Stiffness Tensor Research Articles

Tensorial harmonic bases of arbitrary order with applications in elasticity, elastoviscoplasticity and texture-based modeling

In continuum mechanics, one regularly encounters higher-order tensors that require tensorial bases for theoretical or numerical calculations. By using results from [Formula: see text] representation theory, we present a method to derive tensorial harmonic bases which unifies existing approaches in one framework. Applying this convention to symmetric second-order tensors results in a second-order harmonic basis which, for example, simplifies the depiction and numerical handling of stiffness tensors for most common material symmetries and reduces the computational effort involved in implementations of incompressible elastoviscoplastic material laws. The same convention can be applied to texture analysis to yield tensorial texture coefficients for both polycrystals and fiber-reinforced composites. Rotations of higher-order tensors are particularly efficient in harmonic bases as both material and index symmetries can be exploited. While special attention is given here to the examples of small-strain material laws and texture analysis, the framework is entirely general and can be used to simplify calculations in other physical contexts involving tensors of arbitrary order and symmetry. A Python implementation of the harmonic basis convention defined in this work is available as a Supplementary Material.

Symmetries and symmetry-generated averages of elastic constants up to the sixth order of nonlinearity for all crystal classes, isotropy and transverse isotropy.

Algebraic expressions for averaging linear and nonlinear stiffness tensors from general anisotropy to different effective symmetries (11 Laue classes elastically representing all 32 crystal classes, and two non-crystalline symmetries: isotropic and cylindrical) have been derived by automatic symbolic computations of the arithmetic mean over the set of rotational transforms determining a given symmetry. This approach generalizes the Voigt average to nonlinear constants and desired approximate symmetries other than isotropic, which can be useful for a description of textured polycrystals and rocks preserving some symmetry aspects. Low-symmetry averages have been used to derive averages of higher symmetry to speed up computations. Relationships between the elastic constants of each symmetry have been deduced from their corresponding averages by resolving the rank-deficient system of linear equations. Isotropy has also been considered in terms of generalized Lamé constants. The results are published in the form of appendices in the supporting information for this article and have been deposited in the Mendeley database.

Thermal, mechanical, and electrical properties of Si-stacked nanosheet transistors using machine learning interatomic potentials

Thermal and mechanical properties play a key role in optimizing the performance of nanoelectronic devices. In this study, the lattice thermal conductivity (κL) and elastic constants of Si nanosheets at different sheet thicknesses were determined using recently developed machine learning interatomic potentials (MLIPs). A Si nanosheet with a minimum thickness of 10 atomic layers was used for model training to predict the properties of sheets with greater thicknesses. The training dataset was efficiently constructed using stochastic sampling of the potential energy surface (PES). Density functional theory (DFT) calculations were used to extract
the MLIP, which served as the basis for further analysis. The Moment Tensor Potential (MTP) method was used to obtain the MLIP in this study. The results showed that, at sub-6 nm sheet thickness, the thermal conductivity dropped to ∼ 7 % of its bulk value, whereas some stiffness tensor components dropped to ∼ 3 % of the bulk values. These findings contribute to the understanding of heat transport and mechanical behavior of ultrathin Si nanosheets, which is crucial for designing and optimizing nanoelectronic devices. The technological implications of the extracted parameters on nanosheet field-effect transistor (NS-FET) performance at advanced technology nodes were evaluated using TCAD device simulations.

Assessment of Free Energy Functions for Sand

ABSTRACTThe advantages of constitutive models in energy‐conservation frameworks have been widely addressed in the literature. A key component is choosing an appropriate energy potential to derive the hyperelastic constitutive equations. This article investigates the advantages and limitations of different energy potentials found in the literature based on mathematical conditions to guarantee numerical stability, such as the desired order of homogeneity, positive and non‐singular stiffness within the application range, and equivalent Poisson's ratio from a constitutive modelling standpoint. Potentials meeting the aforementioned criteria are employed to simulate the response envelopes of Karlsruhe fine sand (KFS). Moreover, the performance of the potentials, in conjunction with plasticity theories, is examined. To achieve this, the hyperelastic constitutive equations have been coupled with the bounding surface plasticity model of Dafalias and Manzari to reproduce the soil response in a hyperelastic–plastic frame. Finally, one of the potentials is modified, whereas recommendations for incorporating other appropriate free energy functions into different soil constitutive models are presented. Furthermore, 100 closed elastic strain cycles have been simulated with the bounding surface plasticity model of Dafalias and Manzari considering the original hypoelastic stiffness and hyperelastic–plastic constitutive equations. Using the hypoelastic framework in the simulation led to stress accumulation after 100 closed elastic strain loops, while a reversible response was predicted using the hyperelastic stiffness tensor.

Exploration of the triceps surae muscle in ambulatory children with cerebral palsy using instrumented measurements of stiffness and diffusion tensor magnetic resonance imaging for muscle architecture

BackgroundMusculoskeletal alterations causing reduced range of motion of the ankle joint are common in children with cerebral palsy (CP). Objective measurements of passive joint resistance and three-dimensional skeletal muscle volume and muscle architecture can lead to a comprehensive understanding of which factors influence joint range of motion.Research questionTo investigate the relation between the passive dorsiflexion of the ankle joint, biomechanical contributing factors to the passive joint resistance, and muscular architectural properties of the triceps surae muscle in children with CP.MethodsIn this cross-sectional observational study, 14 children with spastic CP (bilateral: 5, unilateral: 9, Gross Motor Function Classification System (GMFCS) level I:11, II:3) naïve to intramuscular tone reducing treatment, and 14 TD children were included. The passive dorsiflexion of the ankle was measured with a goniometer. Passive joint resistance and related parameters were estimated based on a biomechanical model and measurements using a motorized device, the Neuroflexor. Three-dimensional muscle architecture was quantified with diffusion tensor magnetic resonance imaging (DT-MRI).ResultsIn the CP group, the median [min, max] passive dorsiflexion was decreased in the most affected leg (MAL) compared to the less affected leg (LAL) (2.5° [-25°, 20°] vs. 12.5° [5°, 30°], p = 0.001). The stiffness coefficient (Nm/rad) in the MAL was significantly higher in children with CP compared to TD children (7.10 [3.39, 62.00] vs. 2.82 [1.24, 10.46], p = 0.015). Muscle architecture properties did not differ between CP and TD, except for pennation angle in the medial gastrocnemius (MG) of the MAL (CP 17.64° (2.29) vs. TD 21.46° (3.20), p = 0.017). The stiffness coefficient, in the MAL, correlated negatively to passive dorsiflexion (rs=-0.638) and pennation angle in medial gastrocnemius (rs=-0.964), and the non-linear coefficient (Non-linear 1) correlated negatively to the fascicle length of the medial gastrocnemius (rs=-0.857).ConclusionThis study shows that stiffness of the plantarflexors is related to decreased passive dorsiflexion of the ankle and muscle structure of the MG in high-functioning children with spastic CP. Assessments of how dynamic components as well as microscopic muscle alterations contribute to joint stiffness in the plantarflexors in individuals with CP are warranted.Trial registrationRetrospectively registered in ClinicalTrials.gov, NCT05447299. Observational study. Study start: 2019-01-15, register date: 2022-07-01.

Easy-to-use formulations based on the homogenization theory for vascular stent design and mechanical characterization

Background and objectivesVascular stents are scaffolding structures implanted in the vessels of patients with obstructive disease. Stents are typically designed as cylindrical lattice structures characterized by the periodic repetition of unit cells. Their design, including geometry and material characteristics, influences their mechanical performance and, consequently, the clinical outcomes. Computational optimization frameworks have proven to be effective in assisting the design phase of vascular stents, facilitating the achievement of enhanced mechanical performances. However, the reliance on time-consuming simulations and the challenge of automating the design process limit the number of design evaluations and reduce optimization efficiency. In this context, a rapid and automated method for the mechanical characterization of vascular stents is presented, taking the stent geometry, conceived as the periodic repetition of a unit cell, and material as input and providing the mechanical response of the stent as output. MethodsVascular stents were assumed to be thin-walled hollow cylinders sharing the same macroscopic geometrical characteristics as the cylindrical lattice structure but composed of an anisotropic homogenized material. Homogenization theory was applied to average the microscopic inhomogeneities at the stent unit cell level into a homogenized material at the macro-scale, enabling the calculation of the associated homogenized material tensor. Analytical formulations were derived to relate the stent mechanical behavior to the homogenized stiffness tensor, considering linear elastic theory for thin-walled hollow cylinders and three loading scenarios of relevance for vascular stents: radial crimping; axial traction; torsion. Validation was conducted by comparing the derived analytical formulations with results obtained from finite element analyses on typical stent designs. ResultsHomogenized stiffness tensors were computed for the unit cells of three stent designs, revealing insights into their mechanical performance, including whether they exhibit auxetic behavior. The derived analytical formulations were successfully validated with finite element analyses, yielding low relative differences in the computed values of foreshortening, radial, axial and torsional stiffnesses for all three stents. ConclusionsThe proposed method offers a rapid, fully automated procedure that facilitates the assessment of the mechanical behavior of vascular stents and is suitable for effective integration into computational optimization frameworks.

Planar structured materials with extreme elastic anisotropy

Designing anisotropic structured materials by reducing symmetry results in unique behaviors, such as shearing under uniaxial compression. While rank-deficient materials such as hierarchical laminates have been shown to exhibit extreme elastic anisotropy, there is limited knowledge about the fully anisotropic elasticity tensors achievable with single-scale fabrication techniques. No established upper and lower bounds on anisotropic moduli exist. In this paper, we estimate the range of anisotropic stiffness tensors achieved by single-scale two-dimensional structured materials. We first develop a database of periodic anisotropic single-scale unit cell geometries using linear combinations of periodic cosine functions. The database covers a wide range of anisotropic elasticity tensors, which are then compared with the elasticity tensors of hierarchical laminates. We identify the regions in the property space where hierarchical design is necessary to achieve extremal properties. We demonstrate a method to construct various 2D functionally graded structures using this cosine function representation. These graded structures seamlessly interpolate between unit cells with distinct patterns, allowing for independent control of several functional gradients, such as porosity, anisotropic moduli, and symmetry. When designed with unit cells positioned at extreme parts of the property space, these graded structures exhibit unique mechanical behaviors such as selective strain energy localization, compressive strains under tension, and localized rotations.

Models and criteria of destruction of composite materials at the stage of macrocrack initiation

The analysis was carried out and the limits of the use of modern damage models and criteria for the failure of composite materials (CM) at the stage of macrocrack initiation were established. The classification of CM and the main hypotheses and assumptions used in the development of the condition level are described. The main stage of the life cycle of a CM product is considered - the stage of nucleation and accumulation of scattered fractures at the micro and meso levels. The main principles of continuum damage mechanics, thermodynamics of irreversible processes and mechanics of a solid deformed body are used. The main mechanisms of microfailure of reinforced CM are described. It is shown that for them it is necessary to consider a set of phenomenological parameters that could assess the kinetics of the accumulation of diffused failures in the matrix, reinforcing fibers and the delamination between them. These complex processes in reinforced CMs require phenomenological modeling of damage in the form of tensor quantities with certain assumptions. Thus, the "mixture" hypothesis was widely used. It is shown that each component of the damage parameter is more expedient to determine from the hypothesis of the equivalence of specific energies. The method of conducting basic experiments to specify the regularities of damage accumulation in CM is detailed. A modified CM stiffness tensor is established, taking into account the degradation of the mechanical properties of the CM, regardless of the thermo-force parameters of the operating load. An analysis of modern criteria for the destruction of reinforced CMs and the limits of their use was carried out. The main directions of solving the problem of estimating the limit state of CM and the load-bearing elements made from them are shown.

Fracture modeling in composite structures containing orthotropic materials by incorporating strain orthogonal decompositions into phase-field method of interfacial damage

PurposeThe phase-field method of interfacial damage is used to simulate the damage in composite structures containing the brittle orthotropic materials and their interface.Design/methodology/approachIn the brittle fracture modeling, the strain tensor is decomposed into positive and negative parts characterizing tension and compression behaviors. By requiring an elastic energy preserving transformation involving the elastic stiffness tensor, these two strain parts must satisfy the orthogonality condition in the sense that the elastic stiffness tensor responds as a metric. However, most of the recent phase-field methods for brittle fracture do not verify this orthogonality condition. Additionally, to describe the damage in structures with anisotropic phases, recent studies have used multiple phase-field variables, with each preferential orientation represented by a phase-field variable to describe the bulk damage of component materials. This approach increases the complexity of simulation procedure. These disadvantages motivate the present study aimed at enhancing the simulation method.FindingsThe present study improves the phase-field method of interfacial damage by (1) incorporating the strain orthogonality condition into the phase-field method; (2) using only one phase-field variable instead of multiple phase-field variables to simulate damage in component orthotropic phases; and (3) investigating the interaction between interfacial damage and bulk damage as well as the effect of orientation tensor of preferential orientation in each orthotropic phase and the interfacial parameters on crack branching in composite structures.Originality/valueThrough several simulation examples, the present simulation method is proven to be accurate, effective, and helps the simulation process simpler than previous relevant methods.

A probabilistic machine learning framework for stiffness tensor estimation of carbon composite laminate

The potential uses of carbon composite material are vast, particularly in the civil, mechanical and aero-structures. Nonetheless, the practical utilization of carbon composite faces challenges due to complex experimental methods and imprecise algorithms for inverting material properties. Characterization of full set of stiffness tensor for an anisotropic material in particular orthotropic material can be considered as a complex non-linear inversion problem. The inversion process requires a robust optimization algorithm and forward models. The complexity of this inversion process impedes the practical utility in large-scale automation and in real-time structural health monitoring (SHM) application. To cater this, in this work an advanced probabilistic machine learning framework based on multi-output Gaussian process regression (moGPR) is proposed as an inversion algorithm. The inversion algorithm proposed is based on probabilistic framework that utilises the dispersion of Lamb waves as an input for optimal estimation of stiffness tensor. Experimental measurements of full wavefield data of propagating waves are conducted by scanning laser Doppler vibrometer. Further, an optimal training dataset is generated by employing a forward model based on stiffness matrix method (SMM). In the presence of measurement uncertainties, the effectiveness of the optimal prediction algorithm is showcased through a comparison between the estimated parameter and its ground truth. This comparison reveals that the proposed inversion algorithm is both efficient and robust, delivering satisfactory performance even in the presence of substantial measurement uncertainties.

Characterization of Trabecular Bone Microarchitecture and Mechanical Properties Using Bone Surface Curvature Distributions.

Understanding bone surface curvatures is crucial for the advancement of bone material design, as these curvatures play a significant role in the mechanical behavior and functionality of bone structures. Previous studies have demonstrated that bone surface curvature distributions could be used to characterize bone geometry and have been proposed as key parameters for biomimetic microstructure design and optimization. However, understanding of how bone surface curvature distributions correlate with bone microstructure and mechanical properties remains limited. This study hypothesized that bone surface curvature distributions could be used to predict the microstructure as well as mechanical properties of trabecular bone. To test the hypothesis, a convolutional neural network (CNN) model was trained and validated to predict the histomorphometric parameters (e.g., BV/TV, BS, Tb.Th, DA, Conn.D, and SMI), geometric parameters (e.g., plate area PA, plate thickness PT, rod length RL, rod diameter RD, plate-to-plate nearest neighbor distance NNDPP, rod-to-rod nearest neighbor distance NNDRR, plate number PN, and rod number RN), as well as the apparent stiffness tensor of trabecular bone using various bone surface curvature distributions, including maximum principal curvature distribution, minimum principal curvature distribution, Gaussian curvature distribution, and mean curvature distribution. The results showed that the surface curvature distribution-based deep learning model achieved high fidelity in predicting the major histomorphometric parameters and geometric parameters as well as the stiffness tenor of trabecular bone, thus supporting the hypothesis of this study. The findings of this study underscore the importance of incorporating bone surface curvature analysis in the design of synthetic bone materials and implants.

Ab initio property characterisation of thousands of previously unexplored 2D materials

We perform extensive density functional theory calculations to determine the stability and elementary properties of 4249 previously unexplored monolayer crystals. The monolayers comprise the most stable subset (energy within 0.1 eV/atom of the convex hull) of a larger portfolio of two-dimensional (2D) materials recently discovered using a deep generative model and systematic lattice decoration schemes. The relaxed 2D structures are run through the basic property workflow of the Computational 2D Materials Database (C2DB) to evaluate the dynamical stability and obtain the stiffness tensor, piezoelectric tensor, deformation potentials, Born and Bader charges, electronic band structure, effective masses, plasma frequency, Fermi surface, projected density of states, magnetic moments, magnetic exchange couplings, magnetic anisotropy, topological indices, optical- and infrared polarisability. We provide statistical overviews of the property data and highlight a few specific examples of interesting materials. Our work exposes previously unknown parts of the 2D chemical space and provides a basis for the discovery of 2D materials with specific properties. All data is available in the C2DB.

Fluctuations and failure in granular materials: theory and numerical simulations

We consider a dense aggregate of elastic, frictional particles isotropically compressed and next uniaxial strained at constant pressure. We show how failure can be predicted if fluctuations in the kinematics of contacting particles are introduced. We focus on the second order work and the possibility that at some stressed states it becomes negative under proper perturbations. Our analysis involves both a theoretical model and numerical simulations based upon the distinct element method (DEM). The theoretical model deals with contacting particles with incremental relative displacements that deviate from the average deformation in order to ensure their equilibrium. Because of this, the macroscopic stiffness tensor of the aggregate, that relates increments in stress with increments in strain, does not have the major symmetry. Consequently, in the hardening regime, we predict stressed states in which the second order work vanishes. The model seems transparent, and it makes clear and illustrative the role played by the fluctuations introduced in the kinematics of contacting particles in relation to the vanishing of second order work in an aggregate of compressed particles. The comparison with numerical simulations data supports the model.Graphical Statistical representation of the aggregate: conditional average.

The elastic stiffness tensor of cellulosic viscose fibers measured with Brillouin spectroscopy

Brillouin light scattering spectroscopy is applied to study the micromechanics of cellulosic viscose fibers, one of the commercially most important, man-made biobased fibers. Using an equal angle scattering geometry, we provide a thorough description of the procedure to determine the complete transversely isotropic elastic stiffness tensor. From the stiffness tensor the engineering-relevant material parameters such as Young’s moduli, shear moduli, and Poisson’s ratios in radial and axial fiber direction are evaluated. The investigated fiber type shows that, at ideal conditions, the material exhibits optical waveguide properties resulting in spontaneous Brillouin backscattering which can be used to obtain additional information from the Brillouin spectra, enabling the measurement of two different scattering processes and directions with only one scattering geometry.

Inverse design of irregular architected materials with programmable stiffness based on deep learning

Incorporating biological materials-inspired irregularity into architected materials is potential to unfold a wider property or new function space, while the inverse design of irregular architected materials (IAMs) for targeted properties is still challenging, owing to the intricate structure–property bidirectional relationships. Accordingly, here, a data-driven framework is originally developed to design new irregular architected materials with programmable stiffness. In detail, a robust dataset of IAMs, which are constructed from diverse stretch- and bending-dominated building blocks, is created by a randomized strategy, and an end-to-end deep learning model is established, including a forward and an inverse network. The forward predictions reveal that IAMs exactly achieve a wide space of programmable stiffness, especially, including extremely high or low dominant elastic stiffness C11, C22 and C66. More importantly, the deep learning-driven inverse design flexibly and precisely realizes the generation of IAMs, fulfilling the customized stiffness targets. Especially, various stiffness tensors, including isotropic, orthotropic, and anisotropic types, are successfully achieved. The remarkable advantage of the inverse design is that solely through adjusting the frequency hints of building blocks, without mandatory geometrical symmetry, IAMs present the exclusive ability to hit the targeted stiffness. Overall, this work fills the gap in the inverse design of irregular architected materials with programmable stiffness and paves the way for material design via deep learning.

Multiscale enhanced non‐uniform transformation field analysis

AbstractEnhanced transformation field analysis (E‐TFA), recently proposed for reduced‐order modeling, is here formulated for and applied to multiscale analysis. The approach is able to reproduce a highly complex nonlinear macroscale behavior, resulting from a microstructure with cohesive interfaces embedded in an elasto‐plastic bulk. E‐TFA features a consistent tangent matrix in its solution procedure, which enables a straightforward definition of the upscaled tangent stiffness tensor. Numerical tests show that, compared to FE, the proposed approach yields accurate solutions at a lower computational cost.

Radially transverse isotropic inclusions in isotropic elastic media: Local fields, neutral inclusions, effective elastic properties

Radially transverse isotropic inclusions in homogeneous isotropic elastic host media are considered. Mellin transform method is used for solution of the volume integral equation of the problem for an isolated inclusion subjected to a constant external stress (strain) field. The tensor structure of the solution is revealed with precision to three scalar functions of the radial coordinate, and the system of ordinary differential equations for these functions is derived. For multilayered radially transverse isotropic inclusions with constant elastic coefficients inside layers, explicit solution of these equations is obtained. An efficient numerical algorithm of solution for inclusions with an arbitrary number of the layers is proposed. Neutral inclusions that do not disturb homogeneous external fields applied to the medium are considered. It is shown that an inclusion with an isotropic core and radially transverse isotropic external layer can be weak neutral by appropriate choice of the layer elastic constants. The effective field method is used for determination of the effective elastic stiffness tensor of a homogeneous isotropic medium containing a random set of radially transverse isotropic inclusions.

Three-dimensional strain and stress orthogonal decompositions via an elastic energy preserving transformation

Motivated by the recent development of phase field methods for modeling and simulating the fracture of brittle and quasi-brittle materials, a general approach is proposed to decompose the infinitesimal strain tensor (or Cauchy stress tensor) into a positive part and a negative part which are orthogonal in the sense of an inner product where the elastic stiffness (or compliance) tensor acts as a metric. This approach is based on a strain (or stress) transformation preserving the elastic energy, and gives rise to a generalized Pythagorean theorem. It is valid not only for the isotropic elastic case but also for all the anisotropic ones. Decompositions of the strain (or stress) tensor are given in a coordinate-free way. The main results obtained are illustrated and detailed for the elastic isotropic case in an analytical explicit manner.

Anisotropic Elastic Strain-Gradient Continuum from the Macro-Scale to the Granular Micro-Scale

A multi-scale framework is constructed for the computation of the stiffness tensors of an elastic strain-gradient continuum endowed with an anisotropic microstructure of arbitrarily-shaped particles. The influence of microstructural features on the macroscopic stiffness tensors is demonstrated by comparing the fourth-order, fifth-order and sixth-order stiffness tensors obtained from macro-scale symmetry considerations to the stiffness tensors deduced from homogenizing the elastic response of the granular microstructure. Special attention is paid to systematically relating the particle properties to the probability density function describing their directional distribution, which allows to explicitly connect the level of anisotropy of the particle assembly to local variations in particle stiffness and morphology. The applicability of the multi-scale framework is exemplified by computing the stiffness tensors for various anisotropic granular media composed of equal-sized spheres. The number of independent coefficients of the homogenized stiffness tensors appears to be determined by the number of independent microstructural parameters, which is equal to, or less than, the number of independent stiffness coefficients following from macro-scale symmetry considerations. Since the modelling framework has a general character, it can be applied to different higher-order granular continua and arbitrary types of material anisotropy.

A Novel QCT-Based Deep Transfer Learning Approach for Predicting Stiffness Tensor of Trabecular Bone Cubes

ObjectivesThis study was performed to prove the concept that transfer learning techniques, assisted with a generative model, could be used to alleviate the ‘big data’ requirement for training high-fidelity deep learning (DL) models in prediction of stiffness tensor of trabecular bone cubes. Material and methodsTransfer learning approaches of domain adaptation were used, in which a source domain included 1,641 digital trabecular bone cubes synthesized from a generative model, and a target domain included 868 real trabecular bone cubes from human cadaver femurs. Simulated quantitative computed tomography (QCT) images of both the synthesized and real bone cubes were used as input, whereas the stiffness tensor of these cubes determined using finite element simulations were used as output. Three transfer learning algorithms, including instance-based (TrAdaBoostR2 and WANN) and parameter-based (RNN) methods, were used. Two case studies, one with varying sizes of training dataset and the other with a gender-biased training dataset, were performed to evaluate these deep transfer learning models in comparison with a base deep learning (DL) model trained using the dataset from the target domain. ResultsThe results indicated that these deep transfer learning models were robust both to sample size and to the gender-biased training dataset, whereas the base DL model was very sensitive to such changes. Among the three transfer learning algorithms, the prediction accuracy of the RNN-based deep transfer learning model was the best (0.92-0.96%) and comparable to that of the base DL model trained using the dataset from the target domain. ConclusionThis study proved the proposed concept and confirmed that high fidelity QCT-based deep learning models could be obtained for prediction of stiffness tensor of trabecular bone cubes.