aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n A Shell Frictionally Coupled to an Elastic Foundation
Kavinda Jayawardana ∗ Abstract
In our analysis, we derive a model for a shell that is frictionally coupled to anelastic foundation. We use Kikuchi and Oden’s model for Coulomb’s law ofstatic friction [1] to derive a displacement-based static-friction condition forour shell model, and we prove the existence and the uniqueness of solutionswith the aid of the works of Kinderlehrer and Stampacchia [2]. As far as weare aware, this is the first derivation of a displacement-based friction condi-tion, as only force and stress based friction conditions currently exist in theliterature. For numerical analysis, we extend Kikuchi and Oden’s model forCoulomb’s law of static-friction [1] to model a full two-body contact problemin curvilinear coordinates. Our numerical results indicate that if the shellhas a relatively high Young’s modulus or has a relatively high Poisson’s ratio,and the contact region has a very high coefficient of friction or less curved,then the displacement field of the foundation predicted by both models arein better agreement.
Keywords:
Contact Mechanics, Coulomb’s Law of Static Friction,Curvilinear Coordinates, Elastic Foundations, Mathematical Elasticity,Shell Theory
1. Introduction
Consider a situation where two elastic bodies that are in contact witheach other, where the contact area exhibits friction. A common area wheremodelling of such problems can be found in the field of tire manufacturing[3, 4]. Now, consider the scenario where one of the elastic bodies is verythin and almost planar in a curvilinear sense, in comparison to the other ∗ Corresponding author
Email address: [email protected] (Kavinda Jayawardana)
January 21, 2021 ody. Then the thin body can be approximated by a shell or a membrane,and such models can be used to model skin abrasion caused by fabrics as aresult of friction. Need for valid modelling techniques are immensely impor-tant in fields such as field of incontinence associated dermatitis [5], sportsrelated skin trauma [6] and cosmetics [7]. It is documented that abrasiondamage to human skin in cases such as the jogger’s nipple [8] and dermatitisfrom clothing and attire [9] are caused by repetitive movement of fabrics onskin, and in cases such as pressure ulcers [10] and juvenile plantar dermatitis[11], friction may worsen the problem. Thus, the goal of this reading is topresent a simple but a mathematically valid model to model such problems,i.e. a mathematical model for a shell that is on an elastic foundation whensubjected to a displacement-based friction condition in a static dry-friction [1, 12] setting.
Consider a three-dimensional elastic body that is in contact with a rigidboundary whose contact area is rough [13, 14] (i.e. contact area exhibitsfriction), then, given that we know the pressure experienced on the elasticbody at the contact region in advance, the governing equations that describesthe behaviour at the contact region can be represented by Kikuchi and Oden’smodel for Coulomb’s law of static friction [1], which has the formulation ofthe following form, j ε ( u ) = Z Γ (cid:20) K | u T | − ε (cid:21) ds , if | u T | ≥ ε , Z Γ K | u T | ε ds , if | u T | < ε , (1)where ν F is the coefficient of friction, σ T ( u ) is the normal-tangential stresstensor at the contact boundary, u is the displacement field and u T is thetangential displacement field of the contact boundary, K (units: Nm − ) isthe spring modulus, and ε is the regularisation parameter. If one assumesthat the purely-normal stress, σ n <
0, (i.e. pressure) is no longer an unknown,but it is prescribed, and further assumes that K = − ν F σ n , then the Gˆateauxderivative of j ε ( · ) has the following form, σ T ( u ) = ν F σ n u T | u T | , if | u T | ≥ ε ,ν F σ n u T ε , if | u T | < ε . (2)2ow, assume that we are considering a shell (i.e. two-dimensional representa-tion of a very thin three-dimensional elastic body), then the very idea of nor-mal stress becomes meaningless, because for a shell, we find that σ T ( u ) = and σ n = 0, and thus, Kikuchi and Oden’s [1] model, i.e. equation (2), willfail. Note that for a thorough mathematical analysis of the shell theory, werefer the reader to chapter 4 of Ciarlet [15].To overcome this difficulty, one may employ Gao’s model for a nonlinearbeam on a rigid obstacle with friction [16], which has the formulation of thefollowing form, ξ ,xx + βξ ,yy + (1 + ν ) w ,x w ,xx = 0 ,ξ ,y ( x, ± h ) + w ,x ∓ ¯ q ± ( x ) = 0 , (cid:0) ν ) w ,x + β (cid:1) + 12 h Z h − h ((1 + ν ) ξ ,x w ,x + βξ ,y ) ,x dy + ¯ f ≤ ,w ( x ) − φ ( x ) ≥ , (cid:20)(cid:0) ν ) w ,x + β (cid:1) + 12 h Z h − h ((1 + ν ) ξ ,x w ,x + βξ ,y ) ,x dy + ¯ f (cid:21) × ( w ( x ) − φ ( x )) = 0 , where ¯ q − ( x ) frictional force on the contact surface of the beam, φ ( x ) strictlyconcave functional that describes the rigid obstacle, ¯ f = ( hE ) − (1 − ν ) f ( x ), f ( x ) is an external loading, β = (1 − ν ), and ( ξ ( x, y, t ) , w ( x ) ) E is the dis-placement field, 2 h is the thickness, E is the Young’s modulus and ν is thePoisson’s ratio of the beam. Now, assume that the obstacle that we areconsidering is no longer rigid, but it is now elastic, then Gao’s [16] modelbecomes meaningless as the author’s model only assumes a rigid obstacle.
2. Derivation
Consider an unstrained static three-dimensional elastic body whose vol-ume described by the diffeomorphism ¯ X ∈ C ( ¯Ω; E ), where Ω ⊂ R is aconnected open bounded domain that satisfies the segment condition with auniform- C ( R ; R ) boundary (definition 4.10 of Adams and Fournier [17]), C k ( · ) is a space of continuous functions that has continuous first k partialderivatives in the underlying domain, E k is the k th-dimensional Euclidean3pace and R k is the k th-dimensional curvilinear space. Now, assume that ona part of the elastic foundation’s boundary, with a positive mean-curvaturewith respect to the unit outward normal and described by the injection σ ∈ C (¯ ω ; E ) where ω ⊂ R is a connected open bounded plane that satis-fies the segment condition with a uniform- C ( R ; R ) boundary, lies an elasticshell with the same curvature and the same physical form as the given sur-face. Further assume an overlying shell whose lower-surface is parametrisedby the sufficiently smooth injection σ ( x , x ). Now, consider the following: Assertion 1.
Let the map σ ∈ C (¯ ω ; E ) describes the lower-surface ofan unstrained shell, where ω ⊂ R is a connected open bounded plane thatsatisfies the segment condition with a uniform- C ( R ; R ) boundary. Giventhat the thickness of the shell is h , we assert that ≤ h K < hH ≪ , ∀ ( x , x ) ∈ ¯ ω , i.e. the lower-surface of the shell is non-hyperbolic and it is a surface with apositive mean curvature, and the thickness of the shell is sufficiently small. Note that K = ( F F − F F ) is the Gaussian curvature, H = − F α [II] α is the mean curvature, F [II] αβ = N i ∂ αβ σ i , ∀ α, β ∈ { , } , is the second fundamental form tensor of σ with respect to curvilinear coor-dinates and N = ∂ σ × ∂ σ || ∂ σ × ∂ σ || is the unit normal to the surface σ , ∂ α are partial derivatives with respect tocurvilinear coordinates x α , × is the Euclidean cross product and || · || is theEuclidean norm. Also, note that Einstein’s summation notation (see section1.2 of Kay [18]) is assumed through out, bold symbols signify that we aredealing with vector and tensor fields, we regard the indices i, j, k, l ∈ { , , } and α, β, γ, δ ∈ { , } , and we usually reserve the vector brackets ( · ) E forvectors in the Euclidean space and ( · ) for vectors in the curvilinear space.With assertion 1 and in accordance with the analysis of Jayawardana [19],we can express the energy functional of the shell on the elastic foundation as4ollows, J ( u ) = Z Ω (cid:20) A ijkl E ij ( u ) E kl ( u ) − f i u i (cid:21) d Ω+ Z ω (cid:20) B αβγδ (cid:18) hǫ αβ ( u ) ǫ γδ ( u ) + 13 h ρ αβ ( u ) ρ γδ ( u ) (cid:19) − hf i u i (cid:21) dω − Z ∂ω hτ i u i d ( ∂ω ) , where u is the displacement field, T ij ( u ) = A ijkl E kl ( u ) is second Piola-Kirchhoff stress tensor, E ij ( u ) = ( g ik ¯ ∇ j u k + g jk ¯ ∇ i u k ) is linearised Green-StVenant strain tensor, A ijkl = ¯ λg ij g kl + ¯ µ ( g ik g jl + g il g jk )is the isotropic elasticity tensor g ij = ∂ i ¯ X k ∂ j ¯ X k , ∀ i, j ∈ { , , } , is the covariant metric tensor of ¯ X with respect to curvilinear coordinates, ∂ j is the partial derivative with respect to the coordinate x j ,¯ λ = ¯ ν ¯ E (1 + ¯ ν )(1 − ν ) and¯ µ = 12 ¯ E ν are the first and the second Lam´e’s parameters respectively, ¯ E ∈ (0 , ∞ ) isthe Young’s modulus and ¯ ν ∈ ( − , ) is the Poisson’s ratio of the elasticfoundation, f is an external force density field acting on the elastic foun-dation, and ¯ n is the unit outward normal to the boundary ∂ Ω in curvilin-ear coordinates. Furthermore, τ αβ ( u ) = B αβγδ ǫ γδ ( u ) is the stress tensor, η αβ ( u ) = B αβγδ ρ γδ ( u ) negative of the change in moments density tensor, ǫ αβ ( u ) = 12 ( ∇ α ( u β | ω ) + ∇ β ( u α | ω )) − F [II] αβ ( u | ω )is half of the change in the first fundamental form tensor, ρ αβ ( u ) = ∇ α ∇ β ( u | ω ) − F [II] αγ F γ [II] β ( u | ω ) + F [II] βγ ∇ α ( u γ | ω )+ F [II] αγ ∇ β ( u γ | ω ) + (cid:0) ∇ α F [II] βγ (cid:1) ( u γ | ω )5he change in the second fundamental form tensor, B αβγδ = 2 λµλ + 2 µ F αβ [I] F γδ [I] + µ ( F αγ [I] F βδ [I] + F αδ [I] F βγ [I] )is the isotropic elasticity tensor,¯ λ = νE (1 + ν )(1 − ν ) and¯ µ = 12 E ν are the first and the second Lam´e’s parameters respectively, E ∈ (0 , ∞ ) isthe Young’s modulus and ν ∈ ( − , ) is the Poisson’s ratio of the shell, f is an external force density field acting on the shell, n is the unit outwardnormal vector to the boundary ∂ω in curvilinear coordinates, τ is an externaltraction field acting on the boundary of the shell, and u | ω is in a trace sense(see section 5.5 of Evans [20]). Finally, ¯ ∇ is the covariant derivative operatorin the curvilinear space, i.e. for any v ∈ C ( ¯Ω; R ), we define its covariantderivative as follows, ¯ ∇ j v k = ∂ j v k + ¯Γ kij v i , where ¯Γ kij = 12 g kl ( − ∂ l g ij + ∂ i g jl + ∂ j g li )are the Christoffel symbols of the second kind, and ∇ is the covariant deriva-tive operator in the curvilinear plane, i.e. for any u ∈ C (¯ ω ; R ), we defineits covariant derivative as follows, ∇ β u γ = ∂ β u γ + Γ γαβ u α , where Γ γαβ = 12 F γδ [I] (cid:0) − ∂ δ F [I] αβ + ∂ α F [I] βδ + ∂ β F [I] δα (cid:1) are the Christoffel symbols of the second kind in the curvilinear plane.6ow assume that the shell is coupled to the elastic foundation with fric-tion, where a portion of the foundation is satisfying the zero-Dirichlet bound-ary condition. Also, assume that one is applying forces to both the top andto a portion of the boundary of the shell to mimic compression and shear atthe contact region respectively. Now, if higher the compression, then higherthe normal displacement is towards the bottom, i.e. u | ω + < u α u α ) | ω + >
0, where ω + = lim x → + { ω × [0 , h ) } and by convention ( u α u α ) = √ u u + u u . Now, we consider Kikuchi andOden’s model [1] for Coulomb’s law of static friction for a three-dimensionalelastic body (i.e. not a shell), and once extended to curvilinear coordinatesand after taking the limit ε →
0, we find the following, (cid:2) T β ( v ) + ν F ( g ) ( v α v α ) − v β T ( v ) (cid:3) | ω + ≤ , (3)for T ( v ) | ∂ω + <
0, where v is the displacement field and the volume { ω × [0 , h ] } describes the reference configuration of this elastic body. Just as itis for Coulomb’s friction case (where the bodies are in relative equilibriumgiven that the magnitude of the normal stress is above a certain factor ofthe magnitude of the tangential stress), we assert that the bodies (i.e. theshell and the foundation) are in relative equilibrium given that the normaldisplacement is a below a certain factor of the magnitude of the tangentialdisplacement, i.e. (cid:2) ( u α u α ) + C ( g ) u (cid:3) | ω + ≤ , (4)if u | ω + ≤
0, for some dimensionless constant C . To determine the constant C , consider Coulomb’s law of static friction for the limiting equilibrium case(i.e. at the point of slipping) and rearrange equation (3) to obtain the fol-lowing, (cid:20) ¯ ∇ (cid:18) ( v α v α ) + 2 ν F (cid:18) γ − γ (cid:19) ( g ) v (cid:19) + (cid:18) ν F γ − γ (cid:19) ( g ) ¯ ∇ α v α + v δ ¯ ∇ δ v ( v α v α ) (cid:21) | ω + = 0 . Now, the above equation must hold for all elastic conditions, even under ex-treme conditions such as the incompressible elasticity condition, i.e. ( γ − γ ) ¯ ∇ i v i p ( x , x , x ), where p ( · ) is a finite function [21]. Thus, we may assume thefollowing equation, " ¯ ∇ (cid:16) ( v α v α ) + 2 ν F ( g ) v (cid:17) + v δ ¯ ∇ δ v ( v α v α ) + 2 ν F ( g ) p ( x , x , x ) | ω + = 0 . Now, nondimensionalise the above equation by making the transformations v i = ℓw i , x α = ℓy α and x = hy where ℓ = p meas( ω ; R ), and wheremeas( · ; R k ) is standard Lebesgue measure in R k (see chapter 6 of Schilling[22]), to obtain the following, (cid:20) (cid:18) ℓh (cid:19) (cid:0) g (cid:1) ∂∂y (cid:16) ( w α w α ) + 2 ν F ( g ) w (cid:17) + ( g ) w δ ( w α w α ) (cid:18) ∂w ∂y δ + Γ δi w i (cid:19) + 2 ν F p ( ℓy , ℓy , hy ) (cid:21) | ω + = 0 . (5)As our goal is to study shells, we consider the limit ( h/ℓ ) →
0. Also, as wealso require Coulomb’s law of static friction for the limiting-equilibrium caseto stay finite in this limit, equation (5) implies that h ( w α w α ) + 2 ν F ( g ) w i | { ( ωℓ ) × [0 , } = q ( y , y ) + O (cid:18)(cid:18) hℓ (cid:19) , y (cid:19) , where q ( · ) is a finite function. Furthermore, as we are seeking for a relationof the form of equation (4), we may assume q ( · ) = 0, and thus, we mayassume letting C = 2 ν F is a sound approximation. Finally, assuming that u is continuous on ¯Ω and noting that we have g = 1 in a shell by construction,we arrive at the following hypothesis: Hypothesis 1.
A shell supported by an elastic foundation with a roughcontact area that is in agreement with assertion 1 satisfies the followingdisplacement-based friction condition, (cid:2) ν F u + ( u α u α ) (cid:3) | ω ≤ , where ν F is the coefficient of friction between the shell and the foundation,and u is the displacement field of the shell with respect to the contact region ω . If [2 ν F u + ( u α u α ) ] | ω < , then we say that the shell is bonded to thefoundation, and if [2 ν F u + ( u α u α ) ] | ω = 0 , then we say that the shell is atlimiting-equilibrium. Theorem 1.
Let Ω ⊂ R be a connected open bounded domain that satis-fies the segment condition with a uniform- C ( R ; R ) boundary ∂ Ω such that ω, ∂ Ω ⊂ ∂ Ω with ¯ ω ∩ ¯ ∂ Ω = Ø and meas( ∂ Ω ; R ) > , and let ω ⊂ R bea connected open bounded plane that satisfies the segment condition with auniform- C ( R ; R ) boundary ∂ω . Also, let ¯ X ∈ C ( ¯Ω; E ) be a diffeomor-phism and σ ∈ C (¯ ω ; E ) be an injective immersion with ≤ h K < hH ≪ in ω . Furthermore, let f ∈ L (Ω) , f ∈ L ( ω ) and τ ∈ L ( ∂ω ) . Then thereexists a unique field u ∈ V F ( ω, Ω) such that u is the solution to the minimi-sation problem J ( u ) = min v ∈ V F ( ω, Ω) J ( v ) , where V F ( ω, Ω) = { v ∈ V S ( ω, Ω) | (cid:2) ν F v + ( v α v α ) (cid:3) | ω ≤ . e . } , V S ( ω, Ω) = { v ∈ H (Ω) | v | ω ∈ H ( ω ) × H ( ω ) × H ( ω ) , v | ∂ Ω = , ∂ β ( v | ω ) | ∂ω = 0 ∀ β ∈ { , }} ,J ( u ) = Z Ω (cid:20) A ijkl E ij ( u ) E kl ( u ) − f i u i (cid:21) d Ω+ Z ω (cid:20) B αβγδ (cid:18) hǫ αβ ( u ) ǫ γδ ( u ) + 13 h ρ αβ ( u ) ρ γδ ( u ) (cid:19) − hf i u i (cid:21) dω − Z ∂ω hτ i u i d ( ∂ω ) , and ν F is the coefficient of friction between the foundation and the shell. Note that L k ( · ) are the standard L k -Lebesgue spaces and H k ( · ) are thestandard W k, ( · )-Sobolev spaces (see section 5.2.1 of Evans [20]). Proof.
Note that there exists a unique field u ∈ V S ( ω, Ω) such that u is thesolution to the minimisation problem J ( u ) = min v ∈ V S ( ω, Ω) J ( v ) , V F ( ω, Ω) , J ( · )) ⊂ ( V S ( ω, Ω) , J ( · )) by construction, it issufficient to show that 2 ν F u + ( u α u α ) ≤ . e . in ω is a convex functional,where a . e . means almost everywhere (definition 1.40 of Adams and Fournier[17]).Now, let I ( u ; U ) = R U [2 ν F u +( u α u α ) ] dx dx . By construction I ( u ; U ) ≤ ∀ U ∈ M ( ω ) with meas( U ; ω ) >
0, where M ( · ) is a σ -algebra (definition1.37 of Adams and Fournier [17]). Also, by construction F [I] is positive defi-nite in ¯ ω (see section 5.3 of Kay [18]) and this implies that I ( · ; U ) is a convexfunctional for all U ∈ M ( ω ) with meas( U ; ω ) >
0, i.e. I ( t u + (1 − t ) v ; U ) ≤ tI ( u ; U ) + (1 − t ) I ( v ; U ). Furthermore, I ( t u + (1 − t ) v ; U ) ≤ ∀ U ∈ M ( ω )with meas( U ; ω ) >
0, and thus, our convexity result does not violate the def-inition of the functional I ( · ; U ), i.e. the condition 2 ν F u + ( u α u α ) ≤ . e . in ω is not violated. Now the proof follows form section 2.6 of Kinderlehrerand Stampacchia [2] or section of 8.4.2 Evans [20].Theorem 1 implies that there exists a unique weak solution to our prob-lem. However, due to the free-boundary constraint [2 ν F u + ( u α u α ) ] | ω ≤ . e . , the unique minimiser u may fail be a critical point in ( V F ( ω, Ω) , J ( · )),and thus, one requires the following: Corollary 1.
Let u ∈ V F ( ω, Ω) be the unique solution to the minimisa-tion problem J ( u ) = min v ∈ V F ( ω, Ω) J ( v ) , then we get the following variationalinequality ≤ J ′ ( u )( v − u ) , ∀ v ∈ V F ( ω, Ω) . Proof. ( V F ( ω, Ω) , J ( · )) is a convex space, and thus, the proof follows fromsection 8.4.2 of Evans [20]. We assume that u ∈ C (Ω; R ), u β | ω ∈ C ( ω ), u | ω ∈ C ( ω ) and 2 ν F u +( u α u α ) ≤ ω , and thus, theorem 1 and corollary 1 impliesthat the governing equations of the elastic foundation can be expressed asfollows, ¯ ∇ i T ij ( u ) + f j = 0 , ∀ j ∈ { , , } , boundary conditions of the elastic foundation can be expressed asfollows, u | ∂ Ω = , [¯ n i T ij ( u )] | { ∂ Ω \{ ω ∪ ∂ Ω }} = 0 , ∀ j ∈ { , , } , where ¯ n is the unit outward normal to the boundary ∂ Ω in curvilinear coor-dinates.As for the governing equations of the overlying shell , notice that the set V F ( ω, Ω) is not a linear set as it violates the homogeneity property. How-ever, it can be shown that for any field u ∈ V F ( ω, Ω) there exists a field w ∈ V F ( ω, Ω) \ { u } and a constant ε > u + s w ∈ V F ( ω, Ω), ∀ s ∈ ( − ε, R U [2 ν F ( u + sw )+( u α u α +2 su α w α + s w α w α ) ] dx dx ≤ ∀ U ∈ M ( ω ) with meas( U ; ω ) > ν F u + ( u α u α ) ] | ω < u ∈ V O ( ω, Ω), where V O ( ω, Ω) = { v ∈ V F ( ω, Ω) | [2 ν F v + ( v α v α ) ] | ω O < . e . } and where ω O = { V ∈ M ( ω ) | [2 ν F v +( v α v α ) ] | V < . e ., meas( V ; ω ) > } . Now, given a w ∈ V O ( ω, Ω), thereexists an ε > u + s w ∈ V F ( ω, Ω), ∀ s ∈ ( − ε,
1] where ε < (cid:16) ν F || u || L ( U ) − || ( u γ u γ ) || L ( U ) (cid:17)(cid:16) ν F || w || L ( U ) + || ( w α w α ) || L ( U ) (cid:17) for some U ∈ M ( ω O ). Now, simply let v = u + s w in corollary 4 to obtain0 ≤ J ′ ( u )( s w ), ∀ s ∈ ( − ε,
1] for this w ∈ V O ( ω, Ω). Finally, noticing that0 ≤ J ′ ( u )(sign( s ) | s | w ), ∀ w ∈ V O ( ω, Ω), we get the governing equations forthe bonded case: If (cid:2) ν F u + ( u α u α ) (cid:3) | ω < then ∇ α τ αβ ( u ) + 23 h F α [II] β ∇ γ η γα ( u )+ 13 h (cid:0) ∇ γ F α [II] β (cid:1) η γα ( u ) − h Tr( T β ( u )) + f β = 0 , ∀ β ∈ { , } ,F γ [II] α τ αγ ( u ) − h ∇ α ( ∇ γ η αγ ( u ))+ 13 h F δ [II] α F α [II] γ η γδ ( u ) − h Tr( T ( u )) + f = 0 , T j ( u )) = T j ( u ) | ω and Tr( · ) is the trace operator (see section 5.5of Evans [20]).To find the governing equations for the [2 ν F u + ( u α u α ) ] | ω = 0 case, con-sider a unique minimiser u ∈ V C ( ω, Ω), where V C ( ω, Ω) = { v ∈ V F ( ω, Ω) | [2 ν F v + ( v α v α ) ] | ω C = 0 a . e . } and where ω C = { V ∈ M ( ω ) | [2 ν F v +( v α v α ) ] | V = 0 a . e ., meas( V ; ω ) > } . Now, noticing that u j | ω C are notindependent, but are related by the condition u | ω C = − ν − F ( u α u α ) | ω C , weget δu | ω C = − ν − F ( u α u α ) − ( u γ δu γ ) | ω C . Let V C ( u ; ω, Ω) = { v ∈ V C ( ω, Ω) | ( v , v ) | ω C = ( cu , cu ) | ω C a . e ., ∀ c > , u ∈ V C ( ω, Ω) } , and now, given a w ∈ V C ( u ; ω, Ω) there exists an ε > u + s w ∈ V F ( ω, Ω), ∀ s ∈ ( − ε, ε < || ( w α w α ) || − L ( U ) || ( u γ u γ ) || L ( U ) for some U ∈ M ( ω C ). Now, simply let v = u + s w in corollary 4 to obtain0 ≤ J ′ ( u )( s w | Ω + s ( w , w ) | ω C ), ∀ s ∈ ( − ε,
1] for this w ∈ V C ( ω, Ω). Finally,noticing that J ′ ( u )( w | Ω ) = 0 (this leads to the governing equations in thefoundation) and 0 ≤ J ′ ( u )(sign( s ) | s | ( w , w ) | ω C ), ∀ w ∈ V C ( u ; ω, Ω) ⊂ V C ( ω, Ω), we get the governing equations for the limiting-equilibrium case(adapted from Section 8.4.2 of Evans [20]): If (cid:2) ν F u + ( u α u α ) (cid:3) | ω = 0, then ν F ∇ α τ αβ ( ¯ u ) − u β ( u α u α ) F γ [II] α τ αγ ( ¯ u )+ 23 ν F h F α [II] β ∇ γ η γα ( ¯ u ) + 16 h u β ( u α u α ) ∇ α ∇ γ η αγ ( ¯ u )+ 13 ν F h (cid:0) ∇ γ F α [II] β (cid:1) η γα ( ¯ u ) − h u β ( u α u α ) F δ [II] α F α [II] γ η γδ ( ¯ u ) − ν F h Tr( T β ( ¯ u )) + 12 h u β ( u α u α ) Tr( T ( ¯ u ))+ ν F f β − u β ( u α u α ) f = 0 , ∀ β ∈ { , } , where ¯ u | ω = ( u , u , − ν F ( u α u α ) ) | ω and ( ∂ ¯ u , ∂ ¯ u , ∂ ¯ u ) | ω = ( ∂ u , ∂ u , ∂ u ) | ω . boundary conditions of the overlying shell can be expressedas follows, (cid:2) n α τ αβ ( u ) + 23 h n γ F α [II] β η γα ( u ) (cid:3) | ∂ω = τ β , ∀ β ∈ { , } , − h [ n γ ∇ α η αγ ( u )] | ∂ω = τ ,∂ β ( u | ω ) | ∂ω = 0 , ∀ β ∈ { , } , where n is the unit outward normal vector to the boundary ∂ω in curvilinearcoordinates and τ is an external traction field acting on the boundary of theoverlying shell. Assume that we are dealing with overlying shell with a thickness h that isfrictionally coupled to an elastic foundation, where the unstrained configura-tion of the foundation is an infinitely long annular semi-prism characterisedby the following diffeomorphism,¯ X ( x , x , x ) = ( x , a sin( x ) , b cos( x ) ) E + x ϕ ( x ) ( , b sin( x ) , a cos( x ) ) E , where ϕ ( x ) = ( b sin ( x ) + a cos ( x )) , x ∈ ( −∞ , ∞ ), x ∈ ( − π, π ), x ∈ ( − L, a is the horizontal radius and b is the vertical radius ofthe contact region. Thus, the equations of the foundation can be expressedas follows, (¯ λ + ¯ µ ) ∂ (cid:0) ¯ ∇ i u i (cid:1) + ¯ µ ¯∆ u = 0 , (¯ λ + ¯ µ ) ∂ (cid:0) ¯ ∇ i u i (cid:1) + ¯ µ ¯∆ u = 0 , where u = ( , u ( x , x ) , u ( x , x ) ) is the displacement field, ¯∆ = ¯ ∇ i ¯ ∇ i isthe vector-Laplacian operator in the curvilinear space (see page 3 of Moonand Spencer [23]) with respect to Ω New .Now, eliminating x dependency, express the remaining boundaries as13ollows, ∂ Ω New = ¯ ω New ∪ ∂ Ω New0 ∪ ∂ Ω New f ,ω New = { ( − π, π ) × {− }} ,∂ Ω New0 = { ( − π, π ) × {− L }} ,∂ Ω New f = {{− π } × ( − L, } ∪ {{ π } × ( − L, } . Thus, the boundary conditions one imposes on the foundation reduce to thefollowing, u | ∂ Ω New0 = 0 (zero-Dirichlet) ,u | ∂ Ω New0 = 0 (zero-Dirichlet) , (cid:2) ( ¯ ψ ) ∂ u + ∂ u (cid:3) | ∂ Ω New f = 0 (zero-Robin) , (cid:2) (¯ λ + 2¯ µ ) ∂ u + ¯ λ (cid:0) ∂ u + ¯Γ u + ¯Γ u (cid:1) (cid:3) | ∂ Ω New f = 0 (zero-Robin) , where ¯ ψ = ϕ ( x ) + x ab ( ϕ ( x )) − .Now, consider overlying shell’s unstrained configuration, which is de-scribed by the injective immersion σ ( x , x ) = ( x , a sin( x ) , b cos( x ) ) E ,where x ∈ ( −∞ , ∞ ) and x ∈ ( − π, π ). Thus, we may express the gov-erning equations of the shell as:If [2 ν F u + ψ | u | ] | ω New <
0, then h Λ ∂ ǫ ( u ) + 13 h Λ(2 F ∂ ρ ( u ) + ∂ F ρ ( u )) − Tr( T ( u )) = 0 , − h Λ F ǫ ( u ) + 13 h Λ(∆ ρ ( u ) − F F ρ ( u )) + Tr( T ( u )) = 0 ;where u | ¯ ω New = ( , u ( x , , u ( x , ) is the displacement field of the shelland ∆ = ∇ α ∇ α is the vector-Laplacian in curvilinear plane with respect to ω New . 14f [2 ν F u + ψ | u | ] | ω New = 0, then ν F h Λ ∂ ǫ ( ¯ u ) − h Λ ψ sign( u ) F ǫ ( ¯ u )+ 13 ν F h Λ(2 F ∂ ρ ( ¯ u ) + ∂ F ρ ( ¯ u ))+ 16 h Λ ψ sign( u )(∆ ρ ( ¯ u ) − F F ρ ( ¯ u )) − ν F Tr( T ( ¯ u )) + 12 ψ sign( u )Tr( T ( ¯ u )) = 0 , where ¯ u | ω New = ( , u , − ν − F ψ | u | ) | ω New and ( , ∂ ¯ u , ∂ ¯ u ) | ω New = ( , ∂ u , ∂ u ) | ω New . Note thatTr( T ( u )) = ¯ µ (cid:0) ( ¯ ψ ) ∂ u + ∂ u (cid:1) | ω New , Tr( T ( u )) = (cid:2) ¯ λ (cid:0) ∂ u + ¯Γ u + ¯Γ u (cid:1) + (¯ λ + 2¯ µ ) ∂ u (cid:3) | ω New , and Λ = 4 µ λ + µλ + 2 µ = νE (1 + ν )(1 − ν ) . Now, eliminating x dependency, one can express the remaining bound-aries as follows, ∂ω New = ∂ω New T ∪ ∂ω New T max ,∂ω New T = { } ,∂ω New T max = { π } . Thus, the boundary conditions of the shell reduce to the following, (cid:2) Λ ǫ ( u ) + 23 h Λ F ρ ( u ) (cid:3) | ∂ω New T = τ (traction) , (cid:2) Λ ǫ ( u ) + 23 h Λ F ρ ( u ) (cid:3) | ∂ω New T max = τ max (traction) ,∂ ρ ( u ) | ∂ω New = 0 (zero-pressure) ,∂ u | ∂ω New = 0 (zero-Neumann) . { ( x , x ) | ( x , x ) ∈ [ − π, π ] × [ − L, } . Although theorem 1 is only validfor bounded domains, we show that the reduced two-dimensional problem isnumerically sound.To conduct numerical experiments, we use the second-order accuratefourth-order derivative iterative-Jacobi finite-difference method. The grid de-pendence is introduced in the discretisation of the (reduced two-dimensional)domain in curvilinear coordinates implies that the condition ψ ∆ x ≤ ∆ x , ∀ ψ ∈ { ¯ ψ ( x , x ) | x ∈ [ − π, π ] and x ∈ [ − L, } must be satisfied,where ∆ x j is a small increment in x j direction. For our purposes, we use∆ x = N − π and ψ = ¯ ψ ( π, N = 250. We also keep the values a = 2, L = 1, ¯ E = 10 , ¯ ν = , τ = 1 and τ max = 1 fixed for all experiments. -2-1.5-1-0.5100.511.5 A z i m u t ha l D i s p l a c e m en t × -4 Radius θ -102 12 -18-16-14-12-101-8-6-4-2 R ad i a l D i s p l a c e m en t × -5 Radius θ -102 12 Figure 1: Displacement field of the foundation predicted by the shell modelwith friction.Figure 1 is calculated with the values of τ max = 1, b = 2, h = , E = 8000, ν = , ν F = 1 and with a grid of 250 ×
41 points, and it shows the azimuthal(i.e u ) and the radial (i.e. u ) displacements. The maximum azimuthaldisplacements are observed at ( θ =) x = ± π , with respective azimuthaldisplacements of u = ± . × − . The maximum radial displacement isobserved at x = ± π , with a radial displacement of u = − . × − . Fur-16hermore, in the intervals x ∈ ( − . , − . θ =) x ∈ (0 . , .
3. Comparison Against Kikuchi and Oden’s Work
The most comprehensive mathematical study on friction that we areaware of is the publication by Kikuchi and Oden [1], and thus, we extendtheir model to model two-body friction model in curvilinear coordinates asa benchmark model to compare against our shell model with friction, andwe do so by modifying equation (1) (see section 5.5 (v) of Kikuchi and Oden[1]). Now, assume that an elastic body on a rough rigid surface where the fic-tion is governed by Coulomb’s law of static friction. Given that one is usingcurvilinear coordinates, fix the purely-normal stress at the contact bound-ary as a constant, i.e. ν F T ( v ) | ω + = K . Then, equation (1) implies that j ′ ε ( v ) δ v = − R ω ( g ) T α ( v ) δv α | ω + dω . Note that v | ω + describes the rela-tive displacement between the elastic body and the boundary ω , and thus,if the elastic body is in contact with another rough elastic body, then thedisplacement field one must consider is the relative displacement (due tothe fact that friction apposes relative potential motion). Now, consider atwo-body contact problem where the contact area is rough and the frictionis governed by Coulomb’s law of static friction. Now, let the displacementfields of the overlying body be v and the foundation be u . As purely-normalstress is continuous at the boundary, just as before, fix the purely normalstress as ν F T ( u ) | ω − = ν F T ( v ) | ω + = K , where ω − = lim x → − Ω, andmake the transformation v β | ω + → v β | ω + − u β | ω − in the functional j ε ( · ) tosignify the relative displacement field. Now, collecting all the tangentialterms from the contact boundary (i.e. u β | ω − and v β | ω + terms), one finds j ′ ε ( v − u )( δ v − δ u ) = − R ω ( g ) ([ T α ( v ) δv α ] | ω + − [ T α ( u ) δu α ] | ω − ) dω , where j ε ( v − u ) = Z ω (cid:20) K Φ( v − u ) − ε (cid:21) dω , if Φ( v − u ) | ω ≥ ε , Z ω K ε − (Φ( v − u )) dω , if Φ( v − u ) | ω < ε , Φ( v − u ) = ( ℓ α ( v , u ) ℓ α ( v , u )) , ℓ ( v , u ) = ( v , v ) | ω + − ( u , u ) | ω − .
17s the two bodies are in contact, the normal displacement (of the bothbodies) is identical. Thus, one obtain the extended Kikuchi and Oden’s modelfor Coulomb’s law of static friction for a two-body problem in curvilinearcoordinates, which is described by the following set of equations, v | ω + − u | ω − = 0 ,T ( v ) | ω + − T ( u ) | ω − = 0 ,T β ( v ) | ω + = − ν F ( g ) (cid:0) v β | ω + − u β | ω − (cid:1) Φ( v − u ) (cid:0) T ( v ) | ω + (cid:1) , if Φ( v − u ) | ω + ≥ ε , − ν F ( g ) (cid:0) v β | ω + − u β | ω − (cid:1) ε (cid:0) T ( v ) | ω + (cid:1) , if Φ( v − u ) | ω + < ε ,T β ( u ) | ω − = − ν F ( g ) (cid:0) v β | ω + − u β | ω − (cid:1) Φ( v − u ) (cid:0) T ( u ) | ω − (cid:1) , if Φ( v − u ) | ω − ≥ ε , − ν F ( g ) (cid:0) v β | ω + − u β | ω − (cid:1) ε (cid:0) T ( u ) | ω − (cid:1) , if Φ( v − u ) | ω − < ε , where T β ( v ) = µ (cid:0) ¯ ∇ β v + ¯ ∇ v β (cid:1) ,T ( v ) = λ ¯ ∇ α v α + ( λ + 2 µ ) ¯ ∇ v ,T β ( u ) = ¯ µ (cid:0) ¯ ∇ β u + ¯ ∇ u β (cid:1) ,T ( u ) = ¯ λ ¯ ∇ α u α + (cid:0) ¯ λ + 2¯ µ (cid:1) ¯ ∇ u , and where ν F is the coefficient of friction. Note that given that T ( u ) | ω − is fixed as a positive constant and one is considering Euclidean coordinates,one can see that the above problem simply reduces to Kikuchi and Oden’smodel for Coulomb’s law of static friction [1] in the limit u →
0. Also, notethat in the set { ω × [0 , h ) } , we have g = 1. Furthermore, above extendedKikuchi and Oden’s model can further be simplified by noticing that thecontinuousness of the purely-normal stress at the boundary; however, fromour numerical analysis we find this reduced model is non-convergent in afinite-difference setting. Thus, we insist upon the given formulation.To proceed with our analysis, we numerically model the overlying body asa three-dimensional body and we do not approximate this body as a shell orotherwise. Thus, the displacement at the contact region with this approach18s obtained by the use of the standard equilibrium equations in linear elas-ticity and extended Kikuchi and Oden’s model.In accordance with the framework that is introduced in section 2.2, theoverlying body is restricted to the region x ∈ (0 , h ). Now, we can expressthe governing equations of the overlying body as follows,( λ + µ ) ∂ (cid:0) ¯ ∇ i v i (cid:1) + µ ¯∆ v = 0 , ( λ + µ ) ∂ (cid:0) ¯ ∇ i v i (cid:1) + µ ¯∆ v = 0 , where v = ( , v ( x , x ) , v ( x , x ) ) is the displacement field of the overlyingbody, the perturbed governing equations of the overlying body as follows,( λ + µ ) ∂ (cid:0) ¯ ∇ i δv i (cid:1) + µ ¯∆ δv = 0 , ( λ + µ ) ∂ (cid:0) ¯ ∇ i δv i (cid:1) + µ ¯∆ δv = 0 , where δ v = ( , δv ( x , x ) , δv ( x , x ) ) is a small perturbation of the displace-ment field of the overlying body, and the perturbed governing equations ofthe foundation as follows,(¯ λ + ¯ µ ) ∂ (cid:0) ¯ ∇ i δu i (cid:1) + ¯ µ ¯∆ δu = 0 , (¯ λ + ¯ µ ) ∂ (cid:0) ¯ ∇ i δu i (cid:1) + ¯ µ ¯∆ δu = 0 , where δ u = ( , δu ( x , x ) , δu ( x , x ) ) is the perturbation of the displace-ment field of the foundation. Also, we can express the boundary conditionsof the overlying body as follows, (cid:2) ( λ + 2 µ ) ∂ v + λ (cid:0) ∂ v + ¯Γ v + ¯Γ v (cid:1) (cid:3) | { ∂ω New T × [0 ,h ] } = τ (traction) , (cid:2) ( λ + 2 µ ) ∂ v + λ (cid:0) ∂ v + ¯Γ v + ¯Γ v (cid:1) (cid:3) | { ∂ω New T max × [0 ,h ] } = τ max (traction) , (cid:2) λ (cid:0) ∂ v + ¯Γ v + ¯Γ v (cid:1) + ( λ + 2 µ ) ∂ v (cid:3) | { ( − π, π ) ×{ h }} = 0 (zero-Robin) , (cid:2) ( ¯ ψ ) ∂ v + ∂ v (cid:3) | { ∂ω New × [0 ,h ] }∪{ ( − π, π ) ×{ h }} = 0 (zero-Robin) , boundary conditions if the displacement fields as follows, (cid:2) v − u (cid:3) | ω New = 0 (continuous radial displacement) , (cid:2) T ( v ) − T ( u ) (cid:3) | ω New = 0 (continuous radial stress) , δu | ∂ Ω New0 ∪ ∂ Ω New f = 0 ,δu | ∂ Ω New = 0 ,δv | { ∂ω New × (0 ,h ) }∪{ [ − π, π ] ×{ h }} = 0 ,δv | ¯ ω New ∪{ ∂ω New × (0 ,h ) }∪{ [ − π, π ] ×{ h }} = 0 . Thus, the equations characterising the frictional coupling of the overlyingbody to the foundation can be expressed as:If ¯ ψ | v − u || ω New ≥ ǫ , then (cid:2) µ (cid:0) ¯ ψ ∂ v + ( ¯ ψ ) − ∂ v (cid:1) + ν F sign( v − u ) T ( v ) (cid:3) | ω New = 0 , (cid:2) ¯ µ (cid:0) ¯ ψ ∂ u + ( ¯ ψ ) − ∂ u (cid:1) + ν F sign( v − u ) T ( u ) (cid:3) | ω New = 0 ;If ¯ ψ | v − u || ω New < ǫ , then (cid:2) µ (cid:0) ¯ ψ ∂ δv (cid:1) + ν F ǫ − ¯ ψ ( v − u ) T ( δ v ) + ν F ǫ − ¯ ψ ( δv − δu ) T ( v )+ µ (cid:0) ¯ ψ ∂ v + ( ¯ ψ ) − ∂ v (cid:1) + ν F ǫ − ¯ ψ ( v − u ) T ( v ) (cid:3) | ω New = 0 , (cid:2) ¯ µ (cid:0) ¯ ψ ∂ δu (cid:1) + ν F ǫ − ¯ ψ ( v − u ) T ( δ u ) + ν F ǫ − ¯ ψ ( δv − δu ) T ( u )+ ¯ µ (cid:0) ¯ ψ ∂ u + ( ¯ ψ ) − ∂ u (cid:1) + ν F ǫ − ¯ ψ ( v − u ) T ( u ) (cid:3) | ω New = 0 , where T ( v ) = λ (cid:0) ∂ v + ¯Γ v + ¯Γ v (cid:1) + ( λ + 2 µ ) ∂ v ,T ( u ) = ¯ λ (cid:0) ∂ u + ¯Γ u + ¯Γ u (cid:1) + (¯ λ + 2¯ µ ) ∂ u . To conduct numerical experiments, we use the second-order accurateiterative-Jacobi finite-difference method with Newton’s method for nonlin-ear systems (see chapter 10 of Burden et al. [24]). Also, as a result ofthe grid dependence in the overlying body, we must satisfy the condition ψ ∆ x ≤ ∆ x , ∀ ψ ∈ { ¯ ψ ( x , x ) | x ∈ [ − π, π ] and x ∈ [0 , h ] } ,where ∆ x j is a small increment in x j direction. For our purposes, we use∆ x = N − π and ψ = ¯ ψ ( π, h ), where N = 250.Figure 2 is calculated with the values τ max = 1, b = 2, h = , E = 8000, ν = , ε = 10 − , ν F = 1 and with a grid of 250 ×
41 points, and it shows theazimuthal (i.e. u ) and the radial (i.e. u ) displacements of the foundation.20 × -4 A z i m u t ha l D i s p l a c e m en t Radius θ -102 12 -18-16-14-12-101-8-6-4-2 R ad i a l D i s p l a c e m en t × -5 Radius θ -102 12 Figure 2: Displacement field of the foundation of the extended Kikuchi andOden’s model.The maximum azimuthal displacements are observed at ( θ =) x = ± π withrespective azimuthal displacements of u = ± . × − . The maximum ra-dial displacement is observed at x = ± π with a radial displacement of u = − . × − . Also, in the interval ( θ =) x ∈ ( − π, π ), i.e. the entirecontact region ω , we see that the overlying body is at limiting-equilibrium.Our final goal in this section is to investigate how our shell model with fric-tion predicts the displacement field of the foundation relative to the extendedKikuchi and Oden’s model for the variables ν F , δτ = τ max /τ , δb = b/a , δh = h/L , δE = E/ ¯ E and δν = ν/ ¯ ν . To calculate the relative error betweenthe displacement field of the foundation predicted by each model, we presentthe following metricRelativeError( u i ) = (cid:16)P { kl } || u i shell ( y k , y l ) − u i Kikuchi ( y k , y l ) || (cid:17) (cid:16)P { kl } || u i shell ( y k , y l ) + u i Kikuchi ( y k , y l ) || (cid:17) , where y k = − π + k ∆ x , y l = − L + l ∆ x , 1 ≤ k ≤ N and 1 ≤ l < N . Notethat we assume the default values ν F = 1, δτ = 1, δb = 1, δh = , δE = 8and δν = 1 and ε = 10 − throughout, unless it strictly says otherwise.21 .3 0.4 0.5 0.6 0.7 0.8 0.9 1 ν F A z i m u t ha l R e l a t i v e E rr o r ν F R ad i a l R e l a t i v e E rr o r Figure 3: Relative error for ν F . δτ A z i m u t ha l R e l a t i v e E rr o r δτ R ad i a l R e l a t i v e E rr o r Figure 4: Relative error for δτ .Figure 3 shows that as the coefficient of friction at the contact surface, ν F , increases, the relative error decreases, and this a significant reduction inthe error. This implies that rougher the contact surface is, then closer ourshell model with friction resembles the extended Kikuchi and Oden’s model.This is an intuitive result as the coefficient of friction increases, both modelsresemble the bonded case, and in chapter 3 of Jayawardana [19], it is shownthat the bonded shell model is a better approximation of the overlying body22 .05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 δ h0.010.020.030.040.050.060.07 A z i m u t ha l R e l a t i v e E rr o r δ h0.020.040.060.080.10.120.140.16 R ad i a l R e l a t i v e E rr o r Figure 5: Relative error for δh . δ E0.020.040.060.080.10.120.140.160.18 A z i m u t ha l R e l a t i v e E rr o r δ E0.10.150.20.250.30.35 R ad i a l R e l a t i v e E rr o r Figure 6: Relative error for δE .with respect to Baldelli and Bourdin’s asymptotic method [25].Figure 4 shows that as the traction ratio, δτ , increases, both azimuthaland radial relative errors also increases. This implies that limiting-equilibriumsimplied by each model can be different. We observed this effect in our earliernumerical modelling (compare the analysis of figure 1 and figure 2).23 δν A z i m u t ha l R e l a t i v e E rr o r δν R ad i a l R e l a t i v e E rr o r Figure 7: Relative error for δν . δ b00.010.020.030.040.050.060.07 A z i m u t ha l R e l a t i v e E rr o r δ b00.050.10.150.20.25 R ad i a l R e l a t i v e E rr o r Figure 8: Relative error for δb .Figure 5 shows that as the relative thicknesses of the shell, δh , decreases,both the azimuthal and the radial relative error increases. This is a con-tradictory result as we derived our displacement-based friction condition byconsidering Coulomb’s law to remain valid the limit δh →
0, and thus, weshould expect a better agreement between the two models for smaller valuesof δh . 24igure 6 shows the that as the relative Young’s modulus of the shell, δE ,increases, the relative error decreases. If one assumes the shell is bonded tothe elastic foundation, then this result seems to be consistent with the workof Aghalovyan [26].Figure 7 shows that as the relative Poisson’s ratio of the shell, δν , in-creases, the relative error decreases. This implies that as the shell becomesincompressible, both models would be in better agreement. However, unlikefor Young’s modulus case, one cannot indefinitely reduce the relative error byincreasing Poisson’s ratio as Poisson’s ratio cannot neither attain nor exceedthe value .Figure 8 shows that as the curvature of the contact region, δb , decreases,both azimuthal and radial relative error also decreases. This result is consis-tent with our derivation as our shell equation assumed to be valid for contactregions with small (but non-negative) curvatures (see assertion 1). Note that δb may be interpreted as the curvature as the δb is correlated with the meancurvature (i.e. H ( x ) = ab ( ϕ ( x )) − ) in the set x ∈ ( − π, π ).
4. Conclusions
In our analysis, we derived a model for a shell that is frictionally cou-pled to an elastic foundation. We used Kikuchi and Oden’s model [1] forCoulomb’s law of static friction to derive a displacement-based static-frictioncondition. By construction, this displacement-based friction condition ismathematically sound as we proved the existence and the uniqueness of so-lutions for our shell model with friction with the aid of the works of Kinder-lehrer and Stampacchia [2], and section of 8.4.2 Evans [20]. Note that, as faras we are aware, this is the first derivation of a displacement-based frictioncondition, as only force and stress based friction conditions currently existin the literature.For numerical analysis, we extended Kikuchi and Oden’s model [1] forCoulo-mb’s law of static-friction to model a full two-body contact problemin curvilinear coordinates and the purpose of numerical analysis is to ascer-tain how the displacement field of the foundation behaves when the overlyingbody is modelled with our shell model with friction relative to when the over-lying body is modelled with extended Kikuchi and Oden’s model. The results25ndicate that, if the shell has a relatively high Young’s modulus (i.e. stiff)or has a relatively high Poisson’s ratio (i.e. close to incompressible), and thecontact region has a very high coefficient of friction or less curved, then thedisplacement field of the foundation predicted by both models are in betteragreement. We also observed that both model are in better agreement forthicker shells, which is a contradictory result as it is inconstant with thederivation of our shell model. At this time, it is unclear the inaccuracy is be-ing introduced by our shell model with friction, extended Kikuchi and Oden’smodel or both.From our numerical analysis, the greatest reduction in the error is ob-served for higher coefficients of friction, i.e. as the coefficient of frictionincreases, the bodies behaves as if they are bonded, and thus, the greateragreement between the solutions of the extended Kikuchi and Oden’s modeland our shell model with friction. This is an expected result as our initiallyderived overlying shell model is derived to approximate bonded thin bodieson elastic foundations, and the efficacy of this model is numerically demon-strated in sections 3.5 and 3.6 of Jayawardana [19]. The second greatestreduction in the error is observed for higher Young’s moduli of the shell.This is also an expected result as it seems to be consistent with the asymp-totic analysis of Aghalovyan [26] on shells (given that the shell is bonded tothe elastic foundation).To ascertain the physical validity of our shell model with friction, we con-ducted human trials to measure the frictional interactions that fabrics haveon soft-tissue of human subjects (10 subjects in the first trial and 8 subjectsin the second trial, see Jayawardana et al. [27] or chapter 6 Jayawardana[19]). We discover that there exists a positive correlation: (i) between thedisplacement of soft-tissue and the volume of soft-tissue; and (ii) betweenthe applied tension to the fabric and the volume of soft-tissue. Also, thereexists a negative correlation: (iii) between the displacement of the soft-tissueand the Young’s modulus of soft-tissue; and (iv) the applied tension to thefabric and the Young’s modulus of soft-tissue. Further numerical modellingsof our shell model with friction (where now the shell is approximated with ashell-membrane), we were able to predict the correlations-(i) to (iii). How-ever, our numerical modelling could not predict correlation-(iv).As a closing remark, we remind the reader that the physical validity of26ach model is still an open question as there exist no definitive friction modelthat we can compare our results against, and at no point we claim our modelsdepict how friction behaves in real life.