frictional instabilities in contact problems: numerical ...

CMNE/CILAMCE 2007 Porto, 13 a 15 de Junho, 2007 c

APMTAC, Portugal 2007

FRICTIONAL INSTABILITIES IN CONTACT PROBLEMS: NUMERICAL AND ANALYTICAL COMPUTATION OF INSTABILITY MODES M.A. Agwa

1a∗

, A. Pinto da Costa1

1: Departamento de Engenharia Civil e Arquitectura and ICIST Instituto Superior Técnico Universidade Técnica de Lisboa Avenida Rovisco Pais, 1049 − 001 Lisboa, Portugal e-mails: {agwa,apcosta}@civil.ist.utl.pt, web: http://www.civil.ist.utl.pt a: On leave from the Department of Mechanical Engineering, Zagazig University, Egypt

Keywords: Infinite layer, Coulomb friction, Directional instability, Complementarity eigenproblem, Orthotropy, Finite elements Abstract. We determine the appropriate combinations of stiffness, mass and friction conditions under which a specific divergence type instability phenomenon of a layer in unilateral frictional contact may occur. It is our purpose to study the occurrence of smooth dynamic solutions, beginning arbitrarily close to equilibrium states of infinite orthotropic elastic layers, that take the system away from equilibrium in an exponential non oscillatory way. The problem is treated both from the analytical and the numerical points of view.

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Agwa, Pinto da Costa

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Description of the continuum

We consider an infinite orthotropic transversely isotropic linear elastic layer of height h (see Figure 1). It is possible to study a rectangular section of the layer provided the mechanical properties and the solutions we search for are periodic or do not vary with x1 for each fixed x2 . The upper segment CD has a vanishing prescribed displacement, the lower segment AB is in contact with a plane frictional obstacle. The principal directions of orthotropy are p1 , p2 and e3 and the plane of isotropy of the material is plane p2 − e3 .

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Figure 1: A schematic illustration of a rectangular section of an infinite layer.

Numerical solution

For a finite dimensional mechanical system with N degrees of freedom in the presence of a frictional contact obstacle, the smooth dynamic evolution of the system is governed by the following set of equations and conditions involving static and kinematic variables M¨ u(t) + Ku(t) = f 0 + r(t), rF (t) = 0, p 0 ≥ un (t) ⊥ rnp (t) ≤ 0, |rtp (t)| + µrnp (t) ≤ 0, |u˙ pt(t)|rtp (t) − µu˙ pt (t)rnp (t) = 0,

(1) (2) (3) (4)

where M and K stand for symmetric positive definite mass and stiffness matrices, t denotes time, u(t) and r(t) denote the displacement and obstacle reaction vectors; an equilibrium state (u0 , r0 ) satisfies (1)-(4) with vanishing velocities and accelerations; subscripts F , n and t denote the degrees of freedom that, respectively, are free from any kinematic constraint or are normal or tangent to the obstacle and superscript p is the label identifying the contact candidate particles. It may be shown [1] that the occurrence of dynamic solutions to (1)-(4) of the form u(t) = u0 + eλt v,

r(t) = r0 + eλt w,

(5)

where the vector pair (v, w) define a constant escape direction from the equilibrium state in the displacement–reaction space, constitutes a sufficient condition for the occurrence of divergence instability in the sense of Liapunov. If there are no particles in contact with zero reaction (no particles in grazing contact), a necessary and sufficient condition for an exponentially growing solution of the type (5) to exist is the existence of a λ ≥ 0 and vectors 2

Agwa, Pinto da Costa

(

)

(

)

xf vf x= = 6= 0, xst −Svst

(

)

(

)

yf 0 y= = , yst −(Swst + µwsn )

(6)

0p )), such that the following Mixed Complementarity EigenProblem where S = diag(sign(rst 2 in terms of λ (MCEIP-λ2 ) is solvable: Compute λ ≥ 0 and (x, y) with x 6= 0, such that





λ2 M∗ + K∗ x = y,

(7)

yf = 0, 0 ≤ xst ⊥ yst ≥ 0.

(8) (9)

In the previous set of conditions the subscripts f, sn and st, corresponding respectively to degrees of freedom without any kinematic constraint in the current equilibrium state (free) and degrees of freedom of the particles that are currently in a state of impending slip (normal or tangent to the obstacle). The inequalities in (9) should be interpreted in a componentwise way. Matrices M∗ and K∗ are, in general, non symmetric mass and stiffness linear matrix pencils in the coefficient of friction µ, the structure of which is exemplified next "

#

"

#

0 0 Kf,f −Kf,st S . +µ K∗ = K0 + µK1 = −Ksn,f Ksn,stS −SKst,f SKst,st S

(10)

It is possible to compute the coefficient of friction and the mode corresponding to the onset of instability, i.e., corresponding to a vanishing λ, by solving the Mixed Complementarity EigenProblem in terms of µ (MCEIP-µ): Compute µ ≥ 0 and (x, y) with x 6= 0, such that (K0 + µK1 ) x = y, yf = 0 and 0 ≤ xst ⊥ yst ≥ 0. Figure 2 illustrates the results of a finite element model with 465 four node bilinear finite elements assuming that the lower segment of the elastic layer is in impending slip to the left. The solution of problem MCEIP-µ yielded an onset of instability occurring at µcr = 0.2629238 with a purely distortional mode represented on the figure. For example, for µ = 0.6 > µcr the equilibrium state of impending slip to the left is unstable: the eigenvalue is λ = 36.9974990 and the mode is no longer purely distortional. This value of λ does not coincide with the analytic value (37.1674012) but λ converges to it upon mesh refinement. Previous studies on divergence instabilities include, among others, [1, 2] (frictional case, discretized structures).

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Figure 2: Directional instability mode at the onset of instability corresponding to µ = 0.2629238 (MCEIP-µ) involving the sliding of all nodes for θ = 80o , E1 = 8, E2 = 1, ν12 = 0.4, ν = 0.3, G12 /E2 = 0.1.

Agwa, Pinto da Costa

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Analytical solution for the homogenous layer

Due to the specific set of boundary, periodicity and frictional contact conditions it is possible to determine analytically the minimum value of µ (denoted by µcr ) for which the equilibrium state is directionally unsta44 ble, when it exists: µcr = D for a state of D24 plane strain. Figure 3 shows how µcr depends 1 on θ and E for a state of plane strain. The E2 graph was done for θ ∈ [0, 180o] since µcr (θ) is a function with period 180o. In this figure it is observed that µcr → +∞ when the principal directions of orthotropy are aligned with the directions normal and tangent to the obstacle (for θ → 0, 90o or 180o). It may also be proved that an isotropic layer is never directionally unstable. 4

4 1

0.5 16

2 3

8

4

4

8

µ

2

2

16

1 0.5

1

0 0

30

60

90

θ

120

150

180

Figure 3: Values of the coefficient of friction at the onset of directional instability for ν = 0.3, ν12 = 0, G12 /E2 = 0.1, for several values of the ratio E1 /E2 indicated on the curves (plane strain).

Conclusions • The coefficient of friction necessary for the occurrence of directional instabilities in infinite layers may be small, depending on the material properties. • The continuum problem of an infinite linear elastic orthotropic transversely isotropic homogenous layer has an analytic solution.

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Acknowledgements

This work was fully supported by a scholarship from “Programa de Financiamento Plurianual Programático” of the Portuguese Foundation for Science and Technology (FCT), attributed to the first author. REFERENCES [1] J.A.C. Martins, S. Barbarin, M. Raous and A. Pinto da Costa (1999) Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction. Comput. Methods Appl. Mech. Engrg., 177, 289–328. [2] I.R. Ionescu and R. Hassani (2006) Equilibrium configurations with Coulomb friction. Existence, multiplicity and stability. Solids, Structures and Coupled Problems in Engineering. Proceedings of the III European Conference on Computational Mechanics, Laboratório Nacional de Engenharia Civil, Lisboa, Portugal, 5–8 June. Edited by C.A. Mota Soares et al.. 4