Study On Anisotropy Sheet Instability Under Pressure

The metal sheet under the longitudinal pressure is easy to be instable, instability of the sheet can cause severe damage, the pressure in the moment of instability is called critical pressure. The factors such as sheet size, material constitutive and the constraints on sheet etc can influence critical pressure. Metal sheets are usually made of through rolling, so the orientation distribution of the crystals are orthotropic, and the orientation distribution of the crystals can be described through orientation distribution function(ODF). Because the orthotropic of metal sheet can be described through texture coefficients, and instability of the metal sheet under pressure have close relationship with the mechanical properties of the material, so the purpose of this article is to study critical pressure expressions of the sheet containing the texture coefficients effects.Assume that the thickness of the sheet can not be changed after a bending with small deflection, and the normal of the middle surface after deformation is the same as the one before deformation. In this article, starting with the orthotropic flexibility tensor, we derive the differential equations and study the instability of the rectangular sheet on the simply support on four sides through the condition that the balance between internal forces and load, calculating using the analytical method and the energy method, the two results match with each other. The critical pressure expressions we get contain the texture coefficients effects, the critical pressure can degenerate to isotropic if all the texture coefficients are zero.The wrinkling phenomenon on flange area of circular sheet metal can be seen as sheet plastic instability when deep drawing. The flange area of metal sheet is stretched on radial direction and compressed on hoop direction in the process of deep drawing. The pressure on hoop direction can come about wrinkling on the flange area of metal sheet, we have derived the tangent modulus and the reduced modulus on orthotropic condition. The wrinkling flange part can be seen as a ring, assuming the additional stretch of wrinkle convex surface less than the additional compression caused by increasing pressure when wrinkling, then there is non-partial unload. We can get the smallest tangential stress(critical stress) when wrinkling through the energy method with the tangent modulus, then get the expression of instability critical stress on flange area when deep drawing.