Analysis of a model of elastic plastic mixtures - PowerPoint PPT Presentation

Analysis of a model of elastic plastic mixtures (“Prandtl-Reuss-mixtures”) Project of Josef M´ alek and Jens Frehse Praha 2012

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Outline Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures We refer to a paper J. Kratochv´ ıl, J. M´ alek, K.R. Rajagopal and A.R. Srinivasa: Modelling of the response of elastic plastic materials treated as mixture of hard and soft regions , ZAMP 55 (2004), 500-518 and to Phd-Thesis of Luba Khasina adviced by Kratochv´ ıl-M´ alek-Frehse, where first steps are done to formulate the model of KMRS in the framework of Sobolev spaces nad variational inequalities.

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures KMRS consider a body Ω consisting of soft and hard material and loading. During the loading process, soft material is assumed to be perfectly elastic plastic, the hard material is assumed to satisfy a certain hardening rule in the non elastic region. The beautiness of the theory consists in the fact, that no artificial internal variables enter, the history of the plastic deformation of the soft material replaces the internal variable.

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Purpose of the present talk Presenting a mathematical formulation in the framework of quasi-variational inequalities and Sobolev spaces. The theory turns out to be very similar to the Prandtl-Reuss-law for single materials; all known regularity results hold also for the mixture model.

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures KMRS-model after simplification and choice of example for illustration basic domain ⊂ ❘ 3 Ω [0 , T ] loading interval σ s = σ s ( x , t ) , x ∈ Ω , t ∈ [0 , T ] stress of the soft material stress of the hard material σ h e = 1 2( ∇ u + ( ∇ u ) T ) strain e ps plastic strain of the soft material plastic strain of the hard material e hs α = α ( x , t ) volume fraction of the soft material 1 − α volume fraction of the hard material 0 ≤ α ≤ 1, (later 0 < ε 0 < α < 1 − ε 0 ) α ( x , 0) = 0, (resp. ε 0 ) α ( x , t ) = α ( x , t ; e ps ), α monotone increasing in t

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Governing equations Balance of forces − div( α σ s + (1 − α ) σ h ) = f in Ω Weak formulation with boundary values � p 0 ϕ d o , ∀ ϕ ∈ H 1 , 2 Γ (Ω; ❘ 3 ) ( α σ s +(1 − α ) σ h , ∇ ϕ ) L 2 (Ω) = ( f , ϕ ) L 2 (Ω) + ∂ Ω Symmetry σ = σ T Hooke’s law in the elastic region σ s = C ( e − e ps ) , σ h = C ( e − e ph ) , C elasticity tensor, of course different C s and C h can be used. Plastic incompressibility trace e ps = trace e ph = 0 .

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Yield function and yield condition Let B D = B − 1 3(trace B ) I “Deviator” F s ( σ s ) = | σ s | − κ s F h ( σ h ) = | σ h | − κ h ( t ) , κ h ( t ) = κ h ( t , e ps ) These are v Mises type yield functions. Yield conditions F s ( σ s ) ≤ 0 , F h ( σ h ) ≤ 0 Kuhn-Tucker-conditions σ sD σ hD e s = λ s ˙ e h = λ h ˙ | σ sD | , | σ hD | , λ s , λ h ≥ 0 , λ s F s ( σ s ) = 0 , λ h F h ( σ h ) = 0.

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Model for the volume fraction α � t ℓ = | ˙ e ps | 0 α ( · , t ) = α 0 + (1 − α 0 ) e − c 0 ℓ Model for the hardening rule and yield parameters κ s = const > 0 , κ h = κ s + r 0 ℓ. Later we will work with Lipschitz α and κ h .

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Prandtl-Reuss law Formulation as variational inequality Find σ ∈ L ∞ (0 , T ; L 2 (Ω; ❘ 3 × 3 )) such that ˙ σ ∈ L ∞ ( L 2 ) and  σ (0) = σ 0   σ = σ T    � ∂ Ω p 0 ϕ d o , ∀ ϕ ∈ H 1 , 2 Γ (Ω; ❘ 3 ) ( σ , ∇ ϕ ) L 2 (Ω) = ( f , ϕ ) L 2 (Ω) + ❑  F ( σ ) = | σ D | − κ     F ( σ ) ≤ 0 ( A ˙ σ , σ − ˜ σ ) L 2 (Ω) ≤ 0 ∀ ˜ σ satisfying ❑ Theorem ∃ unique solution σ ∈ L ∞ ( L 2+ δ ) , σ ∈ L ∞ ( H 1 , 2 Regularity results: ˙ loc ) (1992), σ ∈ H 1 / 2 , 2 ( L 2 ) (2011) ˙ Strain and plastic strain can be derived via penalty approximation.

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Prandtl-Reuss law. Penalty approximation. (Penalty) σ , τ ) + ( µ − 1 max[ | σ D | − κ ] + σ D ( A ˙ | σ D | , τ ) = 0 µ → 0 + ∀ τ such that τ = τ T ( τ , ∇ ϕ ) = 0 Initial condition, symmetry, balance of force remain as before, yield condition is replaced by the penalty term.

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures How to prove existence of solutions to the Prandtl-Reuss law. Estimates Step 0 Solve the “ODE” in an abstract Hilbert space setting or use Rothe approximation ⇒ (Penalty) is solvable Step 1 Assume safe load condition, i.e. existence of a “good” ˆ σ satisfying ❑ ⇒ L ∞ ( L 2 ) estimate for σ and L 1 ( L 1 ) estimate for the penalty term. σ − ˙ σ ⇒ L 2 ( L 2 ) estimate for ˙ Step 2 Test by ˙ ˆ σ σ ⇒ L ∞ ( L 2 ) estimate for ˙ Step 3 Test by ¨ σ Step 4 Go to Step 1 once more ⇒ L ∞ ( L 1 ) estimate for penalty term (all estimates uniformly for µ → 0)

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Converge and existence for Prandtl-Reuss Step 5 Use symmetric Helmholtz decomposition ⇒ ∃ v ∈ H 1 , 2 Γ (Ω; ❘ 3 ) such that 1 2( ∇ v + ( ∇ v ) T ) = A ˙ σ + Penalty term with �∇ v + ( ∇ v ) T � L ∞ ( L 1 ) ≤ K uniformly as µ → 0 ⇒ The strain velocities are only measures for µ = 0 1 2( ∇ v + ( ∇ v ) T ) ∈ C ∗ , similar, for the penalty term µ − 1 max[ | σ D | − κ ] + σ D | σ D | ⇀ ˙ Λ ∈ C ∗ (0 , T ; ❘ 3 × 3 ) µ − 1 max[ | σ D | − κ ] + ⇀ λ | σ D | ˙ Λ is the plastic strain velocity If one had σ µ → σ uniformly ⇒ representation as multiplier ˙ Λ = λ σ , λ ≥ 0 , λ F ( σ ) = 0 .

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Theorem σ µ → σ , σ solution of the Prandtl-Reuss variational inequality and 1 σ + ˙ 2( ∇ v + ( ∇ v ) T ) = A ˙ Λ . A similar procedure can be done for the mixture MDE model of KMRS.

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Constitutive law with elastic interaction Let � ˆ � τ s � � τ s τ = , τ = ˆ ˆ τ h τ h and � Q (ˆ τ , τ ) = [ A ss ˆ τ s : τ s + A sh ˆ τ h : τ s + A sh ˆ τ s : τ h + A hh ˆ τ h : τ h ] dx Ω Here A ss , A hh , A sh = A hs are inverse elasticity tensors, say Lame-Navier structure. They model the elastic interaction between the soft and hard material.

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Penalty approximation for Prandtl-Reuss mixtures Find σ s , σ h ∈ L ∞ (0 , T ; L 2 (Ω; ❘ 3 × 3 )) such that ˙ σ h ∈ L ∞ ( L 2 ) σ s , ˙ and σ s (0) = σ s 0 , σ h (0) = σ h 0 σ s = σ T s , σ h = σ T h � ∂ Ω p 0 ϕ d o , ∀ ϕ ∈ H 1 , 2 Γ (Ω; ❘ 3 ) ( α σ s + (1 − α ) σ h , ∇ ϕ ) L 2 (Ω) = ( f , ϕ ) L 2 (Ω) + (Penmix) � � α ˙ σ s , α τ s Q + (1 − α ) ˙ σ h , (1 − α ) τ h � � σ sD αµ − 1 [ | σ sD | − κ s ] + | σ sD | , τ s + � � σ hD (1 − α ) µ − 1 [ | σ hD | − κ h ] + | σ hD | , τ h = 0 for all τ s = τ T s , τ h = τ T h such that div( α τ s + (1 − α ) τ h ) = 0.

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Consequence of (Penmix) Choose τ h = 0 , div( α τ s ) = 0 , α τ s = τ 0 ⇒ � � � σ sD σ h + µ − 1 [ | σ sD | − κ s ] + α A ss ˙ σ s + (1 − α ) A sh ˙ : τ 0 dx = 0 | σ sD | Ω By symmetric Helmholtz decomposition 1 � u s ) T � σ sD + µ − 1 [ | σ sD | − κ s ] + ( ∇ ˙ u s ) + ( ∇ ˙ = α A ss ˙ σ s + (1 − α ) A sh ˙ σ h 2 | σ sD | � �� � � �� � rate of stress acting on soft material approximate plastic strain velocity of the soft material Similarly 1 � u h ) T � σ hD σ h + µ − 1 [ | σ hD |− κ h ] + ( ∇ ˙ u h ) + ( ∇ ˙ = α A sh ˙ σ s +(1 − α ) A hh ˙ 2 | σ hD |

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Theorem Assume α, κ h Lipschitz, Q positively definite and smooth data. Assume a safe load condition with smooth stresses ˆ σ s and ˆ σ h . Let 0 ≤ ε 0 ≤ α ≤ 1 − ε 0 . Then 1. The solutions σ n s , σ n h are uniformly bounded in the norms σ ∈ L 2 ( H 1 / 2 , 2 L ∞ ( L 2 ) , L ∞ ( L 2+ δ ) , H 1 , ∞ ( L 2 ). Furthermore ˙ ). Γ 2. The partial strain velocities and the partial plastic strain velocities are bounded in L ∞ ( L 1 ). 3. The approximate stresses converge weakly h ⇀ σ h ( µ → 0 + ) in H 1 , 2 ( L 2 ) ∩ L 2 ( H 1 , 2 σ n s ⇀ σ s , σ n loc ).

Introduction KMRS model Prandtl-Reuss Prandtl-Reuss mixtures Theorem (continuation) The limit satisfies the variational inequality �� � � �� α ˙ α ( σ s − τ s ) σ s ≤ 0 (V) Q , (1 − α ) ˙ (1 − α )( σ h − τ h ) σ h with respect to the yield conditions and the balance of forces defining the convex set. Since in the application α = α ( t , x ; σ s , σ h ) and similar κ h , inequality (V) is a quasi-variational inequality.