A solid oxide fuel cell (SOFC) is a high-efficient electrochemical device that directly converts the chemical energy of fuels such as hydrogen and natural gas into electrical energy. In various configurations and designs, anode-supported planar SOFCs are widely used due to their higher power density and lower manufacturing cost [1–4].
Residual stresses are developed during the manufacturing process of the cells due to the thermo-elastic mismatch between the cell layers [5–11]. These residual stresses can result in performance degradation as a result of delamination  or the formation of cracks in the cell layers [3, 11] and in the extreme case may lead to the complete deterioration of the cell [12–14].
In the case of the anode-supported cells, much attention has been paid to the residual stress distribution after the cell fabrication process at room temperature and/or during its high temperature operation, especially in the electrolyte and anode layers.
Figure 1 shows a schematic cross-section of the residual stress distribution in an anode-supported cell at room temperature. The stress is compressive in the electrolyte layer and its value does not change significantly through the electrolyte thickness. The anode experiences tensile stress towards the electrolyte and compressive stress towards its free surface. The stress level at the free surface is lower than the stress level at the interface [6, 15, 16].
For conventional anode (Ni/YSZ) and electrolyte (YSZ) materials, depending on the fabrication procedure and cell design, the maximum compressive stress in the electrolyte is as high as 500–700 MPa [6, 12, 15–20]. The maximum tensile and compressive stresses in the anode are respectively about 20–100 MPa and 10–50 MPa [6, 12, 15, 16, 20, 21].
Figure 2 shows a similar trend for residual stress distribution in an anode-supported cell at operating temperature (i.e. 800°C). However, the operating temperature reduces the absolute stress level by about 50% [15, 19, 20] or more .
During cell operation, chemical stresses (e.g., redox and chemical expansion) and thermally induced stresses are also superimposed to the residual stresses. These stresses are the major cause of the failure of the cell [22, 23] and should not exceed the strength of the materials [18, 19, 24, 25]. Since the residual stress significantly affects the magnitude and distribution of stresses in the cell at operating conditions , estimation of residual stress at room temperature would be beneficial to calculate the stress in the anode-supported cells under operating conditions [12, 27, 28].
The effects of different factors such as layers thickness, using an additional layer, reduction and re-oxidation, applying different additives, sintering temperature, cell configurations and fabrication method on the residual stress of anode supported cells have been studied.
Zhang et al.  developed an analytical model to predict the residual thermal stresses in a single solid oxide fuel cell. They investigated the influence of the thickness of each layer on the residual stress distributions in the cell. In a similar study, Fan et al.  calculated the relationship between the residual stresses and the thicknesses of different cell components at room temperature. Laurencin et al.  proposed a numerical model to study the risk of cell failure due to residual stresses and investigated the effect of electrolyte thickness on the risk of anode failure. Severson et al.  developed a structural model for analysis of residual stresses in anode- and electrolyte-supported planar SOFCs and studied residual stress distribution for different thickness combinations.
Malzbender et al.  showed that applying an additional layer, as a support for the anode layer, compensates the cell curvature. However, the average residual tensile stress in the anode increases, which could lead to a larger fracture probability of the anode layer. Sun et al.  compared the effect of thermal cycling on residual stress and distortion in a standard 3-layer cell and one with an additional layer. Charlas et al.  analyzed and discussed the influence of an additional layer on the residual stresses in 4 layers half-cells.
Malzbender et al.  studied the influence of the reduction and re-oxidation on microstructure and residual stresses in the anode-supported cells. Fischer et al.  showed that the reduction of NiO to Ni in the anode reduces the absolute stress level in the electrolyte by 10% at room temperature. Hatae et al.  investigated the effect of anode expansion due to the re-oxidation on the residual stress in the electrolyte. Sun et al.  studied the effect of reduction of the supports on the electrolyte residual stress. Villanova et al.  investigated the change of residual stress in the electrolyte layer, after manufacturing, reduction and re-oxidation of the anode layer, using X-ray diffraction. Wang et al.  studied the residual stress variations during reduction at different temperatures and different operation time intervals. Fan et al.  calculated the residual stresses at room temperature caused by the manufacturing before and after the reduction and concluded that the reduction of NiO to Ni in the anode reduces the absolute stress level in the cell by 20%. Xiang et al.  derived an analytical model to study the residual stress during anode reduction. Shang et al.  described the stress state in electrolyte as a result of the competition between the oxidation-induced stress and thermal residual stress. Frandsen et al.  showed that the residual stresses are relaxed at the point of chemical reduction of NiO-YSZ to Ni-YSZ.
He et al.  showed that adding Al2O3 in NiO–YSZ support materials affected the thermal expansion mismatch and reduced the residual stress in the cell. Cologna et al.  tailored the electrolyte composition by adding a fraction of fine powders to coarse powders to reduce the sintering stresses.
Yakabe et al.  studied the effect of sintering temperature on the calculated residual stress in the electrolyte at room temperature with X-ray measurements. Malzbender et al.  determined residual stress of half-cells with oxidized anode as a function of temperature.
Fujita et al.  and Somekawa et al.  estimated and compared residual stresses in the electrolytes of segmented-in-series solid oxide fuel cells (SIS-SOFCs) and usual anode-supported cells at room temperature by X-ray diffraction. Nakajo et al.  used a model based on the Euler–Bernoulli theory to study the residual stresses in the cell layers. Using the temperature-dependent mechanical properties of materials has enabled the study of the residual stress in several anode-supported SOFC configurations. Menzler et al.  developed a novel route to fabricate anode-supported solid oxide fuel cells, and measured the residual stresses in the electrolyte after sintering, before and after flattening. The stress level is significantly reduced in comparison to the data obtained for half-cells manufactured via the classical route.
Fischer et al.  showed that the flattening procedure of the SOFCs to remove the warp essentially does not change the residual stress level; however, reduces the in-plane fluctuation. Moon et al.  concluded that applying compressive force during co-firing can affect the residual stress distribution. The co-fired cell under optimal pressure showed homogeneous stress distribution. Shin el al. showed that the roll calendaring process can produce cells with lower and more uniform residual stress compared to the conventional uniaxial press.
Many researchers have developed equations for calculating the residual stress and radius of curvature during the sintering process based on fundamental stress-strain relations [15, 29, 36, 42, 43]. Using these equations, the impact of different parameters on residual stress and radius of curvature were investigated, including layer thickness, cell configuration, sintering temperature and material properties. These equations do not take into account the dimensions and geometry of cell and implicitly assume that the size and geometry do not affect residual stress and radius of curvature.
However, some experiments showed that the dimension and geometry of cell affect the radius of curvature. Arias et al.  showed that the increase in cell diameter increases the radius of curvature. Converting deflection data of Moon et al.  and Orui et al.  into the radius of curvature demonstrated that the radius of curvature increases with increasing dimension. Mücke et al.  showed that the radius of curvature of the horizontally sintered specimens is three times higher than the free-hanging samples. This difference is attributed to the effect of gravity and the weight of the samples. Molla et al.  presented an improved model that the effect of weight of the sample (gravity) on the kinetics of distortion is considered. Even under identical conditions, large cells are received more force due to the gravity effect. Malzbender  also investigated the effect of cell geometry on the warpage behavior of cells.
On the other hand, the commercial cells are usually much larger than the samples made in the laboratories or used in the research experiments. So, while researchers use small-size cells to conduct preliminary studies on the effect of various parameters on cell performance, it is still necessary to enlarge the cell size for practical and commercial use. Any changes of the residual stress and curvature state in this scaling up process should be investigated. Regarding the effect of dimension on the radius of curvature and the relationship between curvature and residual stress, we experimentally investigated the potential effects of the shape and size of ceramic cells on the residual stress.
In this work, we report our evaluation of the residual stresses in the electrolyte of anode-supported solid oxide fuel cells, which reveals the effect of the shape and size of cell on residual stresses. All cells have been fabricated by the conventional tape casting and co-sintering method. The conditions of the co-firing and tape-casting processes were identical for all specimens. We used the X-ray diffraction method to measure the residual stresses in the electrolyte of the anode-supported cell. This technique has been widely used to evaluate the residual and thermal stress in solid oxide fuel cells [6, 17–19, 48–50].