Critical time step for a bilinear laminated composite Mindlin shell element.

Abstract

The critical time step needed for explicit time integration of laminated shell finite element models is presented. Each layer is restricted to be orthotropic when viewed from a properly oriented material coordinate system. Mindlin shell theory is used in determining the laminated response that includes the effects of transverse shear. The effects of the membrane-bending coupling matrix from the laminate material model are included. Such a coupling matrix arises even in the case of non-symmetric lay-ups of differing isotropic layers. Single point integration is assumed to be used in determining a uniform strain response from the element. Using a technique based upon one from the literature, reduced eigenvalue problems are established to determine the remaining non-zero frequencies. It is shown that the eigenvalue problem arising from the inplane normal and shear stresses is decoupled from that arising from the transverse shear stresses. A verification example is presented where the exact and approximate results are compared.

Adagio 4.18 user's guide.

“…The strain error convergence measure is based on a strain quantity E I that is computed as the difference between the scaled stress and the unscaled stress and dividing that value by a modulus: 16) where M is an appropriate material modulus that is used to represent the unscaled constitutive response. For the case where the true material response is linear and the scaled stress is computed by stiffening the material response, E I as defined by Equation (3.16) corresponds exactly to the difference between the kinematic strain e I and the strain ǫ I needed in the unscaled constitutive equation to give the scaled stress: 17) where M would be the real material modulus.…”

Adagio 4.14 user<U+2019>s guide.

“…• Chapter 1 presents the overall structure of the input file, including conventions for the command descriptions, style guidelines for file preparation, and naming conventions for input files that reference the Exodus II database [11]. The chapter also gives an example of the general structure of an input file that employs the concept of scope.…”

“…Let us begin with the true material response in model problem I given by 11) where σ I is the true stress corresponding to the kinematic strain e I determined from the nodal displacements, and f (·) is the function representing the constitutive response. Although the true material response has been written for the hyperelastic case employing total stress and strain quantities, the control stiffness algorithms that are being presented can be equally applied to hypoelastic models where the stress rate is written in terms of the strain rate.…”

Laminated composites modeling in ADAGIO/PRESTO.

“…However, when computing a critical time step for PRESTO, these transverse shear correction factors need to be explicitly taken into account. 1 …”

“…A complete discussion of using these shear correction factors is given in Ref. 1. However, for the present discussion, it suffices to point out that 5/6 is the traditional value used to give the same strain energy as a parabolic distribution of transverse shear stresses and strains, whereas the 6h 2 /L 2 value is used to recover Kirchhoff bending behavior for thin shell elements without shear locking the element.…”