Archimedes Center for Modeling, Analysis & Computation. Singular solutions in elastodynamics

Transcription

1 Archimedes Center for Modeling, Analysis & Computation Singular solutions in elastodynamics Jan Giesselmann joint work with A. Tzavaras (University of Crete and FORTH) Supported by the ACMAC project - European Union FP7 June / 27

2 Outline Introduction 1d: Discontinuous solutions to nonlinear second order equations 1d: Energy and admissibility Comparison to a discrete model in 1d 3d: Transfer of the 1d solution concept 3d: Energy and admissibility Summary & Prospects 2 / 27

3 Introduction Picture by Gent and Lindley, In stretched rubber cavities appear at relatively low tensile loads. Study of fracture, shear bands and cavitation in elastic solids. To which extent can these phenomena be described by continuum models. Give a meaning to (very) singular solutions to nonlinear equations. Admissibility criteria / energy rate. Comparison to discrete models. 3 / 27

4 Nonlinear elasticity Search displacements y : R d [0, T) R d such that det( y) > 0 satisfying the wave equation y tt div(τ( y)) = 0 (WAVE). Stress response is hyperelastic τ = W F : Rd d + R d d + ; W : R d d + R. Assume W to be isotropic and frame indifferent, which implies W (F) = Φ(λ 1,..., λ d ) where λ 1,..., λ d eigenvalues of F T F and Φ is symmetric. Assume W to be polyconvex, i.e. Φ convex in {λ i }. 4 / 27

5 Nonlinear elasticity Equivalent system of first order conservation laws for F = y, v = y t : F t v = 0 v t div(τ(f)) = 0. (CONS) Prescribe the following initial and boundary data y(x, t) = λx for x > rt and for t = 0; y t (x, t) = 0 for t = 0. These admit the trivial solution y(x, t) = λx. 5 / 27

6 Continuum approach to cavitation In 1982 Ball studied radially symmetric minimizers of ( W (F) W (λx) dx for W (F) = 1 d d ) λ 2 i +h λ i, d 3. R 2 d Radially symmetric ansatz y(x) = w(r) x R 0 = 1 ( R d 1 d 1 Φ R R (w R, w λ 1 R,..., w ) R ) i=1 i=1 with R = x leads to d 1 R Φ (w R, w λ 2 R,..., w R ). Ball constructed minimizers with discontinuous displacement field s.t. the normal component of the Cauchy stress vanishes on the surface of the cavity. Thus, R d W (F) W (λx) dx is well defined. For λ >> 1 these solutions have less energy than the trivial solution. 6 / 27

7 Continuum approach to cavitation In 1988 Spector and Pericak-Spector considered the corresponding dynamic problem w tt = 1 ( R d 1 d 1 Φ R R (w R, w λ 1 R,..., w ) R ) using the self-similar ansatz d 1 R y(x, t) = w(r, t) x R = r(ξ) ξ x with ξ = R t. Φ (w R, w λ 2 R,..., w R ) The normal component of the Cauchy stress vanishes on the surface of the cavity. Integrals involved in defining weak solutions are well-defined. 7 / 27

8 Continuum approach to cavitation Smooth solutions of (WAVE) satisfy the energy equality ( ) d 1 dt 2 (y t) 2 + W ( y) div (τ( y)y t ) = 0. 8 / 27

9 Continuum approach to cavitation For weak solutions classically the energy inequality is imposed ( ) d 1 dt 2 (y t) 2 + W ( y) div (τ( y)y t ) 0. 8 / 27

10 Continuum approach to cavitation For weak solutions classically the energy inequality is imposed ( ) d 1 dt 2 (y t) 2 + W ( y) div (τ( y)y t ) 0. For the weak solutions constructed by Spector and Pericak-Spector integrals involved in defining the energy of weak solutions are well-defined. Energy dissipation along an outgoing spherical shock wave. No contribution of the opening cavity to the energy rate. For λ >> 1 energy rate criteria favor the cavitating solutions. 8 / 27

11 Continuum approach to cavitation For weak solutions classically the energy inequality is imposed ( ) d 1 dt 2 (y t) 2 + W ( y) div (τ( y)y t ) 0. For the weak solutions constructed by Spector and Pericak-Spector integrals involved in defining the energy of weak solutions are well-defined. Energy dissipation along an outgoing spherical shock wave. No contribution of the opening cavity to the energy rate. For λ >> 1 energy rate criteria favor the cavitating solutions. Bonds need to be broken to create the cavity. Why is this not reflected by the energy? 8 / 27

12 Situation in 1d Consider a longitudinal or shearing motion y tt = (τ(y x )) x, y(x, 0) = λx, y t (x, 0) = 0, y(x, t) = λx for x > rt. Ansatz: ( x ) y(x, t) = ty t with Y ( ξ) = Y (ξ) ξ > 0, lim Y (ξ) > 0. ξ>0,ξ 0 Then (WAVE) amounts to ξ 2 Y = (τ(y )). W τ u u Impose the conditions W > 0 and W < 0. 9 / 27

13 Rankine Hugoniot conditions y t (x, t) = Y (ξ) ξy (ξ) =: V (ξ), y x (x, t) = Y (ξ) =: U (ξ), where ξ = x t. The equations for U, V are ξu + V = 0 ξv + (τ(u )) = 0. ( ) For a shock with speed σ the Rankine Hugoniot conditions read [τ(u )] σ[u ] = [V ] and σ[v ] = [τ(u )] = σ =. [U ] Ensure the conservation at discontinuities of U, V. 10 / 27

14 One dimensional ansatz Thus, we investigate the following ansatz: y(x, t) = ty ( x t ) Y (0) + αξ : 0 < ξ < σ Y (ξ) := Y (0) + αξ : σ < ξ < 0 (1d-ANSATZ) λξ : ξ > σ. with the continuity condition Y (0) + ασ = λσ. x = σt ty(0)+αx t ty(0)+αx x = σt v (α,y(0)) δ-sh 2-sh u (λ,0) λx λx x 1-sh (α, Y(0)) By this construction the initial and boundary conditions are satisfied. The shocks are admissible for α < λ (Lax criterion). The size of the hole is proportional to t. 11 / 27

15 One dimensional ansatz U = 2Y (0)δ ξ=0 + αχ { ξ <σ} + λχ { ξ >σ} V = Y (0)χ {0<ξ<σ} Y (0)χ { σ<ξ<0} U V λ λ Y(0) α α σ σ ξ ξ Y(0) σ σ 12 / 27

16 One dimensional ansatz U = 2Y (0)δ ξ=0 + αχ { ξ <σ} + λχ { ξ >σ} V = Y (0)χ {0<ξ<σ} Y (0)χ { σ<ξ<0} U V λ λ Y(0) α α σ σ ξ ξ Y(0) σ σ What is the meaning of τ(u ) near the origin in this case? 12 / 27

17 General considerations The developing singularity can be viewed as the effect of higher-order physics. The cavity develops in some stable maner. The exact higher order mechanism is unknown. 13 / 27

18 General considerations The developing singularity can be viewed as the effect of higher-order physics. The cavity develops in some stable maner. The exact higher order mechanism is unknown. Instead of introducing higher order terms in the PDE we consider a generic space-time averaging procedure. We will consider solutions such that their averages constitute approximate solutions. Due to the self similar structure of the solutions mollification in space also induces mollification in time. 13 / 27

19 Slic solutions Definition: Slic solution in 1d We call y C([0, T), Lloc 1 (R)) a singular limiting induced from continuum (slic) solution provided for all ψ C0 (R, R + ) satisfying supp(ψ) [ 1, 1], ψ(x)dx = 1, and ψ(x) = ψ( x) the following holds: lim n [yn tt (τ(yx n )) x ] = 0 in D where y n (x, t) = y x nψ(n ). Related to so called δ-shocks for hyperbolic conservation laws, see Danilov, Shelkovic One main difference is that our notion of solution is based on the underlying structure of the 2nd order problem. 14 / 27

20 Are there slic solutions? Slic solutions generalize standard weak solutions Lemma (G., Tzavaras 2013): Let y H 1 ([0, T], L 1 (R)) L 1 ([0, T], H 1 (R)) with essinf y x > 0 satisfy T 0 then y is a slic solution. R y t ϕ t τ(y x )ϕ x dxdt = 0 ϕ C 1 0 ((0, T) R), For sufficiently weak materials our Ansatz yields weak solutions: Lemma (G., Tzavaras 2013): A function y given by (1d-ANSATZ) with Y (0) + ασ = λσ is a slic solution if and only if Y (0) = τ(λ) τ(α) σ and τ(u) lim = 0. u u 15 / 27

21 Energy of slic solutions Let B R have finite volume. Define the energy at time t of a slic solution y inside B as E B (y, t) := lim n Lemma (G., Tzavaras 2013): B 1 2 (yn t (x, t)) 2 + W (y n x (x, t)) dx. Let B contain the whole wave fan at time t and y a slic solution given by (1d-ANSATZ). Then for lim u τ(u) = it holds E B (y, t) = for t > 0 while E B (y, 0) <. for τ = lim u τ(u) < it holds E B (y, t) = B W (λ) + 2Y (0)t(τ τ(α)) }{{}}{{} initial energy energy of the cavity 2t ( σw (λ) σ 2 Y (0)2 σw (α) Y (0)τ(α) ). }{{} energy dissipated at the shocks 16 / 27

22 Energy rate Lemma (G., Tzavaras 2013): W (u) In case τ = lim u u = lim u τ(u) < all slic solutions given by (1d-ANSATZ) satisfy and for n sufficiently large. B d dt E B(y, t) > 0 (τ(y n x )y n t ) x dx = 0 The energy increases while there is no energy influx through the boundary. 17 / 27

23 Discrete, stationary model The stationary 1d problem can be seen as the limit of a discrete model with N masses at points {x i } i=0,...,n with x i < x i+1 and an energy functional only considering nearest neighbor interactions: E[{x i }] := 1 N N 1 i=0 Boundary conditions: x 0 = λ 2, x N = + λ 2. W (N x i+1 x i ) 18 / 27

24 Discrete, stationary model The stationary 1d problem can be seen as the limit of a discrete model with N masses at points {x i } i=0,...,n with x i < x i+1 and an energy functional only considering nearest neighbor interactions: E[{x i }] := 1 N N 1 i=0 W (N x i+1 x i ) Boundary conditions: x 0 = λ 2, x N = + λ 2. For a (strictly) convex W Jensen s inequality implies that the (unique) energy minimizer is given by x i = λ 2 + λ N i. 18 / 27

25 Discrete, stationary model The stationary 1d problem can be seen as the limit of a discrete model with N masses at points {x i } i=0,...,n with x i < x i+1 and an energy functional only considering nearest neighbor interactions: E[{x i }] := 1 N N 1 i=0 W (N x i+1 x i ) Boundary conditions: x 0 = λ 2, x N = + λ 2. For a (strictly) convex W Jensen s inequality implies that the (unique) energy minimizer is given by x i = λ 2 + λ N i. The energy of a solution with one crack { λ ˆx i = 2 + λ N i : i < N /2 λ 2 + c + λ N i : else with c > 0, and c + N λ = N λ satisfies E[{ˆx i }] = 1 N W (cn + λ) + N 1 N N W ( λ) cτ + W ( λ). 18 / 27

26 Discrete, dynamic model Study the energy of the discrete system subject to the continuous motion: On the discrete level the contiuous motion gives rise to ct + α ( m N ) : 2 < m N < σt ˆx i := ct + α ( m N ) : 2 σt < m N < 1 2 λ ( m N ) 1 2 : m N 1 2 > σt What is the difference in energy to a trivial solution at some time t : Choose m N such that m N < 1 m+1 2N + σt < N 2m N c2 (W (α) W (λ) + 2 ) + 1 ( ) W (Nct + α) W (λ) N + 2 ( ( ( W N λ ( m + 1 N N 1 ) ( m ct + α 2 N 1 ) ) }{{ 2 } bounded N 2σt(W (α) W (λ) + c2 2 ) + ctτ. ) ) W (λ) 19 / 27

27 Situation in 3d Aim: Test the weak solutions constructed by Pericak-Spector and Spector against being slic solutions. Consider energies of the form W (F) = 1 2 d ( d ) λ 2 i + h λ i i=1 i=1 with h > 0, h < 0 lim v 0 h(v) = lim v h(v) =. For solutions of the form y(x, t) = w(r, t) x R mollify as follows: Symmetric mollifier φ Cc (R, R + ) with φ = 1, supp(φ) [ 1, 1], φ(x) = φ( x), φ(0) > 0. Let φ n (R) = nφ(nr) and for w L 1 loc (R + R) define w n (R, t) = We define 0 φ n (R R)w( R, t)d R 0 φ n (R + R)w( R, t)d R. y n (x, t) = w n (R, t) x R. 20 / 27

28 Self similar slic solutions for d 3 Definition: Slic solution for d 3 Let y Lloc (R; L1 loc (Rd ; R d )) of the form y(x, t) = w(r, t) x R with w(, t) monotone increasing satisfy y(x, t) = λx for t 0 and for x > rt, t > 0 for some r > 0. The function y is called a singular limiting induced from continuum (slic)-solution if y n satisfies y n tt ψ + τ( y n ) : ψ dxdt 0, as n, R R d holds for all φ C c (R) positive, symmetric, mollifiers with φ(0) > 0, and for ψ C 2 c (R d R, R d ). 21 / 27

29 Existence of slic solutions Theorem (G., Tzavaras 2013): The weak solutions constructed by Pericak-Spector and Spector extended by y(x, t) = λx for t < 0 are slic-solutions provided They are not slic-solutions in case Crucial technical ingredient: h (v d ) lim = 0. v v lim inf v h (v d ) v > 0. det( y n (x, t)) n d t d for x < 1 n. The initial boundary value problem has (at least) two slic solutions: the cavitating and the trivial one. 22 / 27

30 Energy and admissibility Definition: Energy in 3d The energy of a slic-solution y W 1, loc (R; L1 loc (Rd ; R d )) of the form y(x, t) = w(r, t) x R in some bounded domain B Rd and for a.e. t R is defined as 1 E[y, B](t) := lim n 2 yn t (x, t) 2 + W ( y n (x, t)) dx. Proposition (G., Tzavaras 2013): B h(v) If lim v v = the energy of the cavitating solution constructed by Pericak-Spector and Spector in the sense of our definition of energy satisfies E[y, B](t) = for every t > / 27

31 Energy and admissibility h(v) For weaker materials, i.e. lim v v < energy of cavitating solutions becomes finite. Proposition (G., Tzavaras 2013): h(v) Let L := lim v v be finite and let B contain the whole wave fan at time t. Then, the energy of the weak solution found by Pericak-Spector and Spector in the sense of our definition of energy satisfies E[y, B](t) = E[λx, B](t) + td σ d ω d J + td ω d d d r(0)d L, where J := 1 2 w R(tσ, t) 2 + h(w R (tσ, t)λ d 1 ) 1 2 λ2 h(λ d ) + 1 [ w R (tσ, t) + h (w R (tσ, t)λ d 1 )λ d 1 2 is the energy dissipation of the outgoing shock. + λ + h (λ d )λ d 1] (λ w R (tσ, t)) 24 / 27

32 Sign of the energy rate Proposition (G., Tzavaras 2013): h(v) Let λ > 0 be given, lim v v < and y a cavitating solution as computed by Pericak-Spector and Spector. Then d E[y, B](t) > 0 dt for any ball B containing the whole wave fan at time t. At the same time the energy influx accross the boundary is zero τ( y n )yt n ds = 0 for n sufficiently large. B 25 / 27

33 Summary 1d: In 3d Introduced a concept of discontinuous solutions of nonlinear wave equations and a notion of energy for these solutions. Constructed such solutions. Energy of these solutions increases in time. Transferred our concept of slic-solutions from 1d and studied its ramifications on the energy. New notion of energy accounts for energy needed to open the cavity. Criterion when weak solutions constructed by Pericak-Spector and Spector are slic solutions and whether they dissipate energy. (With our definition of energy) their energy increases in time. This is in contrast to the situation when the energy needed to open the cavity is not accounted for. 26 / 27

34 Prospects Our approach can be extended to describe vacuum in a Lagrangian description of fluid dynamics. There, vacuum does not contribute to the energy (rate). Compare our results to discrete considerations, which may be more appropriate to describe fracture and cavitation phenomena. Use more appropriate constitutive functions. Couple discrete and continuous models? 27 / 27