Finite Deformation Analysis Research Articles

Prediction of huge earthquake-induced deformation of in-service embankments using crushed mudstone as a soil material with slaking and proposal of countermeasures

Evaluating the seismic resistance of embankments using crushed mudstone as a geomaterial is an urgent and crucial requirement. In this study, a ground investigation was conducted on the actual embankment. Based on the results, the seismic response of the embankment, simulating progressed slaking, was conducted by elastoplastic finite deformation analysis using two major types of earthquake motion: epicentral and subduction zone earthquakes. Based on the results of the geotechnical investigation, the embankment could be divided into three layers owing to differences in physical properties, and slaking progressed below the groundwater level in the embankment. The embankment did not exhibit large deformation during the epicentral earthquake owing to the short duration. For the subduction zone earthquake, the developed shear strain was from the two large acceleration groups and the subsequent smaller accelerations, resulting in large deformation. Seismic loading caused the gradual loss of the overconsolidation and decay of the structure which reduced the embankment strength. This analysis revealed that shear strain developed at the slope toe and the lower part of the embankment. Furthermore, the analysis after the earthquake was also conducted to examine whether or not countermeasure method is feasible for emergency restoration. The seismic resistance was greatly improved when a combination of ground improvement and replacement/counterweight fill methods were used to reinforce these areas, which is not only during but also after the earthquake. This study can contribute to the understanding of the seismic behavior of soil structures using materials undergoing internal deterioration and to the development of countermeasure methods for such structures.

Elastoplastic finite element simulation of domino fault formation associated with tilting of highly structured ground

Ground deformation on the Earth’s surface layer is strongly affected by the nonlinearity of geomaterials. However, the formation process of such deformation has yet to be described in a unified manner based on mechanics. The present study focuses on the normal faults in a submarine ground with highly developed soil skeleton structures and attempts to reproduce the process of normal fault formation associated with the tilting of a horizontally deposited submarine ground using an elastoplastic finite element simulation. The simulation was conducted using the soil–water coupled finite deformation analysis code GEOASIA, which incorporates an elastoplastic constitutive equation of the soil skeleton based on the modified Cam-clay model and the soil skeleton structure concept. The key findings are as follows:1) Normal faults are formed from the ground surface to depth as shear bands, where shear strain is localized while exhibiting softening behavior with plastic volume compression.2) Multiple normal faults are almost equally spaced and parallel to each other, with the inter-fault blocks rotating backward. The morphology of normal faults formed by the tilting of the ground shows domino-style characteristics.3) The degree of the soil skeleton structure influences the formation of normal faults.This study demonstrates that elastoplastic geomechanics can explain the formation process of ground deformation, which has usually been interpreted from the perspectives of geomorphology and geology.

A B-spline based gradient-enhanced micropolar implicit material point method for large localized inelastic deformations

The quasi-brittle response of cohesive-frictional materials in numerical simulations is commonly represented by softening plasticity or continuum damage models, either individually or in combination. However, classical models, particularly when coupled with non-associated plasticity, often suffer from ill-posedness and a lack of objectivity in numerical simulations. Moreover, the performance of the finite element method significantly degrades in simulations involving finite strains when mesh distortion reaches excessive levels. This represents a challenge for modeling cohesive-frictional materials, given their tendency to experience strongly localized deformations, such as those occurring during shear band dominated failure. Hence, accurate modeling of the response of cohesive-frictional solids is a demanding task. To address these challenges, we present an extension of the material point method (MPM) for the unified gradient-enhanced micropolar continuum, aiming at the analysis of finite localized inelastic deformations in cohesive-frictional materials. The generalized gradient-enhanced micropolar continuum formulation is employed to tackle challenges related to localization and softening material behavior, while the MPM addresses issues arising from excessive deformations. The method utilizes a B-spline formulation for the rigid background mesh to mitigate the well-known cell crossing errors of the MPM. To demonstrate the performance of the method, 2D and 3D numerical studies on localized failure in sandstone in plane strain compression and triaxial extension tests are presented. A comparison with finite element results confirms the suitability of the formulation. Moreover, an efficient numerical implementation of the formulation is presented, and it is demonstrated that the additional MPM specific overhead is negligible.

A generalized peridynamic material correspondence formulation using non-spherical influence functions

Peridynamics theory is a nonlocal continuum mechanics theory. The peridynamic material correspondence formulation provides passage for direct incorporation of any material constitutive models from the classical local continuum mechanics theory. The performance of the material correspondence formulation heavily relies on the influence function whose currently used form is spherical and depends only on the bond length. Together with the way how the nonlocal deformation gradient is constructed based on the material point horizon, this results in the well-known issue of material instability in the conventional peridynamic material correspondence formulation. In this paper, a generalized peridynamic material correspondence formulation that enables use both spherical and non-spherical influence functions is developed. A recently proposed class of parameterized non-spherical influence functions that depend on both the bond length and bond direction is used in the generalized formulation. Numerical examples including wave dispersion, small and finite deformation and linear elastic fracture analyses are studied to check the material stability, convergence characteristic, prediction accuracy and applicability to fracture problems without singularity issue of the proposed formulation. It is found that the proposed formulation is inherently stable, possesses linear convergence rate, yields highly accurate predictions for both small and finite deformation problems, and small horizon can be used to improve computational efficiency.

Tsunami hazard evaluation of river embankment structures incorporating their vulnerability to seismic strong motion

Development of coastal areas in Japan for various land uses since the 1960s has contributed to industrial upgrades and improved the efficiency of transportation networks. However, there are concerns about the vulnerability of developments on alluvial plains and reclaimed lands to geological events, like ground subsidence due to liquefaction during large earthquakes. Realistic assessment of earthquake and tsunami hazards and evaluation of possible countermeasures require accurate estimation of the amount of subsidence that can be expected from liquefaction at coastal and riverside sites supporting various structures. In this study, to evaluate the amount a river embankment structure might be expected to settle as a result of strong motion from an assumed Nankai Trough great earthquake, we conducted a numerical simulation using the soil–water coupled finite deformation analysis code GEOASIA. We then investigated the effect of the estimated embankment subsidence on tsunami inundation, which was simulated by using nonlinear shallow-water equations and a grid spacing as fine as 3.3 m. The influence of urban structures on the inundated area was taken into account by using a structure-embedded elevation model (SEM). The results showed that subsidence of river embankments and the collapse of parapet walls on top of them would increase both the depth and area of inundation caused by a tsunami triggered by a Nankai Trough scenario earthquake. Our findings underscore the importance of evaluating not only earthquake resistance but also vulnerability of coastal and riverside structures to strong motion in tsunami hazard analyses. Furthermore, the importance of tsunami inundation analysis using a SEM for predicting the behavior of tsunami flotsam in urban areas was demonstrated.

An ANS/ATFs‐based unsymmetric solid‐shell finite element algorithm for high‐quality finite deformation analysis of hyper‐elastic shell

AbstractAn effective updated Lagrangian (UL) algorithm is designed for extending the recent distortion‐tolerant unsymmetric 8‐node, 24‐DOF hexahedral solid‐shell element, US‐ATFHS8, to finite deformation analysis of hyper‐elastic shell structures. The distinguishing feature of this unsymmetric element is that two different interpolation schemes are employed for virtual displacement and real stress calculations, respectively. The assumed natural strain (ANS) method with shell assumptions, referring to the current configuration, is introduced to modify the strain tensors derived from the assumed virtual displacement fields in terms of isoparametric coordinates, thereby mitigating shear locking and trapezoidal locking. On the other hand, the analytical trial functions (ATFs) derived from the general solutions of homogenous governing equations for linear elasticity are updated in each increment step to obtain the incremental deformation gradient, which is then utilized for calculating the real stresses for curing the numerical difficulties in large deformation problems. Numerical examples show that the proposed algorithm enables the hyper‐elastic solid‐shell element US‐ATFHS8 to exhibit excellent performance in both regular and distorted meshes and yield considerable results even when other models cannot work.

Comprehension of Seismic-Induced Groundwater Level Rise in Unsaturated Sandy Layer Based on Soil–Water–Air Coupled Finite Deformation Analysis

Immense liquefaction damage was observed in the 2011 off the Pacific coast of Tohoku Earthquake. It was reported that, in Chiba Prefecture, Japan, the main shock oozed muddy water from the sandy ground and the aftershock which occurred 29 min after the main shock intensified the water spouting; thus, the aftershock expanded the liquefaction damage in the sandy ground. For comprehending such a phenomenon, using a soil–water–air coupled elastoplastic finite deformation analysis code, a rise in groundwater level induced by main shock is demonstrated, which may increase the potential of liquefaction damage during the aftershock. The authors wish to emphasize that these results cannot be obtained without soil–water–air coupled elastoplastic finite deformation analysis. This is because the rise in groundwater level is caused by the negative dilatancy behavior (plastic volume compression) of the saturated soil layer which supplies water to the upper unsaturated soil layer, and it is necessary to precisely calculate the settlement of ground and the amount of water drainage/absorption to investigate the groundwater level rise. This study provides insight into the mechanism of ground liquefaction during a series of earthquakes.

A three-dimensional micropolar beam model with application to the finite deformation analysis of hard-magnetic soft beams

The main purpose of this contribution is to develop a three-dimensional (3D) nonlinear beam model based on the micropolar continuum theory. To do so, a kinematic model based on the deformation of three directors and accounting for the micro-rotation tensor of the micropolar theory is introduced. One of the main characteristics of the present beam model is that 3D constitutive equations without any modification can be directly used in the formulation. Furthermore, it is known that a body couple field is induced in hard-magnetic soft materials (HMSMs) when subjected to external magnetic fluxes. Therefore, the stress tensor in HMSMs is asymmetric, in general. Since the asymmetry of stress is one of the main features of the micropolar theory, the present formulation can be used for analyzing the deformation of beams made of HMSMs. Accordingly, the virtual external work of the present model is formulated so that it accounts for the contribution from uniform or constant-gradient external magnetic fluxes on the beam. Moreover, a Total Lagrangian (TL) nonlinear finite element (FE) formulation to provide numerical solutions of the related problems is developed. Several numerical examples are solved to investigate the capability of the developed formulation. It is shown that the present formulation can model the size-dependent behavior of beam-like structures if the material length-scale parameter of the micropolar constitutive model is comparable to the thickness of the beam. Moreover, the proposed model can successfully predict the finite deformation of 3D beams made of HMSMs subjected to magnetic loading.

Modeling failure and instability of a single-jointed rock column based on finite deformation theory

The mechanical behaviour of rock columns is closely tied to factors such as joint development and height-to-width (H/W) ratio. Based on a recently developed finite deformation analysis scheme that can consider both material failure and structural instability of rock columns, uniaxial compression numerical tests of different H/W ratio rock columns containing a single-filled joint were carried out. It analyzes the effects of the strength of joint-filled material (joint strength) and H/W ratio on strength characteristics and failure modes of rock columns. The numerical results indicate that the joint strength controls the joint failure degree, macro-strength, stress distribution, and failure zone range of the rock column. Under different H/W ratios, the macro-strength of the rock column increases in an S-shaped curve with the increase in joint strength. Lower joint strength will accelerate joint failure, resulting in the failure mode of the rock column being joint failure-induced structural instability. As the joint strength and H/W ratio increase, the failure mode of the rock column evolves into the mode of cooperative material-structure-induced collapse and finally develops into structural instability-induced material failure. The research results have a particular reference significance for selecting grout materials for columnar jointed rock mass engineering.

Structural topology optimization of three-dimensional multi-material composite structures with finite deformation

In this work, an explicit three-dimensional (3D) topology optimization approach is presented for multi-material composite structures accounting the finite deformation effect. The proposed method employs different sets of 3D Moving Morphable Voids (MMVs) to identify each phase material, resulting in explicit geometric descriptions of the optimized composite structures and a reduction in the number of design variables. The decoupling between the topology description and finite element analysis of composite structures enables the removal of redundant degrees of freedom, thereby mitigating the convergence issue of finite deformation analysis caused by low-density elements and leading to significant computational savings. Numerical experiments of composite structures with two- and three-phase materials are optimized to validate the accuracy and efficiency of the proposed method. The results demonstrate that, under finite deformation, the distribution of each phase material in optimized 3D composite structures is significantly affected by the amplitude of external loads, and the optimized layout could be quite different from its counterpart under small deformation assumption.

Micromechanics of kink-band formation in bimetallic layered composites

Bimetallic layered composites with ultrathin layers possess high strength and hardness, and excellent shock resistance and radiation damage tolerance. However, they are prone to deformation localization and readily form kink-bands when subjected to layer parallel compression. Following this, we carried out extensive plane strain finite element finite deformation analyses to understand and rationalize the experimentally observed kink-band formation in these composites. In the calculations, both constituent materials were assumed to follow a rate-independent isotropic elastic-plastic constitutive relation with an ad-hoc thickness dependent yield strength. Our results demonstrate that in these composites, layer refinement, together with the strength differential between the layers of the constituent materials, is sufficient to trigger kink-banding. Importantly, this phenomenon occurs even in the absence of elastically stiff layers, geometrical imperfections (such as waviness of the layers), and extreme plastic anisotropy within the layers. Additionally, we performed parametric studies to investigate the individual effects of layer thickness, strength differential between the two constituent materials, and strain-hardenability of the materials on kink-band formation. The outcomes of our parametric studies reveal that the strain-hardenability of the constituent materials stabilizes the formation of kink-bands. Furthermore, it leads to a transition from the abrupt formation of through-width kink-bands to the onset and propagation of stable inclined wedge-shaped kink-bands, similar to the experimental observations.

Level-set-based topology optimization of a morphing flap as a compliant mechanism considering finite deformation analysis

Level-set-based topology optimization of a morphing flap as a compliant mechanism considering finite deformation analysis

Finite deformation analysis of the rotating cylindrical hollow disk composed of functionally-graded incompressible hyper-elastic material

Finite deformation analysis of the rotating cylindrical hollow disk composed of functionally-graded incompressible hyper-elastic material

Finite deformation analysis of the elastic circular plates under pressure loading

As one of the most commonly used engineering components, thin plates usually undergo large deflections when subjected to intense pressure loadings. In addition to the material nonlinearity, such as plastic yielding, the geometric nonlinearity caused by this finite deformation also has a great influence over its total response. In this study, a finite deformation approximation solution is given theoretically for the elastic circular plate, based on the von Kármán equations and the uniform membrane strain assumption. Through comparisons with the corresponding finite element simulations and the existing theoretical solution, the present solution is confirmed to be both accurate and efficient, for thin plate under uniform pressure loading, in both static and dynamic loading conditions. Thereafter, the competing effects and the respective dominant ranges of bending and stretching are investigated.Besides, based on the accurate stress prediction provided by the proposed high-order deformation model, the limit state of the elastic stage is analyzed, including the location that first reached the elastic limit and the corresponding instant. Thus, the present approximate solution will be valuable in future studies of failure criteria and subsequent inelastic behavior.

Finite deformation analysis of electro-active shells

In this contribution, a novel nonlinear formulation for the finite deformation analysis of shell-like structures made of dielectric polymers and subject to electrical loading is developed. The kinematics of mechanical deformation is described via a seven-parameter shell model. Moreover, a second-order Taylor expansion, containing three parameters, is proposed to approximate the electrical potential through the thickness of the shell. The combined mechanical and electrical parameters constitute a ten-parameter electro-shell model. After developing the variational formulation of the problem, a nonlinear finite element formulation for the numerical solution of the coupled electromechanical problem is also developed. The formulation is further enriched by the enhanced assumed strain method to alleviate locking in thin shells. Finally, the applicability of the proposed formulation in several benchmark examples is demonstrated. In particular, it is shown that the formulation can capture the results available in the literature with high accuracy.

Distance minimizing based data‐driven computational method for the finite deformation of hyperelastic materials

AbstractThe distance minimizing based data‐driven solvers are developed for the finite deformation analysis of three‐dimensional (3D) compressible and nearly incompressible hyperelastic materials in this work. The data‐driven solvers bypass the construction of a constitutive equation for the hyperelastic materials by considering a dataset of Green‐Lagrange strain‐second Piola–Kirchhoff stress pairs. They recast the boundary‐value problems into the distance minimization problems with basic kinematical and mechanical constraints. Moreover, the deviatoric/volumetric split of stress and the additional incompressible constraint are further introduced into the solver for the nearly incompressible hyperelastic materials. Several representative three‐dimensional examples are presented and the results demonstrate the good capability and robustness of the proposed data‐driven solvers.

On the generalized nonlinear mechanics of compressible, incompressible, isotropic, and anisotropic hyperelastic membranes

The aim of this paper is to develop a generalized formula for the finite deformation analysis of hyperelastic membranes under various geometry, material, and loading conditions that can be applied to the mechanics of polymer or biological membranes. To do so, the kinematics of deformation in three-dimensional space is formulated and basic kinematic and kinetic quantities are introduced. Set of differential equations governing the deformation of hyperelastic membranes are expressed in a variational framework. Due to inherently nonlinear characteristics of governing equations at large regimes of deformations, a Total Lagrangian (TL) nonlinear finite element formula is developed. The formulation accounts for isotropic as well as anisotropic with transverse isotropy hyperelastic materials. Moreover, both compressible and incompressible cases of membranes are considered. Particular emphasis is on compressible case, where an additional nonlinear equation is solved to obtain the stretch of the membrane in the thickness direction. Furthermore, a study about the effect of participation of fourth and fifth invariants in transversely isotropic strain energy functions is performed. In several cases, introducing transversely isotropic strain energy is unavoidable. As observed in this paper, in the case of biological membranes, the existence of a preferred direction is necessary to predict its behavior in different directions, correctly. Eventually, various examples are solved and excellent results in comparison to those available in the literature are achieved.

Fine-grained heterogeneous parallel direct solver for finite element problems

The improvement of computational effectiveness is a vital issue in the field of large-scale finite element analysis. The performance is fundamentally determined by the efficiency of solving sparse linear system equations using the implicit finite element method. This paper presents a direct linear solver based on heterogeneous hybrid parallel computing on CPUs and GPUs. This can efficiently utilize computing resources of multiple devices to achieve performance improvement. Initially, we partition the elimination tree into several subtrees to accomplish the task decomposition. Based on this, we build a dynamic programming mathematical model to balance the computational load of the various devices. Then, we develop a numerical decomposition strategy by combining node parallelism and tree parallelism for the CPUs. In addition, efficient numerical decomposition is achieved on the GPU through batch processing and maximizing the overlap between computations and data transfers. Numerical experiments show that, compared with MKL PARDISO, the performance of numerical factorization can be improved by up to 10 times by using CPU and dual-path GPU hybrid calculations, and the computation time of simulation can be reduced by one-third for the multicondition analysis of Body In White and by 20% for the large-scale nonlinear finite element deformation analysis.

Extension of the unsymmetric 8‐node hexahedral solid element US‐ATFH8 to 3D hyper‐elastic finite deformation analysis

AbstractThis work extends the recent US‐ATFH8 element to 3D hyper‐elastic finite deformation analysis. Using two sets of shape functions, the new 3D element comprises of 8 nodes and 24 DOFs. The first set of shape functions represent the test functions that come from the conventional isoparametric interpolation, and the second set, representing the trial functions, are constructed from the homogenous solutions for linear elasticity governing equations, termed analytical trial functions (ATFs). This study considers finite deformation for hyper‐elastic materials, but it is assumed that the analytical solutions associated with hyper‐elastic materials can be updated to hold approximately in each incremental step. Moreover, the deformation information required for stress computation is updated by using the incremental deformation gradient, which is constructed from the updated ATFs. Numerical examples show that without additional pressure DOF, the element US‐ATFH8 still behaves well in nearly incompressible hyper‐elastic 3D problems with finite deformation, even when the meshes are extremely distorted.

Risk assessment and support design optimization of a high slope in an open pit mine using the jointed finite element method and discontinuous deformation analysis

Risk assessment and support design optimization of a high slope in an open pit mine using the jointed finite element method and discontinuous deformation analysis