Paul Steinmann

Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting

The concern of this work is a consequent exploitation of the notion of material forces for the application within hyperelastostatic fracture mechanics. Contrary to physical forces, material forces act on the material manifold, thus essentially representing the tendency of defects like cracks or inclusions to move relative to the ambient material. Based on the formulation of the appropriate quasi-static balance laws in the material space we aim at a fresh look onto classical aspects of hyperelastostatic fracture mechanics. Operating throughout within the geometrically nonlinear setting we emphasize on the one hand the duality of the direct and the inverse motion description and on the other hand we re-establish the classical path integrals from elementary equilibrium considerations in the material space.

On the continuum formulation of higher gradient plasticity for single and polycrystals

This paper develops a geometrically linear formulation of higher gradient plasticity of single and polycrystalline material based on the continuum theory of dislocations and incompatibilities. As a result, a phenomenological but physically motivated description of hardening is obtained, which incorporates for single crystals second order spatial derivatives of the plastic deformation gradient and for polycrystals fourth order spatial derivatives of the plastic strains into the yield condition. Moreover, these modifications mimic the characteristic structure of kinematic hardening, whereby the backstress obeys a nonlocal evolution law. For the one-dimensional example of an infinite shear layer the relation between the characteristic length l and the width w of a localized elasto–plastic shear band is examined in detail for both cases.

Views on multiplicative elastoplasticity and the continuum theory of dislocations

The objective of this contribution is a geometrically non-linear formulation of the continuum theory of dislocations within the framework of multiplicative elastoplasticity at finite strains. Thereby, the continuum theory of dislocations is particularly motivated by the kinematic structure of single crystals. Two different views on the continuum theory of dislocations at finite inelastic strains are adopted. Firstly, different dislocation density tensors are introduced from the viewpoint of continuum mechanics as the incompatibility of the so-called plastic intermediate configuration. Secondly, the continuum theory of dislocations is motivated as a Cartan differential geometry where the corresponding torsion tensor is associated to the dislocation density. Finally, as the main outcome of this contribution, the kinematically necessary dislocation density is considered within the exploitation of the thermodynamical principle of positive dissipation. As a result, a phenomenological description of hardening is obtained, which on the one hand incorporates second spatial derivatives of the plastic deformation gradient into the yield condition and on the other hand mimics the characteristic structure of kinematic hardening.

Application of material forces to hyperelastostatic fracture mechanics. II. Computational setting

Abstract The concern of this work is a novel algorithmic treatment of hyperelastostatic fracture mechanics problems consistent to the notion of material forces within the geometrically nonlinear setting of continuum mechanics. To this end, we consider the continuum mechanics of material forces, as outlined in Part I of this work (P. Steinmann, Int. J. Solid Struct. 37, 7371–7391), which act, contrary to the common physical forces, on the material manifold or rather in the material space. In the sequel it is proposed to discretize the corresponding quasi-static balance of pseudo momentum by a standard Galerkin finite element procedure. As a result we obtain global discrete node point quantities, the material node point forces, which prove to be of the same qualitative and quantitative importance for the assessment of fracture mechanics problems as the classical J -integral.

A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity

The aim of this work is to formulate a geometrically exact theory of finite deformation and finite rotation micropolar elastoplasticity to obtain a generalized nonlinear continuum framework. To this end, the classical deformation map is supplemented by an independent rotation field to yield an enhanced configuration space. Thereby, the rotational part of the formulation is consequently parameterized in terms of the rotation (pseudo) vector via the Euler-Rodrigues formula. Then, micropolar hyperelasticity and multiplicative elastoplasticity are conceptionally derived as in the classical Boltzmann continuum. The proposed theory is consequently developed in a modern geometry oriented fashion. Linearization of the kinematics retrofits the well-known structure of the micropolar geometrically linear theory.

Mechanical properties of gray and white matter brain tissue by indentation

The mammalian brain is composed of an outer layer of gray matter, consisting of cell bodies, dendrites, and unmyelinated axons, and an inner core of white matter, consisting primarily of myelinated axons. Recent evidence suggests that microstructural differences between gray and white matter play an important role during neurodevelopment. While brain tissue as a whole is rheologically well characterized, the individual features of gray and white matter remain poorly understood. Here we quantify the mechanical properties of gray and white matter using a robust, reliable, and repeatable method, flat-punch indentation. To systematically characterize gray and white matter moduli for varying indenter diameters, loading rates, holding times, post-mortem times, and locations we performed a series of n=192 indentation tests. We found that indenting thick, intact coronal slices eliminates the common challenges associated with small specimens: it naturally minimizes boundary effects, dehydration, swelling, and structural degradation. When kept intact and hydrated, brain slices maintained their mechanical characteristics with standard deviations as low as 5% throughout the entire testing period of five days post mortem. White matter, with an average modulus of 1.89 5kPa ± 0.592 kPa, was on average 39% stiffer than gray matter, p<0.01, with an average modulus of 1.389 kPa ± 0.289 kPa, and displayed larger regional variations. It was also more viscous than gray matter and responded less rapidly to mechanical loading. Understanding the rheological differences between gray and white matter may have direct implications on diagnosing and understanding the mechanical environment in neurodevelopment and neurological disorders.

On the numerical treatment and analysis of finite deformation ductile single crystal plasticity

Abstract This contribution is concerned with the numerical analysis and simulation of instability phenomena such as shearband localization in plastic flow processes within the framework of ductile single (fcc) crystal viscoplasticity. To this end, a family of implicit constitutive algorithms and their implementation are considered in detail. Issues of accuracy and conservation of plastic incompressibility are examined. Moreover, the implementation within an overall Newton-Raphson finite element solution strategy relies on the correct evaluation of the algorithmic tangent moduli which result from linearizing the stress update integrator. Likewise, these algorithmic tangent moduli are employed within the numerical analysis of the spatial localization tensor. Examples demonstrating the performance of the proposed strategy at the local constitutive level as well as at the global structural level are given.

Constrained integration of rigid body dynamics

In the present paper rigid body dynamics is formulated as mechanical system with holonomic constraints. This approach offers the appealing possibility to deal with finite rotations without employing any specific rotational parameterization. The numerical discretization of the underlying system of differential algebraic equations is treated in detail. The proposed algorithm obeys major conservation laws of the underlying continuous system such as conservation of energy and angular momentum. In addition to that, the constraints on the configuration and momentum level are fulfilled exactly. Two numerical examples are dealt with to assess the performance of the constrained algorithm.

Theoretical and computational aspects of a thermodynamically consistent framework for geometrically linear gradient damage

This paper presents the theory and the numerics of an isotropic gradient damage formulation within a thermodynamical background. The main motivation is provided by localization computations whereby classical local continuum formulations fail to produce physically meaningful and numerically converging results. We propose a formulation in terms of the Helmholtz free energy incorporating the gradient of the damage field, a dissipation potential and the postulate of maximum dissipation. As a result, the driving force conjugated to damage evolution is identified as the quasi-nonlocal energy release rate, which essentially incorporates the divergence of a vectorial damage flux besides the strictly local energy release rate. On the numerical side, besides balance of linear momentum, the algorithmic consistency condition must be solved in weak form. Thereby, the crucial issue is the selection of active constraints which is solved by an active set search algorithm borrowed from convex nonlinear programming. In the examples, we compare the behavior in local damage with the performance of the gradient formulation.

Thermomechanics of Solids With Lower-Dimensional Energetics: On the Importance of Surface, Interface, and Curve Structures at the Nanoscale. A Unifying Review

Surfaces and interfaces can significantly influence the overall response of a solid body. Their behavior is well described by continuum theories that endow the surface and interface with their own energetic structures. Such theories are becoming increasingly important when modeling the response of structures at the nanoscale. The objectives of this review are as follows. The first is to summarize the key contributions in the literature. The second is to unify a select subset of these contributions using a systematic and thermodynamically consistent procedure to derive the governing equations. Contributions from the bulk and the lower-dimensional surface, interface, and curve are accounted for. The governing equations describe the fully nonlinear response (geometric and material). Expressions for the energy and entropy flux vectors, and the admissible constraints on the temperature field, all subject to the restriction of non-negative dissipation, are explored. A particular emphasis is placed on the structure of these relations at the interface. A weak formulation of the governing equations is then presented that serves as the basis for their approximation using the finite element method. Various forms for a Helmholtz energy that describes the fully coupled thermomechanical response of the system are given. They include the contribution from surface tension. The vast majority of the literature on surface elasticity is framed in the infinitesimal deformation setting. The finite deformation stress measures are, thus, linearized and the structure of the resulting stresses discussed. The final objective is to elucidate the theory using a series of numerical example problems.