**5. Design analysis of local stresses in composite structures of the reactor**

The intra-chamber components of thermonuclear reactors, including ITER [3, 6, 11, 12], are subject to high stress levels, and are the most critical elements as they are in direct contact with the plasma. Taking into account p. 2, they are designed to withstand cyclic loads resulting from intense heat flows and volume forces, thermal shocks and dynamic effects during plasma disruptions and abrupt displacements of magnetic axes. The operating conditions to which the materials used are subjected are complicated by exposure to radiation.

When designing intra-chamber components, it is very difficult to find a structural material that is sufficiently resistant to all the above factors simultaneously and can provide the structure with the required operability and service life. The intra-chamber components were therefore constructed using layers of materials with different qualities: beryllium, copper, stainless and austenitic steel.

In view of the high stresses to which the individual layers whose surface is directly in contact with the plasma, are exposed, and their increased brittleness, it is of great importance to apply fracture mechanics methods in order to confirm and ensure their integrity.

The main load factor to which the multilayer elements of the first wall are exposed is the effect of plasma in the form of cyclically repeating high power heat flows and changing electromagnetic loads during plasma disruptions. Design strength analysis needs to be conducted in respect of such elements, taking into account possible changes in their properties during heating and increased brittleness due to radiation [4–8, 11, 13].

In addition to thermal loads caused by varying temperature fields, areas adjacent to the boundaries of the composite layers experience additional loads due to the difference in the thermal expansion coefficients of the dissimilar materials. Moreover, residual stresses caused by the manufacturing process are localized in this area. Due to differences in the materials' physical and mechanical properties, these residual stresses are not relieved during subsequent operational heating.

In effect, the boundary between the layers can be considered as another material, with own initial level of damage and defects, and a specific fracture resistance. For example, one of the problems with obtaining efficient Be/Cu compounds was that combining beryllium with almost any other material results in the formation of brittle intermetallic phases [5, 14, 15].

The operating conditions to which the dissimilar materials used for the structure are subjected are complicated by exposure to radiation, which increases the likelihood of fracture due to brittleness. The fracture resistances of the materials used (beryllium, copper and stainless steel), taking into account increased brittleness due to radiation, are set out in the report [16].

To maintain the proper operational condition of the multilayer element which is in contact with the plasma, it is necessary to ensure both the integrity of the reinforcing material (beryllium) and the proper operational condition of the beryllium-copper joint. Possible damage (failure) scenarios during operation

## *Theoretical and Experimental Analysis of Structural Properties of Load-Bearing Components… DOI: http://dx.doi.org/10.5772/intechopen.94531*

include the initiation of a brittle fracture in the beryllium layer, and delamination at the beryllium-copper boundary.

For the purpose of assessing the damage resistance of a heterogeneous structure, it can be divided into three separate zones: areas remote from the boundary between layers of dissimilar materials; areas near the boundary; and the boundary itself.

To assess the fracture resistance of layers remote from the boundary, fracture mechanics appropriate to a homogeneous material can be applied. The applicable theoretical and practical foundations applicable to this case have been developed [17, 18].

To analyze the resistance to brittle fracture of the materials in the layers located near the boundaries, the methodology appropriate for a homogeneous material is applied, but the corresponding analytical formulae for calculating the criterion parameters need to be adjusted [19]. This applies to the boundary zone where dissimilar materials are joined together, which is, at present, the area in respect of which least research has been done.

The main factors to be considered are as follows.


In view of the above, in accordance with currently accepted design practice the calculation of the durability of multilayer elements in reactors needs to be carried out in two stages.


In the case (**Figure 13**) of a homogeneous plate with a fracture *l* under nominal loads *σ* [5–8, 17–20, 22] the distribution of local stresses *σ*r at a distance *r* from the crack tip is described by a singular equation:

$$
\sigma\_r = \frac{K\_l}{\sqrt{\pi r}} \cdot f\_\kappa = \left(\frac{K\_l}{\sqrt{\pi}} \cdot f\_\kappa\right) r^\lambda \tag{4}
$$

where *K*I is stress intensity factor of at the crack tip ( *K l <sup>I</sup>* = π ), *f*к is a dimensionless function, which depends on the dimensions of the plate and crack, and λ is the singularity index (γ = −0.5).

In accordance with the strength conditions (1) and (2) for a plate without a fracture (*l* = 0) and where a fracture is present as shown in **Figure 12**, the fracture resistance condition is written in the form

$$K\_I \le \frac{K\_{\rm lc}}{n\_{\kappa}} \tag{5}$$

where *K*Iс is the critical stress intensity factor and *n*к is safety factor for fracture resistance (*n*<sup>к</sup> ≤ *n*u).

During the analysis of the design features of the multilayer elements of the first wall of the thermonuclear reactor [14, 23, 24] several main types of singularity sources were identified (**Figure 14**):


In a situation as per scheme 14а, in which a fracture (discontinuity) is completely located in one of the homogeneous materials of a multilayer element,

*Theoretical and Experimental Analysis of Structural Properties of Load-Bearing Components… DOI: http://dx.doi.org/10.5772/intechopen.94531*

#### **Figure 14.**

*Basic types of sources of stress singularities in multi-layer elements of the first wall of reactor. (a) defects (cracks), located in homogeneous material; (b) delamination crack, located at the boundary of heterogeneous materials; (c) crack, abutting on the boundary of a joint; (d) joints of heterogeneous materials Beryllium-Copper 900 –900 ; (e) joints of heterogeneous materials Beryllium- Copper 900 –1800 .*

the procedures for calculating the SSS and durability are well developed ([17, 18, 22]). The level of stress singularity indicator in this case is λ = − 0.5.

For delamination fractures (**Figure 14b**), subject to loads by the mechanisms of normal separation and transverse shear, the characteristic Eq. (4), corresponding to the solution of the characteristic equation that determines the degree of stress singularity - λ has a complex root [17, 19, 25].

$$\lambda = -\mathbf{1} / 2 \pm \beta; \ \beta = \frac{\mathbf{1}}{2\pi} \ln \left( \frac{\mu\_1 + \mu\_2 \kappa\_1}{\mu\_2 + \mu\_1 \kappa\_2} \right) \tag{6}$$

where *μ* is the shear modulus of the materials, *κ* = 3-4 *ν* (for plane deformation), *ν* is Poisson's ratio and the indices 1 and 2 indicate whether the material is the 1st or 2nd in the compound of dissimilar materials.


From (4)–(6) it follows that the asymptotic distribution of stresses at the fracture tip at *r* → 0, β ≠ 0 is singular, with a different λ singularity. In a bulk (three-component) stress state, three models of fracture mechanics (I, II, III) are introduced into the calculation [18–20], and then, based on (4)

$$
\sigma\_r = \frac{K\_I}{\sqrt{\pi}} \cdot f\_{\text{KI}} r^{\lambda\_l} + \frac{K\_{\text{II}}}{\sqrt{\pi}} \cdot f\_{\text{KI}} r^{\lambda\_{\text{II}}} + \frac{K\_{\text{III}}}{\sqrt{\pi}} \cdot f\_{\text{KIII}} r^{\lambda\_{\text{III}}} \tag{7}
$$

The proposed approach to assessing strength of adapters will require additional study of residual technological stresses, which, due to the difference between the physical and mechanical properties of adapter materials, always reach significant magnitudes.
