Abstract
Standard 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 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.
Highlights
It is well known that equations in the field of Continuum Mechanics such as conservation of mass, conservation of momentum, constitutive relations, etc., might be formulated in a component-free1 E.T.S.I
It is well recognised that such a component-free notation is a convenient and concise tool to manipulate most of the relations in Continuum Mechanics
The performance of the component-free approach for large strain isothermal elastodynamics is illustrated by five numerical examples, one in 1D, three in 2D and one in 3D
Summary
It is well known that equations in the field of Continuum Mechanics such as conservation of mass, conservation of momentum, constitutive relations, etc., might be formulated in a component-free (or intrinsic or symbolic or invariant). The authors propose a completely different approach, as compared to the standard Voigt’s notation and to the indicial-tensor notation derived by Gupta and Mohraz [12], to perform the finite element formulation and implementation for isothermal elastodynamics at large strain in the initial configuration. 6, the proposed elastodynamic component-free Lagrangian finite element formulation is considered to reproduce five different benchmarks including a Riemann problem, a plane strain Cook’s membrane under a dynamic load, a plane strain solid block with low stiffness under the action of gravity, a large deformation vibration of a cantilever beam under its own weight and a pinched thick-walled cylinder.
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