AN EXTENDED MULTISCALE FINITE ELEMENT METHOD (EMSFEM) ANALYSIS OF PERIODIC TRUSS METAMATERIALS (PTMM) DESIGNED BY ASYMPTOTIC HOMOGENIZATION

Abstract

ASYMPTOTIC HOMOGENIZATION (AH) AND THE EXTENDED MULTISCALE FINITE ELEMENT METHOD (EMSFEM) ARE BOTH PROCEDURES THAT ALLOW WORKING ON A STRUCTURAL MACROSCALE THAT INCORPORATES THE EFFECT OF AVERAGED MICROSCOPIC HETEROGENEITIES, THUS RESULTING IN COMPUTATIONALLY EFFICIENT STRATEGIES. AH FORMULATION IS BASED ON THE HYPOTHESIS THAT THE MICROSCALE IS INFINITESIMAL WHEN COMPARED TO THE DIMENSION OF THE STRUCTURE ITSELF. ON THE OTHER HAND, THE EMSFEM WORKS DIRECTLY ON COUPLED FINITE MICRO AND MACROSCALES USING NUMERICALLY BUILT DISCRETE INTERPOLATION FUNCTIONS. PERIODIC TRUSS MATERIALS ARE CELLULAR MATERIALS FORMED BY THE PERIODIC REPETITION OF A TRUSS-LIKE UNIT CELL. WHEN THE UNIT CELL IS ENGINEERINGLY TAILORED TO SHOW A GIVEN MACROSCOPIC RESPONSE WE CALL IT A PERIODIC TRUSS METAMATERIAL (PTMM). IN THIS WORK WE ANALYZE THE NUMERICAL BEHAVIOR OF SELECTED PTMMS THAT WERE DESIGNED FOR EXTREME POISSON RATIOS USING AH THEORY. AS A FIRST ISSUE, WE STUDY MACROSCOPIC STRUCTURES MADE OF FINITE UNIT CELLS AND VERIFY HOW CLOSE THEIR AVERAGE BEHAVIOR COINCIDES WITH THE MATERIAL PROPERTIES PREDICTED BY AH. IN THIS REGARD, WE EVALUATE THE POISSON RATIO AND ISOTROPY. FOR COMPARISON, WE SOLVE THE MACROSCOPIC PLANE STRESS ASSOCIATE PROBLEMS THAT EMPLOY THE ELASTIC CONSTITUTIVE TENSOR OBTAINED BY AH. THIS COMPARISON IS AFFECTED BY SIZE EFFECTS, SINCE OUR AH PROCEDURE IS RIGOROUSLY VALID ONLY FOR PERIODIC BOUNDARY CONDITIONS IN AN INFINITE DOMAIN. THE SECOND ISSUE IN THIS PAPER IS CONCERNED WITH THE ABILITY OF EMSFEM TO REPRODUCE THE STRUCTURAL BEHAVIOR OF THE FULL MACRO-MICRO MODEL, WHICH WE NAME FULL BAR MODEL. WE EMPLOY TWO VERSIONS OF THE EMSFEM, WHICH ADOPT LINEAR AND PERIODIC BOUNDARY CONDITIONS TO BUILD THE NUMERICAL INTERPOLATION FUNCTIONS. IN AGREEMENT WITH SCIENTIFIC LITERATURE, THE USE OF LINEAR BOUNDARY CONDIIONS IN THE EMSFEM-LBC IS PRONE TO SUFFER OVERSTIFFENING AND DOES NOT PRODUCE RELIABLE RESULTS. ADOPTION OF PERIODIC BOUNDARY CONDITIONS IS BETTER BUT ALSO NOT FULLY RELIABLE. THE THIRD AND MOST IMPORTANT ASPECT DISCUSSED IN THIS RESEARCH CONCERNS EVALUATION OF THE EMSFEM DOWNSCALED DISPLACEMENT FIELDS. WE OBSERVE THAT ACCORDING TO THE LAYOUT OF THE DESIGNED UNIT CELL, TO THE USE OF LBC OR PBC AND, DEPENDING ON THE BOUNDARY CONDITIONS PRESENT IN THE MACROSCOPIC PROBLEM, SPURIOUS DOWNSCALED DISPLACEMENTS MIGHT OCCUR. SUCH SPURIOUS DISPLACEMENTS ARE DUE TO EXCESSIVE COMPLIANCE OF THE CORRESPONDING UNIT CELL AND CAN BE DETECTED WHEN BUILDING THE NUMERICAL INTERPOLATION FUNCTIONS. THE UNEXPECTED LARGE DISPLACEMENTS ARE CONSEQUENCE OF TAKING INTO ACCOUNT THE FINITE DIMENSION OF THE MICROSCALE, THUS NOT DETECTABLE BY AH MODELING. WE CONCLUDE THAT THE LAYOUT OPTIMIZATION OF PTMM USING AH MUST BE CAREFULLY INTERPRETED AND THAT EMSFEM IS A GOOD TOOL TO DETECT A MACROSCOPIC EXCESSIVELY COMPLIANT RESPONSE AT AN EARLY DESIGN STAGE.