In materials that undergo martensitic phase transformation, macroscopic loading often leads to the creation and/or rearrangement of elastic domains. This paper considers an example involving a singlecrystal slab made from two martensite variants. When the slab is made to bend, the two variants form a characteristic microstructure that we like to call “twinning with variable volume fraction.” Two 1996 papers by Chopra et al. explored this example using bars made from InTl, providing considerable detail about the microstructures they observed. Here we offer an energyminimizationbased model that is motivated by their account. It uses geometrically linear elasticity, and treats the phase boundaries as sharp interfaces. For simplicity, rather than model the experimental forces and boundary conditions exactly, we consider certain Dirichlet or Neumann boundary conditions whose effect is to require bending. This leads to certain nonlinear (and nonconvex) variational problems that represent the minimization of elastic plus surface energy (and the work done by the load, in the case of a Neumann boundary condition). Our results identify how the minimum value of each variational problem scales with respect to the surface energy density. The results are established by proving upper and lower bounds that scale the same way. Themore »
Minimizers for the CahnHilliard energy functional under strong anchoring conditions
We study analytically and numerically the minimizers for the CahnHilliard energy functional with a symmetric quartic doublewell potential and under a strong anchoring condition(i.e., the Dirichlet condition) on the boundary of an underlying bounded domain. We show a bifurcation phenomenon determined by the boundary value and a parameter that describes the thickness of a transition layer separating two phases of an underlying system of binary mixtures. For the case that the boundary value is exactly the average of the two pure phases, if the bifurcation parameter is larger than or equal to a critical value, then the minimizer is unique and is exactly the homogeneous state. Otherwise, there are exactly two symmetric minimizers. The critical bifurcation value is inversely proportional to the first eigenvalue of the negative Laplace operator with the zero Dirichlet boundary condition. For a boundary value that is larger (or smaller) than that of the average of the two pure phases, the symmetry is broken and there is only one minimizer. We also obtain the bounds and morphological properties of the minimizers under additional assumptions on the domain.Our analysis utilizes the notion of the Nehari manifold and connects it to the eigenvalue problem for the negative Laplacian more »
 Award ID(s):
 1913144
 Publication Date:
 NSFPAR ID:
 10288371
 Journal Name:
 SIAM journal on applied mathematics
 Volume:
 80
 Issue:
 5
 Page Range or eLocationID:
 22992317
 ISSN:
 00361399
 Sponsoring Org:
 National Science Foundation
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