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Sandwich theory〔Plantema, F, J., 1966, Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates, and Shells, Jon Wiley and Sons, New York.〕〔Zenkert, D., 1995, An Introduction to Sandwich Construction, Engineering Materials Advisory Services Ltd, UK.〕 describes the behaviour of a beam, plate, or shell which consists of three layers - two facesheets and one core. The most commonly used sandwich theory is linear and is an extension of first order beam theory. Linear sandwich theory is of importance for the design and analysis of sandwich panels, which are of use in building construction, vehicle construction, airplane construction and refrigeration engineering. Some advantages of sandwich construction are: * Sandwich cross sections are composite. They usually consist of a low to moderate stiffness core which is connected with two stiff exterior face-sheets. The composite has a considerably higher shear stiffness to weight ratio than an equivalent beam made of only the core material or the face-sheet material. The composite also has a high tensile strength to weight ratio. * The high stiffness of the face-sheet leads to a high bending stiffness to weight ratio for the composite. The behavior of a beam with sandwich cross-section under a load differs from a beam with a constant elastic cross section as can be observed in the adjacent figure. If the radius of curvature during bending is large compared to the thickness of the sandwich beam and the strains in the component materials are small, the deformation of a sandwich composite beam can be separated into two parts * deformations due to bending moments or bending deformation, and * deformations due to transverse forces, also called shear deformation. Sandwich beam, plate, and shell theories usually assume that the reference stress state is one of zero stress. However, during curing, differences of temperature between the face-sheets persist because of the thermal separation by the core material. These temperature differences, coupled with different linear expansions of the face-sheets, can lead to a bending of the sandwich beam in the direction of the warmer face-sheet. If the bending is constrained during the manufacturing process, residual stresses can develop in the components of a sandwich composite. The superposition of a reference stress state on the solutions provided by sandwich theory is possible when the problem is linear. However, when large elastic deformations and rotations are expected, the initial stress state has to be incorporated directly into the sandwich theory. ==Engineering sandwich beam theory== In the engineering theory of sandwich beams,〔 the axial strain is assumed to vary linearly over the cross-section of the beam as in Euler-Bernoulli theory, i.e., : Therefore the axial stress in the sandwich beam is given by : where is the Young's modulus which is a function of the location along the thickness of the beam. The bending moment in the beam is then given by : The quantity is called the flexural stiffness of the sandwich beam. The shear force is defined as : Using these relations, we can show that the stresses in a sandwich beam with a core of thickness and modulus and two facesheets each of thickness and modulus , are given by
:= -\cfrac we can write the axial stress as : The equation of equilibrium for a two-dimensional solid is given by : where is the shear stress. Therefore, : where is a constant of integration. Therefore, : Let us assume that there are no shear tractions applied to the top face of the sandwich beam. The shear stress in the top facesheet is given by : At , implies that . Then the shear stress at the top of the core, , is given by : Similarly, the shear stress in the core can be calculated as : The integration constant is determined from the continuity of shear stress at the interface of the core and the facesheet. Therefore, : and : |} For a sandwich beam with identical facesheets and unit width, the value of is : If , then can be approximated as : and the stresses in the sandwich beam can be approximated as : If, in addition, , then : and the approximate stresses in the beam are : If we assume that the facesheets are thin enough that the stresses may be assumed to be constant through the thickness, we have the approximation
Hence the problem can be split into two parts, one involving only core shear and the other involving only bending stresses in the facesheets. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「sandwich theory」の詳細全文を読む スポンサード リンク
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