Relations of Microstructural Attributes and Strength-Ductility of Zirconium Alloys with Hydrides

As the first safety barrier of nuclear reactors, zirconium alloy cladding tubes have attracted extensive attention because of its good mechanical properties. The strength and ductility of zirconium alloy are of great significance to the service process of cladding tubes, while brittle hydrides precipitate and thus deteriorate the overall performance. Based on the cohesive finite element method, the effects of cohesive strength, interfacial characteristics, and hydrides geometric characteristics on the strength and ductility of two-phase material (zirconium alloy with hydrides) are numerically simulated. The results show that the fracture behavior is significantly affected by the cohesive strength and that the overall strength and ductility are sensitive to the cohesive strength of the zirconium alloy. Furthermore, the interface is revealed to have prominent effects on the overall fracture behavior. When the cohesive strength and fracture energy of the interface are higher than those of the hydride phase, fracture initiates in the hydrides, which is consistent with the experimental phenomena. In addition, it is found that the number density and arrangement of hydrides play important roles in the overall strength and ductility. Our simulation provides theoretical support for the performance analysis of hydrogenated zirconium alloys during nuclear reactor operation.


Introduction
Zirconium alloys are widely used in fuel cladding tubes in nuclear reactors.They attract more attention than other materials because of a series of advantages such as excellent thermal conductivity, corrosion resistance, and extremely low neutron absorption cross section [1,2].Fuel cladding wraps the fissile radioactive uranium fuel pellets, acts as the first barrier against the release of radioactive elements, and directly contacts the coolant [3,4].As we all know, the integrity of structural materials is usually very crucial [5].During the operation of the reactors, corrosion reactions occur at the interface between the coolant and the nuclear fuel cladding, and hydrogen atoms released by this process diffuse into the zirconium alloys [6].The solid solubility of hydrogen in zirconium alloys is relatively low, and it remains solid solute state till reaching the solid solubility.When the solid solubility is exceeded, brittle hydrides precipitate in the zirconium alloys and then they deteriorate the mechanical properties of cladding materials [7,8].It is well known that hydrogen embrittlement is one of the most important causes for the fracture and premature failure of nuclear fuel cladding tubes during the operation process of nuclear reactors.Therefore, the effects of hydrides on the strength and ductility of hydrogenated zirconium alloys are important to evaluate the structural integrity.
It was found that the hydrides might have defects because the occurrence of hydrogenated α-Zr involves a large volume expansion, about 17% of a unit cell [9].In an oxygen-rich environment, local overabsorption and thermal transport of hydrogen leads to hydride blisters [10,11].The emergence of these hydride blisters is usually related to the stress concentration in the nuclear fuel cladding, and cracks initiate at these hydride blisters [12].The phenomenon that hydrides induce crack propagation in the zirconium alloys is often referred to as the delayed hydride cracking, and the relevant study is crucial to the integrity and reliability of nuclear fuel cladding [13][14][15].
In the operation of nuclear reactors, the influences of hydride phase on the hydrogenated zirconium alloys are not negligible.In order to investigate the adverse effects of hydrides on the overall ductility, fracture toughness, and service life of zirconium alloys, many researchers have carried out extensive study.On the one hand, due to the brittleness of the hydrides, experimental studies have revealed that the mechanical properties of zirconium alloys are degraded significantly.Kim et al. [16] found that the degradation of the performance depended not only on the orientation of the hydrides but also on the concentration of the hydrides in the cladding tubes.Zhang et al. [17] studied the effects of hydrogen concentration and temperature on the fatigue crack initiation and propagation in the zirconium alloy by in-situ scanning electron microscopy.However, due to the limitation of experiments, it is difficult to accurately repeat the key factors such as the geometric characteristics of hydride phase.On the other hand, numerical simulation is an effective approach to study the fracture behavior of hydrogenated zirconium alloys.For instance, Tseng et al. [18] studied the crack through hydrides in the fuel cladding by the finite element method, and found that long hydrides and double radial hydrides had significant influences.By means of the cohesive finite element method (CFEM), Fang et al. [19] numerically simulated the ductile fracture process in hydrogenated zirconium alloy under different stress states and analyzed the influences of geometric characteristics of hydrides.However, the previous numerical simulation lacks the attention to the effects of the hydrides on the strength and ductility of cladding, and so far, theoretical elaboration of experimental phenomenon is far from enough.Therefore, the numerical simulation is used in this work to clarify the above problems comprehensively.
Stemming from the principle of virtual work, the extended finite element method (XFEM) [20], the isogeometric analysis (IGA) combining with the XFEM (known as extended IGA (XIGA)) [21,22], and the CFEM [23][24][25] have been widely used in investigating the fracture processes.The former two have some difficulties in modeling spontaneous multi-crack initiation, branching, and coalescence, while they are often used to study single crack propagation [26].Compared with them, the CFEM has obvious advantages.Specifically, it can effectively simulate the propagation of pre-cracks and also the initiation and propagation of multiple microcracks in the fracture process.Therefore, the CFEM is adopted in this study.
In this paper, microstructural attributes include fracture characteristics of the zirconium alloy and hydrides, interfacial characteristics, and geometric characteristics of hydrides.Based on the CFEM, we simulate the fracture behavior of hydrogenated zirconium alloys and explore the correlation between the microstructural attributes, and overall strength and ductility.

Numerical Framework
The strength and ductility of structural materials are crucial to evaluate the integrity and reliability of engineering structures [27][28][29].In this study, in order to clarify the effects of hydrides on the strength and ductility of hydrogenated zirconium alloys, idealized microstructures are presented in Section 2.1.The hydrides are generally in microns and more brittle, while the zirconium alloys are more ductile.Therefore, a big gap exists between the mechanical properties of hydrides and zirconium alloy.
To study the performance of the above two-phase material, the plastic constitutive laws of two phases are given in Section 2.2, and the bilinear traction-separation law is described in Section 2.3.

Idealized Microstructures
This study aims at a rectangular sample (400 × 300 μm 2 ) of zirconium alloy with hydride(s).The damage evolution in the two-phase material is numerically simulated by the CFEM.The involved element types include solid element and cohesive element, and cohesive elements are inserted between two adjacent solid elements.According to the distribution locations, the cohesive elements in the zirconium alloy matrix are referred to as Zr-Zr elements, those in the hydride phase are H-H elements, while those between the zirconium alloy and hydride are Zr-H elements (i.e., interface elements).Figure 1 shows a typical microstructure (Model-1), in which the red region is the hydride phase (length L h : 200 μm, width L b : 30 μm) and the green region is the zirconium alloy.The yellow region represents the Zr-Zr elements, the blue region represents the H-H elements, while the purple region represents the Zr-H elements.
Among multiple categories of solid and cohesive elements, the linear triangular plane stress solid element and the two-dimensional four nodes cohesive element are used.Displacement boundary conditions are applied to the left and right boundaries of the sample.The upper and lower boundaries are unconstrained.Simulation is carried out by the finite element software package ABAQUS [30].
The number density, spacing, and arrangement of hydrides are important to the fracture process of hydrogenated zirconium alloys.In order to facilitate subsequent analysis, nine microstructures are designed and then divided into three series according to the arrangement of hydrides, as illustrated in Figure 2: (i) One-hydride series (series A) including microstructure A1 (Figure 2a), A2 (Figure 2b), and A3 (Figure 2c).In A1, the hydride is along the tensile direction; in A2, it is perpendicular to the tensile direction; in A3, it is at an angle of 45° with the tensile direction.
In series A, series B, and series C, the length and width of hydrides are 120 μm and 30 μm, respectively, except that in microstructure A3, they are 116.67 μm and 31.82μm.For the two hydrides in microstructures B1− B4, both the spacing along the tensile direction d h and the spacing perpendicular to the tensile direction d s are 20 μm.

Plastic Constitutive Laws
The plastic constitutive law of zirconium alloy should be specified.Table 1 lists the available results from tensile tests and Figure 3 depicts the true stress versus the plastic strain.Here, a moderate plastic constitutive law (theta direction) [33] is adopted for the zirconium alloy.So far, there are relatively few results on the plastic constitutive law of hydrides.Here, the results in Ref. [35] is adopted for the hydrides.

Cohesive Finite Element Method
The behavior of cohesive elements can be characterized by the traction-separation laws [36,37].A bilinear traction-separation law can clearly describe a linear elastic stage of cohesive elements and a linear softening stage after reaching the peak, as shown in Figure 4.When an effective separation δ reaches δ 0 , the traction reaches cohesive strength T max [23].When δ is smaller than δ 0 , the loading and unloading process is fully reversible.When δ is larger than δ 0 , the loading and unloading process becomes irreversible.When δ reaches a failure separation δ f , the traction reduces to zero, the cohesive element fails, and thus the microcrack initiates.In the mixed-mode situation, the damage initiation criterion is very important.When it is satisfied, the material begins to degrade.The criterion adopted in this study is the quadratic nominal stress criterion (Quads), i.e., the damage initiation occurs when where t n and t s are the traction in the normal and shear directions, respectively; �t n � = t n when t n > 0 , and �t n � = 0 when t n ≤ 0 ; t 0 n and t 0 s represent the maximum nominal stress in mode I (opening mode) and mode II (sliding mode) failure, respectively.
In the two-dimensional case, G coh (fracture energy) is the energy required for failure of cohesive element per unit length, and its value is equal to the area of ΔOAB (1)   where T is the effective traction [23].
Table 2 shows the parameters input to the solid and cohesive elements.As listed in the table, the cohesive strength of Zr-Zr and H-H elements are taken as n Zr times of the yield stress of zirconium alloy and n H times of the yield stress of hydride, respectively.In addition, the fracture toughness ( K C ) of zirconium alloy was measured to be 40 MPa √ m [39] and that of hydride 3.3 MPa √ m [40].Therefore, the fracture energy of Zr-Zr elements can be calculated as 16508 J•m -2 by the formula ( ), and that of H-H elements 89 J•m -2 .Unless otherwise stated, for the Zr-H elements, the cohesive strength and fracture energy are taken as the averages of Zr-Zr and H-H elements [23,24].In order to facilitate follow-up discussion, the combination of n Zr and n H is denoted n Zr -n H in this study.

Results and Discussion
In this section, we carry out the qualitative comparisons of simulated results with experiments, and then explore the influences of the microstructural attributes ((i) the cohesive strength of each phase, (ii) the interfacial characteristics (cohesive strength and fracture energy), and (iii) the geometric characteristics (number density, spacing, and arrangement) of hydrides) to the strength and ductility of hydrogenated zirconium alloys.

Comparisons with Experimental Phenomena
Comparisons between simulated results and the experimental phenomena are shown in Figure 5. Specifically, in order to verify our numerical framework by comparing with the tensile fatigue experimental phenomenon in Figure 5a [41], another microstructure with a slender hydride is designed and shown in Figure 5b.It is evident that when n Zr -n H is 1.8-1.2, the fracture morphology and the blunting characteristics around crack tips are quite consistent with the experiment.The number density (2) of microcracks is larger than that of microcracks in the experiment, which might lie in different load conditions and stress states.Furthermore, Figure 5c illustrates the maximum shear strain in the hydrogenated zirconium alloy in the macroscopic tensile test of Wang et al. [42], while Figure 5d shows our equivalent plastic strain of Model-1 (n Zr -n H is 1.8-1.2) before microcrack initiation.As seen in Figure 5c and d, the region with high strain is concentrated both at the interface and in the zirconium alloy matrix, and our simulation in a qualitative agreement with the experimental strain distribution characteristics.Therefore, the above two comparisons (fracture morphology, strain distribution characteristics of twophase material) demonstrate that our numerical framework is reliable.Furthermore, n Zr -n H values can be taken as 1.8-1.2except in Section 3.2.

Influences of Cohesive Strength
The initiation and propagation of microcracks depend heavily on the damage initiation and evolution of cohesive elements.Therefore, in this section, we conduct parametric study on the cohesive strength of zirconium alloy and hydrides to explore their influences on the overall fracture process.On the one hand, Figure 6a shows the stress-strain curves for Model-1 when n Zr is 1.5, 1.8, and 2, while n H is 1.2.It is evident that with the increase in n Zr , both the overall strength and ductility increase significantly.Equivalently, they are very sensitive to the cohesive strength of Zr-Zr elements.On the other hand, Figure 6b depicts the stress-strain curves for Model-1 when n H is 1.1, 1.3, and 1.5, while n Zr is 1.8.It can be seen that the overall strength and ductility increase modestly with the increase in n H .

Influences of Interfacial Characteristics
Formers' studies revealed that when the strength and ductility of matrix and inclusion phase differed a lot, debonding would occur at the interface, which would promote crack propagation and even rapid failure [23,43,44].Therefore, the influences of the fracture characteristics of interface elements (Zr-H elements) on the strength and ductility of Model-1 (shown in Figure 1)  Note that the cohesive strength and fracture energy of Zr-H element gradually increase from interface I to IV. Equivalently, the fracture characteristics of interface I are the worst, while those of interface IV are the best.
Figure 7 shows the stress-strain curves for Model-1 with the four categories of interfaces.The overall strength and ductility associated with interface I are superior to those associated with the other three cases.It can be seen that the overall stress-strain curve is relatively smooth in the case of interface I, and the Zr-H element gets damaged first and then a microcrack initiates, and the entire microstructure fails eventually with the tensile deformation.The overall strength and ductility associated with interface II, III, and IV are essentially the same.In all three cases, the stress-strain curves have obvious fluctuation, as illustrated in the zoomed box in Figure 7.
In order to further explore the influences of interfacial characteristics, Figure 8 shows the fracture morphologies of Model-1 with four categories of interfaces.Although the ratio of interface elements to the total cohesive elements is as low as 0.32%, the interfacial characteristics have great influences on the overall performance.As illustrated in Figure 8a, in the case of interface I, the microcracks occur at the interface, and then continuously and steadily propagate in the zirconium alloy perpendicular to the tensile direction till the eventual failure.As shown in Figure 8b, in the case of interface II, the microcracks initiate at the interface and in the midspan of the hydride as the cohesive strength and fracture energy of the interface elements are the same as those of the hydride.As depicted in Figure 8c and d, in the cases of interface III and IV, the cohesive strength and fracture energy of the interface elements are larger than those of the hydride, so that microcracks occur both near to the interface and in the mid-span of the hydride, which is different from the phenomenon in Figure 8b.Note that the fracture morphologies associated with interface III and IV are consistent with the experimental phenomenon that the fracture initiates in the hydrides of hydrogenated zirconium alloys [12,41].In general, in the case of interface II, both the Zr-H and H-H elements get damaged first, while in the cases of interface III and IV, damage initiates only in the H-H elements.In short, once the hydride is present, no matter of categories of interface, the microcrack initiates either in the hydride or at the interface, which implies larger influences of the hydride on the microcrack initiation.

Influences of Geometric Characteristics of Hydrides
The geometric characteristics of hydride phase cannot be controlled precisely in the experiments.Some studies [18,19] have focused on the effects of geometric characteristics (size, arrangement, quantity, and spacing) of hydrides on the fracture behavior of hydrogenated zirconium.Furthermore, to explore the strength and ductility of hydrogenated zirconium alloys, the effects of geometric characteristics (number density, spacing, and arrangement) of hydrides need to be studied in details.
Studies have shown that when the hydride width L b remains 30 μm, its length L h varies from 100 μm to 200 μm [45].This motivates us to study the effects of the number density of hydrides while keeping the same volume fraction of hydrides.In Figure 2a, the model with a hydride (the length L h : 200 μm, the width L b : 30 μm) is Model-1.In Figure 2d and 2e, the models with two coaxial and uncoaxial hydrides (the length L h : 100 μm, the width L b : 30 μm) are denoted as Model-2 and Model-3, respectively.The distance d h and d s are kept at 60 μm.
Figure 9 shows the stress-strain curves of microstructures with the same volume fraction but the different number density of hydrides.On the one hand, both the strength and ductility of Model-2 are the best, while those of Model-3 are the worst.Specifically, the tensile strength of Model-2 is 598 MPa and the failure strain is 0.07; the tensile strength of Model-3 is 529 MPa and the failure strain is 0.043.Equivalently, the strength of the specimen with the coaxial hydrides is 1.13 times of that with the uncoaxial ones, and the ductility of the former is 1.62 times of that of the latter.It can also be found that when the same volume fraction of hydrides is remained, the microstructure with two hydrides along the tensile direction is slightly more ductile than that with one hydride along the tensile direction.More importantly, the coaxial hydrides lead to much better overall strength and ductility than the uncoaxial ones.On the other hand, no matter in Model-1, Model-2, or Model-3, the microcrack initiation location is close to the interface.Note that in Model-2, the first and second microcracks initiate close to the interfaces near to the tensile ends.Subsequently, the third and fourth microcracks initiate close to the interfaces at the middle position of the microstructure.However, in Model-1 and Model-3, the first and second microcracks initiate near to the interface, and then the third microcrack occurs in the mid-span of each hydride.From the above results, it can be inferred that the specimens with more coaxial hydrides usually have superior performance than those with more uncoaxial ones.
Figure 10 shows the stress-strain curves when the spacing d h and d s vary in a microstructure containing two hydrides (L h : 120 μm, L b : 30 μm).It can be seen that when the d h changes from 20 μm to 80 μm, the strength and ductility of the microstructure are improved modestly.When the d s varies from 20 μm to 80 μm, the overall strength and ductility almost coincide.
Figure 11 shows the stress-strain curves of the microstructures in series A. It can be seen that the arrangement of hydride has obvious influences on the overall performance.Specifically, the strength and ductility of the microstructures with the hydride along the tensile direction are much better than those of the microstructures with the hydride perpendicular to it.This is also consistent with the experimental finding of Li et al. [41] that the radial hydrides perpendicular to the tensile direction are easier to lead to the overall fracture than the circumferential ones.Furthermore, there are three microcracks through the hydride in microstructure A1 and two through the hydride in A2, while the microcracks in A2 are much longer than those in A1.It is noteworthy that in A3, microcracks are also perpendicular to the tensile direction.With the tensile deformation, many microcracks initiate close to the interfaces and then propagate into the matrix phase till the eventual failure.
Figure 12 depicts the stress-strain curves of the four microstructures in series B. It is evident that among the four arrangements, microstructure B1 has the best strength and ductility, and B3 and B4 are relatively weaker, while B4 has the worst ductility.Specifically, when they are along the tensile direction, the coaxial hydrides lead to the better overall performance than the uncoaxial ones.Equivalently, the overall performance is tremendously deteriorated when the hydrides are perpendicular to the tensile direction.
Figure 13a−d shows the fracture morphologies of four microstructures (B1, B2, B3, and B4) in series B. In Figure 13a, three microcracks occur in each hydride.The first microcrack initiates close to the interfaces near to the tensile ends, the second at the interface on the other side, and the third in the mid-span of each hydride.In Figure 13b, two microcracks initiate in each hydride and also close to the interfaces.In Figure 13c, a microcrack initiates in each hydride and also close to the interface.In Figure 13d, a microcrack initiate not in the left hydride but in the right one and it is close to the interface.Again, this reflects our finding that the hydride perpendicular to the tensile direction has larger influences on the overall performance than that along it.
The hydride interconnection phenomenon, as shown in microstructure C1 (Figure 2h) and C2 (Figure 2i), extensively exists in the experimental observations [46], so it is interesting to investigate the fracture behavior of the microstructures in series C. On the one hand, for C1, Figure 14a shows the stress concentration around the nearby vertex of the hydrides.Subsequently, as shown in Figure 14b, bridging occurs in the stress concentration regions around those vertices.In these regions, crack deflection occurs, which is consistent with the finding in Refs.[47,48].On the other hand, for C2, Figure 14c shows the effects of overlapped hydrides on the fracture morphology.The microcrack initiates at the laps and then propagates till C2 fails.

Conclusions
The integrity of zirconium cladding tubes is crucial to the safe operation of nuclear reactors.In this paper, the fracture behavior of two-phase material composed of zirconium alloy and hydride(s) is analyzed based on the two-dimensional cohesive finite element method.The effects of microstructural attributes on the strength and ductility of hydrogenated zirconium alloys are investigated.The main conclusions can be drawn as follows: (1) The overall strength and ductility are more sensitive to the cohesive strength of zirconium alloy than that of hydride.Specifically, the increase in the cohesive strength of zirconium alloy promotes the inhibition on the damage evolution, while the increase in that of hydride has slight effects.
(2) The interface has prominent effects on the overall fracture behavior.When the interfacial cohesive strength and fracture energy are lower (interface I), only debonding occurs.When they are the same as the cohesive strength and fracture energy of the hydride (interface II), the fracture occurs at both the hydride and interface.When they are higher (interface III and IV), fracture initiates in the hydride, which is consistent with the experiments.
(3) The geometric characteristics of hydrides play important roles in the overall strength and ductility.Specifically, under the same volume fraction, the coaxial hydrides can lead to an overall strength 13% larger than the uncoaxial ones, and the former can also lead to an overall ductility 62% larger than the latter.The hydride along the tensile direction leads to the much superior overall performance than that perpendicular to it.
Our simulation can effectively expound many fracture phenomena in the experiments of hydrogenated zirconium alloys and also deepen the understanding of the influence of hydrides on their strength and ductility.

Figure 3 Figure 4
Figure 3 True stress versus plastic strain of zirconium alloy are studied.Four categories of interfaces are introduced according to the fracture characteristics of Zr-H elements: (i) Interface I -Its cohesive strength and fracture energy are half of the corresponding values of H-H elements; (ii) Interface II -Its cohesive strength and fracture energy are the same as those of H-H elements; (iii) Interface III -Its cohesive strength and fracture energy are the corresponding averages of Zr-Zr and H-H elements; (iv) Interface IV -Its cohesive strength and fracture energy are the same as those of Zr-Zr elements.

Figure 5
Figure 5 Comparison between the simulated results and experiments: a Experimental morphology (Reprinted from Ref. [41] with permission from Elsevier Science), b Simulated morphology, c Experimental strain distribution (Reprinted from Ref. [42] with permission from Elsevier Science), d Simulated equivalent plastic strain distribution

Figure 8
Figure 8 Fracture morphologies of the microstructure with four categories of interfaces: a Interface I, b Interface II, c Interface III, d Interface IV (stress in Pa)

Figure 9 Figure 10 Figure 11 Figure 12
Figure 9 Engineering stress-strain curves of microstructures with the same volume fraction but the different number density of hydrides

Figure 13 Figure 14
Figure 13 Fracture morphologies of a microstructure B1, b B2, c B3, and d B4 (stress in Pa)

Table 1
Elastoplastic research of zirconium alloys

Table 2
Constitutive parameters for bulk and cohesive elements