2012년 2학기 강의노트 비선형유한요소 Chapter 4 Continuum Mechanics Incremental Total and Updated Lagrangian Formulations

Basic assumption & concept The solution for time 0, , , … have already been calculated. Hence, in principle, any one of the equilibrium configurations already calculated could be used. ◦ Total Lagrange formulation   All static and kinematic variables are referred to the initial configuration at time 0 ◦ Updated Lagrange formulation   All static and kinematic variables are referred to the last calculated configuration.

Objective of Linearization 선형화를 하는 목적은 다음과 같은 최종적인 형태의 식을 얻기 위함이다. (2.2)식의 형태는 의 상태를 모르기 때문에 미지의 Volume 에 대하여 적분을 할 수 없고, Cauchy stress의 특성상 시간 t 의 값에 증분량을 더할 수가 없기 때문에 쉽게 선형화를 할 수 없다. 참고로 (2.2) 식은 다음과 같으며 시간 의 모든 부분에서 다음의 관계를 만족한다. - Equilibrium   - Compatibility   - The stress-strain law 선형화를 수행하기 위해 기지(旣知)의 reference configuration을 사용하고, energetically conjugate 인 Second Piola-Kirchhoff Stress와 Green-Lagrange strain을 사용하여 (2.2)식을 표현한다.  (2.2), (2.3)으로부터 다음과 같은 기본식을 얻을 수 있으며 (2.2)식과 (4.4), (4.5)식은 완전히 같은 식이다. 

Total and Updated Lagrangian formulation ⅰ) Total Lagrangian ( 4.4 ) ⅱ) Updated Lagrangian ( 4.5 ) 여기서 Total Lagrangian formulation의 선형화 과정은 다음과 같다.  우선 2nd Piola-Kirchhoff Stress 와 Green-Lagrangian strain 는 다음과 같이 증분의 형태로 쓰여질 주 있다.

Linearization of Total Lagrangian formulation ( 4.6 ) : : ( 4.7 ) Known Unknown increments Linear in Nonlinear In 여기서

Linearization of Total Lagrangian formulation Green-Lagrange strain의 variation, 을 증분의 형태로 쓰면 다음과 같다. ( 4.6 ) Make sense physically, because each variation is taken on the d2isplacements at time , with fixed.

Linearization of Total Lagrangian formulation   (4.8), (4.7), (4.6)식을 (4.4)에 대입하여, 정리하면 ( 4.9 ) Highly nonlinear linear known 여기서, is linear in dose not contain is a nonlinear function ( in general ) of is a linear function of Given a variation , the right-handside is known ( is a constant ).  The left-handside contains unknown displacements increments.  All we have done so far is to write the principle of virtual work in terms of and .  The equation of the principle of virtual work is in general a complicated nonlinear function in the unknown displacement increment.  We obtain an approximate equation by neglecting all higher-order terms in (so that only linear terms in remain).  This leads to

Linearization of Total Lagrangian formulation   This process of neglecting  higher order terms is called linearization.   (4.9)식에서 highly nonlinear term인 을 Taylor series를 이용해서 정리하여 선형화 작업을 하면 다음과 같다. linear quadratic known in in   위의 식에서 higher order terms를 무시하고, 와 비교하여 아주 작은 값인 를 무시한다. ( 4.10 ) 여기서,

Linearization of Total Lagrangian formulation ( 4.11 ) when discretized using the finite element method

Linearization of Total Lagrangian formulation 증분형태의 Updated Lagrangian formulation, (4.12), 도 Total Lagrangian의 과정과 같은 방법으로 얻 을 수 있다. ( 4.12 ) An important point is that   because So, we may interpret the right-hand side of (4.11) and (4.12) as an "out-of-valance" virtual work term. ※ Mathematical explanation that   We had If , then the configuration at time is identical to the configuration at time . Hence

Linearization of Updated Lagrangian formulation   It follows that This results makes physical sense because equilibrium was assumed to be satisfied at time .  Hence we can write   We may rewrite the linearized governing equation (4.11) as follows When the linearized governing equation is discretized, we obtain

Linearization of Updated Lagrangian formulation We then use (4.11), (4.12) 관계식은 변위의 증가치를 계산하는데 쓰여지며, 계산된 변위 증가치 는 시간 에서 변위, 변형율, 응력의 근사값을 계산하는데 사용된다. (The relation in (4.11), (4.12) can be employed to calculate an increment in the displacements, which then is used to evaluate approximations to the displacements, strains, and stresses corresponding to time .) 계산된 변위, 응력, 변형율의 근사값을 이용하여 에러, 즉, out of valance virtual work, the difference between internal and external virtual work를 구하면 다음과 같다.   in TL formulation ( 4.13 ) in UL formulation ( 4.14 )

Formulation of Finite Element Matrices   (4.11), (4.12)식을 iteration을 포함한 형태로 쓰면 다음과 같다.   in Total Lagrangian formulation ( 4.15 )   where we used   with initial conditions , ,   in Update Lagrangian formulation ( 4.16 )

Formulation of Finite Element Matrices   where we use   with initial conditions , , ※ Comparison of TL and UL formulations - In the TL formulation, all derivatives are with respect to the initial coordinates whereas in the UL formulation, all derivatives are with respect to the current coordinates. - In the UL formulation we work with the actual physical stresses (Cauchy Stress).

Formulation of Finite Element Matrices   Assuming that the loading is deformation-independent, ( 4.17 )   For a dynamic analysis, the inertia force loading term is ( 4.18 )   If the external loads are deformation-dependent and time step is small enough to have good accuracy ( 4.19 )   and ( 4.20 )

Formulation of Finite Element Matrices   Materially-nonlinear-only analysis : ( 4.21 )   This equation is obtained from the governing TL and UL equations by realizing that, neglecting geometric nonlinearities. ( physical stress )   Implicit time integration ( 4.22 ) Total Lagrange ( 4.23 ) Update Lagrange ( 4.24 ) Material Nonlinear Only ( 4.25 )

Formulation of Finite Element Matrices   The finite element equations corresponding to the continuum mechanics equations are   Materially-Nonlinear-Only analysis Static analysis : ( 4.26 ) Dynamic analysis, implicit time integration : ( 4.27 ) Dynamic analysis, explicit time integration : ( 4.28 )   Total Lagrangian formulation Static analysis : ( 4.29 )

Formulation of Finite Element Matrices Dynamic analysis, implicit time integration : ( 4.30 ) Dynamic analysis, explicit time integration : ( 4.31 ) Updated Lagrangian formulation Static analysis : ( 4.29 ) Dynamic analysis, implicit time integration : ( 4.30 ) Dynamic analysis, explicit time integration : ( 4.31 )

Formulation of Finite Element Matrices The above expression are valid for ∙ a single finite element ( contains the element nodal point displacements ) ∙ a single finite element ( contains all nodal point displacements ) In practice, element matrices are calculated and then assembled into the global matrices using the direct stiffness method.

Formulation of Finite Element Matrices We now concentrate on a single element.   The vector contains the element incremental nodal point displacements. We may write the displacements at any point in the element in terms of the element nodal displacements. ( 4.32 ) Finite element discretization of governing continuum mechanics equations: For all analysis types ( 4.33 ) where we used , is displacements at a point within the element

Formulation of Finite Element Matrices and ( 4.34 ) where Materially-nonlinear-only analysis : Considering an incremental displacements ( 4.35 )

Formulation of Finite Element Matrices where and ( 4.36 ) Total Lagrangian formulation : Considering an incremental displacements ( 4.37 ) where ( 4.38 )

Formulation of Finite Element Matrices where is a matrix containing components of contains components of and ( 4.39 ) where is a matrix containing components of Updated Lagrangian formulation : Considering an incremental displacements ( 4.40 ) where is a matrix containing components of contains components of ( 4.41 )

Formulation of Finite Element Matrices where is a matrix containing components of contains components of and ( 4.42 ) where is a matrix containing components of ※ The finite element stiffness and mass matrices and force vectors are evaluated using numerical integration (as in linear analysis).  In isoparametric finite element analysis we have, schematically, in 2-D analysis ( 4.43 )

Formulation of Finite Element Matrices ( 4.44 ) ( 4.45 )