Modeling of Contact Interfaces using
Segment-to-Segment-Elements for FE Vibration Analysis
Michael Mayer', Lothar Gaul2
Robert Bosch GmbH, Corporate Research and Development,
Dept. of Applied Physics, Michael.Mayer6Qde. bosch.com
University of Stuttgart, Institute A of Mechanics
GaulQmecha.uni-stuttgart .de
Abstract Contact interfaces of jointed structures are often modeled by using simple nodeto-node
spring and dashpot elements. A recently reappearing approach to model contact is to employ Zero
Thickness or Thin Layer Elements, which are segment-to-segment contact elements. This method
originated hom geomechanics and has been recently applied in modal analysis as an efficient way
to define contact stiffness. The main advantage of these contact elements is that both linear and
nonlinear constitutive models for normal and tangential contact behavior can be implemented. For
example, simple linear contact stiffness and viscous contact damping lead to the element stiffness and
damping matrix. This paper gives a short overview of the theory leading to Zero Thickness Elements
and Thin Layer Contact Elements. As a new application of these elements, the implementation of
realistic structural interface models with hysteresis is demonstrated as nonlinear constitutive laws
for the contact element. This method constitutes a new way to simulate multi-degreeof-heedom
systems with structural joints and predict nonlinear stiffness and damping properties. The approach
is validated by simulating the response of a structure coupled by bolted joints.
Column vector; row vector or matrix with capital letter
Three-dimensional diagonal matrix of (-)
Natural coordinate system
Local coordinate system
Displacement vector and displacement components in x, 9, and x directions
Relative displacement vector and components in x,g, and z directions
Strain matrix and vector
Stress vector
Contact traction vector
Matrix of shape functions; shape functions
Strain-displacement matrix
Young's modulus and shear modulus
Constitutive contact matrix
Stiffness and contact stifiess matrix
Jacobian
Virtual work
Weighting functions
Element lengths and thickness
Parameters of the normal contact model
Parameters of the tangential contact model
Internal variables of the tangential contact model
1
Introduction/Motivation
For lightly damped, linear members of a structure, very good estimates of eigenfrequencies, modal
damping values, and corresponding mode shapes can be achieved by Experimental Modal Analysis
(EMA). hrthermore, by model updating of finite element (FE) models of members, very good
predictions of the vibrational behavior up to high frequencies are possible (GAUL,ALBRECHT,
WIRNITZER
[lo])If we now assemble single members into a built-up structure, prediction of the structure’s behavior
can be quite involved, even though the behavior of a single member is well-known. Information
about the details of coupling between members and the contact behavior is still needed.
It is known hom experimental investigations that by assembling members, we get a system which
has higher damping values as compared to that for an individual member. Since this increase of
damping is obviously not caused by the members, energy must be dissipated in the contact areas
between the members. Furthermore, we know that the structural behavior becomes at least slightly
nonlinear, whereas the members behave linearly. This contact nonlinearity is often neglected in
numerical simulations with Finite Element Analysis (FEA) . Instead, linear nodeto-node spring
and dashpot elements are typically introduced in the contact area to model coupling of members.
The choice of linear elements allow for straight-forward solution of the governing equations in the
frequency domain.
The segment-to-segment elements, which will be discussed here, provide significant advantages compared to nodeto-node elements, as is shown below. The complete 3-dimensional contact behavior
between two contacting elements, including linear contact stiffness and damping properties, can be
modeled in a single element. Furthermore, almost arbitrary, nonlinear, orthotropic contact behavior
can be implemented in the constitutive equations for these elements.
As an application example, a simulation of a structure with bolted joints is shown, where a nonlinear
hysteresis law is used to model the tangential contact behavior.
2
2.1
Contact Elements
Thin Layer Elements
The interface between two contacting members is modeled by a continuum with very small but
finite thickness. This continuum is referred to as the thin layer. A typical Thin Layer Element is
depicted in Fig. 1. We focus here on an eight node brick element and will show the similarities and
differences between continuum and Thin Layer Elements.
We start with the discussion of the common continuum element. For a detailed description, see e.g.
BATHE[3]. The displacement field in an eight node brick element is approximated by
{.I
[ H ( I , r ,01 {.}nodal 7
(2)
where {ui}= [uivi wiIT is the displacement vector of node i, and hi are the linear shape functions
=
i
3
Figure 1: Thin Layer Element
c
as formulated in the natural coordinate system t, q , for the element. This choice of coordinate
system simplifies numerical integration of the element stiffness matrix.
Assuming small deformations, we get the strain matrix as follows:
1
[E]
=
(V { u }+
~ {u}VT),
V = [d/& d / d g d/dzIT.
(4)
The strain matrix is symmetric and therefore has only 6 independent components, which we can
write in the vector form
(4 = [ t m t y y
€22
2t5y
2tyz
2
4
T
-
(5)
The strain-displacement relation can thus be cast in the form
{El
= PI
{4-
(6)
We should note that the partial derivatives relating the displacements to the strains are defined in
the local coordinate system 2,8 , z .
For representing the interpolation of the element geometry and the displacement field with the same
shape functions, the isoparametric concept is used , as shown in Fig 2.
Figure 2: Illustration of the Isoparametric Concept in 2-D
The relation between the partial derivatives in the local coordinates and in the scaled natural
coordinate system is given by
where [Jlis the Jacobian. For a rectilinear element, the Jacobian simplifies to
0
d/2
where f1 and f z are the element lengths in s- and y-directions, and d is the element thickness in
x- direction.
It follows horn the balance of moment of momentum that the stress matrix is symmetric. The stress
can then be written in vector notation,
(4= [ r z z r y y uzz r z y r y z r z z l T .
(9)
The next step is to employ a constitutive relation between stresses and strains. For simplicity, we
use the linear elastic isotropic constitutive relation
= [GI ( 4 =
where
[clPI (U}Y
(10)
[ais the isotropic linear elastic material matrix.
The virtual internal work of the element is then given by
Finally, with a 2-point Gaussian quadrature scheme, we are able to solve the integrals numerically,
2
2
[w= >:
2
XLB(&i
%I
b)lT[GI [B(ti~
q j &)I
~
det([J(&, ??j~
&)I) w.$,iwo,j
WC,R-
(13)
i=l j=1 k=1
where wt,wo,wc are the weights of integration.
Considering now the Thin Layer Element, we assume that the thickness d of the element becomes
small in comparison to the lengths [I, f 2 . In the publications of PANDE,
SHARMA
[14], and SHARMA,
DESAI[15] it is shown that for decreasing thickness (d + 0) the in-plane strains (tz, ty, tzy) and
ry,uzy)become negligible when a constitutive law with zero Poisson’s
the in-plane stresses (rz,
uyz)as
ratio ( v = 0) is used. Defining rzas the normal stress component u, of the interface, (uzz,
war),
ez as en and (2ez2, 2eyz) as ( y ~T ~T ~, )the
, constitutive relation
the tangential tractions (wz,
simplifies to
where 23, is the Young’s modulus and GT is the shear modulus. We can see that the normal
contact behavior is decoupled hom the tangential contact behavior. However, if coupling terms are
necessary, they can be easily inserted in the stiffness matrix. Using the constitutive relation in Eq.
(14), we can calculate the element stiffness matrix according to Eq. (12).
For simplicity, the element has been formulated using a linear elastic constitutive relation. But the
formulation also allows for nonlinear elastic, viscoelastic, and elasto-plastic interface behavior. For
example, the conventional Mohr-Coulomb law is very often used in Thin Layer Theory to model
stick-slip behavior in the tangential direction. We can use associated flow rules, which fulfill the
normality condition, as well as non-associated flow rules. For friction phenomena, non-associated
sliding rules should be applied, so as to avoid the unphysical prediction of uplift, as explained by
MICHALOWSKI,
MROZ[13].
The major problem of Thin Layer Elements is that for vanishing thickness (d + 0), the determinant
of the Jacobian tends to zero (det([J1)+ 0). With a determinant of zero, the Jacobian [J1 cannot
be inverted and the strain-displacement matrix [B] can therefore not be calculated. It is shown in
PANDE,
SHARMA[14] and DESAI,ET AL. [6] that the element thickness d must be chosen relative
SHARMA[I41 define the aspect ratio
to the element lengths f1 and f,. PANDE,
A.R. = max(f1, f2)
d
and show that the error for aspect ratios up to a value of 1000 is sufficiently small. DESAI,ET
[6] suggest using elements having an aspect ratio of less than 100.
AL.
For applications of the Thin Layer Elements in geomechanics, see e.g. DESAI,ET AL. [6], SHARMA,
DESAI[15], and PANDE,
SHARMA
[14]. Thin Layer Elements were recently also applied in numerical
modal analysis by AHMADIAN,
EBFLAHIMI,
MOTTERSHEAD,
FRISWELL
[l],and AHMADIAN,
JALALI,
FRISWELL
[2]. It has been shown by the authors that the Thin Layer concept with
MOTTERSHEAD,
a linear elastic constitutive relation (see Eq. (14)) is very well suited for model updating of contact
stfiesses using experimental data.
2.2
Zero Thickness Elements
The concept of Zero Thickness Elements was originated by GOODMAN,
ET AL. [ll]and is discussed
[12]. A typical Zero Thickness Element is depicted in Fig. 3. A Zero Thickness
in detail in HOHBERG
Element consists of two four node quadrilateral elements which face each other. In each quadrilateral
element , the threedimensional displacement field is approximated by
J
tg
tkt
Figure 3: Zero Thickness Element
Again, { u i } = [ui 2ri wiIT is the displacement vector of node i, and hi are the linear shape functions
Since the elements are quadrilaterals, we need only a bilinear approximation of the displacement
field and therefore only two natural coordinates t, 17. Let’s assume that, e.g., the first (bottom)
quadrilateral is connected to the surface of the finite element mesh of the first contacting body, and
the second {top) is connected to the other body. We denote the corresponding displacement fields
as follows
Since these elements are only two-dimensional, the traction vector in each element describes the
interface stresses.
bottom
{t}tO”=
{ :i }
top
Now we can state the virtual internal work for each quadrilateral as follows:
In contact mechanics, we are interested in the relative displacement field of the contacting surfaces.
This is expressed for Zero Thickness Elements through the relative displacement between the top
and the bottom quadrilateral,
A{u} = { u } ~ O P - {u}bottom
=
[H(c,r>l
({u>“,“,“,Ql
-{ u > ~ ~ ~ ~ > .
(23)
Furthermore, we know from Newton’s third law that the traction vectors of the elements in contact
must be equal in magnitude and opposite in direction,
{ t }= {i}t””= - { t } b o t t o m .
(24)
We can now implement a constitutive relation between the contact tractions and the relative displacement field. Again to simplify the discussion, we focus on a linear elastic contact law,
The virtual work of the contact tractions is given by the summation of the virtual internal work for
each element, as given in Eq. (21) and Eq. (22):
swc = sW;O” + sw,bottom =
6’l2
- d{u}bottom)T
{t}dx dg.
Implementing the contact law in the virtual work expression yields a contact stiffness matrix for a
relative displacement or midside quadrilateral element,
To evaluate the integrals, we apply again the isoparametric concept (see Fig. 2) in order to relate
the local coordinate system t o the natural coordinate system. The area element dA = dxdg is
transformed to natural coordinates using the Jacobian
With
dA = dxdg = det([J1)K d q ,
(29)
it follows
Eq. (30) can be integrated numerically using a 2-point Gaussian quadrature scheme. The full
stiffness matrix for the eight node Zero Thickness Element is then
The stiffness matrix [K]is 12-times singular due to its composition, thereby causing 12 zero-energy
modes. A zero-energy mode, or so-called hourglass mode, is a displacement mode that does not
correspond to a rigid body motion, and it produces zero strain energy. As Zero Thickness Elements
are always connected between continuum elements, all 12 zero-energy modes are suppressed. For a
KLEIBER[ 5 ] .
detailed discussion of this subject, see BUCZKOWSKI,
In a Zero Thickness Element, there is no restriction to linear elastic constitutive relations. As
with Thin Layer Elements, non-linear elastic, viscoelastic, and elasto-plastic behavior can be implemented. Furthermore, it is possible to use contact laws fiom contact mechanics in a penalty
approach because of the formulation of the virtual work in terms of the contact traction vector
{ t } and the relative displacement A{u} between the contacting surfaces. A detailed discussion of
different contact laws can be found in GAUL,NITSCHE[9].
For applications of the Zero Thickness Elements in geomechanics, see e-g. BEER[4],BUCZKOWSKI,
KLEIBER[5], and HOHBERG[12].
2.3
Discussion of the contact elements
We showed that both Thin Layer and Zero Thickness Elements are very well suited for treating
contact problems in FE-Analysis. One big advantage of the contact elements is that they are
compatible to the surrounding isoparametric continuum elements and can be formulated as linear or
quadratic elements for connection to linear or quadratic continuum elements. If we would connect
quadratic continuum elements with node-to-node elements, the midside nodes of these elements
remain unconstrained (incompatible mesh).
If we compare Thin Layer with Zero Thickness Elements, we can see that the Zero Thickness
approach has some advantages. It is obvious from Eq. (25) that the contact traction field in
Zero Thickness Elements is approximated with the same order as the displacement field, whereas
the traction approximation in Thin Layer Elements is one order below the approximation of the
displacement field. This means that the traction field in Zero Thickness Elements is one order better
than in Thin Layer Elements. Furthermore, the contact stiffness of Zero Thickness Elements does
not depend on the thickness of the element.
For these reasons, the Zero Thickness approach is found to be more appropriate to model solid-onsolid contact in FE-Analysis. Thin Layer Elements play an important role if a seal having different
constitutive behavior is located between the members, or if spot weld connections need to be defined
JALALI,MOTTERSHEAD,
FRISWELL
[2]).
(see AHMADIAN,
3
Application
Normal and tangential contact laws
For normal contact, the piecewise linear elastic penalty law depicted in Fig. 4 is implemented to
distinguish between contact (Aw 5 go) and separation (Aw > go). Aw is defined as the relative
normal distance (gap) between the two bodies, and go is a reference gap distance at which a contact
pressure p , starts to be transmitted between the bodies. The contact pressure is defined by
0
p , = -t, =
for
c1 (go - Awl
for
c1 go - c2 Aw
for
(c1 + cz) go - c3 (Aw + go) for
Aw
> go
2 Aw > 0
0 2 Aw > -90
Aw I
-go
go
c3
< c2 < c1 < 0.
This is a discretized version of a continuous dependency deduced by WILLNER,
GAUL[16] from
interface physics.
To model the tangential contact behavior the Masing-element depicted in Fig. 5 is used. This
element is able to account for dissipative microslip effects. The governing regularized equations for
the Masing-element are given in rate form as
with the internal variables 6 ' and
~ ~I Y T ~ and the corresponding evolution rules for the internal
variables with stiction limit 6,ax
where
S
1
= -[I
2
+ Sgn(AGgTx+ A V ~ T ~ ) ]
(35)
is the generalized sign function.
go
Aw
Figure 4: Normal contact law
Au
Figure 5: Masing-element in
x-direction
a
au
Figure 6: Hysteresis loop of the Masing model.
(0:n = 2,
x: Exponent n = 20)
The stiffness k1 (Fig. 5) can be represented as the difference between the elastic sticking stiffness
= ,u p n
and the plastic sliding stiffness of the hysteresis, kl = LO - k , (Fig. 6). The stress limit amax
(for simplicity p = patiction= p a l i d i n g ) and exponent n are parameters for adapting the hysteresis
loop to measured data. The hysteresis loop for the Masing model for the cases n = 2 and n = 20
is depicted in Fig. 6. The stiffnesses kl and k , and the stress limit 6,,, depend on the normal
relative distance Aw. Discussion and application of this model can be found in GAUL,BOHLEN
[7]
and GAUL,LENZ[8].
Coupled structure with bolted joints
The application example is depicted in Fig. 7. It consists of a rectangular plate connected to a
cover by four bolted joints.
I
Excitation Force
I
I
!
-A
!
!
I
Figure 7: Assembled structure with bolted joints. It consists of a rectangular plate
(dimensions 118 x 164 x 2 mm3) connected to a cover (external dimensions 122 x
168 mm2, sheet thichess 2 mm) by four bolted joints (M3.5).
Both members are modeled by 8-node brick elements with incompatible modes (BATHE[3]) to
provide a more accurate prediction of bending. The corresponding material parameters of the
members are given in Table 1. The contact interface is modeled by 8-node Zero Thickness Elements,
where the contact laws fiom the previous section have been implemented in the elements constitutive
law. The parameters of the contact laws are listed in Table 2.
Young’s modulus
Poisson’s ratio
Mass density
Cover
E=66GPa
u = 0.3
p = 2700 kg/m3
Plate
E = 27GPa
u = 0.4
p = 2400 kg/m 3
Table 1: Material parameters of the members
Normal contact
c1 = 102N/mm2
c2 = 0.1 c3
c3 = los N/mm2
Tangential contact
= lo4 N/mm2
k , = 0.05
p = 0.1
go=lfim
n=3
Table 2: Parameters of the contact laws
The structure is mounted on four rigid supports by bolts. As shown in Fig. 8, the rigid supports
are modeled by thrust bearings, and the bolts are modeled using a bolt pressure loading.
bolt pressure
I
thrust bearings
rigid support
Figure 8: a) Assembled structure with bolted joints. The complete structure is
shown in Fig. 7. b) The indicated layer between the two members represents the
Zero Thickness Elements.
The fiee and forced response properties of the system having the implemented nonlinear joint damping are computed. For the numerical analysis, the bolt pressure is always applied in the first step
and is held constant during a l l the subsequent steps.
In the first load case, a sinusoidal out-of-plane force with constant fiequency close to the f i s t
eigenfiequency is applied on the cover, as shown in Fig. 7 . In the following dynamic analysis, the
force is suddenly removed, and the damped free vibration response is calculated for a node close to
the excitation. The dynamic response signal consists of over 150 increments and 18 periods of the
lowest eigenfrequency. Fig. 9 shows the corresponding out-of-plane vibration of the response node
on the cover.
0.4
-ti
A
4 2
8
-:
B
B
0.3
0.2
0.1
0
-0.1
-0.2
-0
-3
-0
*
4
0
5
10
15
20
25
Time [ms]
Figure 9: Time history plot of the damped free vibration response. The crosses (x)
represent the calculated, discrete time signal and the line (-) is the corresponding
curvefit.
The envelope lines show the transition between the linear decrement in the slip phase and the
vanishing decrement in the stick phase.
In the second load case, the structure is excited with a sinusoidal force signal at the same location
as before. The excitation frequency is swept in increments of 10 Hz horn 700 Hz up to 800 Hz. The
response is evaluated at a specific node on the cover close to the excitation node. Each response signal
is calculated with 150 increments and includes enough periods to ensure a steady-state response.
The fiequency Response Function (FRF) is then computed by forming the ratio of the Fast Fourier
Transform (FFT) of the displacement response to the FFT of the force input, at the drive hequency
of interest. The computed FRF for the structure is depicted in Fig. 10. The FRF is relatively wide
in the vicinity of the natural hequency, thus indicating significant structural damping (2D= 2.7%).
This finding corresponds to preliminary experimental data.
-l@?O@
720
740
760
Requency
780
800
[&I
Figure 10: Displacement-force frequency response function for a response node
close to the excitation node. The crosses (x) represent the calculated, discrete
amplitudes, and the line (-) is the corresponding curve-fit.
4
Conclusions
The first part of this paper gives an overview and comparison between two different segment-tosegment contact elements, the so-called Thin Layer and Zero Thickness Elements. These segmentto-segment contact elements can have significant advantages compared to node-to-node spring and
dashpot elements. One advantage is that the complete 3-dimensional contact behavior between two
contacting elements can be modeled within one element. Furthermore, it is easier to implement
nonlinear, orthotropic contact behavior in constitutive equations for these elements. The effect of
contact nonlinearities on the overall system behavior can then be readily assessed.
The second part of the paper describes the implementation of special nonlinear normal and tangential
contact laws. The tangential contact law is represented using a hictional Masing element which is
able to model frictional hysteresis due to microslip effects.
With these contact laws built into the finite element model of a test structure, two load cases
were carried out to demonstrate the influence of the hictional tangential behavior on the overall
system response. In the hee vibration simulation, the out-of-plane displacement of the response
node at the beginning of the response followed a linear damping decrement, which is typical for
fiictional damping. After reaching the stiction limit, the response maintained a steady amplitude,
since vibrational energy was no longer dissipated.
A FRF for the structure was then computed using forced harmonic excitation between 700 Hz and
800 Hz. It was found that the introduction of friction in the contact area produced significant
damping in the structural response.
Finally, we conclude that Zero Thickness Elements having built-in phenomenological hysteresis
models, such as the Masing-model, are appropriate for modeling nonlinear stiffness and damping
mechanisms in the normal and tangential directions at solid-to-solid contact interfaces between
jointed members in built-up structures.
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