[...]... austenite and the martensite alone A set of the form SO(3)Ui will in the sequel frequently be called energy well We now describe the framework for the mathematical analysis of martensitic transformations and its connection with quasiconvex hulls 1.1 Martensitic Transformations and Quasiconvex Hulls 3 T > Tc T < Tc Fig 1.2 The cubic to tetragonal phase transformation 1.1 Martensitic Transformations and Quasiconvex... hull of K and the two additional inequalities (x + a)(z − c) ≤ y 2 − b2 , (x − a)(z + c) ≤ y 2 − b2 (2.7) This proves the formula for the polyconvex hull of K In fact, the sum of the two upper and the two lower inequalities in (2.5) implies az ≤ ac and − az ≤ ac, and the sum of the two left and the two right inequalities, respectively, gives cx ≤ ac and − cx ≤ ac Therefore |z| ≤ c and |x| ≤ a and this... based on this definition - separating points from a set by semiconvex functions - will be called the separation method in the sequel Since rank-one convexity is a necessary condition for quasiconvexity and polyconvexity a sufficient one, we have the chain of inclusions K lc ⊆ K rc ⊆ K qc ⊆ K pc , and frequently the most practicable way to obtain formulae for K qc is to identify K lc and K pc There exists... phase transformation For this material, Bladon, Terentjev and Warner derived a closed formula WBTW for the free energy density which depends on the deformation gradient F and the nematic director n, but not on derivatives of n From the point of view of energy minimization, one can first minimize in the director field n and one obtains a new energy W that depends only on the singular values of the deformation... the underlying crystalline lattice onto itself leads to a very degenerated situation with a fluid-like behavior of the material under dead-load boundary conditions The two hypotheses (1.1) and (1.2) have far reaching consequences which we are now going to discuss briefly (see the Appendix for notation and terminology) We focus on isothermal situations, and we assume therefore that W ≥ 0 and that the zero... method, and the splitting method, respectively As a very 1.2 Outline of the Text 9 instructive example for the separation and the splitting method, we analyze a discrete set of eight points Then we characterize the semiconvex hulls for compact sets in 2×2 matrices with fixed determinant that are invariant under multiplication form the left by SO(2) We thus find a closed formula for all sets arising in two-dimensional... in closed form is rather short), it allows us to relate K qc to two more easily accessible hulls of K, the rank-one convex hull K rc and the polyconvex hull K pc which are defined analogously to (1.8) by replacing quasiconvexity with rank-one convexity and polyconvexity, respectively (see Section A.1 for further information) All these hulls will be referred to as ‘semiconvex’ hulls The method for calculating... → 0, while J (Fλ x) > 0 Therefore it is energetically advantageous for the material to form fine microstructure, i.e., minimizing sequences develop increasingly rapid oscillations This argument shows that Fλ ∈ K qc It is clear that this process for the construction of oscillating sequences and elements in K qc can be iterated In fact, if Fλ and Gµ are matrices with the foregoing properties that satisfy... a and this proves that (2.4) and (2.5) imply (2.6) and (2.7) Conversely, if the convex inequalities |x| ≤ a, |z| ≤ c, and |y| ≤ b in (2.6) hold, then x − a ≤ 0, z − c ≤ 0 and −y 2 + b2 ≥ 0 Consequently −(x − a)(z − c) ≤ −y 2 + b2 Similarly, we have x + a ≥ 0, z + c ≥ 0 and thus −(x + a)(z + c) ≤ −y 2 + b2 , as asserted This concludes the proof of the formula for K pc for all parameters a, b, c > 0... Lamination Convex Hull of K for ac − b2 < 0 We now turn towards proving the formula for K lc and we assume first that ac − b2 < 0 We let A = F ∈ conv(K) : |y| = b In this case, none of the matrices in A with y = b is rank-one connected to any of the matrices in A with y = −b, and the assertion follows essentially from the following well-known locality property of the rank-one convex hull Proposition . Dolzmann Variational Metho ds for Crystalline Microstructure - Analysis and Computation 13 Author Georg Dolzmann Department of Mathematics University of Maryland College Par k MD 20742 Maryland,. laws and regulations and therefore free for general use. Typesetting: Camera-ready T E Xoutputbytheauthor SPIN: 10899540 41/3142/ du - 543210 - Printed on acid-free paper A catalog record for. isotropic phase transformation. For this material, Bladon, Teren- tjev and Warner derived a closed formula W BTW for the free energy density which depends on the deformation gradient F and the nematic