Development of a Finite Element Procedure of Contact Analysis for Articular Cartilage with Large Deformation Based on the Biphasic Theory

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1 537 Development of a Finite Element Procedure of Contact Analysis for Articular Cartilage with Large Deformation Based on the Biphasic Theory Xian CHEN,YuanCHEN and Toshiaki HISADA Despite the importance of sliding contact in diarthrodial joints, the contact analysis algorithms presented over the past decade have been limited to cases of infinitesimal deformation and thus cannot reflect the real mechanical behavior of articular cartilage in daily life In this study, a new finite element contact analysis approach allowing a large amount of sliding between articular cartilages is presented based on the biphasic theory, which is an effective model for articular cartilage The geometric constraint condition and the continuity condition of the fluid phase on the contact surfaces are introduced by applying Lagrange multipliers The formulation is carried out by transmitting the contact traction of the tissue and the hydrostatic pressure of the fluid phase equivalently between the contact surfaces by means of integrating virtual work due to contact over the contact area The effectiveness of the proposed algorithm is verified by two numerical examples Key Words: Computational Mechanics, Finite Element Method, Nonlinear Problem, Contact Problem, Biphasic Theory, Articular Cartilage 1 Introduction The extracellular matrix of articular cartilage is composed of proteoglycans and collagen fibers, and protected from excessive deformation and stress by the hydrostatic pressure of the interstitial fluid, which supports about 90% of the external load on the cartilage On the other hand, since mechanical stimulation is regarded a factor that influences the metabolism of chondrocytes, the reduction in deformation and stress of the extracellular matrix may contribute to the maintenance of cartilage function A decrease in the protective function of the interstitial fluid, due to aging or excessive loading, causes destruction of the extracellular matrix and chondrocyte death, thereby leading to joint diseases (1) Therefore, understanding the mechanical behavior of articular cartilage is useful not only for the prevention or treatment of joint diseases but also for the development of artificial tissues Taking the characteristic biphasic structure of cartilage into account, biphasic mixture (2) and poroelastic material (3) mod- Received 23rd May, 2005 (No ) Institute of Environmental Studies, Graduate School of Frontier Sciences, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo , Japan xchen@smlku-tokyoacjp els have been proposed over the past two decades Moreover, since the synovial joint functions as a load transfer mechanism by contact between cartilage surfaces, the finite element method has been applied to contact analyses of articular cartilage based on the biphasic or poroelastic theories (4) (6) However, in these studies, the deformation was assumed to be infinitesimal and the pair of contacting nodes was fixed during the analysis Thus, the sliding that occurs between cartilage surfaces in synovial joints cannot be simulated by these approaches On the other hand, the formulation of a finite element contact analysis for large deformation has been carried out by discretizing the contact surfaces into contact elements and limiting the nodes on the surface of one side (slave surface) to penetrate the surface of the other side (master surface) The so-called one-pass node-to-segment approach (7) has been widely applied due to its ability to reflect a large amount of sliding However, it has been pointed out (8) that this approach may not pass the contact patch test and may lead to errors in the contact traction, since the contact virtual work is evaluated by a concentrated contact force If this approach is applied to a biphasic structure such as cartilage, errors will be produced not only in the contact traction of the tissue but also in the flow and hydrostatic pressure of the interstitial fluid Therefore, JSME International Journal Series C, Vol 48, No 4, 2005

2 538 in order to predict the mechanical behavior of articular cartilage correctly, a new contact analysis algorithm needs to be developed In this study based on the biphasic theory, the impenetrability of the soft tissue and the continuity of the interstitial fluid on the contact surfaces are introduced as constraint conditions by applying a Lagrange multiplier method Furthermore, a new contact algorithm applicable to the contact problems of a biphasic structure with a large amount of sliding is developed, by extending an approach proposed by the authors (9) that can pass a contact patch test developed for single phase structures Furthermore, the validity and effectiveness of the proposed algorithm are verified by two numerical examples 2 Basic Equations of the Biphasic Theory In the biphasic theory, articular cartilage is considered to be a mixture of interstitial fluid as the fluid phase and extracellular matrix as the solid phase Denoting the whole mixture volume as V, the volume fraction of each phase is defined as φ i = dv i /dv i= s, f (1) with superscripts s and f denoting the solid and fluid phases, respectively The saturation condition of cartilage holds as φ f +φ s = 1 (2) For each phase, the apparent density ρ α (mass per unit mixture volume) is related to the true density ρ α T (mass per unit volume of each phase) by ρ α = φ α ρ α T (3) By assuming incompressibility of the solid and fluid phases and combining Eqs (2) and (3) with the mass conservation equation ρ α + (ρ α u α ) = 0 (4) t the following relationship can be obtained (φ f u f +φ s u s ) = (u s +w) = 0 (5) where u s and u f are the velocities of the solid and fluid phases, respectively, w = φ f (u f u s ) gives the relative velocity of the fluid phase with respect to the solid phase Moreover, if the inertial force, volume force and viscosity of the fluid phase are omitted (2), (10), the momentum equation of the mixture and Darcy s law, which governs the motion of the fluid phase in the mixture, are given as follows σ = 0 (6) w = κ p (7) where σ denotes the Cauchy stress tensor of the mixture, the second order tensor κ represents the permeability and p is the hydrostatic pressure of the interstitial fluid By Series C, Vol 48, No 4, 2005 omitting the viscosity of the fluid phase, the Cauchy stress of the mixture can be expressed as σ = pi +σ e (8) where σ e denotes the amount of stress due to elastic deformation of the solid phase and can be determined by applying the constitutive relationship of a hyperelastic material for large deformation problems The kinetic boundary conditions are given by σ T n= ˆt on t (9) p = ˆp on p (10) where ˆt and ˆp are the total external traction of the mixture and the hydrostatic pressure of the external fluid on the portions t and p of the surface, respectively Thus, Eqs (5) (10) and the constitutive relationship of the solid phase compose the boundary value problem of the biphasic theory By multiplying the weight functions δu to Eq (6), δw to Eq (7) and δp to Eq (5), and then integrating them over the volume domain of the solid phaseω, the summation of these integrations leads to δu σdv+ δw ( p+κ 1 w)dv ω ω + δp (u s +w)dv = 0 (11) ω Furthermore, with δu as an admissible displacement field of the solid phase and utilizing Eqs (9) and (10), after integrating Eq (11) by parts followed by use of the divergence theorem and transformation of the volume domain ω and surface domains t, p to the corresponding domains Ω, Γ t and Γ p in the reference configuration, the weak form of the governing equation of the biphasic theory is obtained as (11) δe :(S e pjc 1 )dv + δw κ 1 wjdv Ω Ω ( δw)pjdv + δp (u s +w)jdv Ω Ω = δu tds δw pnds (12) Γ t Γ p where E and C are the Green-Lagrange strain and right Cauchy-Green deformation tensor of the solid phase, respectively, S e is the second Piola-Kirchhoff stress due to deformation of the solid phase, J = det F is the determinant of the solid phase deformation gradient tensor, t and p are the nominal mixture traction and hydrostatic pressure, respectively, referring to the reference configuration, and N is the outward normal unit vector of the surface in the reference configuration 3 Finite Element Formulation for Contact Problems of Articular Cartilage In conventional studies (4) (6), since infinitesimal deformations were assumed or pairs of contacting nodes JSME International Journal

3 539 were fixed in finite element analyses, these approaches cannot be applied to the general case of the large amount of sliding that occurs between cartilage surfaces in synovial joints in daily life On the other hand, although the so-called node-to-segment approach (7), in which sliding is dealt with by considering contact between nodes and contact elements on discretized contact surfaces, is widely adopted due to its simplicity, it has been pointed out that this approach cannot pass the contact patch test and may produce errors of traction in the contact surfaces If this approach is applied to contact analysis of a biphasic structure such as cartilage, it is inevitable that not only errors in the contact traction of the tissue but also errors in the flow and hydrostatic pressure of the interstitial fluid in the contact surfaces will be induced In the present study, in order to overcome this obstacle, a contact analysis algorithm was formulated for articular cartilage with large deformation based on the biphasic theory, by extending an approach proposed by the authors (9) that can pass the contact patch test for a single phase material 3 1 Basic equations for contact analysis Figure 1 shows a model of the contacting surfaces of articular cartilage Considering that the point of cartilage 2 comes into contact with the point of cartilage 1, which is defined as a projection of the point in cartilage 2 to the surface of cartilage 1, the penetration between contact surfaces is defined as follows g = (x 2 x 1 ) n c1 (13) Since the friction coefficient of healthy cartilage surfaces is low, it is assumed that the friction force can be ignored in the formulation By means of the relationship between the outward unit normal vectors in both the contact surfaces n c1 = n c2 (14) the relationship between the contact tractions, t c1 and t c2, of the cartilage surfaces satisfies t c1 = t cn1 n c1, t c2 = t cn2 n c2, t cn t cn1 = t cn2 (15) Therefore, the geometric and kinetic constraint conditions between contacting cartilage surfaces are given as g 0, t cn 0, g t cn = 0 (16) Furthermore, the continuity requirement of an incompressible fluid and the balance condition of hydrostatic pressure between the two sides need (12) Fig 1 Contact surfaces (w c1 w c2 ) n c1 = 0 (17) p c1 = p c2 p c (18) By comparing Eq (17) with Eq (13), it is noted that the penetration defined by Eq (13) specifies the relative location of the contacting cartilage surfaces, while Eq (17) ensures mass balance between the contacting surfaces, which is a characteristic property of a biphasic structure This difference will cause different treatments for these two constraint conditions in the derivation of contact tangent stiffness, as described later By denoting the volume domains of the contacting cartilages in the reference configuration as Ω 1, Ω 2,the surface domains in the reference configuration involving nominal contact tractions t 1, t 2 and nominal hydrostatic pressures p 1, p 2 as Γ t1, Γ t2 and Γ p1, Γ p2, respectively, and the contact boundaries in the current configuration as c1, c2, Eq (12) can be rewritten as δe :(S e pjc 1 )dv + δw κ 1 wjdv Ω 1 Ω 2 Ω 1 Ω 2 ( δw)pjdv + δp (u s +w)jdv Ω 1 Ω 2 Ω 1 Ω 2 = δu tds δw pnds +δw c Γ t1 Γ t2 Γ p1 Γ p2 (19) where δw c = δu t cn1 n c1 ds δw p c1 n c1 ds c1 c1 + δu t cn2 n c2 ds δw p c2 n c2 ds (20) c2 c2 corresponds to the virtual work performed by the contact traction of the mixture and the hydrostatic pressure, and is evaluated in the current configuration for the convenience of introducing continuity condition of interstitial fluid given by Eq (17) 3 2 Discretization of contact surfaces In the context of finite element analysis, the candidate contact articular surfaces are divided into contact elements as shown in Fig 2 In the one-pass approach, the surface of one side (slave surface) is restrained such that it is unable to penetrate the surface of the other side (master surface) Instead of contact between the cartilage surfaces, contact between node x s (slave node) in the slave surface and the element (master element) in the master surface is treated by means of determining the counterpart point x m (master point) of the slave node in the master surface by drawing a line perpendicular to the master element from the slave node Therefore, it becomes possible to treat contact problems with a large amount of sliding between cartilage surfaces by defining the geometric relationship between the node and the element, instead of that between nodes as in the node-to-node approach In Fig 2, ξ 1 and ξ 2 indicate the convected coordinates at the master point x m, while JSME International Journal Series C, Vol 48, No 4, 2005

4 540 Fig 2 Discretization of contact surfaces t m 1, tm 2 and nm are the covariant base vectors and outward unit normal vector, respectively, ie t m 1 = xm ξ 1, tm 2 = xm ξ 2 (21) r m t m 1 tm 2, r m (22) Moreover, the differential area element in the convected coordinate system can be expressed as ds= J m dξ 1 dξ 2 = r m dξ 1 dξ 2 (23) By defining the shape function matrix with shape functions Ni m (i = 1, n) of the contact element as N1 m 0 0 N m n 0 0 [M] = 0 N1 m 0 0 Nn m 0 (24) 0 0 N1 m 0 0 Nn m the weight functions δu, δw and location vector x m at the master point are interpolated as δu m = m] {δu m } (25) δw m = m] {δw m } (26) x m = m] {x m } (27) where {δu m } = { } δu m T 1,1,δum 1,2,δum 1,3,,δum n,1,δum n,2,δum n,3 (28) {δw m } = { } δw m T 1,1,δwm 1,2,δwm 1,3,,δwm n,1,δwm n,2,δwm n,3 (29) {x m } = { x m 1,1, xm 1,2, xm 1,3,, xm n,1, xm n,2, T n,3} xm (30) are vectors consisting of the respective nodal values in the contact element 3 3 Finite element formulation In contrast to evaluating contact virtual work by the concentrated contact force in the conventional one-pass formulation of the finite element contact analysis, the authors have proposed an algorithm (9) that can pass the patch test by performing an integration of the contact virtual work over all the contacting elements The nodes of the master surface are projected onto the slave elements and the contact nodal tractions of the master nodes are transferred from the contact tractions of the projected point, which are obtained by interpolation of the nodal values in the slave elements If this approach is applied to both the contact traction of the tissue and the hydrostatic pressure, nm = rm Series C, Vol 48, No 4, 2005 and Eqs (23) (30) are used, the contact virtual work of Eq (20) in the whole discretized contact surfaces is given by δw c = {δu m } T m m ]T {n m }t m J m dξ 1 dξ 2 +{δu s } T {δw m } T s s ]T {n s }t s J s dξ 1 dξ 2 m m ]T {n m } p m J m dξ 1 dξ 2 {δw s } T s s ]T {n s } p s J s dξ 1 dξ 2 (31) where {δu α }, {δw α } (α = m, s) are defined as the nodal values of the contact element, α similar to Eqs (28) and (29) To perform numerical integration of Eq (31), the contact tractions of the tissue t m, t s and hydrostatic pressures p m, p s at the quadrature point are interpolated from their nodal values and the shape functions of the corresponding contact elements, ie t m = {N m } T {t m }, t s = {N s } T {t s } (32) p m = {N m } T {p m }, p s = {N s } T {p s } (33) {N m } T = {N1 m,,nm n }, {N s } T = {N s 1,,N n} s (34) {t m } = {t1 m,,tm n } T, {t s } = {t s 1,,ts n} T (35) {p m } = {p m 1,, pm n } T, {p s } = {p s 1,, ps n} T (36) On the other hand, by projecting the node of the master element onto the slave element, the nodal contact traction of the tissue ti m and the hydrostatic pressure p m i at node i of the master element are considered to be equal to those at the projected point in the slave element, which can be obtained from interpolation of the nodal values of the slave element, ie ti m = { N s i,1,,n } i,n s p m i = { N s i,1,,n } i,n s t s i,1 t s i,n p s i,1 p s i,n (37) (38) where N s i,l (l = 1, n) is a shape function, and t s i,l and p s i,l are the nodal contact traction of the tissue and the hydrostatic pressure of the projected slave element, respectively With substitution of Eqs (37) and (38) into Eq (31), the contact virtual work for the entire contact surfaces can be expressed as δw c = {δu m } T m m ]T {n m }{N m } T [L]{T s } J m dξ 1 dξ 2 +{δu s } T {δw m } T s s ]T {n s }{N s } T {t s } J s dξ 1 dξ 2 m m ]T {n m }{N m } T [L]{P s } J m dξ 1 dξ 2 JSME International Journal

5 541 {δw s } T s s ]T {n s }{N s } T {p s } J s dξ 1 dξ 2 (39) where N s 1,1,,Ns 1,n [L] = N s n,1,,n n,n s (40) {T s } = {t s 1,1,,ts 1,n ts n,1,,ts n,n} T (41) {P s } = {p s 1,1,, ps 1,n ps n,1,, ps n,n} T (42) denote the shape function matrix, vectors of the nodal contact traction of the tissue and the hydrostatic pressure in all the slave elements related to the nodes in the master surface Furthermore, in this way, only the nodal contact tractions of the tissue and the hydrostatic pressures on the slave surface become the contact nodal variables as Lagrange multipliers In the above formulations, since this research is focused on the patch test problem, the interpolation functions for the contact traction of the tissue and the hydrostatic pressure are chosen to be the same as those of the displacement of the tissue and the relative velocity of fluid phase But it has to be noted that selecting proper interpolations for the contact traction of the tissue and the hydrostatic pressure to satisfy the inf-sup condition is another crucial issue for insuring the stability and optimal convergence of contact analysis of hydrated tissues Since the geometric constraint condition and continuity condition of the fluid phase for the contacting cartilage surfaces need to be satisfied at the contact nodes after discretization, these conditions are discretized by using Eqs (13), (17) and (24) (30) as follows {n m } T ( m ] {x m } [I] { x s }) = 0 (43) {n m } T ( m ] {w m } [I] { w s }) = 0 (44) where { x s } is the location vector, { w s} is the relative velocity vector of the fluid phase of the contact node x s,and[i] is a 3 3 unit matrix To derive the contact tangent stiffness, variations of the unit outward normal vector, the Jacobian transforming the differential area to the convected coordinate system and the covariant base vectors due to deformation are first obtained from Eqs (21) (23) and expressed in matrix form as follows {n α } = 1 J α ( [I] {n α }{n α } T ) {r α } (45) J α = r α = {n α } T {r α } (46) { } t α [ 1 = M α ] {u α } ξ 1 (47) { } t α [ 2 = M α ] {u α ξ 2 } (48) ( [W ] {r α α [ } = 2 M α ] + [ ] W α [ ξ 1 1 M α ] ) {u α } ξ 2 (49) where [M α ]and{u α } are the shape function matrix and the nodal displacement vector of the contact element, respectively, defined similarly to Eqs (24) and (28), and [W1 α] and [W α 2 ] are the matrix expressions of skew-symmetric tensors with the covariant base vectors t α 2 and tα 1 as their axial vectors, ie [ ] 0 t1,3 W α 1 = α t1,2 α t α 1,3 0 t α 1,1 (50) t1,2 α t1,1 α 0 [ ] 0 t2,3 W α 2 = α t2,2 α t2,3 α 0 t2,1 α (51) t α 2,2 t α 2,1 0 In addition, since the nodal contact traction of the tissue and the hydrostatic pressure of the master node are determined from those at the projected point of the slave element, which depends on the relative movement of the contacting surfaces, as long as the variation in the local coordinates of the projected point is computed (13), the contact tangent stiffness can be obtained by differentiating Eq (39) The contributions of the geometric constraint condition and the continuity condition of the fluid phase to the contact tangent stiffness are determined by differentiating Eqs (43) and (44), respectively Since the geometric constraint reflects the relative location of the contacting surfaces, the relationships n m (x m x s ) = 0andn m t m i = 0, (i = 1, 2) considerably simplify the calculation, and differentiation of Eq (43) leads to {n m } T ( m] {u m } [I] { u s } ) = 0 (52) However, for the continuity condition of the fluid phase, similar relationships do not exist, and therefore, considering the variations in the local coordinates and outward normal vector at the master point due to deformation, the differentiation of Eq (44) is derived as {n m } T ( m] {w m } [I] { w s } ) ( +{n m } T m ] ξ 1 + m ] ) ξ 2 {w m ξ 1 ξ 2 } + {n m } T ( m] {w m } [I] { w s } ) = 0 (53) In the above equation, the variations of local coordinates, ξ 1 and ξ 2, have been taken into account, since the master point is defined as the projection of the slave node onto the master element, and is therefore dependent on the relative movement of the contact surfaces 4 Numerical Examples The formulation performed in section 3 is implemented by building the nodal contact traction of the tissue, JSME International Journal Series C, Vol 48, No 4, 2005

6 542 nodal hydrostatic pressure and contact tangent stiffness associated with contacting articular cartilage surfaces into a self-coded biphasic finite element analysis program Next, numerical examples are carried out to assess the validity and effectiveness of the proposed approach using a hexahedral type of element with 8 nodes for displacement and relative fluid velocity and 1 node for hydrostatic pressure In addition, the conventional one-pass approach evaluating the discretized contact virtual work by a concentrated contact force is also performed for comparison 4 1 Contact problem between a cartilage layer and a rigid cylinder To assess the validity of the proposed approach, the contact problem between a cartilage layer and an impermeable rigid cylinder (as shown in Fig 3) is analyzed for comparison with the results (5) of finite element contact analysis of articular cartilage under infinitesimal deformation Since the impermeable rigid cylinder plays the role of imposing a shape on the contact surface, only the part near the surface of the cylinder is modeled and, furthermore, the symmetry is utilized in the modeling All the nodal degrees of freedom of displacement and relative fluid velocity at the bottom of the cartilage layer are constrained by assuming that the cartilage layer is connected to an impermeable rigid bone Furthermore, the nodal degrees of freedom of displacement and relative fluid velocity in the outward direction of the plane are constrained to simulate a 2-D problem In the contact analysis, the impermeable rigid cylinder is set as the master side For comparison with the conventional studies, the geometric nonlinearity is ignored and the solid phase of the cartilage is assumed to be linear elastic with the material constants (5) shown in Fig 3 The load is first applied to the cylinder as a ramp function of time with the maximum value of 1 N per unit length of the cylinder reached within 10e 3 s and then held to induce creep deformation of the cartilage layer The distributions of the contact traction of the tissue and the hydrostatic pressure within the contact area at 100 s after loading obtained by the proposed approach are shown in Fig 4, together with those of the conventional research (5) These results show good agreement, and the small discrepancy near the border of the contact area is considered to be induced by the different finite element meshes of the contact surfaces used in each computation In addition, it is noted that the hydrostatic pressure accounts for about 86% of the contact traction of the tissue, indicating that the hydrostatic pressure supports the majority of the contact load Figure 5 shows the deformation of the cartilage and distribution of the relative velocity of the fluid phase immediately after the load reaches its maximum Since the fluid is prohibited from flowing in and out of the surface of the impermeable rigid cylinder, the continuity condition of the fluid phase on the contact surface results in fluid flow in the tangential direction of the contact surface within the contact area The fluid flow out of the cartilage layer occurs almost at the border of the contact area with a maximum value of 48 µm/s In addition, the contact analysis was also carried out by applying the conventional one-pass approach and, leaving out the details, the results were mostly in agreement with the results of the proposed approach, when the impermeable rigid cylinder was set as the master body in the conventional approach This fact can be explained as follows In the one-pass approach, the nodes in the slave surface, ie the contact surface of the cartilage layer, are constrained such that they are unable to penetrate the master surface Hence, if the master surface is rigid, the nodal displacement in the slave surface is prescribed by the shape Fig 3 Contact model of an impermeable rigid cylinder and a cartilage layer Fig 4 Distributions of the contact and interstitial fluid pressures at 10 s after loading Fig 5 Deformation of cartilage (amplified by 5) and distribution of the relative fluid velocity (arrows) at maximum loading Series C, Vol 48, No 4, 2005 JSME International Journal

7 543 of the master surface In a similar way, the nodal relative fluid velocity in the slave surface is also imposed by means of the continuity condition if the master surface is impermeable Accordingly, the displacement and relative fluid velocity fields, and subsequently the nodal forces in the slave surface, can be evaluated appropriately Consequently, since the conventional one-pass approach involves discretization of the contact virtual work in the slave surface with the concentrated forces at the slave nodes, these appropriately evaluated nodal forces become equivalent to the contact traction of the tissue and hydrostatic pressure in the slave surface in the sense of virtual work However, in real synovial joints, the contact occurs between deformable articular cartilage layers In addition, since it has been pointed out that the conventional one-pass approach for contact problems of a single phase elastic solid cannot pass the patch test due to the nonconforming meshes of the contact surfaces (8), it can be predicted that considerable errors in the contact traction of the tissue and the hydrostatic pressure will be induced by the non-conforming meshes of the contact surfaces From a practical standpoint, the meshes cannot always remain conformable under a large amount of sliding even when they are generated conformably in the initial state Therefore, the effectiveness of the proposed approach for contact problems between deformable bodies with a large amount of sliding will be discussed in the next section 4 2 Contact problem between articular cartilage layers The finite element mesh generated without special attention to the conformability is shown in Fig 6 All the nodal degrees of freedom of displacement and relative fluid velocity at the bottom of the lower cartilage layer are constrained by assuming that the cartilage layer is connected to an impermeable rigid bone The shaded portion of the upper body is also assigned as rigid impermeable bone by specifying a very high stiffness and fixing the degree of freedom of the relative fluid velocity, while the remainder represents another cartilage layer The nodal degrees of freedom of displacement and relative fluid velocity in the outward direction of the plane are constrained to simulate a 2-D problem The solid phase of the two cartilage layers is assumed to be a hyperelastic material of the Saint Venant-Kirchhoff type with the same material constants specified in Fig 6 The analyses are carried out using both the conventional and proposed approaches for comparison by applying a load from 0 1 N within 001 s In the analysis using the conventional approach, due to unevenness of the deformed contact surfaces caused by factors discussed below, certain contact nodes near the edges of the contact elements could not be matched to an unique contact element, ie such contact nodes chattered (14) during the iteration and caused the computation to diverge after the load reached 019 N Therefore, only the results for the two approaches at a load level of 019 N are compared below As shown in Figs 7 and 8, considerable disorder appears in the distributions of the contact traction of the tissue and the hydrostatic pressure using the conventional approach, while smooth distributions of these results are obtained using the proposed approach Consequently, as shown in Fig 9, the conventional approach leads to an unevenly deformed contact surface and a disturbed distribution of the relative fluid velocity, while the proposed approach gives a smooth deformation and regular distribution of the relative fluid velocity From these results, it can be concluded that the proposed approach is applicable to contact problems between deformable cartilages, whereas the conventional approach may cause considerable errors in this case The errors of the conventional approach are consid- Fig 7 Comparison of the total contact pressures obtained by the two approaches at a load level of 019 N Fig 6 FEM model of contacting cartilage layers Fig 8 Comparison of the hydrostatic pressures obtained by the two approaches at a load level of 019 N JSME International Journal Series C, Vol 48, No 4, 2005

8 544 (a) Conventional approach Fig 9 (b) Proposed approach Comparison of the deformations (amplified by 15) and relative fluid velocities (arrows) obtained by the two approaches ered to be contributed by the discretization of the contact virtual work with the concentrated force related to the contact traction of the tissue and the hydrostatic pressure In general, this discretization is not guaranteed to be equivalent to the original virtual work and therefore transfers of the contact traction of the tissue and the hydrostatic pressure between contact surfaces are unable to be carried out appropriately The errors not only influence the deformation of the cartilage but also the elastic stress of the solid phase and the flow of the interstitial fluid, and furthermore, they cannot be reduced by mesh refinement On the other hand, in the proposed approach, the contact virtual work is numerically integrated in both the master and slave elements, and instead of a concentrated force, the nodal contact traction of the tissue and the hydrostatic pressure are transferred between the contact surfaces In this way, the nodal forces with respect to contact can be evaluated equivalently in the sense of virtual work and thus reasonable results are obtained Next, contact analysis of articular cartilage layers with a large amount of sliding is carried out using the proposed approach After the load reaches 1 N at 001 s, a rightward horizontal velocity of 10 mm/s is applied to the upper cartilage to produce relative sliding over the lower cartilage within the zone shown in Fig 6, while the load remains unchanged The distributions of the contact traction of the tissue, hydrostatic pressure and normal stress of the solid phase due to deformation when the upper cartilage moves near the center of the lower cartilage are shown in Fig 10 All the results are obtained smoothly, with the maximum values appearing in the center of the contact area In addition, the compression of the cartilage Series C, Vol 48, No 4, 2005 Fig 10 Distributions of the total contact pressures, hydrostatic pressures and normal elastic stresses on the contact surfaces when the upper cartilage layer slides near the center of the lower cartilage layer due to the contact induces the interstitial fluid to flow toward the outside of the contact area, thereby producing the tensile stress state of the solid phase shown in Fig 10 and the bulge deformation of the tissue shown in Fig 11 near the border of the contact area Moreover, Fig 11 shows the distributions of the vertical displacement of the surface of the lower cartilage when the upper cartilage moves at the left side, the center and the right side of the lower cartilage, respectively Without showing the details of the other locations, smooth deformations are obtained during the whole sliding process In the early stage of sliding (Fig 11 (a)), the leading edge of the contact area is pushed to increase the bulge deformation Since the interstitial fluid flows more easily backward into the area that the upper cartilage has passed, the bulge deformation at the trailing edge of the contact area increases gradually along with the sliding due to the increased amount of fluid flow- JSME International Journal

9 545 Fig 12 Distribution of the relative fluid velocity on the surface of the lower cartilage layer (deformation amplified by 7) (a) Sliding near the left side (time = 01 s) For the prevention and medical treatment of joint diseases, as well as the development of artificial tissues that are close to the living tissues, it is necessary to clarify the mechanical behaviors of articular cartilage under physiological conditions In this study, a finite element contact analysis approach for articular cartilage based on the biphasic theory is proposed by taking into account the large amount of sliding that occurs between cartilage surfaces of synovial joints in daily life The impenetrability of the tissue and the continuity condition of the fluid phase on the contact surfaces are enforced by applying Lagrange multipliers The contact virtual work is integrated in the contact surfaces and the contact forces are transferred between the contact surfaces equivalently in the sense of virtual work The proposed approach is implemented and two numerical examples have been carried out to confirm the validity and effectiveness of the proposed approach References (b) Sliding near the center (time = 05 s) (c) Sliding near the right side (time = 09 s) Fig 11 Vertical nodal displacements on the upper surface of the lower cartilage layer ing backward Eventually, the bulge deformation at the trailing edge of the contact area becomes larger than that at the leading edge (Fig 11 (c)) The distribution of the relative velocity of the fluid phase shown in Fig 12 also demonstrates that a larger relative velocity appears at the trailing edge than at the leading edge Based on the results of the contact analysis for articular cartilage with a large amount of sliding discussed above, the validity and effectiveness of the proposed approach for contact problems of articular cartilage with a large amount of sliding are confirmed, and it is expected that the proposed approach will be a useful tool for actual analyses of joint mechanics 5 Conclusions ( 1 ) Milentijevic, D, Helfet, DL and Torzilli, PA, Influence of Stress Magnitude on Water Loss and Chondrocyte Viability in Impacted Articular Cartilage, Journal of Biomechanical Engineering, Vol125 (2003), pp ( 2 ) Mow, VC, Kuei, SC, Lai, WM and Armstrong, CG, Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression: Theory and Experiments, Journal of Biomechanical Engineering, Vol102 (1980), pp73 84 ( 3 ) Simon, BR, Wu, JSS, Carlton, MW, Evans, JH and Kazarian, LE, Structural Models for Human Spinal Motion Segments Based on a Poroelastic View of the Intervertebral Disk, Journal of Biomechanical Engineering, Vol107 (1985), pp ( 4 ) Macirowski, T, Tepic, S and Mann, RW, Cartilage Stresses in the Human Hip Joint, Journal of Biomechanical Engineering, Vol116 (1994), pp10 18 ( 5 ) Donzelli, PS and Spilker, RL, A Contact Finite Element Formulation for Biological Soft Hydrated Tissues, Computer Methods in Applied Mechanics and Engineering, Vol153 (1998), pp63 79 ( 6 ) Federico, S, Rosa, GL, Herzog, W and Wu, JZ, Effect of Fluid Boundary Conditions on Joint Contact Mechanics and Applications to the Modeling of Osteoarthritic Joints, Journal of Biomechanical Engineering, Vol126 (2004), pp ( 7 ) Wriggers, P, Computational Contact Mechanics, (2002), John Wiley & Sons Ltd ( 8 ) El-Abbasi, N and Bathe, KJ, Stability and Patch Test Performance of Contact Discretizations and a New Solution Algorithm, Computers and Structures, Vol79 (2001), pp ( 9 ) Chen, X and Hisada, T, Development of Finite Element Contact Analysis Algorithm Passing Patch Test, JSME International Journal Series C, Vol 48, No 4, 2005

10 546 Trans Jpn Soc Mech Eng, (in Japanese), (in Press) (10) Ateshian, GA and Wang, H, A Theoretical Solution for the Frictionless Rolling Contact of Cylindrical Biphasic Articular Cartilage Layers, Journal of Biomechanics, Vol28 (1995), pp (11) Chen, X, Chen, Y and Hisada, T, A Study on Mechanical Model of Soft Tissues by Nonlinear Finite Element Analysis Based on Biphasic Theory, Trans Jpn Soc Mech Eng, (in Japanese), Vol70, No697, A (2004), pp (12) Hou, JS, Holmes, MH, Lai, WM and Mow, VC, Boundary Conditions at the Cartilage-Synovial Fluid Interface for Joint Lubrication and Theoretical Verifications, Journal of Biomechanical Engineering, Vol111 (1989), pp78 87 (13) Laursen, TA and Simo, JC, A Continuum-Based Finite Element Formulation for the Implicit Solution of Multibody, Large Deformation Frictional Contact Problems, Int J Numer Methods Eng, Vol36 (1993), pp (14) Puso, MA and Laursen, TA, A 3D Contact Smoothing Method Using Gregory Patches, Int J Numer Eng, Vol54 (2002), pp Series C, Vol 48, No 4, 2005 JSME International Journal