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Comparing Two Algorithms to Add Large Strains to a Small Strain Finite Element Code

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Comparing Two Algorithms to Add Large Strains to a Small Strain Finite Element Code INTERNATIONAL CENTER FOR NUMERICAL METHODS IN ENGINEERING Comparing Two Algorithms to Add Large Strains to a Small Strain Finite Element Code A. Rodríguez Ferran B. A. Huerta Publication CIMNE Nº-91, November 1996 COMPARING TWO ALGORITHMS TO ADD LARGE STRAINS TO A SMALL STRAIN FINITE ELEMENT CODE y Antonio Ro drguezFerran and Antonio Huerta Member ASCE 1 Research Assistant Departamento de MatematicaAplicada I I I ETS de Ingenieros de Caminos Universitat Politecnicade Catalunya Campus Nord C E Barcelona Spain 2 Professor Departamento de MatematicaAplicada I I I ETS de Ingenieros de Caminos Universitat Politecnicade Catalunya Campus Nord C E Barcelona Spain y Corresp onding author email huertaetseccpbupces ABSTRACT Two algorithms for the stress update ie timeintegration of the constitutive equation in large strain solid mechanics are discussed with particular emphasis on two issues the in cremental objectivity and the implementation aspects It is shown that both algorithms are incrementally objective ie they treat rigid rotations properly and that they can be employed to add large strain capabilities to a smal l strain nite element code in a simple way Aset of benchmark tests consisting of simple large deformation paths rigid rotation simple shear extension extension and compression dilatation extension and rotation have been usedto test and compare the two algorithms both for elastic and plastic analysis These tests evidence dierent time integration accuracy for each algorithm However it is demonstrated that the in general less accurate algorithm gives exact results for shearfree deformation paths Keywords nonlinear computational mechanics large strain solid mechanics stress up date algorithms nite elementcodedevelopment large strain b enchmarks i INTRODUCTION Many problems of interest in physics and engineering are nonlinear Focusing on solid mechanics two basic typ es of nonlinearity are encountered material and geometric Material nonlinearity refers to the plastic or more generally inelastic b ehavior shown by many engineering materials such as metals In some pro cesses moreover the solid undergo es such large deformations that the variation in shap e may not b e neglected as in a standard linear computation thus resulting in geometric nonlinearity The two kinds of nonlinearity invalidate in many cases a classical linear elastic analysis From the viewp ointofcontinuum mechanics a convenientway to describ e large strains in a solid is by means of a convected frame whichisattached to the b o dy and deforms with it Since the convected frame follows the b o dy motion it allows for a simple statementand handling of the constitutive equations In nonlinear computational mechanics however the standard approach is to use a xed Cartesian frame like in the linear case This leads to a simpler description of motion b ecause the frame do es not change but on the other hand the treatment of constitutivelaws b ecomes more involved b ecause the frame do es not follow the material Nonlinear material b ehavior is often describ ed by a rateform constitutive equation re lating some measure of the rate of deformation to a rate of stress In a largestrain context the choice of a prop er stress rate is a key p oint b ecause the principle of ob jectivity should b e resp ected the constitutive equation must b e indep endent of the observer This is only achieved when ob jectivequantities are employed It can b e shown that the material derivative of stress is not an ob jective tensor and therefore an alternative ob jective stress rate is needed The ob jectivity of the constitutive equations should b e resp ected by the numerical algorithms employed for their timeintegration This requirement is referred to as incremental ob jectivity Various stress up date algorithms ie algorithms for the numerical timeintegration of the constitutive equations can b e found in the literature see for instance In many cases however employing these algorithms to add largestrain capabilities to a smallstrain FE co de is a cumbersome task b ecause new quantities not employed for a small strain analysis must b e computed This pap er discusses two incrementally ob jective algorithms that allow to transform an existing smallstrain FE co de into a largestrain co de in a simple way Only the par ticular case in which the elastic part of the deformation is mo deled byanhyp o elastic lawa common choice in nonlinear computational mechanics will b e addressed here The rst algo rithm uses the full Lagrange strain tensor including quadratic terms to account for large strains The second algorithm presented in employs the same strain tensor as in a smallstrain analysis but computed in the midstep conguration Various implementation asp ects for the two algorithms are commented It is shown in particular that very few additional features must b e added to a co de with smallstrain and nonlinear material b ehavior to enable its use for largestrain analysis The two algorithms are tested and compared with the help of a set of b enchmark tests consisting of simple deformation paths rigid rotation simple shear uniaxial extension ex tension and compression extension and rotation Moreover it is demonstrated that for some particular deformation paths the in general less accurate algorithm captures the exact solution This pap er is organized as follows Some preliminaries including the basic equations of large strain solid mechanics and the concept of ob jectivity are briey reviewed in Section The two stress up date algorithms are presented in Section After some introductory remarks in Section the notion of incremental ob jectivity is reviewed in Section The two algo rithms are then shown in Section and their implementation in a smallstrain FE co de is discussed in Section Section deals with the numerical examples Finally some concluding remarks are made in Section PRELIMINARIES Basic equations The rst ingredient of continuum mechanics is the equation of motion x xX t which yields the p osition x of material particles denoted by their material co ordinates X at time t If the initial spatial co ordinates are employed as material co ordinates the material displacements can b e dened as ut xt X Once the displacements are dened the kine matical description continues with strain representation The starting p oint is the deformation gradient F x F X Various strain tensors may b e dened by means of F The Lagrange strain tensor for instance is T E F F I where T means transp ose and I is the identity Another tensor representing strain is the spatial gradientofvelo city l This tensor yields relevant tensors if decomp osed into symmetric part rateofdeformation tensor d and skewsymmetric part spin tensor v l d x Avery common simplication in solid mechanics is that of small deformations If dis placements rotations and strains are small enough two imp ortantpoints follow i the relation between displacements and strain is linear and ii the initial conguration of the b o dy can 0 b e used to solvethegoverning equations Because of this a geometrically linear problem results In some other problems on the contrary displacements are large when compared to the initial dimensions of the b o dy The relation b etween displacements and strains is no longer linear and moreover the governing equations must b e solved over the current conguration t at time t not over Since the motion that transforms into is precisely the fundamental t 0 0 unknown a geometrically nonlinear problem is obtained The balance laws of continuum mechanics state the conservation of mass momentum and energy For a wide range of problems in solid mechanics three simplifying assumptions are common i mechanical and thermal eects are uncoupled ii the density is constantandiii inertia forces are negligible in comparison to the other forces acting on the b o dy quasistatic pro cess The mechanical problem is then governed by the momentum balance alone which b ecomes a static equilibrium equation ij b i x j is the Cauchy stress tensor and b is the force p er unit volume Equation mo dels where many problems of practical interest including for instance various forming pro cesses see Stress tensors The most common representation of stress is the Cauchy stress tensor dened in the current conguration and already presented in Eq This tensor has a clear physical t meaning b ecause it involves only forces and surfaces in the current conguration Exp erimen tal stress measures taken in a lab oratory corresp ond to Cauchy stresses also known as true stresses In a large strain context other representations of stress are p ossible and indeed useful The key idea is that and are dierent congurations so tensors dened in each t 0 0 conguration cannot b e combined by op erations such as subtraction and addition Let and t b e the Cauchy stress tensors at the initial time t and current time t resp ectively the incre 0 t 0 ment of stress may not b e dened as b ecause the two tensors are referred to dierent congurations As stress increments will b e needed to up date stresses a prop er denition is required An alternative representation of stress is the second PiolaKirchho tensor S denedas the pullbackof 1 T S J F F where J detF is the Jacobian of the motion which reects the variation of unit volume 1 asso ciated to the deformation and the inverse of the deformation gradient F is employed to transform from to see Figure a Equation is called the pullback
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