By Jože Korelc, Peter Wriggers
New finite parts are wanted in addition in study as in environments for thedevelopment of digital prediction innovations. The layout and implementation of novel finiteelements for particular reasons is a tedious and time eating job, in particular for nonlinearformulations. The automation of this method may help to hurry up this processconsiderably because the iteration of the ultimate computing device code should be sped up by means of order ofseveral magnitudes.This ebook presents the reader with the mandatory wisdom had to hire modernautomatic instruments like AceGen inside of sturdy mechanics in a profitable approach. It covers the rangefrom the theoretical heritage, algorithmic remedies to many various functions. Thebook is written for complicated scholars within the engineering box and for researchers ineducational and commercial environments.
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90) denotes the internal virtual work also called stress divergence term. The last two terms describe the virtual work of the applied loading and the inertia term. 87) is advantageous in the automated finite element method. From P · Grad η = P · δF = P · ∂F δϕ ∂ϕ 22 1 Basic Equations of Continuum Mechanics ∂W the equivalent form follows with the hyperelastic constitutive equation5 P = ∂F for the stress divergence term P · Grad η = ∂W ∂W ∂W ∂F · δϕ = · δϕ = · η. 92) ∂Bσ B This relation simplifies automatic differentiation since only one differentiation of the strain energy function with respect to all displacement variables is needed instead of two differentiations, first to obtain the stress tensor P and then to get the variation of the deformation gradient Grad η, see Sect.
2 Balance Equations 17 Stress rates. The time derivative of stress tensors is of significance for the statement of incremental forms of constitutive equations. g. the second Piola–Kirchhoff stress tensor S) the derivative with respect to time is given by the material time derivative ∂S(X, t) S˙ = . 35) σ˙ = σ ∂σ + grad σ v. g. Truesdell and Toupin (1960), that the material time derivative of the Cauchy stress tensor is not objective, but objectivity is an inevitable prerequisite for the formulation of constitutive equations.
Truesdell and Toupin (1960), that the material time derivative of the Cauchy stress tensor is not objective, but objectivity is an inevitable prerequisite for the formulation of constitutive equations. Hence numerous time derivatives were formulated—so called objective time derivatives—which can be applied to compute stress rates. g. Truesdell and Toupin (1960) or Marsden and Hughes (1983). 44) as Lv τ = F ∂ [ F−1 τ F−T ] ∂t FT . 37). g. Marsden and Hughes (1983). It is equivalent to the Lie derivative of the Kirchhoff stress tensor.
Automation of Finite Element Methods by Jože Korelc, Peter Wriggers