Response to comments on "Material stability of multicomponent mixtures and the multiplicity of solutions to phase-equilibrium equations. 1. Nonreacting mixtures

This paper gives a concise summary of the general theoretical framework suitable to describe shells with solid-like and liquid-like behavior. Thin-shell kinematics are considered and used to derive the equilibrium equations from linear- and angular-momentum balance. Based on the mechanical power balance and the mechanical dissipation inequality, the constitutive equations for the hyperelastic material behavior of constrained shells are derived and their material stability is examined. Various constitutive examples are considered and assessed for their stability. The governing weak form of the formulation is derived and decomposed into in-plane and out-of-plane components. The presented work provides a very general framework for a unified description of solid and liquid shells and illustrates what leads to their loss of material stability. This framework serves as a basis for developing computational shell formulations based on rotation-free shell discretizations. Therefore the full linearization of the formulation is also presented here.

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Simplified phase equilibrium equations for multicomponent systems

Fluid Phase Equilibria ◽

10.1016/0378-3812(87)85038-0 ◽

1987 ◽

Vol 33 (3) ◽

pp. 207-221 ◽

Cited By ~ 15

Author(s):

E.M. Hendriks

Keyword(s):

Phase Equilibrium ◽

Multicomponent Systems ◽

Equilibrium Equations

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Constraints on phase equilibrium equations

Chemical Engineering Science ◽

10.1016/0009-2509(88)80075-7 ◽

1988 ◽

Vol 43 (4) ◽

pp. 803-810 ◽

Cited By ~ 2

Author(s):

Eric Kvaalen ◽

Daniel Tondeur

Keyword(s):

Phase Equilibrium ◽

Equilibrium Equations

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A topological theory of phase diagrams for multiphase reacting mixtures

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1990.0112 ◽

1990 ◽

Vol 430 (1880) ◽

pp. 669-678 ◽

Cited By ~ 9

Keyword(s):

Phase Equilibrium ◽

Phase Diagrams ◽

Topological Invariant ◽

Global Constraint ◽

Multicomponent Mixtures ◽

Phase Rule ◽

Reacting Mixtures ◽

Equilibrium Data ◽

Residue Curve Maps ◽

Integer Equation

Residue curve maps are an effective way of representing phase equilibria in non-ideal multicomponent mixtures. In this representation the phase equilibrium surfaces are replaced by an equivalent flow of trajectories of a vector field. The flow is characterized by a set of singular points that correspond to the pure components and azeotropes present in the mixture. It is shown that the patterns in these maps for reaction mixtures obey a global constraint arising from a topological invariant for the manifold on which they are defined. This constraint is in the form of an integer equation that phase diagrams must obey in addition to the Gibbs phase rule. The main advantage of the method is that certain global statements can be made about the structure of reactive phase diagrams, independently of the details of phase equilibrium data or models.

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Necked states of non-linearly elastic plates

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500018758 ◽

1989 ◽

Vol 112 (3-4) ◽

pp. 277-291 ◽

Cited By ~ 3

Author(s):

Pablo V. Negrón-Marrero

Keyword(s):

Homogeneous Solution ◽

Large Displacements ◽

Elastic Plates ◽

Circular Plates ◽

Equilibrium Equations ◽

Multiplicity Of Solutions ◽

The Stability

SynopsisIn this paper we study the equilibrium equations for axisymmetric deformations of isotropic circular plates in tension. We give results on the global multiplicity of solutions and study the stability of the trivial homogeneous solution for large displacements.

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