Crack nucleation in rod-type nuclear fuel pellet

Abstract

A plane problem of fracture mechanics on crack nucleation in a rod-type nuclear fuel pellet is considered. Nuclear reactor fuel pellets in operation may be damaged in various ways; in particular, crack nucleation. We consider a problem for the case of a heat-releasing fuel pellet with cladding: as the heat release intensity increases, zones of heightened stress are formed in the nuclear fuel pellet. The heightened stress will promote the appearance of prefracture bands that are simulated as zones of weakened interparticle bonds of the material. Interaction of prefracture zone faces is simulated by placing bonds between faces that have a specified deformation pattern. The problem of equilibrium of a fuel pellet with prefracture zones is reduced to the solution of a system of singular integral equations. An analysis of the ultimate state of the zone of weakened interparticle bonds of the material is realized on the basis of the criterion of critical opening of prefracture zone faces.

Keywords

Rod-type fuel pellet cladding prefracture zones crack nucleation plastic flow zones heat release

1. Introduction

Heat-generating assemblies are often [1 –3] bundles of cylindrical rod-type fuel pellets between whose clearances heat-transfer agent flows parallel to the element axis. Different forms of damage to the heat-releasing elements, in particular, crack nucleation in fuel rods, are possible in the process of a nuclear reactor’s operation.

To ensure reliable operation of fuel pellets, it is necessary to study their stress–strain state and fracture. Problems related to strength of solid nuclear fuel pellets are urgent even today, and undoubtedly interest in them will increase in connection with the existing tendency of an increased energy release in the nuclear fuel. It is appropriate to note the use of new kinds of nuclear fuel, such as mixed-oxide fuel in fast (neutron) reactors.

The problem of crack nucleation and growth in constructions and goods is a significant problem of strength theory [1 –26].

The development of design models for studying crack nucleation in fuel pellets is an urgent problem of the mechanics of materials.

Currently, to the author’s knowledge, there are no studies on crack nucleation in rod-type fuel pellets. The considered form of the nuclear system is a bundle of cylindrical rod fuel pellets located in regular lattices (Figure 1). The coolant flows parallel to the pellets’ axes in the gaps between fuel pellets.

Figure 1.

Disposition scheme for fuel elements in the nuclear system of fuel elements and coolant.

In recent years, in the Russia Federation, a fabrication plant for mixed-oxide fuel for fast reactors has been built and commissioned. In other countries, such plants are being built. The nuclear fuel is a mixture of plutonium and uranium dioxides. This fuel is used in BN-800 reactors on fast neutrons in the fourth block of Beloyarsk Nuclear Power Plant. In future, it is expected to use this fuel in more powerful power unit BN-1200 reactors [27]. It should be noted that the considered nuclear fuel is assumed to be an elastic material. Therefore, the proposed model is valid for most fuel elements currently used in nuclear reactors.

The development of methods for studying crack nucleation will promote serviceability increase of heat-releasing elements, by enabling reasonable selection of structural parameters of elements at the design stage.

The goal of this paper is to develop a mathematical model of crack nucleation in heat-releasing fuel pellets in nuclear reactors.

2. Statement of the problem

A plane problem of fracture mechanics is considered for a fuel pellet occupying a domain in the form of a circle of radius R₁. The geometry of the fuel pellet is delimited by a circle of radius R₁ and the cladding is delimited by circles of radii R₁ and R₂. Let the considered heat-releasing element occupy a circle of radius R₁ in the plane z = x+ iy, centered at the origin of the coordinates, bounded from outside by the circumference L₁, and let the cross-section of the cladding occupy, in the same plane, a ring bounded from outside by a circumference L₂ of radius R₂, whose center coincides with the center of the circumference L₁, and bounded from inside with the circumference L₁ of radius R₁.

In the course of operation of the nuclear reactor, different forms of damage to the fuel pellet material are possible; in particular, crack nucleation. Let us consider a problem for a rod-type fuel pellet with cladding for the case when zones of heightened stresses appear as heat release intensity q in the element increases. The zone of heightened stresses will promote appearance of prefracture zones that are simulated as areas of weakened interparticle bonds of the material.

It is considered that the plane deformation conditions are fulfilled. To simplify the problem, we consider the double-layer structure in the form of “fuel rod cladding.”

The initial axially symmetric problem for the fuel rod is violated because of the pressure of the coolant and the presence of the prefracture zones. The location and size of the first prefracture band are unknown beforehand and are found in the process of solving the problem.

A generalized assumption on plane deformation is accepted. Prefracture bands are described by a cohesive zone form of the Leonov–Panasyuk type brittle model [28]. Recall that in Leonov–Panasyuk’s generalized brittle model, the normal and shear cohesive forces are accepted as constant and are equal to two parameters of the material, $σ_{0}$ and $τ_{0}$ , respectively. Here, $σ_{0}$ equals the breaking limit of the brittle strength and $τ_{0}$ is the shear brittle strength of the material.

We will assume that the fuel rod is so large that in its middle part the stress pattern is identical at all its sections perpendicular to the axis of the fuel’s cylinder.

In the great majority of cases, it is attempted to create close contact (adhesion) between the core and the cladding to improve heat conductivity and ensure optimum heat sink conditions. The absence of such contact may lead to the appearance of large temperature gradients and violation of integrity of the cladding and core. However, the creation of contact between the core and cladding almost always creates some difficulties as the core and cladding are manufactured separately from completely different materials. If there is a gap between the fuel and the cladding, the cladding should have considerable strength in order to resist destructive action due to external pressure of the coolant. Thin claddings ensure neutron economy, but they are not strong enough and should have additional support in order to resist the external pressure of the coolant. Under close mechanical contact of fuel and cladding, when friction forces between fuel and cladding are great, the fuel may serve as a support for the cladding. From the side of the fuel, the forces arising in the fuel because of thermal stresses act on the cladding. These forces may be balanced; then fuel pellets are subjected to uniform two-axial stress. This is possible only in the case when the gap between the fuel and cladding is identical along the entire length of the fuel pellet. Thus, we can assume that the plane deformation conditions are fulfilled. The case when there is a gap between the fuel and cladding is not studied.

In the deformation process, at some points of the domain occupied by the material of the fuel pellet there may appear prefracture zones, which are simulated as zones of weakened interparticle bonds of the material. As the indicated zones (interlayers of stressed material) are small compared with other parts of the element, we can mentally remove them and replace them by cuts whose surfaces interact by some law corresponding to the action of the removed material, where the material of the fuel pellet is deformed beyond elasticity. Let the fuel pellet’s material have an arbitrary number N prefracture zones (Figure 2). At the centers of the prefracture zones (plastic flow zones under constant stress) we place the origin of the local system of coordinates x_kO_ky_k, whose axes x_k coincide with the lines of prefracture zones and make angles α_k with the axis x (Figure 2).

Figure 2.

Design scheme of the crack nucleation problem for a rod-type nuclear fuel pellet.

The interaction of prefracture zone faces is simulated by placing the bonds with the specified deformation pattern between the faces. Under the action of external forces in bonds connecting the prefracture zone faces, there will arise constant normal, σ_s, and tangential, τ_s, cohesive stresses. Thus, normal and tangential stresses numerically equal to σ_s and τ_s, respectively, will be applied to the prefracture zone faces. Herewith, the locations and sizes of the prefracture zones are unknown beforehand and should be determined in the course of solution of the problem.

Experimental research [26, 29, 30] on the appearance of domains wherein the material is deformed beyond elasticity show that, at initial loading stages, the prefracture zones have the form of a narrow stretched layer, and then, as the load increases, there suddenly appears a secondary system of domains that contain a material with partially violated bonds. Based on this research [26, 29, 30], at the initial stages of loading, it is accepted that the prefracture zones extend along a straight line. The number of prefracture zones is given in advance. Allowance for the curvilinearity of prefracture zone bands significantly complicates the problem of crack initiation in nuclear fuel, and this is not considered in this paper. For mathematical description of defect nucleation as a crack in the material of the heat-releasing element in the course of operation, in the considered case, we arrive at a plane problem of thermoelasticity for a two-layer cylindrical structure when the element’s material has prefracture zones (plastic flow zones).

The domain of the fuel pellet bounded by the circumference L₁ we denote by S₁; the domain of the cladding by S₀. As is well-known, an increase in the profitability of nuclear reactors is connected with an increase in burn-out of nuclear fuel. The burn-out of the fuel is limited by the changes that occur in the materials of a fuel pellet during operation in the reactor and reduce to fracture of the fuel pellet. The life time of a fuel pellet may vary, for various reasons, for example, corrosive failure of the cladding. The problems associated with corrosive failure of the cladding are solved by selecting a more suitable cladding material, improved construction, etc. Crack initiation in the core material is an important reason for failure of fuel pellets. Crack initiation reduces to a loss of mechanical strength and fracture under the action of internal stresses. We will consider that the materials of the element and the cladding are isotropic and homogeneous with non-identical mechanical properties.

Let us consider a fracture mechanics problem of crack nucleation in a fuel pellet strengthened with cladding provided that the heat release intensity (the specific capacity of internal heat sources) in the element q is uniformly distributed in the volume.

The stresses in the material of the fuel rod are determined by the second derivatives of the function U (which is called the Airy stress function) in the following form:

σ_{x} = \frac{\partial^{2} U}{\partial y^{2}}, σ_{y} = \frac{\partial^{2} U}{\partial x^{2}}, τ_{xy} = - \frac{\partial^{2} U}{\partial x \partial y} .

In the case under consideration, the boundary value problem for stress function U has the form

Δ Δ U = \frac{α Eq}{λ_{c}} for | z | \leq R_{1},

(1)

Δ Δ U_{0} = 0 for R_{1} \leq | z | \leq R_{2},

(2)

σ_{r} - i τ_{r θ} = σ_{r}^{0} - i τ_{r θ}^{0}, v_{r} - i v_{θ} = v_{r}^{0} - i v_{θ}^{0} for | z | = R_{1},

(3)

σ_{r} - i τ_{r θ} = f_{0} (t) for | z | = R_{2};

(4)

on the prefracture zone faces

σ_{y_{k}} - i τ_{x y_{k}} = σ_{s} - i τ_{s} (k = 1, 2, \dots, N) .

(5)

Here the function f₀(t) describes the action of coolant on the external surface of the cladding; α is the factor of linear temperature expansion; E is the modulus of elasticity of the fuel pellet; λ_c is the heat-conduction coefficient; $σ_{r}$ , $τ_{r θ}$ and $σ_{r}^{0}$ , $τ_{r θ}^{0}$ are stress tensor components for the fuel pellet and the cladding, respectively; and $v_{r}$ , $v_{θ}$ and $v_{r}^{0}$ , $v_{θ}^{0}$ are displacement vector components.

To determine the intensity values of the internal heat sources under which crack nucleation happens, the problem statement should be completed with the criterion of crack appearance (rupture of interparticle bonds of the material). As such a criterion, we accept the condition of critical opening of prefracture zone faces:

| (u_{k}^{+} - u_{k}^{-}) - i (v_{k}^{+} - v_{k}^{-}) | = δ_{c} (k = 1, 2, \dots, N),

(6)

where $(v_{k}^{+} - v_{k}^{-})$ and $(u_{k}^{+} - u_{k}^{-})$ are normal and tangential components of the opening of the faces of the kth prefracture zone, δ_c is the characteristic of resistance of the fuel pellet’s material to cracking. The value of δ_c is determined experimentally [26].

This criterion enables us to determine the parameters of a two-layer cylindrical structure for which a crack appears in a fuel pellet.

3. Solution method

Using Kolosov–Muskhelishvili complex potentials [31], the problem under consideration is reduced to finding two pairs of complex functions Φ_j(z) and Ψ_j(z) (j = 0, 1) of the complex variable z = x+ iy analytic in domains S_j (j = 0, 1) and satisfying the boundary conditions:

Φ_{0} (τ_{2}) + \bar{Φ_{0} (τ_{2})} - \frac{τ_{2}^{2}}{R_{2}^{2}} [\frac{R_{2}^{2}}{τ_{2}} Φ_{0}' (τ_{2}) + Ψ_{0} (τ_{2})] = f_{0} (θ) on L_{2},

(7)

\begin{matrix} Φ_{0} (τ_{1}) + \bar{Φ_{0} (τ_{1})} - \frac{τ_{1}^{2}}{R_{1}^{2}} [\frac{R_{1}^{2}}{τ_{1}} Φ_{0}' (τ_{1}) + Ψ_{0} (τ_{1})] \\ = Φ_{1} (τ_{1}) + \bar{Φ_{1} (τ_{1})} - \frac{τ_{1}^{2}}{R_{1}^{2}} [\frac{R_{1}^{2}}{τ_{1}} Φ_{1}' (τ_{1}) + Ψ_{1} (τ_{1})] + \frac{q_{*} τ_{1} {\bar{τ}}_{1}}{32} on L_{1} \end{matrix}

(8)

\begin{matrix} κ_{0} \bar{Φ_{0} (τ_{1})} - Φ_{0} (τ_{1}) + \frac{τ_{1}^{2}}{R_{1}^{2}} [\frac{R_{1}^{2}}{τ_{1}} Φ_{0}' (τ_{1}) + Ψ_{0} (τ_{1})] = \\ = \frac{μ_{0}}{μ_{1}} {κ_{1} \bar{Φ_{1} (τ_{1})} - Φ_{1} (τ_{1}) + \frac{τ_{1}^{2}}{R_{1}^{2}} [\frac{R_{1}^{2}}{τ_{1}} Φ_{1}' (τ_{1}) + Ψ_{1} (τ_{1})] - \frac{q_{*} τ_{1} {\bar{τ}}_{1}}{32}} on L_{1}; \end{matrix}

(9)

on the prefracture zone faces $y_{n} = 0$ , $| t_{n} | \leq l_{n}$ (n = 1, 2, …, N):

Φ_{1} (t_{n}) + \bar{Φ_{1} (t_{n})} + t_{n} \bar{Φ_{1}' (t_{n})} + \bar{Ψ_{1} (t_{n})} + \frac{3 q_{*} t_{n}^{2}}{32} = σ_{s} - i τ_{s} .

(10)

Here, $q_{*} = α Eq / λ_{c}$ ; $τ_{1} = R_{1} e^{i θ}$ ; $τ_{2} = R_{2} e^{i θ}$ ; t_n is the affix of the points of the prefracture zone faces: μ₁ and μ₀ are shear modules of the materials of the fuel and cladding, respectively, and κ₁ and κ₀ are Muskhelishvili constants for the fuel and cladding.

Under the action of loading, at the expense of differences in elastic properties of the materials of the fuel and the cladding, some self-balanced system of forces acts on the fuel pellet from the side of the cladding.

Considering for now this load on the element given formally, we find complex potentials Φ₁(z) and Ψ₁(z). Denote by N₁– iT₁ the left-hand side of boundary condition (9) and expand it on the contour L₁ in Fourier series:

Φ_{0} (t) + \bar{Φ_{0} (t)} - \frac{t^{2}}{R_{1}^{2}} [\frac{R_{1}^{2}}{t} Φ_{0}' (t) + Ψ_{0} (t)] = N_{1} - i T_{1},

(11)

N_{1} - i T_{1} = \sum_{k = - \infty}^{\infty} A_{k} e^{i k θ} .

(12)

According to boundary conditions (8), (9), and (11) for determining the potentials Φ₁(z) and Ψ₁(z), we have, on the contour L₁, the boundary conditions

Φ_{1} (t) + \bar{Φ_{1} (t)} - \frac{t^{2}}{R_{1}^{2}} [\frac{R_{1}^{2}}{t} Φ_{1}' (t) + Ψ_{1} (t)] = \sum_{k = - \infty}^{\infty} A_{k} e^{i k θ};

(13)

on the segments $y_{n} = 0$ , $| x_{n} | \leq l_{n}$ (n = 1, 2, …, N)

Φ_{1} (t_{n}) + \bar{Φ_{1} (t_{n})} + t_{n} \bar{Φ_{1}' (t_{n})} + \bar{Ψ_{1} (t_{n})} = σ_{s} - i τ_{s} - \frac{3 q_{*} t_{n}^{2}}{32} .

(14)

We look for the solution of the boundary value problem (equations (13) and (14)) in the form

Φ_{1} (z) = Φ_{11} (z) + Φ_{12} (z), Ψ_{1} (z) = Ψ_{11} (z) + Ψ_{12} (z);

(15)

\begin{array}{l} Φ_{11} (z) = \frac{1}{2 π} \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} \frac{g_{k} (t)}{t - z_{k}} d t, \\ Ψ_{11} (z) = \frac{1}{2 π} \sum_{k = 1}^{N} e^{- i 2 α_{k}} \int_{- l_{k}}^{l_{k}} [\frac{\bar{g_{k} (t)}}{t - z_{k}} - \frac{{\bar{T}}_{k} e^{i α_{k}}}{{(t - z_{k})}^{2}} g_{k} (t)] d t; \end{array}

(16)

\begin{array}{l} T_{k} = t e^{i α_{k}} + z_{k}^{0}; z_{k}^{0} = x_{k}^{0} + i y_{k}^{0}; z_{k} = e^{- i α_{k}} (z - z_{k}^{0}) \\ \frac{i (κ_{1} + 1)}{2 μ_{1}} g_{k} (x) = \frac{\partial}{\partial x} [u_{k}^{+} - u_{k}^{-} + i (v_{k}^{+} - v_{k}^{-})] (k = 1, 2, \dots, N) . \end{array}

(17)

The potentials Φ₁₂(z) and Ψ₁₂(z) are regular in the entire circle of radius R₁; they should be determined from the solution for an entire circle according to boundary condition (13); to determine the unknown functions g_n(x) (n = 1, 2, …, N), the boundary conditions (14) on the segments $y_{n} = 0$ , $| x_{n} | \leq l_{n}$ should be used.

In what follows, all linear sizes (x, t, $z_{n}^{0}$ , and l_n) are dimensionless quantities, having been divided by radius R₁.

We satisfy by functions (15) and (16) the boundary conditions on the faces of the plastic deformations zones and obtain a system of singular integral equations with respect to unknown functions g_n(x) (n = 1, 2, …, N)

\begin{matrix} \frac{1}{π} \int_{- l_{n}}^{l_{n}} \frac{g_{n} (t)}{t - x_{n}} d t + \frac{1}{π} \sum_{\binom{k = 1}{k \neq n}}^{N} \int_{- l_{k}}^{l_{k}} [g_{k} (t) K_{nk} (t, x_{n}) + \bar{g_{k} (t)} L_{nk} (t, x_{n})] d t + Φ_{12} (x_{n}) + \bar{Φ_{12} (x_{n})} + x_{n} \bar{Φ_{12}' (x_{n})} + \bar{Ψ_{12} (x_{n})} \\ = σ_{s} - i τ_{s} - \frac{3 q_{*} x_{n}^{2}}{32} \end{matrix}

(18)

Here,

\begin{matrix} K_{nk} (t, x) = \frac{e^{i α_{k}}}{2} [\frac{1}{T_{k} - X_{n}} + \frac{e^{- i 2 α_{n}}}{{\bar{T}}_{k} - {\bar{X}}_{n}}], \\ L_{nk} (t, x) = \frac{e^{- i α_{k}}}{2} [\frac{1}{{\bar{T}}_{k} - {\bar{X}}_{n}} + \frac{T_{k} - X_{n}}{{({\bar{T}}_{k} - {\bar{X}}_{n})}^{2}} e^{- i 2 α_{n}}], \\ X_{n} = x e^{i α_{n}} + z_{n}^{0} . \end{matrix}

Boundary condition (13) admits an auxiliary problem for an entire circle:

Φ_{12} (τ) + \bar{Φ_{12} (τ)} - e^{- i 2 θ} [τ \bar{Φ_{12}' (τ)} + \bar{Ψ_{12} (τ)}] = f (τ),

(19)

f (τ) = \sum_{k = - \infty}^{\infty} {\bar{A}}_{k} e^{- i k θ} - \frac{1}{2 π} \sum_{k = 1}^{\infty} \int_{- l_{k}}^{l_{k}} {[\frac{1}{t - z_{k}} - \frac{e^{i 2 α_{k}}}{τ^{2} (t - {\bar{z}}_{k})}] g_{k} (t) - [\frac{1}{t - {\bar{z}}_{k}} - \frac{τ - T_{k}}{τ^{2} {(t - {\bar{z}}_{k})}^{2}} e^{i α_{k}}] \bar{g_{k}' (t)}} d t,

(20)

z_{k} = (τ - z_{k}^{0}) e^{- i α_{k}}, τ = e^{i θ} .

The solution of auxiliary problem (19) is given by Muskhelishvili’s formulas:

Φ_{12} (z) = \frac{1}{2 π i} \int_{L} f (τ) (\frac{1}{τ - z} - \frac{1}{2 τ}) d τ,

(21)

Ψ_{12} (z) = \frac{1}{z^{2}} Φ_{12} (z) + \frac{1}{z^{2}} {\bar{Φ}}_{12} (\frac{1}{z}) - \frac{1}{z} Φ'_{12} (z) .

After substituting function (20) in formula (21) and changing the integration order, as well as using the residue theory, we finally get:

Φ_{12} (z) = \frac{1}{2 π} \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} {(\frac{1}{z {\bar{T}}_{k} - 1} + \frac{1}{2}) \bar{T_{k}} e^{i α_{k}} g_{k} (t) + [\frac{T_{k}}{2} - \frac{z^{2} \bar{T_{k}} - 2 z + T_{k}}{{(z \bar{T_{k}} - 1)}^{2}}] e^{- i α_{k}} \bar{g_{1} (t)}} d t + \sum_{k = 0}^{\infty} a_{k} z^{k},

(22)

\begin{array}{l} Ψ_{12} (z) = \frac{1}{2 π} \\ \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} [\frac{e^{i α_{k}} {\bar{T}}_{k}^{3}}{{(z {\bar{T}}_{k} - 1)}^{2}} g_{k} (t) + (z^{2} {\bar{T}}_{k}^{2} + 4 - 3 z {\bar{T}}_{k} + z T_{k} {\bar{T}}_{k}^{2} - 3 T_{k} {\bar{T}}_{k}) \frac{{\bar{T}}_{k} e^{- i α_{k}}}{{(z {\bar{T}}_{k} - 1)}^{3}} \bar{g_{k} (t)}] d t + \sum_{k = 0}^{\infty} b_{k} z^{k}; \end{array}

a_{0} + {\bar{a}}_{0} = A_{0}; A_{0} = {\bar{A}}_{0};

a_{k} = - {\bar{A}}_{k}, (k = 1, 2, \dots); b_{k} = - (k + 1) {\bar{A}}_{- k - 2} - A_{k + 2}, (k = 0, 1, \dots) .

(23)

Substituting the found values of Φ₁₂(z) and Ψ₁₂(z) into equation (18), we finally get a system of singular integral equations with respect to the unknown functions g_n(x) (n = 1, 2, …, N):

\begin{matrix} \frac{1}{π} \int_{- l_{n}}^{l_{n}} \frac{g_{n} (t)}{t - x_{n}} d t + \frac{1}{π} \sum_{\binom{k = 1}{k \neq n}}^{N} \int_{- l_{n}}^{l_{n}} [K_{nk}^{*} (t, x_{n}) g_{k} (t) + L_{nk}^{*} (t, x_{n}) \bar{g_{k} (t)}] d t = \\ = σ_{s} - i τ_{s} - \sum_{k = 0}^{\infty} [(2 k + 1) A_{- k} + {\bar{A}}_{- k} - (k + 1) A_{- k - 2} - {\bar{A}}_{k + 2}] x_{n}^{k} - \frac{3 q_{*} t_{n}^{2}}{32}, \end{matrix}

| x_{n} | \leq l_{n} (n = 1, 2, \dots, N) .

(24)

Here,

\begin{array}{l} K_{n k}^{*} (t, x) = K_{n k} (t, x) - \frac{e^{i α_{k}}}{2} [\frac{X_{n} {\bar{T}}_{k}^{2}}{1 - X_{n} {\bar{T}}_{k}} + \frac{{\bar{X}}_{n}^{2} T_{k} - 2 {\bar{X}}_{n} + {\bar{T}}_{k}}{{(1 - T_{k} {\bar{X}}_{n})}^{2}} + e^{- i 2 α_{n}} \frac{2 X_{n} (T_{k} {\bar{T}}_{k} - 1) + 4 T_{k} + T_{k}^{2} ({\bar{X}}_{n} + {\bar{T}}_{k}) ({\bar{X}}_{n} T_{k} - 3)}{{(1 - T_{k} {\bar{X}}_{n})}^{3}}] \\ L_{n k}^{*} (t, x_{n}) = L_{n k} (t, x) - \frac{e^{- i α_{k}}}{2} [\frac{T_{k}^{2} {\bar{X}}_{n}}{1 - T_{k} {\bar{X}}_{n}} + \frac{X_{n}^{2} {\bar{T}}_{k} - 2 X_{n} + T_{k}}{{(1 - X_{n} {\bar{T}}_{k})}^{2}} + \frac{T_{k}^{2} (X_{n} - T_{k}) e^{- i 2 α_{n}}}{{(1 - T_{k} {\bar{X}}_{n})}^{2}}] \end{array}

To the system of singular integral equations (24) it is necessary to add the conditions following from the physical sense of the problem:

\int_{- l_{n}}^{l_{n}} g_{n} (t) d t = 0 (n = 1, 2, \dots, N) .

(25)

Knowing the potentials Φ₁(z) and Ψ₁(z), describing the stress–strain state of a fuel pellet, we find the following value:

\begin{matrix} - κ_{1} \bar{Φ_{1} (t)} + Φ_{1} (t) - \frac{t^{2}}{R_{1}^{2}} [\frac{R_{1}^{2}}{t} Φ_{1}' (t) + Ψ_{1} (t)] \\ = \frac{A_{0}}{2} (1 - κ_{1}) - (1 + κ_{1}) {\bar{B}}_{0} + \sum_{k = 1}^{\infty} A_{k} e^{i k θ} - κ_{1} \sum_{k = 1}^{\infty} A_{- k} e^{- i k θ} - (1 + κ_{1}) \sum_{k = 1}^{\infty} {\bar{B}}_{k} e^{- i k θ} + κ_{1} \frac{q_{*} R_{1}^{2}}{32} . \end{matrix}

(26)

Here,

\begin{matrix} B_{n} = a_{n}^{*} + d_{n} + c_{n}; \\ a_{n}^{*} = \frac{1}{2 π} \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} \frac{e^{i α_{k}} g_{k} (t)}{T_{k}^{n + 1}} d t; \\ d_{0} = - \frac{3}{4 π} \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} e^{i α_{k}} {\bar{T}}_{k} g_{k} (t) d t; \\ d_{n} = - \frac{1}{2 π} \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} e^{i α_{k}} {\bar{T}}_{k + 1}^{n + 1} g_{k} (t) d t; \\ c_{n} = {c^{'}}_{n - 2} - 2 {c^{'}}_{n - 2} + c_{n}^{*}, (n = 2, 3, \dots); \\ c_{0} = c_{0}^{*} + c_{*}; \\ c_{1} = 2 {c^{″}}_{0} + c_{1}^{*}; \\ c_{*} = \frac{1}{2 π} \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} \frac{T_{k}}{2} e^{- i α_{k}} \bar{g_{k} (t)} d t; \\ {c^{'}}_{n} = \frac{n + 1}{2 π} \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} e^{- i α_{k}} \bar{g_{k} (t)} {\bar{T}}_{k}^{n + 1} d t, (n = 0, 1, \dots); \\ {c^{″}}_{n} = \frac{n + 1}{2 π} \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} e^{- i α_{k}} {\bar{T}}_{k}^{n} \bar{g_{k} (t)} d t, (n = 0, 1, \dots); \\ c_{n}^{*} = \frac{n + 1}{2 π} \sum_{k = 1}^{N} \int_{- l_{k}}^{l_{k}} e^{- i α_{k}} T_{k} {\bar{T}}_{k}^{n} \bar{g_{k} (t)} d t, (n = 0, 1, \dots) . \end{matrix}

When constructing the solution for a fuel pellet, we considered the coefficients A_k (k = 0, ±1, …) to be given, but still unknown. They should be determined in the course of the solution of the boundary value problem. To find them, we use the solution of an elasticity problem for a cladding. To determine the potentials Φ₀(z) and Ψ₀(z) and the coefficients A_k (k = 0, ±1, …), based on relations (7), (9), and (11), we have the following boundary conditions:

\begin{matrix} Φ_{0} (t) + \bar{Φ_{0} (t)} - \frac{t^{2}}{R_{2}^{2}} [\frac{R_{2}^{2}}{t} Φ_{0}' (t) + Ψ_{0} (t)] = N_{0} + \sum_{k = 1}^{\infty} [N_{k} {(\frac{t}{R_{2}})}^{k} + N_{- k} {(\frac{R_{2}}{t})}^{k}] on L_{2}, \end{matrix}

(27)

Φ_{0} (t) + \bar{Φ_{0} (t)} - \frac{t^{2}}{R_{1}^{2}} [\frac{R_{1}^{2}}{t} Φ_{0}' (t) + Ψ_{0} (t)] = \sum_{k = - \infty}^{\infty} A_{k} e^{i k θ} on L_{1},

(28)

\begin{array}{l} - κ_{0} \bar{Φ_{0} (t)} + Φ_{0} (t) - \frac{t^{2}}{R_{1}^{2}} [\frac{R_{1}^{2}}{t} Φ_{0^{'}} (t) + Ψ_{0} (t)] \\ = \frac{μ_{0}}{μ_{1}} {\frac{A_{0}}{2} (1 - κ_{1}) - (1 + κ_{1}) {\bar{B}}_{0} + \sum_{k = 1}^{\infty} A_{k} e^{i k θ} + κ_{1} \frac{q_{*} R_{1}^{2}}{32} - κ_{1} \sum_{k = 1}^{\infty} A_{- k} e^{- i k θ} - (1 + κ_{1}) \sum_{k = 1}^{\infty} {\bar{B}}_{k} e^{- i k θ}} on L_{1} . \end{array}

(29)

The functions Φ₀(z) and Ψ₀(z) are regular interior to the domain S₀, and therefore may be expanded in Laurent series. Thus, interior to the domain S₀, we will have

Φ_{0} (z) = \sum_{k = - \infty}^{\infty} F_{k} z^{k}, Ψ_{0} (z) = \sum_{k = - \infty}^{\infty} F_{k^{'}} z^{k} .

(30)

Satisfying by functions (30) the boundary conditions (27) to (29) and comparing the coefficients under identical powers of exp(iθ) in the left- and right-hand sides of boundary conditions (27) to (29), after some transformations we get:

2 F_{0} - F_{- 2^{'}} = A_{0}; 2 F_{0} - \frac{F_{- 2^{'}}}{λ^{2}} = N_{0}; F_{0} = {\bar{F}}_{0};

λ = \frac{R_{2}}{R_{1}}; (1 - k) F_{k} + {\bar{F}}_{- k} - {F^{'}}_{k - 2} = A_{k} (k = \pm 1, \pm 2, \dots);

(1 - k) F_{k} λ^{k} + {\bar{F}}_{- k} λ^{- k} - F'_{k - 2} λ^{k - 2} = N_{k};

(31)

F_{0} (1 - κ_{0}) - F'_{- 2} = \frac{μ_{0}}{μ_{1}} {\frac{A_{0}}{2} (1 - κ_{1}) - (1 + κ_{1}) {\bar{B}}_{0} + \frac{q_{*}}{64} (1 + κ_{1})};

(1 - k) F_{k} - κ_{0} {\bar{F}}_{- k} - {F^{'}}_{k - 2} = D_{k} (k = \pm 1, \pm 2, \dots);

D_{k} = \frac{μ_{0}}{μ_{1}} A_{k} (k = 1, 2, \dots); D_{- k} = - \frac{μ_{0}}{μ_{1}} {κ_{1} A_{- k} + (1 + κ_{1}) {\bar{B}}_{k}} .

(32)

From equations (31) and (32) we find

F_{0} = \frac{N_{0} λ^{2} - A_{0}}{2 (λ^{2} - 1)}; {F^{'}}_{- 2} = \frac{λ^{2} (N_{0} - A_{0})}{λ^{2} - 1};

F_{k} = \frac{(1 + k) (λ^{2} - 1) (N_{k} λ^{- k + 2} - A_{k}) - ({\bar{N}}_{- k} λ^{k + 2} - {\bar{A}}_{- k}) (λ^{- k + 2} - 1)}{(1 - k^{2}) {(λ^{2} - 1)}^{2} - (λ^{2 k + 2} - 1) (λ^{- 2 k + 2} - 1)}

(k = \pm 2, \pm 3, \dots);

F_{- 1} = \frac{{\bar{A}}_{1}}{1 + κ_{0}}; {F^{'}}_{- 1} = - \frac{κ_{0} A_{1}}{1 + κ_{0}};

F_{1} = \frac{{\bar{N}}_{- 1} λ^{3} - {\bar{A}}_{- 1}}{λ^{4} - 1} - \frac{2 A_{1}}{(1 + κ_{0}) (λ^{2} + 1)}; {F^{'}}_{k - 2} = - A_{k} + (1 - k) F_{k} - {\bar{F}}_{- k};

A_{0} = \frac{M_{1}}{M_{0}}; M_{1} = \frac{μ_{0}}{μ_{1}} (1 + κ_{1}) (\frac{q_{*}}{64} - {\bar{B}}_{0}) + (1 + κ_{0}) \frac{N_{0} λ^{2}}{2 (λ^{2} - 1)};

M_{2} = \frac{λ^{2}}{λ^{2} - 1} - \frac{1 - κ_{0}}{2 λ^{2} - 1} - \frac{μ_{0}}{μ_{1}} (1 - κ_{1});

A_{k} = - \frac{1 + κ_{0}}{(\frac{μ_{0}}{μ_{1}} - 1)} {\bar{F}}_{- k} (k = 1, 2, \dots);

A_{- k} (1 + \frac{μ_{0}}{μ_{1}} κ_{1}) - (1 + κ_{0}) {\bar{F}}_{k} = - \frac{μ_{0}}{μ_{1}} (1 + κ_{1}) {\bar{B}}_{k} .

The last relations admit expression of the coefficients A_k (k = 0, ±1, …) by the values B_k, which are dependent on the unknown functions g_n(x) (n = 1, 2, …, N). Conversely, the system of singular integral equations contains the unknown coefficients A_k.

If we represent the unknown functions $g_{k} (x_{k})$ in the form

g_{k} (x_{k}) = v_{k} (x_{k}) - i u_{k} (x_{k}),

then from the system of N complex singular integral equations, after separation of real and imaginary parts, for finding $v_{k} (x_{k})$ and $u_{k} (x_{k})$ , we get the system of 2N real singular integral equations. We will use an algebraization procedure of the system of singular integral equations [29, 32 –34].

For the right-hand side of relation (17), we have

(v_{k}^{+} (x_{k}, 0) - v_{k}^{-} (x_{k}, 0)) - i (u_{k}^{+} (x_{k}, 0) - u_{k}^{-} (x_{k}, 0)) = - \frac{1 + κ_{1}}{2 μ_{1}} \int_{- l_{k}}^{x_{k}} g_{k} (x_{k}) d x_{k} (n = 1, 2, \dots, N) .

(33)

Thus, the infinite system of algebraic equations and the system of singular integral equations should be solved jointly.

Now we pass to algebraization of the mentioned system of equations.

4. Algebraization of main resolving equations

By means of change of a variable, we reduce all integration intervals to one segment [−1, 1]. For that we put

t = l_{n} τ, x = l_{n} η, | t | < l_{n}, | x | < l_{n} .

Then the system of singular integral equations (24) and the additional conditions (25) are written in the form

\begin{matrix} \frac{1}{π} \int_{- 1}^{1} \frac{g_{n} (τ)}{τ - η} d τ + \frac{1}{π} \sum_{k \neq n}^{N} \int_{- 1}^{1} [g_{k} (τ) K_{nk}^{*} (l_{k} τ, l_{k} η) + \bar{g_{k} (τ)} L_{nk}^{*} (l_{k} τ, l_{k} η)] d τ \\ = σ_{s} - i τ_{s} - \frac{3 q_{*} η^{2}}{32} - \sum_{k = 0}^{\infty} [(2 k + 1) A_{- k} + {\bar{A}}_{- k} + (k + 1) A_{- k - 2} - {\bar{A}}_{k + 2}] l_{n}^{k} η_{r}^{k} | η | < 1, \end{matrix}

(34)

\int_{- 1}^{1} g_{n} (τ) d τ = 0 (n = 1, 2, \dots, N) .

(35)

In equations (34) and (35), for convenience, for the unknown functions, we preserved the previous designations. We represent the solution of system (34) under conditions (35) in the form

g_{n} (τ) = \frac{ϕ_{n} (τ)}{\sqrt{1 - τ^{2}}} (n = 1, 2, \dots, N) .

(36)

Here, the functions $φ_{n} (τ)$ are Holder continuous on the segment [−1, 1] and the functions $φ_{n} (τ)$ are replaced by an interpolational polynomial constructed by Chebyshev nodes.

By means of Gauss–Chebyshev-type quadrature formulas [32 –34], the system of integral equations (34) under conditions (35) is reduced to the system of N×M algebraic equations with respect to the unknowns $φ_{n} (τ_{m})$ (n, m = 1, 2, …, N)

\begin{matrix} \frac{1}{M} \sum_{m = 1}^{M} \sum_{k = 1}^{N} l_{k} [ϕ_{k} (τ_{m}) K_{nk}^{*} (l_{k} τ_{m}, l_{n} η_{r}) + \bar{ϕ_{k} (τ_{m})} L_{nk}^{*} (l_{k} τ_{m}, l_{n} η_{r})] \\ = σ_{s} - i τ_{s} - \sum_{k = 0}^{\infty} [(2 k + 1) A_{- k} + {\bar{A}}_{- k} + (k + 1) A_{- k - 2} - {\bar{A}}_{k + 2}] l_{n}^{k} η_{r}^{k} - \frac{3 q_{*} η_{r}^{2}}{32}; \end{matrix}

\sum_{m = 1}^{M} φ_{n} (τ_{m}) = 0 (r = 1, 2, \dots, M - 1) .

(37)

Here,

τ_{m} = \cos \frac{2 m - 1}{2 M} π (m = 1, 2, \dots, M); η_{r} = \cos \frac{π r}{M} (r = 1, 2, \dots, M - 1) .

Gauss–Chebyshev-type quadrature formulas admit complete algebraization of the related systems of equations mentioned already, by means of the relations

a_{n}^{*} = \frac{1}{2 π} \sum_{k = 1}^{N} l_{k} \int_{- 1}^{1} \frac{e^{i α_{k}} g_{k} (τ)}{{(T_{k}^{*})}^{n + 1}} d τ = \frac{1}{2 M} \sum_{k = 1}^{N} l_{k} \sum_{m = 1}^{M} \frac{e^{i α_{k}} ϕ_{k} (τ_{m})}{{(T_{k}^{*})}^{n + 1}};

d_{0} = - \frac{3}{4 M} \sum_{k = 1}^{N} l_{k} e^{i α_{k}} \sum_{m = 1}^{M} {\bar{T}}_{k}^{*} ϕ_{k} (τ_{m});

d_{n} = - \frac{1}{2 M} \sum_{k = 1}^{N} l_{k} e^{i α_{k}} \sum_{m = 1}^{M} {({\bar{T}}_{k + 1}^{*})}^{n + 1} ϕ_{k} (τ_{m});

c_{*} = \frac{1}{4 M} \sum_{k = 1}^{N} l_{k} e^{- i α_{k}} \sum_{m = 1}^{M} T_{k}^{*} \bar{ϕ_{k} (τ_{m})};

(38)

{c^{'}}_{n} = \frac{n + 1}{2 M} \sum_{k = 1}^{N} l_{k} e^{- i α_{k}} \sum_{m = 1}^{M} {({\bar{T}}_{k}^{*})}^{n + 1} \bar{ϕ_{k} (τ_{m})} (n = 0, 1, \dots);

{c^{″}}_{n} = \frac{n + 1}{2 M} \sum_{k = 1}^{N} l_{k} e^{- i α_{k}} \sum_{m = 1}^{M} {({\bar{T}}_{k}^{*})}^{n} \bar{ϕ_{k} (τ_{m})} (n = 0, 1, \dots);

c_{n}^{*} = \frac{n + 1}{2 M} \sum_{k = 1}^{N} l_{k} e^{- i α_{k}} \sum_{m = 1}^{M} T_{k}^{*} {({\bar{T}}_{k}^{*})}^{n} \bar{ϕ_{k} (τ_{m})} (n = 0, 1, \dots) .

The mentioned systems are not closed yet.

To preserve the design scheme, the solution of the system of singular integral equations was sought in the form of equation (36), that is, it has root singularities at the end of the prefracture zone. In fact, within the model of the prefracture zone, with interfacial bonds between faces, the stresses in the fuel pellet are everywhere finite, including at the tip of prefracture zone.

It would be well to look for the solution of the problem in the class of everywhere-bounded functions (stresses). The condition of stress limitation at the tip of the prefracture zone helps to determine the location of unknown sizes of prefracture zones. We complete the obtained main systems of algebraic equations by the stress limitation conditions in the vicinity of prefracture zone tips:

\sum_{m = 1}^{M} {(- 1)}^{m} φ_{k} (τ_{m}) \cdot \cot \frac{2 m - 1}{4 M} π = 0 (k = 0, 1, \dots, N);

\sum_{m = 1}^{M} {(- 1)}^{M + m} φ_{k} (τ_{m}) \cdot \tan \frac{2 m - 1}{4 M} π = 0 .

(39)

The system of algebraic equations (31), (32), (37), and (38) is closed. Because of the unknown lengths of the prefracture zones, the system of combined algebraic equations is not linear. Its numerical solution enables determination of the coordinates of the tips (location) and the sizes of the prefracture zones; that is, the values $φ_{n} (τ_{m})$ (n = 1, 2, …, N, m = 1, 2, …, M).

Obviously, having determined the coordinates of the tips of all prefracture zones, using the known formulas of analytic geometry one can find the coordinates $z_{k}^{0}$ of the prefracture zone centers and the angles α_k with the axis x (Figure 2).

Knowing the function g_n(x), one can find the stress–strain state of a fuel pellet and its cladding.

5. Limit level of heat release intensity

After finding the sought-for functions $g_{k} (x_{k})$ (k = 1, 2, …, N), we calculate the opening of the prefracture zone faces by the relation

\begin{matrix} V_{0 k} (x_{k}^{0}) = | v_{k}^{+} (x_{k}^{0}, 0) - v_{k}^{-} (x_{k}^{0}, 0) - i (u_{k}^{+} (x_{k}^{0}, 0) - u_{k}^{-} (x_{k}^{0}, 0)) | \\ = - \frac{κ_{1} + 1}{2 μ_{1}} \int_{l_{k}}^{x_{k}^{0}} g_{k} (x_{k}) d x_{k}, \end{matrix}

where $x_{k}^{0}$ are the coordinates of the point of the plastic flow zone wherein ruptured interparticle bonds of the material are noticeable.

As a result of these calculations, the lengths of the prefracture zones were determined as a function of the intensity of the internal heat sources and the parameter R₁/R₂.

By means of the criterion of critical opening of prefracture zone faces, the condition determining the limit level of the heat release intensity in the element was found for each prefracture zone. The value of heat release intensity that caused appearance of the kth crack is determined from the relation

| - \frac{1 + κ_{1}}{2 μ_{1}} \int_{l_{k}}^{x_{k}^{0}} g_{k} (x_{k}) d x_{k} | = δ_{c} (k = 0, 1, \dots, N) .

(40)

The critical value of the intensity of the internal heat sources in the fuel pellet will be minimal among $q_{* k}$ (k = 1, 2, …, N)

q_{*} = min {q_{* k}} .

Because of the unknown quantities l_k (k = 1, 2, …, N), the combined resolving system of equations becomes nonlinear. To solve it, we used the successive approximations method. We solve the combined system under some certain values $l_{k}^{*}$ with respect to the unknowns $g_{k} (t_{m})$ (m = 1, 2, …, M). The values of $l_{k}^{*}$ and the found quantities $g_{k} (t_{m})$ are substituted in equation (39), that is, to the unused equation of the combined system. The taken values of parameters $l_{k}^{*}$ and appropriate values $g_{k} (t_{m})$ will not, generally speaking, satisfy equation (39). Therefore, by choosing the values of the parameters $l_{k}^{*}$ , we will repeat calculations until equation (39) of the combined system is satisfied to the specified accuracy.

At each approximation, the combined system of equations was solved by the Gauss method, choosing the principal element for different values of M.

The calculation results are given in Figures 3 to 7. The dependences of the prefracture zone length l_k/R₂ (k = 1, 2, 3) on the dimensionless value $q_{*} / σ_{s}$ for different orientation angles α₁ = 15°; α₂ = 30°; and α₃ = 45° are represented in Figure 3. In calculations, M = 30; $κ_{0} / κ_{1} = 1.5$ ; and $μ_{0} / μ_{1} = 0.5$ were accepted.

Figure 3.

Prefracture zone length l_k/R₂ (k = 1, 2, 3) as a function of the dimensionless value of the intensity of internal heat sources for different orientation angles: α₁ = 15°; α₂ = 30°; α₃ = 45°.

Figure 4.

Normal and tangential components of the displacement vector as a function of dimensionless coordinates $x'_{1}$ for orientation angle α₁ = 15°.

Figure 5.

Normal and tangential components of the displacement vector as a function of dimensionless coordinates $x'_{2}$ for orientation angle α₂ = 30°.

Figure 6.

Normal and tangential components of the displacement vector as a function of dimensionless coordinates $x'_{3}$ for orientation angle α₃ = 45°.

Figure 7.

Critical intensity of internal heat sources as a function of the relative opening $δ_{*} / l_{1}$ at the center of the prefracture zone.

Graphs of the distribution of the normal $(v_{k}^{+} - v_{k}^{-}) / R_{2}$ and tangential $(u_{k}^{+} - u_{k}^{-}) / R_{2}$ components of the displacement vector are given in Figures 4 to 6. In calculations, dimensionless coordinates $x'_{k} = x_{k} / l_{k}$ were used. The location of the prefracture zone has a significant influence on the opening of faces of plastic flow zones. When prefracture zones are located close to each other, calculations show an increase in both the sizes of the prefracture zones and the opening of the prefracture zone faces, and also a decrease in the openings of faces and the sizes of the prefracture zones. Different kinds of mutual influence of damages (zones of weakened interparticle bonds of the material) are explained by differences in their locations.

The stress components $σ_{z}$ are determined by the formula $σ_{z} = ν (σ_{x} + σ_{y})$ . Here $ν$ is the Poisson ratio of the material. The bases of the cylindrical fuel pellet are not tension-free; they are subjected to normal forces. More exactly, the normal stress $σ_{z}$ acts on the upper base; the stress $- σ_{z}$ acts on the lower base. Application of these stresses is necessary to maintain the deformation plane.

Finally for stresses $σ_{z}$ for which there are no deformations in the direction of the axis z, we have

σ_{z} = ν (σ_{x} + σ_{y}) - 2 μ \frac{1 + ν}{1 - ν} α T + EC,

where the constant C is calculated from the condition $\int \int_{D} σ_{z} d x d y = 0$ ; D is the domain occupied by the cross-section of the fuel and the cladding; T is a temperature function; $μ$ is the shear modulus of the material; and E is the elasticity modulus of the material. Thus, we find

σ_{z} = ν (σ_{x} + σ_{y}) - 2 μ \frac{1 + ν}{1 - ν} α T + \frac{(1 - 2 ν) α E}{F_{D}} \int \int_{D} T d x d y,

where F_D is the area of the domain D.

The dependence of the critical intensity of the internal heat sources in the material of the fuel pellet $q_{*}^{c} / σ_{s}$ on the relative opening $δ_{*} / l_{1}$ at the center of the prefracture zone is shown in Figure 7. Here, α₃ = 45°; $δ_{*} = π δ_{c} μ_{1} / (1 + κ_{1}) σ_{s}$ .

The obtained algebraic system of equations of the problem allows us to get the solution with any accuracy given beforehand. The convergence of the procedure of numerical solution using singular equations is discussed in many papers [32 –34]. Numerous studies show that the number of collocations M should not be less than 20. Since M = 20, the values of the coefficients of the function of normal displacement are not significantly changed. In calculations, M was equal to 30.

Thus, a joint solution of the obtained algebraic systems and the limiting condition (40) enables (under the given characteristics of the material of the fuel pellet) us to determine the critical value of the intensity of the internal heat sources and the sizes of the plastic flow zone for the limit equilibrium state under which crack nucleation happens.

Through calculation of the results for the case N = 5, $σ_{s} = 450$ MPa, $q_{*} / σ_{s} = 0.5$ , a graphic representation of the prefracture bands is given in Figure 8.

Figure 8.

Calculation results for the prefracture bands (N = 5).

The evolution of the length of the prefracture band on the number of iterations is illustrated in Figure 9. Here, the solid line is the solution for 10 iterations; the dashed line is the solution for three iterations.

Figure 9.

Evolution of the length of prefracture band as a function of the number of iterations.

6. Single checks

No heat source, infinite cohesive stress, same material properties for fuel and cladding

In this case, $μ_{1} = μ_{0}$ , $κ_{1} = κ_{0}$ , and equation (1) becomes homogeneously biharmonic. The addend $q_{*} τ_{1} {\bar{τ}}_{1} / 32$ in boundary conditions (8) and (9), as well as the addend $3 q_{x} t_{n}^{2} / 32$ in boundary condition (10), disappear. In the case of infinite cohesive stress, the locations and sizes of the prefracture zones should be given beforehand. In this case, the solution of the integral equation is sought in the class of unbounded functions (stresses) at the vertices of the prefracture zones. Conditions (39) drop out. In this case, the system of algebraic equations turns out to be linear and is solved by the Gauss law with the choice of the main element.

No heat source, infinite cohesive stress, different material properties for fuel and cladding

In this case, $μ_{1} \neq μ_{0}$ and $κ_{1} \neq κ_{0}$ ; in the rest, what has been said before holds in this case as well. The system of linear algebraic equations is solved by the Gauss method.

Heat source and external pressure, infinite cohesive stress, different material properties for fuel and cladding

In this case, the locations and sizes of the prefracture zones should be given beforehand. The solution is sought in the class of unbounded functions (stresses) in the vertices of the prefracture zones. Conditions (39) drop out. The system of algebraic equations is linear and is solved by the Gauss method. In these particular cases, the results of the solution completely and accurately coincide with the analytic and numerical solutions of these problems reported in the literature.

At some stage of loading, simultaneous existence of the plastic flow zone and crack nucleation in the fuel pellet is possible. The solution method in this case combines the simultaneous account of damages and cracks for the end zones of plastic deformations.

7. Conclusions

In practice, the use of nuclear fuel pellets in nuclear reactors shows that at the design stage it is necessary to take into account the cases when cracks may be initiated in an element. In this circumstance, it is necessary to perform limit analysis of fuel pellets in order to establish limiting intensities of heat release under which crack nucleation occurs in the fuel pellet. The size of the limit zone of the weakened interparticle bonds of the material under which crack nucleation happens should be considered in determining the design characteristics of the material of a fuel pellet. Based on the suggested calculation model, which takes into account the occurrence of damage (plastic flow zones) in the material of a nuclear fuel pellet, a method to calculate the parameters of the heat release element under which crack nucleation happens was developed.

The developed mathematical model of cracking in a fuel pellet, in the author’s opinion, may be generalized in future for more complicated phenomena of the behavior of fuel rods that occur during reactor operation.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Michel

Sercombe

Thouvenin

et al . 3D fuel cracking modelling in pellet cladding mechanical interaction. Eng Fract Mech 2008; 75: 3581–3598.

Marchal

Campos

Garnier

. Finite element simulation of pellet–cladding interaction (PCI) in nuclear fuel rods. Comput Mater Sci 2009; 45: 821–826.

Sumi

Keer

Nemat-Nasser

. Thermally induced radial cracking in fuel element pellets. J Nucl Mater 1981; 96: 147–159.

Bolotin

. Mechanics of the initiation and initial development of fatigue cracks. Mater Sci 1986; 22: 14–19.

Mirsalimov

. Nucleation of crack type defect in the hub of a contact pair. Mathematicheskoye modelirovaniye 2005; 17: 35–45.

Mirsalimov

. The solution of a problem in contact fracture mechanics on the nucleation and development of a bridged crack in the hub of a friction pair. J Appl Math Mech 2007; 71: 120–136.

Mir-Salim-zade

. Crack initiation in a stiffened plate. J Appl Mech Tech Phys 2007; 48: 562–570.

Michel

Sercombe

Thouvenin

et al . 3D fuel cracking modelling in pellet cladding mechanical interaction. Eng Fract Mech 2008; 75: 3581–3598.

Besson

. Continuum models of ductile fracture: A review. Int J Damage Mech 2010; 19: 3–52.

10.

Mir-Salim-zade

. Generation of cracks in a perforated reinforced plate. J Appl Mech Tech Phys 2008; 49: 1030–1039.

11.

Zolgharnein

Mirsalimov

. Nucleation of a crack under inner compression of cylindrical bodies. Acta Polytech Hung 2012; 9: 169–183.

12.

Bian

Cheng

Taheri

. Elasto-plastic analysis of critical fracture stress and fatigue fracture prediction. Acta Mech 2014; 225: 3059–3072.

13.

Hales

Williamson

Novascone

et al . BISON theory manual: The equations behind nuclear fuel analysis. BISON release 1.1. INL/EXT-13-29930 Rev. 1. Idaho Falls: Idaho National Laboratory, 2014.

14.

Mirsalimov

Hasanov

. Nucleation of cracks in an isotropic medium with periodic system of rigid inclusions under transverse shear. Acta Mech 2015; 226: 385–395.

15.

. Effects of fiber cracking on elastoplastic damage behavior of fiber–reinforced metal matrix composites. Int J Damage Mech 2013; 22: 48–63.

16.

Mirsalimov

Hasanov

. Modeling of crack nucleation in covering on an elastic base. Int J Damage Mech 2014; 23: 430–450.

17.

Mirsalimov

Kalantarly

. Cracking in a circular disk under mixed boundary conditions. Acta Mech 2015; 226: 1897–1907.

18.

Mir-Salim-zade

. Modeling of crack formation in a perforated stringer panel. Math Models Comput Simul 2010; 2: 533–541.

19.

Mella

Wenman

. Modelling explicit fracture of nuclear fuel pellets using peridynamics. J Nucl Mater 2015; 467: 58–67.

20.

Gatt

J-M

Sercombe

Aubrun

et al . Experimental and numerical study of fracture mechanisms in UO₂ nuclear fuel. Eng Fail Anal 2015; 47: 299–311.

21.

Mirsalimov

Kalantarly

. Crack nucleation in circular disk under mixed boundary conditions. Arch Mech 2015; 67: 115–136.

22.

Schiavone

Abeygunawardana-Arachchige

Silberschmidt

. Crack initiation and propagation in ductile specimens with notches: Experimental and numerical study. Acta Mech 2016; 227: 203–215.

23.

Hasanov

. Cracking in sheet structural element under non-uniform stress field. Struct Mech Eng Constr Build 2017; 4: 19–28.

24.

Mirsalimov

. Cracking in heat-releasing perforated material. Acta Mech 2017; 228: 201–214.

25.

Oterkus

Madenci

. Peridynamic modeling of fuel pellet cracking. Eng Fract Mech 2017; 176: 23–37.

26.

Panasyuk

. Mechanics of quasibrittle fracture of materials. Kiev: Naukova Dumka, 1991 (in Russian).

27.

Tashlykov

Shcheklein

. Ecological forecasting in the nuclear power of XXI century. Int Sci J Altern Energy Ecol 2015; (8–9): 50–57.

28.

Leonov

Panasyuk

. Development of the smallest cracks in a solid. Prikladnaya Mekhanika 1959; 5(4): 391–401.

29.

Mirsalimov

. Non-one dimensional elastoplastic problems. Moscow: Nauka, 1987 (in Russian).

30.

Rusinko

. Plasticity and creep of metals. Berlin: Springer Verlag, 2011.

31.

Muskhelishvili

. Some basic problems of mathematical theory of elasticity. Amsterdam: Kluwer Academic, 1977.

32.

Panasyuk

Savruk

Datsyshyn

. A general method of solution of two-dimensional problems in the theory of cracks. Eng Fract Mech 1977; 9: 481–497.

33.

Ladopoulos

. Singular integral equations: Linear and non-linear theory and its applications in science and engineering. Berlin: Springer Verlag, 2000.

34.

Savruk

Kazberuk

. Stress concentration at notches. Cham: Springer International Publishing, 2017.