1 Introduction

The mechanical properties like hardness, elasticity, ductility and toughness of materials are as vital as the superconducting properties like critical temperature, critical current density and critical magnetic field for industrial applications in the form of wires and tapes. Engineering applications of high temperature superconducting ceramics are generally restricted because of their brittleness nature; therefore improvement of the mechanical properties of BSCCO is a major research objective and very important for their practical applications. Hardness is the measure of the resistance of a material against a load applied to its surface. Hardness measurements are nondestructive and easy to perform and are commonly used to evaluate the quality of produced materials in industry. Hardness is a mechanical property and it is strongly related to the composition and structure of solids. Therefore, the hardness tests are popular to characterize materials [1–4]. Vickers hardness testing is commonly used to analyze the hardness of most materials and produce convenient results. If the deformation is due to very small applied loads, elastic deformation is observed. Forces above some critical value produce plastic (irreversible) deformation.

It is well known that the microhardness of solids depends on the applied load. Generally, the microhardness value decrease with increase in the applied load, which is known as ISE (Indentation Size Effect). Another case is RISE (Reverse Indentation Size Effect), where the microhardness increases with applied load. In this study, we analyzed the effect of PbSe addition and heat treatment on the mechanical properties of Bi-2212 samples, and we employed various models to extract load independent microhardness values of the samples.

2 Experimental Procedure

PbSe added Bi2Sr2CaCu2(PbSe) x O y samples with x=0,0.1,0.3 and 0.5 were prepared by glass ceramic method using high purity Bi2O3, SrCO3, CaCO3, CuO and PbSe chemicals. The powders are mixed for 1 hour in agate mortar and then melted for 3 hours at 1150 ∘C. The molten phase is quenched between pre-cooled copper plates to produce 0.5–0.8 mm thick glasses. The glass products are sintered at 845 ∘C for 120 hours to obtain the superconducting crystal structure. For simplicity, the glass products PbSe0, PbSe01, PbSe03, PbSe05 will be called A, B, C and D and the superconducting samples FPbSe0, FPbSe01, FPbSe03, FPbSe05 will be called E, F, G and H, respectively.

The phase compositions are characterized by XRD investigation by Bruker D8 Advance XRD with CuK α radiation (λ=1.5418 Å) in the range 2θ=4–70∘ at a scan speed of 3 ∘/min and a step increment of 0.02∘ at room temperature. Phase ratio and lattice parameters a and c of the samples are determined from the high-angle (00l) peaks of the XRD patterns. The relative volume fractions of the Bi-2212 and Bi-2201 phases were determined from the peak intensities of the same particular reflections. Additionally, the average particle sizes of the samples produced are computed with the aid of the Scherrer–Warren approach. The surface morphologies of the sample are studied by using a TESCAN scanning electron microscope (SEM) to determine the grain sizes and possible precipitation at the grain boundaries.

The Vickers microhardness measurements are taken at room temperature with Shimadzu HVM-2 model digital microhardness tester. The applied load (F) varied in the range 0.245–2.940 N and applied for 10 seconds. The indenter was pressed on the polished different surfaces of the samples making sure that the indentations do not overlap. Vickers microhardness results and the previously mentioned microhardness models are used to determine the load independent microhardness values of the samples.

3 Result and Discussion

3.1 XRD Analysis

XRD peaks of the samples are shown in Fig. 1. Sample H has the highest intensity peaks among PbSe added superconducting samples. The increase in the peak intensity can be associated with the increase in grain sizes. The fraction of Bi-2212 phase in sample H is higher than that of sample E (Table 1). There is a small amount of Bi-2201 phase but the dominant phase is Bi-2212 phase. From Table 1 we can conclude that addition of PbSe enhances the formation of Bi-2212 phase. The lattice parameters a and c are calculated using least squares method for orthorhombic structure and the results are tabulated in Table 1. The lattice parameter a decreases monotonically, while the lattice parameter c increases with addition. The results show that the structural properties of superconducting samples improve with addition.

Fig. 1
figure 1

XRD patterns of the E, F, G and H samples

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Table 1 Lattice parameter a and c, volume fraction and grain size of the all samples
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3.2 Grain Size Calculation

The average grain sizes of the samples are calculated from the XRD patterns and using Scherrer–Warren equation.

$$ D = 0.941\lambda/\beta\cos\theta_{\beta} $$
(1)

where D is the grain size, λ is the wavelength of X ray, β is the FWHM (full width at half maximum) of the highest intensity peak, and θ is the corresponding angle of the peak. The results of the calculations are tabulated in Table 1. As seen from the table, the average grain sizes of the superconducting samples increase with addition. This result is consistent with the SEM images of the samples.

3.3 SEM Analysis

So as to determine the grain sizes and possible precipitation at the grain boundaries, the structure of surface morphology of the samples are studied by SEM. Figure 2 shows the SEM images of the samples E, F, G and H. From the images it is seen that with increase in addition, the grain size increase and porosities between grains decrease. The sample H is denser and has less porosity. Since porosity decreases with addition, the grains approach one another and so the grain connectivity enhances. The increase in grain sizes is also confirmed by the grain size calculations using XRD peaks.

Fig. 2
figure 2

SEM micrographs of all samples

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3.4 Vickers Microhardness Measurements

The indentation sizes are measured and load dependent microhardness, elastic modulus, yield strength and fracture toughness values are calculated using (2)–(5).

(2)
(3)
(4)
(5)

The results are tabulated in Table 2. From the table it is clear that for all samples the microhardness values depend on the applied load. For samples A, E, F, G and H the microhardness value decreases with increasing applied load. This kind of nonlinear microhardness—load behavior is called indentation size effect (ISE) in the literature [5, 6].

  • Smaller values of loads exhibit higher values of hardness.

  • Higher values of applied loads exhibit smaller values of hardness. This observation is associated with the weak grain boundaries of the ceramic materials.

For samples B, C and D the microhardness values increase with increasing applied load which is known as reverse indentation size effect (RISE) [7, 8]. The possible cause of RISE behavior may be the specimen cracking [9]. Excessive cracking in the sample decreases the elasticity and thus the microhardness of the material. For these kinds of material there is no elastic recovery, but only plastic deformation is observed.

Table 2 The calculated load dependent H v ,E,Y, and K IC for the samples
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The dependence of Vickers microhardness of the samples on the applied load are displayed in Fig. 3. For all samples exhibiting either ISE or RISE behavior, for higher applied loads the load dependence of microhardness values decreases and the microhardness values converge to a plateau of rather constant microhardness value. The microhardness values at the plateau region are considered to be the load independent (real) hardness values [10]. The microhardness values reaches to that of the plateau region at about 1 N for F, G and H samples and about 2 N for the other samples.

Fig. 3
figure 3

The variations of microhardness with load for the samples

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For glass samples (A, B, C and D) the microhardness increases with increasing addition, but for superconducting samples (E, F, G and H) the microhardness decreases monotonically with addition. For pure samples, the transition from glass phase to superconducting phase after heat treatment causes an increase in microhardness. For the doped samples, the glass—superconductor transition leads to a significant decrease of microhardness values and behavioral transition from RISE to ISE. The decrease in microhardness due to addition may be related to the impurity phases and disorder. These lead to weakening of the bonds among grains and thus the decrease of the microhardness.

The elastic modulus (E) and yield strength (Y) values increases with PbSe addition for glass samples (A, B, C, D) but it decreases significantly for crystalline Bi2212 samples (E, F, G, H). Fracture toughness (K IC) values are negative for PbSe added glass samples, but they are positive for Bi-2212 crystalline samples and decrease with addition.

Since the microhardness depends on the applied load, one may be interested in calculating the load independent (real) microhardness values. In the literature various models are proposed to calculate the real microhardness values and to investigate ISE and RISE behaviors. The results of these proposed models are then compared to the microhardness values of the plateau region. In this part of the study, Meyer’s law, Proportional Sample Resistance (PSR) Model, Elastic/Plastic Deformation (EPD) Model, Hays-Kendall (HK) approach and Indentation-Induced cracking (IIC) Model will be used to calculate the load independent microhardness values of our samples.

3.4.1 Analysis According to Meyer’s Law

It is a simple relation between the applied load and the indentation size:

$$ F = ad^n $$
(6)

where n K is Meyer number which is calculated by fitting the curves of experimental data. Meyer number n K is a measure of ISE behavior. For values of n K greater than 2, RISE behavior is observed; for values less than 2, ISE behavior is observed and for n K =2 microhardness becomes load independent which is Kick’s law [2, 10–12].

In Fig. 4, lnF–lnd curves of the samples are shown. The values of n K calculated from the slope of the curves are less than 2 for samples A, E, F, G and H; but for samples B, C and D the values of n K are greater than 2. The value of Meyer number being less than 2 for samples A, E, F, G and H confirms that their load dependent microhardness has ISE behavior. For samples B, C and D the value of Meyer number is greater than 2, indicating that their microhardness has RISE behavior. These data are summarized in Table 3.

Fig. 4
figure 4

Variation of applied load lnF with diagonal lnd for the samples

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Table 3 Best-fit results of experimental data according to Meyer’s law
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3.4.2 Analysis According to Proportional Sample Resistance Model

Proportional Sample Resistance (PSR) model which is developed by Li and Bradt is successfully used for the analysis of microhardness of materials showing ISE behavior [13]. This model is given in (7).

(7)
(8)

where α and β values are calculated from (F/d)−d graph (Fig. 5). α is the surface energy, and the change in α value is associated with the energy dispersion of the surface cracks [14]. β is a parameter used to calculate the real microhardness value. In PSR model the load independent microhardness value is calculated as

$$ H_{\mathrm{PSR}} + 1854.4 \beta $$
(9)
Fig. 5
figure 5

Plots of F/d versus d for the samples

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The calculated values of α,β and load independent microhardness values are tabulated in Table 4. The value of α is positive for samples A, E, F, G and H which show ISE behavior while it is negative for samples B, C and D obeying RISE behavior. This verifies that there is no elastic deformation for samples obeying RISE behavior. The microhardness values of the samples calculated using PSR model are far from the values of the plateau region [10, 15, 16]. Therefore, PSR model is invalid in computing the load independent microhardness values of our samples.

Table 4 Best-fit results of experimental data according to PSR model
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The values of load independent elastic modulus (E), yield strength (Y) and fracture toughness (K IC) are calculated using load independent H PSR values and are also tabulated in Table 5. When compared to the load dependent values, load independent E 0, Y 0 and K IC values decrease for samples A, E, F, G and H which obey ISE behavior, but they increase for samples B, C and D obeying RISE behavior. The increase in K IC value is associated with the increase in the value of surface energy α. Fracture toughness (K IC), is an important mechanical property of ceramic materials. It is a vital parameter in choosing materials for technological applications. For samples obeying RISE behavior, the microhardness values are higher when compared to other samples because there is no elastic recovery. Since elastic modulus, yield strength and fracture toughness depend on the values of microhardness, the increase in their values is expected. For samples obeying ISE behavior, elastic deformation as well as plastic deformation is observed. Namely, after the indenter is removed there is loosening in the sample surface. This causes a decrease in the microhardness of the sample. Toughness also decreases along with microhardness. The results of our samples are consistent with the results reported in the literature [17].

Table 5 The calculated load independent H v ,E,Y, and K IC for the samples
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3.4.3 Analysis According to Elastic/Plastic Deformation Model

Bull et al. proposed that the dependence of indentation size on the applied load is given as [18, 19]

$$ F = A_2(d_p+d_e)^2 $$
(10)

Here A 2 is a constant and d e is related to d p plastic deformation. The values of A 2 and d e are calculated from F 1/2−d p graph (Fig. 6).

Fig. 6
figure 6

Plots of diagonal length versus square root of applied loads for the samples

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The load independent microhardness value is calculated using;

$$ H_{\mathrm{EPD}}=1854.4A_2 $$
(11)

As seen in Table 6, the calculated d e values for samples B, C and D are negative. That means no elastic deformation is observed. Elastic deformation is observed only for samples A, E, F, G and H obeying ISE behavior. These samples obey ISE behavior because elastic deformation along with plastic deformation is observed in these samples.

Table 6 Best-fit results of experimental data according to EPD model
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3.4.4 Analysis Due to Hays–Kendall Approach

To analyze ISE, Hays–Kendall proposed that only elastic deformation develops in the samples for applied load values below a certain limit, but above the limit plastic deformation is observed [20]. Gane and Bowden observed that the indenter does not penetrate to some materials up to a certain value of applied load, and above the critical value of load it abruptly penetrates [21]. The indentation size does not increase although the applied load increases up to a critical applied load value. Hays-Kendall observed that the indentation size depends on an effective applied load F eff=F−W HK instead of the applied load itself:

$$ F - W_{\mathrm{HK}} = A_{1\mathrm{HK}} d^2 $$
(12)

where A 1 is a constant independent of the applied load. The values of W and A 1 are calculated from F–d 2 graph. The load independent hardness is calculated using

$$ H_{\mathrm{HK}} = 1854.4 A_{1\mathrm{HK}} $$
(13)

The slope of the graph in Fig. 7 gives the value of A 1HK. In Table 7 the calculated values of load independent microhardness H HK, W HK and A 1HK are tabulated.

Fig. 7
figure 7

Applied load vs. the square of the impression semi-diagonal length for the samples

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Table 7 Best-fit results of experimental data according to HK model
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The negative value of W HK can be interpreted as the applied load being adequate to generate plastic deformation but it is inadequate to create elastic deformation [22]. In our study elastic deformation is observed only for samples A, E, F, G and H. No elastic deformation is observed for the other samples.

For the samples obeying ISE behavior, microhardness values computed due to HK model are closer to those at the plateau region when compared to other models. As mentioned before, the load independent microhardness values are expected to be close to those at the plateau region [10, 23, 24]. Therefore, the Hays-Kendall model is the most suitable model for the analysis of the microhardness values and mechanical properties of the samples A, E, F, G and H. For the samples obeying RISE behavior, the models used above did not produce microhardness values close to the ones of the plateau region. Indentation-Induced Cracking (IIC) Model is another model to be used for the analysis of mechanical properties of the samples obeying RISE behavior.

3.4.5 Analysis Due to Indentation-Induced Cracking (IIC) Model

It is a model proposed by Li and Bradt for the analysis of the microhardness of materials obeying RISE behavior [25]. According to this model, the applied load is balanced by the total sample resistance at the maximum depth. The sample resistance is composed of four components

  1. 1.

    The slipping of the sample or the indenter at the interfaces.

  2. 2.

    Elastic deformation.

  3. 3.

    Plastic deformation.

  4. 4.

    Cracks in the sample.

According to Li and Bradt friction (slipping) and elastic effects produce ISE behavior, and the indentation cracking produces RISE behavior. The microhardness value is calculated using the following equation:

$$ H_v = \lambda_1 K_1\biggl(\frac{F}{d^2}\biggr) + K_2\biggl(\frac{F^{5/3}}{d^3}\biggr) $$
(14)

where d is the indentation size, and λ 1,K 1,K 2 are constants. K 2 depends on the applied load, while K 1 depends on the geometry of the indenter.

For an ideal plastic material H v =K 1(F/d 2), λ 1=1 and K 2(F 5/3/d 3)=0. For perfect brittle solids H v =K 2(F 5/3/d 3) and λ 1=0. In the (14) we can take d=7h. This equality is associated with the angle (148 ∘) between the opposite sides of the indenter, and h is the depth of the indentation. For brittle materials the second part of the equation is used. In our PbSe added samples no elastic deformation is observed, which means they are brittle. Therefore, (15) derived from (14) is to be used to calculate the hardness values of our samples. We have

$$ H_v = K \biggl(\frac{F^{5/3}}{d^3}\biggr) $$
(15)

where K and m are load independent constants whose values are calculated from ln(H v )−ln(F 5/3/d 3) graph (Fig. 8a). The value of m is a measure of ISE. For m>0.6 the sample obeys ISE behavior, while for m<0.6 the sample obeys RISE behavior [17, 22].

Fig. 8
figure 8

(a) Variation of lnH V with ln(F 5/3/d 3) according to IIC model for all samples, (b) the correlation between the IIC model parameters K and m for all samples

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Since no elastic deformation is observed in the hardness analysis of samples B, C and D this model produced the best results for samples obeying RISE behavior. The values of m calculated using the slope of the graph are tabulated in Table 8. In the table m<0.6 for samples B, C and D while m>0.6 for samples A, E, F, H, G. We see that the samples B, C and D obey RISE, while samples A, E, F, G and H obey ISE behavior. The value of m increases for both PbSe added Bi-2212 crystallines and glass samples with addition ratio.

Table 8 Best-fit results of experimental data according to IIC model
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The fitting of the graph in Fig. 8b gives an empirical relation between K and m like

$$K = 21.22 e^{11.06m} $$

4 Conclusion

The effect of PbSe addition on the mechanical properties of Bi-2212 superconductor is analyzed. XRD measurements are taken to determine the crystal structure and lattice parameters, SEM micrographs are used to determine surface morphology and grain sizes. Vickers microhardness measurements are made to determine the mechanical properties of the samples. For glass samples the microhardness increases with increasing PbSe addition, while for superconducting samples microhardness decreases with PbSe addition. The experimental results of the microhardness measurements are analyzed using Kick’s law, PSR, EPD, Hays–Kendall and IIC models. After employing the above mentioned models, we conclude that the Hays–Kendall model is suitable for samples obeying ISE behavior (A, E, F, G, H) and IIC model is suitable for samples obeying RISE behavior (B, C, D). These results are summarized in Table 9. XRD results show that the Bi-2212 phase ratio, c lattice parameter and grain size increase in superconducting samples, while the lattice parameter a decrease with addition. The increase in average grain sizes is also verified by SEM analysis. The increase in the grain sizes results in the decrease in porosity among grains and better crystallinity of the samples.

Table 9 The results of load dependent Vickers microhardness at the plateau region and load independent hardness values calculated using PSR, MPSR, EPD and HK models
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