Transverse wrinkles usually emerge in a uniaxially stretched elastic film and can be suppressed upon further tension, which is an instability-restabilization behavior due to the nonlinear competition between stretching energy and bending energy. Here, we show that curvature can effectively and precisely tune the wrinkling localization and amplitude.
Prescribing Surface Strains to change Gauss curvature
To change Gauss curvature K of a surface we need to strain each differential shell element by virtue of Egregium theorem ( K is invariant if strain is zero in isometry mappings).
Can someone help with a geometrical problem where imposed strains are to be defined (at differential shell element level of a right circular cylinder) to obtain a surface of revolution with Gauss curvature +1 or -1 (sphere or pseudosphere) ? Or their isometric equivalents? Cylindrical coordinates may be used.
Amit Acharya Marta Lewicka Mohammad Reza Pakzad
In Nonlinearity, 29, 1769-1797
We study a class of design problems in solid mechanics, leading to a variation on the
classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new
context, we derive a necessary and sufficient existence condition, given through a system of total
diff erential equations, and discuss its integrability. In the classical context, the same approach
yields conditions of immersibility of a given metric in terms of the Riemann curvature tensor.
In the present situation, the equations do not close in a straightforward manner, and successive
diff erentiation of the compatibility conditions leads to a more sophisticated algebraic description
of integrability. We also recast the problem in a variational setting and analyze the infi mum value
of the appropriate incompatibility energy, resembling "non-Euclidean elasticity". We then derive a
Γ-convergence result for the dimension reduction from 3d to 2d in the Kirchhoff energy scaling
regime. A practical implementation of the algebraic conditions of integrability is also discussed.
We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motions of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of the displacement gradient and the existence of stress functions on non-contractible bodies.
We discuss the relevance of non-metricity in a metric-affine manifold (a manifold equipped with a connection and a metric) and the nonlinear mechanics of distributed point defects. We describe a geometric framework in which one can calculate analytically the residual stress field of nonlinear elastic solids with distributed point defects. In particular, we use Cartan's machinery of moving frames and construct the material manifold of a finite ball with a spherically-symmetric distribution of point defects.
In the theory of dislocations, the Burgers vector is usually defined by referring to a crystal structure. Using the notion of affine development of curves on a differential manifold with a connection, we give a differential geometric definition of the Burgers vector directly in the continuum setting, without making use of an underlying crystal structure.
In this paper we formulate a geometric theory of thermal stresses.
Given a temperature distribution, we associate a Riemannian
material manifold to the body, with a metric that explicitly
depends on the temperature distribution. A change of temperature
corresponds to a change of the material metric. In this sense, a
temperature change is a concrete example of the so-called
referential evolutions. We also make a concrete connection between
our geometric point of view and the multiplicative decomposition
This paper studies the invariance of balance of
energy for a system of interacting particles under groups of
transformations. Balance of energy and its invariance is first
examined in Euclidean space. Unlike the case of continuous media,
it is shown that conservation and balance laws do not follow
from the assumption of invariance of balance of energy under
time-dependent isometries of the ambient space. However, the
postulate of invariance of balance of energy under arbitrary
