Stress profile and thermal expansion of layered materials determined from surface stresses

Abstract

A simple method of inferring the stress profile and the effective difference in thermal expansion or strain in an unconstrained elastic multilayer system from a measurement of a limited number of surface stresses as obtained for example using x-ray diffraction or Raman spectroscopy is outlined. Explicit relationships are given for bilayered systems. The analysis procedure is exemplified for literature data of electronics materials, solid oxide fuel cells and thermal barrier systems. Following the outlined procedure, a determination of the stress profile and difference in thermal expansion for composites of alternating layers is also possible. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1759773͔ Layered materials find widespread use in electronic, magnetic, optical, and structural components. Such components can be subjected to residual stresses due to intrinsic processes, such as sintering or crystallization, mismatch in thermal expansion coefficient or mechanical loading. Theories to relate stress, strain and curvature to the mechanical and thermal loading have been established for isotropic materials, bilayered, 1 multilayered composites 2 and materials with property gradients.3 These theories can be used to determine the difference in strain or thermal expansion as well as the residual stresses in a layered composite from the change in deflection or curvature.Residual stresses in the surface of coated and layered materials are often assessed using Raman spectroscopy or x-ray diffraction.4,5 However, a determination of the stress profile requires a large number of measurements. In the current letter it is shown how the stress profile can be determined from a limited number of measurements. Only the more common cases of bilayer ͑coating on a substrate͒ and trilayer composites are analyzed in detail. However, following the outlined procedure the stress profile and thermal expansion for composites of alternating layers can be determined.It is assumed that the stress is measured in the surface of layer 1. It has to be considered that the stress is only representative of the surface if the analyzed depth is small compared to the layer thickness or the stress is relatively constant over the layer thickness, which is the case if the film thickness is much smaller than the substrate thickness. All stresses are in the plane of the laminate. The stress in the surface of layer 2, 2,surface , in layer 2 next to the interface to layer 1, 2,interface , and in layer 1 next to the interface to layer 2, 1,interface , follow from the general relationships for the stress in multilayered composites where t is the thickness and E the elastic modulus ͑if the specimen is a plate E/(1Ϫ 2 ) has to be substituted͒. Since the stress can be a result of differences in thermal expansion but also intrinsic, growth related, the difference in strain is considered. If the strain is a result of thermal expansion ⑀ ϭ␣⌬T and ⌬⑀ϭ⌬␣⌬T. Defining the strain in layer two as ⑀ 2 ϭ⑀ 1 ϩ⌬⑀ 1,2 yieldsHere E and t are the average values during th...

An explicit elastic solution for a brittle film with periodic cracks

“…This mechanism has been used to evaluate the interfacial shear strength (Agrawal and Raj 1989) and to measure the elastic modulus and the fracture toughness of the film (Thouless et al 1992;Wang et al 1998). A number of experiments have been developed to study the dependence of fracture behavior on the geometry of the film, the mechanical properties of the film and substrate, and conditions of the interface (see, for example, Hu and Evans 1989;Bordet et al 1998;Etzkorn and Clarke 2001;Alaca et al 2002;Zhao et al 2002;Malzbender 2004;Tadepalli et al 2008). Fig.…”

Sealing Evaluation of an Alkaline Earth Silicate Glass for Solid Oxide Fuel/Electrolyser Cells

“…The tri-layer assembly can be treated as AISI 441/SABS-0 and SABS-0/ZrO 2 bi-layer assemblies in order to evaluate the interfacial stresses [28][29][30]. In this equation, subscript 1 refers to the thicker layer and subscript 2 refers to the thinner layer in the bi-layers.…”

“…This is because the heat transfer through cell components by conduction requires time to reach uniform temperature in the cell stacks. The temperature gradient due to conductive heat transfer can be calculated from the following equation: [30,32,33]. Three assumptions are made in calculating the temperature gradient.…”

Residual Stress Assessment for Thin 8YSZ Electrolytes Using Focused Ion Beam Milling and Digital Image Correlation

“…The determined strain value of ≈0.2% yields in combination with the biaxial elastic modulus a residual stress of ≈300 MPa, that might be used as a basis to calculate the stress profile of the entire cell as outlined in Ref. [13]. Considering a Da (YSZ-NiO/YSZ) of 1.5 × 10 -6 K -1 [10] a strain level of between 0.15% and 0.18% is derived for a stress free temperature between 1,000 and 1,200°C (viscose and creep effects will relax stresses above 1,000°C, hence the exact stress free temperature cannot be defined), respectively, which is in good agreement with the experimentally obtained value.…”

“…With respect to reliability, the mechanical stability of the substrate material [9,10] and the residual stress state of the cell are of main importance to assess the failure probability [11] and re-oxidation stability [12], both being of strong relevance for the operation of SOFC systems. In fact, a measurement of the stress/strain state of the electrolyte layer can be used to calculate the stress state of the entire SOFC [13].…”