A first order hyperbolic framework for large strain computational solid dynamics. Part II: Total Lagrangian compressible, nearly incompressible and truly incompressible elasticity

AbstractStandard finite element formulation and implementation in solid dynamics at large strains usually relies upon and indicial-tensor Voigt notation to factorized the weighting functions and take advantage of the symmetric structure of the algebraic objects involved. In the present work, a novel component-free approach, where no reference to a basis, axes or components is made, implied or required, is adopted for the finite element formulation. Under this approach, the factorisation of the weighting function and also of the increment of the displacement field, can be performed by means of component-free operations avoiding both the use of any index notation and the subsequent reorganisation in matrix Voigt form. This new approach leads to a straightforward implementation of the formulation where only vectors and second order tensors in $${\mathbb {R}}^3$$ R 3 are required. The proposed formulation is as accurate as the standard Voigt based finite element method however is more efficient, concise, transparent and easy to implement.

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Versatile stabilized finite element formulations for nearly and fully incompressible solid mechanics

Computational Mechanics ◽

10.1007/s00466-019-01760-w ◽

2019 ◽

Vol 65 (1) ◽

pp. 193-215 ◽

Cited By ~ 2

Author(s):

Elias Karabelas ◽

Gundolf Haase ◽

Gernot Plank ◽

Christoph M. Augustin

Keyword(s):

Finite Element ◽

Solid Mechanics ◽

Large Strain ◽

Computationally Efficient ◽

Stabilized Finite Element ◽

Pressure Formulation ◽

Pressure Projection ◽

Bubble Functions ◽

Incompressible Elasticity ◽

Nearly Incompressible Elasticity

Abstract Computational formulations for large strain, polyconvex, nearly incompressible elasticity have been extensively studied, but research on enhancing solution schemes that offer better tradeoffs between accuracy, robustness, and computational efficiency remains to be highly relevant. In this paper, we present two methods to overcome locking phenomena, one based on a displacement-pressure formulation using a stable finite element pairing with bubble functions, and another one using a simple pressure-projection stabilized $$\mathbb {P}_1 - \mathbb {P}_1$$P1-P1 finite element pair. A key advantage is the versatility of the proposed methods: with minor adjustments they are applicable to all kinds of finite elements and generalize easily to transient dynamics. The proposed methods are compared to and verified with standard benchmarks previously reported in the literature. Benchmark results demonstrate that both approaches provide a robust and computationally efficient way of simulating nearly and fully incompressible materials.

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A Total Lagrangian upwind Smooth Particle Hydrodynamics algorithm for large strain explicit solid dynamics

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/j.cma.2018.09.033 ◽

2019 ◽

Vol 344 ◽

pp. 209-250 ◽

Cited By ~ 4

Author(s):

Chun Hean Lee ◽

Antonio J. Gil ◽

Ataollah Ghavamian ◽

Javier Bonet

Keyword(s):

Large Strain ◽

Smooth Particle Hydrodynamics ◽

Particle Hydrodynamics ◽

Solid Dynamics

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Onset of Nonlinearity in the Elastic Bending of Blocks

Journal of Applied Mechanics ◽

10.1115/1.4001282 ◽

2010 ◽

Vol 77 (6) ◽

Cited By ~ 29

Author(s):

M. Destrade ◽

M. D. Gilchrist ◽

J. G. Murphy

Keyword(s):

Nonlinear Elasticity ◽

Fourth Order ◽

Order Term ◽

Maclaurin Series ◽

Third Order ◽

Bending Angle ◽

First Order ◽

The Moment ◽

Incompressible Elasticity ◽

Nonlinear Shear

The classical flexure problem of nonlinear incompressible elasticity is revisited assuming that the bending angle suffered by the block is specified instead of the usual applied moment. The general moment-bending angle relationship is then obtained and is shown to be dependent on only one nondimensional parameter: the product of the aspect ratio of the block and the bending angle. A Maclaurin series expansion in this parameter is then found. The first-order term is proportional to μ, the shear modulus of linear elasticity; the second-order term is identically zero because the moment is an odd function of the angle; and the third-order term is proportional to μ(4β−1), where β is the nonlinear shear coefficient, involving third-order and fourth-order elasticity constants. It follows that bending experiments provide an alternative way of estimating this coefficient and the results of one such experiment are presented. In passing, the coefficients of Rivlin’s expansion in exact nonlinear elasticity are connected to those of Landau in weakly (fourth-order) nonlinear elasticity.

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A STABILISED TOTAL LAGRANGIAN CORRECTED SMOOTH PARTICLE HYDRODYNAMICS TECHNIQUE IN LARGE STRAIN EXPLICIT FAST SOLID DYNAMICS

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016) ◽

10.7712/100016.2408.10124 ◽

2016 ◽

Cited By ~ 1

Author(s):

Giorgio Greto ◽

Sivakumar Kulasegaram ◽

Chun H. Lee ◽

Antonio J. Gil ◽

Javier Bonet

Keyword(s):

Large Strain ◽

Smooth Particle Hydrodynamics ◽

Particle Hydrodynamics ◽

Solid Dynamics

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Complete Riemann Solvers for the Hyperbolic GPR Model of Continuum Mechanics

The Bulletin of Irkutsk State University Series Mathematics ◽

10.26516/1997-7670.2021.35.60 ◽

2021 ◽

Vol 35 ◽

pp. 60-72

Author(s):

U. Ariunaa ◽

◽

M. Dumbser ◽

Ts. Sarantuya ◽

◽

...

Keyword(s):

Continuum Mechanics ◽

Riemann Solver ◽

Governing Equations ◽

Overdetermined System ◽

Riemann Solvers ◽

First Order ◽

Conservative Schemes ◽

Solid Dynamics ◽

First Time ◽

Better Than

In this paper, complete Riemann solver of Osher-Solomon and the HLLEM Riemann solver for unified first order hyperbolic formulation of continuum mechanics, which describes both of fluid and solid dynamics, are presented. This is the first time that these types of Riemann solvers are applied to such a complex system of governing equations as the GPR model of continuum mechanics. The first order hyperbolic formulation of continuum mechanics recently proposed by Godunov S. K., Peshkov I. M. and Romenski E. I., further denoted as GPR model includes a hyperbolic formulation of heat conduction and an overdetermined system of PDE. Path-conservative schemes are essential in order to give a sense to the non-conservative terms in the weak solution framework since governing PDE system contains non-conservative products, too. New Riemann solvers are implemented and tested successfully, which means it certainly acts better than standard local Lax-Friedrichs-type or Rusanov-type Riemann solvers. Two simple computational examples are presented, but the obtained computational results clearly show that the complete Riemann solvers are less dissipative than the simple Rusanov method that was employed in previous work on the GPR model.

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