell elements combined with an elastic–plastic constitutive model accounting for the anisotropy of the aluminium alloy. The properties of the HAZ were established based on tensile test data from the literature [14] and hardness testing. In Ref. [13], non-local regularization was applied to reduce the mesh size sensitivity of the finite element solutions. Pickett et al. [15] investigated the reduced mechanical properties in the HAZ of welded aluminium connections and proposed a failure criterion suitable for crash simulation of automotive components. They also presented simulation tools and techniques enabling the process history of the material in vicinity of the weld to be analyzed and the reduced material properties in the HAZ to be mapped to the structural model for crash analysis. The approach was validated against experimental results for various welded connections and joints. The principal objective of the current work is to evaluate available solutions methods for finite element (FE) simulations of plastic failure (i.e. necking and strain localization in the necked region) in the heat-affected zone (HAZ) of welded aluminium structures. To this end, a uniaxial tensile specimen including a characteristic HAZ is modelled using solid elements, shell elements and cohesive-zone elements. The material is assumed to be the aluminium alloy 6060 in T6 temper. It is assumed that mesh-converged brick element predictions represent the physically correct solution, while similar predictive quality using shell and cohesive elements is sought due to numerical efficiency. The heterogeneous material properties of the HAZ are calculated using welding simulations to obtain the temperature history as input to combined precipitation, yield stress and work-hardening models. Thermal history-dependent material parameters are mapped as field variables to the finite element model to account for their spatial variation inside the HAZ. Convergence studies are performed using solid elements and shell elements with and without non-local regularization. It is shown that very small elements are needed (much less than the plate thickness) to resolve the large strain gradients within the HAZ and to obtain converged solutions. Non-local regularization is vital in shell element analysis to obtain converged solutions with similar ductility (or elongation at failure) as predicted with solid elements. If not, the predicted ductility is grossly underestimated. Two pragmatic approaches to get reasonable estimates of the ductility in welded aluminium connections with coarser meshes are proposed. In the first approach, the shell element size is linked to the length scale of the dominating failure mechanism, which is assumed to be local necking. In the second, the weakest zone of the HAZ is lumped into a cohesive element which is placed between two rows of elements. When loaded, the nodes of the cohesive element separate according to a traction-separation law and the element fails according to a ductile fracture criterion. With this approach, the local necking and fracture in the HAZ are modelled in an efficient way. Cohesive-zone shell elements are not currently available in the finite element code used in this work, and a combination of solid elements and

R1 5

520

70

25.05

Fig. 1. Geometry of welded aluminium specimen.

25

521

C. Dørum et al. / Computers and Structures 88 (2010) 519–528

WELDSIM Thermal model

In Eq. (2), rss is the solid solution hardening potential of the alloy, which is calculated from the solid solution concentrations assuming that the contribution from each alloying element is additive. Since the amount of elements tied up in particles are given from the PSD, the concentration of elements in solid solution can be estimated from a simple mass balance as described in Refs. [1,2,20].

NaMo Precipitation model

Yield strength model Work hardening model

Mechanical Model (LS-DYNA)

Fig. 2. Coupling between the different models used in the present study.

3.2. Implementation of the work-hardening model invoking the yield stress and the work-hardening model, as schematically illustrated in Fig. 3. The theoretical basis for the precipitation, yield stress and work-hardening model has been given in Refs. [1,2,20–23], and no detailed outline of the underlying theory will be given in the present article. The response equations, i.e. the equations connecting the precipitate parameters to ry and Drd, are however, described in the following sections.

The microstructure based work-hardening model proposed in Refs. [22,23] has been implemented as a user-defined material model in LS-DYNA. The work-hardening model predicts the individual evolution of statistically stored and geometrically necessary dislocations, respectively, based on well established evolution laws [24–27]. The implemented work-hardening model also accounts for the effect of large non-shearable precipitates on the generation of geometrically necessary dislocations during plastic deformations [28] up to a critical local strain where decohesion or fracture of the non-shearable particles occurs [29]. The work-hardening model is novel in the way it includes the precipitate structure through the fully integrated NaMo model, as shown in Figs. 2 and 3. This means that any changes in the particle size distribution (PSD) due to heat treatment or welding, will be reflected by a corresponding change in the work-hardening response, as represented by the net contribution from dislocation hardening Drd expressed by the following response equation [22,23]:

3.1. The yield stress model In the yield stress model, the individual contributions to the overall macroscopic yield strength ry are given as follows [1,2]:

ry ¼ ri þ rp þ rss :

ð2Þ

Here, ri is equal to the intrinsic yield strength of pure aluminium. rp is the precipitation hardening contribution, given as:

rp ¼

MF ; bl

ð3Þ

Drd

where the mean interaction force between dislocations and particles F and the mean planar particle spacing along the bending dislocation l are both extracted from the PSD. M is the Taylor factor and b is the magnitude of the Burgers vector.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2   2 u k  kref k2 ep 1 g;0 e : ¼ aMGbt 1  exp þ qref g;s ref k2 2 kg;0 ec

ð4Þ

Here a is a numerical constant and G is the shear modulus, while k1 is a model parameter, expressing the rate of generation of statistically

Precipitation model

Yield strength model Precipitation hardening

Particle size distribution (PSD) Shearing Bypassing

N

F l

Bypassing

l Shearing

Solid solution hardening

rc

r

Ci

Elastic stress field around a dissolved atom

Parameters extracted from PSD

F

l Ci λg

fo

Mean interaction force between dislocations and particles

Work hardening model

Ci

Dynamic recovery

Friedel length Mean solute concentrations in matrix of element i Geometric slip distance Volume fraction of Orowan particles

λg

fo

Storing of geometrically necessary dislocations Multiple slip, decohesion and fracture of particles

Fig. 3. Diagram defining the parameters extracted from the particle size distribution (PSD) and transferred to the yield stress and work-hardening model, respectively.

C. Dørum et al. / Computers and Structures 88 (2010) 519–528

de ¼1 dep

  n

e ec

ð5Þ

:

In the above expressions, four quantities must be defined in order to obtain a unique description of the stress–strain curve. These include the yield stress ry, and the three parameters kg;0 , k2, ec of the work-hardening model, which are all calculated by the NaMo model on the basis of a given chemical composition and thermal history obtained from a weld simulation applying the thermal module of WELDSIM™. The two latter parameters, i.e. k2 and ec, de^ Mg , and the volume pend upon the equivalent Mg concentration, C fraction of non-shearable particles, fo, respectively, through the following relation [23]:

aMGb

k2 ¼ k1

ec

^C ^ Mg Þ3=4 kð

foref ¼ fo

;

ref c :

e

ð6Þ



ep when ep 6 ec ; ec when ep > ec :

200

7.5 mm 9.5 mm 10.5 mm 11.5 mm 12.5 mm 13.5 mm 14.5 mm 15.5 mm

ð7Þ

0 0

0.1

0.2

0.3

0.4

εp Fig. 4. Predicted Cauchy stress vs. logarithmic plastic strain curves for AA6060-T6 after welding.

300

ð8Þ

From Eq. (5) it follows that the local strain e* is equal to the macroscopic plastic strain ep at small deformations, but approaches ec at large deformations for all relevant n values. Thus, in the limiting case, when n = 1, we may write [22,23]:

e ¼

300

100

Here, index ref means a chosen reference alloy. The numerical values of the constants in Eqs. (6) and (7) are adopted from Refs. ^ = 2  108 N/m2 wt%3/4, f ref ¼ 0:0109 and [22,23] as follows: k o ref ec ¼ 0:05. The remaining parameters in Eq. (4) are independent of the thermal history and are given the following values in the simulations [22,23]: a = 0.3, M = 3.1, G = 2.7  1010 N/m2, b = 2.86 13 m2 and kref  1010 m, k1 = 4  108 m-1, qref g;s ¼ 4:93  10 g;0 ¼ 4:06 7 10 m . In the numerical implementation it is convenient to introduce ref ref in Eq. (4), which the parameters x = aMGb and k3 ¼ qref g;s kg;0 =ec then reads:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  2  2  p  k k e e 1 2 : rf ðep Þ ¼ ry þ x 1  exp þ k3 k2 2 kg;0

field variables are uniquely defined by the distance from the weld centre line. The minimum value of the yield stress ry within each element and the corresponding parameters k2, kg;0 and ec are mapped to the integration point. It is noted that elements with one-point Gauss quadrature are used. Fig. 4 shows the predicted Cauchy stress vs. logarithmic plastic strain curves against distance from the weld. It is observed that there are large variations both in yield stress and in work-hardening characteristics depending on the distance from the weld line. The lowest yield stress is found 10.5 mm from the weld line. It is characteristic that low yield stress is accompanied by relatively strong work hardening, which means that the difference between the strengths at different locations decreases with increasing strain. This feature is more clearly demonstrated in Fig. 5, which

σf [MPa]

stored dislocations during plastic straining. The alloy dependent parameter k2 expresses the rate of dynamic recovery of statistically stored dislocations during plastic deformation. kg;o and kref g;o are the geometric slip distances, based on non-shearable particles, of an alloy and of the reference system, respectively. e* and eref c are the local plastic strain and the critical macroscopic strain for the reference system. In Refs. [22,23], a specially dedicated calibration procedure is outref lined for defining the numerical values of the parameters qref g;s ; kg;o ref and ec in Eq. (4), and these values are implemented in the following simulations. The relationship between the macroscopic plastic strain ep and local plastic strain e* is given by Hart [29] by the following differential equation:

ð9Þ

The parameters in Eq. (8), which are assumed to be independent of the thermal history, are [22,23]: k1 = 4  108 m1, x = 7.131 N/ m and k3 = 4  108 m-1. The parameters ry, k2, kg;0 and ec are field variables and depend on the thermal history. The constitutive model used in the subsequent finite element analysis consists of the von Mises yield criterion, the associated flow rule and the isotropic hardening rule defined by Eqs. (8) and (9). The field variables ry, k2, kg;0 and ec are predicted based on welding simulations and the combined precipitation, yield strength and work-hardening models and then mapped to the FE model of the welded connection. Since steady-state conditions are assumed for the temperature distribution during welding, the

200

σf [MPa]

522

100 ε p = 0.10 ε p = 0.20 ε p = 0.30 ε p = 0.40

ε p = 0.0 ε p = 0.01 ε p = 0.02 ε p = 0.05

0 6

8

10

12

14

16

d [mm] Fig. 5. Predicted Cauchy stress vs. distance form weld at different levels of logarithmic plastic strain for AA6060-T6 after welding.

523

C. Dørum et al. / Computers and Structures 88 (2010) 519–528

4. Numerical simulations of plastic failure Numerical simulations of the behaviour of the investigated welded aluminium specimen containing a heat-affected zone were carried out with shell elements, solid elements and cohesive-zone solid elements. Furthermore, mesh sensitivity studies were conducted for each of the chosen element formulations. All of the simulations have been carried out using the explicit solver of the FE-code LS-DYNA [16], assuming quasi-static conditions. The results from the simulations are provided as engineering stress– strain curves, where the engineering strain is obtained from a virtual extensometer with gauge length equal to 44 mm. In the simulations, elements were eroded when a ductile fracture criterion was reached. The Cockcroft–Latham fracture criterion was adopted, and element erosion was assumed when R W = max(r1, 0) dep P Wcr. Here, r1 is the major principal stress and Wcr is the fracture parameter, taken as 200 MPa in the subsequent simulations. This value of Wcr ensured that plastic failure always occurred before element erosion. 4.1. Solid elements The behaviour of the welded specimen was modelled with the default constant-stress solid element in LS-DYNA [16]. To evaluate the mesh sensitivity, numerical simulations were carried out using FE meshes with characteristic element length ranging from 2.0 mm to 0.17 mm. Thus, the specimen was modelled with one solid element through the thickness for the coarsest mesh and twelve elements through the thickness for the finest mesh. The predicted engineering stress–strain curves are shown in Fig. 6. There is not much effect of element size on the force level. However, the ductility decreases markedly with decreasing element size, and a convergent solution with necking occurring at an elongation of approximately 0.11 is reached with element lengths less than 0.33 mm (more than six elements through the thickness). A plot of the necked specimen is shown in Fig. 7a. The result is taken from the converged solution. It is seen that necking initiates in both HAZs, but develops only on one side of the weld. The maximum plastic strain in the HAZ is rather large and equal to 0.62 at the configuration shown in Fig. 7a. Henceforth the converged solid element solution will be taken as the reference solution for the shell and cohesive-zone element simulations. 4.2. Shell elements When modelling strain localization with shell elements, it is important to keep in mind the limitation of the element formulation, namely that the out-of-plane normal stress is assumed to be zero. In shell simulations, the stabilizing tri-axial stress state arising in necking regions is excluded, whereby it is expected that strain localization occurs earlier than in simulations with solid elements. The width of the localized neck that develops in thin-walled materials is typical in the order of the thickness. For shell elements the width of the local neck is independent of the thickness and typ-

250

200

F/A0 [MPa]

shows the flow stress vs. distance from the weld at different levels of plastic strain. Hence, the frequently used method of scaling the stress–strain curve in the HAZ according to the reduced yield stress is not particularly accurate, but should give results somewhat to the conservative side. For comparison between experimental results and predictions obtained with the yield stress model and the work-hardening model presented in Section 3, the reader is referred to Refs. [3,23], which show good correlation for both the yield stress and the work-hardening properties.

150

100

2.00 mm 1.00 mm 0.67 mm 0.50 mm 0.33 mm 0.25 mm 0.20 mm 0.17 mm

50

0 0

0.05

0.1

0.15

0.2

0.25

ΔL/L0 Fig. 6. Engineering stress–strain curves from numerical simulations of welded specimen using solid elements.

ically equal to the width of the elements. Hence, the localized necking becomes very mesh dependent. Wang et al. [13] applied a non-local approach, originally proposed by Pijaudier-Cabot and Bazant [30], to reduce this mesh dependency. In the non-local approach used by Wang et al. [13], the incremental plastic thickness strain in a given element is calculated as a weighted average of the incremental plastic thickness strains of elements within a non-local domain defined by a radius L from the centre of the element. This radius is typically in the order of the thickness of the material. It is noted that only through-thickness integration points in the same layer of the shell are considered in the averaging procedure. The *MAT_NONLOCAL option in LS-DYNA [16] is used to invoke non-local averaging of a given history variable. Henceforth, the non-local averaging of the incremental plastic thickness strain will be referred to as non-local thinning. The behaviour of the welded specimen has been modelled using shell elements both with and without non-local thinning. In both cases, the default shell element in LS-DYNA was chosen, namely the Belytschko–Tsay shell element with one-point Gauss quadrature and two integration points through the thickness. Furthermore, six different meshes, with characteristic element size ranging from 3.1 mm to 0.2 mm, were used to evaluate the mesh size sensitivity. The distribution of the field variable ry, the yield stress, is illustrated in Fig. 8a and b for the mesh with the largest and the smallest element lengths, respectively. It is obvious that the spatial discretization used in the finite element model influences the distribution of the field variables after mapping and thus the material properties of the HAZ. This should be kept in mind when evaluating the results from the convergence studies reported in the following. Fig. 9 shows the predicted engineering stress–strain curves from simulations with shell elements without use of non-local thinning. The force level increases somewhat with decreasing element size for the meshes with largest elements. The reason for this is the mapping procedure in which the minimum yield stress within an element and corresponding work-hardening parameters were assigned to the integration points of the element. The result is that as the element size increases, the domain having the minimum predicted strength level will increase and thus the force level

524

C. Dørum et al. / Computers and Structures 88 (2010) 519–528

Fig. 7. Strain localization (necking) in the HAZ as given by the converged (a) solid element solution and (b) shell (non-local radius = 2 mm) element solution.

decreases. It is further seen that the ductility decreases with decreasing element size and that convergence seems to be obtained for an element size equal to 0.4 mm. The strain localization in the HAZ occurs at an elongation of approximately 0.07. The ductility predicted with solid elements is significantly larger. It is concluded that the shell element simulations run without any form of

regularization are way too conservative with respect to plastic failure. By applying non-local thinning, the resistance of the shell elements towards thinning will be enhanced, depending on the size L of the non-local domain. As a result, the predicted strain localization will occur later compared with the simulations presented in

C. Dørum et al. / Computers and Structures 88 (2010) 519–528

525

Fig. 8. Yield stress distribution in FE model of welded specimen with (a) element length equal to 3.1 mm and (b) element length equal to 0.2 mm.

200

F/A0 [MPa]

160

120

80 3.1 mm 1.6 mm 0.8 mm 0.4 mm 0.3 mm 0.2 mm

40

0 0

0.04

0.08

0.12

0.16

0.2

ΔL/L0 Fig. 9. Engineering stress–strain curves from numerical simulations of welded specimen using shell elements without non-local thinning.

Fig. 9 and thus the ductility increases. To assess the effect of variations in non-local length L, mesh sensitivity studies were carried

out. Since the length scale of the critical failure mode of the specimen, namely local necking in the HAZ, is expected to be of the order of the specimen thickness, values of L equal to 0.5, 0.8, 1.5 and 2.0 mm were applied in the simulations. Fig. 10 shows the predicted engineering stress–strain curves from simulations with shell elements using non-local thinning and L = 0.8 mm, i.e. the diameter of the non-local domain is somewhat less than the plate thickness. It is seen that comparable results are obtained with all element sizes less than 3.1 mm, but the converged solution still gives too low ductility compared with the solid element solution. Fig. 11 shows a comparison between the converged engineering stress–strain curves obtained from numerical simulations using shell elements with varying L and solid elements. As expected, the elongation at necking increases with increasing L for non-local thinning, and for L equal to the specimen thickness (2.0 mm) necking occurs at an elongation of approximately 0.11. Hence, a significant increase of the ductility compared with the simulations without regularization. It is further seen that the converged solution for shell elements and non-local thinning with L = 2.0 mm gives approximately the same prediction of the stress–strain behaviour as the converged solution for solid elements. Hence, it seems that non-local thinning could be used in shell element simulations to obtain results on plastic failure in better agreement to those attained with solid elements. Fringes of plastic strain on the necked specimen as obtained in the converged solutions with solid elements and shell elements (L = 2.0 mm) are compared in Fig. 7. It is seen that the failure mode is very similar for the two element types.

526

C. Dørum et al. / Computers and Structures 88 (2010) 519–528

200

F/A0 [MPa]

160

120

80

3.1 mm 1.6 mm 0.8 mm 0.4 mm 0.2 mm

40

0 0

0.04

0.08

0.12

0.16

0.2

ΔL/L0 Fig. 10. Engineering stress–strain curves from numerical simulations of welded specimen using shell elements with non-local thinning, L = 0.8 mm.

200

F/A0 [MPa]

160

cohesive behaviour is typically described by a cohesive strength and a critical separation between the nodes of the adjacent continuum elements. Alternatively, the work of separation may be taken as a material parameter. This parameter is related to the J-integral in fracture mechanics. The behaviour of a cohesive element is defined by a separation law, which couples the cohesive traction vector with the separation (or displacement jump) vector [31–33]. In this work, the 4node cohesive element (element type 19) available in LS-DYNA Version 971 [16] is used. For the investigated welded uniaxial tensile specimen, the reduced strength of the HAZ compared to the base material is lumped into a cohesive element that is placed between two rows of solid elements in the HAZ. Cohesive elements can be modelled with no initial extension. However, as explicit FE simulations are used in this work, this will dramatically increase the CPU cost. Therefore, the cohesive elements were modelled with element lengths equal to the surrounding mesh, as illustrated in Fig. 12. In the cohesive element, a traction-separation law is given for the normal direction (mode I) and the tangential direction (mode II) according to

t ¼ gðdI Þ;

s ¼ gðdII Þ:

GcI ¼ ATSLC TdFI ;

120

80

0.17 mm L = 2.0 mm L = 1.5 mm L = 0.8 mm L = 0.5 mm local thinning

40

0 0

0.05

0.1

0.15

ΔL/L0 Fig. 11. Comparison between engineering stress–strain curves from numerical simulations of welded specimen using shell and solid elements.

The results of the FE analyses show that a very fine mesh is necessary both for shell and solid elements to get converged solutions for the ductility of a welded aluminium connection owing to the heterogeneous properties of the heat-affected zone. It is therefore interesting to investigate alternative and simplified ways of analyzing the current problem. 4.3. Cohesive solid elements The concept of a cohesive zone ahead of the crack tip is a bearing idea for a class of crack propagation models. Interface (or cohesive) elements are introduced between continuum elements to describe the dissipation of energy during crack propagation. The

ð10Þ

In Eq. (10), t ¼ t=T and s ¼ s=S, where t and s are tractions and T and S are peak tractions in the normal and tangential directions, dII ¼ dII =dFII , where dI and dII respectively. Further,  dI ¼ dI =dFI and  are separations and dFI and dFII are failure separations for mode I and II. The function g defines the normalized traction-separation law, which is assumed to be the same for mode I and II in LS-DYNA. The fracture toughness for the two modes is given by

GcII ¼ ATSLC SdFII ;

ð11Þ

where ATSLC is the (dimensionless) area under the normalized traction-separation curve. The traction-separation law for the HAZ was modelled using *MAT_COHESIVE_GENERAL (material type 186 in LS-DYNA [16]) which opens up for the definition of an arbitrary normalized traction-separation curve. The normalized traction-separation law used in this study is illustrated in Fig. 13, and is obtained by normalizing the converged engineering stress–strain curves from simulations using constant-stress solid elements. With dFI ¼ 0:128  44 mm ¼ 5:63 mm (engineering strain at failure = 0.128, measurement length = 44 mm), ATSLC = 0.878 and T = 195 N/mm2, the fracture toughness for mode I is found to be GcI ¼ 964 N=mm. Mode II failure was neglected in the simulations. This was obtained by giving the peak tangential traction S a large value, leading to very high fracture toughness GcII in mode II. The results from the simulations using cohesive elements to model the behaviour of the HAZ is shown in Fig. 14. It is seen that the obtained engineering stress–strain curve is equal to the converged solution using constant-stress solid element that was used for parameter identification. The figure also demonstrates the mesh insensitivity of the approach using cohesive elements. FE simulations of the welded uniaxial tensile specimen using two different meshes with characteristic element size 1.0 and 2.0 mm give equal prediction of the engineering stress–strain relationship. It should be noted that with the *MAT_COHESIVE_GENERAL, the cohesive element will behave too soft in compression. Thus, numerical simulations of the behaviour of HAZ using cohesive elements require a material model where the mechanical properties can be accurately described for all deformation modes.

C. Dørum et al. / Computers and Structures 88 (2010) 519–528

527

Fig. 12. FE mesh of welded specimen where the HAZ is modelled using cohesive solid elements (element length = 2.0 mm).

4.4. ‘‘Computational cell” approach for shells

1

0.8

t/T

0.6

0.4

0.2

0 0

0.2

0.4

0.6

δI/δ

0.8

1

F I

Fig. 13. Normalized traction-separation law for HAZ.

200

5. Concluding remarks

160

F/A0 [MPa]

The convergence study with shell elements demonstrated that without regularization the converged solution gives much less ductility than the reference solution. The reason for this is that the length scale of the property variations in the HAZ is smaller than the length scale of local necking. Since plane stress is assumed for shells, the smallest length scale will determine the converged solution. In fracture mechanics, computational cells are used to introduce a physical length-scale into the finite element model over which continuum damage occurs. Computational cells are finite elements in the process zone having their characteristic size determined by the physical process under consideration [8]. A similar route may be taken to describe plastic failure in the HAZ of welded connections using shell elements, i.e. the characteristic element length is determined by the length scale of the phenomenon responsible for failure. Assume that the length scale of local necking, i.e. the width of the local neck, is about the sheet thickness. It is then reasonable to expect that a mesh with characteristic element size about equal to the sheet thickness would give good results, provided the minimum strength calculated in the HAZ is represented in the mesh. The result from a simulation with 2.0 mm shell elements without any non-local regularization is shown in Fig. 14. It is seen that this approach gives a quite reasonable but conservative estimate of the engineering strain at failure.

120

80 Cohesive elements, el = 2.0 mm Cohesive elements, el = 1.0 mm Shell elements, el = 2.0 mm Solid elements, el = 0.2 mm

40

0 0

0.04

0.08

0.12

0.16

ΔL/L0 Fig. 14. Engineering stress–strain curves from FE simulations of welded specimen using cohesive solid elements with 1.0 and 2.0 mm size and shell elements with 2.0 mm size. The converged solution for solid elements is shown for reference.

The material properties in the HAZ of the welded aluminium connection were predicted using a multi-scale method in which finite element based welding simulation is coupled with a microstructure based precipitation, yield strength and work-hardening model. The numerical predictions of the mechanical properties in the heat-affected zone show that there are large variations both in yield stress and work hardening depending on the distance from the weld line. Minimum value of the yield stress is found 10.5 mm from the weld line. The predicted stress–strain curves demonstrate that low yield stress is generally accompanied by increased work hardening. The result is that the heterogeneity of mechanical properties within the HAZ is reduced with plastic straining. It follows that down-scaling of the stress–strain curve of the base material using the ratio between the minimum yield stress in the HAZ and the yield stress of the base material most probably will underestimate the work hardening in the HAZ and thus the strain to necking of the welded component. In the FE simulations of plastic failure in the 2-mm thick welded connection, small element size is required to reach convergence, namely 0.4 and 0.33 mm for shell and solid elements, respectively. In general, strain localization occurs earlier for shell elements than for solid elements with the same characteristic length, owing to the plane stress assumption in the former. However, non-local

528

C. Dørum et al. / Computers and Structures 88 (2010) 519–528

regularization can be used to improve the convergence properties of the shell-based FE simulation. In the actual case, a non-local radius equal to the thickness of the specimen gave results comparable to the solid element simulations. If regularization is not invoked, it seems that the characteristic mesh size in the shell element simulations should be of the order of the thickness to properly represent the ductility of the welded connection. The explanation for this is that the width of the local neck (it is assumed that local necking is the failure mode) is typically of the order of the sheet thickness. A numerically efficient and mesh insensitive approach for simulation of strain localization and failure in the HAZ of aluminium alloys can be obtained by lumping the weakest zone of the HAZ into a cohesive element. For industrial use, it is proposed that a cohesive shell element should be implemented [33]. It is expected that simulation of strain localization and failure in the HAZ using either cohesive shell elements or cohesive solid elements would give similar results when properly calibrated. Thus, it would make it possible to model HAZ failure of welded connections with relatively large shell elements in the HAZ. It is believed that the proposed multi-scale method for determining the material properties in the HAZ may be used in practical simulations of welded aluminium connections and joints to reduce the need for extensive mechanical testing. The method provides the stress–strain curve in the HAZ and thus plastic failure may be analyzed. However, it remains to determine the fracture characteristics of the material in the vicinity of the weld and thus to predict ductile fracture of the welded component based on multi-scale modelling. The proposed modelling strategy for prediction of ductile fracture of welded components should be validated against experimental data. To this end, an extensive study involving various material combinations is currently in progress. Acknowledgement The authors would like to acknowledge the support from the Research Council of Norway for their support through SIMLab, the Centre for Research-based Innovation. References [1] Myhr OR, Grong Ø, Andersen SJ. Modelling of the age hardening behaviour of Al–Mg–Si alloys. Acta Mater 2001;49:65–75. [2] Myhr OR, Grong Ø, Fjær HG, Marioara CD. Modelling of the microstructure and strength evolution in Al–Mg–Si alloys during multistage thermal processing. Acta Mater 2004;52:4997–5008. [3] Myhr OR, Grong Ø, Lademo OG, Tryland T. Optimizing crash resistance of welded aluminium structures. Welding J 2009;88(2):42–5. [4] EN 1999-1-1. Eurocode 9, design of aluminium structures – part 1-1: general structural rules; version of February 2007. [5] Soetens F. Welded connections in aluminium alloy structures. Heron 1987;32:1–48. [6] Nègre P, Steglich D, Brocks W, Koçak M. Numerical simulation of crack extension in aluminium welds. Comput Mater Sci 2003;28:723–31. [7] Nègre P, Steglich D, Brocks W. Crack extension in aluminium welds: a numerical approach using the Gurson–Tvergaard–Needleman model. Eng Fract Mech 2004;71:2365–83.

[8] Nègre P, Steglich D, Brocks W. Crack extension at an interface. Prediction of fracture toughness and simulation of crack path deviation. Int J Fract 2005;134:209–29. [9] Ødegard J, Zhang ZL. Prediction of the performance of welded aluminium nodes for car body applications. Body Des Eng 1996;20:17–24. [10] Zhang ZL, Ødegard J, Myhr OR, Fjær H. From microstructure to deformation and fracture behaviour of aluminium welded joints – a holistic modelling approach. Comput Mater Sci 2001;21:429–35. [11] Wang T, Hopperstad OS, Lademo O-G, Larsen PK. Evaluation of a finite element modelling approach for welded aluminium structures. Comput Struct 2006;84:2016–32. [12] Wang T, Hopperstad OS, Lademo O-G, Larsen PK. Finite element modelling of welded aluminium members subjected to four-point bending. Thin-Walled Struct 2007;45:307–20. [13] Wang T, Hopperstad OS, Lademo O-G, Larsen PK. Finite element analysis of welded beam-to-column joints in aluminium alloy EN AW 6082 T6. Finite Elem Anal Des 2007;44:1–16. [14] Matusiak M. Strength and ductility of welded structures in aluminium alloys. PhD thesis, Department of Structural Engineering, Norwegian University of Science and Technology; 1999. [15] Pickett A, Fouinneteav M, Gudgeon S, Robin V, Christlein J. Model characterization and failure analysis of welded aluminium components including process history. Int J Crash 2007;12:449–63. [16] Livermore Software Technology Corporation (LSTC), LS-DYNA Keyword User’s Manual, Vols. I and II, Version 960; 2001. [17] Myhr OR, Klokkehaug S, Fjær HG, Grong Ø, Kluken AO. Modeling of microstructure evolution, residual stresses and distortions in 6082-T6 aluminum weldments. Welding J 1998;77:286–92. [18] Fjær HG, Myhr OR, Klokkehaug S, Holm EJ. Advances in aluminium weld simulations applying weldsim. In: 11th international conference on computer technology in welding. Columbus, Ohio, USA; December 2001. [19] Myhr OR, Grong Ø. Klokkehaug, S., Fjær, H.G. Modelling of the microstructure and strength evolution during ageing and welding of Al–Mg–Si alloys. In: 6th international seminar on numerical analysis of weldability. Graz-Seggau, Austria; 1–3 October 2001. [20] Myhr OR, Grong Ø. Modelling of non-isothermal transformations in alloys containing a particle distribution. Acta Mater 2000;48:1605–15. [21] Myhr OR, Grong Ø. Utilizing a predictive tool for designing welded aluminium components. Welding J 2008;87(5):36–9. [22] Grong Ø, Myhr OR. Combined precipitation, yield strength and work hardening models for Al–Mg–Si alloys. In: Ninth international summer school on aluminium alloy technology, thermo-mechanical processing and forming, Trondheim, Norway; 20–24 August 2007. [23] Myhr OR, Grong Ø, Pedersen KO. A combined precipitation, yield strength and work hardening model for Al–Mg–Si Alloys. Met Trans A, accepted for publication. [24] Kocks UF. Laws for work-hardening and low temperature creep. J Eng Mater Tech 1976;98:76–85. [25] Mecking H, Kocks UF. Kinetics of flow and strain hardening. Acta Metall 1981;29:1865–75. [26] Estrin Y. Dislocation-density-related constitutive modeling. In: Krausz AS, Krausz K, editors. Unified constitutive laws of plastic deformation. New York, NY: Academic Press; 1996. p. 69–106. [27] Cheng LM, Poole WJ, Embury JD, Lloyd DJ. The influence of precipitation on the work-hardening behavior of the aluminum alloys AA6111 and AA7030. Metall Mater Trans 2003;34A:2473–81. [28] Ashby MF. The deformation of plastically non-homogeneous materials. Phil Mag 1970;21:399–424. [29] Hart EW. Theory of dispersion hardening in metals. Acta Metall 1972;20:275–89. [30] Pijaudier-Cabot G, Bazant ZP. Nonlocal damage theory. J Eng Mech ASCE 1987;113:1512–33. [31] Needleman A. A continuum model for void nucleation by inclusion debonding. J Appl Mech 1987;54:525–31. [32] Needleman A. An analysis of decohesion along an imperfect interface. Int J Fract 1990;42:21–40. [33] Scheider I, Brocks W. Cohesive elements for thin-walled structures. Comput Mater Sci 2006;37:101–9.