Günther Meschke

Constitutive modeling of crimped collagen fibrils in soft tissues

A microstructurally oriented constitutive formulation for the hyperelastic response of crimped collagen fibrils existing in soft connective tissues is proposed. The model is based on observations that collagen fibrils embedded in a soft matrix crimp into a smooth three-dimensional pattern when unloaded. Following ideas presented by Beskos and Jenkins [Beskos, D., Jenkins, J., 1975. A mechanical model for mammalian tendon. ASME Journal of Applied Mechanics 42, 755-758] and Freed and Doehring [Freed, A., Doehring, T., 2005. Elastic model for crimped collagen fibrils. Journal of Biomechanical Engineering 127, 587-593] the collagen fibril crimp is approximated by a cylindrical helix to represent the constitutive behavior of the hierarchical organized substructure of biological tissues at the fibrillar level. The model is derived from the nonlinear axial force-stretch relationship of an extensible helical spring, including the full extension of the spring as a limit case. The geometrically nonlinear solution of the extensible helical spring is carried out by an iterative procedure. The model only requires one material parameter and two geometrical parameters to be determined from experiments. The ability of the proposed model to reproduce the biomechanical response of fibrous tissues is demonstrated for fascicles from rat tail tendons, for porcine cornea strips, and for bovine Achilles tendons.

The collagen fibril architecture in the lamina cribrosa and peripapillary sclera predicted by a computational remodeling approach

The biomechanics of the optic nerve head is assumed to play an important role in ganglion cell loss in glaucoma. Organized collagen fibrils form complex networks that introduce strong anisotropic and nonlinear attributes into the constitutive response of the peripapillary sclera (PPS) and lamina cribrosa (LC) dominating the biomechanics of the optic nerve head. The recently presented computational remodeling approach (Grytz and Meschke in Biomech Model Mechanobiol 9:225–235, 2010) was used to predict the micro-architecture in the LC and PPS, and to investigate its impact on intraocular pressure–related deformations. The mechanical properties of the LC and PPS were derived from a microstructure-oriented constitutive model that included the stretch-dependent stiffening and the statistically distributed orientations of the collagen fibrils. Biomechanically induced adaptation of the local micro-architecture was captured by allowing collagen fibrils to be reoriented in response to the intraocular pressure–related loading conditions. In agreement with experimental observations, the remodeling algorithm predicted the existence of an annulus of fibrils around the scleral canal in the PPS, and a predominant radial orientation of fibrils in the periphery of the LC. The peripapillary annulus significantly reduced the intraocular pressure–related expansion of the scleral canal and shielded the LC from high tensile stresses. The radial oriented fibrils in the LC periphery reinforced the LC against transversal shear stresses and reduced LC bending deformations. The numerical approach presents a novel and reasonable biomechanical explanation of the spatial orientation of fibrillar collagen in the optic nerve head.

A computational remodeling approach to predict the physiological architecture of the collagen fibril network in corneo-scleral shells

Organized collagen fibrils form complex networks that introduce strong anisotropic and highly nonlinear attributes into the constitutive response of human eye tissues. Physiological adaptation of the collagen network and the mechanical condition within biological tissues are complex and mutually dependent phenomena. In this contribution, a computational model is presented to investigate the interaction between the collagen fibril architecture and mechanical loading conditions in the corneo-scleral shell. The biomechanical properties of eye tissues are derived from the single crimped fibril at the micro-scale via the collagen network of distributed fibrils at the meso-scale to the incompressible and anisotropic soft tissue at the macro-scale. Biomechanically induced remodeling of the collagen network is captured on the meso-scale by allowing for a continuous re-orientation of preferred fibril orientations and a continuous adaptation of the fibril dispersion. The presented approach is applied to a numerical human eye model considering the cornea and sclera. The predicted fibril morphology correlates well with experimental observations from X-ray scattering data.

A new class of algorithms for classical plasticity extended to finite strains. Application to geomaterials

A recently proposed methodology for computational plasticity at finite strains is re-examined within the context of geomechanical applications and cast in the general format of multi-surface plasticity. This approach provides an extension to finite strains of any infinitesimal model of classical plasticity that retains both the form of the yield criterion and the hyperelastic character of the stress-strain relations. Remarkably, the actual algorithmic implementation reduces to a reformulation of the standard return maps in principal axis with algorithmic elastoplastic moduli identical to those of the infinitesimal theory. New results in the area of geomechanics included a fully implicit return map for the modified Cam-Clay model, extended here to the finite deformation regime, and a new semi-explicit scheme that restores symmetry of the algorithmic moduli while retaining the unconditional stability property. In addition, a new phenomenological plasticity model for soils is presented which includes a number of widely used models as special cases. The general applicability of the proposed methodology is illustrated in several geomechanical examples that exhibit localization and finite deformations.

A new 3-D finite element model for cord-reinforced rubber composites: application to analysis of automobile tires

Abstract A 3-D finite element model for cord-reinforced rubber composites is proposed. It is characterized by superimposing so-called “rebar” elements, consisting of one or more reinforcing cord layers with arbitrary orientation, on corresponding 3-D rubber elements. The rubber material and the different cord materials are represented independently. A Lagrange multiplier method is employed for the large strain analysis of rubber which is modelled by the incompressible Mooney material law. The compressible Neo-Hookean material law is adopted for the cords (rebar elements). It is modified for the special case of uniaxial stress states. The proposed method provides a realistic representation of cord-reinforced rubber composites while minimizing the necessary discretization effort. It is applied to 3-D finite element analysis of an automobile tire, involving determination of the pressure distribution in the contact zone and of the radial load-displacement curve. Very good agreement between analysis predictions and experimental results has been obtained.

Environmentally induced deterioration of concrete: physical motivation and numerical modeling

This paper is concerned with the effects of moisture, heat and chemical dissolution processes on the long-term behavior of concrete structures. Motivated by experimental findings described in this paper, numerical models for durability analysis of concrete structures, taking hygrally, thermally and chemically induced degradation processes into account, are described. As an example for chemically corrosive mechanisms in concrete structures, the material degradation due to calcium leaching and mechanical loading is considered in a coupled chemo-mechanical model. Furthermore, the interactions between mechanically induced damage, moisture and heat transport are taken into account in a coupled hygro-thermo-mechanical model for concrete. Results from two-dimensional simulations of a concrete panel subjected to coupled mechanical, hygral and thermal loading and coupled chemo-mechanical loading, respectively, are included as representative numerical examples.

3D Simulations of Automobile Tires: Material Modeling, Mesh Generation, and Solution Strategies

Abstract The paper addresses finite element formulations for automobile tires considering nonlinear material behavior, large strains, and finite deformations while using efficient computational str...

Large-strain 3D-analysis of fibre-reinforced composites using rebar elements: hyperelastic formulations for cords

The well-known finite element representation of reinforcing bars by means of overlay (“rebar”) elements is recast in the context of finite strain analyses of cord-reinforced composite materials. The variational formulation including the linearized forms is presented on the basis of hyperelasticity. Three material laws including two variants of the Neo-Hooekean model and the quadratic logarithmic model are investigated. An explicit formulation for uniaxial stress states is given for the Neo-Hooekean model. A comparative evaluation with regards to computational efficiency and physical plausibility shows that the logarithmic model is optimally suited for this class of problems and for moderately large strains. The rebar-element concept in conjunction with an incompressible finite element formulation for the representation of a rubber matrix material is applied to comparative finite strain FE-analyses of a cord-reinforced rubber sandwich panel, with different hyperelastic models used for modelling of the ply material.

Numerical modeling of concrete cracking

M. Jirasek: Damage and smeared crack models. - I. Carol, A. Idiart, C. Lopez, A. Caballero: Cracking and fracture of concrete at meso-level using zero-thickness interface elements. - A. E. Uespe, J. Oliver: Crack models with embedded discontinuities. - G. Hofstetter, C. Feist, H. Lehar, Y. Theiner, B. Valentini, B. Winkler: Plasticity based crack models and applications. - N. Moes: Crack models based on the extended finite element method. - G. Meschke, S. Grasberger, C. Becker, S. Jox: Smeared crack and X-FEM models in the context of poromechanics.

A re-formulation of the exponential algorithm for finite strain plasticity in terms of cauchy stresses

Abstract The role of the stress measure to be chosen as the argument in the definition of yield functions is discussed in the context of finite strain plasticity theory. Motivated by physical arguments, the exponential algorithm for multiplicative finite strain plasticity is revisited such that Cauchy stresses are adopted as arguments in the yield function. Using logarithmic strain measures, the return map algorithm is formulated in principal axes. The algorithmic tangent moduli are obtained in a slightly modified, unsymmetric format compared to the standard formulation in terms of Kirchhoff stresses. However, the global structure of the exponential algorithm is unchanged. The algorithm is applied to the re-formulation of the Cam—Clay model in terms of Cauchy stresses. The typical calibration procedure of the Cam—Clay model based on Cauchy stresses is demonstrated. As an alternative, a modification of the Cam—Clay model, which allows re-calibration of Cauchy stress-based test data to be used within the framework of a Kirchhoff-based finite strain model is also discussed. The relevance of the adequate choice of the stress measure is illustrated by means of selected numerical analyses.