Classical Continuum Mechanics Research Articles

An extended continuum-mechanics standard linear solid rheology for fluid-saturated porous rock

SUMMARY In a large body of rock-physics research, seismic wave velocity dispersion and attenuation in fluid-saturated porous rock are studied by constructing analytical or numerical models for time- or frequency-dependent dynamic (effective, or viscoelastic) moduli. A key and broadly used model of such kind is the Zener's, or the standard linear solid (SLS). This model is qualitatively successful in explaining many field and laboratory observations and serves as the key element of many generalizations such as the Burgers model for plastic deformations or the generalized SLS explaining band-limited or near-constant seismic attenuation. However, as a physical model of fluid-saturated porous rock, the SLS has several major limitations: disregard of inertial effects, absence of secondary wave modes and lack of key physical parameters such as porosity and Skempton coefficients. Grainy and porous rock is an unconsolidated material in which the effective density is frequency-dependent, and its effects on wave velocities may exceed those of the dynamic modulus. To overcome these limitations of the empirical SLS, we propose a rigorous rheologic model based on classical continuum mechanics and called the extended SLS, or eSLS. This rheology explains the available attenuation and dispersion observations equally well, but it is also close to Biot's model, honours all poroelastic relations, includes inertial effects, and reveals several new physical properties of the material. Detailed comparison of the eSLS and Biot's models gives a physical-mechanism-based classification of wave-induced fluid flow (WIFF) phenomena. In this classification, the so-called ‘global-scale’ flows occur in Biot's type structures within the material, whereas the ‘local-scale’ WIFF occurs in eSLS-type structures. Combining Biot's and eSLS models gives a broad class of rheologies for linear anelastic phenomena within rock with a single type of porosity. The model can be readily generalized to multiple porosities and different types of internal variables, such as describing squirt flows, wetting or thermoelastic effects. Modelling is conducted with relatively little effort, using a single matrix equation similar to a mechanical form of the standard SLS. By combining the eSLS and Biot's models, observations of dynamic-modulus dispersion and attenuation can be inverted for macroscopic mechanical properties of porous materials.

An ordinary state-based peridynamic model for granular fracture in polycrystalline materials with arbitrary orientations in cubic crystals

Granular fracture holds significant implications in material mechanics. However, the previous studies in ordinary state-based peridynamic (OSB-PD) framework often neglect the internal crystalline structure of particles or only consider limited crystal orientation. To address this gap, a novel OSB-PD model for granular fracture within polycrystalline materials is proposed, in which the periodic functions are incorporated in the PD strain energy density, taking into account the inherent random orientation in cubic crystals. By comparing energy density from PD and the classical continuum mechanics, four PD material parameters are defined. Moreover, the corresponding surface correction method in the global coordinate system is also proposed. Several numerical examples including fracture analysis of polycrystalline materials are conducted to validate the effectiveness of the proposed method. The proposed ordinary state-based peridynamic model offers a fresh perspective for investigating granular fracture behaviors within polycrystalline materials.

Exploring novel mechanical metamaterials: Unravelling deformation mode coupling and size effects through second-order computational homogenisation

Architected materials and mechanical metamaterials are known for their unique macroscopic properties and complex behaviour that often defy conventional continuum mechanics. Therefore, in this contribution, a recent multi-scale second-order computational homogenisation method (Dos Santos et al., 2023) is employed to explore these materials under finite strains. The approach combines a second gradient continuum theory at the macro-scale and a representative volume element (RVE) with classical first-order continuum mechanics at the micro-scale. The Method of Multi-scale Virtual Power ensures a consistent scale transition. The predictive capability and applicability of the second-order computational strategy are evaluated through coupled multi-scale numerical simulations. These simulations involve two- and three-dimensional problems, with a strong focus on the development of novel metamaterials, while also accounting for diverse loading conditions, such as tension/compression-induced undulation, bending, and compression-induced torsion. Comparisons with first-order homogenisation and Direct Numerical Simulations validate the approach. Analysis of homogenised consistent tangents reveals valuable insights into macroscopic properties. Overall, the results highlight the capability of the second-order strategy to capture significant phenomena, including second-order deformation modes, coupling deformation mechanisms, and size effects.

A 3D peridynamic model for fracture analysis of transversely isotropic solids

The classical continuum mechanics (CCM) has limitations in simulating crack initiation and propagation, and the peridynamic (PD) theory overcomes these shortcomings. This study proposed a 3D transversely isotropic PD model based on the micro-beam bond model. Firstly, the micromoduli of the micro-beam bond applied to the transversely isotropic 3D solid model are derived based on the elastic matrix relationship between the different coordinate systems. Secondly, the strain energy density expressions of arbitrary material points are expressed in both the PD theory and the CCM theory using strain and elastic matrix under the assumption of homogenization, respectively. Finally, by equating the strain energy density expressions under the two theories based on the conservation of energy, the constitutive model parameters of the 3D transversely isotropic PD model are solved. Numerical examples of crack propagation demonstrate the effectiveness of the proposed model in addressing dynamic 3D fracture problems.

Constitutive modeling of electro-magneto-rheological fluids before yielding

This article presents a comprehensive investigation into electro-magneto-rheological (EMR) fluids that can exhibit reversible behavior changes when exposed to electrical and magnetic fields. The behavior of EMR fluids changes to a solid-like state under the influence of external fields prior to yielding. This research emphasizes the solid-like behavior of these smart materials when subjected to external fields. To establish a constitutive relation, the study makes use of Ericksen’s analysis in addition to classical continuum mechanics. The shear stress that results from the interaction with the external field is developed further using this relationship. The analytical results are validated with previously published research to ensure their validity and dependability. The findings of the study will also aid in the design and development of EMR fluid-based systems, enabling a range of practical applications.

Adaptive coupling of non-ordinary state-based peridynamics and classical continuum mechanics for fracture analysis

In this study, we proposed a coupled model between non-ordinary state-based peridynamics (NOSPDs) and classical continuum mechanics (CCMs) using the Morphing strategy for both quasi-static and dynamic fractures. We leveraged a novel strategy to approximate the deformation energy of the NOSPD model, which can unify the existing BPD and OSPD cases. We established the unified governing equations for NOSPDs and CCMs and applied the generalized strength criteria in the coupling model to activate the NOSPDs only when necessary. The critical stretch is recalibrated for the NOSPD model, and generalized strength-based bond failure criteria are proposed to determine bond failure. The model was numerically implemented using the finite element method (FEM) supplemented with discrete elements in the NOSPD subdomain. Parallel in-house code is written on graphics processing units (GPU) using the Compute Unified Device Architecture (CUDA). The 1-, 2- and 3-dimensional (D) numerical results demonstrated the accuracy and efficiency of the proposed method. Notably, the relative errors due to coupling during wave propagation in a 1D bar and uniform deformation in a 2D plate were less than 0.0375% and 0.14%, respectively. Further, the observed crack propagation speed aligned with the reference value.

Relativistic constitutive modeling of inelastic deformation of continua moving in space-time

Since Einstein introduced the theory of relativity, several scientific observations have proven that it thoroughly approximates the motion of materials in space-time. Thus, efforts have been made to expand classical continuum mechanics to relativistic continuum models to consider relativistic effects. However, most models consider only the elastic range without accounting for inelastic deformation. A relativistic inelasticity model should provide objective measures of the inelastic deformation for different observers with nonzero relative velocities over a wide speed range, even when the moving speed is close to the speed of light. This study presents a mathematical modeling structure of the relativistic constitutive equations of the inelastic deformation of a material moving over a wide speed range to provide objective measures for observers. In this model, the deformation tensors of classical mechanics are expanded to four-dimensional tensors (referred to relativistic Cauchy–Green deformation tensors) in space-time, based on which constitutive equations of inelastic deformation are introduced. The four-dimensional tensors are objective about the homogeneous Lorentz transformation in the Minkowski space-time, and the material dissipation, determined employing the proposed modeling of inelastic deformation, satisfies the second law of thermodynamics. In addition, an example illustrates the application of this theory by explaining the physical meaning of modeling. Finally, it is also demonstrated that the proposed relativistic inelasticity model collapses into a classical inelasticity model when the speed of motion is much slower than the speed of light.

A continuum mechanical model of cell motion driven by a biphasic traction stress.

The aim of this paper is to place the cell locomotion problem within the general framework of classical continuum mechanics, and while doing so, to account for the deformation of the actin network in the cytoskeleton; the myosin activity on the lamellum including its effect on depolymerization at the trailing edge; model the stress-dependent driving forces and kinetic laws controlling polymerization at the leading edge, depolymerization at the trailing edge and ATP hydrolysis consistently with the dissipation inequality; and, based on the observations in Gardel et al. (Gardel et al. 2008 J. Cell Biol. 183, 999-1005 (doi:10.1083/jcb.200810060)), include a biphasic velocity-dependent traction stress acting on the actin network. While we chose certain specific models for each of these, in part to allow for an analytical solution, the generality of the framework allows one to readily introduce different constitutive laws to describe these phenomena as might be needed, for example, to study some different type of cells. As described in §5, the predictions of the model compare well with observations such as the magnitude of the very different actin retrograde speeds in the lamellum and lamellipodium including their jump at the interface, the magnitude of the cell speed, and the relative lengths of the lamellipodium and lamellum.

Analysis of wave-induced crack propagation through a coupling algorithm of the peridynamics and isogeometric analysis

Wave propagation analysis is important in the fields of defect detection, explosion protection, etc. Numerical methods derived from classical continuum mechanics (CCM) are widely applied in the analysis of wave propagation, but these methods cannot predict the cracks induced by wave propagation. Peridynamics (PD) theory is a nonlocal framework, which applies integral equations and remains valid in the presence of discontinuities such as cracks, so it can simulate fracture and damage. In this study, we develop a coupling model of isogeometric analysis (IGA) and bond-based PD theory to study the cracks induced by wave propagation in linear elastic solids. First, the wave propagation process based on bond-based PD theory is investigated to explore the wave dispersion phenomenon due to the spatial discretization of PD. Second, one-dimensional (1D) and two-dimensional (2D) IGA-PD numerical examples are tested to explore wave propagation in 1D and 2D structures, respectively. Third, the wave-induced fracture of the material and the law of crack propagation are investigated. Compared to the traditional CCM method and the pure PD method, the advantage of the proposed IGA-PD model is that it can simulate the material damage and crack propagation caused by wave propagation and improve the computational efficiency. The results show that the proposed coupling method can be successfully applied to simulate wave propagation and the damage and crack propagation it induces.

Comprehensive scrutiny of surface mechanical parameters of graphyne-based materials for metal-ions and metal-air batteries applications: A perspective and a hybrid atomistic-continuum model

Graphyne family, as a low-dimensional carbon allotrope, is utilized for metal-ion and –air batteries as the anode and cathode materials due to their unique characteristics. These novel carbon nanomaterials can provide more theoretical storage capacity than graphene, while their specific surface mechanical parameters and vibrational frequencies can influence battery performance. However, there needs to be clear information that can explain the effects of these parameters on battery efficiency. In this respect, the influences of the surface elasticity, residual stress, density, vibrational modes, and frequencies of α, β, and γ-graphyne circular monolayers on anode and cathode materials have been investigated in current research using the hybrid molecular dynamics simulation and modified classical continuum mechanics methods. Concerning metal-ion batteries, the results disclose that the suitability sequence for anode materials can be expressed as γ-graphyne > graphene > β-graphyne > α-graphyne monolayers. In the case of metal-air batteries, the competence sequence for cathode materials is in the order of γ-graphyne > β-graphyne ∼ graphene > α-graphyne monolayers. Therefore, γ-graphyne with suitable storage capacity, low volume expansion, good surface elasticity, low surface residual stress, and appropriate specific surface area is the promising anode and cathode material for metal-ion and -air batteries, respectively.

Numerical simulation of impact process between spherical ice and a rigid plate based on the ordinary state-based peridynamics

Ice sphere impacts are an important issue in the study of composite materials, polar engineering, and aerospace engineering. The traditional mechanics numerical methods, rooted in classical continuum mechanics, like the Finite Element Method (FEM), grapples with the challenge of spatial derivatives not existing at the crack tip or crack surfaces, rendering it unsuitable for addressing discontinuity-related problems such as fracture and failure. Developed particle method peridynamics shows great ability in handling such discontinuous problems. This paper establishes an ice sphere impact model using the peridynamics method. To verify the accuracy of our model, we compared the simulated results with the tested results of the high-speed impacting ice sphere and obtained that the predicted impact load and crack propagation of ice spheres are consistent well with the experiment results. An isolated damage zone was found to appear at the top region when the surface crack just extends across the hemispherical plane. This phenomenon happens due to that the impact wave propagating along the sphere center axis reaches the top end faster than the surface wave, making spalling damage occurred at the top region, and the encounter between the reflected wave and the surface wave would make the spalling damage more severe.

Improved numerical integration for locking treatment in the Peridynamic Timoshenko beam model

PurposeIn this paper, the standard Peridynamic Timoshenko beam model accounting for the shear deformation is chosen to describe the thick beam kinematics. Unfortunately, when applied to very thin beam structures, the standard Peridynamics (PD) encounters the shear locking phenomenon, leading to incorrect solutions.Design/methodology/approachPD differs from classical continuum mechanics and other nonlocal theories that do not involve spatial derivatives of the displacement field. PD is based on the integral equation instead of differential equations to handle discontinuities and other singularities.FindingsThe shear locking can be successfully alleviated using the developed selective integration method. In particular, this technique has been implemented in the standard PD, which allows an accurate result for a wide range of slenderness from very thin to thick (10 < L/t < 103) structures. It can also accelerate the computational time for particular dynamic problems using fewer neighboring integration particles. Several numerical examples are solved to demonstrate the effectiveness of the proposed method for modeling beam structures.Originality/valueThe paper highlights the severe shear locking phenomenon in the Peridynamic Timoshenko beam available in the literature, especially for very thin structures. A new alternative for the alleviation of shear locking in the Peridynamic Timoshenko beam, using selective integration. Hence the developed Peridynamic Timoshenko beam model is effective for thin and thick structures. A new peridynamic formulation for the low-velocity impact beam models is presented and validated.HighlightsThe paper highlights the severe shear locking phenomenon in the Peridynamic Timoshenko beam proposed in the literature, especially for very thin structures.The developed Peridynamic Timoshenko beam model based on selective integration is effective for thin and thick structures.A new peridynamic formulation for the low-velocity impact beam models is presented and validated.

Element‐based coupling modeling of peridynamics and classical continuum mechanics for dynamic brittle fracture

AbstractIn this work, the hybrid peridynamic and classical continuum mechanical model based on the element‐type numerical discretization is used to study the dynamic fracture problem of brittle materials. The weak form of the motion equation for the hybrid model is derived by applying Hamilton's principle. An element‐based spatial discretization scheme is set up using the peridynamics‐based finite element method (PeriFEM) (Han and Li). The ‐convergence and m‐convergence of the hybrid model are analyzed to verify the numerical convergence and the mesh dependency. The influence of the area of the peridynamic model subdomain and the hybrid model subdomain on the calculation efficiency and accuracy is discussed. And the relationship between the crack propagation speed and the crack branching phenomenon is also investigated. Good agreements are obtained by comparing the crack path and the crack speed profile predicted by the hybrid model with those in the literature. Numerical results verify the ability and efficiency of the hybrid model for dynamic fracture problem.

Investigation of microstructure evolution accounting for crystal plasticity in the multiphase‐field method

AbstractRegarding microstructured materials, a quantitative prediction of phase transformation processes is highly desirable for a wide range of applications. With respect to polycrstalline materials, the plastic material behavior is commonly investigated using a crystal plasticity (CP) theory, since it accounts for the underlying microstructure, that is, slip systems of the crystal lattice. In classical continuum mechanics, grain boundaries (GBs) are commonly modeled as material singular surfaces. However, the tracking of moving GBs, present during phase transformation processes, is numerically challenging and costly. This can be circumvented by the use of a multiphase‐field method (MPFM), which provides a numerically highly efficient method for the treatment of moving interfaces, considered as diffuse interfaces of finite thickness. In this work, the microstructural evolution is investigated within the MPFM accounting for CP. The implementation of the constitutive material behavior within the diffuse interface region accounts for phase‐specific plastic fields and the jump condition approach. To improve the understanding of the impact of plastic deformation on the phase evolution, a single inclusion problem is analyzed. Within a plastically deformed matrix, the shape evolution of a purely elastic inclusion with a different Young's modulus, referred to as inhomogeneity, is investigated. It is shown, how the anisotropic plastic behavior affects the phase evolution. The resulting equilibrium shapes are illustrated and examined.

Closed-Form Wave Dispersion Relationships for Ordinary State-Based Peridynamics

AbstractPeridynamics (PD) is a new continuum mechanics formulation introduced to overcome limitations of classical continuum mechanics (CCM). This is mainly achieved by using integro-differential equations rather than partial differential equations. Another important difference of PD is its nonlocal nature with respect to local characteristic of CCM. Moreover, it has a length scale parameter, horizon, defining the range of nonlocal interactions between material points. This nonclassical behaviour also shows itself for dispersion relationships. As opposed to linear dispersion relationships for CCM, PD dispersion relationships are non-linear similar to the observed in experiments. In this study, closed-form dispersion relationships are provided for ordinary state-based peridynamics which is one of the most common PD formulations. Finally, derived closed-form solutions are used to demonstrate the dispersion relationships for various material systems including copper, gold, silver and platinum.

Second-order multi-scale modelling of natural and architected materials in the presence of voids: Formulation and numerical implementation

This contribution proposes a second-order computational homogenisation formulation for natural and architected materials in the presence of voids. The macro-scale is described by a second gradient continuum theory in the finite strain regime, and the micro-scale is modelled by the concept of representative volume element (RVE) within the classical first-order continuum mechanics. The Method of Multi-scale Virtual Power is employed to link the two scales, ensuring a variationally consistent scale transition. Unlike the multi-scale model proposed by Rodrigues Lopes and Andrade Pires (2022), the developed formulation allows the consideration of voids in the RVE domain and boundary, which is critical for modelling cellular materials, lattice structures, and metamaterials, among others. This is achieved by defining the kinematical quantities only in the solid domain of the RVE and postulating a new homogenisation relation for the second-order gradient. The kinematic constraints are imposed on the RVE by the Lagrange multiplier method and particularised for minimal (lower bound), periodic and direct (upper bound) conditions. It is demonstrated that the homogenised macroscopic stress tensors can be expressed in terms of the Lagrange multipliers. The finite element method is adopted for the numerical solution of the micro and macro equilibrium. The Newton–Raphson scheme is employed to solve the non-linear systems of equations at both scales and the consistent macroscopic tangents required for the FE2 framework are derived. Several numerical examples of porous solids, lattice structures and metamaterials illustrate the consistency and applicability of the formulation for two and three-dimensional problems.

An immersed peridynamics model of fluid-structure interaction accounting for material damage and failure

This paper develops and benchmarks an immersed peridynamics method to simulate the deformation, damage, and failure of hyperelastic materials within a fluid-structure interaction framework. The immersed peridynamics method describes an incompressible structure immersed in a viscous incompressible fluid. It expresses the momentum equation and incompressibility constraint in Eulerian form, and it describes the structural motion and resultant forces in Lagrangian form. Coupling between Eulerian and Lagrangian variables is achieved by integral transforms with Dirac delta function kernels, as in standard immersed boundary methods. The major difference between our approach and conventional immersed boundary methods is that we use peridynamics, instead of classical continuum mechanics, to determine the structural forces. We focus on non-ordinary state-based peridynamic material descriptions that allow us to use a constitutive correspondence framework that can leverage well-characterized nonlinear constitutive models of soft materials. The convergence and accuracy of our approach are compared to both conventional and immersed finite element methods using widely used benchmark problems of nonlinear incompressible elasticity. We demonstrate that the immersed peridynamics method yields comparable accuracy with similar numbers of structural degrees of freedom for several choices of the size of the peridynamic horizon. We also demonstrate that the method can generate grid-converged simulations of fluid-driven material damage growth, crack formation and propagation, and rupture under large deformations.

Isogeometric bending and free vibration analyses of carbon nanotube-reinforced magneto-electric-elastic microplates using a four variable refined plate theory

The classical continuum mechanics theory is inadequate for modeling the mechanical responses of the microstructures due to its failure to account for size effects. Instead, modified strain gradient theory (MSGT) is considered as one of the most accurate theories due to its ability to adjust three length scale parameters (LSPs). Isogeometric approach (IGA) is a computational technique that is capable of accurately solving complex problems. For those reasons, the size-dependent analysis of carbon nanotube-reinforced magneto-electric-elastic microplates based on the MSGT are firstly proposed. The present approach uses the refined plate theory (RPT) with four variables, the MSGT and IGA. The magnetic and electric potentials are assumed to be a combination of a half-cosine and linear variation to satisfy Maxwell’s equations. Different types of functionally graded carbon nanotubes (CNTs) including UD, FG-X, FG-O and FG-V are reinforced in the magneto-electric-elastic matrix of the microplates. Governing equations are derived using the extended virtual work principle and then solved by IGA to determine the deflection and natural frequency of the microplates. The influence of the LSPs, CNT distributions, CNT’s volume fraction, matrix’s volume fraction, magnetic potential, electric voltage and geometry on the deflection and natural frequency of the microplates are studied and discussed.

Peridynamic simulation for the damage patterns of glass considering the influence of rate-dependence and pre-defects

This study presents a novel rate-dependent bond-based peridynamic model for predicting the mechanical behavior of brittle aluminosilicate glass under quasi-static and dynamic flexure loading. To achieve this, the damage criterion is modified with reference to the concept of strain rate in the classical continuum mechanics (CCM). Additionally, a peridynamic (PD) model with inhomogeneous critical stretch based on the Weibull distribution is proposed to account for the randomly distributed defects in glass and investigate their effect on crack initiation and propagation. The accuracy and rationality of the improved PD model are verified by comparing the calculated strength and fracture modes with the experimental results under quasi-static Ball-On-Ring (BOR) and dynamic low-velocity impact tests. The results show that the rate effect increases the damage threshold of glass, enhancing strength and damage ratio. Notably, flexural failures induced by tensile stress are observed under both static and dynamic loading. The rate-dependent inhomogeneous PD model indicates that glass inhomogeneity reduces strength while increasing crack propagation velocity (CPV). The introduction of pre-existing defects alters the symmetrical crack propagation pattern typically observed in homogeneous materials, resulting in a more randomized crack propagation path. Overall, the results can provide a promising approach for predicting the mechanical behavior of brittle materials.

In Search of a Molecular View of Peptide-Lipid Interactions in Membranes.

Lipid bilayer membranes are often represented as a continuous nonpolar slab with a certain thickness bounded by two more polar interfaces. Phenomena such as peptide binding to the membrane surface, folding, insertion, translocation, and diffusion are typically interpreted on the basis of this view. In this Perspective, I argue that this membrane representation as a hydrophobic continuum solvent is not adequate to understand peptide-lipid interactions. Lipids are not small compared to membrane-active peptides: their sizes are similar. Therefore, peptide diffusion needs to be understood in terms of free volume, not classical continuum mechanics; peptide solubility or partitioning in membranes cannot be interpreted in terms of hydrophobic mismatch between membrane thickness and peptide length; peptide folding and translocation, often involving cationic peptides, can only be understood if realizing that lipids adapt to the presence of peptides and the membrane may undergo considerable lipid redistribution in the process. In all of those instances, the detailed molecular interactions between the peptide residues and the lipid components are essential to understand the mechanisms involved.