Plastic Stress-strain Relations Research Articles

Stress wave effects and their mechanisms on stress-strain curves in the elastic phase of SHPB tests

In Split Hopkinson Pressure Bar (SHPB) tests, the stress wave effects during the elastic phase of the stress-strain curve and their influence mechanisms are crucial. Due to the extremely short duration of the elastic deformation phase, the stress wave effects on the specimen during this stage cannot be ignored. This leads to significant errors in the obtained elastic stress-strain curves. However, the dynamic compressive elastic stress-strain relationship forms the basis for studying the viscoelastic behavior of materials. Accurate determination of elastic yield stress and yield strain is also essential for deriving accurate plastic stress-strain relationships. Quantitative research on the stress wave effects during the elastic compression phase of SHPB tests is fundamental for decoupling and obtaining accurate material elastic curves. This paper conducts a quantitative theoretical analysis of the structural effects caused by stress wave evolution during the elastic compression phase, based on the assumption of plane waves. It studies the deviation characteristics and main factors of the phenomenological engineering stress-strain curves of the specimen compared to the actual material stress-strain curves under different conditions, revealing the influence rules and mechanisms of this deviation. The maximum stress deviation value and its corresponding dimensionless time, as well as the variation trend of the maximum stress deviation value within different fluctuation intervals, are calculated. Additionally, the study investigates the cases where the incident wave is arc-shaped or a combination of arc and linear waves. The findings provide theoretical references for the precise design and accurate data processing of SHPB tests.

Study on constitutive relationship of 6061 aluminum alloy based on Johnson-Cook model

Uniaxial tensile tests are conducted on 6061 aluminum alloy sheets at different strain rates of a certain temperature and different temperatures of a certain strain rate. The true stress-strain curves show that the flow stresses are negatively correlated with strain rates and temperatures. Based on the Johnson Cook (J-C) constitutive model, the modified J-C model is constructed by replacing the strain term with a Voce hardening model and correcting the strain rate sensitivity coefficient and the thermal softening exponent. Compared with the J-C model, the modified J-C model has a smaller average absolute error and more accurately describes the plastic stress-strain relationship of 6061 aluminum alloy under various deformation conditions. Furthermore, the modified J-C model is also used for numerical simulation of uniaxial tension. The good agreement between the experimental and simulated load-displacement curves further verifies the usefulness of the modified J-C model. Finally, the microscopic mechanism by which deformation speeds affect ductility is analyzed by scanning electron microscopy (SEM).

Determination of the Plastic Stress–Strain Relationship of a Rupture Disc Material with Quasi-Static and Dynamic Pneumatic Bulge Processes

Rupture discs, manufactured using a hydraulic or pneumatic bulge process, are widely used to protect vessels from over-pressuring. The stress–strain relationship of the material in the bulge process plays a major role in understanding the performance of rupture discs. Moreover, both the theoretical analyses and numerical simulations of rupture discs demand a reliable stress–strain relationship of the material in a real bulge process. In this paper, an approach for determining the plastic stress–strain relationship of a rupture disc material in the bulge process is proposed based on plastic membrane theory and force equilibrium equations. Pressures of compressed air and methane/air mixture explosions were used for the bulge pressure to accomplish the quasi-static and dynamic bulge processes of tested discs. Experimental apparatus for the quasi-static bulge test and the dynamic bulge test were built. The stress–strain relations of 316L material during bulge tests were obtained, compared, and discussed. The results indicated that the bulge height at the top of the domed disc increased linearly with an increase in bulge pressure in the quasi-static and dynamic bulge processes, and the effective strain increased exponentially. The rate of pressure rise during the bulge process has a significant effect on the deformation behavior of disc material and hence the stress–strain relationship. At the same bulge pressure, a disc tested with a larger pressure rise rate had smaller bulge height and effective strain. At the same effective stress at the top of the domed disc, discs subjected to a higher pressure rise rate had smaller effective strain, and hence they are more difficult to rupture. Hollomon’s equation is used to represent the relationship between the effective stress and strain during bulge process. For pressure rise rates in the following range of 0 (equivalent to quasi-static condition), 2–10 MPa/s, 10–50 MPa/s, and 50–100 MPa/s, the relation of stress and strain is σe = 1259.4·εe0.4487, σe = 1192.4·εe0.3261, σe = 1381.2·εe0.2910, and σe = 1368.4·εe0.1701, respectively.

Study of constitutive relation of rail decarburized layer

Rail decarburized layer has significant impact on the wear and rolling contact fatigue in the wheel-rail interaction. Due to the lack of constitutive relation of rail decarburized layer, it is impossible to analyse accurately the influence of rail decarburized layer on wheel-rail contact. To obtain the constitutive relation of this layer of decarburized non-uniform material, nano-indentation experiments were carried out at different vertical positions in the decarburization layer on a rail specimen with decarburization thickness of 0.55 mm. Young’s modulus and hardness were determined using Oliver and Pharr method. Axisymmetric two-dimensional finite element models were established in ABAQUS to simulate the nano-indentation loading process and the power law elastic–plastic stress–strain relationships were obtained for material at different vertical positions in the rail decarburized layer. A set of dimensionless functions were proposed for Young’s modulus, hardness, yield stress and strain hardening exponent at arbitrary depth in a rail decarburized layer or in a decarburized layer treated by rail pre-grinding. The results show that the Young’s modulus and hardness decrease from the rail bulk to the surface of the decarburized layer. The Young’s modulus and hardness near the surface are 11% and 35% lower than those in the rail bulk, respectively. However, the yield stress exhibits an increasing trend from rail bulk to the decarburized surface, where yield stress is 20% higher than the rail bulk. Analysis shows that the main reason that causes the reduction of hardness at the decarburized layer surface is the reduction of Young’s modulus but not the yield stress.

Experimental determination of the plastic deformation and fracture behavior of polypropylene composites under various strain rates

A strain measurement method is proposed that defines an intrinsic flow stress equation for various strain rates using representative brittle and ductile polypropylene composites. Digital image correlation is used to track strain at the local fracture point. Results are compared with those of the gauge length method. Whereas the strain up to diffuse necking is identical, regardless of the acquisition method, the strain after diffuse necking is completely dependent on the specimen's fracture location; hence, it is underestimated if the fracture is not in the specimen's center. In brittle material, fracture strain deviation is not large because fracture occurs immediately after necking. In ductile material, limits to extrapolating from the stress–strain relationship before necking exist, as stress-softening progresses slowly. The proposed method reduced the fracture strain deviation to 20% of that of the existing method, thereby determining a consistent and quantitative plastic stress–strain relationship up to fracture.

Primary and recency effects based on loading path in classical plasticity

We have established an elastoplastic analysis model to explore the effect of loading path in an incompressible thin-walled tube under the combined action of axial force and torque based on Mises yield condition and isotropic linear hardening assumption. Further, four stress areas (σx, τx) are divided according to the characteristics of the final stress, and the plastic stress-strain relationship of twelve stress paths in different stress areas is derived. The “primary effect” of the stress path on plastic strain is demonstrated, namely, the plastic strain caused by the pre-loaded stress in path A (tensile stress is initially applied, followed by shear stress) is always greater than that caused by the post-loaded stress in path C (shear stress is initially applied, followed by tensile stress) irrespective of the value of final stress. The “recency effect” of the strain path on the stress is also established, which indicates that the stress caused by the post-loaded strain in path A is always greater than that caused by the pre-loaded strain in path C irrespective of the value of final strain. From the perspective of deformation, the “primary effect” of the stress path on the plastic strain and the “recency effect” of the strain path on the stress are unified. These effects are succinct and universal, and they provide useful insights on the plastic stress-strain relationship under different loading paths. Furthermore, they can serve as a useful reference for optimizing the processing technologies and construction procedures.

Visco-plastic deformation analysis of rock tunnels based on fractional derivatives

The squeezing of rock tunnels in soft strata is often encountered during and after excavation, which is closely related to the viscoplasticity of rock mass. However, previous studies involved multiple creep elements in the calculation of visco-plastic deformation, which cost much time to identify creep parameters. In this paper, a novel four-element fractional visco-plastic (FVP) model, which includes two Abel dashpots, one spring and one visco-plastic body, is established using fractional derivatives. Three-dimensional creep equation of the FVP model is first derived using the von Mises equivalent plastic stress-strain relationship. Then the closed-form solution of visco-plastic deformation of the surrounding rock around circular tunnels is derived by the FVP model combining with the non-associated flow rule and the Hoek-Brown criterion. Next, taking rock tunnels that are excavated in three different quality rock salt as examples, parametric investigations of time-dependent parameters and time-independent parameters are carried out. Visco-plastic deformation curves of rock tunnels, which includes (a) the displacement of tunnel wall versus time; (b) the displacement of visco-plastic zone versus depth, are analyzed. Finally, the auxiliary tunnel of Jinping II hydropower station is taken as a typical example for studying the time-dependent deformation of rock tunnels. The analytical results are in good agreement with the experimental and monitoring data. This paper may blaze a new trail to quickly evaluate the visco-plastic deformation of rock tunnels during the early design process.

An elastic–plastic ice material model for ship-iceberg collision simulations

Ship-iceberg collisions are currently a hot topic of research. The modeling of iceberg material is crucial for ice mechanics, and the main objective of the present work is to propose an isotropic elastic–perfectly plastic material model to simulate the mechanical behavior of ice in a ship-iceberg collision scenario for Accidental Limit State conditions. The ‘Tsai-Wu’ yield surface model and a new empirical failure criterion were used to describe the plastic flow of iceberg material, while a cutting-plane algorithm was adopted to address the plastic stress–strain relationship. The proposed iceberg material model was incorporated into the LS-DYNA finite element code using a user-defined subroutine. Calibration of the proposed material model was conducted through a comparison with an abnormal level ice event pressure-area curve. The calculated pressure-area curve was comparable to that recommended by the International Organization for Standardization (ISO) rule. A sensitivity analysis was then conducted, and the proposed ice model was found to be more sensitive to the mesh size than to other parameters. Numerical simulations of iceberg-tanker side and iceberg-ship bow collisions were also analyzed. Moreover, the impact force and energy dissipation were examined. The results from these simulations showed that the proposed isotropic elastic–perfectly plastic iceberg material model can be employed to simulate iceberg behavior in ship-iceberg collisions under Accidental Limit State conditions.

Effect of Strength Anisotropy on Failure Envelope of Offshore Shallow Foundations under Combined Loading

Wu, K.; Fan, Q., and Zheng, J., 2015. Effect of strength anisotropy on failure envelope of offshore shallow foundations under combined loading.The bearing capacity of offshore shallow foundations subjected to combined vertical, horizontal and moment loading is of interest particularly to the offshore geotechnical engineers. This paper reports a finite element (FE) study on the effects of strength anisotropy and foundation embedment depth on the combined bearing capacity. The Tresca failure criterion modified to allow for soil strength anisotropy and coupled with an elasto-perfectly plastic stress-strain relationship is assumed in the analysis which is conducted using the FE software ABAQUS. The undrained bearing capacities of shallow strip foundations under uniaxial vertical (V), moment (M) and horizontal (H) loadings are predicted and compared with available solutions. The V-M-H capacity of embedded foundations under combined loading is represented by failure envelopes in H-M space for various vertical loading levels. The results presented in this paper show that the embedment ratio significantly affects the shape of the failure envelope expressed in terms of loads normalized by the pure moment or pure horizontal load-carrying capacity of the foundation. However, the shape of failure envelope defining the undrained capacity of shallow foundations under horizontal and moment loads is independent of soil strength anisotropy.

A micromechanics elastic–plastic constitutive model for sintered stainless steel fiber felt

Sintered stainless steel fiber felt (SSSFF) is a type of cellular material promising for functional and structural applications due to high porosity, controllable permeability, and high specific surface area, etc. In this study, a micromechanics elastic–plastic constitutive model was proposed for accurate representation of material behavior of SSSFF. Firstly, microscopic geometric structure was investigated and representative volume element (RVE) was established according to the reasonable simplifications and assumptions. Secondly, fundamental equations of elastic theory including equilibrium equations and geometric equations were built. Principle of virtual work and generalized Hook’s law were applied to obtain the elastic behavior. The incremental theory was adopted to derive the plastic stress–strain relations. Thirdly, the elastic and plastic relations were unified by statistical theory considering the fiber length and fiber orientation distribution. Finally, four SSSFF materials were introduced as numerical examples and the numerical results show good agreements with experimental results. The proposed micromechanics constitutive model, also expected to be applicable to other metal fiber sintered materials, is beneficial to predict the mechanical properties of sintered fiber materials and guide the material design.

Elastic–plastic stresses in rotating connection disk under temperature rise and transmitted torque

A connection disk rotating at constant speed under temperature rise and transmitted torque at the inner and outer clamped edges is modeled using a nonlinear elastic–plastic stress–strain relation, and the elastic–plastic effective stress in Von Mises yield criterion is obtained efficiently by the finite difference method with a closed-form initial predicted value. Effects of rotational speed, edge temperature rises, transmitted torque and their couplings on the elastic–plastic stresses of the connection disk are studied. It is seen that the effective stress increases generally with the edge temperature rises, and the outer edge temperature rise increases the maximum effective stress more remarkably than the inner edge one; the maximum elastic–plastic effective stress occurs at either the inner edge or outer edge.

Role of Plastic Deformation of Binder on Stress Evolution during Charging and Discharging in Lithium-Ion Battery Negative Electrodes

We studied the mechanical damage within a lithium-ion graphite-based porous electrode during electrochemical cycling. The effects of charging–discharging rate and the variation in graphite diffusivity on average stress in the electrode cell were investigated. In particular, differences between spatial and average stress evolution in graphite particles were explored. We considered two different microstructures: a) graphite particles connected together with binder bridges and b) graphite particles encased in binder shells. Electrochemical charging–discharging in a composite electrode was simulated by spatially resolving the electrode and electrolyte phases. As indicated by experimental measurements, the binder is assumed to follow an elastic–plastic stress–strain relation. Average stress developed in the electrode was calculated for different binder yield-stress levels and an appropriate yield-stress value was chosen on the basis of experimental findings of the literature. We find the stress in the particles can be of the order of 43 MPa, and can be particularly large in regions where the particles come in close contact with their neighbors. The average stress in the electrodes, however, is the range of 10 MPa and is largely determined by the mechanical properties, in particular the yield stress of the binder. Computed stress profiles were compared qualitatively with experimental measurements using the wafer-curvature method. Elastic stresses and plastic strains predicted by 3D models are shown to be close to those predicted using simpler 2D models of the microstructure.

Evaluation of Strength and Ductility of Thin-Walled Stiffened Box Sectional Steel Bridge Piers Using Fiber Model

A simplified seismic evaluation method for steel members with thin-walled stiffened box sections is developed. First, a finite element model based on the shell element was carried out and its effectiveness was validated by comparison with the experiment. Twelve numerical cases were then simulated by the shell model and the fiber model, respectively, and their accuracies were discussed. Numerical results showed that the proposed method using the fiber model, employing the bilinear kinematic hardening relation, is of good accuracy. If the maximum strength is taken as the ultimate point, the bilinear stress-strain relation, replacing the yield point by the buckling stress, was recommended. If a point corresponding to 95% of the maximum strength after the peak is regarded as the ultimate point, the elastic-perfectly plastic stress-strain relation was suggested.

Chemo-poroplastic analysis of a borehole drilled in a naturally fractured chemically active formation

A coupled chemo-poroplastic model is developed to investigate the change in ion transfer and its effect on pore pressure and effective stresses in chemically active fractured media. The equivalent permeability tensors are calculated using the boundary element method for an arbitrary oriented fractured system, and used in the chemo-poroplastic model. An explicit modified Euler algorithm with substepping including a yield surface correction scheme is used to integrate the plastic stress–strain relation. Super-convergent patch recovery method is also used to accurately evaluate the time dependent nodal stress tensors from the stress tensors of gauss points.From the results of this study it was revealed that the pore pressure drops around the wellbore due to chemical osmotic back flow. The drop in pore pressure, however, becomes a function of fractures' orientation and density. The state of stresses reach the yield strength of the rock in fractured shale, thus forming a plastic zone around the wellbore. The relaxation of effective tangential stresses in the plastic zone, therefore, has a potential to cause stability problems.

Micromechanical behavior study of forged 7050 aluminum alloy by microindentation

In order to investigate the micromechanical behavior of forged 7050 aluminum alloy with different heat treatment conditions, microindentation experiments of 7050-T7451 and 7050-T7452 were performed with various maximum indentation loads and various loading speeds. A non-destructive instrumental approach was applied to indicate mechanical properties such as Young’s modulus E, microhardness H, initial yield stress σy and strain hardening exponent n using the available experimental indentation P–h curves from the microindentation tests. The experimental results revealed that loading speed exerted little influence on measurement of E and H. E decreased rapidly because of the indentation damage accumulation, and H decreased with the increase of indentation load, which showed strong indentation size effects. Dimensionless functions П1 and П2 were used to describe the plasticity constitutive equations and revealed that the heat treatment conditions influenced the micromechanical properties. H and σy tested from the microindentation of 7050-T7451 were higher than those of 7050-T7452, and microindentation plasticity constitutive equations were obtained by using Hookean elastic and power-law plastic stress–strain relation equations, which were verified by finite element method (FEM) simulation well.

Analysis of Pore Pressure and Stress Distribution around a Wellbore Drilled in Chemically Active Elastoplastic Formations

Drilling in low-permeable reactive shale formations with water-based drilling mud presents significant challenges, particularly in high-pressure and high-temperature environments. In previous studies, several models were proposed to describe the thermodynamic behaviour of shale. Most shale formations under high pressure are expected to undergo plastic deformation. An innovative algorithm including work hardening is proposed in the framework of thermo-chemo-poroelasticity to investigate the effect of plasticity on stresses around the wellbore. For this purpose a finite-element model of coupled thermo-chemo-poro-elastoplasticity is developed. The governing equations are based on the concept of thermodynamics of irreversible processes in discontinuous systems. In order to solve the plastic problem, a single-step backward Euler algorithm containing a yield surface-correction scheme is used to integrate the plastic stress–strain relation. An initial stress method is employed to solve the non-linearity of the plastic equation. In addition, super convergent patch recovery is used to accurately evaluate the time-dependent stress tensor from nodal displacement. The results of this study reveal that thermal and chemical osmosis can significantly affect the fluid flow in low-permeable shale formations. When the salinity of drilling mud is higher than that of pore fluid, fluid is pulled out of the formation by chemical osmotic back flow. Similar results are observed when the temperature of drilling mud is lower than that of the formation fluid. It is found that linear elastic approaches to wellbore stability analysis appear to overestimate the tangential stress around the wellbore and produce more conservative stresses compared to the results of field observation. Therefore, the drilling mud properties obtained from the elastoplastic wellbore stability in shales provide a safer mud weight window and reduce drilling cost.

A normalized damage variable for ductile metals based on toughness performance

The normalized damage function based on elastic modulus is proposed to evaluate damage for ductile metals under a unified criterion. An equivalent half-sine model is employed to illustrate plastic stress–strain relationship. Toughness indicator is introduced to describe the toughness character in metals. An exponential rule is employed to calculate toughness indicator, and the parameter of toughness indicator is formulated by conventional mechanical properties of metals. Repeated loading–unloading tensile tests are conducted on stainless steel and aluminium alloy. The good correlation coefficient and dispersion are presented between the experimental values and predicted values of damage variable (within 12%). The damage function with toughness indicator can be applied to predict the effective elastic modulus.

Elastic–plastic analysis of two-dimensional functionally graded materials under thermal loading

The two-dimensional functionally graded materials, (2D-FGMs) have been recently introduced in order to significantly reduce the thermal stresses in machine elements that subjected to sever thermal loading. To the author’s knowledge no work was found that investigates the elastic–plastic stress analysis for 2D-FGMs. In the current work, a 3D finite element model of 2D-FGM plates made of ZrO 2, 6061-T6 and Ti-6Al-4V with temperature dependent material properties has been proposed to perform such analysis. An elastic plastic stress–strain relation based on the rule of mixture of the 2D-FGM has been introduced in the model. Also, a 3D finite element model of conventional FGM plates, of ZrO 2/Ti-6Al-4V and ZrO 2/6061-T6, with temperature dependent material properties has been proposed for the investigation of these plates too. Then, elastic–plastic stress analysis of the considered four plates (two conventional FGMs and two 2D-FGMs) under the same transient cyclic heating and cooling was carried out. It was found that heat conductivity of the metallic constituents of FGM has great effect on the temperature distributions that resulting from the thermal loads. Minimum temperatures variation and minimum stresses can be obtained using ZrO 2/6061-T6/Ti-6Al-4V 2D-FGM. Also, the results indicate that only ZrO 2/6061-T6/Ti-6Al-4V 2D-FGM can stand with the adopted sever thermal loading without fracture or plastic deformations.

Nonlinear analysis of reinforced concrete beam with/without tension stiffening effect

The aim of this paper is to do materially nonlinear failure analysis of RC beam by using finite element method. In the finite element modeling, two different approaches and different tension stress–strain models with/without tension stiffening effect are used by considering two different mesh sizes. In the first approach, the material matrices of concrete and reinforcement are constructed separately, and then superimposed to obtain the element stiffness matrix. In the second approach, the reinforcement is assumed to be uniformly distributed throughout the beam. So, the beam is modeled as a single composite element with increasing the modulus of elasticity of concrete by considering the reinforcement ratio. For these two approaches, elastic-perfectly plastic stress–strain relationship is used for concrete in compression. For the concrete in tension, a stress–strain relationship with/without tension stiffening is used. It is concluded that the approaches and the models considered in this study can be effectively used in the materially nonlinear analysis of RC beams.

Plastic behaviour of two-dimensional regular hexagonal structures with bilinear and uniaxial strength asymmetry in cellular materials

An elasto-plastic micromechanical model of the two-dimensional regular hexagonal structure was developed. General analytical expressions for the incremental constitutive relations were derived in terms of parameters defining the architecture and material of an internal beam. Non-linearity of the structure was introduced by considering the elastic—linear strain hardening behaviour of each internal beam, in which uniaxial strength asymmetry of the cellular material was accounted for. The plastic stress—strain relationship of the structure under any loading conditions can therefore be analysed by localized beam deformation. The results show that the bending deformation of the internal beam dominates under uniaxial stress loading conditions, however, the axial displacement dominates under the uniaxial strain conditions. The structure will present different behaviours under different loading conditions. The corresponding stresses under the uniaxial strain condition are greater than those under the uniaxial stress condition. The analyses also show that the volume fraction is highly correlated with the elastic constants and yield stresses of the structure. The denser the structure, the higher the moduli and yield stresses.