On a consistent dynamic finite-strain shell theory and its linearization

2018 ◽  
Vol 24 (8) ◽  
pp. 2335-2360 ◽  
Author(s):  
Zilong Song ◽  
Jiong Wang ◽  
Hui-Hui Dai
Keyword(s):  
Shell Theory ◽  
Finite Strain ◽  
Circular Cylindrical Shell ◽  
Shell Element ◽  
Momentum Balance ◽  
Orthotropic Materials ◽  
Bottom Surface ◽  
Position Vector ◽  
Field Equations ◽  
Order Error

In this paper, a dynamic finite-strain shell theory is derived, which is consistent with the three-dimensional (3-D) Hamilton’s principle with a fourth-order error under general loadings. A series expansion of the position vector about the bottom surface is adopted. By using the bottom traction condition and the 3-D field equations, the recursive relations for the expansion coefficients are successfully obtained. As a result, the top traction condition leads to a vector shell equation for the first coefficient vector, which represents the local momentum-balance of a shell element. Associated weak formulations, in connection with various boundary conditions, are also established. Furthermore, the derived equations are linearized to obtain a novel shell theory for orthotropic materials. The special case of isotropic materials is considered and comparison with the Donnell–Mushtari (D-M) shell theory is made. It can be shown that, to the leading order, the present shell theory agrees with the D-M theory for statics. Thus, the present shell theory actually provides a consistent derivation for the former one without any ad hoc assumptions. To test the validity of the present dynamic shell theory, the free vibration of a circular cylindrical shell is studied. The results for frequencies are compared with those of the 3-D theory and excellent agreements are found. In addition, it turns out that the present shell theory gives better results than the Flügge shell theory (which is known to provide the best frequency results among the first-approximation shell theories).

On a consistent finite-strain shell theory for incompressible hyperelastic materials

2018 ◽  
Vol 24 (5) ◽  
pp. 1320-1339 ◽  
Author(s):  
Yuanyou Li ◽  
Hui-Hui Dai ◽  
Jiong Wang
Keyword(s):  
Shell Theory ◽  
Finite Strain ◽  
Circular Cylindrical Shell ◽  
Force Balance ◽  
Bottom Surface ◽  
Series Expansions ◽  
Surface Boundary ◽  
Hyperelastic Materials ◽  
Governing System ◽  
Shell Equation

In this paper, a consistent finite-strain shell theory for incompressible hyperelastic materials is formulated. First, for a shell structure made of an incompressible material, the three-dimensional (3D) governing system is derived through the variational approach, which is composed of the mechanical field equation and the constraint equation. Then, series expansions of the independent variables are conducted about the bottom surface and along the thickness direction of the shell. The recursive relations of the coefficient functions in the series expansions can be derived from the original 3D governing system. Further from the top surface boundary condition, a 2D vector shell equation is obtained, which represents the local force-balance of a shell element. The associated edge boundary conditions are also proposed. It is verified that shell equation system is consistent with the 3D variational formulation. The weak formulation of the shell equation is established for future numerical calculations. To show the validity of the shell theory, the axisymmetric deformations of a spherical and a circular cylindrical shell made of incompressible neo-Hookean materials are studied. By comparing with the exact solutions, it is shown that the asymptotic solutions obtained from the shell theory attain the accuracy of O( h2).

scholarly journals On a consistent finite-strain plate theory based on three-dimensional energy principle

2014 ◽  
Vol 470 (2171) ◽  
pp. 20140494 ◽  
Author(s):  
Hui-Hui Dai ◽  
Zilong Song
Keyword(s):  
Finite Strain ◽  
Virtual Work ◽  
Plate Theory ◽  
Three Dimensional ◽  
Force Balance ◽  
Two Dimensional ◽  
Field Equations ◽  
Order Error ◽  
Energy Principle ◽  
Balance Structure

This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional potential energy under general loadings with a fourth-order error. Starting from the three-dimensional nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the three-dimensional field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a two-dimensional virtual work principle. An alternative approach based on a two-dimensional truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a two-dimensional energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Compared with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of loadings, applicability to finite-strain problems and no involvement of non-physical quantities.

scholarly journals Generalizing the coupling between geometry and matter: $$f\left( R,L_m,T\right) $$ gravity

2021 ◽  
Vol 81 (7) ◽  
Author(s):  
Zahra Haghani ◽  
Tiberiu Harko
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
Newtonian Limit ◽  
Gravity Models ◽  
Momentum Balance ◽  
Field Equations ◽  
Gravitational Fields ◽  
Generalized Poisson

AbstractWe generalize and unify the $$f\left( R,T\right) $$ f R , T and $$f\left( R,L_m\right) $$ f R , L m type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R, of the trace of the energy–momentum tensor T, and of the matter Lagrangian $$L_m$$ L m , so that $$ L_{grav}=f\left( R,L_m,T\right) $$ L grav = f R , L m , T . We obtain the gravitational field equations in the metric formalism, the equations of motion for test particles, and the energy and momentum balance equations, which follow from the covariant divergence of the energy–momentum tensor. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equations of motion is also investigated, and the expression of the extra acceleration is obtained for small velocities and weak gravitational fields. The generalized Poisson equation is also obtained in the Newtonian limit, and the Dolgov–Kawasaki instability is also investigated. The cosmological implications of the theory are investigated for a homogeneous, isotropic and flat Universe for two particular choices of the Lagrangian density $$f\left( R,L_m,T\right) $$ f R , L m , T of the gravitational field, with a multiplicative and additive algebraic structure in the matter couplings, respectively, and for two choices of the matter Lagrangian, by using both analytical and numerical methods.

On the Initiation of Necking Modes in Layered Plastic Solids

1987 ◽  
Vol 54 (1) ◽  
pp. 141-146 ◽  
Author(s):  
P. S. Steif
Keyword(s):  
Small Parameter ◽  
Finite Strain ◽  
Relative Rate ◽  
Perturbation Expansion ◽  
Nonlinear Field ◽  
Field Equations ◽  
Regular Perturbation ◽  
Layered Solid ◽  
Interface Imperfections ◽  
Nonlinear Field Equations

The deformation of a solid consisting of alternating material layers with interface undulations is studied analytically. We assess the relative rate of growth or decay of these interface imperfections with straining. The analysis is based on a regular perturbation expansion of the finite strain, nonlinear field equations, the small parameter being the amplitude of the undulations relative to the layer thickness. Attention is focused on the influence of material and geometric parameters on imperfection growth-rates. It is found that significant connections exist between the growth of imperfections and the tendency for undulatory bifurcation modes to emerge from a perfect, layered solid.

Free Asymmetric Vibrations of Composite Full Circular Cylindrical Shells Stiffened by a Bonded Central Shell Segment

2004 ◽  
Author(s):  
U. Yuceoglu ◽  
V. O¨zerciyes
Keyword(s):  
Cylindrical Shell ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Circular Cylindrical Shell ◽  
Cylindrical Shells ◽  
Adhesive Layer ◽  
Shear Stresses ◽  
First Order ◽  
Stress Resultant ◽  
Circular Cylindrical Shells

This study is concerned with the “Free Asymmetric Vibrations of Composite Full Circular Cylindrical Shells Stiffened by a Bonded Central Shell Segment.” The base shell is made of an orthotropic “full” circular cylindrical shell reinforced and/or stiffened by an adhesively bonded dissimilar, orthotropic “full” circular cylindrical shell segment. The stiffening shell segment is located at the mid-center of the composite system. The theoretical analysis is based on the “Timoshenko-Mindlin-(and Reissner) Shell Theory” which is a “First Order Shear Deformation Shell Theory (FSDST).” Thus, in both “base (or lower) shell” and in the “upper shell” segment, the transverse shear deformations and the extensional, translational and the rotary moments of inertia are taken into account in the formulation. In the very thin and linearly elastic adhesive layer, the transverse normal and shear stresses are accounted for. The sets of the dynamic equations, stress-resultant-displacement equations for both shells and the in-between adhesive layer are combined and manipulated and are finally reduced into a ”Governing System of the First Order Ordinary Differential Equations” in the “state-vector” form. This system is integrated by the “Modified Transfer Matrix Method (with Chebyshev Polynomials).” Some asymmetric mode shapes and the corresponding natural frequencies showing the effect of the “hard” and the “soft” adhesive cases are presented. Also, the parametric study of the “overlap length” (or the bonded joint length) on the natural frequencies in several modes is considered and plotted.

scholarly journals An advanced shell model for the analysis of geometrical and material nonlinear shells

2020 ◽  
Vol 66 (6) ◽  
pp. 1353-1376
Author(s):  
F. Gruttmann ◽  
W. Wagner
Keyword(s):  
Shell Model ◽  
Geometrical Nonlinearity ◽  
Local Equilibrium ◽  
Deformation Gradient ◽  
Computing Time ◽  
Strain Tensor ◽  
Shell Element ◽  
Field Equations ◽  
Residual Vector ◽  
Layered Shells

AbstractIn this paper layered shells subjected to static loading are considered. The displacements of the Reissner–Mindlin theory are enriched by a an additional part. These so-called fluctuation displacements include warping displacements and thickness changes. They lead to additional terms for the material deformation gradient and the Green–Lagrangian strain tensor. Within a nonlinear multi-field variational formulation the weak form of the boundary value problem accounts for the equilibrium of stress resultants and couple resultants, the local equilibrium of stresses, the geometrical field equations and the constitutive equations. For the independent shell strains an ansatz with quadratic shape functions is chosen. This leads to a significant improved convergence behaviour especially for distorted meshes. Elimination of a set of parameters on element level by static condensation yields an element stiffness matrix and residual vector of a quadrilateral shell element with the usual 5 or 6 nodal degrees of freedom. The developed model yields the complicated three-dimensional stress state in layered shells for elasticity and elasto-plasticity considering geometrical nonlinearity. In comparison with fully 3D solutions present 2D shell model requires only a fractional amount of computing time.

Free vibration of bi-material cylindrical shells

2016 ◽  
Vol 230 (15) ◽  
pp. 2637-2649 ◽  
Author(s):  
Saeed Sarkheil ◽  
Mahmud S Foumani ◽  
Hossein M Navazi
Keyword(s):  
Exact Solution ◽  
Cylindrical Shell ◽  
Free Vibration ◽  
Boundary Conditions ◽  
State Space ◽  
Thin Shell ◽  
Shell Theory ◽  
Circular Cylindrical Shell ◽  
Space Technique ◽  
Thin Shell Theory

Based on the Sanders thin shell theory, this paper presents an exact solution for the vibration of circular cylindrical shell made of two different materials. The shell is sub-divided into two segments and the state-space technique is employed to derive the homogenous differential equations. Then continuity conditions are applied where the material of the cylindrical shell changes. Shells with various combinations of end boundary conditions are analyzed by the proposed method. Finally, solving different examples, the effect of geometric parameters as well as BCs on the vibration of the bi-material shell is studied.

Two-Dimensional Shell Vibration of Microtubule in Living Cell

2010 ◽  
Author(s):  
E. Ghavanoo ◽  
F. Daneshmand ◽  
M. Amabili
Keyword(s):  
Living Cell ◽  
Shell Theory ◽  
Coupled System ◽  
Eukaryotic Cell ◽  
Wave Number ◽  
Orthotropic Materials ◽  
Two Dimensional ◽  
Polynomial Eigenvalue Problem ◽  
Shell Vibration ◽  
Circumferential Wave

The mechanical behavior of a eukaryotic cell is mainly determined by its cytoskeleton. Microtubules immersed in cytosol are a central part of the cytoskeleton. Cytosol is the viscous fluid in living cells. The microtubules permanently oscillate in the cytosol. In this study, two-dimensional vibration of a single microtubule in living cell is investigated. The Donnell’s shell theory equations for orthotropic materials is used to model the microtubule whereas the motion of the cytosol is modeled as Stokes flow characterized by a small Reynolds number with no-slip condition at microtubule-cytosol interface. The stress field in the cytosol induced by vibrating microtubule is determined analytically and the coupled vibrations of the microtubule-cytoplasm system are investigated. A coupled polynomial eigenvalue problem is developed in the present study and the variations of eigenvalues of coupled system with cytosol dynamic viscosity, microtubule circumferential Young’s modulus and circumferential wave number are examined.

Dynamic Stability of Isotropic or Composite-Material Cylindrical Shells Containing Swirling Fluid Flow

1977 ◽  
Vol 44 (1) ◽  
pp. 112-116 ◽  
Author(s):  
T. L. C. Chen ◽  
C. W. Bert
Keyword(s):  
Shell Theory ◽  
Swirling Flow ◽  
Circular Cylindrical Shell ◽  
Critical Flow ◽  
Fluid System ◽  
Critical Flow Velocity ◽  
Rotating Speed ◽  
Fluid Forces ◽  
Shell Motion ◽  
The Stability

A linear stability analysis is presented for a thin-walled, circular cylindrical shell of orthotropic material conveying a swirling flow. Shell motion is modeled by using the dynamic orthotropic version of the Sanders shell theory and fluid forces are described by inviscid, incompressible flow theory. The critical flow velocities are determined for piping made of composite and isotropic materials conveying swirling water. Fluid rotation strongly degrades the stability of the shell/fluid system, i.e. increasing the fluid rotating speed severely decreases the critical flow velocity.

Traveling Wave Vibration of Rotating Thin Circular Cylindrical Shell

2012 ◽  
Vol 226-228 ◽  
pp. 262-266 ◽  
Author(s):  
Yan Qi Liu ◽  
Fu Lei Chu
Keyword(s):  
Cylindrical Shell ◽  
Frequency Response ◽  
Traveling Wave ◽  
Shell Theory ◽  
Circular Cylindrical Shell ◽  
Radial Excitation ◽  
Forward Wave ◽  
Frequency Responses ◽  
Amplitude Frequency Response ◽  
Love’S Shell Theory

In this paper, the vibration characteristics of the rotating thin circular cylindrical shell subjected to the radial excitation are presented. Based on the Love’s shell theory, the governing equation of the rotating thin circular cylindrical shell is derived by using the Hamilton’s principle. Then, the amplitude-frequency responses for traveling wave vibration of the circular cylindrical shell are investigated. The results indicate that there exists the traveling wave vibration for the rotating thin circular cylindrical shell, namely: the forward wave and the backward wave. The effects of the damping and excitation on the amplitude-frequency response are analyzed.

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