Elastic-perfectly Plastic Research Articles

A model for oblique impacts on material surfaces

Many practical situations of material damage, wear, and erosion involve collisions between small particles and surfaces at inclined angles. While there are many well-validated models of normal incidence impact situations, elastic-plastic models for oblique incidence impact events are lacking. Here the finite element method is used to predict the normal and tangential coefficient of restitution in oblique impacts for hard, elastic spheres impacting an elastic-perfectly plastic material surface. The proposed model covers various impact angles ranging from 0° to 45°, within a limiting impact velocity below which the effects of heating are negligible. The normal coefficient of restitution follows power-laws with respect to normalized values of the impact velocity. Interestingly, the tangential coefficient of restitution follows a linear relationship with impact velocity. Together, these results provide a semi-empirical set of equations predicting oblique impact rebounds (both velocity and trajectory) for a wide range of conditions and material properties, with which experimental results can be rapidly interpreted. Laser-Induced Particle Impact Test (LIPIT) data are also presented for aluminum particles impacting aluminum substrates, at impact angles of 25° and 40°; the results compare favorably with the model and validate the general use of such models for the analysis of experimental data.

Improved mesh-free SPH approach for loose top coal caving modeling

This study presents an innovative model in computational geotechnical engineering by improving the Smoothed Particle Hydrodynamics (SPH) method for simulating loose particle dynamics in coal caving processes. The improved model integrates an elastic-perfectly plastic constitutive model with the Drucker-Prager yield criterion and includes several improvements aimed at boosting accuracy, stability, and efficiency. These improvements include gravity loading coupled with particle damping, first-order stress field smoothing, and kernel gradient correction. A series of numerical experiments validates the effectiveness of the improved SPH model, demonstrating its capability to predict large deformations and track the evolution of the coal-rock interface in coal caving processes. Furthermore, the study analyzes the model's sensitivity to material parameters such as the angle of friction and material density, which aids in configuring the model for distinct coal mining situations. Results show that the non-cohesive elastic-perfectly plastic constitutive model can effectively simulate the flow behavior of granular particles, and the landslide simulation results are in good agreement with the experiments. The improved SPH algorithm with stress smoothing technique solves the problem of numerical noise, and the “double peak” stress distribution around the coal outlet is identified. The established SPH model offers an effective tool for understanding dynamics behaviors of loose top coal. Significantly, the model requires only five material parameters, which can be identified through standard experiments, avoiding the typically arduous process of parameter selection or calibration commonly existing in Discrete Element Method simulations.

Development of an improved limit pressure solution for shape-distorted 90o pipe bends with through-wall circumferential cracks

PurposeThis paper presents the approximate limit pressure solution for shape-imperfect and through-wall circumferential cracked (TWCC) 90° pipe bends at the intrados region. Finite element (FE) limit analysis was used to estimate the limit pressure by considering the small geometrical change effects.Design/methodology/approachThree-dimensional (3D) geometric linear FE methodology was implemented to investigate the limit pressure of structurally deformed TWCC 90° pipe bends. The material considered in the analysis is elastic perfectly plastic (EPP). The limit pressure of TWCC shape-distorted pipe bends was predicted from the corresponding internal pressure when von-Mises stress was equal to or just exceeded the material’s yield strength for all the models. The theoretical solution which was published in the literature was used to evaluate the current FE approach.FindingsOvality Co and TWCC at the intrados region caused a considerable impact on pipe bends, while the thinning? Ct produced a negligible effect and hence was not included in the analysis. With the combined effect, the bend portion of pipe bend experiences substantial influence, and the TWCC effect consequently increases with 45o, 60o and 90o crack angles and decreases the limit pressure of pipe bends. An improved closed-form empirical limit pressure solution was proposed for TWCC shape-distorted pipe bends at the intrados region.Originality/valueIn the limit pressure analysis of 90° pipe bends, the implications of structural irregularities (ovality and thinning) and TWCC have not been examined and reported.

Revamp of the eco-friendly grouting materials by mixing basalt fiber for earthen sites conservation: Compressive performance and constitutive models

Reinforcement techniques, including anchorage and sealing with grout, are frequently employed to protect earthen sites and minimize the impact of fissures, the effectiveness of these approaches relies heavily on the mechanical properties of grout. To enhance the strength and ductility of the conventional grout, an eco-friendly composite grouting material mixture basalt fibers of varying lengths and contents is prepared. The DIC-based UCS tests were conducted to evaluate its compressive performance. Constitutive models of the basalt fiber-reinforced grout, including elastic perfectly-plastic (EPP) and median nonlinear (MNL) constitutive models, were established respectively based on the equal energy rule and cross-sectional data analysis for simplification application and precision evaluation purposes. The results show that the mixture of basalt fibers resulted in a distinct semi-brittle failure mode for the grout, with a significant increase in its compressive strength, ductility, compressive fracture energy, and residual bearing capacity. However, the elastic modulus will be lower when the fibers are relatively longer. Considering the differences in the relations between performance parameters and fiber content, and length, the highest compressive strength was reached at the fiber length of 6 mm and the content of 0.2 wt%, while maximum ductility was achieved at 18 mm and 0.4 wt%. The fracture energy exhibits an approximate positive correlation with fiber content. In addition, the EPP and MNL constitutive models can effectively characterize the compressive mechanical behavior of semi-brittle materials to satisfy different requirements on the precision of engineering analysis. This study contributes valuable insights into the development of high-ductility grouting materials of earthen sites, and the establishment of constitutive models for semi-brittle materials.

The structural strain method for fatigue evaluation of welded components: Analytical treatment of reversed plasticity

The traction structural stress method has been adopted by various industry sectors, in addition to ASME Boiler and Pressure Code (B&PV), for fatigue life evaluation and assessment of welded components, especially for high-cycle fatigue applications. Its effectiveness has been validated and demonstrated by showing good data correlation in the form of the master S–N curve scatter band. For low cycle fatigue with the involvement of elastic-plastic deformation, the structural strain method has been recently developed by simply extending the current elastic-deformation-based structural stress method to more general elastic-plastic deformation. Closed-form structural strain solutions for elastic-perfectly plastic materials and numerical solutions for nonlinear strain hardening materials have been obtained under simple load-controlled loading and unloading cycling conditions. In this paper, the equations of nonlinear cyclic structural stress-strain curves under fully-reversed fatigue loading are analytically derived for the first time with a 3-bar model, which consists of three elastic-perfectly plastic bars. The general trends and patterns of the loading paths of the constructed cyclic structural stress-strain curves with the 3-bar model are validated with finite element analysis (FEA). Based on the insights gained from the 3-bar model and the FEA results, a simple procedure for constructing cyclic structural stress-strain curves from their corresponding monotonic loading curves are developed for both elastic-perfectly plastic model and the modified Ramberg-Osgood nonlinear strain hardening model. The effectiveness of this procedure is demonstrated by correlating low-cycle fatigue data of welded structures under pulsating and fully-reversed loading conditions.

Unloading Model of Elastic-Plastic Half-Space Contacted by an Elastic Spherical Indenter.

A new unloading contact model of an elastic-perfectly plastic half-space indented by an elastic spherical indenter is presented analytically. The recovered deformation of the elastic indenter and the indented half-space has been found to be dependent on the elastic modulus ratio after fully unloading. The recovered deformation of the indented half-space can be calculated based on the deformation of the purely elastic indenter. The unloading process is assumed to be entirely elastic, and then the relationship of contact force and indentation can be determined based on the solved recovered deformation and conforms to Hertzian-type. The model can accurately predict the residual indentation and residual curvature radius after fully unloading. Numerical simulations are performed to demonstrate the assumptions and the unloading model. The proposed unloading model can cover a wide range of indentations and material properties and is compared with existing unloading models. The cyclic behavior including loading and unloading can be predicted by combining the proposed unloading law with the existing contact loading model. The combined model can be employed for low-velocity impact and nanoindentation tests and the comparison results are in good agreement.

On elastoplastic behavior of porous enamel–An indentation and numerical study

The micro/nano pores in natural mineralized tissues can, to a certain extent, affect their responses to mechanical loading but are generally ignored in existing indentation analysis. In this study, we first examined the void volume fraction of sound and caries lesion enamels through micro-computed tomography (micro-CT). A Berkovich indentation study was then carried out to characterize the effect of porous microstructure on the mechanical behavior of the human enamels. The indentation tests were also modeled using the nonlinear finite element analysis technique to simulate indentation load-displacement curves, which showed reasonable agreement with the experimental measurements. From the simulation results, the extent of densification in the plastic zone was identified and the corresponding stress and contact pressure evolutions were quantified. Further, a conventional elastic-perfectly plastic material model without considering micropores was also developed to investigate the compaction effect of the porous structure. The simulation results reveal that conventional elastic perfect-plastic constitutive models become less reliable to model the mechanical behavior of carious lesion enamel with increasing loss of mineral content as it underestimates the yield stress and plastic energy dissipation. This study divulges the importance of compaction of porous enamel structure beneath the indented area. Note that understanding the effect of porous microstructures on plastic behavior is vital as the involved inelastic deformation mechanism associated with irreversible processes, such as wear and localized microcracking, has a significant bearing on wear and fatigue behavior of enamel. Statement of significanceBased on micro-CT and nano-indentation characterization, a numerical model was developed aiming to precisely describe the deformation behavior of naturally porous enamel. Inelastic properties and energy dissipation characteristics of porous enamel were investigated in detail. This work demonstrated that the existence of micro-pores in White Spot Lesions (WSLs) contributes to mechanical stability, which can mitigate the reduction in Young's modulus and fracture toughness resulting from loss of mineral components. The knowledge gained from this study can be used to explain the mechanisms related to irreversible processes, such as contact induced cracking and wear, and strengthen understanding of the mechanical behavior of porous mineralized tissues.

Analysis of impact deformation of elastic-perfectly plastic particles

AbstractMaterial Point Method is used to study the impact deformation of elastic-perfectly plastic spherical particles. A wide range of material properties, i.e. density, Young’s modulus and yield strength, are considered. The method is particularly suitable for simulating extensive deformation. The focus of the analysis is on linking the coefficient of restitution and the percentage of the incident kinetic energy dissipated by plastic deformation, Wp/Wi × 100, to the material properties and impact conditions. Dimensionless groups which unify the data for the full range of material properties have been identified for this purpose. The results show that when the particle deforms extensively, Wp/Wi × 100 and the equivalent plastic strain, are only dependent on the particle yield strength and the incident kinetic energy, as intuitively expected. On the other hand, when the deformation is small, Young’s modulus of the particle also affects both Wp/Wi × 100 and the equivalent plastic strain. Moreover, coefficient of restitution is insensitive to Young’s modulus of the material. Dimensionless correlations are then suggested for prediction of the coefficient of restitution, the equivalent plastic strain and Wp/Wi × 100. Finally, it is shown that the extent to which the particle flattens due to impact can be predicted using its yield strength and initial kinetic energy.

Performance of Pile–Wall System Adjacent to Footings

The performance of a retaining wall is dependent on multiple factors including lateral earth pressure, which results from backfill soils and adjacent footings located behind a retaining wall. The prediction of a retaining wall’s performance in a footing–soil–wall system (FSPS) must incorporate the influences caused by the movement of a retaining wall. This study examines the performance of a retaining wall formed by driven, precast, concrete piles located adjacent to a concrete footing using two- and three-dimensional finite element analysis (2D and 3D FEA) by ANSYS 13.0 software. Both soil and concrete are assumed to behave as non-linear, elastic-perfectly plastic and rate-independent materials in compliance with the upper-bound model of Drucker–Prager yield criterion. Three backfill and foundation soils are considered: kaolin, silty clay, and kaolin–sand. Various conditions of soil type, footing shape ratio, pile width, and footing–pile distance through 180 FEA runs are investigated. The effects of 2D and 3D FEA on the behavior of the pile–wall system are compared. The lateral deflection and pressure distribution profiles along the pile–wall are studied and presented. Two empirical equations predicting lateral deflections at the pile toe and pile head and useful for pile structural design are developed under the ultimate pressure of the adjacent footing.

Practical Considerations in the Design of Passive Free Piles in Sliding Soil

The stabilisation of shallow translational landslides can be carried out by using large diameter concrete piles, with the aim of increasing the available strength along the slip surface. In the following article, 3D numerical models of a free-head flexible pile embedded into a translational type of landslide are studied. The landslide model has a given inclination angle, β, and a thickness, D, while the reinforced concrete pile has a fixed diameter, d, and a length, D + L, in the perspective of studying the failure modes B1, BY and B2 of free-head flexible piles. In this category of piles, collapse is reached with the formation of plastic hinges. Both the soil and the concrete are modelled with simple constitutive models, such as Mohr–Coulomb for soil and the elastic-perfectly plastic for the concrete pile, in order to carry out the design approaches provided by Eurocode, as well as to highlight some practical aspects of soil–structure interaction during the landslide displacements. The results highlight how the achievement of the shear strength in a flexible free-head concrete pile generally precedes the achievement of the ultimate bending moment associated with the development of plastic hinges. Furthermore, the axial load supported by the pile may itself contribute to the overall strength available along the slip surface.

Experimental study on progressive interfacial mechanical behaviors using fiber optic sensing cable in frozen soil

Frost heave and thaw settlement in cold regions pose a significant threat to engineering construction. Optical frequency domain reflectometry (OFDR) based on Rayleigh scattering can be applied to monitor ground deformation in frozen soil areas, where the interface behavior of soil-embedded fiber optic sensors governs the monitoring accuracy. In this paper, a series of pullout tests were conducted on fiber optic (FO) cables embedded in the frozen soil to investigate the cable‒soil interface behavior. An experimental study was performed on interaction effects, particularly focused on the water content of unfrozen soil, freezing duration, and differential distribution of water content in frozen soil. The high-resolution axial strains of FO cables were obtained using a sensing interrogator, and were used to calculate the interface shear stress. The interfacial mechanical response was analytically modeled using the ideal elasto‒plastic and softening constitutive models. Three freezing periods, correlating with the phase change process between ice and water, were analyzed. The results shows that the freezing effect can amplify the peak shear stress at the cable-soil interface by eight times. A criterion for the interface coupling states was proposed by normalizing the pullout force‒displacement information. Additionally, the applicability of existing theoretical models was discussed by comparing the results of theoretical back‒calculations with experimental measurements. This study provides new insights into the progressive interfacial failure behavior between strain sensing cable and frozen soil, which can be used to assist the interpretation of FO monitoring results of frozen soil deformation.

A closed-form structural stress-strain relation and implications on plastic design limits in codes and standards

The deformation of elastic-perfectly plastic beam or plate under combined through-thickness membrane and bending stresses (i.e., structural stress) is analyzed in this paper. Closed-form solutions of “plane-remains-plane” strain (i.e., structural strain) distribution as a function of applied stress and yield strength is derived. The relationship between structural stress and structural strain is also revealed for gaining insights on not only the plastic collapse conditions, but also through-thickness deformation behaviors leading to limit state. Based on the closed-form strain solutions, four deformation modes are identified: (1) elastic, (2) one-side yielding, (3) two-side yielding, and (4) plastic collapse. A membrane-bending stress interaction diagram is constructed to depict four zones that are associated with the four deformation modes in the normalized membrane and bending stress space. Closed-form solutions of residual stress, residual strain, and structural stress range-strain range relation after a loading-unloading cycle are also derived. Based on the parabolic plastic collapse stress limit curve, parabolic design stress limit curves with coherent safety margins for any combinations of membrane and bending stresses are constructed. The new insights gained in this study offer a more consistent definition of design stress limit than those used in the current codes and standards, e.g., ASME BP&V Code.

An efficient second-order cell-centered Lagrangian discontinuous Galerkin method for two-dimensional elastic-plastic flows

An efficient second-order cell-centered Lagrangian discontinuous Galerkin (DG) method for solving two-dimensional (2D) elastic-plastic flows with the hypo-elastic constitutive model and von Mises yield condition is presented. First, starting from the governing equations of conserved quantities in the Euler framework, the integral weak formulation of them in the Lagrangian framework is derived. Next, the DG method is used for spatial discretization of both the weak formulation of conserved quantities and the evolution equation of deviatoric stress tensor. The Taylor basis functions defined in the reference coordinates provide the piecewise polynomial expansion of the variables, including the conserved quantities and the deviatoric stress tensor. The vertex velocities and Cauchy stress tensor on the edges are computed using a nodal solver equipped with a variant of Li's new Harten-Lax-van Leer-contact approximate Riemann solver [Li et al., “An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model,” J. Comput. Phys. 448, 110675 (2022)], in which the longitudinal wave velocity in the plastic state is modified. Then the vertex velocities and Cauchy stress tensor on the edges are used to compute numerical fluxes. A second-order total variation diminishing Runge–Kutta scheme is used for time discretization of both the governing equations of conserved quantities and the evolution equation of deviatoric stress tensor. After solving the evolution equation of deviatoric stress tensor, a radial return algorithm is performed at the Gauss points of each element according to the von Mises yield condition. And then the coefficients of the DG expansion for the deviatoric stress tensor on each element are modified by a least squares procedure using the deviatoric stress tensors at these Gauss points. To achieve second-order accuracy, the least squares procedure is used for piecewise linear reconstruction of conserved quantities and the deviatoric stress tensor, and the Barth–Jespersen limiter is used to suppress the nonphysical numerical oscillation near the discontinuities. After that, the coefficients of the DG expansion are modified through L2 projection using the reconstructed polynomials. Finally, a second-order cell-centered Lagrangian DG scheme is established. Several tests demonstrate that the new scheme achieves second-order accuracy with good robustness, and that the DG method of updating the deviatoric stress tensor has comparable accuracy and much higher efficiency with mesh refinement compared with previous works.

From crack line field analysis method to symmetric line analysis method: Elastic-plastic analysis of symmetry problems

The mathematical difficulty encountered in the elastic–plastic analysis of solids lies in solving the partial differential equations analytically in the theory of plasticity. Such a mathematical dilemma has existed for decades and analytical solutions are still waiting to be obtained. This paper extends the crack line field analysis method and puts forward a symmetric line analysis method, which uses the parity of the Taylor series expansion of the plastic stress field near the symmetric line to transform the problem of solving partial differential equations into ordinary ones, and so obtain the general Taylor series solution of the plastic stress field near the symmetric line. The symmetric line analysis method can give elastic–plastic solutions to plane and anti-plane symmetry problems that have not been analytically solved before, which achieves an essential leap forward in the analytical analysis of elastic–plastic solids in the theory of plasticity. In this paper, three types of symmetry problems are defined, and examples are given for the elastic–plastic analysis and solution of each type of symmetry problems of elastic-perfectly plastic solids with the symmetric line analysis method.

Numerical investigation of the influence of transverse welds on the strength of aluminum alloy I-shaped members – Columns

This paper presents a numerical investigation, using finite element analysis, of the effects of transverse welds on the structural behavior of aluminum-alloy compression members. A non-linear finite element (FE) model was developed, and validated using experimental tests, to simulate the behavior of 6061-T6 aluminum-alloy columns with an I-shaped AW6×4.03 section geometry. The FE model included different material properties for the heat-affected zone (HAZ) that occurs in the proximity of the weld and the remaining base material, and incorporated both geometric and material nonlinearities. A parametric study of 80 columns with and without transverse welds was conducted, which investigated ten unbraced lengths, three weld centerline locations along the member length, and two types of post-yield material stiffness models, including elastic-perfectly plastic and strain hardening materials. The FE results provide clear evidence of the significance of the HAZ strength reduction on the load carrying capacity of welded aluminum-alloy columns. Numerical results were compared with those obtained using the provisions within the Aluminum Association’s Specification for Aluminum Structures, and they were used to develop proposed modifications to these provisions, which can remove the degree conservatism observed. A similar study on beams is presented in a companion paper.

A mechanically-derived contact model for adhesive elastic-perfectly plastic particles, Part II: Contact under high compaction—modeling a bulk elastic response

In Part I of this two part series (Zunker and Kamrin, 2024), we presented a multi-neighbor dependent contact model for adhesive elastic–plastic particles built upon the method of dimensionality reduction that is valid for the elastic and fully-plastic contact regimes. In this Part II, we complete the contact model by proposing a treatment for the bulk elastic contact regime which is characterized by a rapid stiffening in the force–displacement curve as interstitial pore spaces vanish. A simple formulation is presented for an additional bulk elastic force. A novel criterion for triggering this force (i.e. detecting the bulk elastic regime) related to the remaining free surface area of the particle is also given. This bulk elastic force is then superimposed with the force response given in Part I to achieve a contact model capable of capturing a variety of complex loadings. In this way, the methodology for treating the bulk elastic regime presented here stands independent of Part I and could be appended to any contact model. Direct comparison is made to finite element simulations showcasing bulk elastic responses, revealing the accurate predictive capabilities of the contact model. Notably, the contact model is also demonstrated to detect and evolve contacts caused purely by outward displacement of the free surface with good precision. A numerical implementation suitable for the discrete element method is provided.

A mechanically-derived contact model for adhesive elastic-perfectly plastic particles, Part I: Utilizing the method of dimensionality reduction

In this two part series (Zunker and Kamrin, 2024), we present a contact model able to capture the response of interacting adhesive elastic-perfectly plastic particles under a variety of loadings. In Part I, we focus on elastic through fully-plastic contact with and without adhesion. For these contact regimes the model is built upon the method of dimensionality reduction which allows the problem of a 3D axisymmetric contact to be mapped to a corresponding problem of a 1D rigid indenter penetrating a bed of independent Hookean springs. Plasticity is accounted for by continuously varying the 1D indenter profile subject to a constraint on the contact pressure. Unloading falls out naturally, and simply requires lifting the 1D indenter out of the springs and tracking the force. By accounting for the incompressible nature of this plastic deformation, the contact model is able to capture multi-neighbor dependent effects such as increased force and formation of new contacts. JKR type adhesion is recovered seamlessly within the method of dimensionality reduction by simply allowing the springs to ‘stick’ to the 1D indenter’s surface. Because of the mechanics-focused formulation of the contact model, only a few physical inputs describing the interacting particles are needed: particle radius, Young’s modulus, Poisson ratio, yield stress, and effective surface energy. The contact model is validated against finite element simulations and analytic theory—including Hertz’s contact law and the JKR theory of adhesion. These comparisons show that the proposed contact model is able to accurately capture plastic displacement, average contact pressure, contact area, and force as a function of displacement for contacts as well as particle volume within the elastic to fully-plastic regimes.

Application of Clay–rubber Mixtures for the Transportation Geotechnics—the Numerical Analysis

Abstract The use of waste materials (including rubber) in industry is one of the most important issues in terms of environmental protection. One of such applications is the use of soil–rubber mixtures in backfills or lower layers of embankments or road structures. The numerical analyses of the behavior of a clay–rubber mixture layer built into a road embankment are presented in this article. An elastic-perfectly plastic model with a Coulomb–Mohr yield surface was used in the finite element analysis. The parameters of soil–rubber mixtures adopted for the analysis were estimated on the basis of triaxial tests: monotonic (UU—unconsolidated undrained, and CU—consolidated undrained) and cyclic (CU) performed with low frequency (f = 0,001 Hz). The triaxial tests were carried out on mixtures of kaolin (K) and red clay (RC) with the addition of 1–5 mm rubber granulate (G) in the amount of 5–25% by weight. Numerical analyses included a static plate load test (VSS) of a layer made of a rubber–soil mixture built into the embankment and testing the stability of embankments using the c–ϕ strength reduction procedure. The results of laboratory tests confirm the necessity of testing soil–rubber mixtures each time before their use in embankments. The observed overall decrease in shear strength and stiffness of the tested material is variable and depends on the type of soil and the content of rubber waste. Satisfactory results of the analysis were obtained, both in terms of the values of layer stiffness modules and slope safety factors, which allows for the conclusion of the possibility of using soil–rubber mixtures (with the recommended granulate addition up to 30% by weight) in the layers of road embankments and (depending on the road class) in the lower layers of the pavement structure.

Elastic effect on the final deflection of rigid-plastic square plates under pulse loading

Rigid-perfectly plastic (R-PP) models are widely employed in the study of structural impact dynamics. However, due to the complexity of dynamic elastic-plastic behavior and the limitations of numerical simulation, it is of great significance to evaluate the elastic effect on the structural response to pulse loading. This paper examines the validity of the R-PP theoretical solutions on large deflection of pulse-loaded square plates. First, by exploring the characteristic of “plastic membrane” and considering the effect of membrane forces and bending moments on energy dissipation, an energy ratio R and a characteristic time ratio are precisely defined. Then based on the R-PP complete solutions recently developed for square plates under rectangular pressure pulse, the discrepancies in final deflection and in saturated impulse between the R-PP theoretical predictions and the elastic-perfectly plastic (E-PP) simulation results are evaluated and elaborated. It is revealed that the discrepancy in final deflection is dependent on the dimensionless stiffness ζplate and the dimensionless pulse magnitude λ. Moreover, the discrepancies in final deflection are compared between square plates and beams via a thorough analysis of key parameters, indicating that most characteristics of the elastic effect are similar for both cases, i.e., the R-PP solutions for both square plates and beams can reliably provide theoretical predictions for engineering applications if energy ratios R>5, with no limitation on the pulse duration. With the same geometric and material parameters, however, the energy ratio R of square plates is about twice that of beams, the discrepancy in final deflection of plates is larger than that of beams, but the absolute value of the discrepancy in final deflection for the clamped square plates remains in the order of 1/(1.5R) and can be bounded by 1/[4λ(λ-1)].

Elastic perfect plastic rock mass analysis and equivalent GSI determination considering strain-softening behavior and three-dimensional strength

The elastic perfect plastic model (EPP) fails to accurately capture the strain-softening mechanical behaviors of rock masses, while the post-peak constitutive model of elastic strain softening (ESS) has great application limitations in practical engineering for complex numerical algorithms. In this paper, an EPP constitutive approximation of rock mass considering three-dimensional strength and strain-softening characteristics is proposed. The forward analysis method of the equivalent Geological Strength Index (GSIeq) is given considering the impact of in-situ stress, rock mass properties and engineering excavation. Extensive case studies demonstrate that the radial stress and displacement obtained by the proposed method agree well with those of the ESS model, with only small errors in the Ground Response Curve. The rock mass elastic modulus E and the initial stress level p0 significantly affect GSIeq, while the support pressure ps contributes little to that. As Rp,EPP/R0 > 3.0, usually GSI < 25, the residual strength parameter GSIr can be directly used as the calculation parameter of the EPP model, while for a highly qualified rock mass, such as GSI > 75, the peak strength parameter GSIp can be directly used as an approximate value of GSIeq. Furthermore, with the increase of the in-situ stress level, GSIeq gradually decreases to GSIr, and the difference of plastic stress distribution law based on ESS and EPP models gradually diminishes. Moreover, the proportion of plastic deformation to total deformation increases nonlinearly with buried depth. The proposed method offers a new calculation idea for the simplified analysis of strain-softening rock masses.