Dynamic HistoryDependent VariationalHemivariational Inequalities - PowerPoint PPT Presentation

Dynamic History–Dependent Variational–Hemivariational Inequalities with Applications Stanislaw Mig´ orski Jagiellonian University in Krakow, Poland Faculty of Mathematics and Computer Science a joint work with Weimin Han and Mircea Sofonea Control of state constrained dynamical systems September 25-29, 2017, Universit` a di Padova (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 1 / 50

Outline of the talk (1) Part 1: Dynamic frictional nonsmooth contact problem Physical setting of the model Classical model for the viscoelastic contact Variational-hemivariational formulation of the contact problem Weak unique solvability of the contact problem (2) Part 2: Abstract first order variational-hemivariational inequality Existence and uniqueness for variational-hemivariational inequality Existence and uniqueness for inclusion with the Clarke subgradient A fixed point argument (3) Remarks, comments, examples (4) Conclusions (5) References (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 2 / 50

Mathematical tool: convex subdifferential Let E be a Banach space and E ∗ be its dual. Definition (convex subdifferential) Let ϕ : E → R ∪ { + ∞} be a convex function. The (convex) subdifferential of ϕ at x , and is defined by ∂ϕ ( x ) = { x ∗ ∈ E ∗ | ϕ ( v ) ≥ ϕ ( x ) + � x ∗ , v − x � E ∗ × E for all v ∈ E } . Sometimes we refer to ∂ϕ as the subdifferential of ϕ in the sense of convex analysis. Observe that if ϕ ( x ) = + ∞ , then ∂ϕ ( x ) = ∅ . (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 3 / 50

The Clarke subgradient (1983) Definition (Clarke subgradient) Let h : E → R be a locally Lipschitz function on a Banach space E . The generalized directional derivative of h at x ∈ E in the direction v ∈ E is defined by h ( y + tv ) − h ( y ) h 0 ( x ; v ) = lim sup . t y → x , t ↓ 0 The generalized subgradient of h at x is given by ∂ h ( x ) = { ζ ∈ E ∗ | h 0 ( x ; v ) ≥ � ζ, v � E ∗ × E for all v ∈ E } . The locally Lipschitz function h is called regular (in the sense of Clarke) at x ∈ E if for all v ∈ E the one-sided directional derivative h ′ ( x ; v ) exists and satisfies h 0 ( x ; v ) = h ′ ( x ; v ) for all v ∈ E . (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 4 / 50

Part 1 Dynamic frictional nonsmooth contact problem (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 5 / 50

Physical setting of the contact model A viscoelastic body occupies an open, bounded and connected set Ω ⊂ R d , d = 2, 3. The boundary Γ is Lipschitz continuous and Γ = Γ D ∪ Γ N ∪ Γ C with mutually disjoint measurable parts, m (Γ D ) > 0. We denote by ν = ( ν i ) the outward unit normal at Γ. The part Γ C the potential contact surface. (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 6 / 50

Basic notation We suppose the body is clamped on Γ D , volume forces of density f 0 act in Ω and surface tractions of density f N are applied on Γ N . The notation S d stands for the space of second order symmetric tensors on R d . On R d and S d we use the inner products and the Euclidean norms defined by u · v = u i v i , � u � = ( u · u ) 1 / 2 for all u = ( u i ) , v = ( v i ) ∈ R d , σ · τ = σ ij τ ij , � σ � = ( σ · σ ) 1 / 2 for all σ = ( σ ij ) , τ = ( τ ij ) ∈ S d , respectively. (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 7 / 50

Basic notation for the contact problem Given a vector field u , notation u ν and u τ represent its normal and tangential components on the boundary defined by u ν = u · ν and u τ = u − u ν ν . For a tensor σ , the symbols σ ν and σ τ denote its normal and tangential components on the boundary, i.e., σ ν = ( σν ) · ν and σ τ = σν − σ ν ν . Sometimes, we omit the explicit dependence on x ∈ Ω ∪ Γ. (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 8 / 50

Problem (Classical model for the contact process) Find a displacement field u : Ω × (0 , T ) → R d and a stress field σ : Ω × (0 , T ) → S d such that for all t ∈ (0 , T ) , � t σ ( t ) = A ε ( u ′ ( t )) + B ε ( u ( t )) + C ( t − s ) ε ( u ′ ( s )) ds in Ω , 0 ρ u ′′ ( t ) = Div σ ( t ) + f 0 ( t ) in Ω , u ( t ) = 0 on Γ D , σ ( t ) ν = f N ( t ) on Γ N , − σ ν ( t ) ∈ k ( u ν ( t )) ∂ j ν ( u ′ ν ( t )) on Γ C , � � t � � σ τ ( t ) � ≤ F b � u τ ( s ) � ds , 0 � � t � u ′ τ ( t ) τ ( t ) � if u ′ − σ τ ( t ) = F b � u τ ( s ) � ds τ ( t ) � = 0 on Γ C , � u ′ 0 u ′ (0) = w 0 u (0) = u 0 , in Ω . (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 9 / 50

Comments on the model Relation � t σ ( t ) = A ε ( u ′ ( t )) + B ε ( u ( t )) + C ( t − s ) ε ( u ′ ( s )) ds in Ω 0 represents the viscoelastic constitutive law in which A is the viscosity operator, B is the elasticity operator, C is the relaxation tensor, and ε ( u ) denotes the linearized strain tensor defined by ε ij ( u ) = 1 ε ( u ) = ( ε ij ( u )) , 2( u i , j + u j , i ) in Ω . Equation ρ u ′′ ( t ) = Div σ ( t ) + f 0 ( t ) in Ω represents the equation of motion, where ρ denotes the density of mass, and f 0 denotes the density of the time-dependent volume forces. For simplicity, we take ρ = 1. (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 10 / 50

Comments on the model Condition u ( t ) = 0 on Γ D is the displacement homogeneous boundary condition which means that the body is fixed on Γ D . Condition σ ( t ) ν = f N on Γ N is the traction boundary condition with surface tractions of density f N acting on Γ N . Relation − σ ν ( t ) ∈ k ( u ν ( t )) ∂ j ν ( u ′ ν ( t )) on Γ C represents the multivalued contact condition with nonmonotone normal damped response in which ∂ j ν denotes the Clarke subgradient of a given function j ν and k is a damper coefficient which is allowed to depend on the normal displacement. (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 11 / 50

Comments on the model Condition � � t � � σ τ ( t ) � ≤ F b � u τ ( s ) � ds , 0 � � t � u ′ τ ( t ) τ ( t ) � if u ′ − σ τ ( t ) = F b � u τ ( s ) � ds τ ( t ) � = 0 on Γ C � u ′ 0 represents a version of the Coulomb law of dry friction in which F b is a given positive function, the friction bound. The friction bound may depend on the quantity � t S ( x , t ) = � u τ ( x , s ) � ds 0 which is the total slip (or the accumulated slip) at the point x ∈ Γ 3 over the time period [0 , t ]. Considering such a dependence is reasonable from the physical point of view, since it incorporates the changes on the contact surface resulting from sliding. (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 12 / 50

A remark It is well known that a variant of the Coulomb law of dry friction on Γ C can be equivalently formulated as follows � � t � ∂ � u ′ − σ τ ( t ) ∈ F b � u τ ( s ) � ds τ ( t ) � R d on Γ C , 0 where  B ( 0 , 1) , if x = 0   ∂ � x � = x � x � , if x � = 0   for x ∈ R d , B ( 0 , 1) denotes the unit closed ball in R d . (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 13 / 50

A remark J. J. Telega, Variational methods in contact problems in mechanics, Advances in Mechanics 10 (1987), 3-95. “Generalizations of subdifferential boundary conditions − σ ν ∈ ∂ j ν ( u ν ) , − σ τ ∈ ∂ j τ ( u τ ) on Γ C to the case of nonmonotone relations, with the use of mathematical results of Rockafellar (1981), are presented by Panagiotopoulos (1983-1985). These generalizations do not have practical applications so far. This follows from the fact that the existing notion of the generalized subdifferential to the case of nonconvex functions is quite complex”. (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 14 / 50

Variational formulation of the contact problem We use the spaces V = { v ∈ H 1 (Ω; R d ) | v = 0 on Γ D } , H = L 2 (Ω; R d ) , H = L 2 (Ω; S d ) . We denote by v the trace on the boundary of an element v ∈ H 1 (Ω; R d ). On V we consider the inner product and the corresponding norm given by � v � V = � ε ( v ) � H for all u , v ∈ V . ( u , v ) V = ( ε ( u ) , ε ( v )) H , Since meas (Γ D ) > 0, it follows that V is a Hilbert space. The trace operator is denoted by γ : V → L 2 (Γ C ; R d ). The space H is a Hilbert space endowed with the inner product � ( σ , τ ) H = σ ij ( x ) τ ij ( x ) dx , Ω and the associated norm � · � H . (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 15 / 50

Variational formulation of the contact problem Define a space of fourth order tensor fields Q ∞ = { E = ( E ijkl ) | E ijkl = E jikl = E klij ∈ L ∞ (Ω) , 1 ≤ i , j , k , l ≤ d } . This is a real Banach space with the norm � �E� Q ∞ = �E ijkl � L ∞ (Ω) . 0 ≤ i , j , k , l ≤ d (Jagiellonian University in Krakow) Variational–Hemivariational Inequalities 16 / 50