SAINT VENANT COMPATIBILITY EQUATIONS ON A SURFACE APPLICATION TO INTRINSIC SHELL THEORY

2008 ◽  
Vol 18 (02) ◽  
pp. 165-194 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
LILIANA GRATIE ◽  
CRISTINEL MARDARE ◽  
MING SHEN
Keyword(s):  
Displacement Field ◽  
Symmetric Matrix ◽  
Three Dimensional ◽  
Tensor Fields ◽  
Compatibility Conditions ◽  
Quadratic Minimization ◽  
Connected Surface ◽  
Saint Venant Equations ◽  
Matrix Fields ◽  
Curvature Tensors

We first establish that the linearized change of metric and change of curvature tensors, with components in L2 and H-1 respectively, associated with a displacement field, with components in H1, of a surface S immersed in ℝ3 must satisfy in the distributional sense compatibility conditions that may be viewed as the linear version of the Gauss and Codazzi-Mainardi equations. These compatibility conditions, which are analogous to the familiar Saint Venant equations in three-dimensional elasticity, constitute the Saint Venant equations on the surface S. We next show that these compatibility conditions are also sufficient, i.e. that they in fact characterize the linearized change of metric and the linearized change of curvature tensors in the following sense: If two symmetric matrix fields of order two defined over a simply-connected surface S ⊂ ℝ3 satisfy the above compatibility conditions, then they are the linearized change of metric and linearized change of curvature tensors associated with a displacement field of the surface S, a field whose existence is thus established. The proof provides an explicit algorithm for recovering such a displacement field from the linearized change of metric and linearized change of curvature tensors. This algorithm may be viewed as the linear counterpart of the reconstruction of a surface from its first and second fundamental forms. Finally, we show how these results can be applied to the "intrinsic theory" of linearly elastic shells, where the linearized change of metric and change of curvature tensors are the new unknowns. These new unknowns solve a quadratic minimization problem over a space of tensor fields whose components, which are only in L2, satisfy the Saint Venant compatibility conditions on a surface in the sense of distributions.

scholarly journals SAINT VENANT COMPATIBILITY EQUATIONS IN CURVILINEAR COORDINATES

2007 ◽  
Vol 05 (03) ◽  
pp. 231-251 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
CRISTINEL MARDARE ◽  
MING SHEN
Keyword(s):  
Displacement Field ◽  
Symmetric Matrix ◽  
Riemannian Metric ◽  
Curvilinear Coordinates ◽  
Compatibility Conditions ◽  
Explicit Algorithm ◽  
Open Set ◽  
Saint Venant Equations ◽  
Simply Connected ◽  
Linear Counterpart

We first establish that the linearized strains in curvilinear coordinates associated with a given displacement field necessarily satisfy compatibility conditions that constitute the "Saint Venant equations in curvilinear coordinates". We then show that these equations are also sufficient, in the following sense: If a symmetric matrix field defined over a simply-connected open set satisfies the Saint Venant equations in curvilinear coordinates, then its coefficients are the linearized strains associated with a displacement field. In addition, our proof provides an explicit algorithm for recovering such a displacement field from its linearized strains in curvilinear coordinates. This algorithm may be viewed as the linear counterpart of the reconstruction of an immersion from a given flat Riemannian metric.

Donati compatibility conditions for membrane and flexural shells

2015 ◽  
Vol 13 (06) ◽  
pp. 685-705 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare ◽  
Xiaoqin Shen
Keyword(s):  
Minimization Problem ◽  
Elastic Shell ◽  
Middle Surface ◽  
Test Functions ◽  
Tensor Fields ◽  
Compatibility Conditions ◽  
Variational Equations ◽  
Quadratic Minimization ◽  
Membrane Shells ◽  
Curvature Tensors

Donati compatibility conditions on a surface allow to reformulate the minimization problem for a linearly elastic shell through the intrinsic approach, i.e. as a quadratic minimization problem with the linearized change of metric and change of curvature tensors of the middle surface of the shell as the new unknowns. Such compatibility conditions typically take the form of variational equations with divergence-free tensor fields as test-functions. In a previous work, the first author and Oana Iosifescu have identified and justified Donati compatibility conditions for shells modeled by Koiter's equations. In this paper, Donati compatibility conditions are identified and justified for two specific classes of linearly elastic shells, the so-called elliptic membrane shells and flexural shells.

NONLINEAR SAINT–VENANT COMPATIBILITY CONDITIONS AND THE INTRINSIC APPROACH FOR NONLINEARLY ELASTIC PLATES

2013 ◽  
Vol 23 (12) ◽  
pp. 2293-2321 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
SORIN MARDARE
Keyword(s):  
Symmetric Matrix ◽  
Elastic Plates ◽  
Compatibility Conditions ◽  
Korn’S Inequality ◽  
The Neumann Problem ◽  
Nonlinearly Elastic Plate ◽  
Matrix Fields ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Nonlinearly Elastic

Let ω be a simply connected planar domain. First, we give necessary and sufficient nonlinear compatibility conditions of Saint–Venant type guaranteeing that, given two 2 × 2 symmetric matrix fields (Eαβ) and (Fαβ) with components in L2(ω), there exists a vector field (ηi) with components η1, η2 ∈ H1(ω) and η3 ∈ H2(ω) such that ½(∂αηβ + ∂βηα + ∂αη3∂βη3) = Eαβ and ∂αβη3 = Fαβ in ω for α, β = 1, 2. Second, we show that the classical approach to the Neumann problem for a nonlinearly elastic plate can be recast as a minimization problem in terms of the new unknowns Eαβ = ½(∂αηβ + ∂βηα + ∂αη3∂βη3) ∈ L2(ω) and Fαβ = ∂αβη3 ∈ L2(ω) and that this problem has a solution in a manifold of symmetric matrix fields (Eαβ) and (Fαβ) whose components Eαβ ∈ L2(ω) and Fαβ ∈ L2(ω) satisfy the nonlinear Saint–Venant compatibility conditions mentioned above. We also show that the analysis of such an "intrinsic approach" naturally leads to a new nonlinear Korn's inequality.

Nonlinear Donati compatibility conditions on a surface — Application to the intrinsic approach for Koiter’s model of a nonlinearly elastic shallow shell

2017 ◽  
Vol 27 (02) ◽  
pp. 347-384 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Oana Iosifescu
Keyword(s):  
Existence Theorem ◽  
Displacement Field ◽  
Shallow Shell ◽  
Existence Theorems ◽  
Middle Surface ◽  
Compatibility Conditions ◽  
Matrix Fields ◽  
Displacement Approach ◽  
Strain Measures ◽  
Nonlinearly Elastic

An intrinsic approach to a mathematical model of a linearly or nonlinearly elastic body consists in considering the strain measures found in the energy of this model as the sole unknowns, instead of the displacement field in the classical approach. Such an approach thus provides a direct computation of the stresses by means of the constitutive equation. The main problem therefore consists in identifying specific compatibility conditions that these new unknowns, which are now matrix fields with components in [Formula: see text], should satisfy in order that they correspond to an actual displacement field. Such compatibility conditions are either of Saint-Venant type, in which case they take the form of partial differential equations, or of Donati type, in which case they take the form of ortho- gonality relations against matrix fields that are divergence-free. The main objective of this paper consists in showing how an intrinsic approach can be successfully applied to the well-known Koiter’s model of a nonlinearly elastic shallow shell, thus providing the first instance (at least to the authors’ best knowledge) of a mathematical justification of this approach applied to a nonlinear shell model (“shallow” means that the absolute value of the Gaussian curvature of the middle surface of the shell is “uniformly small enough”). More specifically, we first identify and justify compatibility conditions of Donati type guaranteeing that the nonlinear strain measures found in Koiter’s model correspond to an actual displacement field. Second, we show that the associated intrinsic energy attains its minimum over a set of matrix fields that satisfy these Donati compatibility conditions, thus providing an existence theorem for the intrinsic approach; the proof relies in particular on an interesting per se nonlinear Korn inequality on a surface. Incidentally, this existence result (once converted into an equivalent existence theorem for the classical displacement approach) constitutes a significant improvement over previously known existence theorems for Koiter’s model of a nonlinearly elastic shallow shell.

Truss and Beam Finite Elements Revisited: A Derivation Based on Displacement-Field Decomposition

1996 ◽  
Vol 11 (4) ◽  
pp. 371-380 ◽  
Author(s):  
Alphose Zingoni
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Symmetry Group ◽  
Displacement Field ◽  
Three Dimensional ◽  
Beam Elements ◽  
Symmetry Properties ◽  
Mass Matrices ◽  
General Displacement

Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.

Lee waves in three-dimensional stratified flow

1984 ◽  
Vol 148 ◽  
pp. 97-108 ◽  
Author(s):  
G. S. Janowitz
Keyword(s):  
Stream Function ◽  
Displacement Field ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Vertical Displacement ◽  
Function Solution ◽  
Dispersion Surface ◽  
Phase Calculation ◽  
Lee Waves ◽  
The Green Function

The effect of a shallow isolated topography on a linearly stratified, three-dimensional, initially uniform flow in the x-direction is considered. The Green-function solution for the velocity disturbance due to this topography, which is equivalent to that due to a dipole at the origin, is shown to be without swirl, i.e. the velocity disturbance lies strictly in planes passing through the x-axis. Thus this disturbance can be described in terms of a stream function. The asymptotic forms of the wavelike portion of the stream function and the vertical displacement field are obtained. The latter is in agreement with the limited versions due to Crapper (1959). The Gaussian curvature of the zero-frequency dispersion surface is obtained analytically as a step in the stationary-phase calculation. The model is extended to determine the vertical displacement field for an arbitrary shallow topography far downstream. For topographies that are even functions of x and y it is shown that the details of the topography affect the displacement field only in the vicinity of the x-axis. Elsewhere, the amplitude of the displacement is proportional to the net volume of the topography.

Consistent Asymptotic Expansion Multiscale Formulation for Heterogeneous Column Structure

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Dongdong Wang ◽  
Pinkang Xie ◽  
Lingming Fang
Keyword(s):  
Asymptotic Expansion ◽  
Displacement Field ◽  
Three Dimensional ◽  
Axial Direction ◽  
Unit Cell ◽  
Local Unit ◽  
Cell Problem ◽  
Element Discretization ◽  
Free Condition ◽  
Column Structure

A consistent asymptotic expansion multiscale formulation is presented for analysis of the heterogeneous column structure, which has three dimensional periodic reinforcements along the axial direction. The proposed formulation is based upon a new asymptotic expansion of the displacement field. This new multiscale displacement expansion has a three dimensional form, more specifically, it takes into account the axial periodic property but simultaneously keeps the cross section dimensions in the global scale. Thus, this formulation inherently reflects the characteristics of the column structure, i.e., the traction free condition on the circumferential surfaces. Subsequently, the global equilibrium problem and the local unit cell problem are consistently derived based upon the proposed asymptotic displacement field. It turns out that the global homogenized problem is the standard axial equilibrium equation, while the local unit cell problem is completely three dimensional which is subjected to the periodic boundary condition on axial surfaces as well as the traction free condition on circumferential surfaces of the unit cell. Thereafter, the variational formulation and finite element discretization of the unit cell problem are discussed. The effectiveness of the present formulation is illustrated by several numerical examples.

Experimental and numerical determination of mode II fracture toughness of woven composites verified through unidirectional composite test data

2019 ◽  
Vol 27 (9) ◽  
pp. 557-566
Author(s):  
Rowan Healey ◽  
Nabil M Chowdhury ◽  
Wing Kong Chiu ◽  
John Wang
Keyword(s):  
Fracture Toughness ◽  
Displacement Field ◽  
Experimental Testing ◽  
Three Dimensional ◽  
Experimental Results ◽  
Woven Composites ◽  
Mode Ii ◽  
Numerical Comparison ◽  
J Integral ◽  
Mode Ii Fracture Toughness

Due to the increase in prevalence of fibre-reinforced polymer matrix composites (FRPMC) in aircraft structures, the need for adaption of failure prediction tools such as fatigue spectra has become more pertinent. Fracture toughness is an important measure with regard to fatigue, while adequate techniques and an ASTM standard for unidirectional FRPMC exist, there are mixed opinions when investigating woven FRPMC. This study describes a three-dimensional finite element model developed to assist in determining the mode II interlaminar fracture toughness ( GIIc) of fibre-reinforced woven composites, validated by an experimental and numerical comparison of GIIc determination for unidirectional FRPMC. Experimental testing mirroring the ASTM D7905 resulted in a measure of 1176 J m−2for the unidirectional specimen, while comparisons made with the literature achieved an average value of 1459.24 J m−2or the woven specimen. Three numerical methods were employed due to their prominence in the literature: displacement field, virtual crack closure techniques and the J integral. Both the J integral and the displacement field three-dimensional models produced satisfactory unidirectional GIIc estimates of 1284 and 1116.8 J m−2, respectively. Displacement field had a 5% uncertainty in GIIc when compared with experimental results, while J integral had an approximately 8.5% uncertainty. Extending the analysis to the woven specimens, values of 1302.8 and 1465.3 J m−2were obtained from J integral and displacement field methods, respectively, both within 10% of the experimental values. Hence, numerically determined unidirectional GIIc values were verified with experimental results, leading to the successful employment and extension to woven composites which displayed similar agreement.

CERTAIN RESULTS ON THREE-DIMENSIONAL CONTACT METRIC MANIFOLDS

2010 ◽  
Vol 03 (04) ◽  
pp. 577-591 ◽  
Author(s):  
Amalendu Ghosh
Keyword(s):  
Ricci Tensor ◽  
Three Dimensional ◽  
Tensor Fields ◽  
Contact Metric Manifold ◽  
Ricci Operator ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Parallel Ricci Tensor

In this paper we study 3-dimensional contact metric manifolds satisfying certain conditions on the tensor fields *-Ricci tensorS*, h(= ½Lξφ), τ(= Lξg = 2hφ) and the Ricci operator Q. First, we prove that a 3-dimensional non-Sasakian contact metric manifold satisfies. [Formula: see text] (where ⊕X,Y,Z denotes the cyclic sum over X,Y,Z) if and only if M is a generalized (κ, μ)-space. Next, we prove that a 3-dimensional contact metric manifold with vanishing *-Ricci tensor is a generalized (κ, μ)-space. Finally, some results on 3-dimensional contact metric manifold with cyclic η-parallel h or cyclic η-parallel τ or η-parallel Ricci tensor are presented.

3D Simulation of Foundation Excavation in a Subway Station

2012 ◽  
Vol 468-471 ◽  
pp. 781-784
Author(s):  
Jin Lu ◽  
Xiao Fan Liao
Keyword(s):  
Displacement Field ◽  
Three Dimensional ◽  
Deformation Field ◽  
Subway Station ◽  
Failure Zone ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
The Status ◽  
Practical Engineering ◽  
Variation Characteristics

According to an practical construction of an subway station, the foundation excavation three dimensional model was built by FLAC3D from the need of practical engineering. The status and variation characteristics of the displacement field, deformation field and plastic failure zone were obtained. Guiding significance was provided by the simulation.

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