Solid Deformation Research Articles

A FEM for three‐field u–p–η poroelasticity with nonreciprocal interactions

AbstractThe de la Cruz and Spanos (dCS) theory of poroelasticity is a pore‐scale volume averaged formulation and differs from the widely used Biot (BT) theory. A novel Finite Element Method (FEM) is developed for dCS theory to enable the study of nonreciprocal solid–fluid interactions, which are omitted from BT model. Solid deformations are quasi‐static and pore fluid flow is transient; dynamic effects are neglected. The form of the dCS theory chosen includes BT theory as a special case. The governing equations are written in terms of three fields: solid displacement u, fluid pressure p, and porosity η. Fully implicit time integration and a mixed‐element formulation are employed to ensure stability. The convergence rate of the FEM dCS model is shown to be optimal in a one‐dimensional consolidation problem. Examples of a footing and subsurface injection problems in two dimensions further attest the robustness of the implementation and are shown to reproduce BT model results as a special case. The effect of nonreciprocal solid–fluid interactions is studied in all examples and shows a wide range of importance depending on the properties of the porous media (e.g., permeability) and problem‐specific constraints. The developed FEM provides a tool to enable further comparisons between dCS and BT theories and validation in practical applications.

Stress‐hybrid virtual element method on quadrilateral meshes for compressible and nearly‐incompressible linear elasticity

AbstractIn this article, we propose a robust low‐order stabilization‐free virtual element method on quadrilateral meshes for linear elasticity that is based on the stress‐hybrid principle. We refer to this approach as the stress‐hybrid virtual element method (SH‐VEM). In this method, the Hellinger–Reissner variational principle is adopted, wherein both the equilibrium equations and the strain‐displacement relations are variationally enforced. We consider small‐strain deformations of linear elastic solids in the compressible and near‐incompressible regimes over quadrilateral (convex and nonconvex) meshes. Within an element, the displacement field is approximated as a linear combination of canonical shape functions that are virtual. The stress field, similar to the stress‐hybrid finite element method of Pian and Sumihara, is represented using a linear combination of symmetric tensor polynomials. A 5‐parameter expansion of the stress field is used in each element, with stress transformation equations applied on distorted quadrilaterals. In the variational statement of the strain‐displacement relations, the divergence theorem is invoked to express the stress coefficients in terms of the nodal displacements. This results in a formulation with solely the nodal displacements as unknowns. Numerical results are presented for several benchmark problems from linear elasticity. We show that SH‐VEM is free of volumetric and shear locking, and it converges optimally in the norm and energy seminorm of the displacement field, and in the norm of the hydrostatic stress.

An investigation into the softening behavior of cast A356 alloy in the solid and semisolid deformation regimes

The current work deals with the hot deformation behavior of A356 aluminum alloy starting with the cast structure and an emphasize on the capability of flow softening. Toward this end, the hot compression tests were performed at temperatures of 150, 250, 350, 450, 500 and 550 °C in the solid range and 580 °C in the semisolid range under the typical strain rate of 0.01s−1. The materials represent high protentional for the occurrence of softening mechanism in solid state regime. During thermomechanical processing (above 250 °C), after reaching the peak stress, the dynamic recrystallization prevails over the deformation behavior and the stress level reaches a steady state regime. The α-dendritic arms are crushed and refined, and the silicon blades are mechanically fragmented and spread irregularly through the microstructure. Being in the semi-solid temperature range, the eutectic silicon phase is melted and causes a sharp decrease in the resistance against deformation. The penetration of the liquid phase along the developed sub-boundaries give rise to grain partitioning and resembles the occurrence of dynamic recrystallization. The presence of the liquid phase during semisolid forming, in addition to the lubricated flow mechanism, facilitates the deformation by providing proper condition for the occurrence of recrystallization. In the present case, a stable stress level as below as 1 MPa was recorded for the experiment cast structure.

Probabilistic-learning-based stochastic surrogate model from small incomplete datasets for nonlinear dynamical systems

We consider a high-dimensional nonlinear computational model of a dynamical system, parameterized by a vector-valued control parameter, in the presence of uncertainties represented by an uncontrolled parameter modeled by a vector-valued random variable, and possibly with stochastic excitation. The objective is to construct a statistical surrogate model where the input is any deterministic value of the control parameter, and the output is a vector-valued observation of the computational model, which is a random vector whose probability measure is updated using a target dataset. To construct this statistical surrogate model, the stochastic response of the computational model must be built, which is a vector-valued time-discretized stochastic process in high dimension, depending on the control parameter. It is assumed that the computational cost of a single evaluation of the deterministic model is high. For the probabilistic updating, we consider a subset of the components of the observation of the computational model, defined as the “identification observation” of the computational model, for which a small target dataset is available. Therefore, the target dataset is associated with partial observability, corresponding to an incomplete data case. Given a prior probability model of the random control and uncontrolled parameters, a training dataset is constructed, consisting of realizations of the random triplet composed of the stochastic response, the random identification observation, and the random control parameter. Since the computational cost of a single evaluation of the deterministic model is assumed to be large, the training dataset is also of small size. The main challenges in this problem are the high dimensionality, partial observability leading to incomplete data in the target dataset for the identification observation of the computational model (which is not sufficient to identify the computational stochastic responses), and the availability of a small training dataset. To address these challenges, we propose a methodology based on statistical methods for constructing necessary reduced representations, direct probabilistic learning under constraints using probabilistic learning on manifolds (PLoM) constrained by the target dataset, and the use of a weak formulation of the Fourier transform of probability measures. Statistical conditioning is also employed to explore the learned dataset. The constructed predictive statistical surrogate model can be implemented in the context of online computation. We apply this approach to a problem of nonlinear stochastic dynamics in high dimensions within the framework of deformable solids mechanics.

Eulerian framework for contact between solids represented as phase fields

Mechanical contact between solids is almost exclusively modeled in Lagrangian frameworks. While these frameworks have been developed extensively and applied successfully to numerous contact problems, they generally require complex algorithms for contact detection and resolution. These challenges become particularly important when contact appears between solids with evolving boundaries, such as in systems where crystals grow in a constrained space. In this work, we introduce a fully Eulerian finite element framework for modeling contact between elastic solids tailored towards problems including evolving and intricate surfaces. The proposed approach uses a phase-field method that involves a diffuse representation of geometries on a fixed mesh, simplifying the modeling of evolving surfaces. Our methodology introduces a novel volumetric contact constraint based on penalty body forces, efficiently resolving the interpenetration of solids. We showcase the validity and versatility of our method through numerical examples, highlighting its ability to accurately capture complex solid–solid interactions. The Eulerian phase-field formulation greatly simplifies contact detection and its resolution. Furthermore, the framework can be straightforwardly coupled with other physical phenomena through the inclusion of multiple energy terms in the evolution of the phase-field. This enables multiphysics modeling, potentially providing a valuable tool for a wide range of applications involving chemically or physically evolving deformable solids in contact, as they commonly occur in deterioration processes of porous media.

Grid-Characteristic Numerical Method on an Irregular Grid with Extending the Interpolation Stencil

A grid-characteristic numerical method for solving a multidimensional transport equation on an unstructured grid with an order higher than one is proposed; this method does not use auxiliary points on edges and faces. The avoidance of auxiliary points on edges and faces simplifies the topology of the computational grid during its motion, which is important for solving dynamic problems of mechanics of deformable solids. To increase the approximation order, an analog of the grid stencil extension implemented for an unstructured grid is used. Results of testing the proposed numerical scheme for continuously differentiable, continuous, discontinuous solutions are presented.

Fully coupled hydro-mechanical modeling of two-phase flow in deformable fractured porous media with discontinuous and continuous Galerkin method

Discrete fracture and matrix modeling of coupled flow and geomechanics is instrumental for understanding flow in fractured media for various geoengineering applications such as enhanced geothermal energy systems and groundwater remediation. We employ the governing equations for two-phase flow in deformable fractured porous media with a stress-dependent porosity and permeability model of matrix, and variable fracture aperture and the corresponding permeability. A finite element framework is presented, in which fractures are regarded as low-dimensional objects, to discretize the coupled two-phase flow and geomechanics in fractured porous media. A hybrid method, combining discontinuous Galerkin (DG) and continuous Galerkin (CG) finite element methods (FEM), is utilized to solve for the two-phase flow, while the solid deformation is approximated using a discontinuous Galerkin FEM approach. Several benchmark cases are utilized to examine the accuracy of the proposed hybrid DG-CG FEM method. Further validation is performed using more complex, realistic fracture configurations. As the mechanical characteristics of the fracture and the surrounding matrix differ, the simulation results demonstrate that displacement and stress are discontinuous on both sides of the fracture. While the case with low permeability fractures exhibits pressure jump across the fractures, the pressure changes are reasonably smooth for the conduit fractures.

A benchmark study for quasi-static numerical upscaling of seismic wave attenuation and dispersion in fractured poroelastic rocks

Numerical simulation is an effective tool for comprehending fluid flow behavior and solid skeleton deformation within porous media, as well as obtaining valid seismic information. The elastic response of seismic waves, serving as an intermediary linking material properties and seismic attributes, is influenced by a combination of the wave-induced fluid flow (WIFF) effects and stress interactions. However, the impact of stress interactions is frequently overlooked in numerous studies, primarily due to the complex distribution of subsurface fractures makes it difficult to clarify the effects caused by local stress perturbations. Therefore, we present a numerical upscaling procedure as a benchmark process to quantify seismic wave attenuation and dispersion in fractured porous media simulations. The procedure is based on COMSOL Multiphysics and allows for the simulation of the energy dissipation resulting from the compression of a model with arbitrary forms of fluid saturation, geometry, and petrophysical parameters by seismic waves. We analyze the fracture models with different spatial distributions and infills to understand the mechanism. Our simulation results show the efficient and accurate capabilities of the numerical upscaling procedure in simulating the acoustic properties of fractured porous media and demonstrate the physical processes at various frequencies. In addition, our study yielded corresponding results: 1) the diverse spatial distribution of fractures not only gives rise to peak anomalies but also influences both the attenuation peaks and characteristic frequencies of seismic waves, thus reflecting the attenuation phenomena occurring at different scales; 2) various infills exhibit distinct sensitivities to fracture morphology and stress interactions.

A hybrid Finite Volume-Smoothed Particle Hydrodynamics approach for shock capturing applications

A hybrid Finite Volume Method (FVM)-Smoothed Particle Hydrodynamics (SPH) approach for shock capturing in compressible fluids is presented. The Python framework Pyro2 is employed to simulate a coarse FVM mesh, while the Python framework PySPH is utilized to model the fluid in regions with high gradients through SPH particles. New FVM-SPH coupling approaches are explored, including online SPH particle injection and deletion around time persistent SPH regions around shocks. The SPH simulation is coupled to the FVM simulation through boundary particles surrounding the active SPH regions, with FVM values interpolated to SPH boundary particles at each timestep. Values in active SPH regions are similarly interpolated to the overlying FVM mesh at each timestep. The performance of the hybrid FVM-SPH scheme in comparison to the individual FVM and SPH methods on two-dimensional test problems is assessed. Our results indicate that the hybrid approach offers higher computational efficiency than SPH while preserving its accuracy, although it remains slower than the FVM. This hybrid SPH algorithm may enable future compressible fluid simulations with more extensive capabilities than grid-based methods alone, offering potential applications in modeling fluid–structure interaction and solid deformation and fracturing in blast simulations. Further research will focus on applying the hybrid method to three-dimensional blast problems, where the relative computational cost reduction is expected to be more significant.

An arbitrary Lagrangian–Eulerian formulation for the Reynolds equation considering the JFO boundary condition

AbstractIn the realm of lubrication theory, the hydrodynamics of the fluid lubricant are commonly elucidated through the Reynolds equation. Its establishment stems from using the Eulerian description method, restricting the fluid domain to a fixed spatial configuration. However, certain scenarios necessitate coupling the Reynolds equation with solid deformation, wherein the fluid boundary undergoes alterations concurrent with the displacement or deformation of the solid. In such cases, the employment of a movable fluid domain becomes imperative. To address this requirement, this contribution introduces the arbitrary Lagrangian–Eulerian method, which effectively transforms the reference system of the Reynolds equation into a fluid domain amenable to arbitrary movement. Subsequently, the weak formulation associated with the movable fluid domain is derived and synergistically combined with a preexisting cavitation formulation. This integration facilitates an efficient solution of the Reynolds equation utilizing the finite element method while duly accounting for the mass‐conserving cavitation effects. Ultimately, a transient hydrodynamic lubrication example is presented to demonstrate the feasibility of the newly developed formulations.

Arbitrary mesh‐moving velocity‐based space‐time finite element method for large deformation analysis of solids

SummaryThis paper presents a new velocity‐based space‐time finite element method for computing the large deformation of solids over arbitrary Lagrange/Eulerian moving meshes. The proposed method is oriented toward quasi‐static deformation problems with hypoelastic and neo‐Hookean hyperelastic constitutive models. The Cauchy stress in the weak form of the linear momentum equation is explicitly described in terms of deformation velocity through consistent time integration of objective stress rates transferred in the arbitrary moving mesh. Consequently, the resultant system equation only contains the space‐time nodal velocity as the primary unknown. An iterative algorithm for solving the discretized system equation is implemented, whereby the nonlinearities induced by geometrical change in the computational domain as well as nonlinear constitutive equations are computed in an iterative manner updating the stress and the strain over the moving mesh of the deforming domain. Three benchmark problems, such as uniform beam compression, thick‐cylinder compression with superimposed rigid body rotation and extrusion of a column involving material loss, have been solved in order to examine the accuracy of the proposed numerical scheme. The numerical results have revealed the proper computation of the stress and the displacement of the solids undergoing the large deformation and rotation over the arbitrary Lagrange/Eulerian moving meshes, and have shown the validity and applicability of the proposed method with linear convergence of the iterative algorithm.

A Peridynamic-enhanced finite element method for Thermo–Hydro–Mechanical coupled problems in saturated porous media involving cracks

In this paper, a peridynamic-enhanced finite element formulation is introduced for the numerical simulation of thermo–hydro–mechanical coupled problems in saturated porous media with cracks. The proposed approach combines the Finite Element (FE) method for governing heat conduction–advection and fluid flow in the fractured porous domain, and the Peridynamic (PD) method for describing solid phase deformation and capturing crack propagation. Firstly, the consolidation problem of a porous column is simulated by using the proposed approach. The m- and δ-convergence studies are carried out with the isothermal condition to determine suitable discretization parameters for the PD model. Subsequently, non-isothermal conditions are considered, and the accuracy and reliability of the proposed approach are validated by comparing the numerical solutions with those obtained from a FE-only model. Furthermore, several numerical examples focusing on scenarios involving cracks are solved and presented to further highlight the capabilities of the proposed approach in addressing heat conduction–advection problems in fractured saturated porous media, as well as hydraulic fracture propagation problems with considerations of thermo–hydro–mechanical coupled effects.

Finite discrete element modeling of desiccation fracturing in partially saturated porous medium

In this study, a numerical framework is presented for the analysis of desiccation fracturing in variably saturated porous media. The solid phase deformation and fracturing are obtained based on the 2D finite-discrete element method (FDEM), while the fluid flow is characterized by nonlinear Richards’ equation and solved by finite volume discretization. The coupling between phases is realized by Biot’s porous media theory. The capabilities of the model in capturing the unsaturated porous medium flow and coupled hydro-mechanical effects are put to the test against benchmark solutions available in the literature. Furthermore, mechanisms of desiccation cracking in the soil are thoroughly investigated through various numerical test cases. In line with experimental observations, results indicate that the type of mechanical constraints and saturation gradients are two main factors contributing to the generation of cracks in desiccation processes.

Connecting beams and continua: variational basis and mathematical analysis

We present a new variational principle for linking models of beams and deformable solids, providing also its mathematical analysis. Despite the apparent differences between the two types of governing equations, it will be shown that the equilibrium of systems combining beams and solids can be obtained from a joint constrained variational principle and that the resulting boundary-value problem is well posed.

Deformation behaviour of the silicon doped metastable Fe50-xMn30Co10Cr10Six complex concentrated alloy using experiments and crystal plasticity simulations

Deformation behaviour of transformative Fe50Mn30Co10Cr10 complex concentrated alloy doped with 0.2 wt % silicon is studied using ex-situ EBSD accompanied with mean and full field crystal plasticity simulations. On silicon doping, martensitic transformation gets suppressed rather nano twinning is observed with tensile deformation. The considerable rise in strength and tensile ductility is attributed to the solid solution strengthening and deformation twinning respectively. Furthermore, crystal plasticity simulations using mean field viscoplastic self-consistent and full field fast Fourier transform FFT approaches are performed to address the texture evolution, the effect of neighbouring grains of different orientations as well as slip and twin activity. Slip is found to be planar in nature which is verified using crystal plasticity simulations. Combination of mean and full field simulations is used to address the effect of neighbouring orientations on the rotation path of individual grains towards the stable end orientation. The critical role of deformation nano twinning in providing extended strain hardening contributing to significant ductility is established. This study highlights the importance of silicon doping in improving the mechanical properties of the Fe50Mn30Co10Cr10 CCA while offering valuable insights into the deformation mechanisms and crystallographic texture evolution.

Semi-Solid Deformation Behavior, Flow Stress Prediction and Deformed Microstructure of GH3536 Superalloy

In current work, the deformation behavior and deformed microstructures of GH3536 superalloy in semi-solid state were investigated, and the semi-solid flow stress was predicted by artificial neural network (ANN) model. The semi-solid compression deformation was carried out at 1320-1350 °C, and the solid deformation behavior at 1200 °C was tested for comparison. The peak stress under 0.01-1 s-1 semi-solid deformation was 45.6-161.9 MPa. The peak stress decreased with the increase of deformation temperature and the decrease of strain rate. The ANN model could well describe semi-solid flow stress. During semi-solid deformation, the apparent viscosity dropped as shear rate increased. At high temperature and low strain rate, more liquid phase was distributed at grain boundaries. The solid grains coarsened as deformation temperature increased.

Micropolar Elasticity in Physically-Based Animation

We explore micropolar materials for the simulation of volumetric deformable solids. In graphics, micropolar models have only been used in the form of one-dimensional Cosserat rods, where a rotating frame is attached to each material point on the one-dimensional centerline. By carrying this idea over to volumetric solids, every material point is associated with a microrotation, an independent degree of freedom that can be coupled to the displacement through a material's strain energy density. The additional degrees of freedom give us more control over bending and torsion modes of a material. We propose a new orthotropic micropolar curvature energy that allows us to make materials stiff to bending in specific directions. For the simulation of dynamic micropolar deformables we propose a novel incremental potential formulation with a consistent FEM discretization that is well suited for the use in physically-based animation. This allows us to easily couple micropolar deformables with dynamic collisions through a contact model inspired from the Incremental Potential Contact (IPC) approach. For the spatial discretization with FEM we discuss the challenges related to the rotational degrees of freedom and propose a scheme based on the interpolation of angular velocities followed by quaternion time integration at the quadrature points. In our evaluation we validate the consistency and accuracy of our discretization approach and demonstrate several compelling use cases for micropolar materials. This includes explicit control over bending and torsion stiffness, deformation through prescription of a volumetric curvature field and robust interaction of micropolar deformables with dynamic collisions.

Mechanics of Tunable Adhesion With Surface Wrinkles

Abstract Surface wrinkles have emerged as a promising avenue for the development of smart adhesives with dynamically tunable adhesion, finding applications in diverse fields, such as soft robots and medical devices. Despite intensive studies and great achievements, it is still challenging to model and simulate the tunable adhesion with surface wrinkles due to roughened surface topologies and pre-stress inside the materials. The lack of a mechanistic understanding hinders the rational design of these smart adhesives. Here, we integrate a lattice model for nonlinear deformations of solids and nonlocal interaction potentials for adhesion in the framework of molecular dynamics to explore the roles of surface wrinkles on adhesion behaviors. We validate the proposed model by comparing wrinkles in a neo-Hookean bilayer with benchmarked results and reproducing the analytical solution for cylindrical adhesion. We then systematically study the pull-off force of the wrinkled surface with varied compressive strains and adhesion energies. Our results reveal the competing effect between the adhesion-induced contact and the roughness due to wrinkles on enhancing or weakening the adhesion. Such understanding provides guidance for tailoring material and geometry as well as loading wrinkled surfaces for different applications.

Challenges and Prospects of All‐Solid‐State Electrodes for Solid‐State Lithium Batteries

AbstractIn the development of all‐solid‐state lithium batteries (ASSLB), progress is made with solid‐state electrolytes; however, challenges regarding compatibility and stability still exist with solid electrodes. These issues result in a low battery capacity and short cycle life, which limit the commercial application of ASSLBs. This review summarizes the recent research progress on solid‐state electrodes in ASSLBs including the solid–solid interface phenomena such as the interface between electrode materials and electrolytes. The mechanical stability problems in solid electrodes, including fracture, brittleness, and deformation of electrode materials, are also discussed, and corresponding methods to measure the solid electrode stress are provided. In addition, strategies for mitigating stress‐related issues are examined. Finally, the fabrication process of solid electrodes is introduced and their future developments, including the exploration of new electrode materials and the design of more intelligent electrode structures, are proposed.

Optimization of Johnson-Cook Constitutive Model Parameters Using the Nesterov Gradient-Descent Method.

Numerical simulation of impact and shock-wave interactions of deformable solids is an urgent problem. The key to the adequacy and accuracy of simulation is the material model that links the yield strength with accumulated plastic strain, strain rate, and temperature. A material model often used in engineering applications is the empirical Johnson-Cook (JC) model. However, an increase in the impact velocity complicates the choice of the model constants to reach agreement between numerical and experimental data. This paper presents a method for the selection of the JC model constants using an optimization algorithm based on the Nesterov gradient-descent method. A solution quality function is proposed to estimate the deviation of calculations from experimental data and to determine the optimum JC model parameters. Numerical calculations of the Taylor rod-on-anvil impact test were performed for cylindrical copper specimens. The numerical simulation performed with the optimized JC model parameters was in good agreement with the experimental data received by the authors of this paper and with the literature data. The accuracy of simulation depends on the experimental data used. For all considered experiments, the calculation accuracy (solution quality) increased by 10%. This method, developed for selecting optimized material model constants, may be useful for other models, regardless of the numerical code used for high-velocity impact simulations.