Prager Model Research Articles

Finite Element Modeling of Brittle and Ductile Modes in Cutting of 3C-SiC

Machining of brittle ceramics is a challenging task because the requirements on the cutting tools are extremely high and the quality of the machined surface strongly depends on the chosen process parameters. Typically, the efficiency of a machining process increases with the depth of cut or the feed rate of the tool. However, for brittle ceramics, this easily results in very rough surfaces or even in crack formation. The transition from a smooth surface obtained for small depths of cut to a rough surface for larger depths of cut is called a brittle-to-ductile transition in machining. In this work, we investigate the mechanisms of this brittle-to-ductile transition for diamond cutting of an intrinsically brittle 3C-SiC ceramic with finite element modeling. The Drucker–Prager model has been used to describe plastic deformation of the material and the material parameters have been determined by an inverse method to match the deformation behavior of the material under nanoindentation, which is a similar loading state as the one occurring during cutting. Furthermore, a damage model has been introduced to describe material separation during the machining process and also crack initiation in subsurface regions. With this model, grooving simulations of 3C-SiC with a diamond tool have been performed and the deformation and damage mechanisms have been analyzed. Our results reveal a distinct transition between ductile and brittle cutting modes as a function of the depth of cut. The critical depth of cut for this transition is found to be independent of rake angle; however, the surface roughness strongly depends on the rake angle of the tool.

On the effect of plastic model on simulation of adhesive bonded joints with FM94

The stress-strain relationship of adhesive material is important as an input for numerical simulation. In this paper, a series of quasi-static experiments to measure the stress-strain relationship of FM94 adhesive material is reported. The experiments include the bulk tension test to obtain the tensile property and the V-notched beam shear test to obtain the shear property. The FE simulation with the Drucker–Prager model shows good agreement with experimental results. Moreover, further numerical results of adhesive joints show it is not always proper to use the Von Mises model to simulate adhesive joints as some previous studies proposed. The experimental results clearly show that the effective yield stress of adhesive materials is different for tension and shear, which disagrees with the Von Mises model often adopted in simulation.

A constitutive model for elastic–plastic materials using scalar conjugate stress/strain base pairs

A constitutive model for elastic–plastic materials is developed using scalar, conjugate, stress–strain, base pairs in a finite deformation setting. These conjugate base pairs arise from an alternative QR decomposition of the deformation gradient that decomposes its matrix into an orthogonal rotation and an upper-triangular matrix, called the Laplace stretch. This decomposition is particularly useful from an experimental standpoint, as it enables one to directly measure the components of Laplace stretch and its plastic contributions in a specific coordinate system. Moreover, from an experimental standpoint, it is difficult to parameterize current material models due to a covariance between their tensor invariants, traditionally used in their constructions. The use of scalar, conjugate, base pairs are also helpful from that point of view. Interestingly, the multiplicative elastic–plastic decomposition of Laplace stretch leads to an additive decomposition of the total strain attributes into their corresponding elastic and plastic components. Although an additive strain decomposition is commonly used in small-strain theory, here such a decomposition is possible even for finite deformations. An additive decomposition of the strain attributes has a deeper consequence in the construction of our constitutive model. A maximum rate of dissipation criterion has been used in deriving our constitutive equations, as this criterion is valid for a wider class of materials. Two constitutive assumptions – one for a Helmholtz potential, and one for the rate of dissipation function – are required for our constitutive construction. This model does not presuppose the existence of a yield surface. In fact, it is shown that whether a material exhibits a yielding or a creep-like behavior depends upon the differentiability of the rate of dissipation function. Two cases of plastic deformation – volume-preserving and dilatant-pressure dependent deformations – have been considered. To illustrate the proposed model, finite strain versions of classical J2 plasticity and a Drucker–Prager model are derived.

Effects of rate-dependent behavior of sandstone on performance of gas storage in aquifer

Strain rate is one of the main factors affecting the properties of rock masses. The effect of strain rate on performance of gas storage in aquifer is investigated combining laboratory tests and constitutive and numerical models. Triaxial tests on sandstone specimens at different strain rates were carried out and the effect of strain rate on mechanical properties of sandstones was investigated. The results showed that the peak strength, axial strain, elastic modulus, Poisson’s ratio, cohesion and internal friction angle of sandstone specimens increased with strain rate. In addition, the elastic–viscoplastic properties of sandstones have been analyzed. It indicated that the deflection angle in the viscoplastic flow direction was increased with strain rate and viscoplastic volumetric strain grew faster than the viscoplastic deviator strain. A rate-dependent constitutive model was adopted and a method for solving the parameters of the constitutive model was determined based on test data. For the constitutive model, the elastic part is a linear elastic model, the plastic part is a linear Drucker–Prager model, and the viscous part is a power law constitutive function. The model and methodology for parameter determination were validated against the test data of sandstones. The results showed that the model selected in this study can well describe effects of strain rate on the mechanical properties of sandstones. The model and method have been applied in gas injection and production capacity assessment of a gas storage in aquifer and the strain rate of the formation was determined by gas injection pressure and time indirectly in simulation. The simulation results showed that the gas injection and production rate of gas storage in aquifer will increase significantly with the strain rate of formation in the same gas injection pressure or time. It also indicated that the gas production capacity will be greatly improved with the strain rate of formation.

Development of analytical strength estimator for self-piercing rivet joints through observation of finite element simulations

Analytical strength estimators for self-piercing riveted (SPR) metal sheets under the lap-shear and the cross-tension modes are developed. The modeling is based on the local mechanical responses of rivet and sheet metals during the riveting process and joint strength tests, which are investigated using finite element (FE) simulations as virtual experiments. The FE simulations consist of the riveting process model and lap shear test model, and the plastic strain hardening and damage of sheet metals near the riveted zone are transferred between the two models. The cross-sectional geometry of the riveted zone and the load-displacement curve during the lap-shear test are predicted with three different ductile fracture criteria, Cockcroft–Latham, Rice–Tracey, and Drucker–Prager model, to evaluate the accuracy of the FE modeling. The four-parameter joint strength estimator for the lap-shear and cross-tension are proposed based on the observations of the geometrical features and mechanical responses in the FE simulations. The results of prediction are within reasonable accuracy for various combinations of steel and aluminum alloy sheets, particularly 31 among 34 cases showed prediction errors within 25% for the lap shear-tests, and 14 among 18 cases for the cross-tension tests, and average errors for lap-shear and cross-tension mode were 12.1% and 14.4%, respectively.

A modified smoothed particle hydrodynamics for modelling fluid-fracture interaction at mesoscale

This study presents a modified smoothed particle hydrodynamics (SPH) for modelling liquid-fracture interaction at mesoscale. In this approach, an interparticle force that can provide repulsive force in the short-range and attractive force in the long-range is introduced to model the liquid–liquid and liquid–solid interactions. This interparticle force prevents the SPH particles from clustering and having high compressibility when under high pressure. The formation of a droplet, different contact angles and realistic compressibility of liquid are achieved to demonstrate the capability of the proposed SPH based approach. Moreover, the Drucker–Prager model and the Grady–Kipp damage model are implemented in SPH to describe the solid failure. The results show that the mixed-mode can reproduce the fracture patterns of available Brazilian disc tests on rock-like 3D printed material. Finally, a process of hydraulic fracturing is simulated at mesoscale. The results demonstrate that the proposed SPH framework has potential in modelling liquid-fracture interaction with the consideration of interfacial effects at mesoscale and solid fracture.

Analysis of the stress-strain state of a vertical ground electrode system for permafrost soils under pushing loads

The article proposes the design, design scheme and model of a vertical ground electrode system with lobe lugs for permafrost soils. The model was implemented using the ANSYS software. In the design scheme, the soil — ground electrode system is taken into account, the elastic-plastic properties of the soil are taken into account by the Drucker — Prager model. When modeling the work of the foundation soils, the Mises strength condition was adopted, according to which the equivalent stress is calculated under the condition of the material hydrostatic compression. The following boundary conditions are accepted: a cylinder-shaped soil mass is rigidly fixed along the lower face and along the side surface of the cylinder. Calculations are made for 5 standard sizes of grounding conductors. Maps of the distribution of stresses in the metal structure of the ground electrode (the rod and petals-emphasis) are received, the movements of the ground electrode in the soil mass are determined. The dependences between the maximum equivalent stresses in the ground electrode lobes and the value of vertical displacement in the ground base are established, as well as the amount of movement of the earthing pad, at which the effective equivalent voltages reach critical values in the area where the paddles are adjacent to the rod.

Numerical methodology for determining the soil mass surface slump, caused by the tunnel construction with the geodetic monitoring results comparison

The work is aimed at determining the construction effect of a running tunnel on the stress-strain state change of the surrounding soil massif, in particular, on the development of a sediment basin on the Earth’s surface. The soil massif is defined by an elastoplastic material according to the Drucker - Prager model. The paper proposes two numerical methods for solving the problem: with and without the use of contact elements for modeling the construction gap between the tunnel shell and the soil mass caused by the tunnel boring mechanized complex operation. The proposed methods were applied in the construction of Kalininsko-Solntsevskaya line tunnels of the Moscow subway. A comparative analysis of the results obtained with geodetic monitoring of the Earth’s surface slump in the construction influence zone has been carried out.

A Novel Method for Constructing Main‐Aftershock Sequences and Its Application in the Global Damage Accumulation Effects Analysis of Gravity Dams

The traditional linear elastic and Drucker–Prager (DP) models cannot truly reflect the strong nonlinear characteristics of the concrete and rock foundation of the dam under earthquake. Therefore, for comprehensive evaluation of the cumulative damage of the gravity dam structure caused by aftershock, the dynamic damage of the dam body concrete is analyzed by many scholars through the plastic damage mechanics method, but there is little research on rock material at the dam foundation with the method utilized; thus, the simulation of the whole dynamic damage evolution is worthy of investigation of the dam body and dam foundation. According to the randomness of ground motion, the transcendental probability (P) is introduced to express the statistical characteristics of aftershock intensity, and a new method for constructing main‐aftershock sequences of ground motion is proposed in this paper. And then, the law of the damage evolution and energy characteristics of the concrete gravity dam under the combined action of the main shock and aftershock sequences is studied. The results are shown as follows: the smaller aftershocks do not cause further damage to the dam; as the aftershock intensity increases, the energy characteristics of the dam body and foundation have shown different changing rules; when the ratio of peak aftershock acceleration to peak main shock acceleration (∇PGA) approximately equals 0.68, the aftershock will cause larger secondary damage to the dam.

Strength of porous α-SiO2 in a shock loaded environment: Calibration via Richtmyer–Meshkov instability and validation via Mach lens

The strength of brittle porous media is of concern in numerous applications, for example, earth penetration, crater formation, and blast loading. Thus, it is of importance to possess techniques that allow for constitutive model calibration within the laboratory setting. The goal of the current work is to demonstrate an experimental technique allowing for strength assessment of porous media subjected to shock loading, which can be implemented into pressure-dependent yield surfaces within numerical simulation schemes. As a case study, the deviatoric response of distended α-SiO2 has been captured in a tamped Richtmyer–Meshkov instability (RMI) environment at a pressure regime of 4–10 GPa. Hydrocode simulations were used to interpret RMI experimental data, and a resulting pressure-dependent yield surface akin to the often employed modified Drucker–Prager model was calibrated. Simulations indicate that the resulting jet length generated by the RMI is sensitive to the porous media strength, thereby providing a feasible experimental platform capable of capturing the pressurized granular deviatoric response. Furthermore, in efforts to validate the RMI-calibrated strength model, a set of Mach-lens experiments was performed and simulated with the calibrated pressure-dependent yield surface. Excellent agreement between the resulting Mach-lens length in experiment and simulation provides additional confidence to the RMI yield-surface calibration scheme.

MAP123-EPF: A mechanistic-based data-driven approach for numerical elastoplastic modeling at finite strain

Direct numerical simulation based on experimental stress–strain data without explicit constitutive models is an active research topic. In this paper, a mechanistic-based, data-driven computational framework is proposed for elastoplastic materials undergoing finite strain. Harnessing the physical insights from the existing model-based plasticity theory, multiplicative decomposition of deformation gradient and the coaxial relationship between the logarithmic trial elastic strain and the true stress is employed to perform stress-update, driven by two sets of the specifically measured one dimensional (1D) stress–strain data. The proposed approach, called MAP123-EPF, is used to solve several Boundary-Value Problems (BVPs) involving elastoplastic materials undergoing finite strains. Numerical results indicate that the proposed approach can predict the response of isotropic elastoplastic materials (characterized by the classical J2 plasticity model and the associative Drucker–Prager model) with good accuracy using numerically/experimentally generated data. The proposed approach circumvents the need for the several basic ingredients of a traditional finite strain computational plasticity model, such as an explicit yielding function, a hardening law and an appropriate objective stress rate. Demonstrative examples are shown and strengths and limitations of the proposed approach are discussed.

Numerical Analysis of Pipeline Uplift Resistance in Frozen Clay Soil Considering Hybrid Tensile-Shear Yield Behaviors

Pipeline uplift resistance plays an essential role in the design of buried pipelines in permafrost regions or artificially frozen ground. The uplift resistance of a frozen soil is dependent on the soil’s mechanical properties, which are needed to characterize plastic zones in the frozen soil that surrounds pipes. Owing to the presence of unfrozen water, frozen soils (especially frozen clay) display complex mechanical properties that vary with temperature. There are limited amount of research studies with a focus on frozen soil–pipe interactions that consider temperature change within the literature. In this study, numerical modeling is conducted to investigate frozen soil–pipe interactions at different temperatures. A pipe is simulated with vertical displacement-controlled or vertical load-controlled conditions. The temperature-dependent mechanical parameters of frozen clay soil are examined in the simulation. Both tensile yield and shear yield behaviors are included in the modeling with the consideration of temperature-dependent failure criterions. Simulated stress paths in monitoring points are collected and displayed. The results from the Mohr–Coulomb model with the Rankine tensile cut off, and the hyperbolic Drucker–Prager model are compared and analyzed. The results indicate that hyperbolic Drucker–Prager model is effective for analyzing soil plastic zones surrounding a pipe. The applied hyperbolic Drucker–Prager model with reduced strength parameters can produce a more conservative uplift resistance–displacement relation when compared with that from the Mohr–Coulomb model. Thermally induced plastic strain may have a significant impact on the uplift resistance analysis, since a significant thermally induced uplift displacement is expected if there is a heat transfer process from a buried pipe to the surrounding frozen soil.

Effect of using different approximation models to the exact Mohr\u2013Coulomb material model in the FE simulation of Anchor Foundations in sand

Singularities in the Mohr–Coulomb (MC) yield criterion create numerical difficulties when implemented in the finite element model (FEM). In this research, a two-dimensional (2D) and a three-dimensional (3D) elasto-plastic FEM coupled with isotropic strain hardening–softening or simple strain-softening law incorporating non-associated flow rule including shear band, is used to investigate the effect of singularities, embedment, shape, soil frictional as well as dilation angle on vertically uploaded shallow anchor foundations buried in Toyoura sand. Mohr–Coulomb–Drucker Prager (MC–DP) model provides a more appropriate solution than Drucker Prager–Drucker Prager (DP–DP) model. Besides, the peak resistance factor and the settlement at the ultimate capacity have been found as the functions of embedment, shape, and sand density. Also, the numerical model gives satisfactory agreement with the previous studies. In particular, a simple strain-softening model cannot capture the settlement behavior of anchor foundations in sand efficiently.

Prediction of dynamic amplification factor on clay brick masonry assemblage

The dynamic amplification factor plays a vital role in the design of masonry structures under action of dynamic forces causing high strain rate loading. Therefore, the study is aimed to predict the dynamic amplification factor through numerical investigations. The finite element simulations have been performed using micro modelling approach on ABAQUS/Explicit commercial finite element software. The Drucker- Prager model and surface cohesive model were used to predict the dynamic amplification factor and toughness of the clay brick masonry assemblage. The response in terms of stress and strain on compression as well as flexural assemblage were predicted and verified with the experimental results available in literature. Further, the influence of strain rate on compression brick masonry assemblage was studied under varying strain rate in terms of stress, strain and deformations. It was observed that the predicted results were found in good agreement with the experimental results for both compression and flexural assemblage. Also, it was observed that for 1 s−1 strain rate, the increment in toughness was found to be insignificant, i.e. 17%, however the toughness was found to increase by 76% and 150% for 100 and 200 s−1 strain rate respectively, compare to the 4.1 × 10−5 s−1 strain rate. It was also suggested that the finite element model proves very useful in light of identifying the dynamic amplification factor as it is an essential parameter required for designing the structure under high rate of loading.

Integration of Transverse Isotropy in the Instantaneous Behaviour of Geomaterials with Application to Numerical Modelling of Underground Structures

The paper deals with the integration of structural anisotropy in a pre-existing constitutive law for rocks, dedicated to the numerical modelling of underground structures. On the basis of bibliographic study, two approaches were selected and applied to a simple constitutive law to begin with, namely the Drucker–Prager model. The two appraoches considered are the fabric tensor approach and the weakness plane approach. Both approaches are described, the equations are provided and the resulting anisotropic models are implemented in the open source finite element software (Code_Aster in EDF R&D code general pour l’etude du comportement mecanique des structures diffuse sous licence GNU GPL, 2017). The models are then applied to the simplified numerical modelling of an underground facility and the main results are provided. The main advantages and drawbacks of both approaches are highlighted. The main findings of this work is that the fabric tensor approach is the most fitted for the material considered for the future prospects. The future prospects are the integration of anisotropy in a much more complex isotropic model being developed by EDF research team for application to geomaterials.

MAP123-EP: A mechanistic-based data-driven approach for numerical elastoplastic analysis

In this paper, a mechanistic-based data-driven approach, MAP123-EP, is proposed for numerical analysis of elastoplastic materials. In this method, stress-update is driven by a set of one-dimensional stress–strain data generated by numerical or physical experiments under uniaxial loading. Numerical results indicate that combined with the classical strain-driven scheme, the proposed method can predict the mechanical response of isotropic elastoplastic materials (characterized by J2 plasticity model with isotropic/kinematic hardening and associated Drucker–Prager model) accurately without resorting to the typical ingredients of classical model-based plasticity, such as decomposing the total strain into elastic and plastic parts, as well as identifying explicit functional expressions of yielding surface and hardening curve. This mechanistic-based data-driven approach has the potential of opening up a new avenue for numerical analysis of problems where complex material behaviors cannot be described in explicit function/functional forms. The applicability and limitation of the proposed approach are also discussed.

Investigating the Fracture of BSCF Material with the Extended Drucker–Prager Model Through Finite Element Simulations

In the present work, experimental results have been reported to show that for BSCF material, an extremely brittle material, the fracture stresses in three-point bending tests and ring-on-ring tests were consistently (room temperature to 800 °C) different. Several material models have been used to see if such discrepancy can be explained even qualitatively. Prospects of Drucker–Prager (DP) model in both qualitatively and quantitatively explaining this behavior has been discussed in the paper. Power law form of extended DP model, a generalized form of DP model, has been fitted to the experimental data at different temperatures. Finite element simulations are run and the results are compared against the experimental findings.

Numerical simulation of granular mixing in a rotary drum using a generalized interpolation material point method

AbstractNumerical simulations using a generalized interpolation material point method (GIMP) and material point method were introduced into the studying of granular mixing in a rotary drum. A Drucker–Prager model that has an equal size with a Mohr–Coulomb criterion was adopted to predict the failure behavior of granular matter. Experiments and discrete element method simulations were conducted to validate the models, and GIMP was finally used due to its accuracy and the ability for eliminating the crossing boundary noise. Velocity field of bed surface and standard deviation were used to measure the mixing performance. In particular, the influence of mesh size, elastic modulus, cohesion, and filling level were investigated. The results of simulations show that the utilization of GIMP significantly reduces the computing cost of simulation. The analysis of mixing performance shows that mixing performance decreases with the increase of elastic moduli, cohesion of granular matter, and filling levels.

Analytical and Numerical Analyses of Tunnel Excavation Problem Using an Extended Drucker–Prager Model

This paper presents a semi-analytical solution as well as finite element numerical simulations for the tunnel excavation problem, based on an extended Drucker–Prager model and under the drained condition. The adopted plasticity model is capable of accounting for the strain-hardening behavior of soils/rocks through both the internal friction angle and the cohesion. The analytical formulations show that this problem can be reduced to solving a system of ordinary differential equations, with the radial, tangential and vertical stresses, as well as the specific volume being the basic unknowns. Parametric studies are conducted to illustrate mainly the tunnel support pressure–displacement curve and the distributions of stress components, and, in particular, the critical support pressure required to stabilize the tunnel on the basis of three different collapse criteria. With regards to the numerical aspect, an integration algorithm has been developed by employing the return mapping scheme for the extended Drucker–Prager model and subsequently implemented into ABAQUS through the user subroutine UMAT. The agreement between the ABAQUS numerical results and the analytical solution predictions is overall excellent, which demonstrates the accuracy and reliability of the proposed integration scheme.

Soil–Water-Structure Interaction Algorithm in Smoothed Particle Hydrodynamics (SPH) with Application to Deep-Penetrating Problems

A fully coupled soil–water-structure interaction algorithm was presented in the framework of smoothed particle hydrodynamics (SPH). In this algorithm, soil–water interaction was simulated based on the two-phase mixture theory. Each phase of the mixture occupies part of the macroscopic mixture and satisfies its own conservation equations of mass and momentum. The Drucker–Prager model with nonassociated plastic flow rule was used to describe the constitutive behavior of soil. The water was treated as Newtonian fluid. Interaction between soil and water was modeled by the pore water pressure and the viscous drag force. The structure was considered as rigid and the interaction with soil/water was modeled by the frictional sliding contact algorithm. With this algorithm, it is possible to investigate pore water pressure, the effective stress and deformation of the soil undergoing large deformation. Moreover, the effect of the temporal and spatial evolution of soil porosity was taken into consideration. This study first examined the proposed algorithm for a U-tube seepage problem and a two-dimensional consolidation problem. Afterwards, the continuous deep penetrating process of the spudcan, which involved large soil deformation and complex soil–water-structure interaction, was simulated under axisymmetric conditions. The comparison with previous research indicates the robustness and applicability of the proposed algorithm. Furthermore, the proposed approach could be a potentially efficient tool helping to reveal the mechanism of soil failure in geotechnical, costal and ocean engineering.