Solution of S3 Problem of Deformation Mechanics for a Definite Condition and Resulting Modifications of Important Failure Theories
Authors: Ranajay Bhowmick
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.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 330
 R. Chatterjee, “Mathematical Theory of Continuum Mechanics”, Narosa Publishing House.
 H. M. Westergaard, “Theory of Elasticity and Plasticity”, John Wiley & Sons, 1952.
 G. C. Nayak and O. C. Zienkiewicz, “Convenient forms of stress invariants for plasticity”, ASCE, Journal of Structural Division 98, 949 – 954, 1972.
 W. 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. Bhowmick, “Solution of Cubic Equation of Stress Invariants for a Particular Condition”, IRJET, 07, 4423 – 4425, 2020.
 W. F. Chen and A. F. Saleeb, “Constitutive Equations for Engineering Materials”, John Wiley & Sons, 1982.
 D. C. Drucker and W. Prager, “Soil Mechanics and Plastic Analysis or Limit design”, Quarterly of Applied Mechanics, 10, 157 – 165, 1952.
 R. von Mises, “Mechanics of Solid Bodies in the Plastically Deformed State”, Nachr. d. Kgl. Ges. Wiss. Göttingen, Math.-phys. Klasse 4 (1913), 582-592.
 A. Nadai, “Plastic Behaviour of Metals in the Strain – Hardening Range Part I”, Journal of Applied Physics, 8, 205 – 213, 1937.
 W. J. M. Rankine, “A Manual of Applied Mechanics”, Richard Griffin & Company, 1958.