Continuum Mechanics Research Articles

Full Forward Kinematics of Lower-Mobility Planar Parallel Continuum Robots

In rigid lower-mobility parallel manipulators the motion of the end-effector is partially constrained due to a combination of passive kinematic pairs and rigid components. Translational mechanisms, such as the Delta manipulator, are the most common ones among this type of mechanisms. When flexible elements are introduced, as in Parallel Continuum Manipulators, the constraint is no longer rigid, and new challenges arise in performing certain motions depending on the degree of compliance. Mobility analysis shifts from being purely a geometric issue to one that heavily relies on force distribution within the mechanism. Simply converting classical lower-mobility rigid parallel mechanisms into Parallel Continuum Mechanisms may yield unexpected outcomes. This work, making use of a planar parallel continuum Delta manipulator, on the one hand, presents two different approaches to solve the Forward Kinematics of planar continuum manipulators, and, on the other hand, explores some challenges and issues in assessing the resultant workspace for different design alternatives of this kind of flexible manipulators.

Exact solutions of a class of generalized nanofluidic models

Abstract Nanofluid, a significant branch of fluid mechanics, plays a pivotal role in thermal management, optics, biomedical engineering, energy harvesting, and other fields. The nanoparticles present in the fluid render the continuum mechanics ineffective, necessitating the adoption of fractional calculus to elucidate the effects of nanoparticles on the motion properties of the nanofluid. This article applies the modified extended tanh-function technique to solve two classical Schrödinger equations, the fractional Phi-4 model and the conformable fractional Boussinesq model, for nanofluids. Multiple exact solutions are obtained, and the corresponding graphical representations are provided to elucidate the basic properties of the nanofluid. This article provides new research perspectives for the development of nanofluids.

IGA Approximations of Elastic Interfaces and their Defects in an Elastic Medium with Couple Stress

AbstractThe objective of this work is to develop and implement a computational algorithm for calculating stress and couple-stress fields induced by bulk and interfacial line defects such as dislocations and generalized disclinations within phase/grain boundaries. The thermodynamic driving forces on line and planar defects, responsible for coupled plasticity and interface motion, fully coupled to stress, couple-stress and applied boundary conditions are also computed. A continuum approach for small deformations is considered following the formulation outlined in Acharya and Fressengeas (Continuum mechanics of the interaction of phase boundaries and dislocations in solids. In: Differential Geometry and Continuum Mechanics, pages 123–165, 2015), extended herein in the thermodynamics to accommodate physically necessary ingredients that arose in the modeling in Zhang and Acharya (J Mech Phys Solids 119:188–223, 2018), Zhang et al. (J Mech Phys Solids 114:258–302, 2018). Constitutive relations are derived from kinematics, balance laws and from the use of the second law of thermodynamics in global form. One of the challenges presented by this approach is the inclusion of couple stresses, Toupin (Arch Rational Mech Anal 17(2):85–112, 1964), and the consequent treatment of 4th order systems arising from the equations of balances of linear and angular momentum. In order to deal with these equations, the classical FEM approach is replaced by iso-geometric analysis (IGA), as proposed by Hughes et al. (Comput Methods Appl Mech Eng 194(39):4135–4195, 2005). Results on stress and couple stress fields of the various defects involved are computed. Further, the coupling of the dislocation density with the eigenwall allows for the capturing of the shear parallel to the grain boundary, which has been observed experimentally and through molecular dynamics.

State-of-the-Art Review of Continuum Mechanics-Based Modelling of Soil Surface Erosion

AbstractSoil surface erosion can shape the morphography of rivers and estuaries in the natural environment and induce high potential risks to structures in engineering. Numerical simulations based on continuum mechanics theory can provide reliable assessments of the evolution of surface erosion from the perspective of a large-scale view. However, current studies on continuum mechanics-based modelling are still limited. This paper comprehensively reviews such numerical simulations of soil surface erosion. This review begins by discussing the fundamental physical mechanisms of surface erosion. Subsequently, it explores the basic physics-based conservation equations controlling soils and fluids in surface erosion. Then, the empirical formulae depicting the different stages of surface erosion are presented. Building on these mathematical foundations, this paper reviews various numerical methods for surface erosion modelling from a continuum mechanics perspective. Finally, this paper discusses the advantages and limitations of the numerical methods. This work can provide researchers convenience for using numerical models on surface erosion simulations.

Evaluation method of progressive failure of loess slope under shallow coal seam mining

Abstract When mining coal under loess gully, the surface slopes may exhibit significant discontinuous failure characteristics such as local failure and landslides. These characteristics, composed of failure and non-failure regions, form an essentially unstable mechanical system. This makes it difficult to obtain a true solution using continuous medium mechanics, greatly increasing the complexity of slope failure evaluation and management. To accurately evaluate the degree of slope failure during mining, clarify its mechanism, and propose appropriate management measures, the Ningtiaota Coal Mine in northern Shanxi was selected as the study area. Based on the unbalanced thrust method and the probability integral settlement prediction theory, and utilizing numerical calculations, we proposed a method for evaluating and managing slope failure in coal mining under loess gully. This method describes the changes in slope failure during mining through the deformation failure ratio of the slip surface and quantifies the degree of slope failure at different mining times, revealing the full process stability evolution characteristics of points and surfaces as the critical state of the slope progresses. When the deformation failure ratio of the slip surface is ≥1, it indicates overall slope failure and the beginning of slope sliding instability; when the deformation failure ratio of the slip surface is <1, it indicates local slope failure. Based on this, corresponding management methods were proposed according to the mechanical distribution characteristics of the slope slip surface in different stress areas. The research results were applied in the Ningtiaota Coal Mine in northern Shanxi, yielding positive outcomes.

Mechanistic Simulation of Salt‐Affected Soil‐Plant‐Atmosphere Continuum Dynamics in Seasonally Frozen Regions

AbstractThe salt‐affected Soil‐Plant‐Atmosphere Continuum (SPAC) is a dynamic, interactive system that is particularly complex in seasonally frozen regions where salt transport, precipitation‐dissolution, and soil freeze‐thaw processes play crucial, interrelated roles. Understanding these coupled processes and representing them with mathematical models is critical for effective management of SPAC systems. This study presents a new mechanistic approach and an improved model that integrates a chemical equilibrium module within a mechanistic‐based transport computational module (modified SHAW model). The chemical equilibrium module determines salt precipitation‐dissolution using thermodynamic theory and explains the effects of efflorescence and subflorescence on system dynamics. The model enables simultaneous solutions for heat, water, and salt transport with chemical equilibrium throughout non‐freezing and freezing seasons, as well as plant growth dynamics. Assessment of the model using laboratory experiments and field studies showed good performance, with coefficient of determination values exceeding 0.65 for simulated and measured evaporation rate, leaf area index, soil water content, salt content, and temperature. Furthermore, a comparison between simulation results considering and neglecting the impact of salt precipitation‐dissolution highlights potential inaccuracies in soil heat‐water‐salt dynamics and plant water use resulting from the omission of this process in mechanistic models.

Weak form quadrature shell elements based on absolute nodal coordinate formulation

Weak form quadrature elements for moderately thick shells with arbitrary initial configurations are developed under the framework of continuum mechanics and the absolute nodal coordinate formulation (ANCF). Locking problems of shell analysis are discussed. Nonlinear analysis of various shell structures is conducted. The joint constraint equations for shells with discontinuous slopes are established. Five examples encompassing static and dynamic shell analysis, post-buckling analysis of shells, as well as analysis of shells with discontinuous mid-surface slopes are examined to assess the performance of the proposed elements. Satisfactory results are obtained, validating the efficacy of the proposed elements.

Numerical Modeling of Two-Phase Fluid Filtration for Carbonate Reservoir in Two-Dimensional Formulation

This work considers the isothermal process of incompressible viscous fluid filtration in an oil-saturated, fractured-porous reservoir. A study of the pressure and water saturation distribution process is carried out for a case in which a production well is put into operation. For this problem, i.e., a mathematical model in a two-dimensional formulation, a numerical method and a parallel algorithm are proposed. The mathematical model of two-phase filtration is written in accordance with the classical laws of continuum mechanics and Darcy’s law and also includes a function of fluid exchange between low-permeability pores and high-permeability natural fractures within the framework of the Warren–Root model. The numerical solution is based on the finite-difference method and a splitting scheme of physical processes and spatial coordinates. For a split system with respect to piezoconductivity, an implicit finite-difference scheme with fixed saturations is constructed, and with respect to saturation transfer, explicit and implicit difference schemes are constructed. For parallel implementation of the developed numerical approach, a method based on geometric parallelism is selected. Testing of the developed method is performed using the example of calculating liquid mass transfer for a wide range of parameters. To verify the model, the obtained calculated pressure curves are compared with field data recorded by a deep-well measuring device. The results allow for estimation of the distribution of reservoir pressure and water saturation depending on the permeability of the fracture set and the pore part. The obtained results allow for monitoring of well operations, reducing unexpected accident risks and optimizing the development system in order to increase oil production in fractured-porous reservoirs. Computational experiments confirm the efficiency of the developed numerical algorithm and its parallel implementation.

Study on a Pseudo-Elastic Model for High-Damping Rubber.

With advancements in seismic isolation and damping technology, high-damping rubber (HDR) bearings are now widely used. However, significant gaps remain in HDR-analysis model research, with few studies integrating multiple factors, the Mullins effect, and stiffness hardening for more accurate practical predictions. This study classifies the effective behavior of HDR and examines the stress-strain relationships of different behavioral types using more appropriate equations. Mathematical models were established based on pseudo-elasticity theory, which is an extension of continuum mechanics. Subsequently, parameter functions were developed through parameter determination tests and regression analysis, leading to the completion of the pseudo-elastic model for HDR. Finally, the model's effectiveness was validated through validation tests. This study finds that behavior classification effectively examines phenomenological-based HDR stress-strain relationships, as distinct behavioral patterns are not adequately captured by a single approach. Incorporating tests to functionalize material parameters complements theoretical models. Additionally, accurately explaining HDR behavior requires considering the Mullins effect and stiffness hardening, influenced by the coupled effects of temperature, strain amplitude, and compressive stress. Consequently, this HDR pseudo-elastic model offers a comprehensive explanation of HDR behavior, including the Mullins effect and stiffness hardening, under various influencing factors based on clear mechanical principles and explicit computational procedures.

CoupNeRF: Property‐aware Neural Radiance Fields for Multi‐Material Coupled Scenario Reconstruction

AbstractNeural Radiance Fields (NeRFs) have achieved significant recognition for their proficiency in scene reconstruction and rendering by utilizing neural networks to depict intricate volumetric environments. Despite considerable research dedicated to reconstructing physical scenes, rare works succeed in challenging scenarios involving dynamic, multi‐material objects. To alleviate, we introduce CoupNeRF, an efficient neural network architecture that is aware of multiple material properties. This architecture combines physically grounded continuum mechanics with NeRF, facilitating the identification of motion systems across a wide range of physical coupling scenarios. We first reconstruct specific‐material of objects within 3D physical fields to learn material parameters. Then, we develop a method to model the neighbouring particles, enhancing the learning process specifically in regions where material transitions occur. The effectiveness of CoupNeRF is demonstrated through extensive experiments, showcasing its proficiency in accurately coupling and identifying the behavior of complex physical scenes that span multiple physics domains.

Strain energy density maximization principle for material design and the reflection in trans-scale continuum theory

Traditional efforts in the design of damage-tolerant structural materials were largely exercises in optimizing the combination of strength and ductility. However, the simultaneous consideration of these two conflicting mechanical indices, improving one inevitably sacrifices the other, makes the design extremely complex and difficult, due to the dilemma of choosing between them. Here, physically guided by the energy variational expression in trans-scale continuum mechanics theory, we propose a general mechanics principle for material design that involving only one index: towards strong and tough material the strain energy density limit (w) should be maximized, i.e., strain energy density maximization principle, referred to as wmax principle. It aims to guide the attainment of exceptional comprehensive mechanical properties, while circumventing the dual-index dilemma by employing a singular index w. Extensive experimental data analyses prove that (i) the maximum wmax always exists, at a critical dimension of characteristic microstructure dc,micro, and (ii) w can effectively index strength-ductility synergy and the wmax is conjugated with both high strength and high ductility, verifying the validity of wmax principle. The universality, practicality and downward compatibility are also examined. The dc,micro approaches twice the span of strain gradient region around internal boundary, suggesting that the microstructure state with wmax is the critical state with strongest strain gradient. Importantly, the w improvement as a function of the characteristic size of either microstructure or deformation field can be well captured by strain gradient theory, confirming the consistence between the wmax principle, experimental results and trans-scale continuum theories. This principle opens up a new design concept for advanced structural materials from the perspective of microstructure-w-mechanical properties relationship.

CFD investigations of a shape-memory polymer foam-based endovascular embolization device for the treatment of intracranial aneurysms.

The hemodynamic and convective heat transfer effects of a patient-specific endovascular therapeutic agent based on shape memory polymer foam (SMPf) are evaluated using computational fluid dynamics studies for six patient-specific aneurysm geometries. The SMPf device is modeled as a continuous porous medium with full expansion for the flow studies and with various degrees of expansion for the heat transfer studies. The flow simulation parameters were qualitatively validated based on the existing literature. Further, a mesh independence study was conducted to verify an optimal cell size and reduce the computational costs. For convective heat transfer, a worst-case scenario is evaluated where the minimum volumetric flow rate is applied alongside the zero-flux boundary conditions. In the flow simulations, we found a reduction of the average intra-aneurysmal flow of > 85% and a reduction of the maximum intra-aneurysmal flow of > 45% for all presented geometries. These findings were compared with the literature on numerical simulations of hemodynamic and heat transfer of SMPf devices. The results obtained from this study can serve as a guide for optimizing the design and development of patient-specific SMPf devices aimed at personalized endovascular embolization of intracranial aneurysms.

Redesign of a Balance Rehabilitation Device Based on a Parallel Continuum Mechanism

In the present work, a parallel continuum manipulator for trunk rehabilitation tasks for patients who have suffered a stroke was analyzed and redesigned. The manipulator had to perform active assistance exercises for the motor recovery of the patient. Based on this background, a series of requirements were defined, which determined the design framework during the modeling of the manipulator. Finally, an improved prototype was built and tested to verify that the model can properly characterize the behavior of the manipulator. Such tests were carried out using a self-made dummy that replicates the simplifying hypotheses and conditions assumed in the mathematical model.

Continuum Mechanics Applied to the Biomechanics of Motion.

Continuum Mechanics is the discipline that studies the motion and deformation of material bodies, and is at the basis of both Solid Mechanics and Fluid Mechanics. This works aims at highlighting how a basic education in modern Continuum Mechanics can be of fundamental importance not only in the Biomechanics of Tissue, but also in the Biomechanics of Movement. The latter is largely based on Rigid Body Mechanics, which can be considered as a simple particular case of the general theory of Continuum Mechanics. As an example, a straightforward methodology for the extraction of the angular velocity from motion analysis data is illustrated. The method is based on a quantity that is more primitive than the angular velocity: the spin tensor.

Size-dependency and lattice-discreetness effect on fracture toughness in 2D crystals under antiplanar loading

AbstractFracture toughness is the material property characterizing resistance to failure. Predicting its value from the solid structure at the atomistic scale remains elusive, even in the simplest situations of brittle fracture. We report here numerical simulations of crack propagation in two-dimensional fuse networks of different periodic geometries, which are electrical analogs of bidimensional brittle crystals under antiplanar loading. Fracture energy is determined from Griffith’s analysis of energy balance during crack propagation, and fracture toughness is determined from fits of the displacement fields with Williams’ asymptotic solutions. Significant size dependencies are evidenced in small lattices, with fracture energy and fracture toughness both converging algebraically with system size toward well-defined material-constant values in the limit of infinite system size. The convergence speed depends on the loading conditions and is faster when the symmetry of the considered lattice increases. The material constants at infinity obey Irwin’s relation and properly define the material resistance to failure. Their values are approached up to $$\sim 15\%$$ ∼ 15 % using the recent analytical method proposed in Nguyen and Bonamy (Phys Rev Lett 123:205503, 2019). Nevertheless, the deviation remains finite and does not vanish when the system size goes to infinity. We finally show that this deviation is a consequence of the lattice discreetness and decreases when the super-singular terms of Williams’ solutions (absent in a continuum medium but present here due to lattice discreetness) are taken into account.

Multiscale homogenized constrained mixture model of the bio-chemo-mechanics of soft tissue growth and remodeling.

Constrained mixture models have successfully simulated many cases of growth and remodeling in soft biological tissues. So far, extensions of these models have been proposed to include either intracellular signaling or chemo-mechanical coupling on the organ-scale. However, no version of constrained mixture models currently exists that includes both aspects. Here, we propose such a version that resolves cellular signal processing by a set of logic-gated ordinary differential equations and captures chemo-mechanical interactions between cells by coupling a reaction-diffusion equation with the equations of nonlinear continuum mechanics. To demonstrate the potential of the model, we present 2 case studies within vascular solid mechanics: (i) the influence of angiotensin II on aortic growth and remodeling and (ii) the effect of communication between endothelial and intramural arterial cells via nitric oxide and endothelin-1.

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

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

Peridynamic computations for thin elastic rods

AbstractPeridynamics is a nonlocal continuum mechanics formulation well suited for simulating dynamic fracture phenomena. Various peridynamic material formulations have been developed in recent years. These models have some differences, particularly regarding the correct handling of elastic deformation. This study investigates the elastic wave propagation characteristics of bond‐based, ordinary state‐based, continuum‐kinematics‐inspired peridynamics and a local continuum consistent correspondence formulation. Specifically, it examines the behavior of longitudinal pressure waves within thin rods. While all formulations demonstrate adequate wave propagation handling, all except the correspondence formulation are sensitive to incomplete horizons. In contrast, the local continuum consistent formulation exhibits outstanding accuracy in modeling wave propagation. Moreover, the fascinating “spaghetti fracture” phenomenon, where a bent thin rod always breaks into three or more pieces, motivates additional investigations. It is shown that reproducing a different number of fragments using peridynamic simulations is possible. Although initial experiments can be replicated, the sensitivity to numerous parameters necessitates further investigations for a more comprehensive understanding and validation.

Strain-gradient and damage failure behavior in particle reinforced heterogeneous matrix composites

A strain gradient plasticity model with strengthening effect and damage effect (SGSED) is developed under the continuum medium framework for particle-reinforced heterogeneous matrix composites (PRHMCs). Boron carbide (B4C) particle-reinforced ultrafine-grained (UFG)/fine-grained (FG) heterogeneous matrix composites (B4Cp/(UFG/FG)) are fabricated, in which the UFG region consisted of carbon nanotubes (CNTs)/6061 aluminum (Al) flakes with grains in the ultrafine range and the FG region is processed via 6061Al. The SGSED model is written into the user subroutines using commercial finite element (FE) calculation software, and the three-dimensional (3D) FE representative volume element (RVE) for B4Cp/(UFG/FG) composites is established, from which the distribution of the interface-affected-zone (IAZ) formed of the strain gradient caused by the uncoordinated deformation of the UFG-FG heterogeneous matrix and reinforced phase-matrix is calculated. The evolution of the strain gradient in the deformation process of composites and the influence of the strain gradient on the progressive damage and crack evolution of composites are analyzed, and the strain gradient strengthening-toughening mechanism of composites is revealed. It is found that the IAZ has a considerable strengthening-toughening effect on the composites, which can reduce stress concentration at the interface between the reinforced phase and the matrix, and slow down the crack propagation of the matrix.

Research Progress of Three-dimensional Geological Modeling and Multi-field Coupling of Fractured Oil and Gas Reservoirs

Fractured oil and gas reservoirs play an important role in oil and gas exploration and development due to their complex fracture network structure. As the main fluid channel in the reservoir, fractures have a significant impact on the migration, accumulation and exploitation efficiency of oil and gas. This paper summarizes the key geological characteristics of fractured reservoirs, including fracture types, distribution rules and main controlling factors of fracture development, and discusses the influence of fractures on reservoir physical properties. Furthermore, the deterministic and stochastic modeling methods of three-dimensional geological modeling of fractured reservoirs and the latest progress of discrete fracture network modeling technology are introduced in detail. In addition, this paper also discusses in detail the application of multi-field coupling model in the numerical simulation of fluid flow in fractured reservoirs, including continuous medium model and discrete medium model, and their advantages and disadvantages in simulating fluid flow in fractured reservoirs. Finally, this paper summarizes the research progress of multi-physics coupling model of fractured reservoirs, and points out the direction and challenges of future research. Through these studies, it can provide theoretical support and technical guidance for oil and gas exploration and development, in order to achieve more effective development and management of fractured reservoirs.