Elastic Description Research Articles

Mechanics of Materials: Multiscale Design of Advanced Materials and Structures

Materials can now be designed and architectured like structural components for targeted mechanical and physical properties. Structures and microstructures should not be studied independently and their design will benefit from a multiscale approach combining nonlinear continuum mechanics approaches and physical descriptions of elasticity, viscoplasticity, phase transformations and damage of microstructures, at various scales. The aim of the workshop was to gather outstanding junior and senior researchers in the various branches of mathematics, physics and engineering sciences suited to address the question of design of materials and structures by means of multiscale discrete and continuum approaches to their constitutive behavior. Examples include atomic or macroscopic lattices, random or periodic cellular materials, smart materials like shape memory alloys, 3D woven composites, acoustic and electromagnetic metamaterials, etc. Modern continuum mechanics relies on sophisticated constitutive laws for anisotropic materials exhibiting elastoviscoplastic behavior, still a field of intense research with new mathematical concepts. In particular size-dependent properties are addressed by resorting to generalized continua such as gradient or micromorphic and phase field models. The latter are attractive for the simulation of microstructure evolution coupled with mechanics, due to thermodynamic and metallurgical processes and damage. Scale transition and homogenization methods for continuous and discrete systems are required for the determination of effective material and structural behavior. Metamaterials are architectured materials specifically designed to achieve certain propagation and dispersion properties of elastic and plastic waves. Optimization strategies for the design of optimal architectures are involved in the design process. Target functions for optimization are now based on multicriteria (stiffness, strength, thermal expansion, transport properties, anisotropy etc.).

On the experimental identification of equilibrium relations and the separation of inelastic effects in soft biological tissues

The mechanical characterization of vascular tissues has been mainly focused on the measurement of elastic properties, while the investigation of inelastic effects has received comparatively little attention. Even the relatively simple, purely elastic description of the material behavior requires an appropriate set of experimental data that cannot be easily isolated using standard testing procedures. The presence of viscous and damage-related phenomena poses some challenges in the definition of appropriate testing protocols capable of identifying an equilibrium response, which in general does not solely represent the elastic material behavior. The primary goal of the present study is therefore to devise an experimental procedure that can distinguish and evaluate the different constitutive phenomena separately. To this end, we apply methodologies widely used in the mechanical testing of rubber-like materials and transfer them to the field of biomechanics. We performed two types of experiments in equibiaxial extension on porcine thoracic aorta: a continuous cyclic test followed by a single-step relaxation test and a cyclic multi-step relaxation test, each at varying stretch rates. We demonstrate that the approximation of quasi-stationarity through continuous testing at slow rates is inadequate for the identification of an equilibrium relation. Alternatively, a step-wise protocol allows for the separation of equilibrium and viscous effects. This motivates a thermodynamic discussion of the experimental results in terms of energy dissipation and a closer look at the interplay of inelastic phenomena.

Coupling Phase Field Crystal and Field Dislocation Mechanics for a consistent description of dislocation structure and elasticity

This work addresses differences in predicted elastic fields created by dislocations either by the Phase Field Crystal (PFC) model, or by static Field Dislocation Mechanics (FDM). The PFC order parameter describes the topological content of the lattice, but it fails to correctly capture the elastic distortion. In contrast, static FDM correctly captures the latter but requires input about defect cores. The case of a dislocation dipole in two dimensional, isotropic, elastic medium is studied, and a weak coupling is introduced between the two models. The PFC model produces compact and stable dislocation cores, free of any singularity, i.e., diffuse. The PFC predicted dislocation density field (a measure of the topological defect content) is used as the source (input) for the static FDM problem. This coupling allows a critical analysis of the relative role played by configurational (from PFC) and elastic (from static FDM) fields in the theory, and of the consequences of the lack of elastic relaxation in the diffusive evolution of the PFC order parameter.

A FINITE ELEMENT ANALYSES OF THE EFFECT OF HYDROSTATIC PRESSURE IN A THIN ADHESIVE LAYER ON ITS FRACTURE UNDER MIXED-MODE LOADING

The problem of deformation of a sample, which is a composition of orthotropic bodies bound by an adhesive layer, realizing a mixed loading mode I + II for the layer material in the vicinity of a crack-like defect, is considered. A crack-like defect in a composite was considered both in the form of a mathematical section and a physical section, the thickness of which was considered as a linear parameter. A finite element solution with an elastic description of materials in a state of plane deformation is used. To find critical states, the values of specific elastic energy in the tip of a crack-like defect were analyzed. So, for the mathematical section, the values of energy were associated with stress intensity factors and for the physical section with the energy product in the form of the product of a linear parameter and the average specific free energy on the face of a dead-end finite element of the adhesive layer. Finding the characteristics averaged over the layer thickness was determined within the framework of the Neuber – Novozhilov approach. For a small finite value of the linear parameter the convergence of the energy product to the critical value of the specific elastic energy found for the crack model in the form of a mathematical cut is shown for equivalent loading of the sample with a physical cut and the linear parameter tends to zero. For an adhesive layer of finite thickness an energy fracture criterion is considered which takes into account the loosening of the material due to hydrostatic pressure. The loosening of the material was taken into account by the parameter which was determined from the solution of the problem of normal tension of the adhesive layer by the cantilevers of the two-cantilever beam taking into account the critical values of the specific elastic energy. A comparison is made of the calculated critical external load for the formulation of the problem using a layer of finite thickness and the classical representation of a crack-like defect in the form of a mathematical section and a layer of zero thickness under the appropriate failure criteria.

A life off the beaten track in biomechanics: Imperfect elasticity, cytoskeletal glassiness, and epithelial unjamming.

Textbook descriptions of elasticity, viscosity, and viscoelasticity fail to account for certain mechanical behaviors that typify soft living matter. Here, we consider three examples. First, strong empirical evidence suggests that within lung parenchymal tissues, the frictional stresses expressed at the microscale are fundamentally not of viscous origin. Second, the cytoskeleton (CSK) of the airway smooth muscle cell, as well as that of all eukaryotic cells, is more solid-like than fluid-like, yet its elastic modulus is softer than the softest of soft rubbers by a factor of 104-105. Moreover, the eukaryotic CSK expresses power law rheology, innate malleability, and fluidization when sheared. For these reasons, taken together, the CSK of the living eukaryotic cell is reminiscent of the class of materials called soft glasses, thus likening it to inert materials such as clays, pastes slurries, emulsions, and foams. Third, the cellular collective comprising a confluent epithelial layer can become solid-like and jammed, fluid-like and unjammed, or something in between. Esoteric though each may seem, these discoveries are consequential insofar as they impact our understanding of bronchospasm and wound healing as well as cancer cell invasion and embryonic development. Moreover, there are reasons to suspect that certain of these phenomena first arose in the early protist as a result of evolutionary pressures exerted by the primordial microenvironment. We have hypothesized, further, that each then became passed down virtually unchanged to the present day as a conserved core process. These topics are addressed here not only because they are interesting but also because they track the journey of one laboratory along a path less traveled by.

Evaluation of the elastic field in phase‐field crystal simulations

AbstractThe phase‐field crystal model (PFC) describes crystal structures at diffusive timescales through a periodic order parameter representing the atomic density. One of its main features is that it naturally incorporates elastic and plastic deformation. To correctly interpret numerical simulation results or devise extensions related to the elasticity description, it is important to have direct access to the elastic field. In this work, we discuss its evaluation in classical PFC models based on the Swift–Hohenberg energy functional. We consider approaches where the stress field can be derived from the microscopic density field (i.e., the order parameter) and a simple novel numerical routine is proposed. By numerical simulations, we demonstrate that it overcomes some limitations of currently used methods. Moreover, we shed light on the elasticity description conveyed by classical PFC models, characterizing a residual stress effect present at equilibrium. We show explicitly and discuss the evaluation of the elastic fields in prototypical representative cases involving an elastic inclusion, a grain boundary, and dislocations.

Analytical flat belt drive model considering bilinear elastic behaviour with residual strains

In this work an analytical model of a steady-state flat belt drive with multiple driving and driven pulleys is presented. The belt is characterised by a novel bilinear elastic constitutive description with a superposition of residual strain due to viscoelastic effects, such as relaxation, and considering inertia effects. Both prevalent tensioning methods for the drive are examined: the pretensioning of the belt to a target pre-strain, and the utilization of a tension pulley to ensure adequate tension in the drive.The model is discussed based on a case study. The enhanced accuracy in modelling the mechanical response of the belt results in more accurate estimation of the maximal transmissible power as well as the state of tensile strain of the belt. Traditionally, the maximal transmissible power and tensile strain of the belt are overestimated. Consequently, the potential for failures arising from drives operating proximate to their operational thresholds can be effectively mitigated. The more accurate description of the belt state can lead to better utilization of the belt capabilities and a larger design space when dimensioning the belt drive.Furthermore, the advantages of employing a tension pulley for belt drive tensioning are demonstrated under the assumed conditions.

数理・情報科学的手法による高分子のメゾ構造-物性相関解析

Polymers are raw materials for products in our daily life such as plastics, resin, rubber, and organic glasses. The long molecular chains in polymers generate versatile and complex structure in mesoscopic scale, which affects the material property. Therefore, it is hard to deal with the mesostructure-property relationships by simple experiment or theory. Recently, the author and coworkers have developed the mesostructured-property relationships in polymer science based on mathematical and informatics methods. First, description of rubber elasticity of elastomers by complex network analysis are represented. Then, process-mesostructure-property relationships in crystalline polymers by machine learning techniques are mentioned. Furthermore, reverse analysis of biodegradable polymers with both toughness and degradability are included.

Characterising the mechanical properties of soft solids through acoustics and rheology, exemplified by anhydrous milk fat.

Foods vary in their elastic properties over a wide range of behaviours. In the case of mastication, textures vary from hard solid through brittle (chocolate bar) and crispy/crunchy (biscuits) to viscous and extensional flow (syrup) and finally very low viscosity fluid (water). Here we deploy an elastic description of soft solids which embraces all these behaviours to quantify the elastic behaviour of food, in particular through the use of sound. We illustrate the use of this mathematical description in the quantitative characterisation of the elastic and flow properties of food through orthodox measurement techniques and novel ultrasound methods. Measurement is complicated by human sensory capabilities that span the entire range from solid to fluid to gas in an integrated manner, during the appreciation of food. We use acoustic and rheological measurement techniques for the determination of the mechanical properties of soft solids, comparing oscillatory rheometry with acoustic parameters as exemplified by acoustic and oscillatory rheometry measurements in crystallising anhydrous milk fat (AMF). We conclude that acoustic and rheological measurements complement each other with acoustic techniques offering the possibility of inline, in process determination of mechanical and flow properties such as viscosity, rigidity, compressibility and bulk modulus.

Group theoretical approach to elasticity under constraints and predeformations

Respecting deformational constraints and predeformations poses a substantial challenge in the description of nonlinear elasticity. We here outline how group theory can play a beneficial role to overcome this challenge. Specifically, group theory guides us to generalized definitions of nonlinear shear deformation gradients and expressions of generalized elastic moduli in the nonlinear regime. Particularly, such achievements become important in the context of larger deformations under constraints and additional deformations on top of predeformations.

Mathematical model of Alzheimer’s disease with prion proteins interactions and treatment

The accumulation of β-amyloid (Aβ) is one of the most important pathogenic factors in the occurrence of Alzheimer’s disease (AD). Studies have shown that oligomers are more toxic in the process of Aβ aggregation because oligomers can interact with receptors such as prion proteins (Prpc) and the interaction causes Prpc to be misfolded into pathogenic oligomeric prion proteins (Prpol). In this paper, we propose an AD model including two types of Aβ, the interaction of oligomers with Prpc and anti-Aβ drugs treatment. The existence, uniqueness, and non-negativity of the solutions are analyzed. Furthermore, we prove that the model admits a unique globally asymptotically stable equilibrium, which means the drug cannot cure AD completely. Finally, we present some numerical simulations and investigate the relative contributions to AD with two types of Aβ, Aβ40 and Aβ42. In addition, elastic analysis description Aβ42 paranuclei can be used as a therapeutic target.

Predicting plasticity of amorphous solids from instantaneous normal modes.

We present a mathematical description of amorphous solid deformation and plasticity by extending the concept of instantaneous normal modes (INMs) to deformed systems, which allows us to retain the effect of strain on the vibrational density of states (VDOS). Starting from the nonaffine lattice dynamics (NALD) description of elasticity and viscoelasticity of glasses, we formulate the linear response theory up to large deformations by considering the strain-dependent tangent modulus at finite values of shear strain. The (nonaffine) tangent shear modulus is computed from the VDOS of affinely strained configurations at varying strain values. The affine strain, found analytically on the static (undeformed) snapshot of the glass, leads to configurations that are rich with soft low-energy modes as well as unstable modes (negative eigenvalues) that are otherwise completely "washed out" and lost if one lets the system fully relax after strain. This procedure is consistent with the structure of NALD. The INM spectrum of deformed states allows for the analytical prediction of the stress-strain curve of a model glass. Good parameter-free quantitative agreement is shown between the prediction and simulations of athermal quasistatic shear of a coarse-grained polymer glass.

Performance review of locking alleviation methods for continuum ANCF beam elements

The absolute nodal coordinate formulation (ANCF) is a nonlinear finite element approach proposed for the large deformation dynamics analysis of beam- and plate/shell-type structures. In the ANCF approach, elastic forces can be defined using three-dimensional elasticity-based continuum mechanics. This approach is often straightforward, and it makes it possible to use advanced material models in the ANCF framework. However, it has been pointed out in several studies that continuum ANCF-based elements with a full three-dimensional elasticity description can suffer from locking phenomena. In this study, a comparison between various combinations of locking alleviation techniques and their applicability to different ANCF beam variants is studied using numerical examples. Furthermore, the enhanced deformation gradient (EDG) technique, which has been proposed recently in finite element literature, is demonstrated for high-order ANCF beam elements. Based on the numerical tests, none of the currently available techniques are suitable for all types of ANCF elements. The paper also shows that the efficiency and accuracy of the techniques are case-dependent. For the ANCF beam element involving higher-order terms with respect to trapezoidal mode, however, the EDG-based techniques are preferable to reduce locking phenomena.

A computational insight on damage-based constitutive modelling in femur mechanics

The present paper addresses femur failure mechanics, by numerically investigating the influence of brittle/ quasi-brittle bone constitutive description when combined with several failure criteria and different descriptions of bone ultimate parameters. Starting from computed tomography images of an experimentally-tested cadaveric femur, the bone geometry has been reconstructed through a semi-automatic segmentation procedure, and patient-specific material properties have been derived. Loading-induced loss of structural integrity has been simulated through a progressive thermodynamically-based damage model, by introducing different strain- and stress-based damage evolution laws. An in-house displacement-driven incremental approach has been implemented in a finite element framework to mimic the in-vitro experimental procedure. An energy-based regularization technique allowed to obtain results which are mesh independent and therefore physically meaningful. Depending on the adopted modelling strategy, significant differences in terms of yield and failure load, as well as in fracture patterns, have been numerically experienced. Comparisons between the proposed numerical results and the available experimental outcomes have been carried out. For the femur model herein analysed, an elastic quasi-brittle bone description combined with strain-based failure criteria seems to be more effective in predicting the mechanical behaviour up to the fracture. Presented mesh-independent results therefore contribute to justify the need of damage-based approaches for predicting in an effective way failure mechanisms of femurs.

Methodology for Neural Network-Based Material Card Calibration Using LS-DYNA MAT_187_SAMP-1 Considering Failure with GISSMO.

A neural network (NN)-based method is presented in this paper which allows the identification of parameters for material cards used in Finite Element simulations. Contrary to the conventionally used computationally intensive material parameter identification (MPI) by numerical optimization with internal or commercial software, a machine learning (ML)-based method is time saving when used repeatedly. Within this article, a self-developed ML-based Python framework is presented, which offers advantages, especially in the development of structural components in early development phases. In this procedure, different machine learning methods are used and adapted to the specific MPI problem considered herein. Using the developed NN-based and the common optimization-based method with LS-OPT, the material parameters of the LS-DYNA material card MAT_187_SAMP-1 and the failure model GISSMO were exemplarily calibrated for a virtually generated test dataset. Parameters for the description of elasticity, plasticity, tension–compression asymmetry, variable plastic Poisson’s ratio (VPPR), strain rate dependency and failure were taken into account. The focus of this paper is on performing a comparative study of the two different MPI methods with varying settings (algorithms, hyperparameters, etc.). Furthermore, the applicability of the NN-based procedure for the specific usage of both material cards was investigated. The studies reveal the general applicability for the calibration of a complex material card by the example of the used MAT_187_SAMP-1.

Optimization based Feedforward mitigation of Elastic Flexure of Distributed Parameter System due to Reference Tracking

If a large scale structure performs a rigid body motion, this motion is superposed by an elastic flexure due to its inherent elasticity. In order to mitigate the elastic motion feedforward and feedback laws can be applied. By minimizing the L2-norm of the elastic flexure subject to a reference trajectory and the elastic system description we derive a feedforward signal that attenuates elastic motions of the structure. A comparison to simplified solution approaches shows a major improvement for reference signals in the range of the system's natural frequencies.

Elastic Modeling of Two-Step Transitions in Sterically Frustrated 1D Binuclear Spin-Crossover Chains

Among the large family of spin-crossover materials, binuclear systems play an important role due to their specific molecular configurations, allowing the presence of multi-step transitions and elastic frustration. Although this issue benefited from a significant number of spin-based theories, there is almost no elastic description of the spin transition phenomenon in binuclear systems. To overcome this deficiency, in this work we develop the first elastic modeling of thermal properties of binuclear spin-crossover solids. At this end, we investigated a finite spin-crossover open chain constituted of elastically coupled binuclear (A = B) blocks, ⋯A=B−A=B−A=B⋯, in which the considered equivalent A and B sites may occupy two configurations, namely low-spin (LS) and high-spin (HS) states. The sites of the binuclear unit interact via an intramolecular spring and couple to the neighboring binuclear units via other springs. The model also includes the change of length inside and between the binuclear units subsequent to the spin state changes. When injecting an elastic frustration inside the binuclear unit in the LS state, competing interactions between the intra- and the inter-binuclear couplings emerge. The latter shows that according to the intra- and inter-binuclear elastic constants and the strength of the frustration, multi-step transitions are derived, for which a specific self-organization of type (HS = HS)-(LS-LS)-(HS = HS)⋯ is revealed and discussed. Finally, we have also studied the relaxation of the metastable photoinduced HS states at low temperature, in which two relaxation regimes with transient self-organized states were identified when monitoring the elastic frustration rate or the ratio of intra- and intermolecular elastic interactions. These behaviors are reminiscent of the thermal dependence of the order parameters of the system. The present model opens several possibilities of extensions of elastic frustrations acting in polynuclear spin-crossover systems, which may lead to other types of spin-state self-organizations and relaxation dynamics.

Motion of a tightly fitting axisymmetric object through a lubricated elastic tube

We consider the translation of a rigid, axisymmetric, tightly fitting object through a cylindrical elastic tube filled with viscous fluid, using a combination of theory and direct numerical simulations. The intruding object is assumed to be wider than the undeformed tube radius, forcing solid–solid contact in the absence of relative motion. The motion of the object establishes a thin fluid film that lubricates this contact. Our theory couples lubrication theory to a geometrically nonlinear membrane description of the tube's elasticity, and applies to a slender intruding object and a thin tube with negligible bending rigidity. We show using asymptotic and numerical solutions of the theory, that the thickness of the thin fluid film scales with the square root of the relative speed for small speeds, set by a balance of hoop stresses, membrane tension and fluid pressure. While membrane tension is relatively small at the entrance of the film, it dominates near the exit and produces undulations of the film thickness, even in the limit of vanishing speeds and slender objects. We find that the drag force on the intruding object depends on the slope of its surface at the entrance to the thin fluid film, and scales as the square root of the relative speed. The predictions of the lubricated membrane theory for the shape of the film and the force on the intruder are in quantitative agreement with three-dimensional direct numerical simulations of the coupled fluid–elastic problem.

Local micromorphic non-affine anisotropy for materials incorporating elastically bonded fibres

There has been increasing experimental evidence of non-affine elastic deformation mechanisms in biological soft tissues. These observations call for novel constitutive models which are able to describe the dominant underlying micro-structural kinematic aspects, in particular relative motion characteristics of different phases. This paper proposes a flexible and modular framework based on a micromorphic continuum encompassing matrix and fibre phases. In addition to the displacement field, it features so-called director fields which can independently deform and intrinsically carry orientational information. Accordingly, the fibrous constituents can be naturally associated with the micromorphic directors and their non-affine motion within the bulk material can be efficiently captured. Furthermore, constitutive relations can be formulated based on kinematic quantities specifically linked to the material response of the matrix, the fibres and their mutual interactions. Associated stress quantities are naturally derived from a micromorphic variational principle featuring dedicated governing equations for displacement and director fields. This aspect of the framework is crucial for the truly non-affine elastic deformation description. In contrast to conventional micromorphic approaches, any non-local higher-order material behaviour is excluded, thus significantly reducing the number of material parameters to a range typically found in related classical approaches. In the context of biological soft tissue modelling, the potential and applicability of the formulation is studied for a number of academic examples featuring anisotropic fibre-reinforced composite material composition to elucidate the micromorphic material response as compared with the one obtained using a classical continuum mechanics approach.

On the Geometric Description of Nonlinear Elasticity via an Energy Approach Using Barycentric Coordinates

The deformation of a solid due to changing boundary conditions is described by a deformation gradient in Euclidean space. If the deformation process is reversible (conservative), the work done by the changing boundary conditions is stored as potential (elastic) energy, a function of the deformation gradient invariants. Based on this, in the present work we built a “discrete energy model” that uses maps between nodal positions of a discrete mesh linked with the invariants of the deformation gradient via standard barycentric coordinates. A special derivation is provided for domains tessellated by tetrahedrons, where the energy functionals are constrained by prescribed boundary conditions via Lagrange multipliers. The analysis of these domains is performed via energy minimisation, where the constraints are eliminated via pre-multiplication of the discrete equations by a discrete null-space matrix of the constraint gradients. Numerical examples are provided to verify the accuracy of the proposed technique. The standard barycentric coordinate system in this work is restricted to three-dimensional (3-D) convex polytopes. We show that for an explicit energy expression, applicable also to non-convex polytopes, the general barycentric coordinates constitute fundamental tools. We define, in addition, the discrete energy via a gradient for general polytopes, which is a natural extension of the definition for discrete domains tessellated by tetrahedra. We, finally, prove that the resulting expressions can consistently describe the deformation of solids.