Extended framework of Hamilton’s principle for continuum dynamics

Kim, Jinkyu; Dargush, Gary F.; Ju, Young‐Kyu

doi:10.1016/j.ijsolstr.2013.06.015

Cited by 51 publications

(32 citation statements)

References 29 publications

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“…(7). However, this supplementary conceptual condition is necessary for proper use of initial conditions in the EHP (the interested reader is referred to [34,36]).…”

Section: Variational Formulation Of Duffing Equationmentioning

confidence: 99%

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Extended framework of Hamilton's principle applied to Duffing oscillation

Kim

Lee

Shin

2020

Applied Mathematics and Computation

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The paper begins with a novel variational formulation of Duffing equation using the extended framework of Hamilton's principle (EHP). This formulation properly accounts for initial conditions, and it recovers all the governing differential equations as its Euler-Lagrange equation. Thus, it provides elegant structure for the development of versatile temporal finite element methods. Herein, the simplest temporal finite element method is presented by adopting linear temporal shape functions. Numerical examples are included to verify and investigate performance of non-iterative algorithm in the developed method.

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“…(7). However, this supplementary conceptual condition is necessary for proper use of initial conditions in the EHP (the interested reader is referred to [34,36]).…”

Section: Variational Formulation Of Duffing Equationmentioning

confidence: 99%

“…Eqs. (33)-(34) to have a real number solution, respectively. However, it would be problematic if Eq.…”

mentioning

confidence: 99%

Extended framework of Hamilton's principle applied to Duffing oscillation

Kim

Lee

Shin

2020

Applied Mathematics and Computation

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“…The governing equations of transient brittle fracture for a phase‐field crack approximation originate from a Hamiltonian principle. The general procedure involves the Lagrangian energy density

scriptL

scriptL

specifies the amount of total energy density at a material point for time t n and contains energetic contributions of inertia, strain potential, and fracture energy density.…”

Section: Phase‐field Model For Quasi‐viscous Fracturementioning

confidence: 99%

“…The general procedure involves the Lagrangian energy density . 37  specifies the amount of total energy density at a material point for time t n and contains energetic contributions of inertia, strain potential, and fracture energy density. The variation functional generally reads…”

Section: Governing Equations For a Phase-field Model Considering Tranmentioning

confidence: 99%

Formulation and implementation of strain rate‐dependent fracture toughness in context of the phase‐field method

Yin

Steinke

Kaliske

2019

Numerical Meth Engineering

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Summary The phase‐field approach is a promising technique for the realistic simulation of brittle fracture processes, both in quasi‐static and transient analysis. Considering fast loading, experimental evidence indicates a strong relationship between the rate of strain and the material's resistance against fracture, which can be considered by a dynamic increase factor for the strength of the material. The paper at hand presents a novel approach within the framework of phase‐field models for brittle fracture. A rate‐dependent fracture toughness is formulated as a function of the rate of crack driving strain components, which results in higher strength for faster loading. Beside the increased amount of energy necessary to evolve a crack at a high strain rate loading situation, the model incorporates quasi‐viscous stress‐type quantities that are not directly related to the formation of the crack and exist only in the phase‐field transition zone between broken and sound material. The governing strong form equations for a transient simulation are derived and the relevant information for an implementation of the model into a finite element code is outlined in detail. The performance of the model is demonstrated for static and dynamic benchmark simulations and for a comparison to experimental findings.

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“…The concept of this method was originally introduced by [4] and [24]. Nevertheless it was ascertained later that the applicability of this variational principle formulated in the previous paragraph to time finite element modeling is restricted in that form because only one type of initial conditions can be treated appropriately [35]. Therefore this had to be extended from the viewpoint of the numerical utility.…”

Section: Introductionmentioning

confidence: 99%

Multi-field Dual-Mixed Variational Principles Using Non-symmetric Stress Field in Linear Elastodynamics

Tóth

2015

J Elast

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Three types of multi-field dual-mixed variational principles using a priori non-symmetric stress field will be derived for dynamic problems of linearly elastic solids. Starting from the functional of complementary Hamilton's principle it is shown how this variational formulation is modified with applying the Lagrange multiplier technique, to obtain a new four-field dual-mixed functional. The independent fields of this functional will be the displacement vector, the non-symmetric stress tensor, the skew-symmetric rotation tensor and the momentum vector. Then we present how to treat appropriately the six different types of initial conditions in a weak and/or strong sense by adding a new integral expression to the four-field functional. Modifying this functional through a Legendre transformation results in a new complementary energy-based five-field dual-mixed functional which allows the momentum-and velocity vectors to be independently approximated. A complementary energy-based three-field dual-mixed variational formulation is obtained by eliminating the momentum-and velocity fields through the strong enforcement of the kinematic equation and the impulse-velocity relation from the five-field principle.

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Extended framework of Hamilton’s principle for continuum dynamics

Cited by 51 publications

References 29 publications

Extended framework of Hamilton's principle applied to Duffing oscillation

Extended framework of Hamilton's principle applied to Duffing oscillation

Formulation and implementation of strain rate‐dependent fracture toughness in context of the phase‐field method

Multi-field Dual-Mixed Variational Principles Using Non-symmetric Stress Field in Linear Elastodynamics

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