Continuity Conditions in Elastic Shells With Phase Transformation (original) (raw)
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The Nonlinear Theory of Elastic Shells with Phase Transitions
Journal of Elasticity, 2000
We develop the general nonlinear theory of elastic shells with an account of phase transitions in the shell material. Our formulation is based on the dynamically and kinematically exact through-the-thickness reduction of three-dimensional description of the phenomenon to the twodimensional form written on the shell base surface. In this model shell displacements are expressed by work-averaged translations and rotations of the shell cross-sections. All shell relations are then found from the variational principle of the stationary total potential energy. In particular, we derive the new global dynamic continuity condition at the singular surface curve modelling the phase interface. We also discuss particular forms of the local dynamic continuity conditions at coherent and incoherent interface curves. The results are illustrated by an example of a phase transition in an infinite plate with a circular hole. : 74K25, 74N10, 74K35, 74A50, 74N20.
On Continuity Conditions at the Phase Interface of Two-Phase Elastic Shells
The general non-linear theory of elastic shells undergoing stress-induced phase transition of martensitic type is developed. Our formulation is based on the statically and kinematically exact shell model. We also take into account the strain energy density of capillarity type as well as forces and couples applied along the curvilinear phase interface itself. The boundary value problem is formulated in the weak form through the variational principle of stationary, total potential energy. In particular, we derive the refined static continuity conditions at the coherent interface and at the interface incoherent in rotations.
Thermomechanics of shells undergoing phase transition
Journal of the Mechanics and Physics of Solids, 2011
We develop the general nonlinear theory of elastic shells with an account of phase transitions in the shell material. Our formulation is based on the dynamically and kinematically exact through-the-thickness reduction of three-dimensional description of the phenomenon to the twodimensional form written on the shell base surface. In this model shell displacements are expressed by work-averaged translations and rotations of the shell cross-sections. All shell relations are then found from the variational principle of the stationary total potential energy. In particular, we derive the new global dynamic continuity condition at the singular surface curve modelling the phase interface. We also discuss particular forms of the local dynamic continuity conditions at coherent and incoherent interface curves. The results are illustrated by an example of a phase transition in an infinite plate with a circular hole. : 74K25, 74N10, 74K35, 74A50, 74N20.
On the Nonlinear Theory of Two-Phase Shells
Advanced Structured Materials, 2011
We discuss the nonlinear theory of shells made of material undergoing phase transitions (PT). The interest to such thin-walled structures is motivated by applications of thin films made of martensitic materials and needs of modeling biological membranes. Here we present the resultant, two-dimensional thermodynamics of non-linear theory of shells undergoing PT. The global and local formulations of the balances of momentum, moment of momentum, energy and entropy are given. Two temperature fields on the shell base surface are introduced: the referential mean temperature and its deviation, as well as two corresponding dual fields: the referential entropy and its deviation. Additional surface heat flux and the extra heat flux vector fields appear as a result of through-the-thickness integration procedure. Within the framework of the resultant shell thermodynamics we derive the continuity conditions along the curvilinear phase interface which separates two material phases. These conditions allow us to formulate the kinetic equation describing the quasistatic motion of the interface relative to the shell base surface. The kinetic equation is expressed by the jump of the Eshelby tensor across the phase interface. In the case of thermodynamic equilibrium the variational statement of the static problem of two-phase shell is presented.
On quasi-static propagation of the phase interface in thin-walled inelastic bodies
Applying the general non-linear theory of shells undergoing phase transitions, we derive the balance equations along the singular curve modelling the phase interface in the shell. From the integral forms of balance laws of linear momentum, angular momentum, and energy as well as the entropy inequality we obtain the local static balance equation along the curvilinear phase interface. We also derive the thermodynamic condition allowing one to determine the interface position within the deformed shell midsurface. The special case of the pure mechanical theory is also considered.
International Journal of Solids and Structures, 1970
Two-dimensional iterative procedures for the determination of the components of the stress tensor and of the displacement vector in thin anisotropic shells (plates) are derived from the three-dimensional (geometrically) non-linear equations of the elastic continuum theory by means of the method of asymptotic integration. The conditions, both for cases when the main system of equations of the iterative process is linear and for cases when the main system of the zeroth order approximation is non-linear, are given in terms of characteristic quantities, which character& geometric and material properties of the shell (plate), and the intensity and the variability of the surface load. The attention is confined to the interior problem, the discussion of edge effects is omitted. THE problem considered in this paper is the derivation of appro~mate two-dimensional theories of thin plates and shells starting from the three-dimensional geometrically nonlinear equations of elastic continuum theory. The method of derivation adopted here is an extension of the method of asymptotic integration used in a series of articles concerning the linear theory of shells and plates (see, e.g. [l-4]).
Intrinsic equations for non-linear deformation and stability of thin elastic shells
International Journal of Solids and Structures, 2004
We formulate the complete boundary value problem (BVP) for the geometrically non-linear theory of thin elastic shells expressed entirely in terms of intrinsic field variables--the stress resultants and bendings of the shell reference surface. Applying a perturbation technique the corresponding intrinsic buckling equations are also derived. These intrinsic shell and buckling equations are then consistently simplified in three special cases: (a) the almost inextensional bending state, (b) the almost membrane state, and (c) the bending state. For each special case we derive also consistent sets of dynamic and kinematic boundary conditions expressed entirely through the intrinsic field variables. Additionally, the complete BVP for shells with slowly varying curvatures is formulated in terms of stress and deformation functions. The relatively simple BVPs given here are applicable to highly non-linear problems of flexible shell structures without any restriction of shell geometry, displacements, deflections, and/or rotations.
On the homogenization of a nonlinear shell
Mathematics and Mechanics of Solids, 2018
In this paper we propose a multiscale finite-strain shell theory for simulating the mechanical response of a highly heterogeneous shell of varying thickness. To resolve this issue, a higher-order stress-resultant shell formulation based on multiscale homogenization is considered. At the macroscopic scale level, we approximate the displacement field by a fifth-order Taylor–Young expansion in thickness. We take account of the microscale fluctuations by introducing a boundary value problem over the domain of a three-dimensional representative volume element (RVE). The geometrical form and the dimensions of the RVE are determined by the representative microstructure of the heterogeneity. In this way, an in-plane homogenization is directly combined with a through thickness stress integration. As a result, the macroscopic stress resultants are the volume averages through RVE of microscopic stress. All microstructural constituents are modeled as first-order continua and three-dimensional c...
On the dynamics of an elastic-rigid material
We present a model for a class of materials which behave as an elastic solid if the stress exceeds a certain threshold and as a rigid body when the stress is below such a threshold. The constitutive equations for these materials fall in the class of implicit constitutive relations, since the stress is a multivalued function of the strain. The mathematical structure of the constitutive equation is similar to the one of Bingham fluids, the main difference being that above the stress threshold the body behaves like an elastic solid and not as a viscous fluid.
The Theory of Simple Elastic Shells
Theories of Plates and Shells, 2004
In the report the main aspects of the shell theory based on the direct approach are presented. The main attention will be focussed on the establishment of the constitutive equations. It is shown that the deformation energy must be an integral of a system of partial differential equations. Thus it can be expressed as a function of integrals of the characteristic system of the ordinary differential equations. These integrals are called the reduced deformation tensors. In order to found the structure of the elasticity tensors a new theory of symmetry is introduced.