Elasto-viscoplastic Constitutive Law Research Articles

Analysis of thermally-induced fracture of Callovo-Oxfordian claystone: From lab tests to field scale

Argillaceous rocks are a suitable host rock for the deep geological disposal of exothermic High-Level Radioactive Waste (HLW) and Spent Fuel (SF). Excess pore pressures develop in this type of rocks when subject to increased temperatures that, in some circumstances, may lead to the fracturing of the rock. The paper explores this phenomenon by means of coupled numerical analyses carried out within a fully coupled thermo-hydro-mechanical (THM) framework. The general THM formulation is described as well as the anisotropic elastic and anisotropic elastoviscoplastic constitutive laws employed. Thermal extension triaxial tests are simulated as a check on the performance of the numerical formulation and to provide calibration data for the tensile strength of the material. Selected results from a comprehensive set of three-dimensional analyses of a large-scale field heating test, designed to study the possibility of thermal-induced failure in the rock, are presented and discussed. The analyses reproduce satisfactorily the observed patterns of behaviour. The effects of the constitutive law, material parameters and the presence of the excavation damage zone (EDZ) around the main drift and around the heater boreholes are studied. In particular, their effects on the state of the stress in the heated area are examined in the context of the potential for thermal fracturing of the rock.

Effect of anisotropic creep on the convergence of deep drifts in Callovo-Oxfordian claystone

The long term behavior of the lining support in deep drifts is a crucial issue for nuclear waste underground repository project. In order to demonstrate the feasibility of the project, the French Radioactive Waste Management Agency (Andra) has built an Underground Research Laboratory (URL) at Bure. The underground structure is composed of a network of galleries whose drilling generates stress redistribution, damage and fracture in the surrounding medium. The damage and fracturing of this excavation damage zone have a direct effect on the displacement field and on the interaction between the formation and the lining support. The design of the support system request, in order to be as efficient as possible, a precise knowledge of the stress and displacement fields around the drift. For viscoplastic rock formation, the stress field around the drift and its effect on the support system must be determined for at least a couple hundred years. This is why we have studied the long term behavior of drifts using an elasto-viscoplastic constitutive law. The laboratory test data and field observations have shown that the Callovo-Oxfordian claystone has a slightly anisotropic behaviour for its elasticity, failure criterion and also creep properties. This anisotropy is largely increased by the presence of the fractures, which we take into account with transverse isotropic models. Homogenization techniques are used to establish a macroscopic expression of the viscous deformation depending on the behavior of the rock matrix and of the fractures. The macroscopic viscous deformation depends on the orientation and density of fractures, which is why based on investigation campaign, we divide the fractured zone, in our numerical models, into zones with fractures of different density and orientation to describe the behavior of the fractured zone as closely to the reality as possible. The model is then applied to the simulation of the convergences of GCS drift. The so called GCS drift belong to the network of monitored drift of the URL and was excavated in the direction of the major principal horizontal stress.

Interface-resolved simulations of the confinement effect on the sedimentation of a sphere in yield-stress fluids

We perform three-dimensional numerical simulations to investigate the confinement effect on the sedimentation of a single sphere in an otherwise quiescent yield stress fluid, in the presence of finite elasticity and weak inertia. The carrier fluid is modeled using the elastoviscoplastic constitutive laws proposed by Saramito (2009). The additional elastic stress tensor is fully coupled with the flow equation, while the rigid particle is represented by an immersed boundary method. The simulations show the faster relaxation of the fluid velocity and the progressive translation of the location of the negative wake downstream of the sphere as the bounding walls are brought closer to the particle. Moreover, the sphere drag decreases by increasing the particle–wall distance. We show that the confinement ratio (ratio of the gap between rigid confining walls and the sphere radius) reaches a critical value beyond which the wall-effect on the particle and flow dynamics becomes negligible. The key finding here is that the critical confinement ratio and the maximum variation of the Stokes drag with confinement ratio are weakly dependent on the level of material elasticity and plasticity for a certain range of material parameters. Finally, we propose an expression for the Stokes drag coefficient, as a function of material plasticity and confinement ratio.

Influence of initial relative densities on the sintering behavior and mechanical behavior of 316 L stainless steel fabricated by binder jet 3D printing

Compared with other 3D printing methods, binder jet 3D printing (BJ3DP) shows the great potential for obtaining large-scale complex parts with high efficiency and low cost. However, understanding the sintering behavior of powders is one of the challenging topics in BJ3DP. In this work, on the basis of the thermo–elasto–viscoplastic constitutive law model, the sintering behavior of 316 L stainless steel (316 L SS) cuboids prepared by BJ3DP with different initial relative densities (relative densities of green parts), i.e., 49.5%, 51.5% and 53.5%, was investigated via finite-element numerical simulations. It suggested that the shrinkage decreased as the initial relative density increased, resulting from that a higher initial relative density would provide higher sintering stress. Additionally, precise shrinkage and final relative densities of the BJ3DP samples were quantitatively predicted via numerical simulations, and the simulated results were in good agreement with experimental results. According to the experimental results, a final relative density of ~98.7% could be achieved in the BJ3DP 316 L SS sample with an initial relative density of 53.5% after sintering, showing a yield strength of ~308 MPa and an ultimate tensile strength of ~601 MPa along with a total elongation of ~59.4%.

Numerical simulation of underground excavations in an indurated clay using non-local regularisation. Part 2: sensitivity analysis

A sensitivity study is presented to evaluate the influence of different parameters on the simulation of an underground excavation in the Callovo-Oxfordian (COx) argillaceous formation performed in the Meuse/Haute-Marne underground research laboratory. An elasto-viscoplastic constitutive law representing the characteristic behaviour of indurated mudrocks and stiff clays has been employed. It incorporates anisotropy, strain-softening, creep deformations and dependence of permeability on damage. In addition, a non-local formulation, able to simulate localised deformations objectively, has been incorporated in the analyses. The following features affecting the excavation have been studied: initial stress, strength and stiffness anisotropy, strength parameters, hydraulic and hydromechanical parameters, and scale effects. A simulation reported in a companion paper provides the base case for benchmarking. The results are compared in terms of extent and configuration of the excavation fractured zone, vertical and horizontal tunnel convergences, and the development and evolution of pore pressures in the rock. From the comparisons, an enhanced understanding of the hydromechanical mechanisms associated with underground excavations in COx claystone, and other similar argillaceous materials, has been achieved.

Evaluating the subsidence above gas reservoirs with an elasto-viscoplastic constitutive law. Laboratory evidences and case histories

Subsidence of hydrocarbon fields is induced by the depletion of reservoirs and connected aquifers, which causes the compaction of geologic layers. A fundamental requirement of subsidence models is then capturing the mechanical behaviour of such layers with proper constitutive laws.This paper refers to the case of two gas reservoirs from the Adriatic basin, where a time delay of subsidence with respect to production was observed. This phenomenon has been reported for other fields worldwide, also occurring when subsidence rates do not immediately vanish after the end of production. In those cases, numerical simulations based on elasto-plastic constitutive models did not provide a satisfactory match of subsidence history, even though very accurate reconstructions of pore pressure evolution were used. This motivated a further insight into the time-dependent behaviour of the sandy reservoir materials. Thus, the results of an exhaustive laboratory investigation were interpreted with an elasto-viscoplastic model from the literature, implemented by the authors in a commercial Finite Element code. The satisfactory reproduction of the experimental results proved the adequacy of the model for the materials of concern and allowed determining the parameters that were then used in the new one-way coupled field simulations. A very good agreement was finally obtained between simulation results and field monitoring data in terms of reservoir compaction, maximum settlements, settlement history and extent of the subsidence bowl.

An efficient solution scheme for small-strain crystal-elasto-viscoplasticity in a dual framework

Computational homogenization schemes based on the fast Fourier transform (FFT) enable studying the effective micromechanical behavior of polycrystalline microstructures with complex morphology. In the conventional strain-based setting, evaluating the single crystal elasto-viscoplastic constitutive law involves solving a non-linear system of equations which dominates overall runtime. Evaluating the inverse material law is much less costly in the small-strain context, because the flow rule is an explicit function of the stress.We revisit the primal and dual formulation of the unit cell problem of computational homogenization and use state of the art FFT-based algorithms for its solution. Performance and convergence behavior of the different solvers are investigated for a polycrystal and a fibrous microstructure of a directionally solidified eutectic.

Multiple Tensor Train Approximation of Parametric Constitutive Equations in Elasto-Viscoplasticity

This work presents a novel approach to construct surrogate models of parametric differential algebraic equations based on a tensor representation of the solutions. The procedure consists of building simultaneously an approximation given in tensor-train format, for every output of the reference model. A parsimonious exploration of the parameter space coupled with a compact data representation allows alleviating the curse of dimensionality. The approach is thus appropriate when many parameters with large domains of variation are involved. The numerical results obtained for a nonlinear elasto-viscoplastic constitutive law show that the constructed surrogate model is sufficiently accurate to enable parametric studies such as the calibration of material coefficients.

Experimental and numerical analysis of the Lüders phenomenon in simple shear

The Lüders phenomenon is investigated in a low carbon ferritic steel under simple shear loading. Tensile and shear experiments are carried out associated with digital image correlation (DIC) measurements of the local strain field. An elastoviscoplastic constitutive law is identified with special attention paid to the choice of the equivalent stress measure in the yield function. It is then used to simulate the shear experiment using finite element analyses. Several mesh types and sizes are used to illustrate some mesh sensitivity observed in the finite element results. A regularized model based on strain gradient plasticity is then proposed to ensure fully mesh insensitive simulations and properly describe the finite thickness of the Lüders band front. The additional internal length introduced in the regularized model is identified from the DIC strain measurements. Finally, the boundary conditions best–suited for an accurate description of the shear experiment are discussed.

Numerical and experimental investigation of HDPE behavior during 2-ECAP process using 90° and 120° dies

Among various severe plastic deformation methods, equal channel angular pressing (ECAP) is an effective tool to introduce large plastic deformation of simple shear. The principle consists to extrude the sample several times through two intersecting channels with the aim to improve the mechanical properties of the material by altering the morphology of its microstructure. In this work, a numerical and experimental investigation of a typical semi-crystalline thermoplastic polymer (high density polyethylene, HDPE) during two-turn ECAP (2-ECAP) using 90° and 120° dies has been presented. The warping of the extruded samples has been also highlighted experimentally. The material parameters of an elasto-viscoplastic constitutive law were identified using compressive tests at different temperatures and strain rates. The effects of the main parameters, such as, the channel angles, the dimensions of the intermediate channel, the friction and the temperature have been analyzed. Finally, to confirm the numerical results, two dies composed of one elbow (1-ECAP) and two elbows (2-ECAP) have been manufactured and tested. It was found that the warping obtained using 2-ECAP die is more reduced than that of 1-ECAP die even with several passes.

In Situ Measurement and Prediction of Stresses and Strains During Casting of Steel

Modeling the thermo-mechanical behavior of steel during casting is of great importance for the prediction of distortions and cracks. In this study, an elasto–visco–plastic constitutive law is calibrated with mechanical measurements from casting experiments. A steel bar is solidified in a sand mold and strained by applying a force to bolts that are embedded in the two ends of the bar. The temporal evolutions of the restraint force and the bar’s length change are measured in situ. The experiments are simulated by inputting calculated transient temperature fields into a finite element stress analysis that employs the measured forces as boundary conditions. The thermal strain predictions are validated using data from experiments without a restraint. Initial estimates of the constitutive model parameters are obtained from available mechanical test data involving reheated steel specimens. The temperature dependence of the strain rate sensitivity exponent is then adjusted until the measured and predicted length changes of the strained bars agree. The resulting calibrated mechanical property dataset is valid for the high-temperature austenite phase of steel. The data reveal a significantly different mechanical behavior during casting compared to what the stress–strain data from reheated specimens show.

Evolution of stresses in metal injection molding parts during sintering

The evolution of stresses due to inhomogeneity in metal injection molding (MIM) parts during sintering was investigated. The sintering model of porous materials during densification process was developed based on the continuum mechanics and thermal elasto–viscoplastic constitutive law. Model parameters were identified from the dilatometer sintering experiment. The real density distribution of green body was measured by X-ray computed tomography (CT), which was regarded as the initial condition of sintering model. Numerical calculation of the above sintering model was carried out with the finite element software Abaqus, through the user-defined material mechanical behavior (UMAT). The calculation results showed that shrinkages of low density regions were faster than those of high density regions during sintering, which led to internal stresses. Compressive stresses existed in high density regions and tensile stresses existed in low density regions. The densification of local regions depended on not only the initial density, but also the evolution of stresses during the sintering stage.

The dynamic response of fluid-saturated porous materials with application to seismically induced soil liquefaction

The numerical simulation of liquefaction phenomena in fluid-saturated porous materials within a continuum-mechanical framework is the aim of this contribution. This is achieved by exploiting the Theory of Porous Media (TPM) together with thermodynamically consistent elasto-viscoplastic constitutive laws. Additionally, the Finite Element Method (FEM) besides monolithic time-stepping schemes is used for the numerical treatment of the arising coupled multi-field problem. Within an isothermal and geometrically linear framework, the focus is on fully saturated biphasic materials with incompressible and immiscible phases. Thus, one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the solid and fluid momentum balance equations and the algebraic incompressibility constraint. Applying the suggested material model, two important liquefaction-related incidents in porous media dynamics, namely the flow liquefaction and the cyclic mobility, are addressed, and a seismic soil–structure interaction problem to reveal the aforementioned two behaviors in saturated soils is introduced.

Analytic study of the onset of plastic instabilities during plane tension and compression tests on metallic plates

Abstract In this article, we propose an approximate analytic formulation of the growth-rate of plastic instabilities (symmetric and antisymmetric modes with respect to the median plane of the plate) during plane tension or compression tests on metals supposed to satisfy the Von Mises plasticity criterion, valid for any elastoviscoplastic constitutive law, and long and medium wavelengths λ (in comparison with thickness e ) along the loading direction (typically e ≤ λ , and even e /3 ≤ λ in the absence of viscous effects). This work generalizes important published results. For static tests, or dynamic ones in the field of long wavelengths, this formulation retrieves the formulae proposed by Fressengeas and Molinari (Instability and bifurcation in the plane tension test, 1992, Archives of Mechanics 44 (1), 93; Fragmentation of rapidly stretching sheets, 1994, European Journal of Mechanics A/Solids , 13 (2), 251) for the development of plastic necking instabilities during plane tension tests on rigid viscoplastic materials obeying Norton's constitutive law. In the absence of viscosity, for static tests, it retrieves the bifurcation stress proposed by Hill, Hutchinson (Bifurcation phenomena in the plane tension test, 1975, Journal of the Mechanics and Physics of Solids 23 , 239) and Young (Bifurcation phenomena in the plane compression test, 1976, Journal of the Mechanics and Physics of Solids 24 , 77).

Mathematical modeling of residual stresses and distortions induced by gas nitriding of 32CrMoV13 steel

The aim of this paper is to provide a model simulating the nitriding of 32CrMoV13 low alloy steels. The model is based on thermodynamics of irreversible processes and chemical kinetics. The mechanical approach is based on the mechanics of continuous media and considers an elasto-viscoplastic constitutive law. A self-consistent scale transition scheme establishes the link between chemical and mechanical calculations. The simulation describes the diffusion of nitrogen and carbon, the main precipitations with the associated volume changes, the residual stresses developed during treatment, and the distortions of complex structural parts. It enables evaluating the macroscopic volumetric eigenstrains, the thermal strain and the variation of Young’s modulus induced by the treatment. The problem is solved by Finite Element calculations. The unknown inputs of the model are identified through an inverse method using different kinds of experimental data such as nitrogen and carbon mass fractions, distortions and residual stresses. Flat and cylindrical samples are employed for that purpose. These specimens are gas nitrided during 120h at 550°C. The simulation program is finally validated on a sample of complex geometry treated in the same conditions. The distortions predicted by the model are therefore compared to results of measurements carried out by 3D full-field optical scanning.

Thermo-mechanical fatigue behaviour of welded tubular parts made of ferritic stainless steel

Car exhaust manifolds are critical components subjected to cyclic thermo-mechanical fatigue (TMF) during function. To reduce design costs, robust numerical design tools are required to assess their behaviour and lifetime. Manifolds are constructed by welding several ferritic stainless steel tubular parts together. TMF behaviour of a 1.4509 steel in welded and unwelded conditions is assessed under various loading conditions. Unified elasto-viscoplastic constitutive laws are developed. The specific thermo-mechanical behaviour of the heat-affected zone (HAZ) is also taken into account for welded steel. The reliability of the proposed models in predicting the mechanical response, in particular in the welded zone, is investigated. The local strains of the welded area are measured using a digital image correlation technique. Hence, several numerical models are implemented in ABAQUS and different areas are analysed to reproduce the mechanical behaviour of the heat-affected zone. Results are discussed and compared with experiments to validate the proposed model of the mechanical response of a welded component.

A 3D coupled hydro-mechanical granular model for the prediction of hot tearing formation

A new 3D coupled hydro-mechanical granular model that simulates hot tearing formation in metallic alloys is presented. The hydro-mechanical model consists of four separate 3D modules. (I) The Solidification Module (SM) is used for generating the initial solid-liquid geometry. Based on a Voronoi tessellation of randomly distributed nucleation centers, this module computes solidification within each polyhedron using a finite element based solute diffusion calculation for each element within the tessellation. (II) The Fluid Flow Module (FFM) calculates the solidification shrinkage and deformation-induced pressure drop within the intergranular liquid. (III) The Semi-solid Deformation Module (SDM) is used to simulate deformation of the granular structure via a combined finite element / discrete element method. In this module, deformation of the solid grains is modeled using an elasto-viscoplastic constitutive law. (IV) The Failure Module (FM) is used to simulate crack initiation and propagation with the fracture criterion estimated from the overpressure required to overcome the capillary forces at the liquid-gas interface. The FFM, SDM, and FM are coupled processes since solid deformation, intergranular flow, and crack initiation are deeply linked together. The granular model predictions have been validated against bulk data measured experimentally and calculated with averaging techniques.

Time-Dependent Modeling of Tunnels in Squeezing Conditions

The numerical analysis of tunnels in rock masses, which exhibit a squeezing behavior, is considered, with reference to the use of two elastoviscoplastic constitutive laws implemented in finite-difference and finite-element codes. Because of the fact that in the most severe squeezing conditions time dependence is observed during excavation and may last for a long time, both the reinforcement of the tunnel face and the support of the tunnel section are to be considered at the design and construction stages. As a consequence, the need arises to analyze the tunnel excavation-construction sequence in detail and to account for time-dependent behavior. Numerical analyses have been carried with the Saint Martin La Porte access adit, along the Lyon-Turin Base Tunnel, taken as a case study. The interest stems from the performance monitoring data available for a significant tunnel length in the carboniferous formation, which exhibits a severely squeezing behavior. Also of interest is the adoption of a novel yielding support system, which revealed to be successful in coping with the difficult conditions encountered. Following a series of preliminary back analyses, carried out in axisymmetric conditions and meant to calibrate the rock mass parameters, attention has been focused on the support system. The actual excavation-construction sequence has been simulated in detail to investigate the tunnel response and the stress changes in the support. The numerical results are compared with the monitoring data, and the potential of the two different elastoviscoplastic constitutive models in describing the tunnel behavior is assessed in detail.

Simulation of Semi-Solid Material Mechanical Behavior Using a Combined Discrete/Finite Element Method

As a necessary step toward the quantitative prediction of hot tearing defects, a three-dimensional stress–strain simulation based on a combined finite element (FE)/discrete element method (DEM) has been developed that is capable of predicting the mechanical behavior of semisolid metallic alloys during solidification. The solidification model used for generating the initial solid–liquid structure is based on a Voronoi tessellation of randomly distributed nucleation centers and a solute diffusion model for each element of this tessellation. At a given fraction of solid, the deformation is then simulated with the solid grains being modeled using an elastoviscoplastic constitutive law, whereas the remaining liquid layers at grain boundaries are approximated by flexible connectors, each consisting of a spring element and a damper element acting in parallel. The model predictions have been validated against Al-Cu alloy experimental data from the literature. The results show that a combined FE/DEM approach is able to express the overall mechanical behavior of semisolid alloys at the macroscale based on the morphology of the grain structure. For the first time, the localization of strain in the intergranular regions is taken into account. Thus, this approach constitutes an indispensible step towards the development of a comprehensive model of hot tearing.

Modelling of the residual state of friction stir welded plates

This paper presents a steady-state simulation of friction stir welding based on a prior computational method developed in [Bastier, A., Maitournam, M.H., Dang Van, K., Roger, F., 2006. Steady state thermomechanical modelling of friction stir welding. Sci. Technol. Weld. Joining 11 (3), 278–288]. This simulation includes two main steps. The first one uses an Eulerian description of the thermomechanical problem: a 3D-mixed Finite Element Model based on a Computational Fluid Dynamics package is used to establish the material flow and the temperature field during the process. In the second step, a steady-state algorithm based on an elastoviscoplastic constitutive law is used to estimate the residual state induced by the process. The steady-state assumption and the original elastoviscoplastic constitutive law are two key features of the present model. This calculation takes into account the whole thermal, metallurgical and mechanical history of the material since the algorithm is based on an integration along the path lines of the particles. The material considered is a 7050 aluminium alloy. It is observed that the longitudinal residual stress field has a two peaks profile: these two peaks are situated in the zone of high gradient of dissolved precipitates fraction. Finally, a parametric study about the influence of welding and rotational speeds is carried out. This parametric study shows that the higher the welding speed and the lower the rotational speed, the lower the temperatures and the lower residual distortions.