Computational Methods of Solid Mechanics Poject epot Due on Dec. 6, 25 Pof. Allan F. Bowe Weilin Deng
Simulation of adhesive contact with molecula potential Poject desciption In the poject, we will investigate the adhesive contact between a soft tip with ough suface and a igid substate, see Fig. (a). Since the tip size is much smalle than the substate, we take the substate to be semi-infinite. The pofile of the ough suface of tip in the efeence configuation is given by [ ( )] x 2 (x ) = x2 2R + A 2πx cos () λ whee the fist tem is a paabola shape with maximum cuvatue R and the second tem is a sinusoid undulation with wavelength λ and amplitude A. We take λ, A R to epesent the small scale oughness of the suface. The tip mateial is descibed by Neo-Hookean model as discussed in the class. We model the adhesive contact behavio by assuming that the tip and substates mateial points inteact with one anothe though the Lennad-Jones type inteaction potential [ ( σ ) 6 ( σ ) ] 2 V LJ = 4ɛ + (2) whee ɛ and σ ae constants and have units of enegy and length espectively, is the distance between two points. The inteaction foce between the tip and substate mateial point is obtained by diffeentiating the inteaction potential. Since the substate is semi-infinite lage and igid, the body foce acted on the tip mateial point and the wok of adhesion of the mateial can be deived analytically. Due to symmety, the only nonzeo component of body foce is [ ( ) (y) = 4πɛσ 2 σ ρ s ρ t (y) ( ) ] 4 σ (3) 5 y 2 2 y 2 as shown in Fig. (b), whee ρ t and ρ s ae the tip and substate densities. The wok of adhesion is w = 4 ( ) /3 2 πɛρ s ρ t σ 4 (4) 5 Note that the body foce is descibed in the defomed configuation. Rewite eq. 3 in the efeence configuation we have [ ( ) (x) = 4πɛσ 2 ρ s ρ t J σ (x) ( ) ] 4 σ (5) 5 x 2 + u 2 2 x 2 + u 2 whee u 2 is the displacement in ê 2 diection. 2 Govening equations The equilibium equation in the defomed configuation is σ ij y j + b i = in V σ ij n j = t i on S 2 (6) u i = u i on S whee σ ij is the Cauchy stess. Let η i be the test function with η i = u i on S. The weak fom is η i σ ij dv b i η i dv t i η i da = (7) y j V S 2 V 2
(a) e (b) 2 y 2 e.5 y 2.5..5.2 -.5 - Figue : (a) The schematic of adhesive contact between the soft indente and semi-infinite igid substate and (b) the Lennad-Jones type body foce. Map the integal ove the defomed configuation to the efeence configuation we have η i τ ij d b i η i d t i η i da = (8) y j whee τ ij = J σ ij is the Kichoff stess, b i = J b i, and t i is the taction mapped back to the efeence configuation. In the poject, the body foce is dependent on the position as descibed by eq. 5, and fo simplicity, the taction bounday condition is not consideed. Hence the thid tem on the left hand side of the weak fom is zeo. S 2 3 FE implementation Intoduce FE intepolation Substitute into the weak fom we obtain whee u i = N a (x)u a i, η i = N a (x)η a i (9) R a i [u b k] + B a i [u b k] = F a i () Ri a = τ ij [B kl ] V N a d y j Bi a = b i N a d Fi a = t i N a da S 2 Note that the foce vecto Bi a is now a function of displacement since the body foce is nonunifom distibuted given by [ ( ) b i = 4πɛσ 2 ρ s ρ σ t δ i2 ( ) ] 4 σ (2) No summation of i hee. The nonlinea equation will be solved using Newton-Raphson iteation. Stat with initial guess wi a and ty to coect the guess with coection dwa i. Lineaizing eq. in dwa i yields a system of linea equation (Kaibk R + Kaibk)dw F k b = Ri a Bi a + Fi a (3) () 3
to solve fo dwi a, whee the tem KR aibk is the usual one we deived in the class. The new tem KB aibk appeaing in the stiffness matix is Kaibk B = Ba i b i u b = k u b N a d (4) k Eveything is famila except the new tem we need to take cae of. The FE pocedue will be implemented with UEL in ABAQUS. Let us denote f(y i ) = 5 ( σ y i ) ( ) 4 σ (5) 2 y i To avoid f(y i ) goes to infinity as y i =, we can set the body foce to the following fom: { b δi2 c f(y i ), y i i = δ i2 c [h + k (y i )], y i < (6) whee c = 4πɛσ 2 ρ s ρ t, is a vey small positive paamete. Fo example, we can take =.5σ, and h = f yi= = 984 5 k = df yi= = 432 dy i σ (7) Theefoe, fo y i, B a i [ ( ) σ = δ i2 c ( ) ] 4 σ N a d (8) and Fo y i <, and [ Kaibk B = δ 2k c B a i 2σ 4 (x i + u i ) 5 2σ (x i + u i ) ] N a N b d (9) = δ i2 c [h + k (y i )] N a d (2) Kaibk B = δ 2k c k N a N b d (2) 4 Benchmaks In this section, we veify the UEL suboutine by seveal simple test cases. In cases -3, the simple body foces (Fig. 2(b-d)) ae applied to a single element ( in size), which is specified with a igid body motion in the -ê 2 diection (Fig. 2(a)). The body foces have the fom Case : =., > Case 2: =., < <., <, > 3 3. + Case 3: =, 2 < < 3., < < 2., < (22) 4
The paamete is set to be.. The total eaction foce of the element, fo the thee cases, is shown in Fig. 3, which is the same as the exact solution. The total foce is negative since a positive body foce is applied. (a) 4 3 (b) v e 2 2 e (c) (d) 3 - Figue 2: Schematic of element test with thee simple body foces..8 case case 2 case 3 Total eaction foce.6.4.2.2.4.6.8.2.4.6.8 Nomalized displacement Figue 3: The eaction foce F 2 of the element with igid body motion in test cases -3. In test case 4, we compae the Hetz contact simulation of the body foce method with that in ABAQUS/CAE. The contact of a elastic indente of adius 2.5 with igid substate is simulated. A indent depth of.2 is applied. The Young s modulus E and Poisson s atio µ ae set to be 2. and.3. Theoetically, the contacting suface cannot penetating each othe. Hypothetically, it can be viewed as a infinite foce is 5
applied on these suface. Theefoe, the body foce has the fom = {, > k, < (23) whee k is a lage numbe. (a) (b) (c) (d) Figue 4: The compaison of u 2 and σ 22 of ABAQUS/CAE (a,c) and body foce method fo k = 2 (b,d) in the Hetz contact simulation. 8 6 4 2 Hetz contact Body foce k= 2 Body foce k= 8 Body foce k= 2 Reaction foce 8 6 k 4 2 2.5..5.2 Indentation depth Figue 5: The compaison of foce-depth cuves of Hetz contact of ABAQUS/CAE and body foce method. 6
Fig. 4 shows the compaison of u 2 and σ 22 plots calculated fom ABAQUS/CAE and the body foce method with k = 2. As we can see, the distibution of u 2 and σ 22 is the same, but the magnitude is diffeent. The esult of body foce method conveges to that of ABAQUS/CAE as the slope k deceases. This is eflected in the compaison of foce-depth cuves, as shown in Fig. 5. As the slope k deceases fom -2 to -2, the diffeence of foce-depth cuve of ABAQUS/CAE and body foce method becomes smalle and smalle. 5 Adhesive contact In this section, we study the adhesive contact though the method of Lennad-Jones type body foce, as descied in Sec.. As pedicted by JKR theoy, thee exist pull-in and pull-off instabilities duing the loading and unloading of the tip due to the adhesive inteactions. To ovecome the convegence poblem in the numeical simulation, a viscosity tem is added to the body foce, i.e., [ ( ) b σ i = δ i2 c ( ) ] 4 ( σ + η du ) i. (24) dt whee η is a small numbe. Accodingly, the foce vecto Bi a and stiffness matix Kaibk B become [ ( ) Bi a σ = δ i2 c ( ) ] 4 ( σ + η u ) i N a d (25) t and [ 2σ 4 (x i + u i ) 5 2σ (x i + u i ) [ ( ) σ δ 2k c ( ) ] 4 σ Kaibk B = δ 2k c ] ( + η u i t ) N a N b d η t N a N b d In the simulation, a 2D tip is moving in the -ê 2 diection and subjected to the body foce as descibed by Eq. 24. The paametes σ and η ae taken to be nm and., espectively. The paametes and k ae taken to be.5σ and. The mateial is descibed by Neo-Hookean model. The shea modulus is.5 MPa. The wok of adhesion of the mateial is mj/m 2. Equivalently, the constant c is taken to be 3.32 2. The foce-displacement cuve duing the loading is shown in Fig. 6. As we can see, as the the foce deceases suddenly at a citical point when the pull-in instability occus. Then the foce inceases as the indent depth inceases. The contou plot of σ 22 afte the pull-in instability is shown in Fig. 7. Remak: The convegence is a big issue in this poject. The mesh should be vey efined to be able to esolve the body foce field. Although the pull-in instability is captued in the loading pocess, the computation does not convege duing the unloading pocess, hence the pull-off instability is not shown in the esult. A moe plausible way is to do the dynamic simulation by using VUEL suboutine. (26) 7
4 x 2 3 2 Foce (un) -. -.2 -.3 -.4 -.5 -.6 -.7 -.8 Displacement (um) 2 Figue 6: The foce-displacement cuve of the adhesive contact. Figue 7: The contou plot of σ 22 afte the pull-in instability. 8