Modeling strain hardening and texture evolution in friction stir welding of stainless steel

Materials Science and Engineering A 398 (2005) 146–163

Modeling strain hardening and texture evolution in friction stir welding of stainless steel Jae-Hyung Cho, Donald E. Boyce, Paul R. Dawson∗ Sibley School of Mechanical and Aerospace Engineering, Cornell University, 196 Rhodes Hall, Ithaca, NY 14853, USA Received 15 October 2004; received in revised form 4 March 2005; accepted 4 March 2005

Abstract Steady-state friction stir welding of stainless steel has been modeled using an Eulerian formulation that considers coupled viscoplastic flow and heat transfer in the vicinity of the tool pin. Strain hardening is incorporated with a scalar state variable that evolves with deformation as material moves along streamlines of the flow field. The model equations are solved using the finite element method to determine the velocity field and temperature distribution, with a modified Petrov–Galerkin employed to stabilize the temperature distribution. The evolution equation for the state variable for strength is integrated along streamlines using an adaptive procedures to determine step size based on the element size. The intense shearing and associated heating lead to complex behavior near the tool pin. The effect of this complex response is demonstrated with the crystallographic texture, which displays a nonmonotonic strengthening and weakening history along streamlines that pass close to the tool pin. © 2005 Elsevier B.V. All rights reserved. Keywords: FSW; Eulerian formulation; Texture; Yield function

1. Introduction Friction stir welding (FSW) is a solid-state joining technique invented by TWI that is effective for metallic and nonmetallic materials [1]. The process advantages result principally from the fact that temperatures remain below the melting point of the materials being joined. The benefits include the ability to join materials which are difficult to fusion weld, for example 2000 and 7000 aluminum alloys. Other advantages are low shrinkage and distortion, excellent mechanical properties, and low production of fumes. A FSW process commences by inserting a spinning tool pin into the surface of adjoining parts. The tool pin shoulder presses against the workpieces and helps contain the material flow. The tool rotates rapidly, inducing complicated material flow patterns and highly altering the microstructures. The weld microstructures may be placed into two broad categories: the thermomechanically affected zone (TMAZ) and the heat-affected zone (HAZ). The HAZ is similar to that of ∗

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conventional, fusion welding processes, including evidence of precipitate changes, recovery, recrystallization, and grain growth. In contrast to fusion welding, the TMAZ is not believed to experience melting but instead to go through severe shear deformation while remaining solid. FSW is inherently asymmetric about the contact plane of adjoining parts because of the difference between the advancing and retreating sides. A number of studies that address microstructural alterations and flow pattern visualization during FSW have been reported in the literature [2–6]. From electron backscatter diffraction (EBSD) analyses, it has been shown that severe gradients in crystallographic texture exist through the thickness and across the width of parts joined by FSW. These textures can generally be characterized as one or more of the ideal shear textures [2]. Measured hardness profiles across the welded zone in a 6063 aluminium alloy joined by FSW indicate that the hardness is diminished in the weld zone, which has fine recrystallized grains relative to the base material [3]. Flow visualization studies of the FSW process have been used to investigate the influence of the welding parameters on

J.-H. Cho et al. / Materials Science and Engineering A 398 (2005) 146–163

material flow and the shape of the interface between the various zones (HAZ and TMAZ). With a differential etch contrast or marker insert technique (MIT), complex residual flow patterns have been visualized [4–6]. The material transport in FSW comes from the tool pin’s translation and rotation. The tool motion produces deformation heating in the workpiece leading to a reduction in the flow stress. The deformation also induces strain hardening, which is most pronounced around tool pin (mainly in the TMAZ zone). Strain hardening within the TMAZ zone is thought to be greater for high melting temperature materials such as the stainless steel used for this research. Several researchers have reported on aspects of the heat transfer processes during FSW [7–11]. A three-dimensional (3D) transient heat-transfer model for FSW was presented in [9] that combines an Eulerian reference frame and a finite difference method to treat the difficulty of modeling the moving tool. It is pointed out that a preheat is beneficial to increase the temperature of the workpiece in front of the tool pin to make the material easier to weld. A thermomechanical model including the mechanical action of the shoulder was also reported in [10]. The maximum temperature gradients in longitudinal and lateral directions are predicted in a region just beyond the shoulder edge. The longitudinal component of the residual stress is greater than the lateral component at the top surface of the weld. Using a 3D finite element model, parametric studies were conducted to determine the effect of tool speeds on welding temperatures and forces acting on the pin [11]. Increasing the translational speed has the effect of increasing the magnitude of the forces, while increasing the rotational speed has the opposite effect. A complete analysis of a solid under large strain deformations would necessarily include both elastic and inelastic behaviors. Numerical formulations that incorporate elastoplastic material behavior and geometric nonlinearities are available [12,13]. In many forming process, however, the plastic strains are large enough to justify neglecting elastic strains for the purpose of understanding the evolution of microstructure. Consequently, rigid–plastic or rigid–viscoplastic material models have been used for forming analysis [14,15]. In these analyses, the flow stress increases with the deformation. The focus of this paper is the Eulerian modeling of two-dimensional (2D) FSW using a state variable constitutive model that assumes isotropic strain hardening and separately characterizes the rate dependence of the flow strength and saturation limit of the state variable. The model parameters have been evaluated from independent laboratory data for stainless steel (304L). (The state variable represents the material strength or hardness and is a measure of the dislocation density of the material.) The computed state variable distribution and temperatures are compared to a measured Vickers hardness results and temperatures. Calculated pole figures and yield functions are shown across outlet boundary after friction stir welding modeling.

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2. Governing equation for the forming process The motion and temperature of the workpiece are governed by balance laws for mass, momentum and energy [16]: conservation of mass

Dρ + ρ div u = 0 Dt

balance of linear momentum conservation of energy

ρ

div σ T + b = 0

De ˙ =0 + div q − Q Dt

(1) (2) (3)

where ρ is the density, u the velocity, σ the Cauchy stress, b the body force, e the internal energy, q the heat flux and ˙ the volumetric heat source. Inertia has been neglected in Q the momentum balance and the heat generation includes a contribution from viscous dissipation in the energy equation. Boundary conditions are specified for the motion and energy of the workpiece. For the motion, either known tractions or velocities are imposed on the surface, σ · η = T¯ on ST

(4)

u = u¯ on Su

(5)

where η is the surface normal vector, T¯ a known traction vector, and u¯ a known velocity vector. ST and Su are portions of the total surface, S. In the case of sliding friction over a portion of the boundary, the tangential component of the surface traction may be due to frictional contact with the forming equipment, such as a tool pin or die. In this case the traction vector is written as T¯ t = β(utool − u)t

(6)

where β is a coefficient that can depend on the temperature and traction, especially its normal component, and the subscript, t, indicates the tangential component. Boundary conditions for the energy equation include known heat fluxes or temperatures on the surface: k grad θ · η = q¯ on Sq

(7)

and θ = θ¯ on Sθ

(8)

where θ is the temperature, and q¯ and θ¯ the known values of the flux and temperature, respectively. Sθ and Sq the portions of the total surface area, S. Convective losses on the surface are assumed to be proportional to the difference between the surface temperature and an ambient temperature, θ∞ , q¯ = h (θ − θ∞ )

(9)

where h is a film coefficient. It is necessary to specify the initial conditions for the system of equations. This is accomplished in an Eulerian reference frame by prescribing the value of the temperature and state variable as the material enters the spatial domain defined by the Eulerian coordinates.

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Thus, initial conditions are specified on that part of the surface (Sl ) where material enters the Eulerian region: θ = θ¯

and

κ = κ¯ on Sl

(10)

where θ¯ and κ¯ are known values of the temperature and state variable, respectively.

3. Constitutive equations Constitutive equations are required to characterize mathematically the material behavior associated with the internal energy and plastic flow. The internal energy and heat flux are written as functions of the rate of deformation, temperature and state of the material as described by a collection of state variables. Simple linearized forms of internal energy, e, and heat flux, q, are assumed to represent the material adequately, D D (e) = cp (θ) Dt Dt

(11)

σ¯ = τ = τ P + τ V ,

and q = −k grad θ

The flow stress of the constitutive equations depends on the temperature, the rate of deformation, and the deformation history. The change in flow stress is the difference between the hardening and recovery, which are related to accumulation and annihilation of crystal defects (i.e., dislocation), respectively. Hardening and dynamic recovery occur concurrently with deformation, while the static recovery may require much longer times than are typical of period of deformations in an FSW process. Thus, constitutive equations considering hardening and recovery need to accurately characterize the material during deformation. An example of the state variable models is the one proposed by Hart [21] and used by Eggert and Dawson in simulation of solid-state welding (upset welding) [22,23]. Hart’s complete model has both tensor and scalar state variables, but exists in a simplified form with a scalar state variable. The simplified Hart’s model incorporates two contributions to the flow stress, one called a plastic contribution and the other a viscous contribution. In the simplified form, the yield condition may be written as

(12)

In these forms, the specific heat, cp , and conductivity, k, often are specified as functions of temperature. The constitutive equations for the stress are required to describe the plastic flow of the metal workpiece. This paper focuses on plastic behavior that is rate-sensitive since the principal interest is in a regime of warm/hot deformation processes. A phenomenological, state variable model was chosen for this purpose. The particular model we used is similar in form to other constitutive models for the plastic flow of polycrystalline materials [17–19]. Such models permit changing load path, strain rates, and temperatures in a theoretically consistent manner. The constitutive equations assume an isochoric motion and possess a structure consisting of a flow law, a yield condition, and evolution equations for the state variables. There are a number of assumptions made in representing the material behavior with equations of this form. The material is treated as isotropic and the plastic strains are much larger that the elastic strains. The flow law simply states that the deviatoric Cauchy stress, σ  , is proportional to the deviatoric deformation rate, D . The state is represented with a scalar variable, κ, that is equivalent to a back-extrapolated definition of the yield stress. Further, it is assumed that the response of a material to an applied load may be determined uniquely from the material’s current state. The collection of state variables completely defines the material properties and eliminates the need to remember the temperature and deformation history that brought the material to its present condition. When we use a single scalar variable to describe the state of the material, one microstructural feature controls the rate of deformation for a particular combination of stress and temperature. A general review of state variable constitutive equations has been given in [20].

(13)

where σ¯ is the effective Cauchy stress and τ P and τ V the plastic and viscous contributions to the rate and temperaturedependent flow stress, τ. These contributions are written as
1/M ¯ D V (14) τ =G a where



Q a = a0 exp − Rθ

 (15)

and


  b λ τ = κ exp − ¯ D P

where b = b0

 κ N G



Q exp − Rθ

(16)  (17)

¯ is the effective value of deformation rate, D, and κ the D state variable. The effective stress is defined by 23 σij σij and effective deformation rate is given by 23 Dij Dij . At the microstructural level the viscous element represents frictional forces along slip planes which resist dislocation glide. The frictional forces are temperature-dependent and an Arrhenius dependence is assumed. The contribution to the total stress from the viscous element is important at relatively low temperatures or very high deformation rates. The plastic contribution represents flow resistance from dislocation entanglement and is parameterized by a scalar state variable, κ, which we will refer to as the strength. κ can be considered to characterize the density and strength of barriers to the motion of dislocations through the crystal

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Table 1 Material parameters for the simplified Hart’s model for annealed 304 stainless steel

Table 2 Material parameters for the state variable evolution for annealed 304 stainless steel

a0 (s−1 ) b0 (s−1 ) G (GPa) κ0 (MPa) Q (kcal/mol) Q (kcal/mol) λ M N

h0 (MPa) n0 D0 (s−1 ) m0 C (MPa) κss (MPa)

1.36 × 1035 8.03 × 1026 73.1 150.0 98 21.7 0.15 7.8 5.0

structure. Through the form of Eq. (16), κ is the upper limit for τ P , the stress in the plastic element. This limit is approached at low temperature or high strain rate. The material model parameters, G, Q, Q , M, N, λ, b0 , and a0 for 304 stainless steel have been determined using data from large deformation tests and are listed in Table 1. The value of the initial strength for the undeformed material, κ0 , must also be determined from experimental data. R is the universal gas constant. The behavior of the hardening law proposed by Hart specifies that the rate of change of strength is approximately proportional to the deformation rate at low temperatures or high deformation rate. At high temperature or low deformation rate little hardening occurs. In FSW, the deformation rate is high around tool pin. Under these conditions, it is more reasonable to use an evolution equation that specifies a saturation value for the strength as a function of strain rate and temperature. Therefore, the state variable evolution in Hart’s model is replaced by one with Voce-like saturation limit for state variable. The saturation limit for the state variable depends on temperature and deformation rate, and can be written as
m0 C κsat = (18) ϕ with the Fisher factor [24], ϕ, given by
 D0 ϕ = θ ln ¯ D Using these relations, the evolution then is written  κ n0 ¯ D (κ) = h0 1 − sat D Dt κ

100 6 1.0 × 108 2.148 132 200

Fig. 1. Linear regression between Fisher factor and estimates of the saturation flow stress from several experiments.

saturation stresses are estimated reasonably well using 2.148 and 132.0 MPa for m0 and C, respectively. Fig. 2 shows the flow stress for stainless steel computed using the modified form of Hart’s model together with the Voce-like hardening of Eq. (20). The FSW process induces large strains as well as severe temperature changes. We limit the growth in the flow stress from strain hardening by placing realistic limits on the saturation value of the state vari-

(19)

(20)

where h0 and n0 are experimentally determined model parameters. Table 2 lists the material parameters for state variable evolution. κss is the state variable value within the steady state boundary layer around the tool pin. It is necessary to determine the saturation limit of the state variable and thus flow stress, in Eq. (18) based on experiments. Fig. 1 shows estimated saturation stress levels for 304 stainless steel as a function of the Fisher factor during compression, tension and torsion experiments [25,26]. The flow stresses for several temperature and deformation rates of 304 stainless steel reported by Cook also have been used for the estimation of saturation stress [27]. From the regression, the experimental

Fig. 2. Flow stress computed for a modified Hart’s model.

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Fig. 3. Adiabatic heating effects on flow stress computed for a modified Hart’s model: (a) 373 K, (b) 1173 K, (c) 1373 K.

able in FSW. At 1173 K, the flow stress ranges between 95 and 165 MPa according to deformation rate. Lower values of the deformation rate corresponds to lower values of the flow stress. At 1373 K, the flow stress is smaller than that at 1173 K as expected from the kinetics of slip. Fig. 3 shows the effect of adiabatic heating on the flow stress for various initial temperatures. In general, higher deformation rates result in higher peak stress values.

as a weighted residual as

ψ div u dV = 0

4. Numerical solution of the model equations

u(x) = [N(x)]{U}

(23)

The velocity field is determined from a weak form of the equilibrium equation:

p(x) = [Np (x)]{P}

(24)

−

V

tr[σ · grad ] dV +

S

T ·  dS = 0

(21)

where  are the weights, V the workpiece volume, and S its surface in the current configuration. The yield condition and the flow rule are used to eliminate the deviatoric portion of the stress that appears in the first integral. For incompressible deformations, the conservation of mass equation (1) is written

V

(22)

This equation acts as a constraint on the motion allowed by Eq. (21) and is treated by the consistent penalty method [28,29]. Approximating functions for the velocity and the pressure (negative of the mean stress) over a finite element are introduced as

Quadratic, continuous interpolation functions are specified for the velocities and linear, discontinuous interpolation functions are specified for pressure. To increase the accuracy of the consistent penalty method, the penalty parameter is allowed to vary from element to element [30] based on a weighted average of the effective viscosity evaluated at quadrature points. Following standard finite element procedures, the resulting matrix equations from Eq. (21) are given by [Kµ ]{U} + [G]{−P} = {F }

(25)

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and [Mp ]{P} = −* [G] {U} T

(26)

*=

NQP 

λi × Wi

(27)

i=1

Here Wi are the standard Gauss weights and NQP indicates the number of quadrature points per element. More details about the expression, [Kµ ], [G], [Mp ], and {F} are found in [31]. The pressure can be eliminated at an element level, thereby reducing the problem to one involving only nodal velocities. Thus, we obtain [Kµ + Kλ ] {U} = {F }

(28)

where [Kλ ] = * [G] [Mp ]−1 [G]T

(29)

To determine the temperature field, a residual is found as the energy equations assuming steady-state conditions. Using a weighting function, φ and an Eulerian reference frame, the residual may be written as from Eqs. (3), (11) and (12):

˙ φ dV = 0 [div (k grad θ) − ρcp u · grad θ + Q] (30) V

In this form, the specific heat, cp , and conductivity, k, often are specified as functions of temperature, although a constant value is imposed here. The model parameters are given in Table 3. The solution of Eq. (30) may be oscillatory in space when the Peclet number exceeds a critical value. Streamline upwinding by the Petrov–Galerkin method (SUPG) is applied to control the oscillations. A Petrov–Galerkin (PG) formulation is obtained by replacing the trial function φ by φ, which is defined as φ ≡ φ + - u · grad φ

(31)

where the stabilization parameter is given by - = |α|

Euler method, where the characteristic time interval is related to a spatial interval in the Eulerian domain using the velocity: /t =

where

/x

u

(32)

The first term in right side of Eq. (31) is a Galerkin test function and a second discontinuous test function. The parameter α in Eq. (32) is defined as a function of Peclet number. Details about the upwind Petrov–Galerkin formulation may be found in other work [32,33]. The state variable is determined at points throughout the domain by integrating its evolution equation along streamlines of the velocity field. The integral is performed using an Table 3 Material parameters for the thermal response for annealed 304 stainless steel k (W/m K) ρ (kg/m3 ) cp (J/kg K)

31 7940 750

151

/x |u|

(33)

The spatial interval is constrained to be small in comparison to a characteristic size associated with the finite element. A characteristic element size is defined for the 6-noded triangular elements employed here by, 2 + d2 + d2 d13 35 51 , where dchar = 3 (34) dij = (xi − xj )2 + (yi − yj )2 The integration step size is made proportional to a characteristic dimension of the element through which the streamline is passing. In this model, smaller elements are located near the tool pin.

5. Deformation and temperature distributions The FSW process was simulated with the model equations and numerical algorithms presented in the previous sections. The material motion below tool shoulder and around tool pin in the vertical direction implies that a thorough analysis of the process dictates the use of a 3D model. In this paper, however, we focus only on the material rotation around the tool pin and neglect the vertical movement of the material flow. Consequently, we employ a 2D formulation based on a plane strain assumption (no deformation in the plate normal direction). We consider two cases of tool pin and plate geometry as shown in Table 4. The two plates being joined in Case 1 are 500 mm wide and 1000 mm long. The tool pin is located between the plates and has a radius of 100 mm. Case 2 utilizes more realistic dimensions for FSW. The two plates are 50 mm wide and 100 mm long, where the tool pin diameter is 10 mm. The ratio of tool pin radius and plate width is 20% for both cases. The tool pin angular velocities and translational speeds are set in Case 1 to give similarity between the cases. The results of Case 1 are discussed first, followed by those for Case 2. Fig. 4 shows the 2D Eulerian model employed for the simulations. Material moves from left to right and the tool spins clockwise. Therefore the bottom sheet is the advancing side and the top sheet is the retreating side. The two plates are preheated at 373 K. The boundaries along both sides and at the outlet are assumed to be adiabatic. There is severe shearing along tool pin that provides substantial heating. Heat transfer is allowed between tool pin and the workpieces by convection. As pointed before, the traction from sliding friction between the tool and material is proportional to their velocity difference, as given by Eq. (6).

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Table 4 Overall 2D modeling cases and boundary conditions

L u¯ x (mm/s) T0 (K) κ0 (MPa) h (W/(m2 K)) T∞ (K) 3 (rad/s)

Case 1

Case 2

1000 1 373 150 1000, 1500 or 3000 300 or 800 0.1

100 3 or 1.7 373 150 1000 300 5

Note: L is the length of plate, u¯ x is the welding speed, T0 is the initial temperature, κ0 is the initial hardness, h is the heat coefficient, T∞ is the tool temperature, and 3 is the angular velocity of tool.

The 2D model is discretized with 1600 elements (6-noded triangles using a 7-point quadrature rule). Elements are concentrated near the tool surface where gradients of the velocity and temperature are greatest. There are numbers 20 elements along each boundary. One hundred streamlines of the velocity field are given in Fig. 5. The streamlines are numbered from the bottom of the domain to top. Twelve of these are identified in Fig. 5 for later discussion. These are 20, 35, 40, 43, 44, 47, 49, 51, 55, 60, 70 and 80 and are labeled (a) to (l), respectively. The lines show the paths that material particles follow from the inlet to the outlet of the Eulerian region. The longest streamline is number 44 and approaches closer to the tool than any other streamline shown. It traverses some parts of both advancing and retreating sides. Streamline 43 is just below 44 and passes to the lower side of the tool pin, never traversing the retreating side. The interesting features of the flow field occur close to the tool. Streamlines near the top and bottom boundaries exhibit almost straight paths. Near the tool, however, reversals in the direction of flow occur twice on the advancing side. A stagnation point (a point with zero-valued velocity) exists in the vicinity of one reversal. The dotted line shows the material region swept out by the tool pin. Fig. 6 shows velocity vectors of the flow field for a limited number of nodal points. A steady-state boundary layer

Fig. 5. Streamlines of the material flow through the Eulerian domain (Case 1). The large arrows represent the tool rotation direction.

appears to surround the tool pin. This layer is evident in Fig. 6(b), which shows points close to the pin. The velocity vectors along the tool pin interface are tangent to the tool pin surface. This implies that the tool pin interface is itself a streamline of the flow field that is closed within the Eulerian domain. The magnitude of the velocity diminishes rapidly with distance from the pin. High velocity gradients translate into high values of the deformation rate, giving a layer of intense shearing adjacent to the tool pin. Fig. 7 shows the effective stress and mean stress fields. The effective stress levels remain less than 200 MPa everywhere in the region. The highest effective stress is found in the stirred zone of the upper plate. This effective stress distribution is reasonable considering the elevated temperature levels produced by the stirring deformation. The mean (hydrostatic) stress distribution shows compression exists upstream of the tool and tension exists downstream of the tool.

Fig. 4. 2D FE mesh of the FSW model and boundary conditions. WD: the weld direction, AS: the advancing side of the tool: (a) Eulerian mesh, (b) schematic of the FSW geometry.

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Fig. 6. Velocity vectors (Case 1) at selected nodal points (a); an enlarged figure in the vicinity of the tool pin (b).

Fig. 7. Effective stress and mean stress distributions (MPa) over the Eulerian domain (Case 1): (a) effective stress and (b) mean stress.

Fig. 8 shows typical temperature and strength distributions over the Eulerian domain. Material enters the domain at 373 K. Little temperature change occurs prior to entering the stirred region due to the low diffusivity of the stainless alloy. Once material is in the stirred region it heats rapidly.

Considering the direction of the material motion relative to the tool rotation, the advancing side has higher shearing rates than the retreating side. For this reason, there is greater dissipation along the advancing side than along the retreating side. The comparatively high temperature lowers the flow stress

Fig. 8. Temperature and strength distributions (Case 1): (a) temperature (K) and (b) strength, κ (MPa).

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and the saturation limit of the strength. The bottom and top sides far from tool pin experience less heating than material in the TMAZ or HAZ. This results in a comet-like tail in the temperature distribution after welding. In contrast to the temperature distribution, highest values of the state variable, κ, are predicted on the retreating side (Fig. 8(b)). Recall that κ represents the isotropic strength, which is the limit of the flow stress at low temperature or high strain rate. The temperature and deformation rate influence the saturation value through the Fisher factor in opposite directions. The temperature is particularly important considering the substantial variation it exhibits over the domain. Because the strains are large in the stirred region, the strength can readily approach its saturation limit. This is evident in the distribution of the strength downstream of the pin. Along streamlines that pass through the hottest zones, the strength is lower due to the lower saturation limit associated with higher temperature. Material that experiences large strains but at a lower temperature exhibits a higher strength, consistent with a higher saturation limit, as seen for material in the upper part of the domain. The tool temperature will influence the temperature distribution experienced by the workpiece material. Fig. 9 shows temperature profiles for different assumed values

of the tool temperature. With the same heat coefficient of h = 1000 W/(m2 K), a higher tool temperature will result in higher workpiece temperatures around the tool (Fig. 9(a) and (d)). Similarly, higher heat coefficients of h = 1500 and 3000 W/(m2 K) contribute to the lower temperatures around tool pin by providing a more effective means to remove heat (Fig. 9(a)–(c)). Fig. 10 gives the strength distributions corresponding to the cases considered in Fig. 9. When the temperatures are lower, the maximum values of the strength are higher. Conversely, higher temperatures contribute to lower values of the strength downstream of the tool pin. The Eulerian domain can be partitioned roughly into two types of zones indicated in Fig. 5. The upper and lower parts of plates are specified as base metal and the central part where the tool passes is the thermomechanically affected zone (TMAZ). Material on the advancing side experiences more severe shearing than material on the retreating side, generally. In addition, the closer material passes to the tool, the more severe is the deformation. The greater the distance a particle has traveled along its streamline, the greater the strain it experiences. Fig. 11 shows the variations of some thermomechanical quantities along both X = 0 (Y axis) and Y = 0 (X axis) based

Fig. 9. Temperature distributions (Case 1) (K): (a) h = 1000 W/(m2 K) at θtool = 800 K, (b) h = 1500 W/(m2 K) at θtool = 800 K, (c) h = 3000 W/(m2 K) at θtool = 800 K, (d) h = 1000 W/(m2 K) at θtool = 373K.

J.-H. Cho et al. / Materials Science and Engineering A 398 (2005) 146–163

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Fig. 10. Strength distributions (Case 1) (MPa) for various combinations of tool temperature and convective coefficient: (a) h = 1000 W/(m2 K) at θtool = 800 K, (b) h = 1500 W/(m2 K) at θtool = 800 K, (c) h = 3000 W/(m2 K) at θtool = 800 K, (d) h = 1000 W/(m2 K) at θtool = 373 K.

on numerical data taken along streamlines shown in Fig. 5. FSW is an inherently asymmetric process due to the combined effects of the workpiece movement and tool rotation. This results in asymmetric distributions of temperature and effective deformation rate. The effective deformation rate is higher on the advancing side than on the retreating side. It can be seen that the temperatures on the advancing side are also higher than those on the retreating side along X = 0 axis (line with squares). The rapid rise in temperature as material enters the stirred zone is evident upstream of the tool pin, as is the slow cooling of material downstream of the pin along Y = 0 axis (line with circles). The higher temperature downstream of the tool pin is related to continued heating as material flows around tool pin. In contrast to temperature and effective deformation rate, the strength is the same on either side of tool pin because a saturation value is imposed on the streamline lying on the tool pin interface. It was assumed that material in the steady state boundary layer has a constant strength, κss , in Eq. (20). Along the Y = 0 axis, the strength downstream of the pin is higher than along the sides due to the combined characteristics of hardening and heating during FSW. This region also exhibits higher temperatures.

The effective stress shows little difference close to tool pin irregardless of position, but the distributions change abruptly away from the tool pin. The convergence of the solution with the number of finite elements used to discretize the domain is explored in Fig. 12. The evolution of the strength depends strongly on the temperature and effective deformation rate and this is used here to gauge convergence of the full solution. Six different levels of discretization are used and all employ six-noded triangular elements. These meshes have 400, 784, 1024, 1296, 1600, 2304 elements, respectively. The sharp variations of strength around tool pin are captured with the more highly resolved meshes. Case 2 considers the FSW process with dimensions that are smaller by a factor of 10 for the tool pin diameter and workpieces of Case 1. Therefore, tool diameter decreases from 200 to 20 mm and workpiece width from 1000 to 100 mm. Welding speeds are increased to 1.7 and 3 mm/s and the tool pin angular velocity is 5 rad/s. Other model parameters are the same as in Fig. 4(a). Altogether, Cases 1 and 2 exhibit similar thermomechanical histories along streamlines passing near the tool pin. The temperature and strength

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Fig. 11. Effective deformation rate (S −1 ), temperature (K), strength (Pa), and stress (Pa) variations along Y and X axis (Case 1).

distributions are shown in Fig. 13. Increasing the welding speed generally results in increased heat generation and a tendency toward higher temperatures. This is offset, however, by greater heat convection that more rapidly removes the heat from near the tool pin. In this case, the net effect is for the higher welding speed to produce a slightly lower maximum temperature as seen in the difference between Fig. 13(a) and (c). Stronger heat convection from increased welding speeds also results in a narrower region of elevated temperature. Fig. 13(b) and (d) show the strength distributions. The case of higher welding speed in Fig. 13(b) shows higher strength than that seen for lower welding speed shown in Fig. 13(d).

6. Texture and mechanical properties Changes in features of the microstructure can be estimated from knowledge of the thermomechanical history. Particularly important to FSW is texture evolution as the process induces severe deformation. In this section, we briefly outline the equations governing texture evolution. This is followed by an examination of the texture evolution along the outlet boundary. Finally we discuss the texture-dependent

mechanical properties across the outlet of the Eulerian domain. 6.1. Crystal constitutive relations The velocity gradient, L, which has symmetric, D, and skew, W, parts, is obtained by taking the spatial gradient of the velocity field, L=D+W=

∂u ∂x

(35)

This motion at the macroscopic scale is linked to the crystal scale by adopting the extended Taylor hypothesis [34]. Individual crystals are assumed to experience the macroscopic velocity gradient identically. In the case of the high symmetry crystals and large strains considered here, this is a reasonable approximation. As our attention is focused on regions that experience large deformations, elastic effects are neglected and crystals are assumed to deform solely by crystallographic slip. In this situation, the crystal velocity gradient can be decomposed into portions associated with slip and lattice spin,  LC = 3 + (36) γ˙α (bα ⊗ nα ) α

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157

An expression for the crystal stress follows by combining the symmetric part of crystal velocity gradient in Eq. (36) with Eqs. (37) and (38) 



σ C = CC [DC ]

(39) CC

where the stiffness is defined as

α 1/m−1  γ˙0 τ −1

Pα ⊗ Pα CC = α τˆ α ˆ τ α

(40)



and DC is the crystal deviatoric deformation rate. Eq. (39)    may be solved for σ C given DC . DC is known from L assuming the extended Taylor Hypothesis, L = LC [36]. The Taylor assumption is based on strain compatibility and gives  an upper-bound of the stress. With σ C known, one can recover γ˙α and the skew part of crystal velocity gradient in Eq. (36) yields an expression for the lattice spin  3=W− (41) γ˙α Qα α

where W is the macroscopic spin, and Qα = skew (bα ⊗ nα ). For a single crystal, the lattice spin is integrated to compute the lattice reorientation, and thus the texture evolution. 6.2. Texture prediction during FSW

Fig. 12. Dependence of the strength on the mesh discretization (Case 1): (a) along X = 0 axis and (b) along Y = 0 axis.

where 3 is the lattice spin, γ˙α is the shear rate on slip system α, and (bα ⊗ nα ) is the Schmid tensor with bα , the slip direction and nα , the slip normal for slip system α. A viscoplastic constitutive relation is used to relate the shear rate on a slip system to the resolved shear stress [35]:

τ α

τ α

1/m−1 α ˙ γ = γ˙ 0 (37) τˆ τˆ where m is the rate sensitivity, τˆ the slip system hardness (assumed identical for all slip systems), and γ˙ 0 a reference strain rate. Eq. (37) is more simplistic than the equation for the flow stress in Hart’s model, but is sufficient for estimating texture evolution. The resolved shear stress on slip system α is obtained as the projection of the crystal stress onto the slip system τ α = Pα · σ C



where Pα = sym (bα ⊗ nα ) and σ stress.

(38) C

is the deviatoric crystal

Polycrystal aggregates with 1000 discrete crystals are used for consideration of texture evolution during FSW. These single crystal aggregates are chosen initially from a uniform distribution. Using the crystal constitutive equations from Eqs. (35)–(41), texture evolution can be investigated by extracting the thermomechanical histories along streamlines and using these as input to the evolution equations. As stated before, FSW processes involve significant shearing of the material. In general, shear deformation occurs by glide on {111} planes in 110 direction in fcc materials that results in the typical shear texture component [37]. Ideal shear textures are usually represented by two fibers, referred to as the A-fiber, {1 1 1}u v w and the B-fiber, {h k l}110. For shear, two vectors {h k l} and u v w, are aligned with the shear plane normal and shear direction, respectively. Ideal rolling or plane strain deformation modeling can be understood as pure shear by a rotation of 90◦ along RD followed by rotation of 45◦ along TD. It is found that there is some similarity between pure shear and simple shear [36]. Fig. 14 displays 1 1 1 pole figures along outlet boundary for texture evolution during FSW. In Fig. 14, the pole figures in (a)–(l) correspond to the evolved textures along streamlines 20, 35, 40, 43, 44, 47, 49, 51, 55, 60, 70 and 80 in Fig. 5, respectively. Fig. 14(d)–(f) shows skeleton lines similar to those of A and B-fibers in simple shear or torsion textures [37]. They are not exactly the A and B-fibers, however, because the major texture components are rotated relative to a shear plane and shear direction. Streamlines 20 and 80 in Fig. 5(a) and (l) are more distant from the tool pin (more than one

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Fig. 13. Temperature (K) and strength (MPa) distributions for Case 2: (a) temperature (u¯ x = 3 mm/s); (b) strength (u¯ x = 3 mm/s); (c) temperature (u¯ x = 1.7 mm/s); (d) strength (u¯ x = 1.7 mm/s).

pin diameter), and show weak textures consistent with modest degrees of plane strain deformation. From EBSD measurements, it is generally known that gradients in crystallographic texture are present in parts joined by FSW. For FSW of stainless steel, texture gradients were also found through the thickness and across the joints. A more complete discussion of texture evolution is presented in a separate article [38].

nitude. Both isochoric deformation rate and deviatoric stress exist in five-dimensional spaces. The deviatoric deformation rate tensor, D , and the deviatoric Cauchy stress, σ  , are given as 5D vectors by

6.3. Mechanical properties during FSW

(s1 , s2 , s3 , s4 , s5 )

Mechanical properties for polycrystals consisting of more than one textural component can be calculated using an average of the behavior of each texture component. The flow surface of polycrystals can be also computed using averaging of single crystal properties [39]. Piecewise flow surface in stress or deformation rate space provides alternatives to analytical expansions for representing anisotropic strength. An anisotropic flow surface in stress space gives the flow stress directly via its coordinates. For associated flow, the deformation rate is normal to the surface. It is also possible to define a surface in deformation rate space corresponding to a reference deformation rate of fixed mag-

(d1 , d2 , d3 , d4 , d5 ) =

=

 − D ), ((D22 11

 − σ  ), ((σ22 11

√   , 2D , 2D ) 3D33 , 2D23 31 12 √ 2

√

 , 2σ  , 2σ  , 2σ  ) 3σ33 23 31 12 √ 2

(42)

(43)

In the computation of the flow surface, the texture is represented by a set of 1000 orientations. To visualize the flow surface, two sections through the deviatoric stress space are used. Flow sections are obtained by setting the first two deformation rate components to zero (d1 , d2 , d3 , d4 , d5 ) = (0, 0, d3 , d4 , d5 ) or by setting two shear component to zero (d1 , d2 , d3 , d4 , d5 ) = (d1 , d2 , 0, 0, d5 ). Here, we are interested in 2D modeling and so we look mainly at (d1 , d2 , d3 , d4 , d5 ) = (d1 , d2 , 0, 0, d5 ).

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159

Fig. 14. 1 1 1 pole figures along outlet boundary in Fig. 5. The dotted circles show the overall rotation of the textures.

The flow stress obtained for a uniform orientation distribution is shown in Fig. 15(a). Note that the magnitude of the flow stress is not uniform even for the uniform orientation distribution. This is an attribute of the polycrystal plasticity model and is evident in the facet and vertex topology in the stress space representation. The uniform orientation distribution gives rise to two-fold symmetry and axisymmetry in flow surface of (d1 , d2 , 0, 0, d5 ) along the d5 and d2 directions, respectively. Another measure of anisotropy is coaxiality between the deviatoric stress and the associated deformation rate in Fig. 15(b). This is computed as the cosine of the angle between the deviatoric stress and deformation rate vectors. Its value ranges from zero to unity. The closer a value is to unity the greater the fraction of the stress that does work in conjunction with the associated deformation rate. For a material modeled with the Von Mises criterion, these vectors would always be aligned and the coaxiality would be unity. The faceted character of the flow surface, even for a uniform orientation distribution, together with

the normality condition inherent to crystal plasticity, dictates a varying state of coaxiality over the deformation rate sphere. The flow surface and coaxiality for simple and pure shear deformations are also presented for comparison to the uniform distribution in Fig. 16. As pointed out, simple and pure shear deformation are supposed to occur commonly during FSW. The symmetries found in a uniform distribution are not found anymore. The ranges of the coaxiality and flow stress for pure shear are about from 1.00 to 0.80 and 400 to 250 MPa, respectively. For reference, the ranges for simple shear show from 1.00 to 0.92 and 360 to 300 MPa, respectively. When, compared with a uniform distribution in Fig. 15, pure shear deformation shows more anisotropic behavior than simple shear over the deformation rate space. The flow surface and coaxiality for positions (a) and (e) in Fig. 14 are shown in Fig. 17. As found in the pole figure plots of Fig. 14, they show some mixture of simple and pure shear deformation.

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Fig. 15. Flow surface and coaxiality for a uniform distribution: (a) flow stress (MPa) and (b) coaxiality.

Fig. 16. Flow surface and coaxiality for pure and simple shear deformation texture: (a) pure shear flow stress (MPa); (b) pure shear coaxiality; (c) simple shear flow stress (MPa); (d) simple shear coaxiality.

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Fig. 17. Flow surface and coaxiality for some outlet points (points a and e in Fig. 14): (a) flow stress (MPa) at a; (b) coaxiality at a; (c) flow stress (MPa) at e; (d) coaxiality at e.

7. Hardness and temperature measurements The measured hardness profiles along three lines through the weld zone of 304L FSW plate are shown in Fig. 18. The welds were performed using a rotational speed of 600 rpm and a translational speed of 10.2 mm/min. The diameters of tool shoulder and tool pin were 25.4 and 5.6 mm, respectively. The hardness profiles were extracted from a map of hardness values taken over a cross section perpendicular to the welding direction. The elevated hardness is located along stirred welding zone or TMAZ. The hardness in most of the TMAZ spans a range from 200 to 240 HV. In the HAZ, which is supposed to be located between the TMAZ and the base material, the values of hardness are slightly lower. Hardness values for base material are generally lower than 190 HV. Hardness values along the top display the highest values, especially on the advancing side. The top part is expected to have smaller grain size from additional tool shoulder friction and faster

cooling, resulting in the higher hardness values. The asymmetric distribution near the top shows that strain hardening from deformation dominates over the softening that can occur from heat generation. In the modeling reported here, the tool shoulder effects are ignored. The results of the 2D modeling are best compared with middle and bottom parts, which are less affected by the tool shoulder. The retreating side displays a sharp maximum in the hardness value in the center part. Measurements from the bottom part also show higher values in the retreating side than in the advancing side. The measured hardness plot can be compared with the computed strength distribution. Fig. 8(b) shows that the higher values of strength are within the TMAZ and that there is asymmetry in the distribution. The retreating side shows higher values than the advancing side. This trend is similar to the measured hardness profiles. Higher temperatures along the advancing side promote recovery that lowers the hardness.

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8. Conclusions From the simulations and experiments we see following trends:

Fig. 18. Hardness distribution (HV) for one pass FSW of 304L stainless steel measured along thickness direction. (Courtesy of A. Brandemarte, NSWCCD).

Experimentally, the peak temperatures recorded for the welds of 304L stainless steel depend on the tool rotation rate and the distance from the weld centerline. Temperature measurements for FSW of 304L stainless steel are shown in Fig. 19. Thermocouples were set in the midplane of both advancing and retreating plates. The maximum temperature measured occurs close to tool pin on the advancing side and exceeds 1000 K. Farther from the tool pin on the advancing side, 10 mm from the tool center, the peak temperature is closer to 900 K. In contrast, at 10 mm from the tool center on the retreating side, the peak temperature is about 800 K, lower by about 100 K than those observed on the advancing side. This temperature difference between advancing and retreating sides is also predicted from Eulerian model as shown in Figs. 11 and 13(a).

• The temperatures remain relatively cold upstream of the tool, rise quickly near the pin, and exhibit a comet-like tail downstream of the tool. Temperatures are higher on the advancing side than the retreating side by about 100 K in both Eulerian modeling and experiments of 304L. • The strength in the weld zone as modeled with a scalar state variable is predicted to be higher than in the base material. At the exit, material on the retreating side shows higher values of strength than material on the advancing side. • Experimentally, the friction stirred zone is hardened during the FSW process. Microhardness measurements on 304L austenite stainless steel show the increased values over the undeformed base metal. Hardness distributions along the middle and lower parts of the plate show higher values along the retreating side than along the advancing side. • The highest effective stress is found in the stirred zone, higher on the advancing side than on the retreating side; the mean stress is compressive ahead of tool and tensile behind of tool. • Texture evolution is predicted along outlet boundary using the velocity gradients along streamlines of the flow field together with a model for polycrystal plasticity. Textures vary greatly with position. These imply that the mechanical properties also will vary strongly across the weld zone. • Flow surface and coaxiality distributions show anisotropy arising from FSW. Especially, material from the friction stirred zone shows more anisotropy than material passing farther from the tool pin.

Acknowledgements Support for this work has been provided by the Office of Naval Research under contract NOOO14-03-1-0250. The authors would like to thank Professor A. Reynolds, USC, for providing the welded material, Professor C. Sorensen, BYU, for temperature distributions and A. Brandemarte, NSWCCD, for hardness distributions.

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Fig. 19. Temperature distribution for one path FSW of 304L stainless steel measured around tool pin. The maximum temperature is located close to tool pin (Courtesy of Professor C. Sorensen, Brigham Young University).

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