Vector hysteresis models

Transcription

1 Euro. Jnl. Appl. Math. 2 (99), Vector hysteresis models Pavel Krejčí Matematický ústav ČSAV, Žitná Praha, Czechoslovakia Key words: vector hysteresis operator, hysteresis potential, differential inequality. AMS classification: 73 E 5, 34 A 4, 58 C 7 Abstract. Following Krasnosel skii & Pokrovskii 983 we express the constitutive law for the Prandtl- Reuss elastoplastic model in terms of a hysteresis operator and we introduce the vector Ishlinskii model. We investigate some properties (continuity, energy inequalities, dependence on spatial variables) of these operators. Introduction In this paper we will investigate the Prandtl-Reuss elastoplastic model (cf. Duvaut & Lions 972, ečas & Hlaváček 98). We denote by S n the vector space of symmetric tensors n n (we have dim S n = 2n(n+)) and we fix a closed convex (usually unbounded) set Z S n. Its interior represents the elasticity domain and its boundary is the surface of plasticity. The strain tensor ε = {ε ij } is decomposed into two parts ε ij = ε e ij + εp ij. The elastic part ε e satisfies the linear Hooke s law ε e ij = A ijklσ kl, where A = {A ijkl } is a given constant symmetric positive definite matrix with respect to the scalar product < ξ, η >= ξ ij η ij in S n and σ = {σ ij } is the stress tensor. The values of σ cannot leave the domain Z and the plastic strain ε p satisfies the inequality < ε p, σ ϕ > ϕ Z, where the dot denotes the time derivative. Assuming ε to be known we find σ as the solution of the inequality < A σ ε, σ ϕ > ϕ Z with a given initial condition σ() Z. Putting u = A /2 ε, x = A /2 σ, Z = A /2 (Z) we see that x(t) Z satisfies the inequality < ẋ u, x ϕ > ϕ Z. The correspondence u x has been investigated in Krasnosel skii & Pokrovskii 983, where the notion of a hysteresis operator x = f(u) is introduced. The aim of this paper is to derive new properties of the operator f.

2 Using this notation we can express the Prandtl-Reuss constitutive law in the form σ = A /2 f(a /2 ε). Let us consider an elastoplastic body occupying a domain Ω R n and satisfying the Ishlinskii constitutive law σ = A /2 f(a /2 h ε) h dµ(h) = A /2 F (A /2 ε), (cf. (5.3) in Section 5). The strain tensor is given by the formula ε ij (x, t) = 2 ( u i x j + uj x i ), where u = {u i } is the displacement vector. The dynamical behavior of the body is determined by the equation of motion 2 u i t 2 = σ ij x j + h i, where h = {h i } is a given forcing term (which may depend on u, ε and σ), and by appropriate boundary and initial conditions. ( Let us denote (Du) ij = u i ) 2 x j + uj x i, (D σ) i = σ ij x j. The problem consists in solving the equation (.) 2 u t 2 D A /2 F (A /2 Du) = h in Ω (, T ). This is a natural way of transforming evolution differential inequalities into equations with hysteresis (cf. also Krejčí 989). The equation (.) under suitable boundary and initial conditions can be solved by standard methods (e.g Faedo-Galerkin). Let us note that we can derive a priori estimates for the second derivatives of an approximate solution u k (differentiating (.) with respect to t, testing with 2 u k t and using (5.6) (ii) 2 and then testing with D D u k t and using (5.8), cf. Section 5). We can next use the Minty-Browder argument based on the inequality (2.6) in Section 2 (cf. Krejčí 988) and the continuity of F for passing to the limit. A similar existence result has been proved (by different methods) by Visintin

3 Geometry of convex sets We denote by X an -dimensional real vector space with a scalar product <.,. >, x =< x, x > /2 for x X. We assume that Z X is a closed convex set (bounded or unbounded) containing in its interior. We fix a number m > such that (.) x m x Z. We denote by Q the orthogonal projection of X onto Z such that x Qx = min{ x ξ, ξ Z}, P x = x Qx. Let us recall the following properties of P, Q. (.2) Lemma For every x, y X we have (i) < P x, Qx ϕ > ϕ Z, (ii) < P x P y, Qx Qy >, (iii) < P x, x > m P x + P x 2, where m is given by (.), (iv) Q(x + αp x) = Qx for every α. Proof (i) Let us assume < P x, Qx ϕ >< for some ϕ Z and x X. For α [, ] we put ξ α = αϕ + ( α) Qx Z. We have x ξ α 2 = x Qx 2 + α(< P x, Qx ϕ > +α Qx ϕ 2 ) < x Qx 2 for α sufficiently small, and this contradicts the definition of Q. Part (ii) follows immediately from (i) and (iii) is obvious for x Z. For x / Z we find α > such that αp x = m and put ϕ = αp x Z. We have < ϕ, Qx ϕ >, hence < P x, x >= α < ϕ, Qx + P x > α ϕ 2 + P x 2 = m P x + P x 2. (iv) For an arbitrary ϕ Z we have x + αp x ϕ 2 = Qx ϕ 2 + ( + α) 2 P x 2 + ( + α) < P x, Qx ϕ >, hence the minimum of x + αp x ϕ is attained for ϕ = Qx. Let us show some specific properties of unbounded domains. (.3) Lemma Let Z be unbounded. Then there exists a non-empty convex closed cone C Z with vertex such that (i) for every x C and y X, y m we have x + y Z, (ii) the function K(r) = max{dist(z, C), z Z, z = r} is nondecreasing for r sufficiently large and lim K(r)/r =. r Proof Let us denote by M the set of all points x Z such that x = and αx Z for every α. Let {z n } Z be an arbitrary unbounded sequence. We see immediately that every limit point of n z z n belongs to M, hence M is non-void. Put C = 3

4 {αx; x M, α }. The convexity and closedness of C follow from the convexity and closedness of Z. Let us choose x C and y m. For an arbitrary sequence {y n }, y n < m, y n y we choose positive numbers r n > m r m y n >. We have n r n y n < m, hence x + y n = r n (r n x) + ( r n )( r n r n y n) Z and (i) follows easily. Let us denote by Q c the projection of X onto C, P c x = x Q c x. Repeating the proof of (i) and using (.2) (iv) we can easily verify that the set c = {x Z; Q c x = } is bounded. Put r c = max{ x ; x c }. We next prove that for every r 2 > r > r c and for every z Z such that z = r there exists z 2 Z, z 2 = r 2 such that dist(z 2, C) dist(z, C). Put x = Q c z. We have dist(z, C) = z x = P c z, < z, x >. Put z 2 = z + βx, where β > is such that z 2 = r 2. We have ( n )z + n (nβx ) Z for every n, hence z 2 Z. We further put x 2 = Q c z 2. Let us assume z 2 x 2 < z x. For α = β +β > we obtain P c (z + αp c z ) z + αp c z ( + α)x 2 = ( + α) z 2 x 2 < ( + α) P c z, which contradicts (.2) (iv). Thus we have proved that the function K(r) is nondecreasing for r r c. Let us assume lim sup K(r)/r >. Then there exists ε > and a sequence r n such that K(r n ) r εr n. We find z n Z, z n = r n such that dist(z n, C) ε z n. The sequence z n z n has a limit point x M and z n z n x = z n z n z n x z n dist(z n, C) ε which is a contradiction. Lemma (.3) is proved. 2 A differential inequality Let [, T ] be a given interval. We denote by L p, W k,p, C the spaces L p (, T ; X ), W k,p (, T ; X ), C([, T ]; X ) with usual norms.,p,. k,p,.,, respectively. We further introduce the set K = {u L ; u(t) Z a.e.}. (2.) Problem For a given u W, find x W, K such that (i) < x u, x ϕ > a.e. for every ϕ Z, (ii) x() = Qu(). This choice of the initial condition corresponds to the reference state. It enables us to avoid the difficulties in more complex hysteresis models. (2.2) Proposition Problem (2.) has a unique solution. 4

5 Proof Existence. For every fixed n put u j = u( j n T ), j =,..., n x j+ = Q(u j+ u j + x j ), x = Q(u ). Lemma (.2)(i) yields (2.3) < x j+ x j u j+ + u j, x j+ ϕ > ϕ Z, hence (2.4) x j+ x j u j+ u j, x j+ u j+ x j + u j u j+ u j. For t [ j n j+ T, n T ) put u (n) (t) = u j + n ( t T j )( ) uj+ u j, n (2.5) x (n) (t) = x j + n ( t T j )( ) xj+ x j. n We have u (n) W,, x(n) W, K, u(n) u, and x (n) (t) u (n) (t) a.e. By Arzelà-Ascoli theorem and Dunford-Pettis theorem (Edwards 965) there exists x Wn, K and a subsequence {x (m) } of {x (n) } such that x (m) x in L -weak and x (m) x uniformly in C. It remains to verify that x satisfies (2.)(i). Let ϕ Z be arbitrarily chosen. Using (2.3) we obtain for τ ( j j+ nt, n T ) : This yields for every s < t t and (2.)(i) follows easily. < x (n) (τ) u (n) (τ), x (n) (τ) ϕ > n T u j+ u j 2. s < x (τ) u (τ), x(τ) ϕ > dτ Uniqueness. Let u, v W,, be given and let x, y W K satisfy the inequalities < x u, x ϕ >, < y v, y ψ > ϕ, ψ Z. We obtain immediately (2.6) < x y, x y > < u v, x y > a.e., hence u = v implies x = y. (2.7) Remarks (i) We shall see in the next section that in fact x (n) converge strongly to x in W, (Remark (3.3)) 5

6 (ii) Putting ϕ = x(t ± δ) in (2.)(i) we obtain for δ + < x u, x >= a.e. Let us denote ξ = u x. In Krasnosel skii & Pokrovskii 983 ξ and x are called multidimensional play and multidimensional stop, respectively. From the mechanical point of view this corresponds to the decomposition of the strain u into the sum of the plastic strain ξ and elastic strain x. Geometrically, x (t) and ξ (t) are the projections of u (t) on the tangential and normal cones to Z, respectively, at the point x(t). 3 Hysteresis operators in Sobolev spaces Proposition (2.2) enables us to define an operator f : W, W, by the relation x = f(u) x is the solution of (2.). The term hysteresis operator is used by analogy to the one-dimensional case. (3.) Proposition The operator f : W, W, We first prove an easy lemma. is continuous. (3.2) Lemma Let g n, g L (Ω, µ), f n, f L (Ω, µ; X ) be given functions, n =, 2,..., where (Ω, µ) is a measurable space and µ is a positive measure. Let us assume Ω g n g dµ, Then Ω f n f dµ. Proof Put ϕ(x) = if f(x) =, hence Ω < f n f, ϕ > dµ ϕ L (Ω, µ; X ), f n (x) g n (x) µ a.e, f(x) = g(x) µ a.e. Ω ϕ(x) = f(x) f(x), if f(x). We have ϕ L (Ω, µ; X ), < f n, ϕ > dµ f dµ. Put ψ n (x) = g n (x) < f n (x), ϕ(x) >. We have ψ n (x) a.e., Ω ψ n dµ, therefore there exists a subsequence ψ k such that ψ k (x) µ a.e. This means < f k (x), f(x) > f(x) 2 µ a.e. On the other hand, f k (x) f(x) 2 g 2 k(x) + g 2 (x) 2 < f k (x), f(x) > µ a.e., hence (cf. Edwards 965, 4.2) Ω f k f dµ and (3.2) follows immediately. Ω Proof of (3.) Let {u n } be a sequence in W,, u n u,, x n = f(u n ), x = f(u). Put y n = 2x n u n, y = 2x u. We have by (2.7)(ii) y n(t) = u n(t), y (t) = u (t) a.e. and (2.6) yields < x n x, x n x > < u n u, x n x >. Consequently, y n y 6

7 uniformly in C and by Dunford-Pettis theorem y n y in L weak. Using lemma (3.2) for y n = f n, u n = g n we conclude that y n y strongly in W,, hence x n x,. (3.3) Remark The same argument shows that the discrete approximations (2.5) converge to x strongly in W,. (3.4) Corollary. The operator f maps W,p for p <. into W,p for p and is continuous (3.5) Remark For = the operator f is Lipschitz in W,, but not Lipschitz on bounded sets in W,p for p > and not continuous in W, (cf. e.g. Krejčí & Lovicar 99). On the other hand, no proof of the conjecture of Krasnosel skii & Pokrovskii 983 saying that f : W, W, is locally Lipschitz on bounded and smooth characteristics Z has been published. Indeed, the Lipschitz continuity of f : W, C follows trivially from (2.6). 4 Hysteresis operators in the space of continuous functions (4.) Theorem The operator f is closable in C and its closure f : C C is continuous. Remark This result is known for = (then f is Lipschitz) or if Z is bounded (cf. Krasnosel skii & Pokrovskii 983). We extend the method of proof to unbounded characteristics Z. We first prove three lemmas. (4.2) Lemma Let u W, be given, x = f(u), ξ = u x. Let us assume u(t) u(t ) m 2 for t [t, t 2 ], where m is defined by (.). (i) Let Z be bounded. Then t 2 t ξ (t) dt m (diamz)2. (ii) Let Z be unbounded. Then t 2 t ξ (t) dt m K2 ( x(t ) ), where K is defined in (.3)(ii). Proof We have < ξ (t), u(t) ξ(t) ϕ > a.e. for every ϕ Z. We fix x Z such that x = in the case (i) and x C (cf. (.3)) such that x(t ) x K( x(t ) ) in the case (ii). Choosing y = m ξ (t) 2 ξ (t) + u(t) u(t ) for ξ (t) and y = u(t) u(t ) for ξ (t) = we have y m, hence we can put ϕ = x + y. This yields m ξ (t) d dt ξ(t) u(t ) + x 2, hence 7

8 t2 t ξ (t) dt m x(t ) x 2 and (4.2) is proved. (4.3) Lemma Let u n W,, n =, 2,... and u C be given such that u n u,. Put x n = f(u n ), ξ n = u n x n. Then there exist n > and a constant c > depending only on Z and u such that T ξ n(t) dt c for n n. Proof We construct a partition = t < t < < t M = T of [, T ] such that for t [t k, t k+ ] we have u(t) u(t k ) m 6 We fix n such that for n n we have u n u, < m 6. For arbitrary t [t k, t k+ ] and n n we obtain u n (t) u n (t k ) m 2, hence by (4.2) we have (i) tk+ t k ξ n(t) dt m (diamz)2 if Z is bounded, therefore T ξ n(t) dt M m (diamz)2, (ii) tk+ t k ξ n(t) dt m K2 ( x n (t k ) ) if Z is unbounded, hence T ξ n(t) dt M m max K 2 ( x n (t k ) ). k In the case (i) we put simply c = M m (diamz)2. inequality < ξ n, ξ n > < ξ n, u n > and we obtain In the case (ii) we integrate the for each t [, T ]. This yields T ξ n (t) 2 ξ n () u n, ξ n(t) dt x n (t) 2 ( u n (t) + ξ n (t) ) 2 c + c 2 max K 2 ( x n (t k ) ), k hence (.3)(ii) implies x n, c 3, where c, c 2, c 3 are constants depending only on Z and u (we take a larger n, if necessary). Consequently, T ξ n(t) dt const. and (4.3) is proved. (4.4) Lemma Let the assumptions of Lemma (4.3) be fulfilled. Then there exist n > and a constant c > depending only on Z and u such that for every m, n n we have ξ n ξ m, c ( u n u m, + u n u m /2, ). 8

9 Proof The inequality (2.6) for u = u n, v = u m has the form < ξ n ξ m, ξ n ξ m > < ξ n ξ m, u n u m >, hence ξ n (t) ξ m (t) 2 ξ n () ξ m () and (4.4) follows immediately from (4.3). T ( ξ n + ξ m ) dt u n u m, Proof of (4.) Let u C be given. We choose a sequence {u n } W, such that u n u,. We see from (4.4) that the sequence x n = f(u n ) is fundamental in C and its limit is independent of the choice of {u n }. Therefore, we can define f(u) = lim x n. The n continuity of f in C follows from standard considerations. Let {u n } C be an arbitrary sequence, u n u,. The above argument shows that for each n =, 2,.... we can find ũ n W, such that ũ n u n, < n, f(ũ n) f(u n ), < n. We have ũ n u,, hence f(ũ n ) f(u), and f(u n ) f(u),. (4.5) Remarks (i) It follows from Lemma (4.4) that the operator f : C C is 2-Hölder continuous on compact sets. We present two examples showing that this result cannot be in general considerably improved. evertheless, on special characteristics Z the operator f has better continuity properties (it is globally Hölder continuous on strictly convex sets and Lipschitz on polygons, cf. Krasnosel skii & Pokrovskii 983). (ii) The multidimensional play operator I f, where I is the identity, has a particular smoothing property: it maps continuously C -strong into BV (, T ; X )-weak. This follows from the argument of the proof of (4.) applied to (4.3). (4.6) Examples (i) f does not map in general bounded sets in C into bounded sets. Put Z = {(a, b) R 2 ; < a (, b ) log a}, T =. Let us define a sequence rn (t) {u n } in the following way: u n (t) =, where r n is continuous and piecewise linear, ( r 2k ) ( n n =, 2k+ ) rn n =, k =,,..., [ n 2 ], r n(t) = n( ) j for t ( j n, n) j, j =,..., n. In fact, it would be more accurate to shift the domain Z and the functions u n in order to fulfil the assumption IntZ. ( ) αn (t) We determine the vector x n (t) = f(u n )(t) = using (2.7) (ii). Denote αn β n (t) = (, αn k = α 2k ) n n, g(α) = log α. For t (, n ) we have β n(t) = g(α n (t)) and x n(t) is the orthogonal projection of u n(t) on the tangent to Z at the point x n (t), hence x n(t) = r n(t) + g 2 (α n (t)) 9 ( ) g. (α n (t))

10 In other words, α n is the solution of the equation (4.7) α n( + g 2 (αn (t))) = r n, α n () =. We obtain αn = 5 2. For t ( 2k n, ) ( 2k n we have βn (t) = const. = log αn, k α 2k ) n n =. We find τ k ( 2k n, ) 2k+ n such that rn (τ k ) = αn k. We have α n(t) = r n(t), β n(t) = for t ( 2k n, τ ) ( ) k and (4.7) for t τk, 2k+ n, αn (τ k ) = αn, k hence αn k+ is the positive root of the equation ( ) α k+ 2 α k+ n + n αn k =. We see immediately that the sequence α [ n 2 ] n tends to as n, hence the sequence {x n } is unbounded. (ii) The Hölder continuity is optimal if Z is a ball. Put Z = {z X ; z h}, where h > is a given number. In this case the operator f is 2 -Hölder continuous in C (cf. Krasnosel skii & Pokrovskii 983). Let us choose r > h and for t [, ] put r r u n (t) = cos nt h sin nt h., v n(t) = cos nt sin nt., x n = f(u n ), y n = f(v n ). We have indeed y n = v n. The same argument as above shows that x n is the solution of the equation x n(t) = u n(t) h 2 < x n(t), u n(t) > x n (t), x n () = v n (). h h We find out that x n has the form x n (t) = of the equation ϱ + n ( r h cos ϱ ), ϱ() =, hence ϱ(t) = 2arctan ( r h ( r2 h 2 r + h tanh 2h cos(nt + ϱ(t)) sin(nt + ϱ(t))., where ϱ is the solution nt )). We have x n (t) y n (t) 2 = 4h 2 (r h)tanh 2 ( r2 h 2 2h ) nt ( r2 (r + h) + (r h)tanh 2 h 2 2h nt ),

11 hence lim x 2(r h) n() y n () = h n r. 5 Properties of hysteresis operators A. Superposition and inversion formulas. (5.) Proposition For every u W, and α we have f(( + α)u αf(u)) = f(u). Proof Put v = f(u), w = f(( + α)u αv). We have (cf. (.2) (iv)) w() = Q(u() + αp u()) = Qu() = v() and < w (+α)u +αv, w ϕ >, (+α) < v u, v ψ > for every ϕ, ψ Z. Putting ϕ = v, ψ = w we obtain < w v, w v >, hence w = v. (5.2) Corollary For every α >, α + β >, h > we have w = αu + βh f ( h u) u = ˆαw + ˆβĥf( ĥw ), where ˆα = α, ˆα + ˆβ = α+β, ĥ = h(α + β). Proof easily. By (5.) we have f ( ĥw ) = f ( α α+β h u + β α+β f( h u)) = f ( h u) and (5.2) follows B. Vector Ishlinskii operator. Let µ be a positive measure on (, ), µ((, )) < and let f be the multidimensional stop corresponding to Z. We define the vector Ishlinskii operator by the formula (5.3) F (u) = f ( h u) h dµ(h). Let us note that for h > m u, we have f ( h u) (t) = hu(t), hence the integral in (5.3) is always finite. We check easily that the vector Ishlinskii operator F is continuous in C and W,p, p <. C. Energy inequalities (5.4) We define the hysteresis potentials for the operator (5.3) by the following formulas (i) P (u) = 2 ( f ( h u)) 2 h 2 dµ(h) for u C, (ii) P 2 (u) = 2 < F (u), u > for u W,. The Hölder inequality yields immediately (5.5) F (u) 2 2µ((, )) P (u). The following energy inequalities are satisfied.

12 (5.6) Proposition (i) For every u W, (ii) For every u W 2, and for almost all t [, T ] we have P (u) (t) < F (u)(t), u (t) >. and for almost all s < t T we have P 2 (u)(t) P 2 (u)(s) t s < F (u) (τ), u (τ) > dτ. Proof The function h h f ( h u) is continuous and bounded as function R L, hence we can differentiate in (5.3) and (5.4) (i) with respect to t. We obtain P (u) (t) < F (u)(t), u (t) >= by (2.)(i) with ϕ =. For proving (ii) we denote x h (τ) = h f ( h u) (τ). We have < f ( h u) (t) h u (t), f ( h u) (t) > h 2 dµ(h) < x h(τ) u (τ), x h (τ) hϕ >, < x h(τ + δ) u (τ + δ), x h (τ + δ) hψ > for every ϕ, ψ Z. Putting ϕ = h x h(τ + δ), ψ = h x h(τ) we obtain < x h(τ + δ) x h(τ), x h (τ + δ) x h (τ) > < u (τ + δ) u (τ), x h (τ + δ) x h (τ) >. After integration t dτ this yields s 2δ 2 x h(t + δ) x h (t) 2 2δ 2 x h(s + δ) x h (s) 2 t δ 2 < u (τ + δ) u (τ), x h (τ + δ) x h (τ) > dτ, s hence for δ we obtain (cf. Edwards 965, 4.2) 2 x h(t) 2 t 2 x h(s) 2 < u (τ), x h(τ) > dτ for almost all s < t T. s (otice that for u W 2, we have x h W, ). By (2.7) (ii) we obtain x h (t) 2 =< x h (t), u (t) > and (5.6) (ii) follows easily. Remarks (i) The potential P (as well as the operator F ) is rate independent and corresponds to the usual physical notion of energy. On the other hand, the potential P 2 has no natural interpretation and the term energy is maybe not physically justified. We use it in order to emphasize the formal similarity between (5.6) (i) and (ii). 2

13 (ii) In the case =, Z = [, ] we can make use of the simple structure of the hysteresis memory for getting more precise information about the dissipation of energies corresponding to P and P 2. The dissipation of P (u) is characterized by the area of the hysteresis loop in the plane with coordinates u, F (u). Moreover, we can obtain an estimate from below of the dissipation of energy for P 2 (u) which is of the order of u (t) 3 dt. It is shown for instance in Krejčí 988 how this estimate can be used for solving hyperbolic equations with hysteresis operators. D. Dependence on parameters. Let us consider input functions depending on a parameter x J, where J R is a fixed interval and let us assume u(x,.) C for almost every x J. We can define the value f(u)(x, t) of the operator f to be equal to f(u(x,.))(t) for every t [, T ]. We see immediately that the operator f : L p (J, W, ) Lp (J; C ) is Lipschitz for every p [, ]. Let u L (J; W, u ) be given such that x L (J; W, ). We have by (2.6) for x, y J, x > y and s < t T (5.7) 2 f(u)(x, t) f(u)(y, t) 2 2 f(u)(x, s) f(u)(y, s) 2 For s = this yields t f(u)(x, t) f(u)(y, t) 2 Qu(x, ) Qu(y, ) s < f(u)(x, τ) f(u)(y, τ), u t t f(u)(x, τ) f(u)(y, τ) u t u (x, τ) (y, τ) > dτ. t u (x, τ) (y, τ) dτ. t The inequality (.2) (ii) implies Qu(x, ) Qu(y, ) u(x, ) u(y, ) x y u x (ξ, ) dξ, hence x f(u)(x, t) f(u)(y, t) u t x (ξ, ) dξ u (ξ, τ) dξ dτ, x y t x Thus f(u)(., t) is absolutely continuous for all t [, T ] and y t f(u)(x, t) u(x, ) u (x, τ) dτ x x t x a.e. Dividing (5.7) by (x y) 2 we obtain for y x (5.8) 2 x f(u)(x, t) 2 2 t x f(u)(x, s) 2 < s x f(u)(x, τ), 2 u (x, τ) > dτ t x 3

14 for almost every x J. We derive easily analogous inequalities for the Ishlinskii operator. The generalization to a larger number of parameters is obvious. Acknowledgement. The author is much indebted to the referee for valuable remarks and comments. Conclusion This paper presents an elementary approach to the vector hysteresis operators with closed convex characteristics Z introduced by Krasnosel skii & Pokrovskii 983. It contains some new results, namely the continuity in the space of continuous functions and in Sobolev spaces W,p without any assumption on the boundedness and smoothness of Z. We also introduce energy potentials for the Prandtl-Reuss and Ishlinskii hysteresis operators and we prove two energy inequalities. It is shown how an evolution differential inequality can be transformed into an equation with hysteresis operator and how the energy inequalities can be used for solving such problems. References DUVAUT, G. & LIOS. J.-L. 972 Les inéquations en mécanique et en physique. Dunod. EDWARDS, R.E. 965 Functional Analysis Theory and Applications. Holt, Rinehart and Winston. KRASOSELSKII, M.A. & POKROVSKII, A.V. 983 Systems with hysteresis (Russian) auka, Moscow. KREJČÍ, P. 988 A monototonicity method for solving hyperbolic problems with hysteresis. Apl. Mat. 33, KREJČÍ, P. 989 Hysteresis operators - a new approach to evolution differential inequalities. Comment. Math. Univ. Carolinae 3,3, KREJČÍ, P. & LOVICAR, V. 99 Continuity of hysteresis operators in Sobolev spaces. Apl. Mat. 35, EČAS, J. & HLAVÁČEK, I. 98 Mathematical theory of elastic and elasto-plastic bodies: an introduction. Elsevier. VISITI, A. 987 Rheological models and hysteresis effects. Rend. Sem. Mat. Univ. Padova 77,