{"title":"Solution of S3 Problem of Deformation Mechanics for a Definite Condition and Resulting Modifications of Important Failure Theories","authors":"Ranajay Bhowmick","volume":168,"journal":"International Journal of Mechanical and Mechatronics Engineering","pagesStart":568,"pagesEnd":572,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10011636","abstract":"
Analysis of stresses for an infinitesimal tetrahedron leads to a situation where we obtain a cubic equation consisting of three stress invariants. This cubic equation, when solved for a definite condition, gives the principal stresses directly without requiring any cumbersome and time-consuming trial and error methods or iterative numerical procedures. Since the failure criterion of different materials are generally expressed as functions of principal stresses, an attempt has been made in this study to incorporate the solutions of the cubic equation in the form of principal stresses, obtained for a definite condition, into some of the established failure theories to determine their modified descriptions. It has been observed that the failure theories can be represented using the quadratic stress invariant and the orientation of the principal plane.<\/p>\r\n","references":"[1]\tR. Chatterjee, \u201cMathematical Theory of Continuum Mechanics\u201d, Narosa Publishing House.\r\n[2]\tH. M. Westergaard, \u201cTheory of Elasticity and Plasticity\u201d, John Wiley & Sons, 1952.\r\n[3]\tG. C. Nayak and O. C. Zienkiewicz, \u201cConvenient forms of stress invariants for plasticity\u201d, ASCE, Journal of Structural Division 98, 949 \u2013 954, 1972.\r\n[4]\tW. Lode 'Versuche ueber den Einfluss der mitt leren Haupts pannung auf das Fliessen der Metalle Eisen Kupfen und Nickel', Z, Physik, 36,913-39, 1926.\r\n[5]\tR. Bhowmick, \u201cSolution of Cubic Equation of Stress Invariants for a Particular Condition\u201d, IRJET, 07, 4423 \u2013 4425, 2020.\r\n[6]\tW. F. Chen and A. F. Saleeb, \u201cConstitutive Equations for Engineering Materials\u201d, John Wiley & Sons, 1982.\r\n[7]\tD. C. Drucker and W. Prager, \u201cSoil Mechanics and Plastic Analysis or Limit design\u201d, Quarterly of Applied Mechanics, 10, 157 \u2013 165, 1952.\r\n[8]\tR. von Mises, \u201cMechanics of Solid Bodies in the Plastically Deformed State\u201d, Nachr. d. Kgl. Ges. Wiss. G\u00f6ttingen, Math.-phys. Klasse 4 (1913), 582-592.\r\n[9]\tA. Nadai, \u201cPlastic Behaviour of Metals in the Strain \u2013 Hardening Range Part I\u201d, Journal of Applied Physics, 8, 205 \u2013 213, 1937.\r\n[10]\tW. J. M. Rankine, \u201cA Manual of Applied Mechanics\u201d, Richard Griffin & Company, 1958.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 168, 2020"}