Cauchy Born Research Articles

Effect of nonlocality on the dispersion relations of mechanical metamaterials

The advent of mechanical metamaterials, with their unconventional and programmable properties, has revolutionised the field of engineering across length scales. Their design remains an area of interest to researchers requiring careful analysis of unit cells so as to achieve desired mechanical and wave dispersion behaviour. While metamaterials are known to extract their unique properties from the internal geometry of the unit cells, the contributions of the constituent materials are typically accounted for via their local material properties in the form of Young’s modulus and Poisson’s ratio. In this article, we demonstrate that material nonlocality of the constituents can also play a crucial role in modulating the nontrivial properties of metamaterials. Such nonlocality can arise from various sources, such as finer scale inhomogeneity like defects, inclusions etc. Towards this, we propose a nonlocal continuum model by upscaling a discrete potential through a stochastic gradient estimator (SGE) interpreted herein as a generalised Cauchy–Born rule. The nonlocal parameters can also be estimated from experimental data by solving an inverse problem, similar to the estimation of constitutive parameters in classical continuum mechanics. The proposed model accurately captures the dispersion behaviour at the continuum level as compared to its discrete counterpart. We further incorporate the model within the transfer matrix method to derive the dispersion relations of a metabar as a function of the constituent material nonlocality. We envisage that the present study introduces a new handle in the form of constituent material nonlocality for designing architected metamaterials with unprecedented mechanical functionalities.

Violation of the Cauchy–Born rule in multi-principal element alloys

Multi-principal element alloys are a novel class of materials that are formed by combining multiple elements in high concentrations and show exceptional properties compared to conventional alloys. These alloys have high configurational entropy due to inherent atomic disorder. The Cauchy–Born rule is a popular homogenization method for linking atomistic to continuum length scales. In this Letter, we use ab initio density functional theory calculations to report that the Cauchy–Born rule, which holds in conventional alloys in the absence of defects, is not valid in multi-principal element alloys. The violation of the Cauchy–Born rule in these alloys is attributed to the presence of atomic disorder due to which the deformations are inhomogeneous. Our results also show the deviations in deformation are related to the magnitude of the stretch and shear by power laws.

A multiscale FEM-MD coupling method for investigation into atomistic-scale deformation mechanisms of nanocrystalline metals under continuum-scale deformation

This study aims to develop a multiscale bridging method for investigating nanocrystalline metals based on macro-scale deformation. For this purpose, we propose a hierarchical multiscale computational method that can focus on some of the elements in a finite element model for scale bridging to atomistic-scale models. This method assumes that atomistic-scale nanocrystalline models are related to the integration points in a finite element and deform based on the macro-scale deformation. Nanocrystalline aluminum was chosen for the validation of the multiscale method. The finite element method (FEM) and the molecular dynamics (MD) method were used for continuum-scale and atomistic-scale simulations, respectively. We utilized the notion of the CauchyBorn rule (CBR) for communicating deformation information from the continuum scale to the atomistic scale. We studied three different cases with two nanocrystalline models and two loading cases to compare differences resulting from crystal structures and loading. Based on the crystal structure change during relaxation, nonequilibrium grain boundaries (NEGBs) were shown to play a role as deformation mechanisms in the plastic regime and induce the onset and migration of crystal defects, including deformation twins, as reported in the experiment. Furthermore, the crystal orientation dependence of the onset of crystal defects was confirmed by the comparison of the results from the two different nanocrystalline models. The qualitative agreement of the results with experimental observations is also confirmed. The proposed ‘FEM-MD’ method can bridge a large-scale gap, for example, from a nano-scale to a continuum-scale such that an MD model can be coupled to a millimeter or centimeter scale compared to other embedding methods. The present method is ideal for investigating the dislocation behavior of nanocrystalline materials, which contain multi-grained nanostructure at finite temperature, undergoing various loading scenarios at the macro-scale.

Elastic properties and constitutive behaviour of graphene at finite temperature and large deformation

In the earlier reported theoretical studies, the bending modulus of the graphene sheet predicted with the inclusion of the dihedral energy term is observed to be almost double as compared to those reported without considering the dihedral energy term. The bending modulus reported considering dihedral energy term is observed to be more accurate when compared with the quantum mechanical calculations. The findings of these studies for bending modulus (with/without considering dihedral energy terms) are reported without considering the effect of finite temperature. In the present work, a multiscale computational framework incorporating the dihedral energy term and finite temperature in the constitutive model is presented for the first time. A temperature-dependent bond length is expressed in terms of strain and curvature tensor through the temperature-related quadratic-type Cauchy-Born rule. The total Helmholtz free energy of the unit cell is expressed as a sum of the potential energy of the unit cell and thermal energy due to the thermal vibrations of atoms arising from the finite temperature. The atomic interactions are modelled through the Tersoff-Brenner potential considering different sets of empirical parameters, including the authors’ recently proposed set of empirical parameters. The stress/moment tensors and the tangent constitutive matrix are obtained from the derivatives of the Helmholtz free energy density with strains and curvatures. The thermal properties, elastic properties, and tangent stiffness coefficients for armchair and zigzag graphene with/without dihedral energy term at finite strain, curvature, and temperature are investigated and discussed in detail. The proposed multiscale model facilitates the accurate prediction of the constitutive behaviour of the graphene in a thermal environment with remarkable improvements for the prediction of bending modulus due to the inclusion of the dihedral energy term at finite temperature.

Coupling diffusion and finite deformation in phase transformation materials

We present a multiscale theoretical framework to investigate the interplay between diffusion and finite lattice deformation in phase transformation materials. In this framework, we use the Cauchy–Born Rule and the Principle of Virtual Power to derive a thermodynamically consistent theory coupling the diffusion of a guest species (Cahn–Hilliard type) with the finite deformation of host lattices (nonlinear gradient elasticity). We adapt this theory to intercalation materials – specifically Li1−2Mn2O4 – to investigate the delicate interplay between Li-diffusion and the cubic-to-tetragonal deformation of lattices. Our computations reveal fundamental insights into the microstructural evolution pathways under dynamic discharge conditions, and provide quantitative insights into the nucleation and growth of twinned microstructures during intercalation. Additionally, our results identify regions of stress concentrations (e.g., at phase boundaries, particle surfaces) that arise from lattice misfit and accumulate in the electrode with repeated cycling. These findings suggest a potential mechanism for structural decay in Li2Mn2O4. More generally, we establish a theoretical framework that can be used to investigate microstructural evolution pathways, across multiple length scales, in first-order phase transformation materials.

Beyond the Classical Cauchy–Born Rule

Beyond the Classical Cauchy–Born Rule

Emergent elasticity relations for networks of bars with sticky magnetic ends

Magneto-elastic materials have unique mechanical properties arising from the coupling between the magnetic and elastic components, which can be used in many engineering applications, such as waveguide systems, impact mitigation, and soft robotics. Traditional designs of magneto-elastic materials embed magnets or magnetic particles in a monolithic body. Under extreme conditions, the elastic matrices are prone to permanent damage and loss of functionality. An alternative framework was previously proposed by the authors in which 2D magneto-elastic networks were created through vibration-driven self-assembly from a dilute system of elastic bars with sticky magnetic ends. While these networks were demonstrated to fail gracefully under extreme loading and re-assemble under random excitation, how their emergent elasticity depends on magnet and elastic member characteristics remains to be fully understood. In this paper, to calculate the 2D bulk modulus of a given magneto-elastic network, particle dynamics simulations are first performed. An analytical framework is then developed to predict the bulk modulus of derived networks with varying design parameters, such as bar length and magnetic strength, based on the Cauchy–Born approximation. Through dimensional analysis, we demonstrate that 2D bulk modulus is linearly proportional to a composite variable that combines magnet strength, elastic member length, and wall thickness of the magnet holder, as evidenced by the collapse of 120 systems onto a single line. The numerical and analytical investigation of the bulk modulus of this architected magneto-elastic material demonstrates the broad tunability of its emergent elasticity, thereby enabling a myriad of promising applications of these systems.

A computational approach to obtain nonlinearly elastic constitutive relations of strips modeled as a special Cosserat rod

This study presents a computational approach to obtain nonlinearly elastic constitutive relations of strip/ribbon-like structures modeled as a special Cosserat rod. Starting with the description of strips as a general Cosserat plate, the strip is first subjected to a strain field which is uniform along its length. The Helical Cauchy–Born rule is used to impose this uniform strain field which deforms the strip into a six-parameter family of helical configurations — the six parameters here correspond to the six strain measures of rod theory. Two vector variables are introduced to model the position of the deformed centerline of the strip’s cross-section and to model orientation of thickness lines along the strip’s width. The minimization of the strip’s plate energy under the constraint of the aforementioned uniformity in strain field reduces the partial differential equations of plate theory from the entire mid-plane of the strip to just a system of nonlinear ordinary differential equations along the strip’s width line for the above mentioned two vector variables. A nonlinear finite element formulation is further presented to solve this set of ordinary differential equations. This, in turn, yields the strip’s stored energy per unit length as well as the induced internal force, moment and stiffnesses of the strip for every prescribed set of six strain measures of rod theory. The proposed scheme is used to study uniform bending, twisting and shearing of a strip. For the case of uniform twisting and shearing, the strip is also seen to buckle along its width into a more complex configuration which are accurately captured by the presented scheme. We demonstrate that the presented scheme is more general and accurate than the existing schemes available in the literature.

Continuous modelling of single layer 2D carbon nanostructures based on a modified Cauchy–Born rule

AbstractGraphene is the thinnest known structure. It consists of a single layer of carbon atoms and has unique mechanical properties. The carbon atoms form bonds with neighbouring atoms that can be described by interatomic potentials. A simulation of molecular mechanical processes can be performed by applying the formalism of the finite element method to the interatomic potentials. Neglecting local effects, a continuum formulation is an efficient alternative to a discrete model. The continuum formulation requires a connection between the discrete atomic lattice and the continuum. One way of doing this is to use the Cauchy–Born rule. This paper reviews the extensions of the Cauchy–Born rule, in particular, the exponential Cauchy–Born rule, and adds an alternative approach. It is shown that all contributions dealing with the exponential Cauchy–Born rule inevitably lead to shell models with Kirchhoff–Love kinematics. The approach presented here instead uses a shell model with Reissner–Mindlin kinematics, which has the major advantage of a much simpler mesh generation and a simpler application of the boundary condition.

A multiscale two-dimensional finite element incorporating the second-order Cauchy–Born rule for cohesive zone modeling: Simulation of fracture in polycrystalline materials

In this work, a multiscale cohesive zone model is developed by elaborating the second-order Cauchy–Born (CB) rule to simulate crack growth in polycrystalline solids. To do so, a two-dimensional domain is discretized with bulk elements and finite width cohesive zone interphase. As polycrystalline materials are considered, the grain boundaries are meshed with cohesive zone interphase elements, allowing crack to propagate in the boundaries as well as through the grains. The correlation between the micro and the macroscale material properties is specified using CB rule, as the first-order CB is utilized in the bulk elements, and the second-order CB is utilized in the cohesive interphase elements. An extensive numerical study has been performed for understanding the effect of (1) the boundary conditions, (2) the integration scheme, and (3) the mesh refinement. Moreover, a comparison between the proposed method and the one elaborating the depletion potential for cohesive zones calculations has been performed. Also, the applicability of the model to predict fractures due to defects and vacancies is studied, which shows the ability of the model to predict both brittle and ductile fractures. It should be noted that the proposed model elaborates the same constitutive relations for the bulk elements and the cohesive zone elements, making this model a consistent one, unlike most of other cohesive zone models that employ different laws depending on the different element types.

An atomistic entropy based finite element multiscale method for modeling amorphous materials

A new concurrent multiscale method based on the maximum entropy statistical method is proposed for the analysis of amorphous materials. In addition to reducing the number of degrees of freedom, any irregular structure of amorphous materials can be accurately analyzed. The amorphous structure is generated from a solid crystalline structure by a heating/cooling process without the need for any specific independent technique to create such a random structure. The method is expected to perform efficiently because of its entropic and irregular intrinsic. Regions with moderate conditions are discretized by the entropy-based finite element method while the severe parts are simulated by the present atomistic-based multiscale technique. The new proposed approach allows for accurate analysis of amorphous structures across multiple scales and does not suffer from conventional complications such as the standard Cauchy Born rule and consistency of the molecular structure with the standard finite element geometries.The conventional Cauchy-Born rule cannot be directly used due to the non–crystalline microstructure of the material. A remedy is proposed based on the meshfree techniques by constructing a continuous atomic deformation field from the imposed macro deformation gradient. The resultant deformation gradient and the stress field remain consistent in micro scales. In addition, a genetic algorithm-based method, which has less sensitivity to the choice of initial point and number of parameters, is adopted for the maximization of the entropy function.The silicon amorphous structure is considered for MD simulations. It is obtained by quenching from a melted sample. The MD-obtained structure is further analyzed and the predicted displacements and stress contours, as well as the density and radial distribution functions are examined to assess the state of the material. Then, the proposed meshfree technique is applied to construct the continuous form of the deformation gradient on the MD model to improve the accuracy of the solution. The proposed concurrent multiscale method is verified and then employed to simulate an amorphous silicon specimen. Finally, the effects of sample size, strain rate and quenching speed on rupture stress and strain in different 3D tensile simulations are investigated by the proposed multiscale method.

Temperature-dependent multiscale modeling of graphene sheet under finite deformation

The homogenized constitutive models that have been utilized to simulate the behavior of nanostructures are typically based on the Cauchy–Born hypothesis, which seeks the fundamental properties of material via relating atomistic information to an assumed homogeneous deformation field. It is well known that temperature has a profound effect on the validity and size-dependency of the Cauchy-Born hypothesis in finite deformations. In this study, a temperature-related Cauchy–Born formulation is established for graphene sheets and its performance is examined through a direct comparison between the continuum-based constitutive model and molecular dynamics analysis. It is demonstrated that if the temperature increases, the validity surfaces shrink, which is not highly dependent on the size of specimen. At finite strains, graphene sheet exhibits material softening and hardening, which are manifested in the elastic constants. The developed constitutive model is used to study the effect of applied deformations at various temperature levels on the elastic constants of graphene sheets. Finally, a novel multiscale finite element approach is developed for the analysis of graphene sheets in finite strain regime based on the proposed atomistic-continuum model. The method can be used efficiently in the simulation of large specimens where the application of molecular dynamics requires considerable computational efforts. The efficiency and robustness of the proposed multiscale approach are shown through several numerical examples.

A modified applied element model for the simulation of plain concrete behaviour

A modified applied element model to simulate the behaviour of plain concrete continuum structures including discrete cracking is proposed in this study. In the classical applied element model, Poisson effects are fully ignored. To remediate this issue, diagonal elements are introduced to include the Poisson effect, and the constitutive parameters are rigorously determined using the Cauchy–Born rule and the hyper-elastic theory. The formulation is validated for linear elastic problems and the consistency and convergence behaviour of the numerical approach is shown. Tensile softening formulation using the concept of fracture energy is utilised for the nonlinear range. In this range, the approach is validated using the classical benchmark tests with pure tensile, split-tensile, combined shear-tensile and bending dominated push-over loading. The load–displacement behaviour and crack response were captured successfully, showing the proposed methodology can be used to quantify discrete cracks on large systems, such as dam monoliths, from initiation to significant damage levels.

A computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy

We present a computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy. The framework allows the rod’s cross-section to have arbitrary shape and the rod material to obey arbitrary three-dimensional material model for its bulk volume and arbitrary surface energy model for its lateral surface. The kinematics of the Helical Cauchy–Born rule is used to first construct a family of six-parameter (corresponding to the six strain measures of rod theory) helical rod configurations in which the generated three-dimensional strain field is uniform in the rod’s arc-length coordinate. This uniformity allows the three-dimensional nonlinear equations of elasticity to reduce to just the rod’s cross-sectional plane wherein the cross-section’s boundary also experiences traction due to the inherent surface energy in the rod’s lateral surface. The solution of this cross-sectional problem yields the induced in-plane displacement and out-of-plane warping in the cross-section for arbitrarily prescribed strain measures of the rod. We then derive expressions for the induced internal contact force, moment and stiffnesses of the rod in terms of the solution of the deformed cross-section and the prescribed strains of the rod. A finite element formulation is presented to solve the nonlinear cross-sectional deformation problem and further obtain the induced internal contact force, moment and stiffnesses. The presented framework will be useful for modeling of nanorods where surface energy plays a dominant role. Several numerical examples are presented using the presented framework to illustrate the effect of surface energy parameters on the deformation of the nanorod’s cross-section and the nanorod’s bending, torsional, extensional and shearing stiffnesses. We also derive improved analytical formulas for the extensional and bending stiffnesses of isotropic rectangular nanorods in their reference state. These formulas reduce to the existing widely used formulas for a special choice of material parameters, i.e., when the surface Poisson’s ratio and the bulk Poisson’s ratio match thus highlighting the limitation of the existing formulas.

Multiscale analysis method and experiments for the fracture toughness optimization analysis of carbon nanotube-epoxy composites

To better understand the facture mechanics and optimize the fracture toughness of carbon nanotube (CNT)-epoxy composites, a multiscale analysis method was established in this paper. The multiscale analysis method includes three scales: (1) at the microscale, an interfacial model was established based on the Cauchy-Born rule to deduce the relationship between the interfacial stress tensor and CC bond vector; (2) at the mesoscale, the transition condition of the failure mode from the CNT pullout to CNT fracture was established based on the shear-lag theory. And the relationships between the pulling force and CNT pullout displacement were given corresponding to these two failure modes; (3) at the macroscale, the analytical solution of fracture toughness increment was given to conduct the fracture toughness optimization analysis. By validating this method by the experiment, we found that the optimal CC bonds volume density and interfacial length are reached under the transition condition of the failure mode from the CNT pullout to CNT fracture. At last, we proposed a method to approximately estimate the optimal CC bonds volume density and CNT length.

Dynamic Response of Some Noncarbon Nanomaterials Using Multiscale Modeling Involving Material and Geometric Nonlinearities

Abstract Nonlinear dynamic response of some noncarbon nanomaterials, involving material and geometric nonlinearities under different types of dynamic loads, is investigated using computationally efficient multiscale modeling. Multiscale-based finite element model is developed in the framework of the Cauchy–Born rule, which couples the deformation at the atomic scale to deformation at the continuum scale. The Tersoff–Brenner type interatomic potential is employed to model the atomic interactions. The governing finite elemental equations are derived through Hamilton's principle for a dynamic system. The linearization of nonlinear discrete equations is done using Newton–Raphson method and are solved using Newmark's time integration technique. The effects of material and geometric nonlinearities, inherent damping, different types of dynamic loads, and initial strain on the transient response of noncarbon nanosheets with clamped boundary conditions are reported in detail. The present results obtained from the multiscale-based finite element method are compared with those obtained from molecular dynamics (MD) simulation for the free vibration analysis, and the results are found to be in good agreement. The present results are also compared with the results of those obtained from Kirchhoff plate model for some cases.

Homogenization in a simpler way: analysis and optimization of periodic unit cells with Cauchy–Born hypothesis

Homogenization in a simpler way: analysis and optimization of periodic unit cells with Cauchy–Born hypothesis

Plasticity in Amorphous Solids Is Mediated by Topological Defects in the Displacement Field

The microscopic mechanism by which amorphous solids yield plastically under an externally applied stress or deformation has remained elusive in spite of enormous research activity in recent years. Most approaches have attempted to identify atomic-scale structural "defects" or spatiotemporal correlations in the undeformed glass that may trigger plastic instability. In contrast, in this Letter we show that the topological defects that correlate with plastic instability can be identified, not in the static structure of the glass, but rather in the nonaffine displacement field under deformation. These dislocation-like topological defects (DTDs) can be quantitatively characterized in terms of Burgers circuits (and the resulting Burgers vectors) that are constructed on the microscopic nonaffine displacement field. We demonstrate that (i)DTDs are the manifestation of incompatibility of deformation in glasses as a result of violation of Cauchy-Born rules (nonaffinity); (ii)the resulting average Burgers vector displays peaks in correspondence of major plastic events, including a spectacular nonlocal peak at the yielding transition, which results from self-organization into shear bands due to the attractive interaction between antiparallel DTDs; and (iii)application of Schmid's law to the DTDs leads to prediction of shear bands at 45° for uniaxial deformations, as widely observed in experiments and simulations.

Finite temperature atomistic‐informed crystal plasticity finite element modeling of single crystal tantalum (α‐Ta) at micron scale

AbstractIn this work, we have developed a temperature‐dependent higher‐order Cauchy–Born (THCB) rule for multiscale crystal defect dynamics (MCDD) of crystalline solids based on the harmonic approximation. As a template, we employed the THCB rule to develop an atomistic‐informed constitutive model for the body‐centered cubic (BCC) single crystal tantalum (‐Ta). Considering the effect of strain gradients in different process zone elements, the corresponding higher order stress are used to model crystal plasticity of single crystal ‐Ta. Different from face‐centered cubic crystals, BCC crystals are strongly influenced by temperature. It is shown in this article that the developed finite temperature atomistic‐informed crystal plasticity finite element method is able to capture the temperature‐dependent dislocation substructure and hence crystal plastic deformation. The main contributions and novelties of the present work are highlighted by following findings: (1) A THCB rule and an atomistic‐informed strain gradient theory have been developed, and the corresponding temperature‐related higher‐order stress and elastic tensor formulations are derived; (2) The finite temperature MCDD provides an atomistic‐informed crystal plasticity finite element method that can simulate anisotropic crystal plasticity in any orientation within stereographic triangle at micron scale and above; (3) The developed MCDD is able to capture the non‐Schmid effects of BCC single crystal tantalum (‐Ta); (4) The developed MCDD is able to capture the size effect of single crystal plasticity; and (5) The finite temperature MCDD can simulate the temperature dependent dislocation substructure, and it captures cross‐slip in single crystal tantalum at low temperature (∼20 K) and captures dislocation cell structure at high temperature (∼500 K).

Multiscale Modelling and Mechanical Anisotropy of Periodic Cellular Solids with Rigid-Jointed Truss-Like Microscopic Architecture

This paper investigates the macroscopic anisotropic behavior of periodic cellular solids with rigid-jointed microscopic truss-like architecture. A theoretical matrix-based procedure is presented to calculate the homogenized stiffness and strength properties of the material which is validated experimentally. The procedure consists of four main steps, namely, (i) using classical structural analysis to determine the stiffness properties of a lattice unit cell, (ii) employing the Bloch’s theorem to generate the irreducible representation of the infinite lattice, (iii) resorting to the Cauchy–Born Hypothesis to express the microscopic nodal forces and deformations in terms of a homogeneous macroscopic strain field applied to the lattice, and (iv) employing the Hill–Mandel homogenization principle to obtain the macro-stiffness properties of the lattice topologies. The presented model is used to investigate the anisotropic mechanical behavior of 13 2D periodic cellular solids. The results are documented in three set of charts that show (i) the change of the Young and Shear moduli of the material with respect to their relative density; (ii) the contribution of the bending stiffness of microscopic cell elements to the homogenized macroscopic stiffness of the material; and (iii) polar diagrams of the change of the elastic moduli of the cellular solid in response to direction of macroscopic loading. The three set of charts can be used for design purposes in assemblies involving the honeycomb structures as it may help in selecting the best lattice topology for a given functional stiffness and strength requirement. The theoretical model was experimentally validated by means of tensile tests performed in additively manufactured Lattice Material (LM) specimens, achieving good agreement between the results. It was observed that the model of rigid-joined LM (RJLM) predicts the homogenized mechanical properties of the LM with higher accuracy compared to those predicted by pin-jointed models.