**5. Numerical simulation of the interface behaviour**

intermixing phenomena. Note that the vortex instability evolves at high strain rate and excessive deformation so that the plastic work heating enables to melt the aluminium during the swirling phenomenon. The molten phase is quickly solidified due to confinement of the heating and the rapid heat dissipation is facilitated by the good thermal conductivity of both copper and aluminium. These two factors promote a high cooling rate in the range of 104-6 K/s that prevents the atomic structural changes such as crystallisation during slow thermal kinetic. The hyperquenching freezes the random allocation of atoms within the initial molten phase and produces an intermetallic phase which was proven to be amorphous [37]. Hence, the dissimilar Al/Cu combination is conducive to intermetallic formation, within the VAZ and along the interface that alters the physical nature of the weld. The intermetallic phases may appear within discontinuous pockets (**Figure 8b**) or as a continuous layer with a non-uniform thickness (**Figure 8c**). In any case, the presence of the intermediate intermetallic media introduces a new weld variance that is particularly suggested for dissimilar metal joints

**Figure 8.** Typical dissimilar magnetic pulse welds with (a) vortex development, (b) intermetallic pockets and (c) con‐

The formation of a permanent bonding becomes difficult with the accumulation of intermedi‐ ate intermetallic compounds (IMCs). The fast shrinkage during the solidification stage involves cold cracking phenomenon governed by the heterogeneous heat conduction com‐ bined with the incompatibilities of the thermal expansion coefficient. For the case of thin IMC layer, light microscope observations at low magnification reveal transverse cracks across the thickness without further propagations outside the intermetallic zone (**Figure 9a**). These microcracks randomly coalesce and formarbitrary multidirectional crack propagation. The intermetallic media facilitates the cracks to fragment which lead to a catastrophic failure of the joint (**Figure 9b**). It was evidenced that the thickening of the intermetallic phase is favourable to form numerous cracks so that its occurrence can strongly impair the weld integrity. Eventually, the interface can completely break due to a propagation of a macro crack within the intermetallic zone along the interface. According to experimental analyses, this major fracture is attributable to the development of predominant shrinkage stresses during the intermetallic solidification or to a detrimental shearing stress arises from the liquid or solid

**4.6. Fracture within intermetallic phases and detrimental welds**

produced in MPW.

254 Joining Technologies

tinuous intermetallic layer at the interface [33].

This section begins with a literature survey regarding the numerical simulation of the interface behaviour during impact welding, prior to the suggestion of a suitable method, namely Eulerian simulation to compute the collision and weld generation. This method is applied for both similar and dissimilar metal combinations (Al/Al and Al/Cu) to show the convincing predictions of typical interfacial features including the wavy morphology and formation of defects.

#### **5.1. A brief literature review of impact welding simulations**

The numerical simulations of the weld generation during impact welding processes can be classified into five distinct methods known as Lagrangian, Adaptive Lagrangian-Eulerian (ALE), Eulerian, smooth particle hydrodynamic (SPH) and molecular dynamics (MD). Generally, Lagrangian computation fails during the development of excessive interfacial shearing. Due to the large strain produced by the kinematic instability of the interface, the mesh experiences strong flattening and distortion that eventually aborted the computation [38–48]. A wavy interface is difficult to perform even the onset of jet kinematics can be detected using this method. Alternatively, ALE method is suggested to improve the mesh quality by utilizing a node relocation algorithm during the computation [39–41, 46, 49], but its real capability to produce persuasive wavy patterns has to be demonstrated. ALE method also suffers from the bad mesh quality due to the interfacial shearing and jetting. To overcome mesh issues, particle based methods were investigated among which the MD computations allow for the accurate simulation of complex interfacial morphologies but they meet a scale limitation [50–52]. MD method is rather appropriate for small scale, in the range of 10–100 nm unlike the SPH method that enables the computation of the interfacial jetting and wavy morphology at a large scale [42, 43, 53, 54]. However, the accuracy of the SPH method is discussed regarding the consideration of the dissipative terms [55]. The method becomes unsuitable if such physical phenomenon prevails.

. ( ) 1 1 ( ) *<sup>v</sup> u grad u div <sup>u</sup> <sup>F</sup>*

. () : *<sup>e</sup> u grad e D*

+ =

r uuuuuuur uur r uur

+ =+

y

 y

Magnetic Pulse Welding: An Innovative Joining Technology for Similar and Dissimilar Metal Pairs

ruuuuuuuuur (6)

¶ ¶ (7)

¶Æ <sup>=</sup> ¶ (8)

¶ ¶ (9)

(5)

257

http://dx.doi.org/10.5772/63525

s

s

where, respectively denotes density, velocity vector, Cauchy stress, internal body force vector, strain rate tensor and specific internal energy which is assumed to be an

The split strategy consists of an operation that decomposes each conservation equations into two parts in order to separately compute advection variables. In a one dimensional form, the generic conservation equation is given by (Equation 7) and the splitting gives the pair of

> *<sup>x</sup> u F t x* ¶Æ ¶Æ + =

> > *F t*

<sup>0</sup> *<sup>x</sup> <sup>u</sup> t x* ¶Æ ¶Æ + =

Equation (8) represents the usual dynamic equation of solid which is solved by the usual Lagrangian method, i.e. the mechanical computation of spatio-temporal motion given by the mesh deformation depending on the material behaviour. This is called the Lagrangian step. Afterwards, the converged state variables given by the Lagrangian step are advected using Equation (9) on spatial fixed mesh. This is called the Eulerian step during which the advection computes the material flow. The material interface is calculated by the volume of fluid (VOF) front tracking method. The sequential Lagrangian/Eulerian computation eliminates the problems of mesh distortion and flattening, thus enables to reproduce the complex kinematics

Virtual tests are performed under the same conditions as for the experimental results provided in Section 4. **Figure 10** shows the results obtained from a 2D simulation. First computation

*t*

*t*

¶

¶

enthalpy to compute the heating due to the plastic work .

equations (Equations 8 and 9).

of the interface during the collision progression.

**5.3. Virtual testing of the wavy interface generation**

¶

¶

At present, Eulerian computation offers a possibility to reproduce the kinematics of the interface during the collision. Generally, Eulerian method is used in computational fluid dynamics but can be applied to a solid to simulate a material flow, with acceptable results using a Johnson-Cook constitutive model for describing the material [44, 45, 56, 57]. In the literature, some simulations of weakly shaped interfaces were presented [42, 43, 50–53]. The method merits to be further explored to compute the full development of the wavy morphol‐ ogy as well as the defects' formation. Sections 5.2–5.4 include a description of the Eulerian procedure and convincing simulation results encouraging the enactment of such method in MPW.

#### **5.2. Eulerian method**

Eulerian method is mainly used in computational fluid dynamics that solves the conservation equation of mass, momentum and energy. The history of state variables is computed at any point (M) of a domain. The method uses a fixed computational grid and the time dependant variation of these variables on each grid describes the fluid flow. This is called pure Eulerian computation applied to Newtonian and non-Newtonian fluids. The major difference between fluids and solids relies on the description and treatment of mechanical behaviour. For the peculiar case of solid dynamic, the high strain rate dependency of the stress governs the mechanical behaviour and a constitutive law, the Johnson-Cook law, is generally used. For a suitable numerical treatment of this law, a split method is suggested to solve the governing equations of Eulerian method whose differential forms are expressed as follows:

$$\frac{\partial \Psi}{\partial t} + \overline{u} \, \overline{grad(\Psi)} = -\psi \, d\dot{\nu} \left(\vec{u}\right) \tag{4}$$

Magnetic Pulse Welding: An Innovative Joining Technology for Similar and Dissimilar Metal Pairs http://dx.doi.org/10.5772/63525 257

$$\frac{\partial \vec{u}}{\partial t} + \overline{u}. \,\,\underline{\text{grad}(\vec{u})} = \frac{1}{\nu} \,\,\overline{\text{div}(\underline{\sigma})} + \frac{1}{\nu} \,\,\overline{F\_{\text{v}}}\tag{5}$$

$$\frac{\partial e}{\partial t} + \overline{u \, \overline{grad(e)}} = \underline{\sigma} \colon \underline{D} \tag{6}$$

where, respectively denotes density, velocity vector, Cauchy stress, internal body force vector, strain rate tensor and specific internal energy which is assumed to be an enthalpy to compute the heating due to the plastic work .

The split strategy consists of an operation that decomposes each conservation equations into two parts in order to separately compute advection variables. In a one dimensional form, the generic conservation equation is given by (Equation 7) and the splitting gives the pair of equations (Equations 8 and 9).

$$\frac{\partial \widetilde{\mathcal{Q}} \mathcal{Q}}{\partial t} + \mu\_x \frac{\partial \mathcal{Q}}{\partial x} = F \tag{7}$$

$$\frac{\partial \mathfrak{Q}}{\partial t} = F \tag{8}$$

$$\frac{\partial \mathfrak{Q}}{\partial t} + \mu\_x \frac{\partial \mathfrak{Q}}{\partial \mathfrak{x}} = 0 \tag{9}$$

Equation (8) represents the usual dynamic equation of solid which is solved by the usual Lagrangian method, i.e. the mechanical computation of spatio-temporal motion given by the mesh deformation depending on the material behaviour. This is called the Lagrangian step. Afterwards, the converged state variables given by the Lagrangian step are advected using Equation (9) on spatial fixed mesh. This is called the Eulerian step during which the advection computes the material flow. The material interface is calculated by the volume of fluid (VOF) front tracking method. The sequential Lagrangian/Eulerian computation eliminates the problems of mesh distortion and flattening, thus enables to reproduce the complex kinematics of the interface during the collision progression.

#### **5.3. Virtual testing of the wavy interface generation**

shearing. Due to the large strain produced by the kinematic instability of the interface, the mesh experiences strong flattening and distortion that eventually aborted the computation [38–48]. A wavy interface is difficult to perform even the onset of jet kinematics can be detected using this method. Alternatively, ALE method is suggested to improve the mesh quality by utilizing a node relocation algorithm during the computation [39–41, 46, 49], but its real capability to produce persuasive wavy patterns has to be demonstrated. ALE method also suffers from the bad mesh quality due to the interfacial shearing and jetting. To overcome mesh issues, particle based methods were investigated among which the MD computations allow for the accurate simulation of complex interfacial morphologies but they meet a scale limitation [50–52]. MD method is rather appropriate for small scale, in the range of 10–100 nm unlike the SPH method that enables the computation of the interfacial jetting and wavy morphology at a large scale [42, 43, 53, 54]. However, the accuracy of the SPH method is discussed regarding the consideration of the dissipative terms [55]. The method becomes unsuitable if such physical

At present, Eulerian computation offers a possibility to reproduce the kinematics of the interface during the collision. Generally, Eulerian method is used in computational fluid dynamics but can be applied to a solid to simulate a material flow, with acceptable results using a Johnson-Cook constitutive model for describing the material [44, 45, 56, 57]. In the literature, some simulations of weakly shaped interfaces were presented [42, 43, 50–53]. The method merits to be further explored to compute the full development of the wavy morphol‐ ogy as well as the defects' formation. Sections 5.2–5.4 include a description of the Eulerian procedure and convincing simulation results encouraging the enactment of such method in

Eulerian method is mainly used in computational fluid dynamics that solves the conservation equation of mass, momentum and energy. The history of state variables is computed at any point (M) of a domain. The method uses a fixed computational grid and the time dependant variation of these variables on each grid describes the fluid flow. This is called pure Eulerian computation applied to Newtonian and non-Newtonian fluids. The major difference between fluids and solids relies on the description and treatment of mechanical behaviour. For the peculiar case of solid dynamic, the high strain rate dependency of the stress governs the mechanical behaviour and a constitutive law, the Johnson-Cook law, is generally used. For a suitable numerical treatment of this law, a split method is suggested to solve the governing

equations of Eulerian method whose differential forms are expressed as follows:

y

¶

¶

*u grad div u* . () ( ) *<sup>t</sup>*

 y

ur uuuuuuuuur <sup>r</sup> (4)

y

+ =-

phenomenon prevails.

256 Joining Technologies

**5.2. Eulerian method**

MPW.

Virtual tests are performed under the same conditions as for the experimental results provided in Section 4. **Figure 10** shows the results obtained from a 2D simulation. First computation considers a similar Al/Al combination with a flyer thickness of 1.5 mm and an air gap of 2 mm. A Johnson-Cook law is used to describe the material behaviour with the assumptions of isotropic hardening, a von Mises yield surface and a geometric non-linearity.

**5.4. Computation of the thermomechanical phenomena**

contribute thereby to the process proficiency.

ena in b–d.

The plastic work due to the interfacial shearing plays a significant role in the interfacial behaviour. It produces localised heating phenomenon at the vicinity of the excessive sheared zone where a temperature change arises. With the accurate computation of the interfacial deformation, Eulerian simulation captures the interfacial heating while providing accurate predictions in terms of heating location and shape of the heat affected zone (HAZ). **Figure 12** presents a comparison between a computed HAZ and experimental observations. The simulation results exhibit particular HAZ where the highest temperature indicates a potential site of defect's formation. This zone undergoes a thermomechanical softening which is conducive to a development of failure when detrimental conditions (critical strain, stress, damage parameters etc.) occur. Experimentally, along the wavy pattern, formation of voids is observed within the numerically predicted extremely heated sites. Experimental observation of the voids' shapes also concur the shapes of the high temperature distribution during the numerical simulations for the Al/Al joint (**Figure 12a,b**). In the case of Al/Cu combination, Eulerian simulation also reproduces an interfacial heating that corroborates the corresponding experimental observations (**Figure 12c–e**). The computed heating clearly indicates the devel‐ opment of a confined heated layer for Al/Cu joint. The temperature distribution of the layer reveals a convincing shape and size that leads the possibility to predict the formation of intermediate intermetallic phases at the interface. Furthermore, the ejection of significant quantity of material due to the strong interfacial shearing combined with the relatively softer material of the copper than that of aluminium (**Figure 12c–e**) causes a potential detrimental phenomenon that can explain the large experimental cracks (**Figure 12d**). These overall predictions demonstrate the capability of Eulerian simulation to investigate the weldability conditions for MPW by a direct computation of the interfacial behaviour during the collision. Such approach will significantly facilitate the accurate depiction of welding conditions and

Magnetic Pulse Welding: An Innovative Joining Technology for Similar and Dissimilar Metal Pairs

http://dx.doi.org/10.5772/63525

259

**Figure 12.** Typical simulation results of the weld natures for (a) similar and (e) dissimilar materials, predicting wavy interface and localised thermal effects. Experimental observations of wave formation in (b) and detrimental phenom‐

The Eulerian finite element computation can provide accurate predictions with respect to the weld generation and the singular feature found in impact welding including the development of interfacial wavy morphology during the collision progression. In **Figure 10**, the collision propagates from the right to the left. The bonding begins with a straight interface as previously described in the Section 4.1. Although, this situation can be computed successfully by a usual Lagrangian method based on a mesh deformation, it fails to produce the wavy interface. In contrast, Eulerian computation enables the simulation of the weak or big waves' formation along the interface as clearly shown in **Figure 10**. The complex kinematics of jetting is repro‐ duced with a fine description of the material interface. **Figure 11** shows three types of typical jetting stages ahead of collision points. Developments of successive inverted jets [downward (**Figure 11a**) and upward (**Figure 11b**, **c**)] are evidenced. The material flow governed by this jetting phenomenon forms the progressive wavy morphology. Ejection of particles is also evidenced in the simulation. In the numerical point of view, this result arises from the advection procedure that calculates the temporal evolution of each state variable over the computational grid. Material flows occur where they are expected to appear. Note that the ejection is a physically realistic phenomenon in MPW. It was evidenced that solid fragments or aggregates are ejected from the interface during the collision [32, 33].

**Figure 10.** Eulerian simulation showing the development of wavy pattern at the welded interface.

**Figure 11.** Eulerian simulation of jetting kinematics during the collision propagation with downward jet in (a) and up‐ ward jets in (b) and (c).

#### **5.4. Computation of the thermomechanical phenomena**

considers a similar Al/Al combination with a flyer thickness of 1.5 mm and an air gap of 2 mm. A Johnson-Cook law is used to describe the material behaviour with the assumptions of

The Eulerian finite element computation can provide accurate predictions with respect to the weld generation and the singular feature found in impact welding including the development of interfacial wavy morphology during the collision progression. In **Figure 10**, the collision propagates from the right to the left. The bonding begins with a straight interface as previously described in the Section 4.1. Although, this situation can be computed successfully by a usual Lagrangian method based on a mesh deformation, it fails to produce the wavy interface. In contrast, Eulerian computation enables the simulation of the weak or big waves' formation along the interface as clearly shown in **Figure 10**. The complex kinematics of jetting is repro‐ duced with a fine description of the material interface. **Figure 11** shows three types of typical jetting stages ahead of collision points. Developments of successive inverted jets [downward (**Figure 11a**) and upward (**Figure 11b**, **c**)] are evidenced. The material flow governed by this jetting phenomenon forms the progressive wavy morphology. Ejection of particles is also evidenced in the simulation. In the numerical point of view, this result arises from the advection procedure that calculates the temporal evolution of each state variable over the computational grid. Material flows occur where they are expected to appear. Note that the ejection is a physically realistic phenomenon in MPW. It was evidenced that solid fragments

isotropic hardening, a von Mises yield surface and a geometric non-linearity.

or aggregates are ejected from the interface during the collision [32, 33].

**Figure 10.** Eulerian simulation showing the development of wavy pattern at the welded interface.

ward jets in (b) and (c).

258 Joining Technologies

**Figure 11.** Eulerian simulation of jetting kinematics during the collision propagation with downward jet in (a) and up‐

The plastic work due to the interfacial shearing plays a significant role in the interfacial behaviour. It produces localised heating phenomenon at the vicinity of the excessive sheared zone where a temperature change arises. With the accurate computation of the interfacial deformation, Eulerian simulation captures the interfacial heating while providing accurate predictions in terms of heating location and shape of the heat affected zone (HAZ). **Figure 12** presents a comparison between a computed HAZ and experimental observations. The simulation results exhibit particular HAZ where the highest temperature indicates a potential site of defect's formation. This zone undergoes a thermomechanical softening which is conducive to a development of failure when detrimental conditions (critical strain, stress, damage parameters etc.) occur. Experimentally, along the wavy pattern, formation of voids is observed within the numerically predicted extremely heated sites. Experimental observation of the voids' shapes also concur the shapes of the high temperature distribution during the numerical simulations for the Al/Al joint (**Figure 12a,b**). In the case of Al/Cu combination, Eulerian simulation also reproduces an interfacial heating that corroborates the corresponding experimental observations (**Figure 12c–e**). The computed heating clearly indicates the devel‐ opment of a confined heated layer for Al/Cu joint. The temperature distribution of the layer reveals a convincing shape and size that leads the possibility to predict the formation of intermediate intermetallic phases at the interface. Furthermore, the ejection of significant quantity of material due to the strong interfacial shearing combined with the relatively softer material of the copper than that of aluminium (**Figure 12c–e**) causes a potential detrimental phenomenon that can explain the large experimental cracks (**Figure 12d**). These overall predictions demonstrate the capability of Eulerian simulation to investigate the weldability conditions for MPW by a direct computation of the interfacial behaviour during the collision. Such approach will significantly facilitate the accurate depiction of welding conditions and contribute thereby to the process proficiency.

**Figure 12.** Typical simulation results of the weld natures for (a) similar and (e) dissimilar materials, predicting wavy interface and localised thermal effects. Experimental observations of wave formation in (b) and detrimental phenom‐ ena in b–d.
