Total Lagrangian Continuum Formulation

In this section, we will describe the finite element procedure undertaken in the present study and highlight the important aspects of the beam element used in the computational model. The following presentation closely follows the works of Belytschko et al. [104] and Crisfield [98].

The partial differential equations governing the motion of the beam and the boundary conditions are collectively referred to as the strong form of the

problem. Finite element discretization of the strong form is not possible and therefore the weak form, which is an integral expression of the governing equa- tion and the boundary conditions, is needed. The weak form is equivalent to the strong form and requires “less smooth” (C0 _{vs. C}2 _{continuous) solution}

functions. In solid mechanics, it is also called principle of virtual work. In order to obtain the weak form, the governing differential equation is multi- plied by an arbitrary function called “test (weight) function” and integrated over the domain. The test function is required to vanish on the prescribed displacement boundary (essential boundary condition). For the solution of weak form, a set of smooth functions called “trial functions” are considered. The trial function satisfying the essential boundary condition is the solution of the weak form [105].

The continuum-based (CB) beam element is formulated in a total La- grangian framework. Accordingly, Green strain E and second Piola-Kirchhoff (PK2) stress S are used as strain and stress measures, and the motion of the element is described with respect to initial (undeformed) configuration. We now summarize important concepts pertaining to a continuum finite element which will be subsequently used in the formulation of CB beam element. The equation of motion (i.e., conservation of linear momentum) of a body (con- tinuum) in the undeformed configuration can be expressed as (p.550 [106], p.194 [104]):

∇0· (S · FT) + ρb = ρ

D2_u

Dt2, (3.2.1)

where ∇0 is the gradient operator with respect to initial (material) coordinates

X, S is the second Piola-Kirchhoff stress tensor, F is the deformation gradient tensor, ρ is the density, b is the vector of body forces per unit mass, u is

the vector of displacements (x = X + u), and D()_Dt denotes the material time derivative. The deformation gradient is defined as F = ∂x

∂X and it relates

current configuration x to initial configuration X. F is related to the Green strain according to E = 1₂(FT _{· F − I), where I is the identity matrix. Note}

that while the numerical simulations include linear viscous damping, and the numerical implementation of this will be discussed later in this section, Eq. (3.2.1) does not include a viscous damping term.

In order to obtain the weak form, we multiply Eq. (3.2.1) by test function (i.e., virtual displacement) δu and integrate over the initial domain Ω0 of the

body, to obtain:

δWint_{− δW}ext+ δWkin = 0, (3.2.2) where δWint_{, δW}ext_{, δW}kin _{are the virtual works associated with internal,}

external, and inertial forces, respectively, and are defined as (p.197 [104], p.108 [107]): δWint= Z Ω0 S : δE dΩ0, (3.2.3a) δWext = Z Ω0 ρδu · b dΩ0+ nSD X i=1 Z Γ0 ti (δu · ei)(ei· ¯t0i) dΓ0, (3.2.3b) δWkin= Z Ω0 δu · ρ¨u dΩ0. (3.2.3c)

In Eqs. (3.2.3)a and b, the symbol “:” denotes double contraction,† _n

SD stands

for number of space dimensions, Γ0

ti denotes initial boundary over which trac-

tions are prescribed, ¯t0

i represents prescribed tractions, ei denotes the unit

normal of the boundary over which the traction is prescribed, and Γ0 is the

initial boundary. Equation (3.2.2) constitutes the weak form equivalent of the

†_{If A and B are second-order symmetric tensors, A : B = tr(A}T

B) = P3

i,j=1aijbij,

momentum equation in a total Lagrangian frame.

We can now perform finite element discretization which amounts to dis- cretizing the domain Ω0 into a set of subregions (elements) connected appro-

priately at their nodes and then approximating the unknown displacement field u of the element domain in terms of the unknown nodal displacements uI

at these nodes by using interpolation functions NI (called shape functions in

the finite element literature). For the 4-node quadrilateral continuum element shown in Figure 3.2, nodes nI are denoted by 1−, 2−, 2+, 1+; thus, I = 1−,

2−_{, 2}+_{, 1}+_{. The finite element approximation of the trial and test functions}

are given as:

u = uI(t)NI, (3.2.4a)

δu = δuINI, (3.2.4b)

where summation over the range of repeated index is implied. For the pur- pose of describing local approximation over each element, the elements can be considered disjoint (see pp. 73-77, [108]) and hence we can focus our develop- ment to a typical element with domain Ωe

0. In what follows we will drop the

superscript e but it should be understood that all integrals are over an element domain.

At this point we define element nodal forces. Accordingly, the virtual works done by the element nodal forces in moving through virtual nodal displace- ments δuI are expressed as:

δWint _{= δu}T

IfIint, (3.2.5a)

δWkin = δuTIfIkin, (3.2.5c)

where fint

I , fIext, and fIkin are internal, external, and inertial nodal forces, re-

spectively.

Substituting Eqs. (3.2.5) into Eq. (3.2.2) and enforcing the arbitrariness of the test functions δuI yields the semi-discretized (in space) equations of

motion at the element level:

MIJu¨J + fIint= fIext. (3.2.6)

The expressions for the quadrilateral element nodal forces can be ob- tained by equating the virtual works, Eqs. (3.2.3), to the virtual works of the nodal forces, Eqs. (3.2.5), and utilizing the finite element approximations Eqs. (3.2.4). Combining Eqs. (3.2.3b) and (3.2.5b) and using Eq. (3.2.4b) yields:

δWext = δuT_If_Iext= Z Ω0 ρδu · b dΩ0+ Z Γ0 ti (δu · ei)(ei· ¯t0i) dΓ0 = δuT_I ( Z Ω0 ρNT_Ib dΩ0+ Z Γ0 ti NT_I¯t0_i dΓ0 ) , (3.2.7)

and arbitrariness of the test function δuI gives the external nodal forces acting

on the quadrilateral element:

f_Iext = Z Ω0 ρNT_Ib dΩ0+ Z Γ0 ti NT_I¯t0_i dΓ0, (3.2.8)

where the first and second terms represent the body forces and prescribed tractions (e.g., applied surface pressure), respectively.

considered. Also, the effect of gravitational force, which acts in the vertical (z) direction, on the motion of the beam is minimized by setting the flapping motion in the horizontal plane (xy-plane in Figures 3.1a and 3.2). Therefore, further details on the derivation of the external nodal forces fext

I will not be

given as they are not included in the computational model.

To determine the inertial nodal forces, we combine Eqs. (3.2.3c) and (3.2.5c) and use Eqs. (3.2.4a) and (3.2.4b):

δWkin= δuT_If_Ikin = Z Ω0 δu · ρ¨u dΩ0 = δuT_I Z Ω0 ρNINJdΩ0u¨J  , (3.2.9)

and consider the arbitrariness of the test function δuI to obtain:

f_Ikin = Z

Ω0

ρNINJdΩ0u¨J = MIJu¨J. (3.2.10)

The mass matrix given in Eq. (3.2.10) is referred to as a “consistent” mass matrix as it results from a consistent derivation from the weak form [104]. The fact that the consistent mass matrix is not a diagonal matrix makes its use computationally prohibitive in certain circumstances. Therefore, diago- nal mass matrices called “lumped” mass matrices have been developed based upon various procedures such as row-sum technique, physical lumping, HRZ lumping, and optimal lumping [100]. The mass matrix of the beam element is obtained from the consistent or lumped form of the mass matrix (given in Eq. (3.2.10)) of the continuum element via master-slave transformation [104]. However, the transformation does not yield a diagonal mass matrix even if the lumped form of the quadrilateral element mass matrix is used. Therefore, the

following diagonal mass matrix given for the CB beam element is used in the computational model [104]: MIJ = ρ0t0l0w0 420                 210 0 0 0 0 0 0 210 0 0 0 0 0 0 ₂₄1t2 0 0 0 0 0 0 0 210 0 0 0 0 0 0 210 0 0 0 0 0 0 1 24t 2 0                 , (3.2.11)

where ρ0, t0, l0, and w0 are the density, thickness, length, and width of the

beam element in the initial configuration, respectively.