Thermo-mechanical coupling of transversely isotropic materials using high-order finite elements

Constitutive modeling and numerical analysis of the behavior of anisotropic materials, particularly transversely isotropic and orthotropic materials, attained increasing attention in the last few years. The attention is motivated by the wide range of applications of these materials in engineering industries and biomedical technologies. This work aims to develop a constitutive model for transversely isotropic materials undergoing thermo-mechanically coupled finite deformations. The model is based on the idea of multiplicative decomposition of the deformation gradient. Furthermore, making use of high-order finite elements, the capability of the model to simulate the behavior of transversely isotropic material under isothermal and thermo-mechanically coupled loadings is demonstrated by performing some numerical experiments. First of all, a constitutive model for the case of isothermal transversal isotropy is formulated. The proposed model is an extension of the volumetric/isochoric decoupling of the deformation gradient, where the isochoric part is decomposed into two parts, one part containing only the deformation along the preferred direction, while all remaining deformations are included in the other part. This formulation has the advantage that it leads to a clear split of the stress-state, i.e., the stress along the preferred direction is splitted from the remaining stresses. Additionally, the proposed model overcomes the obstacle related to the application of volumetric/isochoric decomposition to anisotropy. The formulation is, then, extended to the case of thermo-mechanically coupled problem, where a thermodynamically consistent constitutive model for transversal isotropy is developed. Moreover, a directionally dependent, i.e. transversely isotropic heat flux vector is derived, which takes into consideration the anisotropy in heat conductivity. The proposed model is implemented into a high-order finite element code, in which the p-version finite element method (p-FEM) and the high-order diagonally implicit Runge-Kutta (DIRK) methods are used for the spatial and time discretizations, respectively. In p-FEM the accuracy of the solution is improved by increasing the polynomial degree of the elements, and this makes p- FEM more convenient for the analysis of thin structures, like in the case of laminated composites. Thus, computations are carried out in order to investigate the behavior of the proposed model with different numerical examples. To this end, the influence of different factor, namely, existence of anisotropy, orientation of the preferred direction, anisotropic thermal expansion as well as anisotropic heat conductivity, on the response of transversely isotropic material under isothermal and/or thermo-mechanical loadings is discussed. Furthermore, the efficiency of the p-version implementations is demonstrated by comparing it with two different h-version finite element implementations

A virtual element method for hyperelasticity

This thesis studies the approximation of plane problems of hyperelasticity, using a loworder virtual element method (VEM). The VEM is an extension of the finite element method (FEM). It is characterised by considerable freedom with regard to element geometry, permitting arbitrary polygonal and polyhedral elements in two and three dimensions respectively. Furthermore, the local basis functions are not known explicitly on elements and take the simple form of piecewise-linear Lagrangian functions on element boundaries. All integrations are performed on element edges. The VEM formulation typically involves a consistency term, computed via a projection, and a stabilization term, which must be approximated. Problems concerning isotropic and transversely isotropic hyperelastic material models are considered. Examples of transversely isotropic materials, which are characterised by an axis of symmetry normal to a plane of isotropy, range from simple fibre-reinforced materials to biological tissues. To date, in the context of hyperelasticity, investigation of the performance of VEM has primarily focused on problems involving the isotropic neo-Hookean material model. Furthermore, there has been limited investigation into the behaviour of the VEM in the nearly incompressible and nearly inextensible limits. In this thesis a VEM formulation with a novel approach to the construction of the stabilization term is formulated and implemented for problems involving isotropic and transversely isotropic hyperelastic materials. The governing equations of hyperelasticity are derived and various isotropic and transversely isotropic constitutive models are presented. This is followed by presentation of the virtual element formulation of the hyperelastic problem and a possible approach to its practical implementation. Through a range of numerical examples, the VEM with the proposed stabilization term is found to exhibit robust and accurate behaviour for a variety of mesh types, including those comprising highly non-convex element geometries, and for problems involving severe deformations. Furthermore, the versatility of the proposed VEM formulation is demonstrated through its application to a range of popular isotropic and transversely isotropic material models for a wide variety of material parameters. Through this investigation the VEM is found to exhibit locking-free behaviour in the limiting cases of near-incompressibility and near-inextensibility, both separately and combined

Polymeric and Microrheological Characterization of the Staphylococcus epidermidis Biofilm Polysaccharide.

In this dissertation, we characterize the polymeric and rheological properties of polysaccharide intercellular adhesin (PIA) present within the matrix of extracellular polymers (EPS) in biofilms formed by Staphylococcus epidermidis. Biofilms are viscoelastic soft matter consisting of bacterial aggregates embedded within the EPS. The EPS predominantly contains polysaccharides, in addition to proteins and DNA. S. epidermidis biofilms frequently contaminate medical implants resulting in blood stream infections. In such cases, biofilm formation by S. epidermidis and resistance to blood shear stresses is attributed to the presence of PIA. Using techniques of chromatography, light scattering, microrheology and colloidal physics, we understand the contribution of PIA towards biofilm viscoelasticity. We identified that PIA exhibits self-associations and complexation with proteins in dilute solutions. At concentrations found within shaker grown biofilms, PIA displayed a viscoelastic rheology. We extracted concentration dependent scaling laws for zero-shear viscosity and the creep compliance of PIA. A polymeric composite consisting of PIA, bovine serum albumin and lambda DNA, simulating the biofilm EPS, was 50-fold more elastic than PIA. However, we found that the EPS polymers, on their own, do not generate the elasticity of mature biofilms. To understand this gap, we report the self-assembly of artificial biofilms using planktonic cells and the polysaccharide chitosan as a proxy for PIA. We report that the elasticity or the long time plateau in creep compliance, in the artificial biofilms is mediated by pH induced phase instability of chitosan. Using this finding, we showed that increasing the pH of the solvent environment resulted in softening of a S. epidermidis biofilm within 4 hours. To support development of medical procedures aimed at biofilm softening, we provide a theoretical derivation to extend the applicability of cavitation rheometry, to measure elastic modulus of materials as small as few microliters, similar to biofilms formed in infection sites. Collectively, the study provides structural and rheological characterization of biofilm exopolymers that supports multi-component constitutive modeling of biofilms and development of treatment techniques that target biofilm disassembly through physicochemical changes to the growth environment.PHDChemical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111461/1/maheshg_1.pd

Least-squares mixed finite elements for geometrically nonlinear solid mechanics

The computation of reliable results using finite elements is a major engineering goal. Under the assumption of a linear elastic theory many stable and reliable (standard and mixed) finite elements have been developed. Unfortunately, in the geometrically nonlinear regime, e.g. applying these elements in the field of incompressible, hyperelastic materials, problems can occur. A possible approach to circumvent these issues might be the least-squares mixed finite element method. Therefore, in this thesis, a mixed least-squares formulation for hyperelastic materials in the field of solid mechanics is provided, investigated and valuated. To create a theoretical basis the continuum mechanical background is outlined, the necessary physical quantities are introduced and the construction of suitable interpolation functionsis shown. Furthermore, the general procedure for the construction of a least-squares functional is described and applied for hyperelastic material laws based on a free energy function. Basis for the proposed least-squares element formulation is a div-grad first-order system consisting of the equilibrium condition, the constitutive equation and a stress symmetry condition, all written in a residual form. The solution variables (displacements and stresses) are, dependent on the element type, interpolated using different approximation spaces. The performance of the provided elements is investigated and compared to standard and mixed Galerkin elements by extensive numerical studies with respect to e.g. bending dominated problems, incompressibility, stability issues, convergence of the field quantities and adaptivity. Furthermore, the crucial influence of weighting is discussed. Finally, the results are evaluated and the used elements are assessed.Ein Hauptziel im Bereich des Ingenieurwesens ist die Berechnung vertrauenswürdiger Ergebnisse mit Hilfe der Methode der finiten Elemente. Unter Annahme einer linear elastischen Theorie wurden hierzu bereits viele stabile und zuverlässige standard und gemischte finite Elemente entwickelt. Es hat sich jedoch herausgestellt, dass bei einigen dieser Elemente, unter anderem angewandt auf inkompressible, hyperelastische Materialien, Probleme auftreten. Ein möglicher Ansatz um diese Probleme zu umgehen ist möglicherweise die gemischte least-squares finite Elemente Methode. Daher wird in Rahmen dieser Arbeit eine gemischte least-squares Formulierung für hyperelastische Materialien vorgestellt, untersucht und bewertet. Um eine theoretische Basis zu schaffen wird zuerst ein kontinuumsmechanischer Rahmen geschaffen, die nötigen physikalischen Größen werden eingeführt und die Konstruktion geeigneter Interpolationsfunktionen wird gezeigt. Im Folgenden wird das allgemeine Vorgehen zur Konstruktion eines least-squares Funktionals beschrieben und angewandt auf hyperelastische Materialien in der Festkörpermechanik basierend auf freien Energiefunktionen. Die Basis für die least-squares Formulierung stellt ein div-grad System erster Ordnung dar, bestehend aus der Gleichgewichtsbedingung, einem Materialgesetz und einer zusätzlichen Bedingung für die Einhaltung einer Spannungssymmetrie. Die Gleichungen liegen hierbei in einer residualen Form vor. Die Lösungsvariablen sind, im Rahmen dieser Arbeit, die Verschiebungen und die Spannungen welche, abhängig vom Elementtyp, mit unterschiedlichen Interpolationsfunktionen interpoliert werden. Die Performanz der entwickelten Elemente wird im Folgenden mit extensiven numerischen Studien untersucht, welche sich unter anderem mit biegedominierten Problemen, Inkompressibilität, Untersuchung von Stabilitätspunkten und der allgemeinen Konvergenz der Lösungsvariablen beschäftigen. Zur Bewertung der Ergebnisse werden diese mit Lösungen verglichen, welche durch standard und gemischte Galerkin Elemente berechnet wurden. Darüber hinaus wird der starke Einfluss der Wichtungsfaktoren auf die Qualität der Lösungen diskutiert. Abschließend werden die Ergebnisse ausgewertet und die entwickelten Elemente bewertet

Peridynamics review

Peridynamics (PD) is a novel continuum mechanics theory established by Stewart Silling in 2000 [1]. The roots of PD can be traced back to the early works of Gabrio Piola according to dell'Isola et al. [2]. PD has been attractive to researchers as it is a nonlocal formulation in an integral form; unlike the local differential form of classical continuum mechanics. Although the method is still in its infancy, the literature on PD is fairly rich and extensive. The prolific growth in PD applications has led to a tremendous number of contributions in various disciplines. This manuscript aims to provide a concise description of the peridynamic theory together with a review of its major applications and related studies in different fields to date. Moreover, we succinctly highlight some lines of research that are yet to be investigated

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

A variational framework for mathematically nonsmooth problems in solid and structure mechanics

This dissertation presents a new paradigm for addressing multi-physics problems with interfaces in the field of Additive Manufacturing and the modeling of fibrous composite materials. The unique process of adding the material layer by layer in the AM techniques raises the issue about the stability of the interfaces between the layers and along the boundaries of multi-constituent materials. A stabilized interface formulation is developed to model debonding in monotonic loading, fatigue effects in cyclic loading, and thermal effects at interfaces which severely impact the functional life of those materials and structures. The formulation is based on embedding Discontinuous Galkerin (DG) ideas in a Continuous Galerkin (CG) framework. Starting from a mixed method incorporating the Lagrange multiplier along the interface, a pure displacement formulation is derived using the Variational Multiscale Method (VMS). From a mathematical and computational perspective, the key factor influencing the accuracy and robustness of the interface formulation is the design of the numerical flux and the penalty or stability terms. Analytical expressions that are free from user-defined parameters are naturally derived for the numerical flux and stability tensor which are functions of the evolving geometric and material nonlinearity. The proposed framework is extended for debonding at finite strains across general bimaterial interfaces. An interfacial gap function is introduced that evolves subject to constraints imposed by opening and/or sliding interfaces. An internal variable formalism is derived together with the notion of irreversibility of damage results in a set of evolution equations for the gap function that seamlessly tracks interface debonding by treating damage and friction in a unified way. Tension debonding, compression damage, and frictional sliding are accommodated, and return mapping algorithms in the presence of evolving strong discontinuities are developed. This derivation variationally embeds the interfacial kinematic models that are crucial to capturing the physical and mathematical properties involving large strains and damage. The framework is extended for monolithic coupling of thermomechanical fields in the class of problems that have embedded weak and strong discontinuities in the mechanical and thermal fields. Since the derived expressions are a function of the mechanical and thermal fields, the resulting stabilized formulation contains numerical flux and stability tensors that provide an avenue to variationally embed interfacial kinetic and kinematic models for more robust representation of interfacial physics. Representative numerical tests involving large strains and rotations, damage phenomena, and thermal effects are performed to confirm the robustness and accuracy of the method. Comparison of the results with both experimental and numerical results from literature are presented.Ope

Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods

Discontinuous Petrov-Galerkin (DPG) finite element methods have garnered significant attention since they were originally introduced. They discretize variational formulations with broken (discontinuous) test spaces and are crafted to be numerically stable by implicitly computing a near-optimal discrete test space as a function of a discrete trial space. Moreover, they are completely general in the sense that they can be applied to a variety of variational formulations, including non-conventional ones that involve non-symmetric functional settings, such as ultraweak variational formulations. In most cases, these properties have been harnessed to develop numerical methods that provide robust control of relevant equation parameters, like in convection-diffusion problems and other singularly perturbed problems. In this work, other features of DPG methods are systematically exploited and applied to different problems. More specifically, the versatility of DPG methods is elucidated by utilizing the underlying methodology to discretize four distinct variational formulations of the equations of linear elasticity. By taking advantage of interface variables inherent to DPG discretizations, an approach to coupling different variational formulations within the same domain is described and used to solve interesting problems. Moreover, the convenient algebraic structure in DPG methods is harnessed to develop a new family of numerical methods called discrete least-squares (DLS) finite element methods. These involve solving, with improved conditioning properties, a discrete least-squares problem associated with an overdetermined rectangular system of equations, instead of directly solving the usual square systems. Their utility is demonstrated with illustrative examples. Additionally, high-order polygonal DPG (PolyDPG) methods are devised by using the intrinsic discontinuities present in ultraweak formulations. The resulting methods can handle heavily distorted non-convex polygonal elements and discontinuous material properties. A polygonal adaptive strategy was also proposed and compared with standard techniques. Lastly, the natural high-order residual-based a posteriori error estimator ingrained within DPG methods was further applied to problems of physical relevance, like the validation of dynamic mechanical analysis (DMA) calibration experiments of viscoelastic materials, and the modeling of form-wound medium-voltage stator coils sitting inside large electric machinery.Computational Science, Engineering, and Mathematic

A hp-adaptive discontinuous Galerkin finite element method for accurate configurational force brittle crack propagation

Engineers require accurate determination of the configurational force at the crack tip for fracture fatigue analysis and accurate crack propagation. However, obtain- ing highly accurate crack tip configuration force values is challenging with numer- ical methods requiring knowledge of the stress field around the crack tip a priori. In this thesis, the symmetric interior penalty discontinuous Galerkin finite element method is combined with a residual based a posteriori error estimator which drives a hp-adaptive mesh refinement scheme to determine accurate solutions of the stress field about about the crack. This facilitates the development of a novel method to calculate the crack tip configurational force that is accurate, requires no a priori knowledge of the stress field about the crack tip with, its error bound by an error estimator which is calculated a posteriori. Benchmark values of the crack tip con- figurational force are presented for problems containing multiple mixed mode cracks in both isotropic and anisotropic materials. Additionally, the hp-adaptivity is com- bined with a mathematical analysis of the stress field at the crack tip to critique the convergence and limitations of other methods in the literature to calculate the crack tip configurational force. Two methods for staggered quasi-static crack prop- agation are also presented. An rp-adaptive method which is simple to implement and computationally inexpensive, element edges aligned with the crack propagation path with the exploitation of the discontinuous Galerkin edge sti↵ness terms exist- ing along element interfaces to propagate a crack. The second method is denoted the hpr-adaptive method which combines the accurate computation of the crack tip configuration force with r-adaptivity to produce a computationally expensive but accurate method to propagate multiple cracks simultaneously. Further, for indeter- minate systems, an average boundary condition that restrains rigid body motion and rotation is introduced to make the system determinate