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World Academy of Science, Engineering and Technology 50 2011

Finite Element Modeling of Viscoelastic Stress Analysis under Moving Loads

Fei Qin, Yue Yu, and Thomas Rudolphi

Abstract-- In this paper a complete modeling and calculation for steady state viscoelastic stress analysis are presented utilizing Finite Element Method (FEM). The studied case is the deformation and stress distribution from a rigid cylinder body moving on the surface of a viscoelastic strip. Indentation rolling resistance occurs due to the viscoelastic nature of the bottom cover of the belt. The finite element analysis presented here provides an effective means to calculate the indentation rolling resistance. A coordinate system of reference moving with the load is applied in the modeling process. The generalized viscoelastic constitutive relations are incorporated into the modeling and treated specifically. This paper first describes the algorithm formulation of one dimensional case and later extends that to a generalized viscoelastic case. The numerical examples using this approach are given for both the one-dimensional and the twodimensional case. The analysis over the outcome yields the promising aspect of our model. The effects of the load speed, material properties and pressure distributions at the contact surface are carefully studied with the framework presented in our research. Keywords--FEM, Rolling Contact, Stress Analysis, Viscoelastic

Material

Fig. 1 Moving contact problem for Viscoelastic body

I. INTRODUCTION

R

OLLING olling contact problems, in which a viscoelastic material is loaded by cylindrical hard roller, as shown in Fig. 1, are frequently seen in industrial applications. Problems in viscoelastic rolling contact are of great importance in applications in the fields of textile, mining, paper and plastic processing and machine design. These problems have been attacked on a limited scale both theoretically and experimentally. [4,5,6,12,16]. There have been extensive publications on the problem of viscoelastic rolling contacts. Contact problems in which one of the bodies is made of a homogeneous linear viscoelastic material and the other is taken to be rigid have been studied by the finite element method by S. C. Hunter, L. W. Morland and Francis Lynch.[7] These studies have entailed the contact of a viscoelastic cylinder with a rigid plane or that of a rigid cylinder with a viscoelastic half-space. Various transformation methods have been used to reduce the problem to that for an elastic case.

Fei Qin was with Department of Aerospace Engineering, Iowa State University, Ames, IA 50011 USA. He is now with Department of Operations and Business Analytics, University of Cincinnati, Carl H. Lindner Hall, Cincinnati, Ohio 45221 USA (email: [email protected]) Yue Yu is with Department of Applied Mathematics, School of Science, Nantong University, Nantong, Jiangsu 226007 P. R. China Thomas Rudolphi is Professor in Department of Aerospace Engineering, Howe Hall, Iowa State University, Ames, IA 50011 USA (email: [email protected]) * Corresponding Author

Since the stress and strain around a contact region shows remarkable time and temperature-dependent behavior, and some boundary conditions vary a great deal with time, considerations based on the viscoelasticity of the material subjected to loading condition are therefore required to analyze these types of complicated problems.[9,10] Steadystate rolling contact type of problems can be analyzed by using coordinates associated with the load acting on the viscoelastic body. The triangles become a 'control volume' whose internal stress depends on the strain in the 'control volumes' through which the material has already flowed. Then the stress in this given control volume is influenced by the strain in each control volume through which the material previously passed. This allows the asymmetric pressure distribution within the contact region to be effectively predicted. This paper discusses a numerical methodology that enables calculation of the time-dependent viscoelastic stress and strain with finite element method. This method is a variation to the finite element technique used for elasticity analyses. The algorithm illustration starts for a simple one dimensional case where the viscoelastic constitutive law can be integrated analytically. Then a complete modeling of plain stress problem is demonstrated in a similar way while the viscoelastic relation will be treated numerically. In our viscoelastic stress analysis, the measured creep or relaxation functions are used directly. Stress analysis problems for linear viscoelastic material behavior are solved on the basis of integral operator stress-strain relations. We analyze the steadystate rolling process and consequently we are able to transform the dependency on the history of the strain to the spatial variation of the strain in the stress-strain constitutive law. Thus, the resulting stiffness matrix has a bandwidth, which is considerably larger than that used to solve an elastic problem. Detailed numerical examples are given and results are compared by the FEA codes in Matlab. For both 1D and 2D cases a simple Maxell constitutive model is used in the

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numerical example. A rectangular viscoelastic strip with a load traveling along a straight edge is analyzed as a two dimensional plane stress problem in this paper. II. MODEL DEVELOPMENT [1] One Dimensional FEA Matrices Formulation

(t )

R(t ) (0)

t 0

R(t

) ( )d

R(0) (t )

t 0

R(t

) ( )d

(5)

x

X

V0

Here R is the stress relaxation function. Since we are studying the steady-state rolling process and consequently it is possible to express the dependence on the past history of the strain rate in the stress-strain law on the spatial variation of the strain. According to (2), in the coordinates of the control volume X, the time integral can be rearranged using a new spatial variable h instead of by

b h, h v0

So t

a X1 X2 b

[ X , b ],

[t ,0 ]

(6)

b v0

X and now the constitutive relation (5) becomes

Fig. 2 Moving load at speed V0 in fixed Xi-coordinate system

The FEM method which satisfies the steady-state viscoelastic stress analysis under a moving load or displacement function is based on the finite element matrices technique in linear elastic and structural analysis. The kinetics of the problem is the statement of equilibrium or linear momentum balance for the static or dynamic problem, respectively. Fig-2 describes a viscoelastic strip under a moving load or displacement with a speed of V0. In terms of stress, the equation of motion in one dimension is given by d d 2u (1) f dx dt 2 The above equation is in a fixed coordinate system x. A new coordinate system X moving with the load is introduced via the relation corresponding to the following coalescence at time t = 0.

X X a X h X a h (7) ) R ( 0) ( ) ) ( )dh R( b v0 v0 v0 v0 The above equation can be further simplified by considering the property of isotropic linear viscoelasticity and variable substitution, X h X (8) ( X ) R ( 0) ( X ) R( ) ( h )dh b v0

(

a

With

du ( X ), ( h ) dX term of displacement u, (X )

du ( h ), (8) is expressed in dX

X b

(X )

R ( 0)

du (X ) dX

R(

h

X du ) ( h )dh v0 dX

(9)

x

X

v0 t

(2)

The historical strain at X in the viscoelastic constitutive law is now totally expressed in the spatial strain on the reverse direction of the flow speed V0 side. Substitute the linear viscoelastic constitutive (9) into (4), we obtain X 2 dN ( X ) X du h X du R(0) R( (X ) ) (h )dh dX X1 b (10) dX dX v0 dX

Then in the new moving coordinate X, for the steady-state motion (1) becomes d (3) f 0 dX We then use the usual finite element method to solve the above equation which is a variation of the sub domain method and a slightly modified Galerkin method. Introducing the shape function N(X), we get

X2 X1

N(X ) (X )

Let p

X2 X1

X2 X1

N ( X ) f ( X )dX

X2 X1

N(X ) (X )

, f

X2 X1

N ( X ) f ( X ) dX

and

u

X2 X1

[ N ] u , (10) becomes

dN( X ) dN( X ) dX u R(0) dX dX

X2 X1

dN( X ) dX

X b

R(

h X dN )[ (h)]dh u dX v0 dX

dN ( X ) dX

( X )dX

N(X )

(X )

X2 X1

X2 X1

N ( X ) f ( X )dX

(4) A material is viscoelastic if its stress response consists of an elastic part and viscous part. Upon application of a load, the elastic response is instantaneous while the viscous part occurs over time. Generally, the stress function of a viscoelastic material is given in an integral form. Within the context of small strain theory, the constitutive equation for an isotropic viscoelastic material in one dimension can be written as

p f (11) Comparing (11) with elastic FEA formulation, the second integral is the only addition that differentiates itself from the elastic stress analysis. Introduce the matrix [ B] dN , (11) is

dX

then written as

X2 X1

[ B ] T R ( 0 )[ B ]dX

u

X2 X1

[ B ]T

X b

R(

h v0

X

)[ B ( h )]dh u dX

p

f

(12)

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If there are m other elements between X and b, the integral from b to X can be rearranged as following,

X2 X1

m k 1 X2 X1

Applying the divergence theory

A

[ B ]T R (0)[ B ]dX

k X2

u

k

Xj

(

ij

wi )dA

S

(

ij

wi ) n j dS , (16) becomes

[ B]

T

h X R( )[ B ( h)]dh u X 1k v0

dX

A

ij

wi dA Xj

S

(

ij

wi )n j dS

A

wi f i dA

(19)

(13) Now put the above equation into vectors and stiff matrices, using

p

f

Formula (17) is the starting point of normal FEA element matrix establishment. In term of deviatoric and volumetric part of strain or stress

[K ]

[J ]

X2 X1

X2 X1

T

[ B ]T R (0)[ B ]dX

k X2

(14) (15)

Eij

eij

1 3

ij

E kk ,

ij

sij

1 3

ij

kk

(20)

[ B]

k X1

R(

h v0

X

Where Eij = Cauchy Strain tensor,

)[ B(h)]dh dX

eij = deviatoric strain,

(13) can be written in vectors as

m

Ekk = volumetric strain.

p f

(16)

ij

[K ] u

k 1

[J ] u

= Cauchy Stress tensor,

k

(14) represents the solution to the differential (3) in algebraic form for viscoelastic constitutive material. It relates the nodal forces and displacements to the applied force. This completes the FEA stiff matrix formulation of one dimensional viscoelastic stress analysis under a moving load of speed V0. Comparing this equation with the element stiffness equation of elastic material, the summarization of [J] matrix is the element stiffness matrix due to the viscoelasticity and need to be added to the whole assembly process of global stiffness matrix as well. [2] Generalized Viscoelastic FEA Formulation The numerical method described previously is ready to be extended to the general viscoelastic FEA case. Similarly, our approach for the steady-state rolling contact problem is also a modification of the finite element matrix displacement technique in structural analysis with elasticity. As we have already observed in one dimensional case, this modification will only apply to the stiff matrix K and account for the strain history in the constitutive relations within the viscoelastic material. Let's start from the equilibrium equation. Steady state rolling contact problem is analyzed in utilizing the coordinate system fixed to the load. Those elements now become control volumes while their internal stress depends on the strains in the control volumes through which the material has already flowed. In the moving coordinate X the equilibrium equation in tensors is demonstrated as

ij

sij = deviatoric stress,

kk

= volumetric stress

Under small deformation Cauchy Strain can be shown in term of displacements as

Eij

1 ui ( 2 Xj

uj Xi

)

(21)

The isotropic viscoelastic material within the context of small strain theory will then exhibit the constitutive relation as

ij

(t )

t 0

G (t

)

deij d

d

t 0

K (t

)

dEkk d d

(22)

X

fi

0

(17)

Here G (t) is the shear relaxation kernel function while K (t) is the stretch or bulk relaxation module. The deviatoric and volumetric parts of the stress can be assumed to follow different relaxation behavior. With the viscoelastic constitutive law, formula (22) is then put back into (19) to provide an overall stiffness of a viscoelastic structure. Since the stress response consists of an elastic part and a viscous part the relaxation kernel functions can also be separated into elastic (instantaneous) module and viscoelastic counterparts according to different viscoelastic models. So when the constitutive convolution integrals are sent to (17) to form the stiffness matrix the stiffness will contains an elastic part similar to stiffness matrix of elastic material while it also includes the matrix part from integrating the viscoelastic relaxation module. In the stiffness matrix formulation the integration of G (t), shear relaxation function and K (t), the stretch relaxation function, can be handled in the same way. For General Maxell Model,

t

j

t

sG

By the weak form of a weighted residual formulation of integrating over the volume (which is arbitrary), it becomes

G (t )

G0

i 1

Gi e

, K (t )

K0

i 1

Ki e

sK

(23)

wi (

A

ij

Xj

f i )dA 0

(18)

After the integration of this weak form of the equilibrium equations in term of displacements, we get

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World Academy of Science, Engineering and Technology 50 2011

m

[K ] u

k 1

[J ] u

k

p

f , which is in the matrix

[K ]

X2 X1

[ B ]T R ( 0)[ B ]dX

k X2 k X1

R ( 0) 1 1 l

1 1

(26)

form ready to be utilized for calculating the nodal solutions. For different elements and viscoelastic relaxation modules the integration method will be very different. In most cases some numerical analysis on forming the stiffness matrix components needs to be applied. The detailed technique will be involved in the following numerical verifications. We will start with a one dimensional Maxwell viscoelastic material FEA using linear element. Then we use above method in two dimension to analyze the moving load situation illustrated in Fig1. The nodal solutions are given in Matlab and both of them come from the previous described variation of FEA stress analysis of elastic materials. III. NUMERICAL VERIFICATION The FEA nodal solutions are analyzed and compared in this section to verify the effectiveness. The one dimensional situation is first solved and followed by the rigid body rolling on the viscoelastic strip case. In both cases the viscoelastic material is characterized as Maxell modal. The relaxation kernel functions thus can be expressed as Prony series. This representation has a prevailing physical meaning, that is corresponds to the solution of the classical differential model, parallel Maxwell model of viscoelasticity. [1] One Dimensional FEA Example

[J ]

X2 X1

X2 X1

[ B]T

R(

k X2 k X1

h X )[ B(h )]dh dX v0

h X )dh dX v0

K X X2 v0s

[ B]T [ B]

T

R(

X2 X1

[ B] [ B]E1v0 (e

2 kl v0s

e

e

K X X1 v0s

)dX

1 1 1 1

(27)

E1sv0 (2e l2

e

(1 k ) l v0s

(1 k ) l v0s

)

Furthermore, once the nodal values on each element are known, the internal forces within the element p(x) are also determined. Similarly, the force vector f(x) can also be obtained. Thus for linear element under constant loading force, the viscoelastic element governing equation (16), becomes

[K ]

u2 u1

m

[J ]

k 1

k u2 u1k

p1 p2

f 0l 1 2 1

(28)

Fig. 3 Maxwell Model of Elasticity

For the case described in Fig.2 the viscoelasticity is expressed in relaxation function. Schematic representation of this Standard Linear Solid model is shown below,

R(t )

E0

E1e

t

s

,

R (t )

X2 l

E1 e s

X

t

s

(24)

For the linear element where N 1 l is the element length, we have

, N2

X l

X1

Here [K] is identical to the stiffness matrix for elastic material and with the matrix [J] the resulting stiffness matrix will have a bandwidth considerably larger than that required to solve an elastic problem. The displacement of the modals within the control volume is then investigated. In the following examples for the material's viscoelasticity we let the relaxation function expressed in Prony series. The control volume coordinates X starts at left end with 0 to the right end at 5. The moving load is taken to be a symmetrical force distribution between 20% and 80% of the control volume system. Here the boundary conditions are set for the displacements to be zero at the both ends of the material. We used 200 linear elements for X between 0 and 5. The Matlab codes are based on linear elements and nodal solutions are compared for different moving speeds in Fig.4-Fig.6. It is observed that the model predictions agree well with the experimental results in relevant experimental papers and capture the material creep with time. The elasticity and viscosity of the material are showing differently in the deformations under different speeds and relaxation times. First for a given relaxation function our model can show the effects of the moving speed to the deformations. For example let relaxation time s=2, E0=20 and E1=80. Thus relaxation function R(t ) 20 80e 2 and all other conditions are kept the same. V equals to 125, 5 and 0.2 separately. From FEA displacement solutions we can observe the characteristic of viscoelastic materials behavior clearly. When the relaxation function is given including the relaxation modules and the relaxation time, the creep of the deformation gradually develops as the flowing speed increases. Due to linear viscoelasticity of the material the steady-state deformation will be non-symmetrical. The instant elastic

t

[B]

dN 1 dN 2 , dX dX

1 [ 1 1] . l

(25)

Using stiffness Matrix expression (14) and (15) the stiffness matrix for linear viscoelastic element in then obtained analytically as following

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World Academy of Science, Engineering and Technology 50 2011

deformation will be obtained under fast speed while an asymptotic level of deformation is reached under a slow speed.

speeds. The FEA analysis results are showing the difference from the results of the steady-state moving elastic stress analysis or steady-state viscoelastic stress analysis. Next for a constant flow speed let's check the effects of relaxation time s. R(t ) 20 80e S and all other conditions are kept the same. V is set to be 10 and s equals to 0.5, 5 and 50 separately.

t

Fig. 4 Nodal Displacements from the FEA modal at V=125

Fig.7 Nodal Displacements from the FEA modal S=0.5

Fig.5 Nodal Displacements from the FEA modal at V=5

Fig.8 Nodal Displacements from the FEA modal S=5

Fig.6 Nodal Displacements from the FEA modal at V=0.2

The stress relaxation is reflected on strain as the creep. The material undergoes an increased deformation when the flow speed increases until an asymptotic level of strain is reached. So the steady-state moving load situation the viscoelasticity of the material will still behave differently under various moving

Fig.9 Nodal Displacements from the FEA modal S=50

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World Academy of Science, Engineering and Technology 50 2011

The deformations of the system for different relaxation time s are illustrated from Fig. 6 to Fig. 9. Under this same flowing speed the time dependent deformation development is increasing with the increase of the relaxation coefficient s. The strain creep reaches its limit much faster with larger relaxation time s. The results are in good accordance with the property of the relaxation modals in describing viscoelasticity.

Fig. 10 FEM model of viscoelastic strip and rigid roller (Nodes: 205; elements: 160)

[2] FEA Numerical Solutions to Rigid Roller Sliding on Viscoelastic Strip Now with all the models and algorithms discussed previously, we are ready to use FEA to solve the steady-state stress for viscoelastic sliding problem. The difficulty in realizing the FEA technique lies in the integration of the viscoelastic constitutive relation in (22). The relaxation kernel functions include elastic modulus, which will form a part of the stiff matrix. This part will be identical to the stiff matrix elements in an elastic analysis. There are various ways of completing the integration of Prony components numerically in (22). In this paper we used the method illustrated in Lynch's paper, which is one of the early FEA analyses of viscoelastic rolling stress analysis. Finite difference is adopted in this technique. For steady-state linear viscoelastic material the integration is transformed from traveling period domain to the spatial domain traveled. The time variable will be converted to a spatial position on the relaxation module curve. This technique is similar to the integration in previous one dimensional analysis and can be applied to any other relaxation modals or to the modulus solely from experimental data. The equilibrium equations for the steady state viscoelastic analysis are then completed by joining the control volumes at their nodal points and assembling a steady state viscoelastic structure. As has been seen in one dimensional example the stiffness matrix for this problem is asymmetric and has a much larger bandwidth as compared to that for the corresponding problem in which the material is modeled as an elastic material. A finite element computer program based on equations (19), (22) and (28), which employs four node isoperimetric elements with 2x2 Gauss integration rule, has been developed. The codes in the paper are based on the finite element solution of the equations of elasticity in two dimensions. The numbers of nodes and elements are 160 and 205, respectively. The element used is a four-node plane strain quadrilateral one as a two-dimensional viscoelastic strip. The displacement at the bottom of the strip plate was fixed in the x and y directions. The relaxation modulus was approximated with a simple Prony series similar to the 1D simulation. The relaxation function in uniform shear deformation is G(t ) 80 160e In uniform compressive deformation it

t 0.2

We now discuss the parameter dependence of deformation in viscoelastic strip. We first review viscoelastic effects on strains under different speed of the rigid roller considered. The nodal displacements {u} are solved by solving the algebraic equation regarding the stiffness matrix. The coordinates flow speed is set to be 0.2, 10, 50 and 200 respectively.

Fig.11 Deformation of the viscoelastic material with V=0.2

. is

. By using (19) for the FEM analysis we can solve the FEM model and its boundary conditions shown in Fig. 8. The contact of rigid roller and viscoelastic strip is modeled as a uniform distributed pressure in the contact area.

K (t )

200 400e

t

0.4

Fig.12 Deformation of the viscoelastic material with V=20

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World Academy of Science, Engineering and Technology 50 2011

Fig.13 Deformation of the viscoelastic material with V=50

Although in our study we have applied even force distribution in the assumed contact area to illustrate the algorithm, it is more common for contact problems that contact area may not be known in advance and attempts to find the solution compatible with geometries and boundary conditions may be necessary. In our assumed contact region, the vertical surface displacement of the sheet is known to conform to the hard roller surface. And the external tangential loads acting on the sheet are zero for frictionless contact. Given the initial displacements of the nodes within the contact zone, the vertical forces acting on the boundary nodes in the contact zone are calculated. As a result of the viscoelastic response of the material, the contact length will not be symmetric about the centerline of the roller. This model can be also combined with other existing contact mechanics model in determining the appropriate contact pressure and area. The horizontal force generated by the load acting within the contact region contact zone is calculated from the sum of the non-symmetric force distribution. The total horizontal force acting at the interface between the strip and the rigid roller is equivalent to the rolling resistance force. The steady-state problem of linear viscoelastic strip in smooth contact with a rigid roll has therefore been solved by the contact mechanics with finite element method of above. The computed solution exhibits the behavior expected intuitively. Also, results computed from Matlab codes for two cases are compared favorably with those obtained by using different computer codes and experimental results developed in previous literatures.[16-18] All these indicate that the developed FEA methodology to solve the title problem does yield reliable results. IV. CONCLUSION

Fig-14 Deformation of the viscoelastic material with V=200

From Fig.11-14, we can see surface displacement versus speed. The deformations go from symmetric at slow flow speed to asymmetric as speed increases and then switch back to symmetric as flow speed continuing increasing. This is similar to the results in the one dimensional example and due to the viscoelastic property of the material. The integral function in (22) can reveal the elastic behavior at the limits of very slow and very fast load. Here,

G0

i 1

Gi and K 0

i 1

K i are, respectively, the shear

and bulk module at the fast load limit (i.e. the instantaneous module). G0 and K0 are the modules at the slow limit. The elasticity parameters input correspond to those of the fast load limit. Moreover by admitting (23), the deviatoric and volumetric parts of the stress are assumed to follow different relaxation behavior. So the small flow speed and fast flow speed actually correspond to the viscous deformation extent of the material.

A computational method for steady state viscoelastic stress analysis has been constructed for mixed boundary value problems. The method has been programmed for use on viscoelastic processing problem. The differences in principal strains were computed for the case of viscous-elastic rolling problem. In this paper a finite element modeling is built upon the traditional elastic stress analysis. A coordinate system of reference moving with the load is introduced and the viscoelastic constitutive law is represented in Maxell Model. The time-dependence of the strain is then transformed into spatial coordinates and solved either analytically or numerically. A detailed analysis of the viscoelastic material with moving load revealed the characteristic non-symmetric displacement, stress and strain fields associated with viscoelastic contact problems, as well as the apparent friction caused by viscoelastic dissipation within the rolled strip. The programmed analysis can serve as a useful stress analysis tool. With slight modifications the program could handle an even wider range of applications such as the more complex strain field, different constitutive curves and other boundary conditions.

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World Academy of Science, Engineering and Technology 50 2011

Future research on this method of analysis should fall mainly into two categories: First the application of the existing method to steady state problems other than rolling contact. Second the modification of the analysis to include behavior other than linear steady state viscoe1asticity. This paper has presented an overview of a linear viscoelastic finite element model to calculate the steady-state. The finite element method models the viscoelastic memory effect of the bottom cover of the strip and calculates the asymmetric pressure distribution within the contact region. The results obtained from the finite element method compared well with the trend in experimentally measured values for a range of variables in relevant papers.

REFERENCES

[1] B. Perssona, O. Albohr, C. Creton and V. Peveri, "Contact Area between a Viscoelastic Solid and a Hard, Randomly Rough, Substrate", Journal of Chemical Physics, Volume 120, Number 18, May 2004. Costantito Creton and Ludwik Leibler, "How Does Tack Depend on Time of Contact and Contact Pressure", Journal of Polymer Science: Part B: Polymer Physics, Vol. 34,545-554, 1996. Chpistensern, "Theory of Viscoelasticity - An Introduction", New York, Academic Press 1982. C. N. Bapat and R. C. Batra, "Finite Plane Strain Deformations of Nonlinear Viscoelastic Rubber-Covered Rolls", International Journal for Numerical Methods in Engineering, Vol 20, 191 1-1927 (1984). C. Wheeler, "Indentation Rolling Resistance of Belt Conveyors-A Finite Element Solution", Bulk Solids Handling, Vol. 26 No. 1 41-43, 2006. David E Hall, J. Cal Moreland, "Fundamentals of Rolling Resistance", Rubber Chemistry and Technology", Vol. 74, 525-539, 2002. Francis De S. Lynch, "A Finite Element Method of Viscoelastic Stress Analysis With Application to Rolling Contact Problems", International Journal for Numerical Methods in Engineering, Vol. 1, 339-394 (1969). J. Greenwood, J. Williamson, 1966, "Contact of Nominally Flat Surfaces," Proceedings of the royal society of London, A295, pp. 300319. J. Gotoh, M. Shiratori, S. Yoneyama and M. Takashi, "Viscoelastic Stress Analysis of a Strip Plate under Moving Contact with Dry Friction", Mechanics of Time-Dependent Materials 4: 43-56, 2000. J. Gotoh, Q. Yu, M. Takashi and M. Shiratori, "Experimental/Numerical Analyses of a Viscoelastic Body under Rolling Contact", Mechanics of Time-Dependent Materials 3: 245261, 1999. K. L. Johnson, 1985, "Contact mechanics," Cambridge: CUP. L. Nasdala, M. Kaliske, A. Becker, H. Rothert, "An efficient viscoelastic formulation for steady-state rolling structures", Computational Mechanics 22, 395-403, 1998. Mysore N.L. Narasimhan, "Principles of continuum mechanics", New York Wiley, 1993. Release 10.0 Documentation for ANSYS, Contact Technology Guide, 2004 S. Krenk, L. Kellezi, S.R.K. Nielsen and P.H. Kirkegaard, "Finite Elements and Transmitting Boundary Conditions for Moving Loads". Proceedings of the 4th European Conference on Structural Dynamics, Eurodyn'99, 447-452, 1999 S. Yoneyama, M. Takashi and J. Gotoh, "Photoviscoelastic Stress Analysis near Contact Regions under Complex Loads", Mechanics of Time-Dependent Materials 1: 5165, 1997. S. Yoneyama, J. Gotoh and M. Takashi, "Experimental Analysis of Rolling Contact Stresses in a Viscoelastic Strip", Experimental Mechanics, Vol. 40, No. 2, 203-210, June 2000. S. Yoneyama, Y. Morimoto, T. Nomura, M. Fujigaki, and R. Matsui, "Real-time Analysis of Isochromatics and Isoclinics Using the Phaseshifting Method", Experimental Mechanics, Vol. 43, No. 1, 83-89, March 2003. Thomas Rudolphi, "Introduction to Finite Element Analysis", Text Notes, 2004

[2]

[3] [4]

[5] [6] [7]

[8]

[9]

[10]

[11] [12]

[13] [14] [15]

[16]

[17]

[18]

[19]

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