Higher order hierarchical plates and curved shell elements based on p-approximation in the thickness direction

Surana, Karan S.; Sorem, Robert M.

doi:10.1016/0895-7177(90)90302-4

Cited by 6 publications

(8 citation statements)

References 15 publications

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“…We clearly see that enhancement of mathematical model used in reference [39] [40] to (96) -(100) using energy methods or principle of virtual work is not always possible (as shown here) due to limitations of energy methods and principle of virtual work. 4) In this paper, CPT and FSDT mathematical models for BVPs are used as representative plate/shell mathematical models that are based on reversible mechanical small deformation without thermal effects.…”

Section: Final Mathematical Model (Thermoelastic)mentioning

confidence: 89%

“…8) It is important to point out that the formulation of p-version hierarchical 0 C plate/shell finite element formulation (for reversible deformation without temperature effects and linear constitutive theory) was originally derived by Surana et al [39] [40] using principal of virtual work. Exactly similar derivation as in ref [39] [40] is also possible using energy method. This approach was also kinematic assumption free, but has the following major shortcomings and limitations.…”

Section: )mentioning

confidence: 99%

“…This approach was also kinematic assumption free, but has the following major shortcomings and limitations. a) Principle of virtual work as used in reference [39] can only be used for reversible mechanical deformation, hence cannot be extended to include physics of dissipation and memory mechanisms.…”

Section: )mentioning

confidence: 99%

“…c) Due to absence of energy equation, rate of work conjugate pairs needed for constitute theories cannot be established. The energy methods and principle of virtual work used in ref [39] [40] assumes a constitutive theory as there is no mechanism to derive it.…”

Section: )mentioning

confidence: 99%

“…Secondly, consideration of thermal effects requires consideration of energy equation (98) which is not possible in energy methods or principle of virtual work. Nonetheless, if we limit the physics to conform to what has been considered in reference [39] [40], then the mathematical model (96) -(100) obviously reduces to (101) and (102).…”

Section: Final Mathematical Model (Thermoelastic)mentioning

confidence: 99%

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Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Surana¹,

Mathi²

2020

AJCM

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Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler's equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic as

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Section: Final Mathematical Model (Thermoelastic)mentioning

confidence: 89%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Final Mathematical Model (Thermoelastic)mentioning

confidence: 99%

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Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Surana¹,

Mathi²

2020

AJCM

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p‐version hierarchical three dimensional curved shell element for elastostatics

Surana

Sorem

1991

Numerical Meth Engineering

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SUMMARYThis paper presents a hierarchical three dimensional curved shell finite element formulation based on the p-approximation concept. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell (c, q) and the transverse direction (i). The curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders pr, pq and p s and the corresponding nodal variable operators for each of the three directions (, q and [ and then taking their products (sometimes also known as tensor product). This procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders p s , p,, and p<. Both the element displacement functions and the nodal variables are hierarchical; therefore, the resulting element matrices and the equivalent nodal load vectors are hierarchical also, i.e. the element properties corresponding to the polynomial orders pg, p, and pC are a subset of those corresponding to the orders (ps + l), (p, + 1) and (pc i-1). The formulation guarantees Co continuity or smoothness of the displacement field across the interelement boundaries.The geometry of the element is described by the co-ordinates of the nodes on its middle surface ([ = 0) and the nodal vectors describing its bottom (i = -1) and top (i = + 1) surfaces. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is equally effective for very thin as well as very thick plates and curved shells. In fact, in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as analytical solutions.

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Completely hierarchical p-version curved shell element for laminated composite plates and shells

Surana

Sorem

1991

Computational Mechanics

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Higher order hierarchical plates and curved shell elements based on p-approximation in the thickness direction

Cited by 6 publications

References 15 publications

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

p‐version hierarchical three dimensional curved shell element for elastostatics

Completely hierarchical p-version curved shell element for laminated composite plates and shells

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