**Wolfgang Blum 1,\*, Jiˇri Dvoˇrák 2, Petr Král 2, Philip Eisenlohr <sup>3</sup> and Vaclav Skleniˇcka <sup>2</sup>**


Received: 5 September 2019; Accepted: 21 October 2019; Published: 26 October 2019

**Abstract:** The influence of the grain structure on the tensile deformation strength is studied for precipitation-strengthened Cu-0.2%Zr at 673 K. Subgrains and grains are formed by equal channel angular pressing (ECAP) and annealing. The fraction of high-angle boundaries increases with prestrain. Subgrains and grains coarsen during deformation. This leads to softening in the quasi-stationary state. The initial quasi-stationary state of severely predeformed, ultrafine-grained material exhibits relatively high rate-sensitivity at relatively high stresses. This is interpreted as a result of the stress dependences of the quasi-stationary subgrain size and the volume fraction of subgrain-free grains.

**Keywords:** Cu-Zr; ECAP; deformation; quasi-stationary; subgrains; grains; coarsening

## **1. Introduction**

Deformation of single crystals and large-grained polycrystals leads to generation of dislocations and subgrains with low-angle boundaries (LABs). Dynamic recovery counteracts the defect generation. The state at which the rate *ρ*˙ <sup>+</sup> of increase of density *ρ* of free dislocations approximately equals the rate *ρ*˙ − of *ρ*-decrease,

$$
\dot{\rho}^+ \approx \dot{\rho}^-,
\tag{1}
$$

is usually called steady state, stationary state or saturation stage (of flow stress) in the literature. We prefer the term quasi-stationary (qs) state [1], because slow changes of microstructure parameters (boundaries [2,3], particles, ...) lead to a drift of the quasi-equilibrium of the qs state that in turn causes slow changes of the qs deformation strength. By definition, there is a spectrum of qs states, depending on the microstructural parameters other than *ρ*. The qs states are important as upper limits of deformation strength at given structure of grains and particles.

The present work deals with the strength of a Cu-Zr alloy that is stabilized by particles and processed by equal channel angular pressing (ECAP). It is well known that ECAP processing leads into a qs state with refined grains and saturated hardness (compare [4–7]). This holds also for Cu-Zr [8–10]. When the ECAP-processed material is subjected to creep at elevated temperature of 673 K, new qs states are developed. The creep ductility is significantly enhanced compared to the coarse-grained (cg) alloy while the creep rate remains fairly low and has relatively low stress sensitivity. In the following section, we treat the evolution of deformation strength and microstructure of ECAP-processed Cu-Zr at 673 K in detail on updated data basis with the goal to better understand how grain refinement influences the qs deformation strength. We discuss the proposition that the qs strength of ultrafine-grained (ufg)

Cu-Zr significantly depends on the stress-dependent qs fraction of subgrain-free grains. There are indications that the variation of this fraction with stress may explain the relatively low stress sensitivity (high rate sensitivity) of the qs strength of grain-refined ufg Cu-Zr.

## **2. Experimental Details**

Processing, microstructure, and creep testing have been described before [10]. The starting material was (cg) Cu-0.2 mass%Zr. Its material parameters are approximated by those of pure Cu provided in the data compilation of Frost and Ashby [11]: Burgers vector *<sup>b</sup>* = 2.56 × <sup>10</sup>−<sup>10</sup> m, elastic shear modulus *<sup>G</sup>* = 3.58 × <sup>10</sup><sup>4</sup> MPa, melting point *<sup>T</sup>*<sup>m</sup> = 1356 K. The cg material was homogenized for 24 h at 1073 K and hot rolled. Billets of this material were solution treated at 1233 K for 1 h to give an initial grain size of 3.5 × <sup>10</sup>−<sup>4</sup> m. The billets were predeformed by *<sup>p</sup>* passes of equal channel angular pressing (ECAP, Figure 1) at room temperature on route BC to refine the grains. The predeformed material is called *p*Cu-Zr. Due to particle hardening the grain size of the material is stable against static annealing at temperatures exceeding the test temperature 673 K [12]. The thermal treatment consisted of storage of the predeformed material at room temperature for up to several years followed by heating to and holding at *T* for a few hours before deformation.

**Figure 1.** From ECAP billet to tensile creep specimen (numbers: lengths/mm).

Flat tensile creep specimens with gauge length *l*<sup>0</sup> and cross section *S*<sup>0</sup> were prepared as illustrated in Figure 1. Deformation in creep mode occurred in tension at temperature *T* = 673 K = 0.5 *T*<sup>m</sup> upon applying a load *F*, corresponding to an engineering stress *σ*eng = *F*/*S*0. The change Δ*l* of the length was measured using LVDTs (from Hottinger Baldwin Messtechnik GmbH, Darmstadt, Germany) attached to the tensile rods near the specimen. The "true" strain results as ln((*l*<sup>0</sup> + Δ*l*)/*l*0) (Elastic contribution to Δ*l* are neglected here because it is relatively small and nearly constant at constant *F* so that the elastic contribution ˙el to the measured rate ˙ is generally negligible). Due to volume constancy and under uniform deformation of the gauge length *l* = *l*<sup>0</sup> + Δ*l*, the cross section *S* varies as *S* = *S*<sup>0</sup> *l*0/*l* = *S*<sup>0</sup> exp(−) and the "true" stress *σ*, given by *F*/*S*, increases with strain as

$$
\sigma = \sigma\_{\text{erg}} \, \exp(\epsilon). \tag{2}
$$

Observations of the grain structure were made by electron microscopy. Orientation maps derived from electron backscatter diffraction (EBSD, device from Oxford Instruments, High Wycombe, United Kingdom) were used to determine the grain size *d* as mean value of the spacings *di*, *i* = 1, 2, 3, ..., of HABs between crystallites with misorientations ≥ 15◦ determined along test lines. Transmission electron microscopy (TEM) images (microscope from JEOL Ltd., Tokyo, Japan) were

used to analogously determine the subgrain size *w* as mean value of the spacings *wi* of boundaries of any kind.

#### **3. Results**

#### *3.1. Influence of Predeformation*

With increasing predeformation *p* the deformation strength of *p*Cu-Zr saturates, because the areal fraction *f*HAB of high-angle boundaries (HABs) increases up to a saturation level [9] where a qs state is established due to dynamic equilibrium of storage and dynamic recovery of dislocations. This statement is consistent with the observation that the Vickers hardness saturates with increasing predeformation *p* at *HV* ≈ 180 [9]; using the formula *σ*/MPa ≈ 3 *HV*, this corresponds to a saturation flow stress of 540 MPa in ECAP. An increase of *f*HAB enhances storage as well as recovery of dislocations because HABs are stronger dislocation obstacles and better dislocation sinks than LABs. The net effect on deformation strength depends on whether storage or recovery are the dominant factors (compare, e.g., [13]).

Figure 2 shows exemplary curves of the evolution of deformation resistance with strain at constant engineering stress *σ*eng = 150 MPa for different amounts *p* of ECAP predeformation. Deformation starts with application of the load *F*, and continues in creep mode when the creep stress *σ*eng has been reached. The starting phase of the tests is connected with work hardening as is visible from the sharp decrease of strain rate ˙ with . After strains of 0.02 to 0.05 the rate ˙ increases. A trivial reason for ˙-increase lies in the increase of *σ* with (Equation (2)) causing the creep rate to increase in the qs state as ˙ ∝ *σn*qs where *n*qs is the qs value of the stress exponent. The dotted line in Figure 2 shows that for severely predeformed Cu-Zr with *p* ≥ 8 practically all of the ˙-increase can be attributed to the increase of *σ* when *n*qs = 6 (as reported before [9]). This means that at *σ*eng = 150 MPa


At about half the fracture strain the ˙-increase accelerates. This indicates formation of local stress concentrations by external and internal necking during the fracture process.

**Figure 2.** ˙ versus for *p*Cu-Zr at 673 K, *σ*<sup>0</sup> = 150 MPa in tension at constant load; dotted line: increase of ˙ expected from increase of *σ* for *n*qs = 6.

The less strongly predeformed specimens 4Cu-Zr and 2Cu-Zr exhibit a distinctly steeper increase of ˙ with . To find the reason for that, it is useful to display the creep process in the qs region near the minimal creep rate, i.e., the creep between the work hardening period and the fracture period, in the log ˙ − log *σ*-field (Figure 3). According to Equation (2), log *σ* increases linearly with . The scale-bars in Figure 3 show the extension of a strain interval Δ of 0.5. Figure 3e confirms that most of the curves for 8Cu-Zr have slopes near *n*qs = 6 in the qs range above 150 MPa and follow a common trend given by the dashed black line. For 12Cu-Zr this behavior is even more perfectly pronounced, indicating better homogeneity of the initial microstructure.

**Figure 3.** ˙ versus *σ* for *p*Cu-Zr at 673 K in tension at constant load.

For lower stresses *σ* < 150 MPa and for *p*Cu-Zr with *p* ≤ 4 the slopes in Figure 3 are distinctly higher and different curves do not merge. One reason is tensile fracture. Figure 3a shows that fracture occurs in cg Cu-Zr (i.e., *p*Cu-Zr with *p* = 0), before a qs region has been developed. This means that the qs deformation strength cannot be measured in tension. Thus, the experimental values of the minimal creep rate are determined by fracture, and therefore constitute only upper bounds of the qs deformation strength. The same holds for predeformed *p*Cu-Zr with *p* = 1 to 4. (To determine the qs deformation strength, compression tests would be required in these cases in order to delay or suppress fracture). For ufg 8/12Cu-Zr the strains are too large to explain the non-merging of the curves at low *σ* by fracture processes. Therefore this effect must be due to softening during creep. In Section 3.3 we provide an explanation of the softening in terms of dynamic coarsening of the microstructure.

In summary, one sees from Figure 3 that the qs deformation strength of Cu-Zr decreases at the test temperature 0.5 *T*<sup>m</sup> when the material is severely predeformed, the grains are refined, and the areal fraction *f*HAB of boundaries with high-angle character increases during creep.

#### *3.2. Microstructure of 8Cu-Zr*

#### 3.2.1. Before Creep

Heating to and holding at test temperature before creep took a few hours. Figures 4 and 5a illustrate the microstructure of 8Cu-Zr after annealing for a few hours at the test temperature. So these microstructures are representative for the microstructure at start of creep. From a micrograph, such as Figure 4, the initial grain size *<sup>d</sup>*<sup>0</sup> of 8Cu-Zr was determined as 4.7 × <sup>10</sup>−<sup>7</sup> m. From the 8Cu-Zr sample of Figure 5a the initial subgrain size *<sup>w</sup>*<sup>0</sup> was determined as 1.5 × <sup>10</sup>−<sup>7</sup> m. Noting that these are local values that may vary from place to place and from specimen to specimen, the agreement with previously published data [10] and with data from the literature for severely predeformed ufg Cu-Zr [12,14,15] data is considered to be satisfactory. From the empirical formula

$$w\_{\rm qs}^{\rm cg} \approx 14 \,\text{b} \,\text{G}/\sigma \tag{3}$$

for cg Cu [16] and the estimate 540 MPa for the saturation strength in ECAP given above one gets a qs subgrain size of 2.4 × <sup>10</sup>−<sup>7</sup> m. In view of the uncertainty of the empirical factor 14 entering Equation (3), the agreement with the measured *w*<sup>0</sup> is considered satisfactory, too. This means that *w*<sup>0</sup> may be interpreted as the qs value of the subgrain size in ufg 8Cu-Zr before creep, that is only marginally influenced by the heating to test temperature.

**Figure 4.** Grain structure map of 8Cu-Zr after 10 h of annealing at 673 K; thin and thick lines: traces of boundaries with misorientation < and > 15◦, respectively. Insert: color code for crystallographic grain normals in standard triangle.

**Figure 5.** TEM of subgrains and Cu5Zr-precipitates in 8Cu-Zr after (**a**) annealing for 8 h at 673 K, (**b**) creep at 673 K, *σ*eng = 150 MPa to a fracture strain of ≈0.5 (compare curve for *p* = 8 in Figure 2).

The Zr containing particles are essential in restricting grain coarsening [10,12,14]. They are present in homogeneous distribution after SPD ([14] and Figure 5). Figure 6 shows electron diffraction patterns of (a) the Cu matrix with particles and (b) a large particle of composition Cu5Zr. A rough estimate based on Equations (7)–(13) in [17] (using a mean particle radius of 4 nm, number of particles per area of TEM foil equal to 10, typical foil thickness of 200 nm) yields an Orowan stress for particle hardening of about 150 MPa at room temperature. However, at the test temperature the tiny particles are easily overcome by local and general climb of dislocations so that the particle hardening term is greatly reduced [17]. In the following, we neglect the direct influence of particles on the stress sensitivity of the deformation strength.

**Figure 6.** Diffraction patterns of (**a**) Cu matrix (strong reflections) with Cu5Zr particles (weak reflections) and (**b**) a large Cu5Zr particle.

The spacing plot of Figure 7 shows *w*<sup>0</sup> and *d*<sup>0</sup> together with the qs spacings ≈ *bG*/*σ* of dislocations and *w*cg qs of boundaries in cg Cu. We assume that the qs spacings for cg Cu are valid also for Cu-Zr. It is interesting to note that at low stresses the (dotted) qs dislocation spacing would be similar to the initial boundary spacing *w*0, meaning that the subgrains will tend to be virtually dislocation-free in the beginning of creep. This will increase the mobility of boundaries. As *<sup>w</sup>*<sup>0</sup> is smaller than *<sup>w</sup>*cg qs(*σ*) in the whole stress interval of creep (Figure 7), the subgrains will tend to coarsen with strain toward the quasi-stationary value *w*cg qs as long as the grains provide enough space, i.e., as long as *w* is smaller than

*d*. The grey area marks the range of possible subgrain coarsening under the simplifying assumptions that all grains are equiaxed and keep the same size *d* = *d*<sup>0</sup> throughout deformation. The solid black line marks the qs boundary spacing in ufg 8Cu-Zr under these conditions. If this line were exactly valid, all grains would become subgrain-free during deformation for stresses < 270 MPa. However, that is unrealistic because of (i) the wide distribution of HAB spacings (Figure 9) and (ii) dynamic grain growth (see below).

**Figure 7.** Initial spacings *w*<sup>0</sup> (all boundaries) and *d*<sup>0</sup> (HABs only) in *p*Cu-Zr with *p* ≥ 8 in comparison to the qs spacings *w*cg qs of subgrains (Equation (3)) and ≈ *b G*/*σ* of free dislocations expected in coarse-grained Cu-base material; circles: HAB-spacings after deformation, triangles: HAB-spacings in regions with relatively fine and coarse grains, respectively; grey area: range of subgrain coarsening for *d* = *d*0.

#### 3.2.2. After Creep

Figure 8 shows grain structures after creep at low *σ*. There is a mixture of large and fine grain sections. The largest ones are much coarser than in the initial state (Figures 4 and 7). Coarsening is also seen from the TEM micrograph of Figure 5b after fracture at *σ*eng=150 MPa in comparison to Figure 5a taken before creep. However, this is probably mainly due to subgrains growing from *w*<sup>0</sup> (Figure 5a) in the attempt to attain the qs spacing *w*cg qs. Figure 9 shows the distributions of individual HAB-spacings determined from Figures 4 and 8. We will use them as rough experimental input to the model presented in Section 4.

Comparison of Figure 5a,b shows no indications for significant coarsening of particles during creep. So we assume that the stabilization of the grain structure by particles will essentially persist in creep.

**Figure 8.** Coarsened grains after slow creep at *σ*eng = (**a**) 75 MPa and (**b**) 50 MPa (grey triangles in Figure 7); slight color variations in some coarsened grains indicate subgrain formation during creep; insert: color code for crystallographic grain normals in standard triangle.

**Figure 9.** Individual HAB-spacings *dl* (*l* = 1, 2, ...) versus cumulative spacing fraction *Fl* evaluated for 8Cu-Zr in annealed state from Figure 4 and after slow creep from Figure 8a,b. Grey dashed line: linear approximation for annealed state.

#### *3.3. Strength Evolution of 8/12Cu-Zr with Strain*

Figure 10 displays the ˙-evolution of ufg 8Cu-Zr from the overview of Figure 3e in greater detail as function of strain . For comparison, the grey dotted lines show the increase of ˙ with expected for qs deformation with *n*qs ≈ 6 at constant *σ*eng. The scatter in the series of tests at *σ*eng = 250 MPa indicates that the thermomechanical pretreatment was not perfect and led to some variations of microstructures and strengths. The intermediate *σ*eng-increase to 83 MPa in the test starting at 75 MPa causes a jump in the qs level of ˙ at ≈ 0.15 (see black squares) which corresponds to *n*qs = 5.3. This is in reasonable agreement with *<sup>n</sup>*qs ≈ 6. At high (true) stresses *<sup>σ</sup>* > 270 MPa and strain rates > <sup>10</sup>−<sup>4</sup> <sup>s</sup>−<sup>1</sup> the exponent *n*qs appears to gradually increase with *σ* beyond 6. This is seen from the test of Figure 11 with a series of load reductions compensating the increase of *σ* with (Equation (2)). When the abscissa in Figure 11a is exchanged for *σ* in Figure 11b, all the branches with *σ*eng < 250 MPa shift to the left and superimpose. A common ˙– line describes both the superimposed curves and their (dashed) connection to the final branch with *σ*eng = 250 MPa. The slope of the dashed connection is given by *n*qs = 10.4 at *σ* > 275 MPa.

**Figure 10.** ˙ of 8Cu-Zr as function of at constant *σ*eng; black circles: identified qs rates; grey: after *σ*eng-reduction of from 250 MPa; see Figure 8 for microstructure at black arrows.

**Figure 11.** ˙-evolution of 8Cu-Zr as function of (**a**) , (**b**) *σ* = *σ*eng exp() in test with *σ*eng-reductions of increasing magnitude to keep *σ* = *σ*eng exp() about constant.

At low *σ*<150 MPa the qs strength evolution can no longer be explained only in terms of the slow increase of *σ* at constant *σ*eng and fracture (see Figures 3 and 10. Therefore changes in the subgrain/grain structures must be invoked. This corresponds to the observed dynamic coarsening of subgrains and grains (Figure 7, Sections 3.2.1 and 3.2.2).
