Equivalent Plastic Strain Increment Research Articles

Mechanical parameter evolutions and deterioration constitutive model for ductile–brittle failure of surrounding rock in high-stress underground engineering

The deep surrounding rock is usually in the true triaxial stress state, and previous constitutive models based on the understanding of uniaxial and conventional triaxial test results have difficulty characterizing the degradation and fracture process of rock ductile–brittle failure under true triaxial stress state. Therefore, this study conducted a series of true triaxial tests to obtain the understanding of the ductile–brittle behaviour of rock, and then combined the test results and the Mogi–Coulomb strength criterion, and proposed calculation methods for the elastic modulus E, cohesion c and internal friction angle φ and the evolution functions of E, c and φ of rock under true triaxial stresses. With the decreasing of the minimum principal stress σ3 or increasing of the intermediate principal stress σ2, the marble post-peak stress drop rate gradually increases, the ductility gradually weakens, and the brittleness significantly strengthens. The calculation method and evolution function of rock E, c and φ under true triaxial stress were proposed. E decreased at first and then tended to remain stable with the increasing of equivalent plastic strain increment dεp. c and φ slowly increased at first and then rapidly decreased. With a method of parameter degradation rate to realize post-peak stress drop rate to reflect the ductile–brittle characteristics, a new three-dimensional ductile–brittle deterioration mechanical model (3DBDM) was established. The proposed model can accurately characterize the influence of σ2 and σ3 on mechanical parameters, the ductile–brittle behaviour of rock under true triaxial stresses, and the asymmetric failure characteristics of surrounding rock after excavation of deep underground engineering. The proposed model can be reduced to elastic–perfectly plastic, elastic–brittle, cohesion weakening friction strengthening (CWFS), Mohr–Coulomb, and Drucker–Prager models.

Study on welding crack sensitivity assessment method based on thermal elastic plastic finite element analysis

This study presents a new welding crack sensitivity assessment method with the equivalent plastic strain increment as an indicator, and the reliability is verified by comparing the computed results and experimental results of bead-on-plate joint under different welding conditions. Then the proposed method is applied to assess the welding crack sensitivity of multi-pass joints in the bridge rehabilitation engineering. The results show that the equivalent plastic strain increment increases with increasing heat input and constraint degree, and decreases with increasing preheat temperature. The equivalent plastic strain increment can be used as an indicator for assessing welding crack sensitivity assessment of joints. With regard to multi-pass welded joints, the welding crack sensitivity decreases with the increase of number of weld passes in terms of equivalent plastic strain increment. The welding crack sensitivity assessment method proposed in this study is reliable and can provide a practical tool for assessing the performance of welded joints.

拘束効果を考慮した極低サイクル疲労下におけるき裂進展条件クライテリオンの提案

The prediction of fracture behavior under extremely low cycle fatigue due to excessive loading is necessary for the life assessment of structures. ΔJ criterion has been used to assess the life under low cycle fatigue. However, the applicability of this method is not well confirmed under extremely low cycle fatigue. The reason is that fracture toughness is dependent on the loading conditions and geometry due to the constraint effect. In the previous study, the crack propagation criterion was proposed for a 1T-CT specimen of SUS316, focusing on the stress triaxiality and the equivalent plastic strain increment. However, no comparison has been performed for specimens of different shapes or materials, and more detailed verification is required. In this study, we performed generation phase and application phase analyses on 1.5T-CT and MT specimens. The results of both the generation and application phase simulation are in good agreement with the experimental results.

Investigation of Yield Surfaces Evolution for Polycrystalline Aluminum after Pre-Cyclic Loading by Experiment and Crystal Plasticity Simulation.

This study aims at introducing the back stress of anisotropic strain-hardening into the crystal plasticity theory and demonstrating the rationality of this crystal plasticity model to describe the evolution of the subsequent yield surface of polycrystalline aluminum at the mesoscopic scale under complex pre-cyclic loading paths. By using two different scale finite element models, namely a global finite element model (GFEM) as the same size of the thin-walled tube specimen used in the experiments and a 3D cubic polycrystalline aggregate representative volume element (RVE) model, the evolution of the subsequent yield surface for different unloading cases after 30 pre-cycles is further performed by experiments and numerical simulations within a crystal plasticity finite element (CPFE) frame. Results show that the size and shape of the subsequent yield surfaces are extremely sensitive to the chosen offset strain and the pre-cyclic loading direction, which present pronounced anisotropic hardening through a translation and a distortion of the yield surface characterized by the obvious “sharp corner” in the pre-deformation direction and “flat” in the reverse direction by the definition of small offset strain, while the subsequent yield surface exhibits isotropic hardening reflected by the von Mises circle to be distorted into an ellipse by the definition of large offset strain. In addition, the heterogeneous properties of equivalent plastic strain increment are further discussed under different offset strain conditions. Modeling results from this study show that the heterogeneity of plastic deformation decreases as a law of fraction exponential function with the increasing offset strain. The above analysis indicates that anisotropic hardening of the yield surface is correlated with heterogeneous deformation caused by crystal microstructure and crystal slip. The crystal plasticity model based on the above microscopic mechanism can accurately capture the directional hardening features of the yield surface.

Cyclic Response and Constitutive Model Study of G20Mn5QT Cast Steel

A series of cyclic loading tests of G20Mn5QT cast steel are carried out at the strain ratio of − 1. The results of tests show that the G20Mn5QT cast steel has the similar tendency in cyclic softening with different constant strain amplitude. A slight strain softening is observed, and then the stress states keep steady during the remaining cycles. A new damage variable is proposed considering the effects of strength degeneration within two inner variables which are the maximum equivalent plastic strain and the equivalent plastic strain increment. The damage variable provides better insight into the multiaxial cyclic loading and gives a reasonable prediction of failure under the loading history of constant strain amplitude. The proposed damage variable is further introduced in the cyclic constitutive model by the user subroutine interfaces VUMAT by the software ABAQUS. The relationships between the equivalent plastic strain and the damage variable, Young’s modulus and yield strength are finally put forward as it is expedient to get the equivalent plastic strain in the numerical analysis.

A new approach to finite element modelling of cyclic thermomechanical stress-strain responses

Modern finite element based structural analyses of thermomechanically loaded structures require accurate simulations with low computational times. However, increasing the complexity of material models capable of modelling cyclic phenomena of engineering materials usually also increases the computational time. Here we present the implementation of the Prandtl operator approach into a finite element solver as a new material model for the study of the stress-strain response of solids subjected to thermomechanical loading. The main advantage of this model is its high computational speed, due mainly to the implicit consideration of the Masing and memory rules by variable temperatures, either during a single load cycle or during a complex thermomechanical load history. The model enables temperature-dependent elastoplastic stress-strain modelling using the von Mises yield function, associated flow rule and multilinear kinematic hardening. The commonly used elastic predictor–plastic corrector procedure now contains an improvement in the calculation of the equivalent plastic strain increment. This includes modelling of the true stress by the time-efficient temperature-dependent spring-slider model. The second advantage of the approach is a reduced number of material parameters per temperature required by the Ramberg–Osgood-type description of the cyclic curve. These material parameters can be obtained from either uniaxial strain controlled low cycle fatigue tests or uniaxial incremental step tests. The model has been validated on several load cases of both a thermomechanically loaded single finite element under tension-compression and shear loads, and a cantilever beam subjected to bending loads. Comparisons with reference material models show almost identical behaviour of the new and the Besseling model, but with the advantage of having up to 35 percent shorter computation times.

A two parameter elasto-plastic formulation for hardening pressure-dependent materials

A new pressure-dependent yield function is proposed by introducing a plastic Poisson's ratio within the theoretical formulation of the plastic potential. In analogy with other classical models, an equivalent stress and an equivalent plastic strain increment are defined. Then, according to these definitions, the equivalent stress–strain curve is derived and an exponential hardening law is introduced. The advantage of the proposed formulation over alternative approaches relies in explicit closed-form expressions of the flow rules and of the plastic multiplier.

Fracture prediction with a material model based on stress-rate dependency related with non-associated flow rule

In this paper, a novel material model for anisotropic metals is proposed. Fracture prediction using the proposed model is based on stress-rate dependency related with non-associated flow rule in which an arbitrary order function for both the yield and plastic potential functions are used. In addition, an explicit formulation of the equivalent plastic strain increment that is plastic-work-conjugated with the equivalent stress is also presented. Then, fracture prediction analyses are conducted by using the 3D local bifurcation theory.

Crack deviation during trimming of aluminium automotive sheets

Crack propagation and deviation during trimming of aluminium sheet metal have been analysed experimentally and modelled by a finite element analysis (FEA). The experimental results may be summarized as follows. Two crack propagation mechanisms operate in aluminium sheet metal, i.e. either between adjacent voids or by multiple coalescence of micro-cracks. A fracture process-zone may be identified for both mechanisms. Outside this fracture process-zone, no damage is observed. The particular crack propagation mechanism does not influence the crack path. A mechanical model based on a two steps analysis is presented. In the first step, the overall stress strain field prior to cracking is determined by an explicit Arbitrary Lagrangian Eulerian (ALE) formulation based FEA. In the second step, dedicated to crack propagation, no remeshing is used. A cumulative damage model based on the equivalent plastic strain increment is shown to be very efficient. The damage variable is interpreted as bifurcation in the direction of the maximum plastic strain rate. Finally, the applications show the possibility of trimming aluminium sheet metal without sliver generation.

Two Efficient Algorithms of Plastic Integration for Sheet Forming Modeling

A fast method called the “inverse approach” for sheet forming modeling is based on the assumptions of the proportional loading and simplified tool actions. To improve the stress estimation, the pseudo-inverse approach was recently developed: some realistic intermediate configurations are geometrically determined to consider the deformation paths; two new efficient algorithms of plastic integration are proposed to consider the loading history. In the direct scalar algorithm (DSA), an elastic unloading-reloading factor γ is introduced to deal with the bending-unbending effects; the equation in unknown stress vectors is transformed into a scalar equation using the notion of the equivalent stress, thus the plastic multiplier Δλ can be directly obtained without iterative resolution scheme. In the γ-return mapping algorithm, the equivalent plastic strain increment estimated by DSA is taken as the initial solution in Simo’s return mapping algorithm, leading to a stable, efficient, and accurate plastic integration scheme. The numerical experience has shown that these two algorithms give a considerable reduction of CPU time in the plastic integration.

A new iterative method for the elastic-plastic stress analysis by the finite element method.

This report proposes a new iterative method for the elastic-plastic stress analysis by the finite element method (FEM), of which scheme simultaneously satisfies the normality condition of plastic strain increments and two other conditions : the conditions that the stress point must move on the yield surface during plastic deformation and that the plastic strain increments relate to the hardening parameters. The combined hardening rule is used in this FEM program. In order to evaluate the accuracy and the computational time for the proposed new iterative method, we analyze an axisymmetric problem that is a round bar with a U groove loaded in the uni-axial direction. Good accuracy is obtained by using the incremental displacement (load) which produces the equivalent plastic strain increment within 0.01 in each step.

Numerical Evaluation of Stiffening Effect of Tubular Joints

Stiffening effect on the X‐type welded tubular joint is evaluated by a numerical procedure. Tubular joints with three kinds of stiffeners are analyzed by finite element technique using an elasto‐plastic isoparametric thick shell element. Two parameters are introduced to the evaluation of the stiffening effect; they are a mean equivalent plastic strain increment and a plastic work increment per unit volume. The parameters are calculated in the three cases of the stiffening types, and are compared with the experimental results prepared by the writers. It is concluded both of them are effective, and the former is recommendable. The calculation is stood on a static load condition, nevertheless the results can be applied to a fatigue problem because the fatigue resistance of the stiffened joint is in proportion to its static strength. Finite element analysis is also applied to the estimation of an optimum shape of the st.

A Plastic-Strain, Mean-Stress Criterion for Ductile Fracture

We describe a computer model for predicting ductile-fracture initiation and propagation. The model is based on plastic strain. Fracture starts or a crack extends when the integrated product of the equivalent plastic-strain increment and a function of the mean stress exceeds a critical value over a critical length. This critical length is characteristic of the microstructure of the material. The computer fracture model is calibrated by computer simulation of simple and notched round-bar tension tests and a precracked compact tension test. The model is then used to predict fracture initiation and propagation is the standard Charpy V-notch specimen. The computed results are compared with experiments. The model predicts fracture toughness from tests of standard surveillance specimens from nuclear-reactor pressure vessels and can be applied to fracture calculations for these vessels.

Construction of the Equivalent Plastic Strain Increment

Strain hardening plastic deformation of a material possessing a yield locus (f(σij)) which may be written as a homogeneous function of the stress components (σij), and which obeys the classical associated flow rule for metals (e.g. Bishop and Hill, 1951) is considered. The material may be anisotropic and may display plastic dilatation. A method is given for constructing the equivalent plastic strain increment in such a way that the increment of plastic work is always equal to the product of the equivalent plastic strain increment and the equivalent yield stress. Construction of the equivalent plastic strain at a corner in the yield locus is given. The method given here is implied in classical treatments of hardening (e.g. Hill, 1950) but seems not to have been given explicitly heretofore.

A note on construction of the equivalent plastic strain increment

A note on construction of the equivalent plastic strain increment