Comput Mech (2008) 41:391–405 DOI 10.1007/s00466-007-0195-5

ORIGINAL PAPER

Numerical analysis of strain path dependency in FCC metals E. M. Viatkina · W. A. M. Brekelmans · M. G. D. Geers

Received: 25 January 2007 / Accepted: 21 May 2007 / Published online: 25 July 2007 © Springer-Verlag 2007

Abstract Strain path dependency in FCC metals is often associated with the anisotropy induced by the dislocation cell structure in deformed metals. In this paper, the mechanical behaviour of metals under various nonuniform deformation paths is studied with the use of a recently developed dislocation cell structure model. It is shown that this model correctly captures the essential features of strain path change effects for moderate strain path changes, i.e. the anisotropy and the dependency on the amount of prestrain. Next, a numerical analysis is performed to assess the micromechanical origin of the modified reloading yield stress and the transient hardening after strain path changes. The separate influence of anisotropic effects due to the cell structure morphology and residual internal stresses are thereby addressed and illustrated. The transient hardening behaviour after a strain path change is related to the adjustment of the internal stresses to the new loading. Results obtained are consistent with related experimental findings reported in literature. Keywords Dislocation cell structure · Internal stress · Strain path change effect · Plasticity anisotropy · Metal forming

1 Introduction Most of the industrial metal forming processes are characterised by a non-monotonic strain history, in which different E. M. Viatkina (B) · W. A. M. Brekelmans · M. G. D. Geers Department of Mechanical Engineering, Section of Materials Technology, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] W. A. M. Brekelmans e-mail: [email protected]

processing steps rapidly succeed each other. The induced changes in the strain path have a significant effect on the mechanical response of the metal involved and therefore influence the material behaviour of a blank during subsequent forming, thereby affecting the properties of the final product. The complexity and significance of strain path change effects require a careful investigation of the associated physical origins to enable an adequate modelling of the resulting mechanical behaviour. Macroscopically the effect of a strain path change manifests itself in an altered reloading yield stress, followed by transient hardening with a decreased hardening rate and, eventually, hardening recovery [4,14,19,20,23]. The effect depends on the amount of prestrain and the orientational difference between successive deformation modes. Experimental investigations dealing with strain path changes revealed that the effect of the texture on the reloading behaviour is weak at moderate deformations [10,15] and that slip anisotropy alone does not provide a satisfactory explanation for the observed strain path change effect [3,10,11]. Therefore, it was concluded that the effect of strain path changes originates from the anisotropy induced by dislocation structures. This was also experimentally confirmed by correlating the presence or absence of dislocation structures with the macroscopic behaviour after a strain path change [3]. During ongoing deformation, dislocations in FCC metals tend to pattern into 3D cell structures, with areas of low dislocation density in the cell interiors and high dislocation density in the cell walls [1,5,13,22]. This nonuniform distribution of dislocations provides an additional source of inhomogeneity and anisotropy in the material. The morphology of the dislocation structure developed after a particular loading path is clearly depending on the loading characteristics. After a strain path change, the resistance and adaptation of the existing dislocation structure to the loading in a new direction is

123

392

typically accompanied by an increased reloading yield stress and transient hardening. Recent investigations [2,21] emphasise the importance of the internal stress developed in the cell structure and its resulting effects on the deformation behaviour of the material. Much of the present understanding of the influence of the long-range internal stresses on the material hardening has been acquired from observations made in fully reversed plastic deformation [7,8]. According to these studies, during prestraining, the long-range internal stresses within the cell interiors act in a direction opposite to that of the applied stress. When the direction of the applied stress is reversed, these internal stresses act in the same direction as the newly applied stress, causing reversed plastic deformation to be initiated at an applied stress level lower than that attained at the end of the prestrain. Thus, long-range internal stresses in the cell structure are intrinsically associated to the decreased reloading yield stress, commonly known as the Bauschinger effect. After the reloading yield stress has been reached, the plastic deformation in the new direction leads to a reorientation of the internal stress state. Wilson et al. [21] studied the effect of internal stresses on the hardening behaviour of aluminium for various strain path changes. It was concluded that the re-orientation of the long-range internal stresses acting in the cell interiors constitutes a source of an increased hardening rate at the early stages of deformation after a strain path change. In recent work, a continuum model of the dislocation cell structure has been proposed in Viatkina et al. [18] that incorporates internal stresses according to a physically based concept. The material with the dislocation cell structure is modelled as a composite consisting of a periodic array of two phases: hard cell walls and soft cell interiors. The internal stress is derived as a natural result of plastic deformation incompatibility. The approach also incorporates the morphology and orientational anisotropy of the cell structure in an idealised manner. In the present paper, this cell structure model is used to predict the mechanical behaviour of metals under various nonuniform deformations. First, the model is verified by comparing predicted and experimentally determined reloading yield stresses after various strain path changes. Next, the deformation behaviour of copper under various nonuniform strain paths is analysed. The effect of the internal stresses, cell geometry and geometrical anisotropy on the macroscopic behaviour is investigated. The model is shown to provide a clear insight into the anisotropy behaviour of FCC metals due to dislocation cell structures.

Comput Mech (2008) 41:391–405

Fig. 1 Model geometry of the cell structure and its components

structure model introduced in [18]. A schematic representation of the cell structure model is given in Fig. 2. The main features of the model are shortly summarised below. An FCC metal with a cell structure is idealised by a composite consisting of four uniform components: the cell interiors and three mutually perpendicular sets of cell walls. The cell wall components represent the phases with a high dislocation density and the cell interior component represents the area with a low dislocation density. The cell structure is modelled as a 3D periodic configuration of cuboid cells formed by three mutually perpendicular sets of planar cell walls (Fig. 1). Two types of dislocations are introduced into the model: geometrically necessary dislocations (GNDs) and statistically stored dislocations (SSDs). The SSDs influence the local deformation behaviour by means of short-range interactions with gliding dislocations. The GNDs create long-range internal stresses in the material. The statistically stored dislocations inside the cell and wall components are assumed to be distributed uniformly, along with the Cauchy stress tensor σ and deformation gradient tensors F inside each component:
{σ (x), F(x)} =

{σ c , F c }, {σ wi , F wi },

x ∈ cell interior (1) x ∈ wall i, i = 1, 2, 3.

Traction continuity and compatibility of deformations are enforced on the interfaces between the cell interiors and the walls: (σ c − σ wi ) · nit = 0

(2)

2 Cell structure model

(F c − F wi ) · (I − ni0 ni0 ) = 0 with i = 1, 2, 3

(3)

The deformation behaviour of FCC metals with a dislocation cell structure is modelled here with the use of the cell

where n1 , n2 and n3 are the normal vectors to the cell-wall interfaces, and the superscripts 0 and t indicate the vectors

123

Comput Mech (2008) 41:391–405

393

in their initial and current configuration, respectively. The ˜ σ˜ } of the composite is calaverage mechanical response { F, culated from the relative contributions of the local responses: σ˜ = f w1 σ w1 + f w2 σ w2 + f w3 σ w3 +(1 − f w1 − f w2 − f w3 )σ c

(4)

˜ = f w1 F w1 + f w2 F w2 + f w3 F w3 F +(1 − f w1 − f w2 − f w3 )F c

(5)

with f w1 , f w2 and f w3 the volume fractions of the corresponding wall components.1 The local mechanical response of the individual components i of the composite, where the subscript i stands for c, w1, w2 and w3, is described according to the classical theory of elasto-plasticity for finite deformations. The deformation gradient tensor F i of the current state with respect to the reference configuration, is multiplicatively decomposed into p a plastic part F i , defining the deformation of the stress-free intermediate state with respect to the reference state, and an elastic part F ie , representing the deformation of the current configuration with respect to the intermediate state, according to: p

F i = F ie · F i .

(6)

The velocity gradient tensor L i in each phase is given by: ˙ i · [F i ] Li = F

−1

.

(7)

The symmetric and skew-symmetric parts of L i are the rate of deformation tensor Di and the spin tensor W i , respectively. By substitution of Eq. (6) into Eq. (7) these tensors can also be decomposed (additively) in an elastic and a plastic part. It is well-known that the decomposition in Eq. (6) is not unique because rotational effects can be assigned either to F ie or to p F i . Uniqueness is restored by the extra condition that the p plastic deformation occurs spin-free: W i = 0. As a consequence of this choice, additional rotations superimposed to the original deformation process are fully attributed to the elastic deformation gradient tensor F ie , while the plastic p deformation gradient tensor F i is not effected. This implies that the elastic deformation gradient tensor F ie is appropriate to determine the stress in the current configuration. The intermediate configuration defined by the plastic p deformation gradient tensor F i is considered as the reference state for the elastic behaviour, implicitly assuming that the elastic constitutive response is not affected by plastic slip. The second Piola–Kirchhoff stress measure τ i is 1

The mechanical responses of the wall intersections are approximated by the response of the neighbouring walls. Consequently, the wall volume fractions include the contributions of the intersections.

defined with respect to that configuration and supposed to be linearly expressed in the elastic Green–Lagrange strain tensor associated with F ie , according to Hooke’s law (isotropic hyper-elastic material behaviour). As the elastic deformations will be small, the second Piola–Kirchhoff stress tensor is suitable to be applied for the plastic part of the deformation. The local plastic deformation of the individual components i of the composite is described using Von Mises J2 elastoplasticity with the following yield function Ψi : √ Ψi = s¯i − α M Gb ρi  3 d d s : si and si = τ i + β i with s¯i = 2 i

(8)

with G the shear modulus and b the magnitude of the Burgers vector, while α and M are additional material parameters. The isotropic hardening is here defined by the local density ρi of the statistically stored dislocations whereas the kinematical hardening results from an internal stress β i , created by the geometrically necessary dislocations. The rate of plasp tic deformation Di is assumed to be governed by an assop ciated flow rule. Exploiting W i = 0, the plastic part of the p deformation represented by the tensor F i then follows from: ˙ ip = D p · F p F i i

(9)

using an adequate integration scheme. The geometrically necessary dislocations (GNDs) are naturally present in a material with heterogeneous plastic deformation and hence also in a material with a cell structure. Under external loading, a nonuniform field of plastic strains is present in the cell structure. In the cell interior phase, plastic slip is easier compared to plastic slip the cell walls, where the deformation is impeded by a high density of tangled dislocations. To ensure compatibility of the plastic deformation across the interface between the hard and soft phases, polarised layers of geometrically necessary dislocations will develop at this interface. In a discrete approach the number of GNDs is coupled to the discontinuity of the plastic deformation gradient in the cells and the cell walls. The field of GNDs will induce long-range self-equilibrating stresses in the cell structure. This internal stress field has to be determined by the solution of a well-defined elastic incompatibility problem. Within this context, the well-established method of Green’s functions can be used, to determine the internal stress of a complete field of GNDs [18]. After calculation of the internal stress field an averaging procedure has to be performed to restore consistency with the point of departure in the present modeling approach as expressed by Eq. (1). The long-range stresses β, see Eq. (8), created by the interface dislocations in a material with a cell structure have been evaluated in a previous work [18].

123

394

Comput Mech (2008) 41:391–405

Fig. 2 Schematic representation of the cell structure model

The evolution of the density of the statistically stored dislocations (SSDs) in the cell interiors ρc , and inside the three mutually perpendicular walls ρw1 , ρw2 , ρw3 is described by the following equations: M √ M √ (I ρc − Rρc )˙εc − C I ρc ε˙ c (10a) b b M √ (I ρwi − Rρwi )˙εwi = b 1 − f w1 − f w2 − f w3 M √ +C I ρc ε˙ c , i = 1, 2, 3 b 3 f wi (10b)

ρ˙c = ρ˙wi

0 = ρ 0 = ρ 0 for t = 0, implying with initial conditions ρwi c that the undeformed material is assumed to be homogeneous. The quantities ε˙ c and ε˙ wi are the equivalent plastic strain rates in the cell interiors and walls, respectively. The material parameters I and R are related to the dislocation creation and dislocation annihilation, respectively. The last terms in

123

both equations describe the dislocation flux from the cell interior to the cell walls, with C a material dependent parameter. The incorporation of dislocation fluxes between cells and cell walls seems to be inconsistent with the assumption of a spatially constant dislocation density in these components. Indeed, at microscopic level, a density gradient is required to support these fluxes while a density jump would lead to an infinite flux. However, within the context of the averaging character of the modelling approach pursued, this continuum point of view is not applicable anymore. The dislocation flux triggers the creation and development of the dislocation structure and manifests itself through an increasing difference between the dislocation densities in the cell and the walls. The relations in the above modelling description are schematically represented in Fig. 2. The outline shows how to determine, e.g. the average stress for a given average deformation. More details on the derivation of the equations can be found in [18].

Comput Mech (2008) 41:391–405

395

(a) 300

3 Model verification

200

100

0 0

0.1

ε

(b)

cells walls GND

3 2.5 2 1.5 1 0.5 0

0

0.1

ε

0.2

(c) y σw /σ

VM y

(11)

VM

σi /σ

σ˜ i j = 0, for i j = 11.

0.3

Simulated dislocation density.

2

F˜11 = 1 + ε˙ 11 t;

0.3

0.2

Macroscopic Von Mises stress, simulated result and experimental data [15].

ρ (1015 m−2 )

To perform simulations, the model parameters are first identified to describe the mechanical behaviour of copper. These eventual values are given in Table 1. The cells are assumed to be cubic with equally thick cell walls. Moreover, the geometry of the cells is assumed fixed under deformation (unlike the dislocation densities). These simplifications may deviate from experimental observations at large deformations, however they are sufficiently accurate for the present analysis which concentrates on the internal stresses induced by the cell structure. The values chosen for the cell size D and the wall size w are typical for a cell structure observed at small to moderate deformations [13]. The value M = 2.5 corresponds to lattice orientations providing symmetrical multiple slip and α = 0.4 is obtained for multiple slip involving mutual intersection of dislocations of different slip systems [12]. These values are adopted for symmetric deformation modes as analysed further in this paper. The evolution of the statistically stored dislocations is defined by (10) and the material parameters I , R, C and ρ 0 . These parameters are identified here to fit the macroscopic stress-strain diagram for monotonic tension of copper given in Schmitt et al. [15], see Fig. 3a. To simulate uniaxial tension the macroscopic deformation F˜11 is applied with a strain rate ε˙ 11 equal to 5 × 10−4 [1/s]. All other directions remain stress-free, i.e. the following components of the macroscopic stress and deformation gradient tensors apply:

σV M (MPa)

3.1 Application to copper

During this simulation, the orientation of the cell structure was fixed such that the tensile axis coincides with the local cell direction [111]. This choice for this structure orientation

y σc /σ

1

VM

0.5 0

Table 1 Parameters for copper Parameter

Symbol

Unit

Value

Shear modulus

G

(GPa)

41.7

Poisson’s ratio

ν

(−)

0.34

Length of the Burgers vector

b

(nm)

0.257

Coefficient

α

(−)

0.4

Mean orientation factor

M

(−)

2.5

Cell size

D

(µm)

2.5

Wall thickness

w

(µm)

0.45

Initial dislocation density

ρ0

(m−2 )

9 × 1013

Creation rate parameter

I

(−)

0.1

Annihilation rate parameter

R

(nm)

2

Dislocation flux constant

C

(−)

0.6

0.1

ε

0.2

0.3

Local stress, experimental data for a Cu monocrystal [13]. Fig. 3 Monotonic uniaxial tension, Cu. a Macroscopic Von Mises stress, simulated result and experimental data [15]. b Simulated dislocation density. c Local stress, experimental data for a Cu monocrystal [13]

corresponds to the formation of a cell structure along the macroscopic planes with maximum shear as suggested by experimental observations [24]. The macroscopic stress evolution under monotonic deformation, however, does not enable to identify C in a unique way. The parameter C is associated with the material

123

396

Comput Mech (2008) 41:391–405

inhomogeneity and should be identified by, for instance, the dislocation density difference between cell and walls or the stress inhomogeneity or the quantitative level of the internal stresses. Unfortunately, the paper of [15] does not supply quantitative data on the microstructure. Therefore, information on mesoscopic inhomogeneities developing in copper under uniaxial tension was taken from another source. Mughrabi et al. [13] provided a detailed description of the dislocation cell structure in a [001]-orientated copper single crystal and the mechanical response of this material under uniaxial tension. This symmetric orientation of the single crystal allows for deformation by multislip and, therefore, it is expected that a cell structure is formed that is comparable to the structure developing in polycrystals. Figure 3c shows the evolution of the local yield stress σ y in the cell and wall phases, simulated with C = 0.6. The local yield stress as measured by [13] in a copper single crystal is also shown. The level of the stress inhomogeneity is captured adequately, particularly since only a single parameter has been introduced to this purpose. Figure 3b shows the corresponding evolution of the dislocation density. By extension of the model description, e.g. by selecting a more sophisticated expression for the yield function in Eq. (8) or for the evolution of the SSDs in Eq. (10), a better fit of the computational results to the experimental data might be at reach. However, the objective of this paper was to show the qualitative capabilities of a relatively simple approach that directly fits in the computational mechanics analysis of the materials of interest. Note that, at this stage, there is a clear lack of sufficient experimental data to support a more detailed quantitative comparison of theory and experiment. Next, with application of the identified parameters, tension–tension tests have been simulated and compared to the experimental data reported by Schmitt et al. [15]. The initial tension load is modelled according to (11) until a certain amount of macroscopic strain defined as

ε pr e =

T 

ξ

T1

Fig. 4 Tension–tension test

(a) T1

ξ

θ e2

With respect to a macroscopic basis. [001]

(b)

[111]

ξ

T1

[010]

e3

ξ

[11¯ 0]

[111¯ ]

[100]

˜ :D ˜ dt, D

˙˜ · F ˙˜ T ) ˜ −1 + F ˜ −T · F ˜ = 1 (F D 2

T2

e3

θ

With respect to the cell structure.

0

(12)

has been reached. Then the elastic unloading, defined as the state with a zero macroscopic stress, was simulated, followed by a second loading step in which the uniaxial tension was applied in another direction, see Fig. 4. A tension–tension strain path change is given by two angles {ξ, θ } that define the orientation of the second tensile axis T2 with respect to a macroscopic basis {T1 , e2 , e3 } associated with the first tensile step, see Fig. 5a. A dislocation cell structure develops during the first tensile stage, which is

123

T2

Fig. 5 Definition of the strain path change angles ξ and θ for tension– tension tests. a With respect to a macroscopic basis. b With respect to the cell structure

assumed to orient with the [111] cell direction2 along the ¯ was fixed in the tensile axis, Fig. 5b. The cell direction [110] simulations along the macroscopic direction e3 . Clearly, both strain path change angles influence the macroscopic reloading effects since they define different orientations of the cell structure with respect to the new loading axis. Experimental Here and further on [abc] denotes direction with respect to the cubic cell structure, in the coordinate system depicted in Fig. 5b.

2

Comput Mech (2008) 41:391–405

(a)

397

1.20 0.06 0.12 0.18

1.16 1.12 1.08 1.04 1.00 0

(b)

15

30

45

60

75

90

30

45

60

75

90

1.20 0.06 0.12 0.18

1.16 1.12 1.08 1.04

are shown in Fig. 6b. The predicted values of the reloading yield stress are well in the experimental range. The dependencies on the amount of prestrain and the angle between the successive tensile axes are also captured. The prediction, however, also demonstrates some deviations from the experiment. Considering the effect of the strain path change angle, the maximum reloading yield stress is found in a tension– tension test at ξ ≈ 55◦ , a so-called cross test, while the experiment shows the maximum effect in a test at ξ = 45◦ . Furthermore, the predicted effect of the amount of prestrain is qualitatively correct for small to medium strain changes only. For angles ξ larger than 75◦ the computed reloading yield stress decreases for larger prestrains, which is not consistent with the available experimental data. It is expected that the main reason for these deviations is a more complex actual dislocation evolution than the one taken into account by the present simplified cell structure model. Nevertheless, the proposed “continuum” cell structure model seems to capture the essential features of strain path change effects adequately. This indicates that the internal stresses and the cell geometry, which are the main components of the model, indeed constitute the major elements triggering a strain path change effect.

1.00 0

15

3.2 Application to aluminium Fig. 6 Strain path change effect for Cu in a tension–tension test. Reloading yield stress as a function of the strain path change ξ (θ = 0◦ ). a Experimental results [15]. b Simulation results

data supplied in Schmitt et al. [15] include information on the reloading yield stress as a function of the strain path change angle ξ , which is the angle between two successive tensile axes. The second angle θ is typically fixed by experimental settings but not given in Schmitt et al. [15]. Therefore, simulations with different values of θ were performed and θ = 0 was found to provide the best correspondence to the experimental data. This angle θ = 0 is further used for the tension–tension simulations, if not specified otherwise. The strain path change effect is evaluated by comparing the reloading yield stress σ y to the stress that is reached at the same total deformation ε pr e in a monotonic test, further denoted σmon . The reloading yield stress is defined here as the macroscopic Von Mises stress at which the macroscopic hardening rate after reloading drops to the level of the macroscopic hardening rate in a monotonic test, i.e. where dσ y/dε = dσmon/dε. Figure 6a shows the reloading yield stress experimentally determined from the tension–tension experiments [15]. The effect depends on the angle ξ between the tensile axes (with θ = 0) and the amount of prestrain. The simulation results

To demonstrate the capability of the cell structure model to predict strain path change effects in other cases, two additional evaluations next are performed. Tension–tension and rolling-tension tests have been simulated for aluminium and the results are compared to experimental data, reported by Li and Bate [11] and Jensen and Hansen [10], respectively. The material parameters defining the cell geometry and elastic behaviour were chosen typical for aluminium. The rest of the parameters were identified in the same manner as previously outlined for copper. Since experimental data on the inhomogeneity evolution in aluminium was not directly available, the parameter C has been defined to fit the reloading yield stress in a tension–tension test with ξ = 90◦ , and ε pr e = 0.1. The same cell geometry was used. The identified parameters are given in Table 2. The tension–tension tests were simulated in the same manner as for copper, with the same orientation of the cell structure. The strain path change angle θ = 30◦ was chosen to provide the best fit to the experimental data. Figure 7 shows the reloading yield stress as a function of the strain path change angle ξ . The predictions are in qualitative agreement with the experimental data. Comparing the data in Fig. 6a and in Fig. 7a, the effects found in copper by [15] and in aluminium by [11] clearly differ. The difference in the macroscopic behaviour is likely due to the different values used for w D and θ , implying a possible difference between the

123

398

Comput Mech (2008) 41:391–405

Table 2 Parameters for aluminium

T

Parameter

Symbol

Unit

Value

Shear modulus

G

(GPa)

26

Poisson’s ratio

ν

(–)

0.35

Length of the Burgers vector

b

(nm)

0.286

Coefficient

α

(–)

0.4

Mean orientation factor

M

(–)

2.5

Cell size

D

(µm)

2.5

Wall thickness

w

(µm)

0.375

Initial dislocation density

ρ0

(m−2 )

5.408 × 1012

Creation rate parameter

I

(–)

0.0513

Annihilation rate parameter

R

(nm)

4.55

Dislocation flux constant

C

(–)

0.15

(a) 1.12

εpre 0.10 0.05

σy /σmon

1.1 1.08 1.06 1.04 1.02 1 0

20

40

ξ

60

80

Tension-tension tests. Comparison with the experiment [11].

(b)

1.1

σy /σmon

1.08 1.06

ξ

RD

Fig. 8 Rolling-tension test

Next, rolling-tension experiments for aluminium are simulated and compared to the experimental data from Jensen and Hansen [10]. Monotonic rolling is simulated by applying the following evolution of the deformation gradient: ˜ F(t) = (1 + ε˙ t)e1 e1 + e2 e2 +

1 e3 e3 . 1 + ε˙ t

(13)

Here the basis vectors e1 , e2 and e3 define the rolling, transverse and longitudinal directions respectively. After ε pr e = 0.2, elastic unloading was applied to obtain a macroscopically stress-free configuration. The uniaxial tension (11) was simulated in the directions 0◦ , 15◦ , 30◦ , 45◦ , 60◦ , 75◦ and 90◦ with respect to the rolling direction (RD), see Fig. 8. The reloading yield stresses in the tension paths were calculated in the same manner as before. In the simulations, the orientation of the cell structure was fixed again along the planes with maximum shear, inclined ±45◦ to the rolling direction in the longitudinal plane and perpendicular to the rolling direction in the rolling plane. This choice is also supported by experimental observations [10], where traces of the cell walls were found at 45◦ with respect to the RD in the longitudinal plane and perpendicular to RD in the rolling plane. Figure 7b shows the calculated results and the experimental data for the rolling-tension experiments. The simulation results match the experimental data quite well. The model predicts the increased reloading yield stress and its dependency on the amplitude of the strain path change.

1.04

4 Numerical assessment of the structure–property behaviour

1.02 1 0

20

40

60

80

ξ

Rolling-tension tests. Comparison with the experiment [10], ε pre = 0.2. Fig. 7 Strain path change effect for aluminium. Reloading yield stress as a function of the strain path change angle ξ . a Tension–tension tests. Comparison with the experiment [11]. b Rolling-tension tests. Comparison with the experiment [10], ε pr e = 0.2

geometries and orientations of cells developed in the corresponding experiments of [11] and [15]. This issue is discussed in more detail in the next section.

123

In this section, various deformations are analysed numerically with the use of the cell structure model. The relationship between the macroscopic behaviour and the micromechanical mechanisms included in the model is discussed. 4.1 Monotonic deformation Under monotonic deformation the evolution of the statistically stored dislocations (Fig. 3b) tends to increase the material inhomogeneity. As a result, the local yield stress in the cells is lower than the yield stresses in the walls, and the difference increases with ongoing deformation, see Fig. 9a.

Comput Mech (2008) 41:391–405

399

(a) 500

cells walls

s11 (MPa)

400 300 200 100 0 0

(b)

500

0.2

ε Local stress s11 = τ11 + β11 .

0.3

Fig. 10 The internal stresses in the cell structure after 10% of monotonic tension

cells walls total

200

providing half of the local yield stress. The value of the internal stress in the cell interior is significantly lower, less than 25% of the applied stress. Note that this distribution of the internal stress is confirmed by experimental estimations of internal stresses in materials with a cell structure [6,16,17]. Figure 10 schematically visualises the state of internal stress state developed after 10% of deformation. During monotonic tension the principal axis of the internal stress coincides with the principal direction of the applied macroscopic stress. In the directions perpendicular to the tensile axis, the internal stress and the applied stress are compressive in both phases, with values of ∼1 MPa and ∼60 MPa for the cell interior and walls, respectively. In the direction of the tensile axis, however, the internal stress is tensile in the cell walls, ∼160 MPa, but compressive in the cell interior, ∼25 MPa. This result qualitatively agrees with the experimental study of [9], who emphasised the tensorial character of the internal stresses in nickel under tension.

100

4.2 Reloading yield stress

400

τ11 (MPa)

0.1

300 200 100 0 0

0.1

0.2

ε Applied stress τ11 .

0.3

(c) 500

cells walls

β11 (MPa)

400 300

0 0

0.1

0.2

ε Internal stress β11 .

0.3

Fig. 9 Stress evolution in the monotonic uniaxial tension of copper (11-components). a Local stress s11 = τ11 + β11 . b Applied stress τ11 . c Internal stress β11

The increased inhomogeneity of the yield stress leads to an increase of the plastic deformation incompatibility and, hence, triggers the evolution of the internal stresses, depicted in Fig. 9c for uniaxial tension. For uniaxial tension, a positive internal stress assists the applied stress inside the cell walls, while a negative internal stress counteracts the applied stress in the cell interior. Moreover, the internal stresses in the walls grow to a level approximately as high as the applied stress,

This section reports details of the mechanical behaviour of a cell material after a strain path change. The dependency of the reloading yield stress on the direction and magnitude of the strain path change is the result of the anisotropy developed in the material under deformation. The cell structure model includes two sources of anisotropy, the plastic inhomogeneity associated to the geometry of the cells and the residual stresses present in the material after unloading. Both effects are considered in more detail here. This analysis relies on the previously introduced example of tension–tension tests for copper. 4.2.1 Plastic inhomogeneity In the composite model, the compatibility of the deformation and the traction continuity on cell-wall interfaces, introduce

123

400

4.2.2 Residual stress Internal stresses develop in the plastically inhomogeneous material with a dislocation structure resulting from prestrain stage. Immediately after a strain path change the internal stresses remain in the material, unaltered during the elastic unloading, leading to significant residual stresses. Figure 12 shows the components of the residual internal

123

(a) 350 45

300

σV M

60 30 90

0

250 200 150 100 50 0 0

0.02

0.04

0.06

ε

0.08

0.1

0.12

Stress-strain diagram.

(b) 1.12

σ y (ξ)/σ y (0)

1.1 1.08 1.06 1.04 1.02 1 0.98 0

10

20

30

40

50

60

70

80

90

70

80

90

ξ(◦ )

Yield stress, θ = 0◦ .

(c) 1.12 1.1 1.08 σ y (ξ)/σ y (0)

a dependency of the stress and strain distributions on the orientation of the cubic cells. Besides, the internal stresses are intrinsically dependent on the cell geometry. As a result, the macroscopic response of the material is anisotropic and determined by the orientation of the cells with respect to the loading axis. Under monotonic deformation the dislocation structure constitutes a material heterogeneity with a geometry that is expected to be (at least partially) energetically favourable with respect to the current loading. A strain path change implies a change in the orientation of the cell geometry with respect to the loading axis, which triggers a change in the mechanical response of the material due to the anisotropy of the cells. Naturally, the response depends on the orientation of the new loading with respect to the cells. Figure 11a shows the stress-strain diagram for uniaxial tension applied to a material with a cell structure. The loading is applied along different directions with respect to the cells. The result exhibits a pronounced anisotropy of the yield stress. The initial structure used for these calculations corresponds to a cell structure obtained after 12% of tensile prestrain. Thus, the results in Fig. 11a reflect the second stage of the tension–tension experiments. But the second tension starts here from completely unloaded state without residual stress. Figure 11 shows that accounting for the plastic inhomogeneity due to the cell geometry only (i.e. without the residual stress contribution) introduces a significant anisotropy of the mechanical response. Figure 11b shows the yield stress as a function of the cell orientation (equal to the angle between the successive tensile axes). This result can be interpreted as the contribution of the plastic inhomogeneity to the strain path change effect σ y/σmon . It is clear that this contribution already provides a significant effect with qualitative tendencies as observed in the experiments, see Fig. 7b. The major part of the predicted effect is attributed to the incorporation of the internal stresses in the model. Figure 11a, b show the anisotropy effect obtained in tension–tension tests with θ = 0◦ . Variation of the strain path change angle θ defines different orientations of the cells with respect to new loadings and leads to different mechanical responses. Figure 11c, for instance, shows the same effect but obtained with θ = 30◦ .

Comput Mech (2008) 41:391–405

1.06 1.04 1.02 1 0.98 0

10

20

30

40

50

60

ξ(◦ )

Yield stress, θ = 30◦ .

Fig. 11 Uniaxial tension applied to a material with a developed cell structure (but without residual stresses). a Stress–strain diagram. b Yield stress, θ = 0◦ . c Yield stress, θ = 30◦

stresses with respect to the new loading axis (tension–tension experiments). In the figure, ξ = 0◦ corresponds to reloading in the same direction, during which the residual stress components are equal to the components of the internal stresses developed upon prestraining. If reloading is performed in a direction perpendicular (ξ = 90◦ ) to the prestrain direction, the 11 and the 22 components of the residual stress are interchanged compared to the ξ = 0◦ case.

Comput Mech (2008) 41:391–405

(a)

401

200

cell wall1 wall2 wall3

150

β11

100

50

0

-50 -100 0

10

20

30

40

50

60

70

80

90

70

80

90

◦

ξ( ) 11-component. (b)

200

cell wall1 wall2 wall3

150

β22

100

50

0

-50 -100 0

10

20

30

40

50

60

ξ(◦ ) 22-component. (c)

20 0

β12

-20 -40 -60 -80

cell wall1 wall2 wall3

-100 -120 0

10

20

30

40

50

60

70

9 80

90

◦

ξ( ) 12-component. Fig. 12 Components of the residual internal stresses at the beginning of the second tensile step versus the strain path change angle ξ (θ = 0◦ ). a 11-Component. b 22-Component. c 12-Component

In the general case of complex deformation, the nonuniform distribution of the applied stresses provides another source of residual stresses that retain in the material after elastic unloading. In tension–tension tests, however, the applied stresses develop almost uniformly during the first loading test, see Fig. 9c, and therefore create negligible residual stresses after unloading. The internal stresses, therefore, are the main source of the residual stresses in the tension–tension tests. To study the effect of the residual stress on the reloading behaviour, uniaxial tension is simulated first along the ¯ cell [111] cell direction, succeeded by tension in the [111] direction, i.e. {ξ ∼ 70◦ , θ = 30◦ }. These loading modes are equivalent with respect to the cell structure. Consequently, the geometrical anisotropy of the cell structure does not affect the reloading behaviour, see Fig. 11c. Thus, the effect of the strain path change observed in this test is caused by the residual stress only. Figure 13 shows the stress evolution during the tension– tension and monotonic tension tests. Only the stress components in the tensile direction are depicted here for clarity. During monotonic tension, a high tensile internal stress assists the applied stress in the walls, while the compressive internal stress resists the applied stress in the cells. After the strain path change, the prestrained material reflects the presence of the previously developed residual stress: in the direction of the new tensile axis it is compressive in the walls and tensile in the cell interiors, see Fig. 12a for ξ ∼ 70◦ . Consequently, the applied stress in the walls is counteracted by the residual stress, see Fig. 13d. The externally applied stress should therefore grow much higher to create the same local yield stress, in comparison with monotonic deformation, Fig. 13c. Opposite in cells, the residual stress assists the applied stress, where, a smaller externally applied stress is needed to initiate plastic deformation. Both, the increase of the applied stress in the walls and the decrease of the applied stress in the cells contribute to the macroscopic response. As a result an increased reloading yield stress is observed after the strain path change (Fig. 13a). This example demonstrates that the reloading yield stress depends on the difference between the residual stresses and the internal stresses that is “native” to the reloading deformation mode. Figure 12 shows that, in the tension–tension test, the deviation of the residual stress from the monotonic internal stress (at ξ = 0) increases with the angle between the successive tensile axes. (The residual stress in the walls has a major effect, since it provides the increased reloading yield stress.) In conclusion, the macroscopic reloading yield stress due to dislocation induced residual stresses is expected to increase with the angle of the strain path change.

123

402

Comput Mech (2008) 41:391–405

(b) 500

250

400

200

s11 (MPa)

σV M (MPa)

(a) 300

150 100

cells walls

300 200 100

50

0 0

(c)

0

0.1

0.15

0.2

ε Macroscopic Von Mises stress.

0

0.25

(d)

500 cells walls total

300 200

0.15

0.2

0.25

0.1

0.15

0.2

0.25

ε L ocal stress s11 = τ11 + β11 .

cells walls

300 200 100

0

0

0.05

0.1

500

100

0

0.05

400

β11 (MPa)

400

τ11 (MPa)

0.05

0.1

0.15

0.2

ε Applied stress τ11 .

0.25

0

0.05

ε Internal stress β11 .

Fig. 13 Stress evolution after a strain path change with {ξ ∼ 70◦ , θ = 30◦ }, ε pr e = 0.12. With respect to the cell structure the tensile ¯ orientation. The axis changes from the [111] orientation to the [111]

monotonic response for uniaxial tension (along the [111]-cell direction) is also shown. a Macroscopic Von Mises stress. b Local stress s11 = τ11 + β11 . c Applied stress τ11 . d Internal stress β11

4.2.3 Strain path change effect

a linear increase with the angle. Note that, for the tension– tension tests considered here, the residual stresses are almost invariant with respect to θ . The same qualitative change of the effect occurs if the volume fraction of the walls is reduced. In a cell structure with thin walls the contribution of the plastic inhomogeneity is found to be smaller and the overall response is dominated by the effect of residual stresses. The reloading yield stress in such materials grows linearly with the angle of the strain path change, independently of the imposed reorientation. The influence of the cell geometry on the strain path change effect has already been observed in the previous section, where the same type of tension–tension tests were simulated for copper and for aluminium. Different values for the volume fraction of the walls w D and for the cell orientation θ were used. Simulations with copper are strongly affected by the plastic inhomogeneity and therefore demonstrate a maximum reloading yield stress at an angle corresponding to a cross test. In the simulations with aluminium, the effect of the residual stresses dominates and a linear growth of the reloading yield stress with the angle is observed. Recapitulating, the analysis of the reloading yield stress based on the cell structure model suggests that the material

Figure 14 shows the reloading yield stress in different tension–tension simulations. For tension–tension experiments in general, the effect of the residual stress is inevitably combined with the effect of the plastic inhomogeneity (Fig. 11). The result in Fig. 14 is therefore a collective effect of the anisotropy due to the cell geometry and due to the residual stresses. Both effects are superimposed on each other. In Fig. 14a, θ = 0◦ , the effect of the plastic inhomogeneity appears to govern the overall tendency: an increase of the reloading yield stress until the angle ξ equals approximately 55◦ , representing a cross test, and a decrease of the effect for higher angles. The effect of the residual stresses provides an additional increase of the reloading yield stress, almost linearly growing with the angle ξ (the cross test is not specially distinguished by this mechanism). The contributions of both effects are of comparable magnitude. Figure 14b shows the reloading yield stress after a strain path change in another direction with θ = 30◦ . This result deviates from the response in Fig. 14a. The contribution of the plastic inhomogeneity is small here and the total effect demonstrates the typical influence of the residual stresses, i.e.

123

Comput Mech (2008) 41:391–405

403

(a) 1.4

(a) 1.25

1.2

1.2

1

σ/σ

mon

1.15 1.1

0.8 0.6 0.4

1.05

0.2

1 0

0.95

0

20

40

60

80

θ = 0◦ .

0.002

0.004

0.006

0.008

0.01

εp E volution of the macroscopic stress upon reloading, scaled with the monotonic stress.

(b)

(b) 1.25

0

cells walls

300 200

β11 (MPa)

1.2 1.15 1.1

100 0 -100

1.05 -300

1 0

0.95

0

20

40

60

80

θ = 30◦ . Fig. 14 Reloading yield stress as a function of the strain path change. Total effect of the residual stresses and the geometrical anisotropy (solid lines) and the contribution of the geometrical anisotropy (dashed lines). a θ = 0◦ . b θ = 30◦

anisotropy due to the plastic inhomogeneity and the residual internal stresses both determine the reloading macroscopic yield stress. Their relative contributions, and therefore the resulting overall effects, depend on the cell geometry and the cell orientation. This theoretical result can be used to explain the different types of reloading behaviour found in the experiments of Schmitt et al. [15] and Li and Bate [11]. This behaviour was explained by Li and Bate [11] by the difference in the initial texture and it was concluded that the texture is responsible for the strain path change effect, rather than a dislocation structure. The analysis based on the present cell structure model shows that the difference in the experimental results might also be due to a difference in the cell geometry, enforced by the initial texture. Reversed loading is another type of strain path change associated with the well-known Bauschinger effect, i.e. a decreased reloading yield stress followed by a decreased hardening and hardening recovery [7]. The Bauschinger effect is often attributed to the presence of a negative inter-

0.05

0.1

0.15

0.2

0.25

ε Internal stress evolution.

Fig. 15 Compression-tension test. a Evolution of the macroscopic stress upon reloading, scaled with the monotonic stress. b Internal stress evolution

nal stress in the cell interior that remain in the material after the forward loading. The calculations with the cell structure model showed that the residual stresses indeed provide a decreased reloading yield stress. Figure 15 shows that the resulting macroscopic stress is initially decreased but grows quickly again thereafter as the result of the internal stress evolution. This consideration allows to conclude that the internal stresses alone do not explain the engineering Bauschinger effect quantitatively. However, they do cause earlier plastic deformation in the cell interiors. Cell dissolution, initiated by early cell slip, has to be taken into account reducing the macroscopic stress as well. This will be incorporated in forthcoming work. 4.3 Transient hardening Another phenomenon observed after strain path changes is transient hardening. Upon deformation beyond the reloading yield stress, the macroscopic response evolves towards the value of the monotonic loading case. This type of transient hardening is observed, e.g. in Fig. 13. The increased

123

Comput Mech (2008) 41:391–405

σV M (MPa)

404 300

[22¯ 1]

5 Conclusions

250

[111]

The cell structure model has been applied and assessed by comparing predicted and experimentally determined reloading yield stresses after various strain path changes. The mechanical behaviour of copper and aluminium under various deformations has been analysed to this purpose. The model correctly captures the essential features of strain path change effects for moderate strain path changes, i.e. the anisotropy of the effect and its dependency on the amount of prestrain. The developed cell structure model has been proven to be a powerful tool for the theoretical assessment of the role of the internal stresses and their influence on the macroscopic behaviour. It has been shown that the application of the cell structure model enables:

200

[111] → [22¯ 1]

150 100 50 0

0

0.05

0.1

ε

0.15

0.2

0.25

Fig. 16 Cross test {ξ ∼ 55◦ , θ = 30◦ }. Responses corresponding to monotonic deformations with the prestrain mode (along the [111]-cell ¯ direction) and the reloading mode (along the [221]-cell direction) are also given

reloading yield stress is caused here by negative residual stresses in the cell walls. As soon as plastic deformation begins after the strain path change, the plastic incompatibility is redistributed and the internal stresses are adjusted to the new loading direction (Fig. 13c). The internal stresses in the walls grow towards a value that is typical for the current deformation under monotonic loading. As a result, the macroscopic stress approaches the monotonic stress. The predicted transient hardening lasts up to 10% of deformation. This hardening evolution is in agreement with experimental observations. The cell structure model adequately predicts the transient hardening as a result of an adjustment of the internal stresses to the new deformation mode. The example in Fig. 13c, however, considers a special case. In the general case of a complex deformation the geometrical anisotropy also influences the macroscopic hardening process. Figure 16 shows yet another example, i.e. a cross test (ξ ≈ 55◦ ). The cross test involves loading along the [111] ¯ direction. cell direction followed by loading along the [221] Two stress-strain responses for monotonic tension along the ¯ cell directions are shown as well. It can [111] and the [221] be seen that after the strain path change the stress approaches ¯ cell orientation, which the stress corresponding to the [221] is higher than the stress in the monotonic, [111], case. This result deviates from experimental observations reporting that the macroscopic stress recovers to the level of the monotonic test. The discrepancy may be attributed to the fixed orientation of the cell structure assumed in the simplified model. It is also confirmed by experiments that after a strain path change the cell structure reorients itself accommodating the new loading. To improve the prediction of transient hardening, the model should be extended to account for the reorientation of the dislocation structure. This is a subject for future research.

123

– An adequate assessment of the internal stress state in an existing dislocation structure. – An experimentally justified prediction of the reloading yield stress for moderate strain path changes. – A clear insight in the physical sources of anisotropy, resulting from two different origins, i.e. the cell geometry and residual stresses. It is thereby shown that the combined contribution of both successfully predicts observed differences related to various strain path change angles. – The establish a relation of the transient hardening behaviour after a strain path change and the adaptation of the internal stresses to the new loading. Moreover, the dislocation cell model has been elaborated at the level of a ‘material point’, thereby providing the required simplicity needed for engineering computations. Acknowledgments This research was carried out under project number MC2.00079 in the framework of the Strategic Research Programme of the Netherlands Institute for Metals Research (http://www.nimr.nl).

References 1. Barker I, Hansen N, Ralph B (1989) The development of deformation substructures in face-centred cubic metals. Mater Sci Eng A 113:449–454 2. Barlat F, Ferreira Duarte J, Gracio J, Lopes A, Rauch E (2003) Plastic flow for non-monotonic loading conditions of an aluminium alloy sheet sample. Int J Plast 19:1215–1244 3. Bate P (1993) The effect of combined strain-path and strain-rate changes in aluminium. Metall Trans A 24A:2679–2689 4. Bate P, Wilson D (1986) Analysis of the Bauschinger effect. Acta Metall 34(6):1097–1105 5. Bay B, Hansen N, Kuhlmann-Wilsdorf D (1992) Microstructural evolution in rolled aluminium. Mater Sci Eng A 158:139–146 6. Borbely A, Maier H, Renner H, Straub S, Ungar T, Blum W (1993) Long-range internal stresses in steady-state subgrain structure. Scr Metall Mater 29:7–12

Comput Mech (2008) 41:391–405 7. Christodoulou N, Woo O, MacEwen S (1986) Effect of stress reversals on the work hardening behaviour of polycrystalline copper. Acta Metall 34(8):1553–1562 8. Hasegawa T, Yakou T, Karashima S (1975) Deformation behaviour and dislocation structures upon stress reversal in polycrystalline aluminium. Mater Sci Eng 20:267–276 9. Hecker M, Thiele E, Hoste C (2002) Investigating of the tensor character of mesoscopic internal stresses in tensile-deformed nickel single crystal by X-ray diffraction. Acta Mater 50:2357–2365 10. Jensen D, Hansen N (1990) Flow stress anisotropy in aluminium. Acta Metall Mater 38(8):1369–1380 11. Li F, Bate P (1991) Strain path change effect in cube textured aluminium sheet. Acta Metall Mater 39(11):2639–2650 12. Mughrabi H (1987) A two-parameter description of heterogeneous dislocation distributions in deformed metal crystals. Mater Sci Eng 85:15–31 13. Mughrabi H, Ungar T, Wilkens W (1986) Long-range internal stresses and asymmetric X-ray line-broadening in tensile-deformed [001]-oriented copper single crystals. Philos Mag A 53(6): 793–813 14. Pedersen O, Brown L, Stobbs W (1981) The Bauschinger effect in copper. Acta Metall 29:1843–1850 15. Schmitt J, Fernandes J, Gracio J, Vieira M (1991) Plastic behaviour of copper sheets during sequential tension tests. Mater Sci Eng A 147:143–154 16. Straub S, Blum W, Maier H, Ungar T, Borbely A, Renner H (1996) Long-range internal stresses in cell and subgrain structures

405

17.

18.

19.

20. 21.

22.

23.

24.

of copper during deformation at constant stress. Acta Mater 44(11):4337–4350 Ungar T, Mughrabi H, Ronnpagel D, Wilkens M (1984) X-ray line-broadening study of the dislocation cell structure in deformed [001]-oriented copper single crystals. Acta Metall 32(3):333–342 Viatkina E, Brekelmans W, Geers M (2007) Modelling of the internal stress in dislocation cell structures. Accepted for publication in European Journal of Mechanics-A/Solids (available on line) Vieira M, Schmitt J, Gracio J, Fernandes J (1990) The effect of strain path change on the mechanical behaviour of copper sheets. J Mater Process Technol 24:313–322 Vieira M, Fernandes J, Chaparro B (2000) Yield stress after double strain-path change. Mater Sci Eng A 284:64–69 Wilson D, Zandrahimi M, Roberts W (1990) Effect of changes in strain path on work-hardening in cp aluminium and an al-cu-mg alloy. Acta Metall Mater 38(2):215–226 Young C, Headley T, Lytton J (1986) Dislocation substructures formed during the flow stress recovery of high purity aluminum. Mater Sci Eng 81:391–407 Zandrahimi M, Platias S, Price D, Barrett D, Bate P, Poberts W, Wilson D (1989) Effect of changes in strain path on work hardening in cubic metals. Metall Trans A 20A:2471–2482 Zhu Q, Sellars C (2001) Evolution of microbands in high purity aluminium-3magnesium during hot deformation testing in tension– compression. Scr Mater 45(1):41–48

123