Plane Strain Assumption Research Articles

Study of the performance of a 3D modular geosynthetic reinforced soil bridge abutment under overloading

The objective of this study was to investigate the response characteristics of three-dimensional geosynthetic reinforced soil (GRS) bridge abutments subjected to overload conditions in terms of deformation, soil stresses, and reinforcement strains. To this end, a 1:2 scale model of a GRS bridge abutment was constructed and subjected to a detailed analysis when the vertical load was increased in incremental steps. The results showed that the GRS abutment performed well and did not reach the plastic state at six times the service limit state (SLS) load. The settlement at the top of the abutment increases with increasing load, and the settlement results in the central bulging of both the front and wing walls, which is inconsistent with the assumption of plane strain. Therefore, a novel three-dimensional facing displacement model that outperforms traditional calculation methods is presented. The incremental vertical soil stresses in the center of the abutment increased significantly, and the U.S. Federal Highway Administration (FHWA) method better predicted the vertical soil stresses on the GRS bridge abutments under SLS loads; however, this method underestimates the incremental vertical soil stresses under overloads. The maximum reinforcement strain occurred in the middle of the abutment, and the deformation of the reinforcement was not uniform under overload, indicating that an overload may lead to the destruction of the reinforcement under the beam seat. The results of this study can serve as a reference for the design and application of GRS bridge abutments.

3D seismic bearing capacity of rectangular foundations near rock slopes using upper bound method

The analysis of foundation's bearing capacity is generally conducted under the assumption of plane strain. However, the damage of rectangular foundation usually presents obvious three-dimensional (3D) effect. This article considers this 3D effect, which, for the first time, provides a theoretical framework for assessing the bearing capacity of rectangular foundations placed on rock slopes under seismic condition. In order to apply the kinematic method of limit analysis, a new 3D kinematically admissible collapse mechanism is first constructed. Owing to the point-to-point discretized technique, this 3D mechanism can avoid complex surface integration and effectively reduce spatial coordinate iterative calculations while maintaining high accuracy. The pseudo-static method is adopted to simulate the effects of earthquakes. Generalized multi‑tangential technique is employed to derive the Mohr-Coulomb constants from Hoek-Brown criterion. The comparison between the present results and results of previous literatures proves the validity of the theoretical framework in this paper. Shape factor and reduction factor are introduced in the parametric study, showing that the 3D effect is more obvious when the rectangular foundation aspect ratio is reduced. As the rectangular foundation location moves away from the slope, its bearing capacity gradually converges towards that of the horizontal foundation. The critical collapse mechanism is also explored, demonstrating that the collapse extent expands with increasing internal friction angle of the rock.

The role of the interfaces and cross-links on the mechanical behavior of mineralized collagen fibrils. A numerical approach

Mechanical properties of bone tissue are highly dependent on its hierarchical structure. The presence of microcracks and diffuse damage in lamellar bone is correlated with the failure of the collagen-mineral interface in mineralized collagen fibrils (MCF). The main goal of this work is to evaluate the mechanical behavior of the interfaces and quantify the stiffness loss of the MCF associated with different failure mechanisms, under controlled in-plane displacement. Additionally, we aim to study the role of the cross-links on the fibril mechanical response, beyond the interface failure. Inter- and intra-microfibrilar cross-links are analyzed. In order to address the first issue, a detailed representative volume of the MCF is analyzed by means of the finite element method, under the assumption of plane strain and periodic boundary conditions. In this model the interfaces between constituents are modeled with an exponential cohesive law. Enzymatic cross-links, located at the molecular terminals connecting each 4D (D=67nm) staggered molecules, are represented by non-linear springs. Three in-plane controlled deformations are applied. The results of this work provide the anisotropic stiffness loss of the tissue involved in the different failure mechanisms at the nano-scale length. The initiation of microcracks and the presence of damage zones are compatible with the failure mechanisms observed at interfaces. Interface failure entails a progressive stiffness loss, bringing a non-linear behavior of bone. The strength obtained for the longitudinal maximum deformation is more than 20 times the transverse strength and 3.5 times the shear strength. The quantification of the reduction percentage in the elastic moduli and the shear stiffness when the fibril is damaged, has a potential application in improving failure criteria based on degradation of elastic constants. When longitudinal elongation is applied, the mechanical contribution of the cross-links in delaying the failure initiation of the interface is shown. Likewise, results of this work confirm the scarce influence of the cross-links in the strain range analyzed. Additionally, a three-dimensional numerical model of several microfibrils is defined with the aim of analyzing the mechanical relevance of inter- and intra-microfibrilar cross-links, beyond the interface failure. Results confirm that cross-links transfer the load when strain increases, being highlighted the mechanical competence of the trivalent cross-links.

A new hybrid local radial basis function collocation method for 2.5D thermo-mechanical modelling of continuous casting of steel

This study presents a new strong-form meshless method to solve the thermo-mechanical problem of the solidification process in the continuous casting of steel. A two-dimensional slice that travels in the casting direction is modelled in the Lagrangian system. The newly developed mechanical model is one-way coupled to the thermal model, where the heat flux due to the mould, sprays, rolls and radiation are imposed to solve heat transfer in the strand. The resulting temperature and metallostatic pressure govern the Norton-Hoff visco-plastic model used for computing shrinkage of the solid shell and induced residual stresses. The results are used to estimate critical areas susceptible to hot-tearing formation. The mechanical model uses a generalised plane strain assumption that includes linear strains perpendicular to the slice and enables the computation of the straightening of the strand. The thermo-mechanical model is spatially discretised with a local radial basis function collocation method (LRBFCM). The mechanical part includes a new hybrid method that combines LRBFCM with finite differences for increased stability. The presented work shows how the developed model is used to assess the impact of casting velocity on the solid shell shrinkage and the probability of hot-tearing occurrence in the continuous casting of square billets.

Consequences of plane-strain and plane-stress assumptions in fully coupled chemo-mechanical Li-ion battery models

In Li-ion battery research, it is common to simulate chemo-mechanical phenomena in reduced dimensions (e.g., 2-D) as opposed to fully resolve these complex physics in 3-D. It is common to assume either (1) the out-of-plane strain is negligible (commonly referred to as plane-strain), or (2) the out-of-plane stress is negligible (commonly referred to as plane-stress). However, there is typically little consideration as to the quantitative consequences of these approximations. Furthermore, the influence of these out-of-plane assumptions can be compounded and convoluted when chemo-mechanics models implement so-called “fully coupled” formulations, where the local species concentrations influence the stress-state and the stress-state influences the local species fluxes. The present manuscript explores the implications of using plane-stress and plane-strain assumptions in 2-D as compared to simulating a full 3-D electrode particle. This comparative study includes simulating both isotropic and anisotropic particle intercalation where the particles can be surrounded by either a liquid or solid electrolyte. Additionally, common Li-ion battery-model metrics such as the state-of-stress, intercalation fraction distribution, and specific capacity are compared, while also considering the effects of particle size and C-rate. As alternatives to the pure plane-strain and plane stress approximations, two modified plane-strain assumptions are found to better approximate the fully coupled chemo-mechanical 3-D behavior.

Simulation Analysis of Ti6Al4V Cutting Under Liquid Nitrogen Cryogenic Cooling

Liquid nitrogen cryogenic cooling technology has great advantages in cutting machining. A thermodynamic finite element model of liquid nitrogen cryogenic cooling based on the plane strain assumption is established to observe the changes in cutting force and tool temperature corresponding to different cutting parameters (feed rate and cutting speed) under liquid nitrogen cryogenic cooling conditions. Existing tests verified the accuracy of the simulation model, and the prediction error of the simulation model was controlled within 10%. Analyzing the data obtained from the simulation model, it is found that a lower tool temperature is obtained when the cutting parameters are selected to be lower values; under the conditions of liquid nitrogen cryogenic cooling and dry cutting machining, the effect of the cutting speed on the cutting force is not obvious, while the effect of the feed volume on the cutting force is more significant; under the conditions of different cutting parameters, the tool temperature as well as the cutting force are lower under the conditions of liquid nitrogen cryogenic cooling compared to the dry cutting machining method.

A hybrid radial basis function-finite difference method for modelling two-dimensional thermo-elasto-plasticity, Part 1: Method formulation and testing

A hybrid version of the strong form meshless Radial Basis Function-Finite Difference (RBF-FD) method is introduced for solving thermo-mechanics. The thermal model is spatially discretised with RBF-FD, where trial functions are polyharmonic splines augmented with polynomials. For time discretisation, the explicit Euler method is employed. An extension of RBF-FD, the hybrid RBF-FD, is introduced for solving mechanical problems. The model is one-way coupled, where temperature affects displacements. The thermo-elastoplastic material response is considered where the stress field is generally non-smooth. The hybrid RBF-FD, where the finite difference method is used to discretise the divergence operator from the balance equation, is shown to be successful when dealing with such problems. The mechanical model is introduced in a plane strain and in a generalised plane strain (GPS) assumption. For the first time, this work presents a strong form RBF-FD for GPS problems subjected to integral form constraints. The proposed method is assessed regarding h-convergence and accuracy on the benchmark with heating an elastoplastic square. It is proven to be successful at solving one-way coupled thermo-elastoplastic problems. The proposed novel meshless approach is efficient, accurate, and robust. Its use in an industrial situation is provided in Part 2 of this paper.

On the generalized plane strain assumption for pressurized membranes

We revisit the problem of translation-invariant pressurized membranes that are squeezed without friction between several planes, all parallel to the axis of translation-invariance (such problem involves material and geometric nonlinearities, including contact). Quite remarkably, it was shown by De Simone and Luongo (2013) that such problems simplify considerably under the plane strain assumption. Indeed, the complex initial boundary-value problem reduces to a simple set of non-linear, algebraic equations. We argue that in many practical cases, the plane strain assumption does not hold. Instead, we introduce the generalized plane strain assumption, that is necessary to account for the longitudinal equilibrium of the membrane. We show how the equations of De Simone and Luongo (2013) are modified, while remaining extremely simple. We thus define an extended class of problems that become (nearly) tractable analytically.

A method for defining tissue injury criteria reveals that ligament deformation thresholds are multimodal.

Soft tissue injuries (such as ligament, tendon, and meniscus tears) are the result of extracellular matrix damage from excessive tissue stretching. Deformation thresholds for soft tissues, however, remain largely unknown due to a lack of methods that can measure and compare the spatially heterogeneous damage and deformation that occurs in these materials. Here, we propose a full-field method for defining tissue injury criteria: multimodal strain limits for biological tissues analogous to yield criteria that exist for crystalline materials. Specifically, we developed a method for defining strain thresholds for mechanically-driven fibrillar collagen denaturation in soft tissues, using regional multimodal deformation and damage data. We established this new method using the murine medial collateral ligament (MCL) as our model tissue. Our findings revealed that multiple modes of deformation contribute to collagen denaturation in the murine MCL, contrary to the common assumption that collagen damage is driven only by strain in the direction of fibers. Remarkably, hydrostatic strain (computed here with an assumption of plane strain) was the best predictor of mechanically-driven collagen denaturation in ligament tissue, suggesting crosslink-mediated stress transfer plays a role in molecular damage accumulation. This work demonstrates that collagen denaturation can be driven by multiple modes of deformation and provides a method for defining deformation thresholds, or injury criteria, from spatially heterogeneous data. STATEMENT OF SIGNIFICANCE: Understanding the mechanics of soft tissue injuries is crucial for the development of new technology for injury detection, prevention, and treatment. Yet, tissue-level deformation thresholds for injury are unknown, due to a lack of methods that combine full-field measurements of multimodal deformation and damage in mechanically loaded soft tissues. Here, we propose a method for defining tissue injury criteria: multimodal strain thresholds for biological tissues. Our findings reveal that multiple modes of deformation contribute to collagen denaturation, contrary to the common assumption that collagen damage is driven by strain in the fiber direction alone. The method will inform the development of new mechanics-based diagnostic imaging, improve computational modeling of injury, and be employed to study the role of tissue composition in injury susceptibility.

Dynamic Interaction Factor of Pipe Group Piles Considering the Scattering Effect of Passive Piles

Based on the plane–strain assumption, a calculation model of pile–soil–pile vertical coupling vibration response considering the scattering effect of passive piles is established in this paper. Using this model, the vertical displacement expressions of pile core soil and pile surrounding soil, soil displacement attenuation function, and longitudinal complex impedance are obtained. Then, based on the strict pile–soil coupling effect, the displacement of the active pile under vertical load and scattering effect, as well as the displacement of the passive pile under incident waves, are solved separately. A new type of dynamic interaction factor for pipe group piles is derived by introducing scattering effect factors. A numerical example shows that the degenerate solution in this paper is in good agreement with the existing solution, which verifies the rationality of the solution. Considering the scattering effect is helpful in improving the accuracy of vibration response analysis of pile groups. The variation in parameters such as slenderness ratio, pile spacing, and outer diameter has significant effects on the interaction factor, and compared with the solid pile, the influence of parameter change on the pipe pile is smaller.

Analysis of Flexoelectric Solids With a Cylindrical Cavity

Abstract Flexoelectricity, a remarkable size-dependent effect, means that strain gradients can give rise to electric polarization. This effect is particularly pronounced near defects within flexoelectric solids, where large strain gradients exist. A thorough understanding of the internal defects of flexoelectric devices and their surrounding multiphysics fields is crucial to comprehend their damage and failure mechanisms. Motivated by this, strain gradient elasticity theory is utilized to investigate the mechanical and electrical behaviors of flexoelectric solids with cylindrical cavities under biaxial tension. Closed-form solutions are obtained under the assumptions of plane strain and electrically impermeable defects. In particular, this study extends the Kirsch problem of classical elasticity theory to the theoretical framework of higher-order electroelasticity for the first time. Our research reveals that different length scale parameters of the strain gradient and bidirectional loading ratios significantly affect the hoop stress field, radial electric polarization field, and electric potential field near the inner cylindrical cavity of the flexoelectric solid. Furthermore, we validate our analytical solution by numerical verification using mixed finite elements. The congruence between the two methods confirms our analytical solution’s accuracy. The findings presented in this paper provide deeper insights into the internal defects of flexoelectric materials and can serve as a foundation for studying more complex defects in flexoelectric solids.

Theoretical model for investigating three-dimensional effect in integrity test of open-end pipe piles

An innovative theoretical model of non-axisymmetric vibration of pipe pile (NVPP) is proposed in this paper to investigate the three-dimensional effect in the pile integrity test (PIT) for open-end pipe piles. A new dynamic equilibrium equation of the pipe pile is established based on the stress simplification assumption. It overcomes the inaccuracies of existing three-dimensional models in predicting the longitudinal wave velocity of pile material. The surrounding soil and soil plug conform to the classical plane strain assumption, while the pile end soil is simulated by the fictitious soil annular pile developed from the traditional fictitious soil pile. Theoretical solutions for the velocity response of each point on the pipe pile top are deduced and validated against existing solutions and FEM results. The NVPP model is then applied to analyze the characteristics of testing signals of PITs for thick-wall and thin-wall pipe piles. Problems concerned in actual PITs for pipe piles are settled and practical recommendations for acquiring high-quality testing signals with minimal interference are provided.

Geometrically consistent nonlinear plane strain and stress constitutive models: Application to soft-material oscillations

Position-gradient constraints are used in this study to develop new geometrically consistent plane strain and stress constitutive models for the nonlinear large-displacement analysis of soft materials. The position gradients are used to define strain and stress constraints that enter into the formulation of the deformation tensor, strain-energy density function, and new constitutive models. The new approach is applied to three nonlinear hyperelastic constitutive models for incompressible materials: Neo-Hookean, Mooney Rivlin, and Yeoh Models. The underlying geometric plane-stress and plane-strain assumptions are preserved by using the finite elements (FE) of the absolute nodal coordinate formulation (ANCF). Limitations of widely used incompressibility penalty functions that produce stresses at initial stress-free configurations are discussed. Several numerical examples developed using planar ANCF shear-deformable beam and rectangular elements are used to assess the proposed formulation and verify its results using spatial solid elements implemented in commercial FE software. It is demonstrated that the ANCF plane-stress and plane-strain formulations for the incompressible materials predict accurately soft-material oscillations, demonstrating that stiffer behavior can be attributed to material-model inconsistencies and cannot always be attributed to locking.

Dispersion curves of acoustoelastic borehole waves: the perturbation method with the correct formulation of the stresses around the borehole

SUMMARY The acoustoelastic model has been widely used to investigate the influence of formation stresses on the dispersion curves of borehole waves. The analytical perturbation method (PM), the finite-difference time-domain (FDTD) and the semi-analytical finite element (SAFE) are three common-used methods to calculate the dispersion curves. However, due to different interpretations of the PM and plane strain assumptions, the obtained dispersion curves are incompatible among existing PMs, which may misguide the interpretation of formation stresses. It is therefore necessary to untangle the applicability and limitations of PM. Considering that the conventional PMs are usually inaccurate at the low frequency or inconsistent with Hamilton’s principle, we develop a revised PM to obtain the dispersion curves of borehole waves propagating along a borehole surrounded by the triaxially stressed formation assumed as a monoclinic medium. The revised PM is more accurate, reasonable and logical than existing PMs. When the formation is subjected to low stresses, our finding is of great benefit for quickly computing dispersion curves, since the revised PM is much more efficient than the FDTD method; and there are small discrepancies between the flexural dispersions obtained by the revised PM and those obtained by the FDTD method. Nevertheless, the revised PM has two limitations. The first limitation is that the revised PM cannot be used to compute the Stoneley dispersion curves, which have been validated by comparison with SAFE and FDTD methods. The second limitation is that flexural dispersion curves show significant discrepancies in the high-frequency domain when the low-stress assumption does not hold, as compared to those obtained by the FDTD method.

A perturbation method for upper bound analysis of stability of slopes based on rigid translational/rotational moving elements

Collapse mechanisms consisting of translational or rotational rigid blocks are widely used as the basis for computing the upper bound on the factor of safety in slope stability problems. Constructing admissible velocities for these blocks mechanism often relies on experience, and optimizing these blocks can be cumbersome work, especially for layered slopes containing both hard and soft layers. In this paper, we present a numerical technique for evaluating and optimizing mechanisms composed of an arbitrary number of spline rigid translational/rotational moving elements, assuming plane strain material obeys the Mohr-Coulomb yield condition. In the proposed method, coordinates defining the nodes of the blocks are treated as unknowns, and the optimal geometry is found by successively perturbing node coordinates and block velocities, starting initially from a user-specified structured mesh. Two different definitions of the factor of safety are considered and the corresponding analysis procedures are introduced. The proposed method is applied to four examples related to slope stability analysis, and these examples illustrate that the proposed method is an efficient and reasonable way to analyze the stability of slopes.

Bayesian approach to micromechanical parameter identification using Integrated Digital Image Correlation

Micromechanical parameters are essential in understanding the behavior of materials with a heterogeneous structure, which helps to predict complex physical processes such as delamination, cracks, and plasticity. However, identifying these parameters is challenging due to micro-macro length scale differences, required high resolution, and ambiguity in boundary conditions, among others. The Integrated Digital Image Correlation (IDIC) method, a state-of-the-art full-field deterministic approach to parameter identification, is widely used but suffers from high sensitivity to boundary data errors and is limited to identification of parameters within well-posed problems. This article employs Bayesian approach to estimate micromechanical shear and bulk moduli of fiber-reinforced composite samples under plane strain assumption, and to improve handling of boundary noise. The main purpose of this article is to quantify the effect of uncertainty in the boundary conditions in the stochastic setting. To this end, the Metropolis–Hastings Algorithm (MHA) is employed to estimate probability distributions of bulk and shear moduli and boundary condition parameters using IDIC, considering a fiber-reinforced composite sample under plane strain assumption. The performance and robustness of the MHA are compared to two versions of deterministic IDIC method, under artificially introduced random and systematic errors in kinematic boundary conditions. Although MHA is shown to be computationally more expensive and in certain cases less accurate than the recently introduced Boundary-Enriched IDIC, it offers significant advantages, in particular being able to optimize a large number of parameters while obtaining statistical characterization as well as insights into individual parameter relationships. The paper furthermore highlights the benefits of the non-normalized approach to parameter identification with MHA (leading, within deterministic IDIC, to an ill-posed formulation), which significantly improves the robustness in handling the boundary noise.

Double-porosity poromechanical models for wellbore stability of inclined borehole drilled through the naturally fractured porous rocks

Fractured porous rock comprises the matrix and fracture systems with double-porosity and double-permeability. This paper treats the fracture system as a porous media due to the fracture-occupied space filled by the depositional minerals or assumes it to be fully saturated with the formation fluid. Thus, one considers the double-porosity poromechanical models and provides the semi-analytical dual-poroelastic solutions to the field variables around arbitrally inclined boreholes subjected to non-hydrostatic stresses. The plane strain assumption holds for deriving solutions to field variables. The field variables include displacements, stresses, and two pore-pressures in the matrix system and fracture system. This work conducts a quantitative study of the relation between fracture spacing related to fracture density, fracture width, and the safe pressure ratios window (SPRW). One defines that the upper limit of SPRW is related to the ratio of fracture pressure to the original formation pore pressure, and the lower limit is referred to as the ratio of collapse pressure to formation pore pressure. The main results show that the dual-porosity medium displays a higher wellbore instability potential than the single-porosity one. The fracture system that is fully saturated with formation fluid has a narrower SPRW than that of the minerals-filled one referred to as a porous medium. When the coupled hydro-mechanical model is considered, the drilling mud-pressure window narrows with increasing time. It suggests that the drilling engineer considers fracture spacing, and fracture width in the predrilling design of the time-dependent SPRW of naturally fractured porous rocks.

Extension–bending coupling phenomena and residual hygrothermal stresses effects on the Energy Release Rate and mode mixity of generally layered laminates

This article presents a novel analytical solution to the delamination problem of interface deformable generally layered (asymmetric and unbalanced) composite laminates. The governing equation describing the interface forces and moments is obtained for the reduced form of the first-order shear deformation theory of plates (FSDT) under plane strain assumptions and accounts for the extension–bending coupling phenomena and residual hygrothermal stresses, both of which have been neglected in the past. The interface forces and moments are included in a novel generalization of the J-integral which is used for the estimation of the Energy Release Rate (ERR). The analytical results of the ERR and the mode mixity exhibit a great agreement with the results of Finite Element (FE) models developed using the Cohesive Zone Method (CZM) and the Virtual Crack Closure Technique (VCCT) for a Double Cantilever Beam (DCB) test of a generally layered fibre metal laminate.

Integrated digital image correlation for micro-mechanical parameter identification in multiscale experiments

Micromechanical constitutive parameters are important for many engineering materials, typically in microelectronic applications and material design. Their accurate identification poses a three-fold experimental challenge: (i) deformation of the microstructure is observable only at small scales, requiring SEM or other microscopy techniques; (ii) external loadings are applied at a (larger) engineering or device scale; and (iii) material parameters typically depend on the applied manufacturing process, necessitating measurements on material produced with the same process. In this paper, micromechanical parameter identification in heterogeneous solids is addressed through multiscale experiments combined with Integrated Digital Image Correlation (IDIC) in conjunction with various possible computational homogenization schemes. To this end, some basic concepts underlying multiscale approaches available in the literature are first reviewed, discussing their respective advantages and disadvantages from the computational as well as experimental point of view. A link is made with recently introduced uncoupled methods, which allow for identification of material parameter ratios at the microscale, still lacking a proper normalization. Two multiscale methods are analysed, allowing to bridge the gap between microstructural kinematics and macroscopically measured forces, providing the required normalization. It is shown that an integrated experimental--computational scheme provides relaxed requirements on scale separation. The accuracy and performance of the discussed techniques are analysed by means of virtual experimentation under plane strain and large strain assumptions for unidirectional fibre-reinforced composites. The robustness against image noise is also assessed.

Numerical study on fracture control blasting using air–water coupling

Fracture control blasting produces rock fractures in the desired direction, which is significant for the stability of excavation structures in rock engineering. The present study proposes a new method of fracture control blasting using air–water coupling. This method utilizes the difference of explosion stress transfer between air and water, which guides the explosion energy consumption in fracturing rock on water-coupling side, i.e. rock in the excavation zone, and thus the rock in the excavation zone is properly fragmented, and the reserved rock is well protected from damage. Based on the plane strain assumption, the transmission and propagation of explosion stress in the excavation and reserved rock with this method are first theoretically analyzed. Then, fracture control blasting using air–water coupling is numerically studied utilizing the LS-DYNA program. The numerical model is first developed and calibrated in the simulation with the laboratory-scale air-coupling and water-coupling blasting tests. Then, the successive processes of pressure attenuation, fracture evolution and energy consumption under air–water coupling blasting are numerically investigated. Thereafter, the effects of the air–water ratio and decoupling ratio on the performance of fracture control are numerically investigated. The theoretical and numerical results show that good performance of fracture control can be obtained using air–water coupling blasting.