A numerical solution for contact problem with finite ... - CyberLeninka

Ain Shams Engineering Journal (2012) 3, 141–151

Ain Shams University

Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com

MECHANICAL ENGINEERING

A numerical solution for contact problem with finite deformation in nonlinear Schapery viscoelastic solids F.F. Mahmoud, A.G. El-Shafei, A.A. Abdelrahman, M.A. Attia

*

Department of Mechanical Design and Production, College of Engineering, Zagazig University, Zagazig 44511, Egypt Received 11 September 2011; revised 7 December 2011; accepted 17 December 2011 Available online 3 February 2012

KEYWORDS Frictionless contact; Finite deformation; Schapery’s model; Nonlinear viscoelasticity; Updated Lagrange; Finite element method (FEM)

Abstract The present paper presents an incremental finite element model for the solution of the two dimensional quasi-static frictionless nonlinear viscoelastic contact problems with large deformations. The nonlinear viscoelasticity is simulated using the Schapery’s single-integral viscoelastic model with stress-dependent properties. The updated Lagrangian formulation is used to model the material and geometrical nonlinearity of the problem. To eliminate the need to store the entire strain histories, the constitutive equations are transformed into an incremental recursive form. The contact constraints are enforced by using the Lagrange multiplier method where the normal contact forces are treated as independent variables. The resulting nonlinear equilibrium equations are solved by the Newton–Raphson method in an incremental-iterative procedure. The capability of the developed numerical incremental-iterative solution procedure is demonstrated by analysing two different contact problems with different nature.  2011 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved.

1. Introduction

* Corresponding author. Tel.: +20 116503002, +2 0105662379. E-mail addresses: [email protected] (F.F. Mahmoud), [email protected] (A.G. El-Shafei), [email protected] (A.A. Abdelrahman), [email protected] (M.A. Attia). 2090-4479  2011 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved. Peer review under responsibility of Ain Shams University. doi:10.1016/j.asej.2011.12.006

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Contact problems are rather complicated from both the theoretical and numerical points of view. Such problems are characterized by a geometric and material discontinuity at the contact interface instead of the usual continuity property holding in solid mechanics. Consequently, contact problems are inherently nonlinear, involve variational inequalities and constrained minimizations. Since viscoelastic materials have a time-and stress-dependent characteristics resulting in a nonlinear response, the contact problem becomes more tedious when the contacting bodies behave as nonlinear viscoelasticity and undergoing finite deformations. Several finite element algorithms using hereditary integral models have been developed for analyzing viscoelastic constitutive models because of their capability in predicting

142 time-dependent material responses under more general loading histories than those of differential equation models. The recursive numerical algorithm which was first developed by Taylor et al. [1] allows formulating current stress tensor as a function of history variables stored in the previous time step and the current strain increments. Rogers and Lee [2], and Holden [3] developed a direct finite numerical method for solving the nonlinear integro-differential equations which arise in the finite deflection of thin linear viscoelastic beams due to constant load history. Shen et al. [4] derived an incremental variational equations based on the total and updated Lagrangian formulations for nonlinear viscoelastic large deformation problems. The FE formulation is developed for three-dimensional analysis and material nonlinearity is presented by strain softening function. Crochon et al. [5] presented two new strategies for implementing Schapery-type nonlinearly viscoelastic constitutive theories into finite element codes. The first strategy used the original differential equations that led to the integral formulation of Schapery-type constitutive theories and finite difference schemes. The second strategy was an improvement of recursive strategies, used by many authors, based on the integral formulation of the constitutive theory. Further studies on the modeling of linear/nonlinear viscoelasticity problems with large mechanical deformations, can be found in Kawahara [6], Christenson [7], Lin [8], and Bonet [9]. In the literature, the analytical treatments handling small deformation frictionless contact problems in linear viscoelastic materials were investigated have been developed by many authors, among of them Lee and Radok [10], Yang [11], Hunter [12] and Ting [13,14]. These treatments were based on the elastic–viscoelastic correspondence principle. Based on the correspondence principle, Fu [15] considered the frictionless contact problem of a rigid axisymmetric indenter of a polynomial profile with a viscoelastic half-space. Argatov and Mishuris [16] derive the general solutions to the problems of elliptical frictionless contact between thin compressible or incompressible layers of arbitrary viscoelastic materials. The obtained analytical solution is valid for monotonically increasing loading conditions. Considering finite element modeling of contact problems in linear viscoelastic solids, Barboteu et al. [17] developed a model for the frictionless contact of a deformable body and an obstacle. The Kelvin–Voigt constitutive law was used to describe the viscoelastic behavior of the body. The contact problem was modeled as the Signorini problem in a form with a zero gap function. Based on the backward Euler scheme, Fernandez et al. [18] studied the frictionless contact between a viscoelastic body and a deformable foundation. Quasistatic loading conditions were assumed while the contact is modeled by normal compliance. Fernandez and Sofonea [19] considered a mathematical model for the quasistatic viscoelastic frictionless contact problem. The material response was simulated using the Kelvin–Voigt viscoelastic model, while contact is modeled with a general normal damped response condition. A variational formulation of the problem including the existence of a unique solution to the model was presented. Fernandez and Martinez [20] introduced a numerical approximation of a viscoelastic frictionless contact problem. A fully discrete scheme was introduced, by using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize time derivatives. Mahmoud et al. [21] developed an incremental adaptive procedure capable of

F.F. Mahmoud et al. detecting the quasi-static response of linear viscoelastic contact systems. The linear response of viscoelastic materials is modeled using the Wiechert model. The Lagrange multiplier method was used to incorporate the contact conditions. This work was extended by Mahmoud et al. [22,23] to account for the friction and thermal effect for layered viscoelastic contact systems, the non-classical friction law was adopted. For more details about the different computational formulations and solution strategies for contact problems with different material characteristics are presented in [24,25]. In this paper, a numerical incremental procedure is developed for the solution of two dimensional quasi-static viscoelastic frictionless contact problems exhibiting material nonlinearity and finite deformations. The Schapery’s nonlinear single-integral viscoelastic model whose properties are stress-dependent is used for nonlinear viscoelastic behavior simulation. The constitutive equations are transformed into an incremental recursive form, which allows bypassing the need to store entire strain histories. The updated Lagrangian formulation is used to model the material and geometrical nonlinearities. The Lagrange multiplier method in which the normal contact forces are treated as independent variables is adopted to enforce the contact constraints. The resulting nonlinear equilibrium equations are solved by the Newton– Raphson method in an incremental-iterative procedure. Two contact problems with different geometry, loading conditions and contact configuration are solved to validate and verify the capability of the developed model. 2. The mathematical model Consider two deformable bodies, at least one of them behaves as nonlinear viscoelasticity, come into contact under the action of an external load as shown in Fig. 1. Assume that both contacting bodies occupy a bounded domain X in RN, N 6 3. The boundary C is assumed to consist of three disjoint measurable parts Cd, Cf and Cc. Cd and Cf are the portions of the boundary on which displacement and mechanical load are prescribed, respectively. Cc is the portion of the boundary that contains material surfaces which candidate to come into contact upon the application of loads. In addition to the boundary conditions that prescribed on the boundary Cd, a set of contact conditions should be imposed throughout the contact interface Cc. With the application of load, boundary conditions throughout the contact interface change continuously. Therefore, the F

f

Body 1

i1 gi c

i2 nc d

Figure 1

d

Body 2

Contact of two deformable bodies.

A numerical solution for contact problem with finite deformation in nonlinear Schapery viscoelastic solids contact region is advancing or receding according to the type of contact. Such problems belong to a class of nonlinear variational initial boundary value problems having inequality types of constraints. In the context of continuum mechanics, a concise statement of this problem can be stated as follows: If the applied loads vary slowly with time, the inertia term may be neglected, leading to a quasi-static problem. For such a condition with small deformations, the equilibrium equations can be defined by: rji;j ðuÞ þ fi ¼ 0 ðin X; tÞ;

ð1Þ

where rij(u) is the induced Cauchy stress tensor which is a function of the displacement vector u and fi is the body force per unit volume. The fact that the configuration of the body changes continuously in large deformation analysis is dealt with an alternative representation of the stress and strain tensors. This is done by using the 2nd Piola–Kirchhoff stress and Green strain tensors. Then Eq. (1) can be expressed as, Bathe [26],   @ @xi þ fi ¼ 0 ðin X; tÞ: SJI ðuÞ ð2Þ @xoI @xoJ For large deformation, the Green strain-displacement relation can be expressed as: 1 eij ¼ ðui;j þ uj;i þ uk;i uk;j Þ: 2

ð3Þ

The following boundary and initial conditions should be imposed; ui ¼ ui ðon Cd ; tÞ and Sij ðuÞnj ¼ Fi ðon Cf ; tÞ; t ui ¼ u0i and t Sij ¼ S0ij ðat t ¼ 0Þ;

ð4Þ ð5Þ

where ui is the prescribed displacement and nj is the unit outer normal vector at the point of interest and Fi is the prescribed surface traction. u0i and S0ij are the initial values of displacement and 2nd Piola–Kirchhoff stresses, respectively. The constitutive relationship for nonlinear homogeneous non-aging viscoelastic material can be expressed by the adopted Schapery’s creep model, Schapery [27], 2 t

eij ¼ ð1 þ mÞ4D0 t g0 t Sdij þ t g1

Z

t

DD 0

þ

ðt Ws WÞ

h i d s g2 s Sdij ds

3 ds5

  Z t ð1  2mÞ d½s g2 s Skk  t s dij D0 t g0 t Skk þ t g1 ds ðin X; tÞ; DDð W WÞ 3 ds 0

ð6Þ where teij is the Green strain vector, tSij is the 2nd Piola– Kirchhoff stress, D0 and DD (w) are the instantaneous and transient components of creep compliance respectively, and w is the reduced time (effective time) which is defined by: Z t Z s df df t s ; : ð7Þ W  WðtÞ ¼ W  WðsÞ ¼ f f 0 að Sij Þ 0 að Sij Þ The nonlinear material parameters are modeled as functions of current stress tSij. The parameter g0 measures the reduction or increase in elastic compliance due to stress and temperature. The parameter g1 measures the nonlinearity effect in the transient compliance. The parameter g2 accounts for the loading rate effect on the time-dependent response. The superscript denotes a dependent time variable. The transient component of creep compliance based on the use of the Prony series exponential series can be expressed as;

N X

DDðt WÞ ¼

143

Dn ð1  exp ½kn t WÞ;

ð8Þ

n¼1

where Dn is the nth coefficient of the Prony series and kn is the nth reciprocal of the retardation time, Dn and kn are assumed to be stress-independent. In addition to the boundary conditions, defined by Eq. (4), the following contact conditions must be imposed for all contact points throughout the contact interface: Gn ¼ un  gn 6 0; rn ðuÞ 6 0; rt ðuÞ ¼ 0 ðon Cc ; tÞ;

rn ðuÞGn ¼ 0 ðon Cc ; tÞ; ð9aÞ ð9bÞ

where Gn is the normal gap function. For the ith contact pair throughout the contact interface, uni = (ui1-ui2)ni is the component of the relative normal displacement vector un, ni is the outward normal vector, gni is the current gap measured along the outward normal, and the subscripts 1 and 2 refer to the first and second bodies, respectively. In Eq. (9a), the first inequality constraint represents the kinematical compatibility condition, where the two contacting surfaces cannot be interpenetrated, while the second one represents the kinetical condition, which states that no tensile normal stresses can develop along the contact interface. The third condition states that the contact pressure can only be nonzero when there is contact. On the other hand, Eq. (9b) expresses the frictionless contact condition, i.e. no tangential contact stress, rt, can be induced throughout the contact interface. 3. The recursive incremental form for the constitutive equations Working directly on the constitutive equation, defined by Eq. (6), leads to the requirement of solving of a set of integrals. However, rather than incorporating the relation (1) directly into the FE formulation, the constitutive equations will be incrementalized to be quite amenable to the implementation in a displacement-based FE modeling. Starting with the development of the uniaxial incremental constitutive relation, and then formulation for multiaxial is extended. Substitution of Eq. (3) into Eq. (1) yields the following uniaxial constitutive relation at time t + Dt: Z tþDt tþDt e ¼ tþDt g0 D0 tþDt S þ tþDt g1 0 N X

Dn ð1  exp½kn ðtþDt W  s WÞÞ

n¼1

d s s ½ g Sds: ds 2

ð10Þ

  Assuming that the term s gs2 S is to be linear over the current time increment, hence, integration of Eq. (10) yields tþDt

e ¼ tþDt g0 D0 tþDt S þ tþDt g1 tþDt g2 tþDt S

N X

Dn  tþDt g1

n¼1 N X



Dn tþDt qn ;

ð11Þ

n¼1

where tþDt

qn ¼

Z 0

tþDt

exp½kn ðtþDt W  s WÞ

d s s ½ g Sds: ds 2

ð12Þ

In order to get the recursive form of Eq. (12), the integration will be divided into two parts; the first part represents the history up to the previous time step, i.e. integration limits

144

F.F. Mahmoud et al.

[0, t], while the second part represents the current time step, i.e. integration limits [t, t + Dt], where t + Dt is the current time. Assuming that the shift factor, a, is not directly a function of time, the following recursive form of Eq. (12) is obtained, tþDt

qn ¼ exp½kn tþDt DWt qn   1  exp½kn tþDt DW tþDt tþDt þ ½ g2 S  t g2 t S: kn tþDt DW

e ¼ tþDt DtþDt S  tþDt Q;

ð13Þ

ð14Þ

with  1  exp½kn tþDt DW ; tþDt kn DW n¼1   N X 1  exp½kn tþDt DW tþDt : D ¼ tþDt g0 D0 þ tþDt g1 tþDt g2 Dn 1  kn tþDt DW n¼1 tþDt

Q ¼ tþDt g1

N X



Dn

tþDt

qn  tþDt g2 tþDt S

¼

1 2

þ

tþDt

g0 J0 þ tþDt g1 tþDt g2

N 1 tþDt X g1 Jn 2 n¼1



t

g2 t Sdij

ð16Þ

 ! N X 1  exp½kn tþDt DW tþDt d Jn 1  Sij kn DtþDt W n¼1

 1  exp½kn tþDt DW  exp½kn tþDt DWt qij;n ; tþDt DW kn ð19Þ

 # N X 1 tþDt 1  exp½kn tþDt DW tþDt tþDt tþDt tþDt ekk ¼ g0 B 0 þ g1 g2 Bn 1  Skk 3 kn tþDt DW n¼1   N X 1 1  exp½kn tþDt DW kn tþDt t  exp : þ tþDt g1 Bn t g2 t Skk DW q kk;n 3 kn tþDt DW n¼1 ð20Þ

The relation for the incremental deviatoric and volumetric Green strains can be obtained from Eqs. (19) and (20), respectively; Dedij ¼ tþDt JtþDt Sdij  t Jt Sdij  þ

N 1t t d X g2 Sij Jn 2 n¼1



tþDt

N 1X Jn ½tþDt g1 exp½kn tþDt DW  t g1 t qij;n 2 n¼1

g1

The above equations are used to determine the unknown stress increments for a given strain increments and the previous history rate, i.e. hereditary integrals. The main difficulty is that the nonlinear stress functions at the current time are not known. Therefore, an iterative procedure is needed for the stress correction. An expressions for the approximate or trail incremental stresses can be obtained from Eqs. (21) and (22) by assuming tþDt

"

tþDt

Then the total incremental Green strain tensor is given by: 1 tþDt Deij ¼ tþDt Dedij þ dij tþDt Dekk : ð23Þ 3

ð15Þ

Eq. (15) allows for the incremental strain–stress calculation for a time increment Dt, which will be added to the total strain from the previous time step (t). The obtained recursive formulation for uniaxial nonlinear viscoelasticity can be generalized to derive the recursive formula for multiaxial constitutive relations. This can be performed by splitting the deviatoric and volumetric strain–stress relations. Stress components are chosen as the independent state variables. The formulation further assumes that the total strains are known for each time increment. Hence, the formulation is carried out with a constant incremental strain rate. This is consistent with many nonlinear constitutive models implemented within a displacement-based FE. The numerical integration results for the uniaxial formulation, presented by Eqs. (14)–(16), can be exploited to express the deviatoric and volumetric strain components, respectively, in the general forms, such that tþDt d eij

N 1X Bn ½tþDt g1 exp½kn tþDt DW  t g1 t qkk;n 3 n¼1   N X 1 1  exp½kn tþDt DW t 1  exp½kn t DW : þ t g2 t Skk Bn tþDt g1 g 1 3 kn tþDt DW kn t DW n¼1

Dekk ¼ tþDt BtþDt Skk  t Bt Skk 

ð22Þ

The expression tqn in the above equation refers to the hereditary integral at the end of the previous time t for every term in the Prony exponential series. This expression represents a history variable that needed to be stored at the end of each time increment. Substituting Eq. (13) into(11) gives the following expression for the current total Green strain tþDt

tþDt

 1  exp½kn tþDt DW t 1  exp½kn t DW ; g1 tþDt t DW kn kn DW ð21Þ

a ¼ t a;

tþDt

DW ¼ t DW;

tþDt

gj ¼ t gj ;

j ¼ 0; 1; 2;

ð24Þ

which lead to the following form of the approximate incremental 2nd Piola–Kirchhoff stresses tþDt

tþDt DSij ¼ tþDt DScr DShij : ij þ

ð25Þ

where tþDt

DScr ij ¼

  1 1 tþDt m tþDt Deij þ dij Dekk ; ðtþDt g0 D0 þ tþDt g1 tþDt g2 tþDt l ð1  2mÞð1 þ mÞ Þ ð1 þ tÞ

ð26Þ

  N X g1 1 tþDt DShij ¼ tþDt Dn ðexp½kn tþDt DW  1Þ t qij;n þ dij t qkk;n : 3 ð g0 D0 þ tþDt g1 tþDt g2 tþDt l Þ n¼1 tþDt

ð27Þ with tþDt

¼ l

N X n¼1

  1  e½kn tþDt DW : Dn 1  kn tþDt DW

ð28Þ

4. The finite element model For the formulation of the contact algorithm based on the incremental procedure, Bathe [26], the contact conditions can be imposed into the overall equilibrium equations by considering the total potential of the contact forces. Hence, the variation of the total virtual work can be expressed as: dUtot ¼ dU  d

X

WK ¼ 0;

ð29Þ

K

where dU is the incremental virtual work, excluding contact forces, within the contacting bodies leading to incremental equilibrium equations without contact conditions, and P W is the incremental potential of the contact forces. K K This term can be interpreted as a Lagrange multiplier contribution to impose the contact conditions. The finite element equilibrium equations for the contact system can be derived as follows. Firstly, the equilibrium equations without contact are developed based on the virtual work principle. Secondly, the contact conditions are enforced directly into the obtained equilibrium equations by adding the virtual work P done by the normal contact force, W . K K

A numerical solution for contact problem with finite deformation in nonlinear Schapery viscoelastic solids 4.1. The finite element equations without contact According to the updated Lagrangian, U.L., formulation in which all static and kinematic variables are referred to the configuration at time t, the virtual work principle yields the following weak form of the equilibrium equations at time t + Dt, assuming that the externally applied loads are configuration dependent, Z Z TtþDt tþDt tþDt tþDt dU ¼ de t SðuÞ dX  duTtþDt f tþDt dX t t tþDt X tþDt X Z tþDt duTtþDt F tþDt dC ¼ 0; ð30Þ  t tþDt C f

where e is the Green-strain vector corresponding to the virtual displacements, given by Eq. (3). The first term in the righthand side of Eq. (30) is virtual work due to internal stresses, while the second and third terms represent the external virtual work due to body force and surface traction and du is the virtual displacement vector. To derive an incremental form for the equilibrium equations given by (30), it is necessary to write an expression for the virtual work during time increment from t to t + Dt. The incremental solution can be obtained in terms of an incremental displacement Du, which will satisfy the virtual work expression. Assuming that the solution for the configuration at time t has been obtained and (k  1) iterations have been performed to obtain the solution at time t + Dt, thus the stress, strain, displacement, surface force, and body force at the end of time t + Dt can be defined as; tþDt f t

¼ tt f þ tþDt f; t ¼ tt u þ Du;

tþDt F t tþDt e t

¼ tt F þ tþDt F; tþDt u t t

¼ tt e þ tþDt De; t

¼ tt S þ tþDt DS ¼ tt r þ tþDt DS; t t tþDt Df, tþDt DF, t t

tþDt S t

ð31Þ

tþDt DS t

where and are the incremental values of the body force, the surface traction, and the 2nd Piola–Kirchhoff stress vectors, respectively, tt r is the Cauchy stress vector. The constitutive law that relates the incremental Green strain to the incremental 2nd Piola–Kirchhoff stress tensors, Eq. (25), can be rewritten as: tþDt DS t

¼ tþDt C t ne

ne tþDt De t

þ tþDt DSh ; t

ð32Þ

C is the nonlinear viscoelastic stress–strain relation where tþDt t matrix and tþDt DSh is the stress increment which expresses the t material history, defined by Eq. (27). For any iteration (k) within time t + Dt, the variational form of the equilibrium equations, defined by Eq. (30) is reformulated utilizing the FEM. Substitution of Eqs. (31) and (32) into the first Eq. (30) with linearization yields: Z ðk1ÞT tþDt neðk1Þ tþDt ðk1Þ tþDt dDuTtþDt BL C B L DuðkÞtþDt dX t t t t tþDt Xðk1Þ Z ðk1ÞT ðk1Þ tþDt ðk1Þ tþDt dDuTtþDt B NL tþDt r B NL DuðkÞtþDt dX þ t t t t tþDt Xðk1Þ Z ðk1ÞT tþDt ðk1Þ tþDt tþDt dDuTtþDt BL r dX þ t t t tþDt Xðk1Þ  Z ðk1Þ ðk1ÞT tþDt tþDt dDuTtþDt BL DSh tþDt dX þ t t t tþDt Xðk1Þ Z tþDt duTtþDt f tþDt dX ¼ t tþDt Xðk1Þ Z tþDt duTtþDt F tþDt dC; ð33Þ þ t tþDt Cðk1Þ f

145

where BL and BNL are the linear and nonlinear components of the strain-displacement matrix, respectively. Both of the first two integrals in the left-hand side of Eq. (33) represents the virtual work of the incremental strain energy, while the third and fourth integrals represent the virtual work due to internal stresses and material damping, respectively. For the whole domain of the contacting bodies, the equilibrium equations without contact, Eq. (33), can be written in the following form:  ðk1Þ tþDt ðk1Þ KL þ tþDt K NL tþDt fDugðkÞ t t ðk1Þ

¼ tþDt R ext  tþDt F int t t tþDt ðk1Þ KL t

ð34Þ

;

tþDt ðk1Þ K NL t

where and the represent overall linear and nonlinear components of the stiffness matrix, tþDt R ext is the t external load vector due to body forces and surface tractions ðk1Þ and tþDt F int is the overall internal force vector, due to both t the internal stresses and material damping. 4.2. Incorporation of frictionless contact conditions The contact conditions, defined by Eq. (9) should be imposed throughout the contact interface. The displacements and the contact forces should be calculated within each contact zone. The node-to-segment contact approach is adopted for modeling the contact interface, Bathe and Chaudaury [28]. In this approach one body is assumed to be a contactor, while the other is a target. The contact interface is composed of contactor nodes that are candidates to come into contact with the target segments. The contactor nodes should not penetrate the target segments. The non-penetration condition, defined by the first inequality condition of Eq. (9a), is tested for each contactor node on the contact interface. The contactor node K is assumed to be in contact with the target segment Sl. To satisfy the non-penetration condition, the node K should be located at a point P, where P is the physical contact point. Fig. 2 shows how node K has come into contact with the target segment formed by nodes A and B. To incorporate the contact condition into the equilibrium equation, the variation of WK for a generic node K on the contactor surface (and of the corresponding nodes on the target surface) should be evaluated. In the incremental iterative solution procedure, it is assumed that, the response at time t has been calculated and that (k  1) iterations have been performed to calculate the solution at time t + Dt. The formulation of the governing equations is achieved by establishing WK for the next iteration (k). Hence, the work done due to contact forces at the contactor node K and target segment nodes A and B is evaluated as,  ðkÞ ðkÞT ðkÞ ðk1Þ tþDt WK ¼ tþDt kK tþDt DuK þ tþDt GK ðkÞT tþDt

þ tþDt kA

ðkÞ

ðkÞT tþDt

DuA þ tþDt kB

ðkÞ

DuB ;

ð35Þ

tþDt ðkÞ kK

where is the total contact force at contactor node K at ðkÞ ðkÞ the end of iteration (k) at time t. Also, tþDt kA and tþDt kB are the total contact forces (reactions) at target nodes A and B, respectively. The contact force at the contactor node K should be balanced by the equivalent forces at nodes A and B, leading to  ðkÞ ðkÞ ðkÞ tþDt ðkÞ kA ¼  1  tþDt nP tþDt kK and tþDt kB ðkÞ tþDt ðkÞ kK ;

¼ tþDt nP

ð36Þ

146

F.F. Mahmoud et al. Contactor boundary K t+Δt t+Δt

t+Δt

Finally, combination of Eqs. (34) and (41) yields the following overall equilibrium equations of a contact system at kth iteration at time t+Dt λ nA

"

tþDt ðk1Þ KL t

λ nK A

λ nB

P

t+Δt

tþDt

(

ξp

¼

Target boundary

B

Contact forces at the contact interface.

Figure 2

with tþDt ðkÞ nP tþDt ðkÞ kK

ðk1Þ

¼ tþDt nP

tþDt ðk1Þ kK

¼

ðkÞ

þ DnP : þ

tþDt

ð37Þ

ðkÞ DkK :

Cðk1Þ n

tþDt R ext t





T

tþDt

Cðk1Þ n 0

#

DuðkÞ DkðkÞ n



ðk1Þ ðk1Þ tþDt ðk1Þ F int þ tþDt C n tþDt kn t t t ðk1Þ tþDt fGg t

) ð43Þ

:

Solution of the equilibrium equations, defined by Eq. (43), yields both the incremental displacement and normal contact force vectors Du and Dkn, respectively. Based on the Lagrange multiplier method and by treating both displacement and Lagrange multiplier fields as independent variables, the contact conditions are satisfied exactly.

ð38Þ

ðkÞ tþDt DkK

is the change in the contact force vector at the where contactor node K (incremental Lagrange multiplier). Substituting of Eqs. (36)–(38) into Eq. (35), and linearzing Eq. (37) ðk1Þ tþDt ðkÞ nP  tþDt nP , it follows of,



o n

T



ðkÞ ðk1Þ

ðkÞ

tþDt WK ¼  tþDt kK þ tþDt DkK tþDt nðk1Þ n h ðkÞ ðk1Þ ðk1Þ tþDt ðkÞ tþDt  1  tþD nP DuK þ tþDt GK DuA io ðk1Þ ðkÞ ; ð39Þ þ tþDt nP tþDt DuB T

ðkÞ

ðk1Þ

þ tþDt K NL t

where tþDt DkK tþDt nðk1Þ is the change in the normal component of the contact force at the contactor node K. The negative sign of the normal vector t+Dtn(k1) is attributed to the increasing of the normal contact force is acting into the opposite direction of the normal vector t+Dtn(k1). It is also noticed that, the first term of Eq. (39) contributes to the force vector, while the second term generates the contact constraints in the incremental equilibrium equations. Variation of the potential of a generic contactor node K, defined by Eq. (39) with respect to ðkÞ ðkÞ the unknown vectors tþDt DuK and tþDt DkK , yields  h  i T ðk1Þ ðkÞ ðk1Þ ðk1Þ  tþDt kK þ tþDt DkK tþDt nðk1Þ 1  1  0:5tþD nP  0:5tþD nP ¼ 0:

ð40aÞ

5. The solution algorithm This section is devoted to present the FE solution procedure for predicting the frictionless viscoelastic contact configuration considering both material and geometrical nonlinearities. In the framework of the U.L. formulation, equilibrium equations, defined by Eq. (43), are constructed and solved by the Newton–Raphson method in an incremental-iterative procedure. The space domain of the problem will be discretized into FEs. Element equations are formulated by using Gauss-quadrature integration scheme. The time domain is divided into time steps, each time step is also divided into load increments. The choice of size of the time step, and the number of load increments within each time step depend mainly on the nature of the problem to be analyzed. The size of the time step should be small enough to obtain a converged solution during the iterative scheme. Now, assuming that the equilibrium configuration is completely known at time t, and we seek the solution at time t + Dt. Within the time t + Dt, it is postulated that (k  1) iterations have been performed to obtain the solution at this time. The solution procedure to obtain the state for the kth iteration and hence the equilibrium configuration at time t + Dt is depicted as follows:

and T

 tþDt nðk1Þ

n T

 tþDt nðk1Þ

tþDt

tþDt

 o ðkÞ ðk1Þ tþDt ðkÞ ðk1Þ ðkÞ DuK  1  0:5tþDt nP DuA  0:5tþDt nP tþDt DuB ðk1Þ

GK

¼ 0:

ð40bÞ

Rewriting Eq. (40) in the matrix form; ) " #( ðkÞ ) ( tþDt ðk1Þ tþDt ðk1Þ tþDt ðk1Þ 0 Cn Cn kn DuK ¼ ; ðkÞ tþDt ðk1ÞT tþDt ðk1Þ Cn 0 GK DknK

1. Construct the linear and nonlinear components of the element stiffness matrix; Z T tþDt eðk1Þ tþDt ðk1Þ tþDt ðk1Þ tþDt ðk1Þ tþDt KL ¼ BL C BL dXe : t t t t tþDt Xeðk1Þ

tþDt eðk1Þ K NL t

¼

ð44Þ

Z

T

tþDt Xeðk1Þ

tþDt ðk1Þ tþDt ðk1Þ tþDt ðk1Þ tþDt B NL t r B NL dXe : t t

ð41Þ

ð45Þ

where is an operator vector, mainly depends on the type of the discrete model, working on the active constraints. This operator matrix includes the constraints of compatible boundary displacements due to contact after (k  1) iterations. From Eq. (40a), 2 3 T tþDt nðk1Þ  6 tþDt ðk1ÞT ðk1Þ 7 tþDt ðk1Þ 7:  n 1  tþDt nP Cn ¼6 ð42Þ 4 5

2. Construct the element internal vector, due to both the internal;

tþDt

Cðk1Þ n

tþDt nðk1Þ

T

tþDt ðk1Þ nP

tþDt eðk1Þ F int t

¼

Z tþDt Xeðk1Þ

T tþDt ðk1Þ BL t



ðk1Þ tþDt ðk1Þ r þ tþDt DSh t t



tþDt

dXe :

ð46Þ

3. Construct the element external force vector due to body forces and surface tractions based on configurationdependent external loads;

A numerical solution for contact problem with finite deformation in nonlinear Schapery viscoelastic solids 147 Z T e functions of the equivalent Von Mises stress only, tþDt e R ext ¼ Ne tþDt f tþDt dXe tþDt ðkÞ t t S e , such that, tþDt Xeðk1Þ t Z * +i T e nj tþDt ðkÞ X þ Ns tþDt F tþDt dCe ; ð47Þ S e  ro ðkÞ ðkÞ t t tþDt i tþDt Ceðk1Þ gj ¼ 1 þ jj ; tþDt a t t ro i¼1 where Ne and Ns are the element and line shape function * +i n tþDt ðkÞ X matrices, respectively. S e  ro i t ¼1þ d j ¼ 0; 1; 2; ð58Þ 4. Construct the global stiffness matrix and the global ro i¼1 force vector for the contactor domain. ðk1Þ where the function ÆYæ equals to Y if Y > 0 and equals to 0 if 5. With tþDt fP as the parameter which defines the posit Y 6 0. r0 is the effective stress limit, experimentally detertion of each physical contact point P at the start of iterðk1Þ mined, that indicates the transition from linear to nonlinear ation (k), construct the operator matrix tþDt C , t n viscoelastic behavior of the material. Moreover, values of the NC X tþDt ðk1Þ tþDt ðk1Þ coefficients jij and di can be calibrated from the creep and C ¼ C ; ð48Þ t t n ni recovery tests. i¼1 15. Calculate the incremental and updating the reduced where NC is the total number of contact constraints. time, 6. If the target is rigid, go to step 8. Dt t ðkÞ 7. Repeat steps from 1 to 5 for the target domain tþDt DWðkÞ ¼ ; tþDt W ¼ : ð59Þ t t tþDt ðkÞ tþDt ðkÞ 8. Solve the incremental equilibrium equations, defined by a a t t Eq. (43), to get both the incremental displacement and 16. Evaluate and the incremental stress vector due to matenormal contact force vectors Du(k) and DkðkÞ n , respectively. ðkÞ rial history, tþDt DS h according to Eq. (27). 9. Updating the displacement and the nodal contact force t ðkÞ 17. Evaluate and update the material history vector, tþDt qn , vectors, respectively, t based on Eq. (13) ðk1Þ tþDt ðkÞ u ¼ tþDt u þ DuðkÞ : ð49Þ t t tþDt ðkÞ k t DWðkÞ tþDt ðk1Þ tþDt ðkÞ k ni t

ðk1Þ

¼ tþDt k ni t

þ DkðkÞ ni ;

i ¼ 1; . . . ; NC:

ð50Þ

t

qn ¼ e

¼

tþDt ðk1Þ g ni t

þ

DuðkÞ ni ;

i ¼ 1; 2; . . . ; NCT:

ð51Þ

is the displacement vector in normal where tþDt DuðkÞ ni direction and NCT is the total number of candidate contact points. 11. Update the contact nodes status: 11.1 based on the new value of the normal gap value;  6 0: ) Node i is in contact tþDt ðkÞ g ¼ ð52Þ t ni > 0: ) Node i is out of contact 11.2 based on the new value of the normal contact force;  P 0: ) Out of contact; tension release tþDt ðkÞ k ¼ t ni < 0: ) Still in contact: ð53Þ 12. Evaluate the incremental and updating the Green strain vector, respectively; tþDt DeðkÞ ¼ tþDt ðBL t t tþDt ðkÞ tþDt ðk1Þ e ¼t e t

ðkÞ

þ BNL Þ DuðkÞ : þ

qn

þ

1e kn t DWðkÞ

! h

tþDt ðkÞ tþDt ðkÞ g2 t S t

ðk1Þ tþDt ðk1Þ S t

 ttþDt g 2

i :

ð60Þ

18. Evaluate the deformation gradient matrix, evaluate the Cauchy-stress vector; tþDt ðkÞ r t

¼ tþDt @ t

ðkÞ tþDt tþDt ðkÞ St @ : t

tþDt ðkÞ @ , t

then ð61Þ

19. Repeat steps 1–18 until the displacement convergence is satisfied. 20. Once the convergence is satisfied, update all system variables; tþDt u t

tþDt ðkÞ u ; t

tþDt S t

tþDt ðkÞ S ; t

tþDt DW t tþDt a t

tþDt k ni t

tþDt ðkÞ k ni ; t

tþDt G ni t

ðkÞ tþDt DSh ; t ðkÞ tþDt DWðkÞ ; tþDt q n tþDt qn ; t t t ðkÞ tþDt ðkÞ a ; tþDt g j tþDt gj : t t t

tþDt DSh t

tþDt s t

tþDt ðkÞ G ni ; t tþDt ðkÞ s ; t

ð62Þ 21. Initiate a next time step and repeat from steps 1 to 20 until finishing the considered time domain.

ð54Þ

tþDt DeðkÞ : t

ð55Þ

13. Evaluate the incremental and updating the 2nd Piola– Kirchhoff stress vector, respectively; ðkÞ tþDt DSðkÞ ¼ tþDt C tþDt DeðkÞ : t t t ðk1Þ tþDt ðkÞ S ¼ tþDt r þ tþDt DSðkÞ t t t

t

kn t DWðkÞ

10. Update the current gap for all candidate contact nodes, i, such that tþDt ðkÞ g ni t

n

ð56Þ ðk1Þ

þ tþDt DSh t

:

ð57Þ

14. Calculate and update the nonlinear stress-dependent material parameters. The stress-dependency of the nonlinear functions is assumed to be general polynomial

6. Applications The developed numerical recursive-incremental solution procedure is implemented into a two dimension time-dependent nonlinear displacement-based FE model. Due to the complexity of the large deformation contact problem, in addition to the nonlinear viscoelastic response, there is no analytical closed form solution for such type of problems. Therefore, comparison of the numerical results with the elastic–viscoelastic correspondence principle-based solutions becomes not valid. Naghieh et al. [29,30] and Mahmoud et al. [21,23] adopted

148

F.F. Mahmoud et al.

an alternative comparison scheme, in which only the instantaneous contact pressure of the linear viscoelastic problem is compared with the corresponding equivalent elastic solution. Based on this comparison procedure, solution of the large deformation contact problems in nonlinear viscoelastic solids will be verified. To demonstrate the applicability of the proposed model for solving large deformation contact problems with nonlinear viscoelastic response, two different problems, with different geometries, loading conditions and contact configurations, will be analyzed. The contact status, contact area, and contact stress distribution, is completely detected at different instants of time. Throughout the following analysis, the used viscoelastic material is simulated using nine terms of the Prony exponential series, the characteristics of this series is given in Table 1. Fourth-order polynomials are sufficient for calibrating the nonlinear stress-dependent functions. The nonlinear stress-dependent functions are assumed to follow Eq. (58). The parameters describing these functions, Eq. (58), assuming fourth order polynomials are given in Table 2, Haj-Ali and Mulina [31]. When the material behaves as nonlinear elasticity, the same constitutive relationship given by Eq. (6) is instantaneously applied at t = 0.

Table 2 Nonlinear stress dependent parameters in the Schapery model. j1r

j2r

j3r

j4r

g0 g1 g2

0.183 0.067 0.773

0.567 0.133 9.707

1.067 2.133 5.787

0.533 2.133 8.533

ar

d1r 0.373

d2r 2.580

d3r 5.227

d4r 3.520

R=100 mm P

6.1. Contact of a nonlinear viscoelastic cylinder with a rigid substrate

120

Hertz theory NLEL, FEM P =200 N P =400 N P =600 N P =800 N

90 Contact pressure, MPa

Consider a plane strain contact problem of a cylinder that pressed against a rigid substrate by a line force directed along its center (creep response) as shown is Fig. 3. The effective stress r0 that determines the limit of material linear response is taken to be 55 MPa. The creep loading history is given by P = 800 N. Taking the advantage of symmetry and frictionless contact conditions, one-quarter of the cylinder is needed to be modeled. This problem is modeled with 717 four-nodes quadrilateral and constant strain three-nodes elements with 642 nodes. The contact interface is modeled with 80 nodes. The time step used in the analysis is 0.25 s. Following the verification methodology mentioned above, solution of contact of nonlinear elastic, NLEL, cylinder behavior is obtained using the proposed model. Fig. 4 shows the variations of the NLEL contact pressure along contact interface at various load levels. The distribution deviates from the Hertzian theory. This deviation increases as the load increases

A deformable viscoelastic cylinder on a rigid substrate.

Figure 3

60

30

0 0.0

1.5

3.0

4.5

6.0

Half contact length, mm

Figure 4 Variations of nonlinear elastic contact pressure along the contact surface at various extents of load. Table 1 Elastic compliance and the Prony coefficients of the material. n

Dn · 106, MPa1

1 2 3 4 5 6 7 8 9

23.6 5.6602 14.8405 18.8848 28.5848 40.0569 60.4235 79.6477 162.179

D0 = 270.9 · 106, MPa1. kn = 10(n1), s1. m = 0.3.

due to the sources of material and geometrical nonlinearity. Such obtained results are in agreement with Zhong and Sun [32] and Chandrasekaran et al. [33]. Then the problem is resolved with nonlinear viscoelastic, NLVE, cylinder behavior. The NLVE contact pressure distribution throughout contact length at different time instants is illustrated in Fig. 5. It is depicted that the instantaneous NLVE contact pressure distribution, t = 0, is equivalent to the NLEL distribution, this result coincides with Rogers and Lee [2] and Naghieh et al. [29,30]. Also, when time evolves, the contact pressure decreases and the contact area increases due to hereditary nature of viscoelastic material.

A numerical solution for contact problem with finite deformation in nonlinear Schapery viscoelastic solids

149

Dp

120

50 mm

NLEL NLVE, Time= 0. sec NLVE, Time= 50. sec NLVE, Time= 500. sec

Contact pressure, MPa

90

100 mm

60

Figure 7

A viscoelastic block on a flat rigid foundation.

30

120 0

0.0

1.5

3.0

4.5

6.0

Figure 5 Variations of nonlinear viscoelastic contact pressure along the contact surface at various time instants.

105

Central contact pressure, MPa

LVE NLVE

Contact pressure, MPa

Half contact length, mm

105 NLEL NLVE, Time= 0. sec NLVE, Time= 50. Sec NLVE, Time= 500. sec

90

100 75 0

10

95

20 30 Half contact length, mm

40

50

Figure 8 Variations of nonlinear viscoelastic contact pressure along the contact surface at various time instants. 90 120 LVE NLVE

85 100

200

300

400

500

Time, s

Figure 6 Relaxation of central contact pressure for LVE and NLVE cylinder responses.

The effect of viscoelastic material nonlinearity on the relaxation of the central contact pressure is presented in Fig. 6. It is indicated that the nonlinear viscoelastic contact pressure falls to lower values rapidly more than that of linear viscoelastic, LVE, contact pressure. This is due to presence of the nonlinear stress-dependent parameters, which accounts for the decrease in the system stiffness.

Max. contact pressure, MPa

0

105

90

75

0

100

200

300

400

500

Time, s

6.2. A viscoelastic block pressed against a rigid foundation To detect the viscoelastic contact configuration under constant prescribed displacement (relaxation response), Dp = 1.75 mm, a plane stress contact problem of a viscoelastic block resting on a rigid foundation is considered. The geometry of the block and loading conditions are shown in Fig. 7. Furthermore, due to the condition of symmetry and frictionless condition, only

Figure 9 Relaxation of contact pressure for LVE and NLVE responses.

one-half of the block is needed to be modeled. The FE grid of this problem is composed of 800 four-noded quadrilateral elements and 861 nodes. The contact interface is modeled with 41 nodes, while the time step is 0.1 s. Fig. 8 illustrates NLVE

150 contact pressure distribution at different instants of time. It is clear that, contact pressure distribution is uniform along the contact interface. Moreover, the instantaneous NLVE contact pressure distribution coincides with NLEL distribution. Although the contact area has a very slight increase when time evolves, the contact pressure decreases due to material relaxation. In addition, as depicted from Fig. 9, the relaxation of the maximum contact pressure for both LVE and NLVE responses. 7. Conclusions A nonlinear recursive-iterative FE model is developed for investigating the contact configuration in nonlinear viscoelastic solids with geometrical nonlinearity. The nonlinear Schapery’s creep model, the response of isotropic nonlinear viscoelastic material is simulated. The model is derived based on implicit stress integration solutions for large mechanical deformations. The developed recurrence formula allows the bypassing of the need to store entire strain histories. The updated Lagrangian formulation was adopted to account for geometrical and material nonlinearities. The contact constraints are enforced into the equilibrium equations throughout the Lagrange multiplier method in which the normal contact forces are treated as independent variables. The resulting nonlinear equilibrium equations are solved using the Newton–Raphson method in an incremental-iterative procedure. Finally, two different examples with different geometry, loading history and contact configuration are presented showing the validity and versatility of the proposed computational model. References [1] Taylor RL, Pister KS, Goudreau GL. Thermomechanical analysis of viscoelastic solids. Int J Numer Meth Eng 1970;2:45–59. [2] Rogers TG, Lee EH. On the finite deflection of a viscoelastic cantilever. In: Proceedigns of the 4th US national congress of applied mechanics. ASME; 1962. p. 977–87. [3] Holden JT. On the finite deflection of thin viscoelastic beams. Int J Numer Meth Eng 1972;5:211–75. [4] Shen YP, Hasebef N, Lee LX. The finite element method of threedimensional nonlinear viscoelastic large deformation problems. Comput Struct 1995;55(4):659–66. [5] Crochon T, Scho¨nherr T, Li C, Le´vesque M. On finite-element implementation strategies of Schapery-type constitutive theories. Mech Time-Depend Mater 2010;14:359–87. [6] Kawahara M. Large strain, visco-elastic numerical analysis by means of finite element method. Proc JSCE 1972;24:141–9. [7] Christenson RM. A nonlinear theory of viscoelasticity for application to elastomers. J Appl Mech 1980;47:762–8. [8] Lin RC. On a nonlinear viscoelastic material law at finite strain for polymers. Mech Res Commun 2001;28(4):363–72. [9] Bonet J. Large strain viscoelastic constitutive models. Int J Solids Struct 2001;38:2953–68. [10] Lee EH, Radok RM. The contact problem for viscoelastic bodies. J Appl Mech 1960;27(3):438–44. [11] Yang WH. The contact problem for viscoelastic bodies. J Appl Mech 1966;33(2):395–401. [12] Hunter SC. The Hertz problem for a rigid spherical indenter and a viscoelastic half-space. J Mech Phys 1960;8:219–34. [13] Ting TCT. The contact stresses between a rigid indenter and a viscoelastic half-space. J Appl Mech 1966;33(4):845–54.

F.F. Mahmoud et al. [14] Ting TCT. Contact problems in the linear theory of viscoelasticity. J Appl Mech 1968;33(4):248–54. [15] Fu G. A theoretical study of complete contact indentation of viscoelastic materials. J Mater Sci 2004;39:2877–8. [16] Argatov I, Mishuris G. Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates. Appl Math Modell 2011;35:3201–12. [17] Barboteu M, Han W, Sofonea M. A frictionless contact problem for viscoelastic materials. J Appl Mech 2002;2:1–21. [18] Fernandez JR, Han W, Sofonea M. Numerical simulations in the study of frictionless contact problems. Math Comp Sci Ser 2003;30:97–105. [19] Fernandez JR, Sofonea M. Numerical analysis of a frictionless viscoelastic contact problem with normal damped response. Comput Math Appl 2004;47:549–68. [20] Fernandez JR, Martinez R. A normal compliance contact problem in viscoelasticity: an a posteriori error analysis and computational experiments. J Comput Appl Math 2011;235(12):3599–614. [21] Mahmoud FF, Elshafei AG, Attia MA. An incremental adaptive procedure for viscoelastic contact problems. J Tribol 2007:129; 305–313. [22] Mahmoud FF, Elshafei AG, Al-Shorbagy AE, Abdelrahman AA. Effect of material parameters on layered viscoelastic frictional contact systems. Adv Tribol 2010;1. [23] Mahmoud FF, El-Shafei AG, Attia MA. Analysis of thermoviscoelastic contact of layered bodies. Finite Elem Anal Des 2011;47:307–18. [24] Wriggers P. Computational contact mechanics. 2nd ed. Berlin: Springer; 2006. [25] Laursen T. Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. 1st ed. Berlin: Springer; 2003. [26] Bathe KJ. Finite element procedures. Upper Saddle River, New Jersey, USA: Prentice-Hall Inc.; 1996. [27] Schapery RA. On the characterization of nonlinear viscoelastic materials. Polym Eng Sci 1969;9(4):295–310. [28] Bathe KJ, Chaudaury A. A solution method for planar and axisymmetric contact problems. Int J Numer Meth Eng 1985;21:65–88. [29] Naghieh GR, Jin ZM, Rahnejat H. Contact characteristics of viscoelastic bonded layers. Appl Math Modell 1998;22:569–81. [30] Naghieh GR, Jin ZM, Rahnejat H. Characteristics of frictionless contact of bonded elastic and viscoelastic layered solids. Wear 1999;232:243–9. [31] Haj-Ali RM, Muliana HA. Numerical finite element formulation of the Schapery nonlinear viscoelastic material model. Int J Numer Meth Eng 2004;59(1):25–45. [32] Zhong WX, Sun SM. A parametric quadratic programming approach to elastic contact problems with friction. Comput Struct 1989;32(1):37–43. [33] Chandrasekaran N, Haisler WE, Goforth RE. A finite element solution method for contact problems with friction. Int J Numer Meth Eng 1987;24(3):477–95. Fatin Faheem Mahmoud emeritus professor of engineering and computational mechanics, faculty of engineering, Zagazig University. He has more than 120 research papers published in international journals and conferences. His areas of interest include contact mechanics, nonlinear solid mechanics, nanomechanics, nonlocal elasticity.

A numerical solution for contact problem with finite deformation in nonlinear Schapery viscoelastic solids Ahmed Gad El-Shafei associate professor of mechanical design, faculty of engineering, Zagazig University. His areas of interest include computational solid mechanics, Contact mechanics of deformable solids, optimal design, Stress analysis of composite structures, modeling of nonlinear thermo-viscoelastic and thermo-viscoplastic materials.

Alaa Ahmed Abdelrahman assistant lecturer, department of mechanical design and engineering, faculty of engineering, Zagazig University. He obtained his M.Sc. degree from the same university in 2006. His areas of interest include computational solid mechanics, rolling contact mechanics of deformable solids, modeling of nonlinear viscoelastic materials.

151

Mohamed Adly Attia assistant lecturer, department of mechanical design and engineering, faculty of engineering, Zagazig University. He obtained his M.Sc. degree from the same university in 2006. His areas of interest include computational solid mechanics, contact mechanics of deformable solids, modeling of nonlinear thermo-viscoelastic and thermoviscoplastic materials.