J. Makowski

Buckling equations for elastic shells with rotational degrees of freedom undergoing finite strain deformation

Abstract A rigorous theory of small deformation superimposed on finite deformation is developed within a fully general theory of elastic shells. The mathematical structure of the configuration space and its associated tangent space is examined for the underlying shell model. Essential features of the theory are examined in the context of applications to the buckling analysis of specific problems.

Finite axisymmetric deformation of shells of revolution with application to flexural buckling of circular plates

SummaryA new theory is presented for shells of revolution undergoing axisymmetric arbitrarily large strain deformations. The material of the shell is assumed to be hyperelastic incompressible. The formulated theory is applied to analyse the flexural buckling of circular plates under uniform radial loads.ÜbersichtEine neue Theorie zur Berechnung achsensymmetrischer Verformungen von Rotationsschalen unter Berücksichtigung beliebig großer Dehnungen wird angegeben. Das Material der Schale wird hyperelastisch und inkompressibel angenommen. Die Theorie wird zur Berechnung des Beulens von Kreisplatten unter gleichmäßiger Radialbelastung angewandt.

On the derivation and comparative analysis of large rotation shell theories

SummaryFor the geometrically nonlinear first approximation theory of elastic shells three energy-consistent large rotation shell variants are constructed. The governing shell equations are derived as Euler-Lagrange equations of an associated variational principle of stationary total potential energy. The numerical applicability is considered for a highly nonlinear shell problem. To incorporate the presented theories into the frame of shell models published in the literature a comparative analysis is carried out for a large number of shell equations.ÜbersichtIm Rahmen der geometrisch-nichtlinearen ersten Schalenapproximation werden drei energiekonsistente Schalentheorien für große Rotationen hergeleitet. Die Schalengleichungen werden als Euler-Lagrange Gleichungen eines zugehörigen Variationsprinzips vom stationären Wert der potentiellen Gesamtenergie hergeleitet. Die numerische Anwendbarkeit wird anhand eines stark-nichtlinearen Schalenbeispiels nachgewiesen. Um die in dieser Arbeit hergeleiteten Theorien entsprechend einordnen zu können, wird für eine größere Zahl von in der Literatur angegebenen Schalengleichungen eine vergleichende Untersuchung mit numerischer Auswertung durchgeführt.

Thermodynamically based concept for the modelling of continua with microstructure and evolving defects

Abstract The aim of the paper is to present a thermodynamically based concept for the analysis of defect evolution in continua with microstructure. In classical models of continua with microstructure, balance laws for linear and angular momentum are formulated for macroforces and microforces, which can be called deformational macroforces and microforces. Characteristic feature of macrodefects and microdefects such as cracks and voids is the fact, that they can migrate relative to the moving body and this relative motion of defects is caused by so-called configurational forces. Therefore, in a continuum with microstructure and evolving macrodefects and microdefects we have to deal with deformational and configurational macroforces and microforces. To formulate a reliable theory for material bodies with microstructure and migrating macrodefects and microdefects, a continuum is considered, where each particle is equipped with an arbitrary number of deformable directors. We distinguish then between directors undergoing a convective deformation leading to deformational (physical) microforces, and directors describing a set of defects migrating relative to the underlying lattice, where this migration is caused by configurational (material) microforces. For deformational and configurational macroforces and microforces corresponding balance laws are presented and the macro- and micro-Eshelby stress tensors are derived. Next, deformational and configurational heatings and entropy fluxes are introduced and the first and second law of thermodynamics for the microcontinuum with evolving macrodefects and microdefects are formulated. Finally, we present the most general form of the first order constitutive equations satisfying the second law of thermodynamics for migrating defects.

Weakly Nonlocal Theories of Damage and Plasticity Based on Balance of Dissipative Material Forces

It is shown that the concept of material forces together with associated balance laws, besides the classical laws of linear and angular momentum in the physical space, provide a firm theoretical framework within which weakly nonlocal (gradient type) models of damage and plasticity can be formulated in a clear and rigorous manner. The appropriate set of balance laws for physical and material forces as well as the first and second law of thermodynamics are formulated in integral form. The corresponding local laws are next derived and the general structure of thermodynamically consistent constitutive equations is formulated.

Gradient theories of ductile and brittle damage: A thermodynamical consistent framework and computational issues

Publisher Summary The aim of this chapter is to derive a thermodynamically consistent gradient-damage theory for ductile and brittle damage. It is assumed that evolving microdefects have inertia and that their motion is caused by microforces that have to satisfy their own dynamical balance laws. In addition to the balance laws of macro and microforces, the chapter formulates the first and second law of thermodynamics for macro and meso levels. Assuming the constitutive equations in general form, it is shown that macro and microforces consist of two parts, a nondissipative part, which can be derived from a free energy potential, and a dissipative part that can be considered as dissipative driving force. In contrast, the physical nature of microdefects can be difficult to identify for a specific brittle or ductile material. In addition, some computational issues are discussed in brief. To overcome these difficulties, various regularization techniques were proposed in the literature, using gradient enhancements of damage and hardening parameters and of strain measures.