Assessment of nonlinear exact geometry sampling surfaces solid‐shell elements and ANSYS solid elements for <scp>3D</scp> stress analysis of piezoelectric shell structures

Куликов, Г. М.; Плотникова, С. В.; Glebov, A. O.

doi:10.1002/nme.6382

Cited by 10 publications

(14 citation statements)

References 59 publications

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“…In the orthonormal basis e i , the relations between the electric field and electric potentials of SaS of the n th layer 60 are expressed as

\begin{array}{l} E_{α}^{(n) i_{n}} = - \frac{1}{A_{α} c_{α}^{(n) i_{n}}} φ_{, α}^{(n) i_{n}}, \\ E_{3}^{(n) i_{n}} = - \sum_{j_{n}} M^{(n) j_{n}} (θ_{3}^{(n) i_{n}}) φ^{(n) j_{n}} . \end{array}

…”

Section: Higher‐order Thermoelectroelastic Theory Of Laminated Shellsmentioning

confidence: 99%

“…The finite element formulation is based on a simple interpolation of displacements and electric potentials of SaS of the n th layer by a GeX four‐node solid‐shell element 52,60

u_{i}^{true(n true) i_{n}} = \underset{r}{false\sum} N_{r} u_{italicir}^{true(n true) i_{n}},

φ^{(n) i_{n}} = \sum_{r} N_{r} φ_{r}^{(n) i_{n}},

where

u_{italicir}^{true(n true) i_{n}}

and

φ_{r}^{(n) i_{n}}

are the displacements and electric potentials of SaS of the n th layer at element nodes; ξ 1 and ξ 2 are the normalized curvilinear coordinates θ 1 and θ 2 depicted in Figure 2; the nodal index r runs from 1 to 4.…”

Section: Hybrid‐mixed Thermopiezoelectric Solid‐shell Elementmentioning

confidence: 99%

“…The extended ANS method 60 can be also applied to interpolate the electric field of SaS of the n th layer

{boldE}^{true(n true) i_{n}} = \underset{r}{false\sum} N_{r} {boldE}_{r}^{true(n true) i_{n}},

where

{boldE}_{r}^{true(n true) i_{n}} = {[E_{1 r}^{(n) i_{n}} E_{2 r}^{(n) i_{n}} E_{3 r}^{(n) i_{n}}]}^{normalT}

are the electric field vectors of SaS of the n th layer at element nodes defined as

{boldE}_{r}^{(n) i_{n}} = - {boldB}_{φ r}^{(n) i_{n}} Φ,

where

{boldB}_{φ r}^{(n) i_{n}}

are the constant matrices of order 3 × 4 N SaS given in Appendix B; Φ is the electric potential vector of the solid‐shell element:

Φ = {[Φ_{1}^{T} Φ_{2}^{T} Φ_{3}^{T} Φ_{4}^{T}]}^{normalT},

{boldΦ}_{r} = ([], φ_{r}^{true[0 true]})

…”

Section: Hybrid‐mixed Thermopiezoelectric Solid‐shell Elementmentioning

confidence: 99%

“…Such choice of unknowns with the use of Lagrange polynomials of degree I n − 1 in the through‐the‐thickness approximations of temperature, temperature gradient and heat flux vector, displacements, strains and stresses, electric potential, electric field and electric displacement vector, as well as the entropy density of the n th layer leads to a reliable higher‐order layer‐wise shell formulation, in which all basic variables are related to SaS. Recently, the SaS formulation was utilized to develop GeX four‐node solid‐shell elements for the geometrically linear and nonlinear analysis of piezoelectric shell structures 59‐61 . Here, these studies are extended to analyze coupled thermo‐electro‐mechanical fields in laminated piezoelectric shells.…”

Section: Introductionmentioning

confidence: 99%

“…It is impossible within the framework of the isoparametric hybrid‐mixed shell element formulation 62,63 . The important feature of the GeX thermopiezoelectric solid‐shell element is the use of effective analytical integration throughout the finite element using the extended assumed natural strain (ANS) method 51,59,60 …”

Section: Introductionmentioning

confidence: 99%

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Coupled thermoelectroelastic analysis of thick and thin laminated piezoelectric structures by exact geometry solid‐shell elements based on the sampling surfaces method

Куликов

Плотникова

2021

Numerical Meth Engineering

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A hybrid‐mixed exact geometry four‐node thermopiezoelectric solid‐shell element through the sampling surfaces (SaS) formulation is proposed. The SaS formulation is based on the choice of an arbitrary number of SaS within layers parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the temperatures, displacements and electric potentials of these surfaces as basic shell unknowns. Due to the variational formulation, the outer surfaces and interfaces are also included into a set of SaS. Such choice of unknowns with the use of Lagrange polynomials in the through‐thickness approximations of temperatures, temperature gradient, displacements, strains, electric potential, and electric field leads to a very compact higher‐order thermopiezoelectric shell formulation. To implement efficient analytical integration throughout the solid‐shell element, the extended assumed natural strain method is employed for all components of the temperature gradient, strain tensor, and electric field vector. The developed hybrid‐mixed four‐node thermopiezoelectric solid‐shell element is based on the Hu–Washizu variational principle and shows a superior performance in the case of coarse meshes. It can be useful for the three‐dimensional thermoelectroelastic analysis of thick and thin doubly curved laminated piezoelectric shells under thermal loading, since the SaS formulation allows one to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach exact solutions of the theory of thermopiezoelectricity as a number of SaS tends to infinity.

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“…In the orthonormal basis e i , the relations between the electric field and electric potentials of SaS of the n th layer 60 are expressed as

\begin{array}{l} E_{α}^{(n) i_{n}} = - \frac{1}{A_{α} c_{α}^{(n) i_{n}}} φ_{, α}^{(n) i_{n}}, \\ E_{3}^{(n) i_{n}} = - \sum_{j_{n}} M^{(n) j_{n}} (θ_{3}^{(n) i_{n}}) φ^{(n) j_{n}} . \end{array}

…”

Section: Higher‐order Thermoelectroelastic Theory Of Laminated Shellsmentioning

confidence: 99%

“…The finite element formulation is based on a simple interpolation of displacements and electric potentials of SaS of the n th layer by a GeX four‐node solid‐shell element 52,60

u_{i}^{true(n true) i_{n}} = \underset{r}{false\sum} N_{r} u_{italicir}^{true(n true) i_{n}},

φ^{(n) i_{n}} = \sum_{r} N_{r} φ_{r}^{(n) i_{n}},

where

u_{italicir}^{true(n true) i_{n}}

and

φ_{r}^{(n) i_{n}}

Section: Hybrid‐mixed Thermopiezoelectric Solid‐shell Elementmentioning

confidence: 99%

“…The extended ANS method 60 can be also applied to interpolate the electric field of SaS of the n th layer

{boldE}^{true(n true) i_{n}} = \underset{r}{false\sum} N_{r} {boldE}_{r}^{true(n true) i_{n}},

where

{boldE}_{r}^{true(n true) i_{n}} = {[E_{1 r}^{(n) i_{n}} E_{2 r}^{(n) i_{n}} E_{3 r}^{(n) i_{n}}]}^{normalT}

are the electric field vectors of SaS of the n th layer at element nodes defined as

{boldE}_{r}^{(n) i_{n}} = - {boldB}_{φ r}^{(n) i_{n}} Φ,

where

{boldB}_{φ r}^{(n) i_{n}}

are the constant matrices of order 3 × 4 N SaS given in Appendix B; Φ is the electric potential vector of the solid‐shell element:

Φ = {[Φ_{1}^{T} Φ_{2}^{T} Φ_{3}^{T} Φ_{4}^{T}]}^{normalT},

{boldΦ}_{r} = ([], φ_{r}^{true[0 true]})

…”

Section: Hybrid‐mixed Thermopiezoelectric Solid‐shell Elementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Coupled thermoelectroelastic analysis of thick and thin laminated piezoelectric structures by exact geometry solid‐shell elements based on the sampling surfaces method

Куликов

Плотникова

2021

Numerical Meth Engineering

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A mixed hexahedral solid‐shell finite element with self‐equilibrated isostatic assumed stresses for geometrically nonlinear problems

Liguori,

Zucco,

Madeo

2024

Numerical Meth Engineering

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Mixed Finite Elements (FEs) with assumed stresses and displacements provide many advantages in analysing shell structures. They ensure good results for coarse meshes and provide an accurate representation of the stress field. The shell FEs within the family designated by the acronym Mixed Isostatic Self‐equilibrated Stresses (MISS) have demonstrated high performance in linear and nonlinear problems thanks to a self‐equilibrated stress assumption. This article extends the MISS family by introducing an eight nodes solid‐shell FE for the analysis of geometrically nonlinear structures. The element, named MISS‐4S, features 24 displacement variables and an isostatic stress representation ruled by 18 parameters. The displacement field is described only by translations, eliminating the need for complex finite rotation treatments in large displacements problems. A total Lagrangian formulation is adopted with the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor. The numerical results concerning popular shell obstacle courses prove the accuracy and robustness of the proposed formulation when using regular or distorted meshes and demonstrate the absence of any locking phenomena. Finally, convergences for pointwise and energy quantities show the superior performance of MISS‐4S compared to other elements in the literature, highlighting that an isostatic and self‐equilibrated stress representation, already used in shell models, also gives advantages for solid‐shell FEs.

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Nonlinear analysis of laminated composite shells with piezoelectric patches under displacement‐dependent pressure loads

Kulikov,

Plotnikova

2024

Z Angew Math Mech

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In this paper, the three‐dimensional stress field in nonlinear laminated composite shells with piezoelectric patches subjected to displacement‐dependent loads is analyzed. To solve the problem, the geometrically exact (GeX) hybrid‐mixed four‐node solid‐shell element has been developed. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms are taken exactly at the element nodes. The laminated solid‐shell element formulation is based on the choice of an arbitrary number of sampling surfaces (SaS) parallel to the middle surface and located at the Chebyshev polynomial nodes within the layers, in order to introduce the displacements and electric potentials of these surfaces as unknown functions. The outer surfaces and interfaces are also included into a set of SaS. Due to analytical integration utilized to evaluate the tangent stiffness matrix, the proposed GeX piezoelectric solid‐shell element under nonconservative loading exhibits superior performance in the case of coarse meshes and allows the use of only one load step in all considered benchmark problems. Therefore, it can be recommended for the large deformation analysis of doubly curved laminated composite shells with piezoelectric patches, since the SaS formulation allows one to find three‐dimensional solutions of electroelasticity with a prescribed accuracy.

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Assessment of nonlinear exact geometry sampling surfaces solid‐shell elements and ANSYS solid elements for 3D stress analysis of piezoelectric shell structures

Cited by 10 publications

References 59 publications

Coupled thermoelectroelastic analysis of thick and thin laminated piezoelectric structures by exact geometry solid‐shell elements based on the sampling surfaces method

Coupled thermoelectroelastic analysis of thick and thin laminated piezoelectric structures by exact geometry solid‐shell elements based on the sampling surfaces method

A mixed hexahedral solid‐shell finite element with self‐equilibrated isostatic assumed stresses for geometrically nonlinear problems

Nonlinear analysis of laminated composite shells with piezoelectric patches under displacement‐dependent pressure loads

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