Higher order finite elements (FEs) based on the absolute nodal coordinate formulation (ANCF) may require the use of curvature vectors as nodal coordinates. The curvature vectors, however, can be difficult to define at the reference configuration, making such higher order ANCF FEs less attractive to use. It is the objective of this investigation to use the concept of the mixed-coordinate ANCF FEs to ensure that the gradient vectors are the highest spatial derivatives in the element nodal coordinate vector regardless of the order of the interpolating polynomials used. This concept is used to convert the curvature vectors to nodes, called position nodes, which have only position coordinates. These new position nodes can be defined at a preprocessing stage, leading to two different sets of nodes: one set of nodes has position and gradient coordinates, while the second set of nodes has position coordinates only. The new position nodes can be used to obtain better distribution of the forces, including contact forces. Higher degree of continuity, including curvature continuity, can still be achieved at the element interface by using, at a preprocessing stage, linear algebraic equations that can reduce significantly the model dimension and ensure higher degree of smoothness. The procedure proposed in this investigation also allows for the formulation of mechanical joints at arbitrary points and nodes using linear algebraic constraint equations. The difficulties that arise when formulating these joint constraints using B-spline and NURBS (Nonuniform Rational B-Splines) representations are discussed. In order to explain the concepts introduced in this paper, low and high order ANCF thin plate elements are used. For the high order thin plate element, the curvature vectors at the interface nodes are converted to internal nodes with position coordinates only, leading to a mixed-coordinate ANCF thin plate element. This element preserves the desirable ANCF features including a constant mass matrix and zero Coriolis and centrifugal forces. Kirchhoff plate theory is used to formulate the element elastic forces. The equations of motion of the structure are formulated in terms of an independent set of structure coordinates. The resulting mass matrix associated with the independent coordinates remains constant. Numerical examples are presented in order to demonstrate the use of the mixed-coordinate ANCF thin plate element when the continuity constraints are imposed.
Semi-analytical static analysis of nonlocal strain gradient laminated composite nanoplates in hygrothermal environment
