Based On The Couple Stress Theory Of The Finite Element Method

It has been proved by some experiments focused on micro materials, which are done by Fleck, Stolken, Evans and others, that metal may be affected by size when its character length is in the level of micron. There is no any characteristic length in the traditional elastic and plastic theories, which can not explain the size effect of materials. There are parameters that mean the length dimensions in the strain gradient theory, which can explain the size effect of materials. Couple stress theory is one of the strain gradient theories, there are only a few questions on couple stress that have analytical solutions, so it become the effective way to solve the questions on couple stress that get the arithmetical solutions. There is a constraint condition in Couple Stress theory that macro rotations must be equal to the micro rotation. In the finite element calculation on couple stress that is done recently, the constraint condition is introduced by a penalty function technique. The penalty function technique has some disadvantages, such as the penalty function is difficult to identify. In order to overcome these difficulties of the penalty function technique, the constraint condition is introduced by Lagrange multiplier method, which its feasibility and efficiency are proved by examples solved in this method.The main research content is stated as follows:The mechanics character of couple stress theory is described systematically.As the couple stress theory has special demand of c~1 continuity, incompatible element and hybrid element are introduced to meet it. In order to improve the two-degree elements and enhance its precision, Wilson came up with the element that displacements without inside nodes are added to the displacement insert functions of elements, which was called incompatible finite element. Hybrid element is the element that the stress is balanced inside the element and the border displacements is compatible between adjacent elements. Although hybrid element method is derived through multivariate variational principle, when the solution is done the node displacements are the variations only. The Lagrange multiplier is introduced to amend the functional of these two elements mentioned above, which the constraint condition that macro rotations must be equal to the micro rotations is added to the functional.The problems of the circular hole in a field of uniform tension or in a field of...