Two-Phase Deformation Fields and Ellipticity of Nonlinear Elastic Materials

Transcription

1 Two-Phase Deformation Fields and Ellipticity of Nonlinear Elastic Materials Alexander B. Freidin Institute of Mechanical Engineering Problems Russian Academy of Sciences Bolshoj pr., 61, V.O., St.Petersburg, , Russia Conditions on the equilibrium phase boundary in an isotropic hyperelastic material are analyzed. Possible jumps of strains on the phase boundary are described. Special attention is given to the consideration of phase boundaries in the context of ellipticity examination. 1 Introduction Solid materials exhibiting phase transformations under the process of deformation have received increased attention in recent years. Theoretical and experimental studies in this field are directed toward the development and practical use of new materials (so called smart materials) that predictably and nontrivially respond to external actions (mechanical, thermal, electrical). Phase transformations lead to shape memory phenomena, when material recovers large deformations during unloading or recovers residual strains through thermal action. Rheological behavior features such as strain localization due to martensite transformations in metals, localized orientation transformations in polymers, and brittle-ductile transitions also can be connected with structural transformations. Although attempts to examine these problems had been undertaken much earlier, it was only by the end of the 1970s early 1980s that results were obtained which allowed researchers to speak about phase boundaries in deformable solids within the framework of continuum mechanics. In this approach new phase areas generated in the process of deformation are the areas bounded by the surfaces of discontinuity in deformation gradient at continuous displacement and two-phase deformation fields are the deformation fields with discontinuous deformation gradients. Conditions on the equilibrium phase boundary in a nonlinear elastic material include an additional thermodynamic condition (the Maxwell relation) (Grinfeld, 1980, James, 1981, Truskinovsky, 1982, Gurtin, 1983, Fosdick & Hertog, 1989) and can be considered as the system of equations that determine the normal to the boundary as well as the strains in one phase as a function of strains in the other. This system of equations has nontrivial solutions not for all hyperelastic materials and not at all deformations. The first circumstance leads to restrictions on constitutive equations, the second one leads to the concept of phase transition zone (PTZ) (Freidin & Chiskis, 1994) formed by all strains which can coexist on the equilibrium interface in a hyperelastic material. The PTZ boundary acts as a phase diagram or yield surface in strain-space. In this paper we analyze the conditions on the equilibrium phase boundary in an isotropic hyperelastic material. We describe possible jumps of strains which are

2 compatible with displacement and traction continuity conditions and discuss the role of an additional thermodynamic condition. Equilibrium deformations with discontinuous gradients can exist in elastic bodies only if ellipticity fails at some deformations (Knowles & Sternberg, 1978, see also [1, 10, 21, 22]). The non-ellipticity zone by necessity crosses the phase transition zone [6, 7]. That is why we devote special attention to the consideration of phase boundaries in the context of Baker Ericksen inequalities examination. 2 Preliminaries. Conditions on the phase boundary, ellipticity In absence of body forces, we are interested in equilibrium deformation fields such that the displacements are twice differentiable everywhere in a body besides a continuously differentiable surfaces (shock surfaces) at which the deformation gradient suffers a jump at continuous displacement. Let G be the shock surface in a reference (undeformed) configuration of a body, m U is the normal to G, U denotes a set of unit vectors. The following conditions have to be satisfied on the equilibrium shock surface [F] = f m, (2.1) [S]m = 0, (2.2) where F is the deformation gradient, brackets [ ] = ( ) + ( ) denote the jump of a function across G, super- or subscripts and + identify the values on different sides of the shock surface, S is the Piola (nominal) stress tensor related with Cauchy stress tensor as T =J 1 SF T, J = det F > 0. The kinematic condition (2.1) follows from the continuity of the displacement [24], the vector f = [F]m (2.3) is called the amplitude. The traction continuity condition (2.2) follows from equilibrium considerations. If the shock surface is considered as an equilibrium phase boundary, it has to satisfy an additional thermodynamic condition [9, 11, 25, 10, 5, 20] [W ] = f S ± m, (2.4) where W is the strain energy per unit reference volume, S = W F (F). By (2.2) and (2.3), (2.4) can be rewritten as m [M]m = 0, (2.5) where M = W I F T S is the energy momentum Eshelby tensor [4] (the chemical potential tensor [9]), I denotes the unit tensor. By (2.1) (2.3), [M]m = ([W ] [F T ]S ± )m = ([W ] f S ± m)m. Thus, a scalar condition (2.5) is equivalent to the vector condition [M]m = 0. Note also that, by (2.1) and (2.2), the condition (2.5) is represented as [W ] = S : [F] tr (S[F T ]).

3 The conditions (2.1), (2.2), (2.4) can be rewritten as F + F = a F T n, (2.6) [T]n = 0, (2.7) [W ] = J a Tn (2.8) where n is the normal to the shock surface in the actual (deformed) configuration, n = F T m 1 F T m, m = F T n 1 F T n, (2.9) amplitudes f and a are related as a = N 1/2 1 f, where N 1± = n B ± n, B = FF T is the left Cauchy-Green strain tensor. We imply the correspondence between upper and lower indices ±. It is known [14, 10] that the equilibrium deformation fields with the shock surfaces can exist (i.e. can satisfy (2.1) and (2.2) at f 0) only if ordinary ellipticity of the equilibrium equations fails at some deformation. This is a reason to recall the notions of ellipticity [24, 12, 13, 17, 19]. The equilibrium equations are said to be ordinary elliptic at F if det Q(F, n) 0 n U, (2.10) where Q (F, n) is the acoustic tensor, Q ik (F, n) = C ijkl (F) n j n l, C (F) = W FF is the elasticity tensor, n j (j = 1, 2, 3) are components of n. An elastic material is said to be elliptic at F if satisfies the Legendre-Hadamard condition e Q(F, n)e 0 e, n U. (2.11) The material is strongly elliptic if (2.11) is satisfied as the strict inequality. Further we consider shock surfaces in isotropic materials. In this case the elastic potential depends on F only through the strain invariants: W = W (I 1, I 2, J), I 1 = λ λ λ 2 3 = tr B, I 2 = λ 2 1λ λ 2 1λ λ 2 2λ 2 3 = J 2 tr B 1, I 3 = λ 2 1λ 2 2λ 2 3 = J 2, where λ i > 0 (i = 1, 2, 3) are the principal stretches, and the Cauchy stress tensor is the isotropic function of B: T = µ 0 I + µ 1 B + µ 1 B 1, (2.12) µ 0 = W 3 + 2J 1 I 2 W 2, µ 1 = 2J 1 W 1, µ 1 = 2JW 2, (2.13) where W 1, W 2, W 3 denote W/ I 1, W/ I 2 and W/ J respectively. Alternatively, W can be expressed in terms of the principal stretches λ i : W = W (λ 1, λ 2, λ 3 ). The principal Cauchy stresses τ k can be calculated as τ k = 1 λ i λ j Wk = 2λ2 k J (W 1 + W 2 (I 1 λ 2 k)) + W 3 (k = 1, 2, 3), (2.14) where W k = W / λ k. If a material is isotropic, then at given B the necessary conditions for the ellipticity can be derived if take the acoustic tensor at n = e k (k = 1 or 2 or 3), where e i is

4 one of the eigenvectors of B. In this case eigenvectors of Q are the eigenvectors of B. The eigenvalues of Q are equal to [17] Q k = λ k τ k (λ 1, λ 2, λ 3 ) λ k, Q i = λ 2 τ k τ i k λ 2 k, Q j = λ 2 τ k τ j λ2 k i λ 2 k λ2 j (k i j k) If λ k λ i then τ k τ i λ 2 k = 2 ( ) W1 + λ 2 λ2 i J jw 2 Thus, if a material is strongly elliptic at (λ 1, λ 2, λ 3 ) then τ k λ k > 0, W 1 + λ 2 i W 2 > 0, W 1 + λ 2 jw 2 > 0 (k i j k). (2.15) The first inequality in (2.15) means that principal Cauchy stress increases if corresponding principal stretch increases (the tension extension inequality), the second and the third BE inequalities express Baker Ericksen criteria sgn(τ k τ m ) = sgn(λ k λ m ) m = i, j. Note that BE-inequalities (2.15) 2,3 remain to be the necessary condition for the strong ellipticity even if λ k = λ i (Simpson & Spector, 1983). One can express F + through F and f and m and substitute into the conditions (2.2) and (2.4). After that, at given F, the conditions (2.2) and (2.4) can be considered as the system of four equations for five unknowns which determine the amplitude f and the unit normal m. Strains for which the system of equations can be resolved form the phase transition zone (PTZ) [6]. The solution in general case has a form of an one-parameter family. Definition. The phase transition zone is formed by all deformation gradients which can coexist on the locally equilibrium phase boundary. The elements of the PTZ are the PTZ-boundary, nonellipticity subzone, critical points (Figure 1). The importance of the PTZ construction is motivated by the following reasons: Deformations outside PTZ cannot coexist on the interface, whatever the loading conditions. The PTZ boundary acts as a phase diagram or yield surface in strain-space. A comparison between PTZ and two phase deformation fields in different two phase configurations of an elastic body at the same boundary conditions allows to choose thermodynamically preferential configuration. 3 Orientation invariants Further we will reformulate the jump conditions in terms of orientation invariants. We will see that the normal n appears in relationships through the orientation invariants (Freidin & Chiskis, 1994) N k = n B k n, k = ±1, ±2,... (3.1) and the amplitude a can be decomposed using the vectors t k = PB k n, k = ±1, ±2,...

5 where P = I n n is a projector. This is a reason to examine properties of N k and t k. First, note that N k and t k are isotropic functions of B and n: Q Orth + : N k (QBQ T, Qn) = N k (B, n), t k (QBQ T, Qn) = Qt k (B, n). (3.2) Due to the Caley Hamilton theorem only two of the orientation invariants and only two of vectors t k are independent. For example, N 2 = I 1 N 1 I 2 + J 2 N 1, N 2 = 1 J 2 (N 1 I 1 + I 2 N 1 ). (3.3) If λ 1 λ 2 λ 3 λ 1 then at given B a couple of invariants (N s, N t ) (t s) determines the normal n in the basis of eigenvectors of B by the system of equations n 2 i = 1, n 2 i λ 2k i = N k, n i = n e i (k = s, t) (3.4) which is linear with respect to n 2 i. If λ 1 = λ 2 λ λ 3 then any two of orientation invariants N l, N m (l m) are linearly dependent N l = For example, 1 ( ) λ 2m λ 2m 3 λ 2m 3 λ 2l λ 2l 3 λ 2m + (λ 2l 3 λ 2l )N m, Nm [λ 2m, λ 2m 3 ]. (3.5) N 1 = λ 2 + λ 2 3 λ 2 λ 2 3 N 1 = I 2 J 1 2 λ λ2 2 J N 1, N 2 1 [ ] λ 2, λ 2 3. Any N m (λ 2, λ 2m 3 ) determines the family of normals with components n 2 3 = N m λ 2m λ 2m 3 λ, 2m n2 1 + n 2 2 = λ2m 3 N m λ 2m 3 λ. 2m If λ 1 = λ 2 = λ 3 λ then N l = λ 2l at any n. Since the solution of the system (3.4) for n 2 i has to be non negative, the admissible values domain D for the orientation invariants N l, N m is a triangle D = D lm with vertexes ( ) λ 2l i, λ 2m i (i = 1, 2, 3) lying on the skeleton curve N m = N m/l l (3.6) (Figure 2). The vertexes ( ) λ 2l i, λ 2m i correspond to n = ei. On the i j side of the triangle n k = 0 (k i, j), n lies in the i j principal plane of B, N l = λ 2m i 1 λ 2m j ( λ 2m i λ 2l j λ 2l i λ 2m j + (λ 2l i λ 2l ) j )N m, Nm [λ 2m i, λ 2m j ]. (3.7) The triangle D lm degenerates into the segment (3.5) if λ 1 = λ 2 λ λ 3. The endpoints ( λ 2l, λ 2m) and ( ) λ 2l 3, λ 2m 3 correspond to n3 = 0 and n = e 3. If λ 1 = λ 2 = λ 3 λ then D lm is a point (λ 2l, λ 2m ) on the skeleton curve.

6 For example, if N 1 and N 1 are taken as independent then ( ) λ 2 i, λ 2 i (i = 1, 2, 3) are the vertexes on the N 1, N 1 plane, the skeleton curve and the i j side of D 1 1 are determined by N 1 = N 1 1, N 1 > 0, N 1 = λ 2 i + λ 2 j λ 2 i λ 2 j N 1 = I 2 J 1 2 λ 2 k λ2 k J 2 N 1, N 1 [ ] λ 2 i, λ 2 j. (3.8) Further we will use a couple of vectors t k as basis vectors and refer to the following Lemma 3.1.The basis degenerates to zero t l = PB l n, t m = PB m n (l m, l, m = ±1, ±2...) t l = t m = 0 (3.9) if and only if n is an eigenvector of B, i.e. if and only if any of the following conditions is met: (a) n = e i (i = 1, 2 or 3), (b) B = λ 2 (I e k e k ) + λ 2 k e k e k, n k = 0 (c) B = λ 2 I. The vectors t l, t m are colinear if and only if either the condition (no sum), (3.10) t l t m 0 (3.11) (d) n j = 0, n i n k 0, λ i λ k, (e) B = λ 2 (I e k e k ) + λ 2 k e k e k (no sum), n k 0, n e k λ λ k (3.12) is met. As (3.12 d) takes place, t l = λ2l i λ 2m i t m = (λ 2m i λ 2l k λ 2m k t m, (3.13) λ 2m k )n k (n k n e k ) (no sum). (3.14) In a case (3.12 e) one has to replace λ i in (3.13), (3.14) by λ. The proof is an exercise in algebra. Since n k n e k = n (n e k ), one can represent t m at n j = 0 as t m = (λ 2m i λ 2m k )n i n k n (e i e k ) = (λ 2m i λ 2m k )n i n k e ikj n e j (no sum) (3.15) where e ikj is the alternating symbol. The degenerations take place only on the shock surfaces which correspond to the boundaries of D. The vectors t m = 0 if and only if the normal n corresponds to the points of D lying on the skeleton curve (3.6), i.e. to the vertexes of the triangle D if λ 1 λ 2 λ 3 λ 1 or to the endpoints of the segment D if λ i = λ j λ k or when λ 1 = λ 2 = λ 3. The sides of the triangle D except the vertexes (if the principal stretches are different) as well as the segment except the endpoints (if two of the stretches are equal) correspond to t l t m 0.

7 4 Kinematic compatibility Further we put a a. It follows from (2.6) that, F + = (I + a n) F, (4.1) B + = B + a B n + B n a+n 1 a a, (4.2) B 1 + = B 1 (J /J + ) (n B 1 a + B 1 a n)+ (J /J + ) 2 ( a B 1 a ) n n. (4.3) Relationships (4.1) (4.3) lead to the following relations [6] between the strain and orientation invariants on different sides of the shock surface: [ I2 J + = 1 + a n, (4.4) J [I 1 ] = 2a B n + N 1 a a, (4.5) ] ( ) 2 J = a B 1 a 2 J a B 1 n. (4.6) J + J 2 J + Also from (4.2), (4.3) and (4.4) follows that [ ] [ ] N1 I2 = 0, J 2 J N 2 1 = 0, (4.7) J 1 + t + 1 = J 1 (t 1 + N 1 h), J + t + 1 = J (t 1 PB 1 h). (4.8) For the plane case one can find (4.7) 1 and in [14, 2]. Conditions (4.7) have evident geometrical meaning. One can verify that if da and dx denote a surface element and length of a tangent line elements on the shock surface in the deformed configuration then (4.7) means that [da] = 0 and [dx ] = 0. By (4.1) ( ) J+ a = 1 n + h, (4.9) J and kinematic relationships (4.5), (4.6) can be rewritten as [I 1 ] = N 1 J [J 2 ] + 2h t N1 h h, (4.10) ( ) I2 [I 2 ] = J N ( 2 1 [J 2 ] + J 2 h B 1 h 2h t 1). (4.11) Keeping in mind the kinematic relationships (4.7) we do not show indexes ± when meet ( ) N 1 I2 and J 2 J N 2 1. One can also rewrite (4.2) in the form B + = B + a g + g a, g B n+ 1 2 N 1 a (4.12) Now we are interested in all a, g R 3 such that (4.12) determines B + that can coexist with B on some shock surface.

8 Proposition 4.1. A symmetric positive tensor B and a tensor B + can be left Cauchy Green tensors compatible with (4.1) at some n if and only if there exist a, g R 3 such that (4.12) holds and (a B 1 a) 1/2 (g B 1 g) 1/2 a B 1 g < 1. (4.13) To prove necessity one can verify that if B > 0, the vector g is given by (4.12) 2, and a n 1, then (4.13) is satisfied. To prove sufficiency one can show that (4.12) 2 is resolvable for n : a n > 1 if and only if (4.13) holds. The above formulation was stimulated by the inequality similar to (4.13) obtained by Ericksen (1991) for the right Cauchy - Green tensor. The differ way we used can be also used for the derivation of the Ericksen s result. 5 Traction continuity condition Projecting the traction condition (2.7) onto the normal n and plane tangent to the shock surface and taking into account (2.12) we get [µ 0 ] + [µ 1 N 1 ] + [µ 1 N 1 ] = 0 (5.1) [µ 1 t 1 ] + [µ 1 t 1 ] = 0 (5.2) By (2.13), the condition (5.1) for the normal component of the traction can be rewritten as [W 3 ] = 2 [JW 1 ] N ( ) 1 J + 2 [JW I2 2 2 ] J N 2 1. (5.3) In a case of incompressible materials the condition for the normal component of the traction becomes the formula which determines the jump of the reaction p in dependence on deformation and orientation invariants [p] = 2 [W 1 ] N 1 2 [W 2 N 1 ] = 2 [W 1 ] N 1 2 [W 2 ] N 1 2W + 2 [I 2 ]. (5.4) The equation (5.2) for the tangent component of the traction takes the form of the equation for h : Ah = J 2 [W 1 ] t 1 + [W 2 ] t 1, A N 1 J 2 W + 1 I+W + 2 PB 1, (5.5) which leads to the theorem [6] analogous to the theorem on the representation of the Cauchy stress tensor as an isotropic tensor function (2.12). Representation Theorem. Assume that the material is isotropic and on the shock surface kinematic (4.1) and traction (2.7) conditions are satisfied and D = N ( ) 1 ( ) W + 2 I2 J J N 2 1 W 1 + W ( ) W (5.6) holds. Then the amplitude a is an isotropic function of the strain tensor B and the normal n and can be decomposed as follows: ( ) J+ a = 1 n + αt 1 + βt 1 (5.7) J

9 where t 1 = PB n, t 1 = PB 1 n, and the coefficients α and β are uniquely given as functions of orientation and strain invariants by the system of linear equations N 1 J W W 2 + J α 2 [W 1 ] N ( ) 1 J W + N J W + I J N =. (5.8) 2 1 W 2 + β [W 2 ] The representation (5.7) remains true for incompressible materials, one needs only to set J 1. By (5.7), h is represented as The representation (5.9) remains true if t l (3.13)) or if t l = t 1 = 0. Since the function D = N + 1 J 2 + h = αt 1 + βt 1 (5.9) t 1 0 (one have to take into account ( ( ) W + 2 I + ) N + J+ 2 1 W 1 + W ( ) W is linear in N + 1, N + 1, its maximum and minimum values are reached on the boundary of D On the i j-side of the triangle D D is linear in N + 1 : D = ( W λ 2 k+w + 2 In the vertex associated with n = e + i ) ( N 1 + W + J+ 2 1 ) + λ 2 k+ W 2 +, N 1 + [ λ 2 i+, λj+] 2. D = λ2 i+ J 2 + ( W λ 2 j+w + 2 ) ( W λ 2 k+w + 2 ). (5.10) Thus, the maximum and minimum values of D are among the values (5.10). Then at given I + 1, I + 2, J +, D 0 N + 1, N + 1 D (5.11) if and only if W λ 2 k+w + 2 > 0 (k = 1, 2, 3) or W λ 2 k+w + 2 < 0 (k = 1, 2, 3). (5.12) Proposition 5.1. Assume that the material is isotropic and on the shock surface kinematic (4.1) and traction (2.7) conditions are satisfied and the Baker Ericksen inequalities W λ 2 k+w 2 + > 0 (k = 1, 2, 3) (5.13) hold. Then the amplitude is represented by (5.7) (5.8) at any n. There are physical and mathematical reasons to believe that the deformations cannot be observed if the Legendre-Hadamard condition fails. That is why one may don t consider strain fields if at least one expression in BE-criteria is negative. The case when BE-inequalities hold is given by the representation theorem. Now let us suppose that one of the BE-inequalities (5.13) becomes the equality and the others hold.

10 Let Then in the case under consideration By (5.15) and (3.8) λ 1+ < λ 2+ < λ 3+, W + 2 > 0. W λ 2 1+W 2 + = 0, W λ 2 k+w 2 + > 0 (k = 2, 3) (5.14) ( ( ) N1 I2 D = J 2 λ2 1+ J N ) λ 2 λ 1+(W ) 2 (5.15) 1+ min D = 0 N + 1,N + 1 D+ 1 1 on the 2 3-side of the triangle D In other points of D + 1 1, including the sides, D > 0 and the representation theorem can be used. Thus, we have to consider shock surfaces such that n 1+ = 0, and (5.14) hold. Proposition 5.2. Assume that on the shock surface W ± 1 + λ 2 k±w ± 2 0 (k = 1, 2, 3), W ± 1 W ± 2 0, D = 0, λ 1+ < λ 2+ < λ 3+ and W + 2 > 0. Then the jump, if exist, has the following properties: n e + 1 = e 1, e h/ h = n e + 1, λ 1+ = λ 1 λ 1, W λ 2 1W + 2 = 0, (5.16) and either or W 1 + λ 2 1W 2 0 and n = e 2 or n = e 3, (5.17) W 1 + λ 2 1W 2 = 0. (5.18) If W + 2 < 0 one has to replace e ± 1 by e ± 3 and λ 1± by λ 3± in (5.16) (5.18). (5.16) means that plane strain state takes place on the shock surface, (5.18) is possible only if there exist λ 1+ = λ 1 λ 1, λ 2±, λ 3± such that ordinary ellipticity fails on the both sides of the shock surface, W1 W2 = W + 1 W + 2 = λ 2 1. (5.19) In a case (5.17) the normal n has to coincide with the eigenvector e 2 of eigenvectors of B rotates round e 1 on the shock surface. or e 3. Trihedron 6 Thermodynamic condition. Phase transition zone By (2.12), the thermodynamic condition (2.8) takes the form [W ] = µ 0 [J ] + 2W 1 a B n 2J 2 W 2 a B 1 n and, after substituting (2.13) and (4.9), can be rewritten as [W ] = τ n [J ] + 2W 1 h t 1 2J 2 W 2 h t 1 (6.1)

11 where the normal component of the traction ( ( ) ) N1 τ n = n Tn = 2J J W I J N 2 1 W 2 + W 3 can be calculated at any side of the phase boundary. In a case of incompressible materials one has to put J = J + 1 in (6.1). Using (4.10) and (4.11) to eliminate h t 1 and h t 1 from (6.1) we find one more form of the thermodynamic condition [W ] =W1 [I 1 ] + W2 [I 2 ] + W3 [J ] ( ( ) ) ( ) N1 J W I J N 2 1 W2 [J ] 2 J 2 N1 J W 2 1 h h+w2 h B 1 h. (6.2) If α and β are given by the system (5.8), then substituting the representation (5.9) into (4.10) and (4.11) gives two equations for five unknowns I 1 +, I 2 +, J +, N1, and N 1. The equation (5.3) is a third one. So, the jump solution compatible with kinematic and traction conditions, if exists, has a form of a two-parameter family. In a case of incompressible materials (5.4) gives two-parameter family of jumps of the reaction p. If also substitute (5.9) into (6.1) or (6.2) we obtain the system of four equation to determine an one-parameter family of the solutions {I 1 +, I 2 +, J +, N1,, N 1} which, at given I1, I2, J, satisfy the local phase equilibria conditions. We emphasize that the representation (5.9) is a form of the tangent traction continuity condition which is substituted into the kinematic and thermodynamic conditions in the developing procedure. If resolve three of the equations for I 1 + = I 1 + (N1, N 1 I1, I2, J ), I 2 + = I 2 + (N1, N 1 I1, I2, J ) and J + = J + (N1, N 1 I1, I2, J ), then the forth equation takes the form of the equation for the one-parameter family of the orientation invariants: Ψ(N1, N 1 I1, I2, J ) = 0. (6.3) The one-parameter family of orientation invariants is presented on the N 1, N 1 -plane by crossing the line (6.3) (the line ab on Figure 2) with the triangle D. The phase transition zone in λ 1, λ 2, λ 3 -space is formed by all principal stretches at which the intersection exists. Some examples of the PTZ construction are given in [6, 7, 8, 18]. Here we consider one of cases when the amplitude is expressed especially simply. If W = W (I α, J) (α = 1 or 2) then ( ) J+ a = 1 n [W α ] J W α + N1 t 1 (6.4) [I 1 ] = N ( ) 1 W J [J 2 2 ] + α 1 L 2 Wα+ 2 1 (N1, N2 ) (6.5) ( ) ( ) I2 W [I 2 ] = J N 2 1 [J 2 ] + J 2 2 α 1 L 2 (N1, N 1) (6.6) W 2 α+ L 1 (N 1, N 2 ) N1 1 t 1 t 1 = N1 1 N 2 N 1, L 2 (N 1, N 1 ) N1 1 t 1 t 1 = N 1 N1 1

12 Let W = W (I 1, J). Then, by (5.3), (6.1) and (6.4) and 2 N 1 J 2 [JW 1 ] = [W 3 ] (6.7) [W ] = W 1 W W + 1 W 3 W 1 + W W 1 W + 1 W 1 + W + 1 [I 1 ] (6.8) Relationships (6.8), (6.7) and (6.5) are three equation to determine four unknowns J +, I +, N 1 and N 2. If resolve (6.8) and (6.7) with respect to J + = J + (N 1, J, I 1 ), I + 1 = I + 1 (N 1, J, I 1 ) (6.9) and substitute (6.9) into (6.5), we get an equation in a form Ψ(N 1, N 2, J, I 1 ) Ψ 1 (N 1, J, I 1 )N 1 + Ψ 2 (N 1, J, I 1 )N 2 = 0. (6.10) The equation (6.10) determines a curve on the N1, N2 -plane. The crossing this curve with the triangle D12 gives an one parametric family of orientation invariants on the equilibrium phase boundary. Corresponding values of J + and I 1 + are calculated by (6.9). The invariants J and I1 have to satisfy inequalities Ψ min (J, I 1 ) 0 Ψ max (J, I 1 ), Ψ min (J, I1 ) = min Ψ(N1, N2, J, I1 ), N 1,N 2 D 12 Ψ max (J, I1 ) = max N 1,N 2 D 12 Ψ(N1, N2, J, I1 ) Since Ψ(N 1, N 2, J, I 1 ) is linear in N 2, its maximal and minimal values are reached at the boundary of D 12, i.e. when the normal lies in the principal plane of B. If, for example, λ 1 < λ 2 < λ 3 then one of these values is reached on the 1 3-side of the triangle D 12. It can be proved that plane jumps take place if the normal lies in the principal plane of the strain tensor and D 0. Thus, if at least one of the the PTZ boundaries is determined by the equation Ψ min (J, I 1 ) = 0 or Ψ max (J, I 1 ) = 0, then this PTZ boundary is associated with plane jumps. Returning to a general dependence of strain energy function on the deformation invariants note that the relative position of the line (6.3) and the triangle D changes when (λ 1, λ 2, λ 3 ) change. One can expect that if (λ 1, λ 2, λ 3 ) belong to the PTZ boundary then the line passes through a single point of D, i.e. passes through the vertex or externally touches the side of the triangle (the lines pq and cd on Figure 2). In these cases the normal coincides with the eigenvector of B or lies in the principal plane of B, one-parameter feature is exhausted.

13 7 Shocks at continuous deformation invariants. Twinning Obviously, [J ] = [I 1 ] = [I 2 ] = 0 if a = 0. On the other hand, if [I 1 ] = [I 2 ] = [J ] = 0 and D = N ( ) 1 J W 2 I J N 2 1 W 1 W 2 + W2 2 0, then, by the representation theorem, a = 0. In particularly, if BE-inequalities hold, then [J ] = [I 1 ] = [I 2 ] = 0 a = 0. Now we study the possibility that on the shock surface [J ] = [I 1 ] = [I 2 ] = 0 and a 0. In this case from the kinematic relationships (4.9) (4.11), (4.2) (or (4.12)) and (4.3) follows that where a = he, h 0, (7.1) 2ν 1 + N 1 h = 0, (7.2) 2ν 1 bh = 0, (7.3) [B] = h(e g + g e), [B 1 ] = h(n q + q n) (7.4) ν 1 = e t 1 = e B n, ν 1 = e t 1 = e B 1 n, b = e B 1 e, (7.5) g = B n N 1he, and, by (4.7) 1, N 1 = N 1 = N 1+. From (7.6), (7.2) and (7.3) follows that q = B 1 e + 1 bhn, (7.6) 2 g e = 0, q n = 0. (7.7) The jumps at which strain invariants remain continuous mean that that the trihedron of eigenvectors of B rotates about some axis or/and reflects about some plane. It can be proved that the rotation is determined by the restrictions which follow from the tangent traction continuity condition. (The condition (5.3) for the normal component of the traction and thermodynamic condition are fulfilled as identity.) Ptoposition 7.1. Assume that the material is isotropic and on the shock surface kinematic (4.1) and traction (2.7) conditions are satisfied, and Then [J ] = [I 1 ] = [I 2 ] = 0, W 1 W 2 0, a 0. a = he n, (7.8) where the one-parameter family of normals is determined by the equation ( ) N 1 J W 2 I J N 2 1 W 1 W 2 + W2 2 = 0, (7.9)

14 e is an eigenvector of PB 1 ± such that: and h is calculated by the formulas PB 1 ± e = N 1W 1 J 2 W 2 e, (7.10) h = 2ν 1 N 1 = 2ν 1, (7.11) b where ν 1 = e B n, ν 1 = e B 1 n, b = e B 1 e = (N 1 W 1 )/(J 2 W 2 ). The left Cauchy-Green tensors on the shock surface are related as B + = QB Q T, (7.12) where Q = 2e e I is the 180 -rotation about the axis e. The Cauchy and Piola tractions are perpendicular to e : e T ± n = e S ± m = 0. Thus, the shock surfaces can be only the surfaces of twins if strain invariants are continuous. We are not interested in cases when the Legendre-Hadamard condition (2.11) fails. On the other hand we need D = 0. Keeping in mind Proposition 5.2, let us consider an opportunity of twinning if W 1 W 2 0, λ 1 < λ 2 < λ 3, and one of the BE-inequalities becomes equality on the both sides of the surface and the other inequalities remain true. Assume W 2 > 0. Then, by (5.16), W λ 2 1W + 2 = 0, n e + 1 = e 1 e 1, e = n e 1, λ 1+ = λ 1 λ 1, (7.13) and, by (7.6) 1 and (7.7) 1, g e 1 = 0, g = N 1 n. Then, by (7.4) 1 From (2.12), (2.13) and (7.10) follows that Thus on the twin surface [B] = hn 1 (n e + e n). (7.14) [T] = 2h ( N 1 J 1 W 1 + bjw 2 ) (n e + e n) = 0. T + = T (7.15) Continuity of the stress tensor does not contradict to the fact that T + = QT Q T, B + B, (7.16) because in this case principal axes of stress do not necessary coincide with principal axes of strain. Substituting W 1 = λ 2 1W 2 into (2.14) gives τ ± 2 = τ ± 3 = 2J λ 2 1 W 2 + W 3 τ, τ 1 ± = 2J (I λ 2 1 2λ 2 1)W 2 + W 3 τ 1. 1

15 Thus, T ± = τ(i e 1 e 1 ) + τ 1 e 1 e 1 (e 1 = e 1 = e + 1 ) is the axially symmetric tensor, whereas B ± are not necessary axially symmetric. This feature provides the opportunity to simultaneously satisfy (7.15) and (7.16). By (7.11) 1, (3.15) and (7.14) one-parameter families of h and [B] are given by h = 2 t 1 /N 1 = 2 (λ 2 2 λ 2 3)n 2 n 3 /N 1 = 2(λ 2 3 λ 2 2)/(λ 2 3k + λ 2 2k 1 ), (7.17) [B] = (λ 2 3 λ 2 2) sin (2φ 2 )(n e + e n), (7.18) where k = tan φ 2, φ 2 is the angle between n and e 2. If n e i (i = 2, 3), then h 0. If W 2 < 0, then n 3 = 0, e + 3 = e 3 = e 3 and e = ±e 3 n. Acknowledgment. This work was supported by Russian Foundation for Basic Research (Grant N ). References [1] R. Abeyaratne, Discontinuous deformation gradients in plane finite elastostatics of incompressible materials. J. Elasticity 10 (1980) [2] R. Abeyaratne and J.K. Knowles, Equilibrium shocks in plane deformation of incompressible elastic materials. J. Elasticity 22 (1989) [3] J.L. Ericksen, On kinematic conditions of compatibility. J. Elasticity 26 (1991) [4] J.D. Eshelby, The elastic energy-momentum tensor. J. Elasticity 5 (1975) [5] R.L. Fosdick and B. Hertog, The Maxwell relation and Eshelby s conservation law for minimizers in elasticity theory. J. Elasticity 22 (1989) [6] A.B. Freidin and A.M. Chiskis, Phase transition zones in nonlinear elastic isotropic materials. Part 1: Basic relations. Izv. RAN, Mekhanika Tverdogo Tela (Mechanics of Solids) 29 (1994) No. 4, [7] A.B. Freidin and A.M. Chiskis, Phase transition zones in nonlinear elastic isotropic materials. Part 2: Incompressible materials with a potential depending on one of deformation invariants. Izv. RAN, Mekhanika Tverdogo Tela (Mechanics of Solids) 29 (1994) No. 5, [8] A.B. Freidin, Small strains approach to the theory on solid-solid phase transformations under the process of deformation. Studies on Elasticity and Plasticity (St. Petersburg State University) 18 (2000) (in Russian). [9] M.A. Grinfeld, On conditions of thermodynamic equilibrium of the phases of a nonlinear elastic material. Dokl. Acad. Nauk SSSR 251 (1980) [10] M.E. Gurtin, Two phase deformations of elastic solids. Arch. Rat. Mech. Anal. 84 (1983) [11] R.D. James, Finite deformation by mechanical twinning. Arch. Rat. Mech. Anal. 77 (1981) [12] J.K. Knowles and E. Sternberg, On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elasticity 5 (1975) [13] J.K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Rat. Mech. Anal. 63 (1977)

16 [14] J.K. Knowles and E. Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J. Elasticity 8 (1978) [15] J.K. Knowles, On the dissipation associated with equilibrium shocks in finite elasticity. J. Elasticity 9 (1979) [16] A.I. Lurie, Ellipticity criteria for equilibria equations in nonlinear elasticity. Mekhanika Tverdogo Tela (Mechanics of Solids) (1979) No. 2, [17] A.I. Lurie, Nonlinear Theory of Elasticity (in Russian). Nauka, Moscow (1980), In English: North-Holland Series in Appl. Math. and Mech. 36 (1990). [18] N.F. Morozov and A.B. Freidin, Phase transition zones and phase transformations of elastic solids under different stress states. Proceedings of the Steklov Mathematical Institute 223 (1998) [19] R.W. Ogden, Non-Linear Elastic Deformations. Ellis Horwood, Chichester, and Jonn Wiley (1984) [20] V.G. Osmolovsky, Variational problem on phase transitions in mechanics of solids. St. Petersburg State University, St. Petersburg (2000) (in Russian). [21] P. Rosakis, Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Rat. Mech. Anal. 109 (1990) [22] P. Rosakis and Q. Jiang, Deformations with discontinuous gradients in plane elastostatics of compressible solids. J. Elasticity 33 (1993) [23] H.C. Simpson and S.J. Spector, On copositive matrices and strong ellipticity for isotropic elastic materials. Arch. Rat. Mech. Anal. 84 (1983) [24] G. Truesdell, A First Course in Rational Continuum Mechanics. The Johns Hopkins Univercity, Baltimore, Maryland (1972). [25] L. Truskinovsky, Equilibrium interphase boundaries. Dokl. Acad. Nauk SSSR 265 (1982) [26] L. Zee and E. Sternberg, Ordinary and strong ellipticity in the equilibrium theory of incompressible hyperelastic solids. Arch. Rat. Mech. Anal. 83 (1983)