Anisotropic Continuum Research Articles

Anisotropic Continuum-Molecular Models: A Unified Framework Based on Pair Potentials for Elasticity, Fracture and Diffusion-Type Problems

This paper presents a unified framework for continuum-molecular modeling of anisotropic elasticity, fracture and diffusion-based problems within a generalized two-dimensional peridynamic theory. A variational procedure is proposed to derive the governing equations of the model, that postulates oriented material points interacting through pair potentials from which pairwise generalized actions are computed as energy conjugates to properly defined pairwise measures of primary field variables. While mass is considered as continuous function of volume, we define constitutive laws for long-range interactions such that the overall anisotropic behavior of the material is the result of the assigned elastic, conductive and failure micro-interaction properties. The non-central force assumption in elasticity, together with the definition of specific orientation-dependent micromoduli functions respecting material symmetries, allow to obtain a fully anisotropic non-local continuum using a purely pairwise description of deformation and constitutive properties. A general and consistent micro-macro moduli correspondence principle is also established, based on the formal analogy with the classic elastic and conductivity tensors. The main concepts presented in this work can be used for further developments of anisotropic continuum-molecular formulations to include other mechanical behaviors and coupled phenomena involving different physics.

A high-efficient anisotropic continuum model for the optimization of heat transfer and chemical reaction in a packed-bed water gas shift reactor

The packed bed reactor with the high-temperature shift reaction has been a competitive industrial device for the production of hydrogen fuel. While the temperature distribution in the packed bed plays an important role in improving the reaction efficiency, and numerical simulation is a powerful way to recover the heat transfer inside the reactor. Nevertheless, the current simulation methods are either high computational cost or low accuracy for the high-temperature shift. To accomplish the effective computational cost and accuracy simultaneously, we developed a two-dimensional anisotropic continuum model with modified space-dependent effectiveness factor and anisotropic thermal conduction based on the pseudo continuum model. Meanwhile, the simulation results were compared with those by the particle-resolved 3D computational fluid dynamics to validate the model accuracy. Due to the overestimation of the radial thermal conductivity by the pseudo continuum model, the axial temperature rise in the packed bed was only 10 K, but a significant temperature rise of 50 K was presented by the other two methods, suggesting the coordinate computational accuracy of the anisotropic continuum model. However, the computational time consumed by the particle-resolved 3D computational fluid dynamics was up to 1320 min, which exceeds a thousand times more than that in current advanced model. All the comparisons indicate the anisotropic continuum model can predict the heat transfer through the packed bed with sufficient computational accuracy and efficiency, making a powerful and time-saving method to conduct the effective heat management in packed bed reactors for actual industrial applications.

An FEM-based homogenization method for orthogonal lattice metamaterials within micropolar elasticity

Lattice metamaterials are widely used in weight-critical applications and gain more design freedoms with the development of additive manufacturing technology, which requires new accurate homogenization methods to model the mechanical behavior of complex microstructures. In the present work, a new homogenization method based on finite element methods was proposed within the frame of the anisotropic micropolar continuum for the metamaterial, which provided a general approach for both homogeneous and conformal metamaterials of orthogonal lattices. The mapping relation between lattice and finite element mesh as well as edge effects were clarified and modeled. The homogenization approach is verified based on detailed modal analysis and shows high numerical accuracy for engineering applications. Computational experiments confirmed that the micropolar model provides much more accurate results than the classical continuum.

Resonant ultrasound spectroscopy measurement and modeling of additively manufactured octet truss lattice cubes

Advancement of additive manufacturing technological readiness requires high throughput evaluation capabilities that can keep pace with the development of complex parts. Resonant ultrasound spectroscopy (RUS) is an acoustic technique that provides rapid holistic probing of a part by tracking fundamental mechanical resonance modes. In this work, the RUS responses of additively manufactured Ti-5553 octet truss lattice cubes were characterized using experimental measurements and three-dimensional finite element models. Varying percentages of missing struts were designed into the lattices as controlled defects and were verified using X-ray computed tomography. Experimental measurements of density and Young’s modulus were treated as input parameters in a homogenous anisotropic continuum model. The continuum model was compared with experimental RUS measurements, thus evaluating the potential for a simplified approximation of the octet truss lattice. [This work was supported by US DOE LLNL-LDRD 20-SI-001 and was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.]

Analysis of damage and fracture behavior in ductile metal sheets undergoing compression and shear preloading

The present paper comprises the experimental and numerical analysis of damage and fracture behavior of ductile metals under non-proportional loading with compression and shear preloading. For this purpose, biaxial experiments with the H-specimen using a pneumatic downholder for compression loading followed by failure under a tension stress state and corresponding numerical simulations are performed and analyzed. A thermodynamically consistent anisotropic continuum model is presented. It takes the effects of micromechanical damage mechanisms on the macro scale behavior as a motivation for a phenomenological description of damage behavior as a function of stress state. The experimental findings are compared with the results of tests with proportional load paths and corresponding numerical simulations. Thereto, strain fields of the critical areas of the specimens are analyzed by means of digital image correlation (DIC) technique. Compression and shear preloading can lead to significantly reduced ductility of the material with initiation of failure processes in the small damage strain range. Evolution of numerically predicted plastic and damage equivalent strains illustrates their stress state dependence. The numerical results are also confirmed by images of fracture surfaces taken by scanning electron microscopy (SEM). This experimental-numerical methodology is therefore an efficient tool to develop and validate general modeling approaches, and it is characterized by almost arbitrarily definable load paths with changes in stress states occurring in forming processes.

An Anisotropic Continuum Damage Mechanics Model for the Simulation of Yield Surface Evolution and Ratcheting

A new anisotropic continuum damage model was presented on the basis of irreversible thermodynamics. Second rank symmetric damage tensor, kinematic hardening tensor, and damage potential surface were introduced. The yield surface in this study has been modeled to distort in the stress space by the evolution of the damage tensor. The new model was applied in yield surface characterization and led to a well description of the subsequent yield surfaces. This model is then applied to predict experimental results of the cyclic nonproportional loading paths obtained by Kang et al. [2004], Hassan et al. [2002] and Taleb and Cailletaud [2011]. It is demonstrated that the new model can predict the yield surface distortion and the ratcheting strains of multiaxial nonproportional cyclic loading with acceptable accuracy and by using less material constants with respect to the previous models.

Grad-Shafranov Equation in Fractal Dimensions

The main aim of this paper is to discuss the influence of fractal dimensions on the behavior of the solutions of the Grad-Shafranov equation. Our study is based on the product-like fractal measure approach constructed by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal continuum media. The fractal Grad-Shafranov equation gives the possibility to analyze, in a toroidal fusion reactor, the plasma equilibrium in fractal dimensions. Examples of the exact equilibrium solution are given for both the vacuum case outside the plasma and the toroidally shaped spheromak. Note: PACS numbers 05.45.Df: Fractals; 28.52.−s: Fusion reactors; 52.30.Cv: Magnetohydrodynamics; and 52.55.Ip: Spheromaks.

An anisotropic localizing gradient damage approach for failure analysis of fiber reinforced composites

This article presents an anisotropic gradient-enhanced continuum damage model developed within the finite element method framework to address complex fracture phenomena in anisotropic layered materials with unidirectional fiber-reinforced composites as the primary material examples. The main objective of the work is to model damage anisotropy due to progressive intra-laminar fracture at mesoscale in transversally isotropic composite laminae using distinct damage variables associated with different in-plane failure modes. Departing from the conventional gradient enhancements, the model adopts an improved spatial nonlocal description to ensure correct localized damage bandwidths using a single internal length scale. The coupled system of equations is decoupled using an operator-split (staggered) methodology to ensure a robust and straightforward computational implementation without compromising accuracy using lower order finite elements. The proposed damage model is tested on experimental results of fracture response in a single-edge notched tension, center notched tension, and open-hole tension fiber-reinforced composite laminae, where the numerical results were consistent with experimental observations.

Data-driven Bayesian model-based prediction of fatigue crack nucleation in Ni-based superalloys

This paper develops a Bayesian inference-based probabilistic crack nucleation model for the Ni-based superalloy René 88DT under fatigue loading. A data-driven, machine learning approach is developed, identifying underlying mechanisms driving crack nucleation. An experimental set of fatigue-loaded microstructures is characterized near crack nucleation sites using scanning electron microscopy and electron backscatter diffraction images for correlating the grain morphology and crystallography to the location of crack nucleation sites. A concurrent multiscale model, embedding experimental polycrystalline microstructural representative volume elements (RVEs) in a homogenized material, is developed for fatigue simulations. The RVE domain is modeled by a crystal plasticity finite element model. An anisotropic continuum plasticity model, obtained by homogenization of the crystal plasticity model, is used for the exterior domain. A Bayesian classification method is introduced to optimally select informative state variable predictors of crack nucleation. From this principal set of state variables, a simple scalar crack nucleation indicator is formulated.

Numerical analysis of damage and fracture in steel sheets undergoing non-proportional loading paths

The paper deals with the numerical analysis of damage and fracture mechanisms in steel sheets. Newly developed H-specimens are taken from thin sheets and are tested under different biaxial loading conditions. Results of numerical simulations of a series of biaxial experiments are presented showing also the effect of non-proportional loading paths on inelastic deformation behavior and on damage processes. In this context, an anisotropic continuum damage model is presented based on yield and damage conditions as well as evolution laws for plastic and damage strain rates. Different stress-state-dependent branches of the damage criteria are taken into account corresponding to various damage and failure processes on the micro-scale depending on the stress triaxiality and the Lode parameter. Experiments with the biaxially loaded H-specimen have been performed. Results for proportional and corresponding non-proportional loading histories are discussed. During the experiments strain fields in critical regions of the specimens are analyzed by digital image correlation (DIC) technique. Numerical simulations of the experiments have been performed and numerical results are compared with experimental data. Furthermore, based on the numerical analysis evolution of stress variables is examined and stress distributions in critical specimens areas are detected allowing prediction of stress-state-dependent damage and fracture mechanisms. The numerical results also demonstrate the efficiency of the experimental program and the new specimens geometries covering stress states in the shear-tension regime as well as the effect of the loading histories on inelastic deformation and damage behavior in steel sheets.

Static Analysis of Doubly-Curved Shell Structures of Smart Materials and Arbitrary Shape Subjected to General Loads Employing Higher Order Theories and Generalized Differential Quadrature Method

The article proposes an Equivalent Single Layer (ESL) formulation for the linear static analysis of arbitrarily-shaped shell structures subjected to general surface loads and boundary conditions. A parametrization of the physical domain is provided by employing a set of curvilinear principal coordinates. The generalized blending methodology accounts for a distortion of the structure so that disparate geometries can be considered. Each layer of the stacking sequence has an arbitrary orientation and is modelled as a generally anisotropic continuum. In addition, re-entrant auxetic three-dimensional honeycomb cells with soft-core behaviour are considered in the model. The unknown variables are described employing a generalized displacement field and pre-determined through-the-thickness functions assessed in a unified formulation. Then, a weak assessment of the structural problem accounts for shape functions defined with an isogeometric approach starting from the computational grid. A generalized methodology has been proposed to define two-dimensional distributions of static surface loads. In the same way, boundary conditions with three-dimensional features are implemented along the shell edges employing linear springs. The fundamental relations are obtained from the stationary configuration of the total potential energy, and they are numerically tackled by employing the Generalized Differential Quadrature (GDQ) method, accounting for nonuniform computational grids. In the post-processing stage, an equilibrium-based recovery procedure allows the determination of the three-dimensional dispersion of the kinematic and static quantities. Some case studies have been presented, and a successful benchmark of different structural responses has been performed with respect to various refined theories.

A mixed finite element formulation for energy‐momentum time integrations of composites with fiber bending stiffness

AbstractWe aim at dynamic simulations of 3D‐fiber‐reinforced materials in light‐weight structures, which are reinforced by fiber rovings comprised of thousands of filaments. Each roving possesses a separate bending stiffness, which has a large influence on the material behaviour on the macro scale. For the description of this material behavior, we extend a hyperelastic anisotropic Cauchy continuum formulation, which models only the macro‐scopic behaviour of a reinforcement by filaments, by a higher‐order gradient of the deformation mapping as an independent field. The new mixed finite element formulation is based on well‐known mixed finite elements for anisotropic polyconvex material formulations, but include new additional mixed fields concernd with the fiber bending stiffness. As we aim at dynamic long‐term simulations, we apply higher‐order accurate time integrators. Here, we include the new mixed finite element formulation in a variational‐based time finite element method. As numerical examples serve a simple cantilever beam. Here, we first analyze the influence of the fiber bending stiffness.

An energy‐momentum scheme for extended continuum models with rotational degrees of freedom

AbstractIn mechanical engineering, the finite element method is well‐known in the simulation of homogeneous materials without microstructures. But, in particular, microstructures can improve materials with respect to their applicability in engineering. Examples are 2D‐ and 3D‐fiber‐reinforced materials, which are manufactured in the microscale or mesoscale, respectively. An efficient finite element simulation of these materials provides an anisotropic continuum formulation. However, 3D‐fiber‐reinforced materials demand continua with extended kinematics. The introduction of such extended continua is possible by mixed finite element methods. In this paper, we introduce internal rotational degrees of freedom to model also a stiffness with respect to fiber flexure and twist. The kinematic formulation can be explained by using a covariant continuum formulation based on curvilinear convected coordinates. In this context, new objective deformation measures and power‐conjugate stress tensors are derived. Therefore, a corresponding energy‐momentum scheme is obtained and simulates numerical examples stable and efficient in accordance with discrete balance laws. These examples demonstrate the additional material stiffness with fiber curvature‐twist strain energy functions of Kauderer‐type by parameter studies with respect to different fiber directions.

Quench in a pancake coil wound with REBCO Roebel cable: model and validation

Roebel cables assembled with high temperature superconducting tapes are a promising technology for several AC and DC applications. Their mechanical flexibility and compact design, combined with the capability of REBCO tapes to carry high transport currents in intense magnetic fields make them suitable for the application both in power devices and high field magnets. In this paper, an electro-thermal finite-element model developed at the University of Bologna (Italy) is described. The model allows computing the current and heat redistribution between the strands of the cable and from turn to turn inside a winding through non-uniform distributed thermal and electrical resistances between strands. The tape is ‘homogenized’ so as to create an anisotropic continuum model through a previously developed homogenization technique. The model is validated by comparison with quench tests performed on a well instrumented 7-turn pancake coil wound with a 2 m long Roebel cable composed of 15 REBCO tapes. The experiments were performed at the University of Southampton (UK) in the frame of the R&D activities of the EuCARD-2 project. The quench decision time, the temperature and electric potential evolution, the current and heat redistribution between strands in the event of a quench are analysed and discussed in the present study.

Mechanical characterisation for numerical simulation of extrusion-based 3D concrete printing

Extrusion-based 3DCP is experiencing exponential advancement in process, control, material, and fresh-state analysis technologies, strategically poising 3DCP to become an industrial manufacturing process for infrastructure development. To elucidate the mechanical characterisation and numerical simulation of 3DCP elements synergy among experimental and computational mechanics is required. Such activities should characterise the unique material properties of printed components to permit limit state design procedures and the correct prescription of finite element (FE) model parameters. In this research, mechanical characterisation procedures that experimentally evaluate the anisotropic material characteristics of a fibre-reinforced printable concrete (FRPC) are presented. These procedures are comprised of direct tensile, uniaxial compression, Young's modulus, and four-point bending (FPB) crack mouth opening displacement (CMOD) tests. The results portrayed anisotropic non-linear and similar elastic behaviour (in terms of Young's modulus) in all experimental tests conducted. From the mechanical characterisation tests two elastic parameters, seven strength parameters and five inelastic parameters are ascertained. The experimental findings relating to the material characteristic parameters are validated via supplementary numerical evaluation, and suitable constitutive relations are selected. The mechanical parameters are implemented in an anisotropic Rankine-Hill continuum multi-surface plasticity FE model, and the FPB-CMOD fracture response of 3DCP specimens is simulate with respectable agreement.

Thermal transport equations in porous media from product-like fractal measure

In this study, we have used the concept of product-like fractal measure to analyze the fractal heat transfer in anisotropic media. This concept was introduced by Li and Ostoja-Starzewski in order to study anisotropic fractal elastic and continuum media. The theory is characterized by extended fluid, mass and heat transfer fractal equations besides the emergence of a damping heat term in the fractal advection-diffusion equation. Several applications are discussed and a number of features are revealed.

Directional rock mass rating (DRMR) for anisotropic rock mass characterization

Rock mass classification systems are used to categorize and estimate the role of the most significant parameters influencing rock mass behavior to represent an equivalent continuum. Generally, in these methods, the impact of the related parameters is considered uniform in all directions. Hence, these systems describe the rock mass as an isotropic medium. In some cases, in particular, for layered strata and systematically fractured rocks, the assumption of isotropic behavior may not provide realistic results. Hence, in this paper, to characterize the rock mass anisotropy, a directional rock mass rating (DRMR) has been proposed based on the well-known rock mass rating (RMR). So, DRMR provides a three-dimensional rock mass rating, quantitatively. By means of statistical distribution of DRMR, a classification for the degree of anisotropy of rock mass has been proposed. Furthermore, a criterion to identify the prominent rock mass, isotropic/transversely isotropic/anisotropic situation, is presented. Then the stereonet has been used to demonstrate an all-round graphical representation of DRMR. This with DRMR provides an illustrative insight into the actual rock mass condition and can assist to select the appropriate method for further analysis. Also, the method can be used as a basis to develop practical solutions for many rock engineering issues, such as characterizing the mechanical parameters of rock mass as an anisotropic equivalent continuum, selective directional rock mass improvement, and proper selection of rock tunnel and rock cavern alignments. Finally, the results of some practical applications of the method, based on field data, are presented.

Coupled continuum damage mechanics and crystal plasticity model and its application in damage evolution in polycrystalline aggregates

In the present study, damage initiation and growth in a polycrystalline aggregate are investigated. In this regard, an anisotropic continuum damage mechanics coupled with rate-dependent crystal plasticity theory is employed. Using a thermodynamically consistent procedure, a finite deformation formulation is derived. For this purpose, the damage tensor is incorporated in the crystal plasticity formulation for a cubic single crystal. The damage evolution is considered to be dependent on the history of damage, equivalent plastic strain, stress triaxiality, and Lode parameters. This material model is implemented in the commercial finite-element code Abaqus/Standard by developing a user material subroutine (UMAT). Using the available experimental tests of 316L single crystal in the literature, the crystal plasticity hardening and damage parameters are calibrated considering the stress–strain curve before and after necking, respectively. The damage sites in a single-phase polycrystalline aggregate are also considered using a polycrystalline model consisting of grains with random sizes and orientations. The results show that the damage arises at the grain boundaries and triple junctions. Moreover, growth of the damage mostly occurs in the grains with higher Schmid factor compared to the neighboring grains. The presented model manifests capacity for determination of damage initiation sites and damage evolution in polycrystalline models.

A new anisotropic continuum traffic flow model with anticipation driving behavior

Based on the anticipation driving car-following model, a new macro traffic flow model is established in this paper by considering the relationship between micro and macro variables. Therefore, the evolution law of traffic flow with anticipation driving effect can be studied from macroscopic level. By using approaches of linear stability analysis, the linear stability discriminant condition of the new macro model to keep the traffic flow stable against small disturbance is obtained. Numerical experiments verify that the model can not only simulate the unique shock wave, rarefaction wave mutation and the dynamic propagation process of small disturbance, but also improve the stability of traffic flow by introducing the information of anticipation driving behavior.

Estimation of elastic properties of an additively manufactured lattice using resonant ultrasound spectroscopy.

Development of improved approaches in the characterization of additively manufactured structures continues to be a topic of interest for the advanced manufacturing community. This article will investigate an approach using resonant ultrasound spectroscopy (RUS) to determine the effective elastic constants of an orthotropic lattice structure. The evaluation is performed on a cube shaped 316 L stainless steel test specimen, constructed using selective laser melting techniques. The approach uses RUS techniques in conjunction with the assumption that in the frequency regime of interest, the wavelength of the diagnostic ultrasound is greater than the discrete structural features of the unit cell of the lattice; thus, the AM structure can be treated as an anisotropic continuum with effective material properties and symmetry inherited from the unit cell. The RUS analysis estimates the nine elastic coefficients associated with orthotropic sample symmetry, which, in turn, are used to determine the elastic moduli and Poisson ratios. Current results show good agreement between experiments and modeled data. Comparisons to published results are also in good agreement, indicating the potential applicability of this characterization technique for estimating the linear elastic properties of innovative additive manufactured metal lattice structures.