An Eulerian model for nonlinear waves in elastic rods, solved numerically by the CESE method

Lowe, Robert L.; Lin, Po-Hsien; Yu, Sheng‐Tao; Bechtel, Stephen E.

doi:10.1016/j.ijsolstr.2016.04.016

Cited by 6 publications

(3 citation statements)

References 91 publications

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“…Moreover, Chen et al [27] performed CESE simulations to study planar-wave expansion from a point source in an anisotropic solid of cubic symmetry (gallium arsenide) and Yang et al [28] extended the CESE simulations to stress waves in solids of hexagonal symmetry, illustrating wave propagation in a heterogeneous solid composed of three blocks of beryl with different lattice orientations. Lately, Lowe et al [29] conducted a comprehensive study of nonlinear longitudinal waves (including both weak and strong shocks, rarefactions, and contact discontinuities) in tapered elastic rods using the CESE method as a numerical tool. In the above studies, the CESE simulations effectively captured waves in a variety of solid materials and provided results consistent with the available analytical solutions.…”

Section: Solid Dynamicsmentioning

confidence: 99%

Other Applications

Wen

Jiang

Shi

2023

Engineering Applications of Computational Methods

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The CESE method has been applied to a wide range of scientific and engineering problems since its inception in the 1990s. Although solving CFD problem is the primary goal of the CESE method, this general approach is actually applicable to a variety of PDE systems with physical backgrounds different from fluid dynamics. This chapter mainly introduces the application highlights of CESE in several representative fields.

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Section: Solid Dynamicsmentioning

confidence: 99%

Other Applications

Wen

Jiang

Shi

2023

Engineering Applications of Computational Methods

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“…So did the CESE method. it has been employed to solve many dynamical and vibration problems in solid structures, for instance, [51][52][53][54][55][56][57][58]. Since the application we are interested in is ultrasonic welding process, thermo-effect is ignored which was demonstrated by experiments.…”

Section: Introductionmentioning

confidence: 99%

Simulation of Wave in Hypo-Elastic-Plastic Solids Modeled by Eulerian Conservation Laws

Yang¹,

Lowe²

2023

Preprint

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This paper reports a theoretical and numerical framework to model nonlinear waves in elastic-plastic solids. Formulated in the Eulerian frame, the governing equations employed include the continuity equation, the momentum equation, and an elastic-plastic constitutive relation. The complete governing equations are a set of first-order, fully coupled partial differential equations with source terms. The primary unknowns are velocities and deviatoric stresses. By casting the governing equations into a vector-matrix form, we derive the eigenvalues of the Jacobian matrix to show the wave speeds. The eigenvalues are also used to calculate the Courant number for numerical stability. The model equations are solved using the Space-Time Conservation Element and Solution Element (CESE) method. The approach is validated by comparing our numerical results to an analytical solution for the special case of longitudinal wave motion.

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“…Hamilton's principle accommodates Eulerian kinematics (e.g., Irschik and Holl, 2002) and is the overwhelmingly popular choice for deriving the equations of motion and boundary conditions for rotating rings and other convecting continua; see, for instance, Archibald and Emslie (1958) for the application of Hamilton's principle to axially moving strings. Conversely, Newtonian approaches employing an Eulerian description of the kinematics and kinetics (e.g., Lowe et al., 2016) – the standard perspective in fluid mechanics – are less common in solid mechanics, particularly for modeling the vibration of translating and rotating systems. Notable exceptions include the work of Vu-Quoc and Li (1995) and Vetyukov (2018), who employ Eulerian or hybrid Eulerian–Lagrangian kinematics to investigate the dynamics of translating strings and beams.…”

Section: Introductionmentioning

confidence: 99%

A Newtonian mechanics formulation for the vibration of translating and rotating elastic continua

Lowe

Cooley

2019

Journal of Vibration and Control

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In this paper, we present a Newtonian mechanics formulation for modeling the vibration of a convecting elastic continuum, i.e., a system characterized by mean kinematic translation or rotation with small superposed vibrations. The proposed Newtonian approach complements customary energy-based techniques and serves as a convenient means to validate and physically interpret their results. We develop the equations of motion and matching conditions in a continuum mechanics setting with respect to a stationary inertial reference frame. Interaction of the convecting continuum with discrete space-fixed elements (e.g., springs and dampers) is enabled, without introducing time-dependent coefficients, through the use of Eulerian kinematics and kinetics. Kinematic discontinuities inherent in these interactions are accommodated by employing a global (or integral) form of balance of linear momentum applied to a space-fixed control volume. A generalized form of Navier's equation of elastic wave propagation is derived, with unsteady, Coriolis, centripetal, and convective contributions to the inertia. The resulting formulation is applied to a broad class of translating and rotating systems – including spinning rings, axially moving strings and beams, and general three-dimensional elastic structures – and shown to successfully reconcile with existing energy-based derivations in the literature.

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An Eulerian model for nonlinear waves in elastic rods, solved numerically by the CESE method

Cited by 6 publications

References 91 publications

Other Applications

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Simulation of Wave in Hypo-Elastic-Plastic Solids Modeled by Eulerian Conservation Laws

A Newtonian mechanics formulation for the vibration of translating and rotating elastic continua

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