These were formulated as concretely as possible.
The problem is formulated as three separate domains: the elastic regime, mixed elastic plastic behavior, and unconstrained (fully plastic) flow.
As an original contribution, the potential energy of the saturated materials weakened by closed frictional microcracks is determined and formulated as a sum of an elastic part and a plastic part, the latter entirely induced by frictional sliding of microcracks.
Similar to error-free cases, we show that the sparse principal component for the latent variables can be formulated as a bias-corrected lasso (elastic net) regression problem based on the observed surrogates, efficient algorithms are also developed to implement the procedure.
For thin cracks, the effective elastic constants are formulated as a function of the crack density parameter (O'Connell and Budiansky 1974).
Elastic (Euler) buckling is formulated as an eigenvalue problem with ([K] + lambda_{i} [S]){ varPsi_{i} }, =, 0. (1).
In the simple cases of waves in a straight beam and axisymmetric waves in a thin elastic cylindrical shell, these conditions are formulated as identities in an explicit form via wavenumbers.
The secondary elastic user rate allocation problem is formulated as the following optimization problem, which maximizes a certain function of secondary link transmission rate: max P i, R i f ( R i ), (1) s.t.
With the ground modeled as a uniform elastic half-space, the inverse analysis of elastic waves scattered by a three-dimensional void is formulated as a task of minimizing the misfit between experimental observations and theoretical predictions for an assumed void geometry.
The NPR metamaterial design problem is recast as a variational problem, where the effective elastic properties of the spatially periodic microstructure are formulated as the strain energy functionals under uniform displacement boundary conditions.
The tumor organoid culture medium was formulated as described before38.
