A three-scale compressible microsphere model for hyperelastic materials

Dal, Hüsnü
MIEHE, Christian
This paper presents a seminumerical homogenization framework for porous hyperelastic materials that is open for any hyperelastic microresponse. The conventional analytical homogenization schemes do apply to a limited number of elementary hyperelastic constitutive models. Within this context, we propose a general numerical scheme based on the homogenization of a spherical cavity in an incompressible unit hyperelastic solid sphere, which is denoted as the mesoscopic representative volume element (mRVE). The approach is applicable to any hyperelastic micromechanical response. The deformation field in the sphere is approximated via nonaffine kinematics proposed by Hou and Abeyaratne (JMPS 40:571-592,1992). Symmetric displacement boundary conditions driven by the principal stretches of the deformation gradient are applied on the outer boundary of the mRVE. The macroscopic quantities, eg,stress and moduli expressions, are obtained by analytically derived pointwise geometric transformations. The macroscopic expressions are then computed numerically through quadrature rules applied in the radial and surface directions of the sphere. A three-scale compressible microsphere model is derived from the developed seminumerical homogenization framework where the micro-meso transition is based on the nonaffine microsphere model at every point of the mRVE. The numerical scheme developed for the derivation of macroscopic homogenized stresses and moduli terms as well as the modeling capability of the three-scale microsphere model is investigated through representative boundary value problems.


A quasi-incompressible and quasi-inextensible element formulation for transversely isotropic materials
Dal, Hüsnü (Wiley, 2019-01-06)
The contribution presents a new finite element formulation for quasi-inextensible and quasi-incompressible finite hyperelastic behavior of transeversely isotropic materials and addresses its computational aspects. The material formulation is presented in purely Eulerian setting and based on the additive decomposition of the free energy function into isotropic and anisotropic parts, where the former is further decomposed into isochoric and volumetric parts. For the quasi-incompressible response, the Q1P0 ele...
A unified approach for the formulation of interaction problems by the boundary element method
Mengi, Y; Argeso, H (Wiley, 2006-04-30)
A unified formulation is presented, based on boundary element method, in a form suitable for performing the interaction analyses by substructure method for solid-solid and soil-structure problems. The proposed formulation permits the evaluation of all the elements of impedance and input motion matrices simultaneously at a single step in terms of system matrices of the boundary element method without solving any special problem, such as, unit displacement or load problem, as required in conventional methods....
A family of second order time stepping methods for the Darcy-Brinkman equations
Cibik, Aytekin; Demir, Medine; Kaya Merdan, Songül (Elsevier BV, 2019-04-01)
This study presents an efficient, accurate, effective and unconditionally stable time stepping scheme for the Darcy-Brinkman equations in double-diffusive convection. The stabilization within the proposed method uses the idea of stabilizing the curvature for velocity, temperature and concentration equations. Accuracy in time is proven and the convergence results for the fully discrete solutions of problem variables are given. Several numerical examples including a convergence study are provided that support...
An analysis of a linearly extrapolated BDF2 subgrid artificial viscosity method for incompressible flows
Demir, Medine (Elsevier BV, 2020-10-01)
This report extends the mathematical support of a subgrid artificial viscosity (SAV) method to simulate the incompressible Navier-Stokes equations to better performing a linearly extrapolated BDF2 (BDF2LE) time discretization. The method considers the viscous term as a combination of the vorticity and the grad-div stabilization term. SAV method introduces global stabilization by adding a term, then anti-diffuses through the extra mixed variables. We present a detailed analysis of conservation laws, includin...
A two-grid stabilization method for solving the steady-state Navier-Stokes equations
Kaya Merdan, Songül (Wiley, 2006-05-01)
We formulate a subgrid eddy viscosity method for solving the steady-state incompressible flow problem. The eddy viscosity does not act on the large flow structures. Optimal error estimates are obtained for velocity and pressure. The numerical illustrations agree completely with the theoretical results. (C) 2005 Wiley Periodicals, Inc.
Citation Formats
H. Dal and C. MIEHE, “A three-scale compressible microsphere model for hyperelastic materials,” INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, pp. 412–433, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/41155.