MÉTODO DOS ELEMENTOS FINITOS POSICIONAL … · MÉTODO DOS ELEMENTOS FINITOS POSICIONAL ... In this section both vector and index notations ... The strain measure to be adopted in

ISSN 1809-5860

Cadernos de Engenharia de Estruturas, São Carlos, v. 11, n. 51, p. 111-130, 2009

MÉTODO DOS ELEMENTOS FINITOS POSICIONAL APLICADO À NÃO LINEARIDADE GEOMÉTRICA DE SÓLIDOS

Daniel Nelson Maciel1 & Humberto Breves Coda2

Resumo Neste trabalho uma formulação não linear geométrica alternativa do método dos elementos finitos é apresentada. É chamada de elementos finitos posicional por conta que as variáveis nodais são posições, ao invés de deslocamentos. Outra novidade da formulação proposta é a aplicação da medida de deformação de engenharia, facilitando a compreensão do problema não linear em face das aplicações na engenharia. Elementos tetraédricos isoparamétricos de ordem cúbica são empregados evitando assim o fenômeno de travamento. Vários exemplos são mostrados a fim de se validar a formulação aqui proposta. Palavras-chave: Não linearidade geométrica. Sólidos. Elementos finitos.

POSITIONAL FINITE ELEMENT METHOD APPLIED TO GEOMETRIC

NONLINEAR TRIDIMENSIONAL PROBLEMS

Abstract An alternative nonlinear finite element approach is presented herein in this work. It called positional finite element method because it is considered nodal positions as variables of the non linear system instead of displacements. Moreover, the proposed approach applies the engineering strain measure which turns the nonlinear problem familiar with linear engineering problems. Isoparametric tetrahedral finite elements with cubic polynomial interpolation of positions and stresses are applied in order to avoid locking. Many examples are shown and compared with results obtained in the specialized literature.

Keywords: Nonlinear geometric. Solids. Finite Elements.

1 INTRODUCTION

The good numerical representation of solids exhibiting large displacement and large strain is a continuous activity of a considerable number of research works. It is well known (SIMO & ARMERO, 1992) that a lot off effort has been invested in the development of low order finite elements to obtain robust and non-locking three and two-dimensional elements to solve geometrically and physically non-linear problems of solid mechanics. To solve the so called locking, related to low order finite elements, strategies based on mixed stress and strain formulations have been proposed as one can see in SIMO et al (1985), SIMO & TAYLOR (1985), SUSSMAN & BATHE (1987), BRINK & STEIN (1996), ZIENKIEWICZ & TAYLOR (2000) and SANSOUR & KOLLMANN (2000) for example. Hopefully, works as the ones proposed by DÜSTER et al (2003) and RANK et al (2003) revels that high order finite elements are locking free and less sensible to mesh geometry distortion, desirable properties for geometrically non-linear and large strain analysis. As mentioned by these authors, SURI (1996) and SZABOO & BABUSKA (1991) have shown that, for small strains, high order elements are locking free. High order polynomials elements have been successfully used for hyper-elastic material modeling in the works of DÜSTER et al (2003) and RANK et al (2002).

1 Doutor em Engenharia de Estruturas - EESC-USP, [email protected] 2 Professor do Departamento de Engenharia de Estruturas da EESC-USP, [email protected]

Daniel Nelson Maciel & Humberto Breves Coda


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This study intends to contribute with the overall research effort dedicated to geometrically non-linear analysis investigating the use of the non-linear engineering strain measure as in CODA & GRECO (2004) to simulate three-dimensional solids subjected to large displacements. A constitutive relation for compressive materials is proposed for the engineering strain measure. Moderated high order finite elements are used avoiding locking and taking advantage of mesh generation packages. For three-dimensional problems an isoperimetric tetrahedron with cubic approximation element is adopted. The positional formulation i.e., nodal parameters are positions instead of displacements, will be proposed here for solid analysis. To solve large systems of equations the MA27 solver is used, resulting in a robust method able to consider thin solids and to solve complicated geometries by using any structured or non-structured mesh generation for hexahedral finite elements. Some examples are explored to show the versatility and accuracy of the proposed procedure.

2 STRAIN MEASURE

In this section both vector and index notations are used. The first is used to clarify the illustrations and the second is preferred to be used in computational implementations. The strain measure to be adopted in this study is called engineering strain due to its relation with the so called "strain gates" used at laboratory to study deformed specimen. This strain measure is the first object of study of almost all structural engineering courses. It is important to mention that the strain measure adopted in text books of linear elasticity is not the engineering strain, but its primary description. At undergraduate curses for example, before the linear simplifications, it is exactly the engineering strain concept, see HIGDON et al (1978), for instance. However, the primary concept is not enough to develop the proposed formulation. In order to do so its mathematical description, given for example in OGDEN (1984), should be resumed. Figure 1 describes two infinitesimal vectors dx and dy fixed in the initial and actual continuum

configurations, respectively. Vector dx can be written as a function of its modulus and direction, called u , as follows:

dx u dx udx= = (1a)

or for index notation:

i i j j idx u dx dx u dx= = (1b)

The same can be done for dy , as:

dy v dy vdy= = (2a)

i i j j idy v dy dy v dy= = (2b)

where v is a unit vector.

x

e1

e2

x0

B

y0

e3RB

xΔ f ( x ) Δy

y

B0

B1

Figure 1 – Engineering strain definition.

Método dos Elementos Finitos Posicional aplicado à não linearidade geométrica de sólidos


113

The vector dy is the vector dx after the configuration change, represented by the vector

function f . This change of configuration is usually called deformation function; this name is avoided here due to misunderstandings generated among Latin speakers. As it can be seen at figure 1 letter x is used to represent coordinates of initial configuration and letter y to represent coordinates of actual configuration. Relation among these infinitesimal vectors is done directly by the gradient of the change of configuration function, represented by letter A, given as follows:

A Grad( f )= (3a)

iij

j

fAdp∂

= (3b)

Where capital letter indicates Lagrangean operation. dy Adx= (4a)

i ij jdy A dx= (4b) substituting equations (1) and (2) into equation (4) one writes:

dyv dxAu= (5a)

i ij jdyv dxA u= (5b) Making the internal product of equation (5) by itself, one writes:

2 2 t tdy dx u A Au= (6a) 2 2

k ki ij jdy dx u A A u= (6b)

where the index t means transpose. It should be mentioned that tC A A= is usually called the right Cauchy-Green stretch tensor and it is positive definite. From equation (6) one achieves the stretch at any point of any fiber following a general direction u in the initial configuration, as:

{ }1/ 2t tdy( u ) u A Au

dxλ = = (7a)

( ) { }1/ 2

k ki ij jdyu u A A udx

λ = = (7b)

It is important to remember that expression (7) does not mean derivative, but the relation between two infinitesimal lengths, and that ( u )λ is a strictly positive quantity. For more detailed description one can see OGDEN (1984). The Lagrangean longitudinal engineering strain at a point, following direction u , is defined as:

{ }1/ 2t tdy dx( u ) u A Au 1 ( u ) 1

dxε λ−

= = − = − (8a)

( ) { } ( )1/ 2

k ki ij jdy dxu u A A u 1 u 1

dxε λ−

= = − = − (8b)

For the sake of completeness, one can make the same for an Eulerian representation: 1dx A dy−= (9a) t

i ij jdx B dy= (9b) 2 2 t 1t 1dx dy v A A v− −= (10a) 2 2 1t

k ki ij jdx dy v A A v−= (10b)

{ }1/ 2t 1t 1dx 1( v ) v A A v

dyλ

λ− −′ = = = (11a)

( ) { }1/ 2

k ki ji jdxv v B B vdy

λ′ = = (11b)



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Where ( v )λ′ is the stretch of a fiber initially in direction u and finally in direction v , related to the final position, i.e., Eulerian stretch. In equations (9) and (11) the tensor tB is the inverse of A . The Eulerian engineering strain is therefore:

dy dx( v ) 1 ( v )dy

ε λ−′ ′= = −)(u

11λ

−= (12)

To complete the engineering strain components it is necessary to define distortion. The Lagrangean engineering distortion is the difference between the final angle defined by two fibers θ and the initial angle defined by them, / 2π , see figure 2.

( 1 ) ( 2 )u u 2πγ θ= − (13)

Figure 2 – Definition of distortion.

The angle θ is easily calculated by the internal product of the final vectors, as:

( 1 ) ( 2 )cos( ) v vθ = (14a)

( 1 )i ( 2 )icos( ) v vθ = (14b) Using equation (5), these final vectors are given by:

( 1 )( 1 )

( 1 )

Auv

( u )λ= ; ( 2 )

( 2 )( 2 )

Auv

( u )λ= (15a)

ij ( 1 ) j( 1 )i

( 1 )

A uv

( u )λ= ; ij ( 2 ) j

( 2 )i( 2 )

A uv

( u )λ= (15b)

From equations (14) and (15) one achieves:

2)u()u(uAAucosa

21

1tt

2uu 21

πλλ

γ −⎟⎟⎠

⎞⎜⎜⎝

⎛= (16a)

( 1 ) ( 2 )

( 2 )k ki ij ( 1 ) ju u

( 1 ) ( 2 )

u A A ua cos

( u ) ( u ) 2πγ

λ λ⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

(16b)

It is simple to write a similar expression for Eulerian distortion, however the relation among nominal stress and true stress (Cauchy) will be done following the principal directions (polar decomposition theorem) avoiding this calculation. For a three dimensional representation one needs six independent strain components to complete its physical representation. They are calculated choosing 1 1u e= , 2 2u e= and 3 3u e= , where

( )1 2 3e ,e ,e is the orthonormal Cartesian base. From these considerations one finds 1ε , 2ε , 3ε ,

12 21γ γ= , 13 31γ γ= and 32 23γ γ= .



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For the sake of completeness it is necessary to recall that tC A A= or kj ki ijC A A= is the right Cauchy-Green stretch tensor. It is positive definite, symmetric and has six independent values, OGDEN (1984). Any strain measure written as a function of the Cauchy-Green stretch is objective, i.e., invariant regarding rotations and translations and complete, see CAMPELLO et al (2003) and GREEN & NAGHDI (1979) for instance. To simplify notation, in the next developments, it is interesting to call distortion ( ijγ ) for i j≠ and

longitudinal strain, for i j= , simply by ijε . It is important to mention that ijε does not respect usual tensor rotation; it is necessary to apply it over the Cauchy-Green stretch and, after that, calculates expressions (8) and (16). The value ijε is then called pseudo-tensor of engineering strain.

3 THE SPECIFIC STRAIN ENERGY AND NOMINAL STRESS

From this section to the end of the paper, only index notation will be employed. As described by various references like OGDEN (1984), MALVERN (1969), CIARLET (1993) etc., a general hyper-elastic material can be fully described by a specific strain energy potential eu written as a function of any objective strain measure. A classical strain energy potential for hyperelastic material will be discussed in the appropriate section. In our case we choose the engineering strain and write the specific energy potential as follows.

( ) ( ) ( ) ( )2 2 32e kl ( a )klij ij kl ( b )klij ij kl ( c )klij ij kl ( d )klij ij

1u ( ) C b C c C d C ....2

ε ε ε ε ε ε ε ε ε= + + + + (17)

The letters (a), (b), (c), (d),… are coefficients that can be calculated via inverse analyses to adequate the specific energy strain to the physical behavior of the body. The values Cα are second order tensors that establish the interdependence of strains for a specific material. As it will be seen later, ( )C α establishes the relation between nominal stress and engineering strain. The simplest material, the one to be implemented here, is the linear elastic one and, as a consequence, equation (17) reduces to:

e kl ( a )klij ij1u ( ) C2

ε ε ε= (20)

where ( a )C is the simple Hokes' Law tensor for isotropic materials given by:

a

K(1 ) K K 0 0 0K K(1 ) K 0 0 0K K K(1 ) 0 0 0

C0 0 0 G 0 00 0 0 0 G 00 0 0 0 0 G

ν ν νν ν νν ν ν

⎡ ⎤−⎢ ⎥−⎢ ⎥⎢ ⎥−

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(21)

where ν is the Poisson ratio and the other constants are given by, EG

2(1 )ν=

+ (22)

EK(1 )(1 2 )ν ν

=+ −

(23)

with E being the well known Young modulus. The bulk modulus is then achieved by K Kν= . It is easy to observe that the proposed strain measure has much more meaning to the ones accustomed to linear elasticity than any other proposed in literature to solve non-linear problems. To find the energy conjugate stress of the engineering strain measure, i.e., the nominal stress, one needs only to differentiate equation (20) regarding each strain component, i.e.:



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kl ( a )klij ijCσ ε= (24)

The pseudo-tensor klσ is called nominal engineering stress. By a simple comparison between equation (24) and the Hokes' Law for linear elasticity one concludes that each component of the nominal engineering stress has exactly the same physical meaning of the True Stress, Cauchy Stress, but referred to the initial configuration. This is the reason it is called nominal engineering stress. This stress measure and the engineering strain are physical quantities that can be easily evaluated at any laboratory of material characterization.

4 POLAR DECOMPOSITION THEOREM AND TRUE STRESS

If 1u , 2u and 3u are the principal directions of the Cauchy-Green stretch at the initial configuration, then, from equation (16), one concludes that 12 13 23 0γ γ γ= = = . By the Polar

Decomposition Theorem one knows that the corresponding directions for 1u , 2u and 3u in the actual

configuration ( )1 2 3v ,v ,v are also orthogonal and are the principal directions of true strain and true

stress, see for example OGDEN (1984). The relation between the principal nominal stress and the principal true stress is captured directly by the initial and final areas relation, i.e.:

true ii

j k

σσλ λ

= (25)

where i,j,k follow cyclic permutation rule. Expression (25) informs that the area initially orthogonal to a principal direction at initial configuration remains orthogonal to the 'new' principal direction at the actual configuration. It also informs that the ratio between nominal principal stress and principal true stress is the inverse to the ratio between initial and actual principal areas. This is an important property of the nominal stress and it brings an easy physical meaning to this Lagrangean quantity. To confirm the assumed relation one writes the strain energy for initial and final configurations as a function of principal stress and strain as follows:

1 2 3

0 0 0

0e 1 1 0 2 2 0 3 3 00 0 0

V V V

U d dV d dV d dVε ε ε

σ ε σ ε σ ε⎧ ⎫⎪ ⎪= + +⎨ ⎬⎪ ⎪⎩ ⎭∫ ∫ ∫ ∫ ∫ ∫ (26)

true true true1 1 1true true true true true true true

e 1 1 2 2 3 30 0 0V V V

U d dV d dV d dVε ε ε

σ ε σ ε σ ε⎧ ⎫⎪ ⎪= + +⎨ ⎬⎪ ⎪⎩ ⎭∫ ∫ ∫ ∫ ∫ ∫ (27)

where 0eU and true

eU are, respectively, the strain energy written for Lagrangian and Eulerian descriptions. For the principal axes the Jacobian of the change of configuration is easily written, resulting

0 1 2 3 0dV JdV dVλ λ λ= = (29) For any conservative mechanical system, the elastic energy should be the same despite the adopted material description, so:

0 truee eU U= (30)

Considering that the strain energy is arbitrary, using equations (25) trough (30) one achieves the differential relation among principal engineering and true strain, i.e:

true ii

( i )

dd εελ

= (31)

For a Lagrangian referential one can integrate equation (31) achieving truei iln( )ε λ= .

This expression corresponds exactly to the true longitudinal strain definition mentioned, for example, in OGDEN (1984) and CRISFIELD (1991).



117

5 KINEMATICAL APPROXIMATION AND POSITIONAL MAPPING

To build the positional finite element method the engineering strain should be calculated for approximated configurations. Lets' define a general solid finite element. For initial configuration any point position belonging to this element can be calculated by the following usual mapping, see figure 3.

0i i 1 2 3 i 1 2 3 if x ( , , ,X ) ( , , )Xξ ξ ξ φ ξ ξ ξ= = (40)

where ix is the ith coordinate of the desired point, iX is the ith coordinate of node , φ is the shape function associated with node and 1 2 3( , , )ξ ξ ξ are non-dimensional Gauss coordinates. The vector

function 0f can be understood as a fictitious change of configuration from the non-dimensional space to the initial one. For actual configuration a similar expression is written, changing x by y, i.e.:

1i i 1 2 3 i 1 2 3 if y ( , , ,X ) ( , , )Yξ ξ ξ φ ξ ξ ξ= = (41)

The vector function 1f can be understood as a fictitious change of configuration from the non-dimensional space to the actual one.

ξ1

ξ2 B 0 B1

A0 A1

Figure 3 – Positional mapping.

The real mapping ( f ), from initial configuration to the actual one is given by:

1 0 1f f ( f )−= (42)

where the symbol means that function 1f is applied over the image of 0 1( f )− . The domain of the

total function is 0B and its image is 1B . Function f is the change of configuration described in the previous section, applied here to the adopted FEM approximation. Its gradient is easily computed as follows. 1 0 1A A ( A )−= (43) where

j j j1 1 1

1 2 3jj j j

j ji2 2 2ik i ,k

1 2 3 kj j j

3 3 3

1 2 3

f f f

ff f fA f

f f f

ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ

⎡ ⎤∂ ∂ ∂⎢ ⎥∂ ∂ ∂⎢ ⎥⎢ ⎥ ∂∂ ∂ ∂

= = =⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥

⎢ ⎥∂ ∂ ∂⎢ ⎥∂ ∂ ∂⎢ ⎥⎣ ⎦

(44)

In equation (44) j means initial or actual (0 or 1) i means direction, k means one of the non-dimensional variables and comma means partial differentiation. Equation (44) can be written from equations (40) and (41) as follows: 0 0

ik i ,k ,k 1 2 3 iA f ( , , )Xφ ξ ξ ξ= = (45)



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1 1ik i ,k ,k 1 2 3 iA f ( , , )Yφ ξ ξ ξ= = (46)

It is important to mention that equation (43) is calculated numerically for each Gauss station. As a consequence, it is possible to calculate trials (for arbitrary actual positions) for the engineering strain components using equations (8) and (16).

6 THE NUMERICAL POSITIONAL PROCEDURE

In this section the simplicity of the technique will become clear, as no additional considerations regarding kinematics needs to be done to solve the geometrical non-linear problem. The principle of minimum potential energy can be written, for a conservative elastostatic problem, using position considerations (not displacements) as follows:

eUΠ Ρ= − (47) Where Π is the total potential energy, eU is the strain energy and Ρ is the potential energy of the applied forces.

Figure 4 – Total potential energy written for a body in two different positions.

Adopting linear constitutive relation for elastic materials, the specific strain energy is the one shown in equation (20) and, as it is a Lagrangean quantity, the whole strain energy stored in the body is written for the reference volume V0 as:

0 0

e e 0 kl ( a )klij ij 0V V

1U u dV C dV2

ε ε= =∫ ∫ (48)

The potential energy of conservative applied forces is written as

k kP Y F= (49)

In equation (41) kY is the actual position occupied by all points of the body, and kF is its corresponding force. It is interesting to note that the potential energy of the applied forces may not be zero in the reference configuration. The total potential energy is written as

0

kl ( a )klij ij 0 k kV

1 C dV F Y2

Π ε ε= −∫ (50)

From equations (45), (46), (8) and (16) it is possible to write equation (50) as a function of actual positions and applied external forces. From this reasoning the problem of achieving the equilibrium of the elastic system is the determination of the minimum of total potential energy regarding positions. It is



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done by deriving equation (50) regarding positions and making it equal to zero. These steps are done as follows,

( )0

j kl ( a )k lim im 0 jj jV

1g C dV FY 2 YΠ ε ε∂ ∂

= = −∂ ∂∫ (51)

In equation (51) ig is a vector which for an exact situation assume null value. Splitting the derivative of the specific strain energy, one writes:

( ) ( )kl k lim im kl k lim im im imj j j j

1 1C C C2 Y 2 Y Y Y

αβ αβ αβαβ αβ

αβ

ε ε εε ε ε ε ε σ

ε∂ ∂ ∂∂ ∂

= = =∂ ∂ ∂ ∂ ∂

(52)

Substituting equation (52) into equation (51), results

0

intj im im 0 j j j

jV

g C dV F F F 0Y

αβαβ

εε

∂= − = − =

∂∫ (53)

where intjF is the first gradient vector of the strain energy potential, understood as internal force.

Equation (53) means that if the internal force vector is equal to the applied one the solid is at equilibrium. If not, vector jg can be understood as the unbalanced force of the mechanical system.

It is important to remember that in this study the applied forces are conservative. Non-conservative forces can be introduced directly in equation (53) if desired. As mentioned before the actual position is the unknown of the problem, so it is necessary to solve equation (53). As the vector function jg ( (Y ))αβε is non-linear regarding nodal parameters it is necessary to expand it from an

initial trial solution, called here 0Y , as follows:

0

0 2

( )

( ) ( ) 0jj j k j

k Y

gg Y g Y Y O

Y∂

= + Δ + =∂

(54)

Remembering that forces are conservative one concludes that

=∂

∂=

∂

∂

)Y(k

intj

)Y(k

j

00 YF

Yg

00

0im im 0 kj

k jV Y

C dV HY Y

αβαβ

εε

⎛ ⎞∂∂=⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

∫ (55)

where the matrix 0kjH is the Hessian matrix of the total potential energy.

The derivative inside the integral term of equation (55) is written as: 2

imim im im im im

k j k j j k

C C CY Y Y Y Y Y

αβ αβ αβαβ αβ αβ

ε ε εεε ε⎛ ⎞ ⎛ ⎞∂ ∂ ∂∂∂

= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ (56)

Neglecting higher order errors ( 2iO ) equation (54) is rewritten as:

( ) ( ) ( ))Y(FFH)Y(gHY 0intjj

10kj

0j

10kjk −=−=

−−Δ (57)

resulting into the Newton-Rapson procedure to solve non-linear system of equations.

The Newton-Rapson procedure is summarized as: One chooses a trial position 0Y and

calculates the unbalanced force vector )Y(g 0j according to equation (51). By applying equation (57)

one finds the variation of position kYΔ to correct 0Y . With this new position vector, one repeats the

procedure until kYΔ or )Y(g 0j becomes small. The error, that is, the final unbalanced force

(residuum) is applied into the next load step if the analyzed problem is divided into load steps, if not the problem is solved.



120

To finalize the technique description the derivative of strain regarding actual nodal positions should be done. To be short it will be shown the derivative of the Cauchy-Green stretch tensor and the rest of arithmetic operations are left for the reader. Recalling that the Cauchy-Green stretch tensor is given by:

AAC t= (58) and omitting, for simplicity, extra indices, one applies the positional FEM mapping and writes:

0 t 1 1 t 1 0 1i iC [( A ) ] ( A ) (Y )A (Y )( A )− −= (59)

Remembering that 0A is constant regarding the actual position, the desired derivative is performed as: 1 t 1

0 t 1 1 1 0 t 1 1 t 0 1i ii 0 i

j j j

( A ) (Y ) A (Y )C [( A ) ] A (Y )( A ) [( A ) ] ( A ) (Y ) ( A )Y Y Y

− − − −∂ ∂∂= +

∂ ∂ ∂ (60)

and, from equation (46), one has:

j)i(321k,j

i321k,i321k,

jj

1 ),,(YY),,()Y),,((

YYA δξξξφξξξφξξξφ =

∂∂

=∂∂

=∂∂

(61)

where the Kronecker delta j)i(δ assumes one if the global degree of freedom j is equal to the local (to the finite element) degree of freedom i and zero otherwise. The variable is the element node and i is the degree of freedom. It is important to mention that the present technique can be applied to any strain measure that is based on the Cauchy-Green stretch.

Equation (61) indicates that the proposed procedure can be operated by means of creating Hessian matrix and internal forces for finite elements and composing the global matrix and internal force vector by summation of coincident degrees of freedom, as it is done for usual FEM procedures. One should remember that all operations are done in the global system of equation avoiding the use of rotations.

7 NUMERICAL EXAMPLES

The proposed formulation is tested mainly regarding its ability of representing large displacement situations, using the engineering strain measure together with the positional mapping. One example is also provided to show the possibilities of the technique regarding hyperelastic materials.

7.1 Cantilever beam subject to a transversal load applied at free end

The structure is discretized into 48 finite elements, totalizing 319 nodes or 957 degrees of freedom. The analytical solution achieved by reference MATTIASSON (1981) is used to compare results at figure 4. In the numerical analysis the load is divided into 100 steps, in order to show the results at different load levels. This example is similar to the one proposed by SURI (1996) and solved using high order finite elements by DÜSTER et al (2003). The beam has a square cross section of 1x1m and a length of 10m. The adopted Young modulus is 71.27 10E x Pa= and Poison Ratio 0.0ν = .

Following the convention of Fig. 5, in Fig. 6 and 7 the responses UX, UY are compared with the analytical solution. In Fig. 7 two deformed shapes for selected load levels are presented.



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Figure 5 – Loading, discretization and convention.

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

Tip

Dis

plac

emen

t U

y/L

Load Parameter PL2/EI

Numerical Result Mattiasson

Figure 6 – Comparison of results for vertical displacements.

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

Numerical Result Mattiasson

Tip

Dis

plac

emen

t U

x/L

Load Parameter PL2/EI Figure 7 – Comparison of results for horizontal displacements.



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As one can see the results are in perfect agreement, showing that the formulation is able to describe large displacements for this simple structure. It is interesting to mention that the adopted constitutive relation is the linear engineering one, (expression (20)).

2 4 6 10 Figure 8 – Some selected configurations for reference load 2 /PL EI (no scaling).

7.2 Clamped column

This example is introduced to show the ability of the proposed formulation to capture instability configuration and its corresponding critical load. A clamped column subjected to a growing compressive load is analyzed, see figure 9. The adopted discretization uses sixty tetrahedral finite elements, as depicted in figure 10. The length of the column is L 2m= . The cross section is square with sides a 0.13m= . An initial lateral eccentricity of L/100 is assumed at the top of the column. The adopted material properties are: E 210GPa= and 0ν = .

P

2 m

Figure 9 – Analyzed column.

Figure 10 – Finite element discretization.



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The lateral displacements for the loaded point is depicted in figure 11 and compared with results obtained using 10 cubic frame elements based on Reissner kinematics.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80

5000

10000

15000

20000

25000

30000

35000

40000

FEM 3D Reissner Kinematics Pcr - First Mode

P (kN)

Lateral displacement (m) Figure 11 – Lateral displacement versus applied load.

From figure 11 it is possible to identify the larger flexibility of the 3D model when compared with the 2D Reissner kinematics formulation. The first critical load is depicted in figure 11 and shows the accuracy of the proposed technique in capturing the instability of the system. At figure 12, for completeness, some configurations for specific load steps are depicted.

Figure 12 – Some deformed configurations (no scaling).

7.3 Lateral instability of a clamped beam

This example shows the ability of the formulation on reproducing coupled instability modes. In this example the lateral instability of a clamped beam is studied, see figure 13. The critical load for this problem has analytical solution ( 24.013 /crP EIGJ L= ), given by TIMOSHENKO & GERE (1961). It is

important to mention that 3I bh / 12= and 3J hb / 3= . For this example it has been adopted: E 100000= , G 50000= , L 100= , b 1= and h 10= , where b and h are, respectively, the thickness and the height of the beam. Applying the analytical solution one achieves crP 47.3= . This value is depicted in figure 14 as a reference value for the numerical response. To run this problem 74



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finite elements (543 nodes) are employed and an initial defect for torsion angle, 0.16radϕ = , is adopted.

Figure 13 – geometry of the analyzed problem.

The z direction is along the beam axis and y direction is the vertical one. As it can be seen, the discretized structure behaves as expected passing trough the critical load region with more flexibility.

0 10 20 30 40 500

20

40

60

80

100

Load

Displacement

-UY -UZ UX PCR

Figure 14 – Displacements versus applied force for the free end of the bar.

At figure 15 one can see some one selected configuration for the analyzed problem.

x

y

z

Figure 15 – Selected deformed configuration (no scaling)



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7.4 Ringed-plate

This example is introduced to test the formulation regarding locking when leading with thin structures. This benchmark is used by PETCHSASITHON & GOSLING (2005) and other cited therein to investigate the performance of finite rotation formulations for shell structures. The ring plate is clamped at one edge and loaded at the other, where these edges are divided by cutting (or splitting) a circular (doughnut) plate (e.g. along the line AB in figure 16). The geometry, loading and material properties of the plate are also given in figure 16. The adopted discretization is depicted in figure 17 and the deformed shape of the ringed plate subjected to a load factor of 80 is illustrated in figure 18. The number of finite elements and nodes are: 420 and 2803, respectively. The relationship between the incremented load (load factor) and de displacement at points A, B and C (defined in figure 15) are plotted and compared with those obtained by PETCHSASITHON & GOSLING (2005) at Fig. 19. From solutions, it is inferred that the proposed formulation is more flexible than the presented by PETCHSASITHON & GOSLING (2005) revealing the absence of locking for this example.

Figure 16 – Geometry, physical parameters and loading [33].

Figure 17 – Discretization.



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Figure 18 – Deformed configuration for load factor 80 (no scaling).

0 20 40 60 80-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Vert

ical

Dis

plac

emen

t

Load Factor

Present Work: Point A Point B Point C

Reference No.[23]: Point A Point B Point C

Figure 19 – Displacement versus load factor.

7.5 Pinched cylinder with rigid diaphragms

A more complex thin shell is solved this time. A cylinder with rigid diaphragms is pinched by concentrated loads at two opposite points at its top and bottom, see figure 20. Two discretizations are employed to run this problem (figure 20). Taking advantage of symmetry only an octant of the cylinder is discretized. The adopted constants are: R 100= , h 1= , 4E 3x10= , L 200= and 0υ = . At figure 21 the results are compared with reference SANSOUR & KOLLMANN (2000) that employed an enhanced strain based shell finite element.



127

Figure 20 – Pinched cylinder geometry, loading and discretizations for 7092 DOF and 19893 DOF respectively.

For completeness, at figure 22 two deformed configurations are depicted.

-20 -10 0 10 20 30 40 50 60 70 80 900

2000

4000

6000

8000

10000

12000

Load

Displacement

Present Work: Node A (7092 DOF) Node B (7072 DOF) Node A (19893 DOF) Node B (19893 DOF)

Reference [32]: Node A Node B

Figure 21 – Displacements for points A and B.

Figure 22 – Selected configurations (Intermediates and Final Load) no scaling.



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From the results one understands that a more fine mesh is required and also a better distribution of it, especially by refining where big curvatures occur during deformation.

8 CONCLUSIONS

The positional FEM for geometrical non-linear analysis of solids has been presented and successfully implemented. The theory is simple and consistent. The engineering strain measure, based on the Cauchy Stretch Tensor, is objective and complete. The nominal stress measure is the conjugate of the proposed engineering strain, it is symmetric and posses an understandable physical meaning. Six examples demonstrate the good behavior of the formulation. In particular, the ringed plate example demonstrates the ability of the formulation on treating thin solids. However, the pinched cylinder case presents a result that, even being better than expected, shows the limitation of using solids to simulate shells exhibiting complicated deformations. Very good convergence for the iterative process has been achieved and the stress profile is reliable. Some interesting features of the formulation should be detached. As it is a strain based formulation it can be directly used for non-linear constitutive relations. The adopted stress-strain conjugate is precisely the one used at engineering laboratories, making easy the necessary calibration of constitutive relations for any material. A rubber-like constitutive relation has been implemented to demonstrate the possibilities of the formulation for future applications. Further developments as position control and arch length control are required to achieve good solutions at unstable bifurcations, snaps troughs and snap backs. The positional formulation deserves further studies and developments to incorporate shell elements.

9 AKNOWLEDGEMENTS

The Authors would like to thank CAPES and CNPq for the financial support.

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