Mathematical Foundations of Limit Criterion for Anisotropic Materials

Abstract

In the paper a new proposition of limit state criteria for anisotropic solids exhibiting different strengths at tension and compression is presented. The proposition is based on the concept of energetically orthogonal decompositions of stress state introduced by Rychlewski. The concept of stress state dependent parameters describing the influence of certain stress modes on the total measure of material effort was firstly presented by Burzyński. The both concepts are reviewed in the paper. General formulation of a new limit criterion as well as its specification for certain elastic symmetries is given. It is compared with some of the other known limit criteria for anisotropic solids. General methodology of acquiring necessary data for the criterion specification is presented. The ideas of energetic and limit state orthogonality are discussed -their application in representation of the quadratic forms of energy and limit state criterion as a sum of square terms is shown.Keywords: anisotropy, strength hypothesis, material e?ort, yield condition, plastic potential, strength differential effect, elasticity, plasticity Artykuł przedstawia nową propozycję kryterium stanu granicznego dla anizotropowych ciał stałych wykazujących różną wytrzymałość przy rozciąganiu i ściskaniu. Propozycja ta opiera się na koncepcji energetycznie ortogonalnych rozkładów stanu naprężenia zaproponowanej przez Rychlewskiego. Pomysł wykorzystania zależnych od stanu naprężenia parametrów określających wpływ konkretnych form naprężenia na całkowitą miarę wytężenia materiału po raz pierwszy został przedstawiony przez Burzyńskiego. W pracy omówione są obydwie koncepcje. Przedstawione jest ogólne sformułowanie nowego warunku granicznego, jak również jego specyfikacja dla wybranych symetrii sprężystych. Kryterium to zostało porównane z niektórymi innymi spośród znanych kryteriów granicznych dla ciał anizotropowych. Przedstawiono ogólną metodologię pozyskiwania danych wymaganych do specyfikacji kryterium.

Energy-Based Yield Criteria for Orthotropic Materials, Exhibiting Strength-Differential Effect. Specification for Sheets under Plane Stress State

“…However, most of the proposed criteria are merely mathematical approximations or interpolations using linear transformations leading to the simplest polynomials or power-laws, lacking any physical motivation or even interpretation. Contrary to such an approach, a new proposition was developed [20,[27][28] which is based on elastic energy density decomposition for linear elastic anisotropic solids -introduced first by Rychlewski [21] -modified by the use of certain stress-state-dependent weighting functions, the idea suggested independently by Burzyński [7], Schleicher [23] and Mises [19].…”

“…Such an additive decomposition (which is not an obvious thing for quadratic functions) is unique for given elasticity tensors. The weighted combination of components of this decomposition will be considered as a limit state criterion [20].…”

Archives of Metallurgy and Materials

“…Mises in his fundamental paper, [15], has presented the thorough study of limit criteria for anisotropic solids. An array of works following the original ideas of Mises was discussed in more detail in [25], where the yield criteria by [8,9] and [37], as well as, [4] were considered.…”

“…The formulation of energy-based hypothesis of material effort for anisotropic solids revealing SDE was attempted by [24,25,35] and [34]. The study is based on the concept of energyorthogonal decompositions of stress state introduced by [31,32] and [23].…”

“…The concept of stress state dependent functions describing the influence of certain stress modes on the total measure of material effort was firstly presented by [2,3]. General formulation of a new limit criterion as well as its specification for certain elastic symmetries was given in [25] and general methodology of acquiring necessary data for the criterion specification was presented. The general formulation was specified for orthotropic materials and formulated both for two-dimensional and threedimensional cases by [34].…”