A BEAM ELEMENT FOR PIEZOELECTRIC PLANAR FRAMES UNDER SMALL STRAINS BUT LARGE ROTATIONS USING THE TOTAL LAGRANGIAN DESCRIPTION

Abstract

SMALL STRAINS ARE CONSISTENTLY INCORPORATED INTO A LARGE ROTATION PIEZOELECTRIC BEAM THEORY, WHERE THE DISPLACEMENT IS ASSUMED TO VARY IN ACCORDANCE WITH THE TIMOSHENKO ASSUMPTION AND THE ELECTRIC POTENTIAL HAS LINEAR VARIATION THROUGH EACH PIEZOELECTRIC LAYER THICKNESS. A FINITE ELEMENT FOR PLANAR FRAMES, BASED ON THE TOTAL LAGRANGIAN DESCRIPTION, IS THEN DEVELOPED. THE DISPLACEMENTS AND ROTATION ARE LINEARLY APPROXIMATED OVER THE ELEMENT AND THE VOLTAGE IN A PIEZOELECTRIC SENSOR LAYER IS TAKEN TO BE CONSTANT. IN ADDITION TO THE WELL-KNOWN MEMBRANE AND SHEAR LOCKINGS, IT IS DISCLOSED THE PRESENCE OF LOCKING IN THE APPLICATION OF GAUSS LAW TO THE PIEZOELECTRIC SENSOR LAYERS. ALL THE MECHANICAL AND ELECTRICAL CONTRIBUTIONS TO THE INTERNAL LOAD VECTORS ARE EVALUATED USING A REDUCED ONE-POINT GAUSSIAN QUADRATURE TO MAKE THIS SIMPLE ELEMENT EFFICIENT. NO SPURIOUS MODES ARE INTRODUCED IN THIS PROCESS. AN INCREMENTAL-ITERATIVE APPROACH, BASED ON THE NEWTON-RAPHSON ALGORITHM, IS EMPLOYED IN THE SOLUTIONS OF THE NUMERICAL EXAMPLES TO ILLUSTRATE THE LARGE ROTATION CAPABILITY OF THE DEVELOPED MODEL.