مهندسی عمران شریف (May 2019)
LAGRANGE METHOD FOR NONLINEAR SOLUTION OF HYPER ELASTIC MATERIAL BY ISOPARAMETERIC ELEMENT
Abstract
In this article complete numerical formulation for nonlinear analysis of planar structures made of hyper elastic materials with application in civil engineering is presented. Structural problems can be solved by finite element method using complete lagrange formulation.The presented formulation is for large strain problems and is applicable for a wide range of hyper elastic nonlinear materials. Generally in nonlinear finite element analysis large deformation and disordered mesh (distorted) is seen. As using disordered elements, isoparametric elements would have low precision. Hyper elastic materials like rubber are one of the most effective materials in engineering application.The high axial strength and large deformation capacity of these materials make them suitable for many applications. Theory of hyper elastic materials is proposed by references [13,14].These theories explain hyper elasticity behavior in complicated formulations. Therefore we are interested in studying planar hyper elastic elements with large deformation like rubber, leather etc.Eventually planar nonlinear elements formulation with hyper elastic behavior is presented for structures having large deformations and complete lagrange formulation is used to analyze the structure. This formulation has been developed to analyze quadrilateral finite element models, which comparing to isoparametric 4-node elements is less sensitive to distorted meshing and doesn't have shear locking problem generated by geometrically distorted meshing.In order to deploy 2QACM benefits in nonlinear applications, complete lagrange formulation 3Q4HY is used, numerical examples in geometric nonlinear problems have shown presented formulation's ability to prevent loss of precision in highly distorted meshing for 4-node hyper elastic elements. To show effectiveness of presented elements two numerical examples are presented. Q4HY elements efficiency in nonlinear analysis of planar elements made of hyper elastic materials is significantly obvious.The program is written in matlab environment for nonlinear analysis of hyper elastic problems. Two formulations are proposed and results have been compared with references results. Examples of rubber like problems have shown these formulations ability to analyze large strain structures.Numerical examples express four node element has less sensitivity to distortion in meshing in nonlinear analysis and can be used with good precision when element is diagonal, while calculated responses using 4 node isoparametric element may be inappropriate.4 noded elements effect on developing a simple, applicable and valid nonlinear geometric analysis is significantly observed. Furthermore, a complete lagrangian formulation with a particular formulation by applying integration techniques to effectively establish stiffness matrix is presented. According to presented examples, the investigated elements have great potential in solving Hyper elastic problems.
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