MODEL OF ORTHOTROPIC MATERIALS WITH THE FINITE-DIFFERENCE METHOD

THIS PAPER PRESENTS A MATHEMATICAL MODEL WITH FINITE DIFFERENCE METHOD FOR THE INTEGRATION OF DIFFE RENTIAL EQUATIONS WI TH PARTIAL DERIVATES WHICH DESCRIBES THE ONE-DIMENSIONAL DISPLACEMENT OR STRESS OF ORTHOTROPIC MATERIALS. WE START FROM THE CL ASSICAL IDEA OF AIRY STRESS FUNCTION , WHICH DESCR IBES THE SECOND ORDER PARTIAL DIFFERENTIAL OF THE FIELD OF TENSIONS AN D WITH THE HELP OF THE TENSIONS AND THE MATERIAL EQUATIONS C AN BE DETERMINED THE SPECIFIC DEFORMATIONS BY ANALOGY WE DISCOV ERED THAT CAN BE USE D A "POTENTIAL FUNCT ION" OF THE DISPLACEMENT, WHICH MAKES POSSI BLE THE USAGE OF MIX ED BOUNDARY CONDITIONS. THE PARTIAL DERIVATIVES OF THIS FUNCTION GIV E THE DISPLACEMENT I N THE DIRECTIONS OF COORDINATE AXES. DIS PLACEMENT DERIVATIVES, THE DERIVATIVES O F SUPERIOR ORDER OF THE DISPLACEMENT FUN CTION GI VE SPECIFIC DEFORMAT IONS AND USING THE M ATERIAL EQUATIONS, THESE SUP ERIOR ORDER DERIVATI VES WILL LEAD TO TEN SION FIELDS. THEREFORE BECOMES PO SSIBLE TO WRITE THE BOUNDARY CONDITIONS AS PRESCRIBED TENSIONS (THE DISTRIBUTED LOAD SHAPE), THERE IS A DIRECT RELATIONSHIP (DIFFERENTIAL EQUATIONS) BETWEEN D ISPLACEMENTS AND STR ESSES. THESE RELATIO NSHIPS ARE APPROXIMATED BY FINITE DIFFERENCES

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