Solid Deformation Research Articles

Penalty 4-Node Quadrilateral Element Formulation for Axisymmetric Couple Stress Problems

To address the issue of size effects in axisymmetric deformation of small-scale solids, this work proposes a 4-node 12-DOF element for axisymmetric problems based on the consistent couple stress theory (CCST), following the framework of the unsymmetric finite element method. With the use of the penalty function method, an independently assumed rotational field is introduced into the virtual work principle to approximate the physical rotation, ensuring the satisfaction of the C1 continuity requirement of the CCST in a weak form. As a benefit, the enriched C0 isoparametric-based interpolation is employed to construct the test functions for displacement and rotation. Furthermore, the force-stress field that satisfies the equilibrium equations related to axisymmetric deformation is employed as the element’s force-stress trial function. In order to circumvent locking issues, reduced integration is employed in the penalty stiffness integration process. The numerical results demonstrate that the new element exhibits high computational accuracy and convergence rate in both static and modal analysis problems, effectively capturing size-dependent phenomena.

Elastodynamic analysis of rotating solids by novel nodal position finite element method

This study develops a novel 8-node isoparametric hexahedral element using the Nodal Position Finite Element Method (NPFEM) for elastodynamic analysis of rotating solids. The element also incorporates the flexural modes directly into its element shape function to alleviate the shear locking when modeling the bending deformation of solids. Unlike conventional displacement-based finite element methods, which require the decoupling of elastic deformation from rigid-body motions, the NPFEM eliminates this process by directly representing strain and kinetic energies through nodal position coordinates, which avoids potential approximation errors in the decoupling process. To validate the accuracy and efficacy of this new NPFEM solid element, numerical simulations of a beam under static and dynamic loads are conducted and benchmarked against the theoretical solutions. Then, dynamic analysis of a rotating blade demonstrates that the NPFEM element can directly account for the centrifugal stiffening effect and superharmonic resonance of rotating blades without resorting to conventional methods. The successful implementation of this NPFEM element in complex simulations highlights its potential to provide significant advancements in computational mechanics.

Observation of converse flexoelectric effect in topological semimetals

A strong coupling between electric polarization and elastic deformation in solids is an important factor in creating useful electromechanical nanodevices. Such coupling is typically allowed in insulating materials with inversion symmetry breaking as exemplified by the piezoelectric effect in ferroelectric materials. Therefore, materials with metallicity and centrosymmetry have tended to be out of scope in this perspective. Here, we report the observation of giant elastic deformation by the application of an alternating electric current in topological semimetals (V,Mo)Te2, regardless of the centrosymmetry. Considering the crystal and band structures and the asymmetric measurement configurations in addition to the absence of the electromechanical effect in a trivial semimetal TiTe2, the observed effect is discussed in terms of a Berry-phase-derived converse flexoelectric effect in metals. The observation of the flexoelectric effect in topological semimetals paves a way for a new type of nanoscale electromechanical sensors and energy harvesting.

Study on the Stress Distribution Characteristics of Rock in the Bottomhole and the Influence Laws of Various Parameters Under the Impact of a Liquid Nitrogen Jet

This study presents research on the stress distribution characteristics of rock in the bottomhole and the influence laws of various parameters under the impact of liquid nitrogen jet. A multi-field coupled numerical model considering transient flow field, conjugate heat transfer, and nonlinear solid deformation was established to investigate the damage-induced fracturing mechanism of rock under liquid nitrogen jet. The study compares the impact effects of liquid nitrogen jet and water jet on rock and analyzes the variations in the stress field under different parameters. Due to its extremely low temperature, the liquid nitrogen jet creates a strong thermal stress gradient in a short time, significantly increasing the maximum principal stress and Mises stress in the rock compared to a water jet. Solid parameters, particularly the confining pressure and elastic modulus of the rock, have a more significant impact on stress distribution, while fluid parameters such as outlet pressure and fluid temperature have a smaller and more volatile effect. An increase in confining pressure inhibits tensile failure in the rock, while a higher elastic modulus enhances both tensile and shear failure. The initial rock temperature significantly affects the stress distribution, with optimal tensile failure observed at intermediate temperatures. The liquid nitrogen jet achieves a higher maximum velocity and overflow velocity than the water jet, contributing to more effective rock fracturing. The results provide a theoretical basis for the optimization of liquid nitrogen jet drilling parameters, which can help improve drilling efficiency.

A critical state μ(I)-rheology model for cohesive granular flows

The dynamic behaviour of granular media can be observed widely in nature and in many industrial processes. Yet, the modelling of such media remains challenging as they may act with solid-like and fluid-like properties depending on the rate of the flow and can display a varying apparent friction, cohesion and compressibility. Over the last two decades, the $\mu (I)$ -rheology has become well established for modelling granular liquids in a fluid mechanics framework where the apparent friction $\mu$ depends on the inertial number $I$ . In the geo-mechanics community, modelling the deformation of granular solids typically relies on concepts from critical state soil mechanics. Along the lines of recent attempts to combine critical state and the $\mu (I)$ -rheology, we develop a continuum model based on modified cam-clay in an elastoplastic framework which recovers the $\mu (I)$ -rheology under flow. This model permits a treatment of plastic compressibility in systems with or without cohesion, where the cohesion is assumed to be the result of persistent inter-granular attractive forces. Implemented in a two- and three-dimensional material point method, it allows for the trivial treatment of the free surface. The proposed model approximately reproduces analytical solutions of steady-state cohesionless flow and is further compared with previous cohesive and cohesionless experiments. In particular, satisfactory agreements with several experiments of granular collapse are demonstrated, albeit with shear bands which can affect the smoothness of the surface. Finally, the model is able to qualitatively reproduce the multiple steady-state solutions of granular flow recently observed in experiments of flow over obstacles.

Statistical Modeling and Probable Calculation of the Strength of Materials with Random Distribution of Surface Defects

Based on the solutions of deterministic fracture mechanics and the methods of probability theory, the algorithm for calculating the probabilistic strength characteristics of plate elements of structures with an arbitrary stochastic distribution of surface defects is outlined. On the plate surface, there are uniformly distributed cracks that do not interact with each other, the plane of which is normal to the surface, and the depth is much less than its length on the surface. The cracks’ depth and angle of orientation are random values, and their joint distribution density is specified. Plates made of this material are under the influence of biaxial loading. The probability of failure, along with the mean value, the dispersion, and the variation coefficient of the plate’s strength, taking into account the surface defects under different types of stress, were determined. Their dependence on the type of loading, the size of the plate, and the surface structural heterogeneity of the material were studied graphically. Joint consideration of the influence of the interrelated properties of real materials, such as defectiveness and stochasticity, on strength and fracture, opens up new opportunities in creating a theory of strength and fracture of deformable solids.

Numerical Simulation of Hydraulic Fracture Propagation in Unconsolidated Sandstone Reservoirs

In order to comprehensively understand the complex fracture mechanisms in thick and loose sandstone formations, we have carefully developed a coupled finite element numerical model that captures the complex interactions between fluid flow and solid deformation. This model is the cornerstone of our future exploration. Based on this model, the crack propagation problem of hydraulic fracturing under different engineering and geological conditions was studied. In addition, we conducted in-depth research on the key factors that shape the geometry of hydraulic fractures, revealing their subtle differences and complexities. It is worth noting that the sharp contrast between the stress profile and mechanical properties between the production layer and the boundary layer often leads to fascinating phenomena, such as the vertical merging of hydraulic fracture propagation. The convergence of cracks originating from adjacent layers is a recurring theme in these strata. Sensitivity analysis clarified our understanding, revealing that increased elastic modulus promotes longer crack propagation paths. As the elastic modulus increases from 12 GPa to 18 GPa, overall, the maximum crack width slightly decreases, with a less than 10% reduction rate. The increased fluid leakage rate will significantly shorten the length and width of hydraulic fractures (with a maximum decrease of over 70% in fracture width). The increase in viscosity of fracturing fluid causes a change in fracture morphology, with a reduction in length of about 32% and an increase in fracture width of about 25%. It is worth noting that as the leakage rate of fracturing fluid increases, the importance of the viscosity of fracturing fluid decreases relatively. Strategies such as increasing fluid viscosity or adding anti-filtration agents can alleviate these challenges and improve the efficiency of fracturing fluids. In summary, our research findings provide valuable insights that can provide information and optimization for hydraulic fracturing filling and fracturing strategies in loose sandstone formations, promoting more efficient and influential oil and gas extraction work.

A novel hybrid PD-FEM-FVM approach for simulating hydraulic fracture propagation in saturated porous media

To enhance the computational efficiency of fluid–solid coupling in peridynamics (PD), a hybrid modeling approach based on the classical Biot theory is proposed for simulating hydraulic crack propagation in saturated porous media. The deformation and damage of solids are described by the coupling of the finite element method (FEM) and PD. Based on Darcy’s law, the finite volume method (FVM) is used to describe fluid seepage and calculate pore water pressure. The mutual transfer of fluid pressure and solid deformation is realized through the transition layer between the solid layer and the fluid layer. Firstly, the effectiveness of the proposed method is verified by a porous media seepage simulation example. Secondly, the ability and efficiency of this method to simulate crack propagation in saturated porous media are verified by several examples of hydraulic fracturing of rock with a single pre-existing crack. Finally, the synchronous hydraulic fracturing process of rock with double cracks is simulated. The ability of this method to simulate the simultaneous propagation of multiple fractures in the rock under fluid–solid coupling is further illustrated. The aforementioned studies demonstrate that the novel hybrid PD-FEM-FVM approach not only ensures computational accuracy and effectiveness but also significantly enhances computational efficiency.

Modeling pore fluid pressure and deformation in unsaturated soils under gravitational compaction and cyclic loading

Cyclic loading of fluid-bearing soils often induces excess pore water pressure, leading to accumulation of strain in the solid matrix. Additionally, variations in material density and volumetric fraction due to gravitational effects generate momentum forces, which subsequently cause deformation in the soil. In this paper, we present a mathematical framework of poroelasticity that generalizes the simultaneous action of gravitational body forces and cyclic loading to analyze solid deformation and fluid pressure dissipation in partially-saturated porous media. To account for the effect of gravity, the gradient of the vertical displacement needs to be treated as an additional dependent variable alongside pore air and water pressures.We numerically solve a mixed strain-controlled and pressure-prescribed boundary-value problem using a finite difference scheme as a representative example to quantify the impact of gravitational forces. Our results show that gravitational compaction affects both total settlement and excess pore water pressure, with its influence being particularly significant at lower water saturations, regardless of excitation frequency, soil depth, or type. The effect of gravity is more pronounced in soils with higher compressibility but appears insensitive to excitation frequency. Notably, the gravitational effect correlates linearly with the depth of the soil deposit and, therefore, should be well integrated into the consolidation theory of poroelasticity.

Inertia effect of deformation in amorphous solids: A dynamic mesoscale model

Shear transformation (ST), as the fundamental event of plastic deformation of amorphous solids, is commonly considered as transient in time and thus assumed to be an equilibrium process without inertia. Such an approximation however poses a major challenge when the deformation becomes non-equilibrium, e.g., under the dynamic and even shock loadings. To overcome the challenge, this paper proposes a dynamic mesoscale model for amorphous solids that connects microscopically inertial STs with macroscopically elastoplastic deformation. By defining two dimensionless parameters: strain increment and intrinsic Deborah number, the model predicts a phase diagram for describing the inertia effect on deformation of amorphous solids. It is found that with increasing strain rate or decreasing ST activation time, the significant inertia effect facilitates the activation and interaction of STs, resulting in the earlier yield of plasticity and lower steady-state flow stress. We also observe that the externally-applied shock wave can directly drive the activation of STs far below the global yield and then propagation along the wave-front. These behaviors are very different from shear banding in the quasi-static treatment without considering the inertia effect of STs. The present study highlights the non-equilibrium nature of plastic events, and increases the understanding of dynamic or shock deformation of amorphous solids at mesoscale.

Microstructure evolution by thermomechanical processing and high-temperature mechanical properties of HCP-high entropy alloys (HEAs)

The solid solution strengthening and deformation mechanisms of the hcp-high entropy alloys (HEAs) have not been elucidated yet because of the complexity introduced by the martensite structure, particularly in alloys undergoing phase transformation from bcc to hcp structures. This study attempted to fabricate an equiaxed hcp microstructure for TiZrHf, TiZrHfAl, and TiZrHfAlSc using the hot rolling process and subsequent heat treatment. The equiaxed hcp phase was successfully formed in TiZrHfAl, but the martensite structure remained in TiZrHf and TiZrHfAlSc. The 0.2 % proof stress at 400 and 600°C increased with the increase of the average atomic size misfit δ and the mixing entropy . The low activation volume and high-stress exponent of TiZrHfAl with equiaxed hcp phase at room temperature suggest minor obstacles such as cluster or short-range order inhibit the movement of dislocation and leading to the significant solid-solution strengthening. However, the effect diminished at 600 °C.

A thermodynamics-based coupled model for multi-phase flowback in deformable dual-porosity shale gas reservoirs

Over the past decades, there has been growing interest in modelling multi-phase flowback in shale gas reservoirs during hydraulic fracturing, primarily due to the significant risk of groundwater pollution. However, the lack of fully coupled simulations and a unified theoretical model has hindered the study of coupled adsorptive hydro-mechanical behaviour and its influence on fluid flowback prediction. In this paper, we have extended the non-equilibrium thermodynamics-based Mixture Coupling Theory to derive the constitutive relationship and governing equation for adsorptive multi-phase fluid flows in deformable dual-porosity media. Helmholtz free energy density has been used to establish the relationship between solids and fluids. The interaction of adsorptive fluids in the pore and fracture space has been fully considered, leading to a new form of constitutive equation, which obtained the molecular adsorptive effect on the rock by introducing adsorption entropy production. The proposed governing equations determine the fully coupled evolution of solid deformation and multi-phase flow, with the consideration of adsorption as well as pore and fracture porosity change. Numerical simulation is then performed to study the flowback of a real shale gas well and the influence of coupled behaviour on model prediction. The simulation shows that the flowback flux is mainly by the fracture (more than 99.9 %) and the overall cumulative volume is 2427.1 m3 within 225 h which accounts for about 5.3 % of the total injected fracturing fluid, and the sensitivity analysis reveals that the manual fracture porosity is the most sensitive factor to the cumulative flowback. This work provides new insights into the flowback process of shale reservoirs in a fully coupled view and the proposed theoretical tool should contribute to the modelling of other adsorptive multi-phase flow problem in deformable dual-porosity media such as carbon storage and coalbed methane production.

Green’s Tensor of BIOT’s Equations by Stationary Oscillations

Various mathematical models of deformable solids mechanics are used in the study of seismic processes in the earth's crust. The processes of waves propagation are most studied in elastic media. But these models do not take into account many real properties of the ambient array. These are, for example, the presence of groundwater, which complicates the construction and operation of surface and underground structures, affect the magnitude and distribution of stresses. Models, which take into account the water saturation form the earth's crust structures, the presence of gas bubbles, etc., are multi-component medium. A variety of multicomponent media, the complexity of the processes associated with their deformation, lead to a large difference in the methods of analysis and modelling used in the solution of wave problems. In this paper the processes of wave propagation in a two-component Biot medium under the action of periodic forces of various forms are considered. Using the Fourier transform of generalized functions, fundamental solutions are constructed - the Green's tensor of the Biot equations and its properties are studied. This tensor describes the process of propagation of harmonic waves of a fixed frequency in spaces of dimension N = 1,2,3 under the action of power sources concentrated at the origin of coordinates, described by a singular delta function. On its basis, generalized solutions of these equations are constructed under the action of various sources of periodic disturbances, which are described by both regular and singular generalized functions. For regularly acting forces, integral representations of solutions are given, which can be used to calculate the stress-strain state of a porous water-saturated medium

Numerical calculation for streaming potentials caused by solid deformation

When a solid is deformed by an external force, it will cause the liquid in its internal pores to flow and generate a streaming potential. To understand the streaming potentials in the pores, a numerical analysis model of solid–liquid-streaming potential coupling was proposed. To validate the model, an experiment was designed and the reliability of the model was demonstrated by comparing with the results of the experiment and a theoretical analysis respectively. Then the model was applied to numerically calculate the streaming potentials in tubes with tapered-through-hole and curved holes with different curvatures respectively. The results show that under the same external loading, the streaming potentials in a straight tapered tube increases with the decrease of tube outlet diameter and increase of the thin tube length. Another result is that the curvature of a thin bent tube or hole can also cause the increment of the streaming potential. The loading frequency also affect the streaming potential. Relatively high loading frequency results in increasing the amplitude of the streaming potential.

A coarse-grained approach to modeling gas transport in swelling porous media

In many engineering applications, understanding gas adsorption and its induced swelling in nanoporous materials is crucial. In this study, we propose a novel coarse-grained molecular dynamics (CGMD) model with gas-gas, solid-solid, and gas-solid interactions explicitly controlled to achieve the coupling between gas transport and solid deformation at the microscale. The CGMD model has the capability to recover solid and gas properties, including density, Young's modulus of the solid, and viscosity of the gas to generate a broad range of swelling ratios relevant to nanostructures by using the innovative bead-spring chain networks. A comparison is made between gas transport through deformable and non-deformable nanochannels of varying sizes (35.4–123.9 nm), which is also compared with the macroscopic Hagen-Poiseuille equation. The proposed model has been further tested in a simplified nanoporous medium composed of four randomly distributed spherical solids. The Kozeny-Carman equation can generally describe the relationship between permeability and porosity, but small deviations are observed in the case of swelling porous media. Our results justify the effect of swelling on reducing gas permeability and provide a new approach to modeling gas transport in swelling porous media at the microscale within the framework of CGMD, with potential applications spanning nanofluidics, energy storage technologies, and environmental nanotechnology.

(De)hydration Front Propagation Into Zero‐Permeability Rock

AbstractHydration and dehydration reactions play pivotal roles in plate tectonics and the deep water cycle, yet many facets of (de)hydration reactions remain unclear. Here, we study (de)hydration reactions where associated solid density changes are predominantly balanced by porosity changes, with solid rock deformation playing a minor role. We propose a hypothesis for three scenarios of (de)hydration front propagation and test it using one‐dimensional hydro‐mechanical‐chemical models. Our models couple porous fluid flow, solid rock volumetric deformation, and (de)hydration reactions described by equilibrium thermodynamics. We couple our transport model with reactions through fluid pressure: the fluid pressure gradient governs porous flow and the fluid pressure magnitude controls the reaction boundary. Our model validates the hypothesized scenarios and shows that the change in solid density across the reaction boundary, from lower to higher pressure, dictates whether hydration or dehydration fronts propagate: decreasing solid density causes dehydration front propagation in the direction opposite to fluid flow while increasing solid density enables both hydration and dehydration front propagation in the same direction as fluid flow. Our models demonstrate that reactions can drive the propagation of (de)hydration fronts, characterized by sharp porosity fronts, into a viscous medium with zero porosity and permeability; such propagation is impossible without reactions, as porosity fronts become trapped. We apply our model to serpentinite dehydration reactions with positive and negative Clapeyron slopes and granulite hydration (eclogitization). We use the results of systematic numerical simulations to derive a new equation that allows estimating the transient, reaction‐induced permeability of natural (de)hydration zones.

On effective surface elastic moduli for microstructured strongly anisotropic coatings

The determination of surface elastic moduli is discussed in the context of a recently proposed strongly anisotropic surface elasticity model. The aim of the model was to describe deformations of solids with thin elastic coatings associated with so-called hyperbolic metasurfaces. These metasurfaces can exhibit a quite unusual behaviour and concurrently a very promising wave propagation behaviour. In the model of strongly anisotropic surface elasticity, strain energy as a function of the first and second deformation gradients has been introduced in addition to the constitutive relations in the bulk. In order to obtain values of surface elastic moduli, we compare dispersion relations for anti-plane surface waves obtained using the two-dimensional (2D) model and three-dimensional (3D) straightforward calculations for microstructured coatings of finite thickness. We show that with derived effective surface moduli, the 2D model can correctly describe the wave propagation.

MPeat2D – a fully coupled mechanical–ecohydrological model of peatland development in two dimensions

Abstract. Higher-dimensional models of peatland development are required to analyse the influence of spatial heterogeneity and complex feedback mechanisms on peatland behaviour. However, the current models exclude the mechanical process that leads to uncertainties in simulating the spatial variability in the water table position, vegetation composition, and peat physical properties. Here, we propose MPeat2D, a peatland development model in two dimensions, which considers mechanical, ecological, and hydrological processes together with the essential feedback from spatial interactions. MPeat2D employs poroelasticity theory that couples fluid flow and solid deformation to model the influence of peat volume changes on peatland ecology and hydrology. To validate the poroelasticity formulation, the comparisons between numerical and analytical solutions of Mandel's problems for two-dimensional test cases are conducted. The application of MPeat2D is illustrated by simulating peatland growth over 5000 years above a flat and impermeable substrate with free-draining boundaries at the edges, using constant and variable climate. In both climatic scenarios, MPeat2D produces lateral variability in the water table depth, which results in the variation in the vegetation composition. Furthermore, the drop in the water table at the margin increases the compaction effect, leading to a higher value of bulk density and a lower value of active porosity and hydraulic conductivity. These spatial variations obtained from MPeat2D are consistent with the field observations, suggesting plausible outputs from the proposed model. By comparing the results of MPeat2D to a one-dimensional model and a two-dimensional model without the mechanical process, we argue that mechanical–ecohydrological feedbacks are important for analysing spatial heterogeneity, shape, carbon accumulation, and resilience of peatlands.

Geometry of plastic deformation in metals as piecewise isometric transformations

Deformation mechanisms of crystalline solids has been the subject of research for more than two centuries. The theory of dislocations dominates modern views but still has significant gaps demanding the introduction of additional concepts for the coherent quantitative description of physical phenomena. In this work, we propose a coherent geometric description of motion and deformation in crystalline solids as piecewise isometric transformations (PWIT). The latter only includes operations that, similar to interatomic spacing in crystalline lattice, do not alter distances between reference points, i.e. translations, rotations and mirror reflections. The difference between solid-body translations and plastic deformations is that the isometric transformations have discontinuities that in real-life materials realise through dislocations (termination of shifts), disclinations (termination of rotations), and twins (mirror reflections). The conceptual description of plastic deformations as PWIT can be useful for the better description of physical phenomena, proposing new hypothesis, and for developing predictive analytical models. In this paper, the use of this conceptual description enables proposing new hypothesis about the nature of such interesting phenomena in severe plastic deformation as (i) stationary ‘solid state turbulence’ stage in high pressure torsion, and (ii) rate of mass transfer (mechanically assisted diffusion) in simple-shear deformation.

MPMNet: A Data-Driven MPM Framework for Dynamic Fluid-Solid Interaction.

High-accuracy, high-efficiency physics-based fluid-solid interaction is essential for reality modeling and computer animation in online games or real-time Virtual Reality (VR) systems. However, the large-scale simulation of incompressible fluid and its interaction with the surrounding solid environment is either time-consuming or suffering from the reduced time/space resolution due to the complicated iterative nature pertinent to numerical computations of involved Partial Differential Equations (PDEs). In recent years, we have witnessed significant growth in exploring a different, alternative data-driven approach to addressing some of the existing technical challenges in conventional model-centric graphics and animation methods. This article showcases some of our exploratory efforts in this direction. One technical concern of our research is to address the central key challenge of how to best construct the numerical solver effectively and how to best integrate spatiotemporal/dimensional neural networks with the available MPM's pressure solvers. In particular, we devise the MPMNet, a hybrid data-driven framework supporting the popular and powerful MPM, to combine the comprehensive properties of MPM in numerically handling physical behaviors ranging from fluid to deformable solids and the high efficiency of data-driven models. At the architectural level, our MPMNet comprises three primary components: A data processing module to describe the physical properties by way of the input fields; A deep neural network group to learn the spatiotemporal features; And an iterative refinement process to continue to reduce possible numerical errors. The goal of these special technical developments is to aim at involved numerical acceleration while preserving physical accuracy, realizing efficient and accurate fluid-solid interactions in a data-driven fashion. The extensive experimental results verify that our MPMNet can tremendously speed up the computation compared with the popular numerical methods as the complexity of interaction scenes increases while better retaining the numerical accuracy.