Plastic Constitutive Relations Research Articles

Evaluation of Accuracy for 2D Elastic–Plastic Analysis by Embedded Force Doublet Model Combined with Automated Delaunay Tessellation

An embedded force doublet (EFD) model is proposed to express the presence of permanent strain in body force method (BFM). BFM is known as a boundary type method for elastic stress analysis based on the principle of superposition. In EFD model, the permanent strain is replaced by distributed force doublets. In an actual elastic–plastic analysis, the plastic region whose shape is not clear a priori, have to be discretized into elements where the magnitude of embedded force doublets is unknown to be determined numerically. In general, the determination process of magnitude of EFD is considerably difficult due to nonlinear nature of yield criterion and plastic constitutive relations. In this study, by introducing the automated Delaunay tessellation scheme for discretizing the prospective plastic region, appreciable reduction in input data was realized. Adding to this, in order to improve the computational efficiency, influence coefficients used for determining the magnitude of EFD are stored in a database. The effectiveness of these two inventions was examined by computing the elastic–plastic problem of an infinite medium with circular hole subjected to uniform internal pressure. The numerical solution was compared with Nadai’s closed form solution and found a good agreement.

A New Yield Function of Isotropic Metals Included One to Six-order Plastic Tensors

The yield function is very important in establishing the plastic constitutive relation and analyzing the plastic deformation. Hence, expanded on the general yield function in its Taylor to six-order plastic tensors, this article gives a new yield function on isotropic metal. Based on independent properties of the hydrostatic pressure for metal yield, such as derivation character, mechanical property invariance of isotropic metal etc ., these plastic tensors are analyzed as traceless, totally symmetric and objective. And the yield function for isotropic metals can be degenerated to the one for identical and different property of tension-compression yield. Finally, by means of the results of Lode test, it is proved that the new yield function is quite suitable for those metal materials having both the identical and different properties of tensioncompression yield. There are many merits of this yield function, such as it includes only 2 parameters for material, i.e. simple form and generality etc . And this yield function will lay a theoretical foundation for analyzing mechanical properties of metal materials.

Plastic limit load of Grade 91 steel pipe containing local wall thinning defect at high temperature

This paper aims to predict the plastic limit load of Grade 91 pipe at high temperature by experimental and numerical analysis methods. The elastic–plastic constitutive relation of P91 considering creep damage at high temperature is proposed and the effectiveness is verified by high-temperature instantaneous tensile test after creep of P91 sample. Then, using the elastic–plastic constitutive relation considering creep damage, the plastic limit loads of P91 pipe containing local wall thinning (LWT) defect at high temperature have been calculated by finite element (FE) method and the accuracy is checked by high-temperature burst experiment after creep of T91 pipe containing LWT. Finally, the fracture analysis and metallographic analysis of Grade 91 sample and pipe containing LWT defect are performed. The results show that the elastic–plastic constitutive relation of P91 considering creep damage proposed by this paper can well describe the elastic–plastic behavior of P91 under creep condition and is useful to calculate the plastic limit load of P91 pipe containing LWT. The failures of P91 tensile samples and T91 pipes containing LWT defect are both plastic fracture. This research would provide an optional method for structure integrity analysis of the pipe containing LWT defect at high temperature.

Evaluation of stress integration algorithms for elastic–plastic constitutive models based on associated and non-associated flow rules

This paper presents in-depth analyses of four stress integration algorithms for finite deformation elastic–plastic constitutive relations. In particular, the four integration schemes were developed for both non-Associated Flow Rule (non-AFR) and Associated Flow Rule (AFR) in the continuum level of plasticity theory. The four integration schemes are (1) fully implicit backward Euler, (2) semi explicit convex cutting plane, (3) fully explicit classical forward Euler and (4) fully explicit forward Euler named Next Increment Corrects Error (NICE-1), which were implemented into the user material subroutines of finite element software ABAQUS. Analysis on numerical accuracy was carried out by using uniaxial tensile simulations at various orientations and time increments. The same analysis was also conducted for uniaxial tension/compression tests to investigate the effect of time increment on the calculated hardening curve. Finally, cylindrical cup deep drawing (manufacturing process which stamps a cylindrical cup) simulations were performed to compare computation time and accuracy for both AFR and non-AFR schemes for the evaluation of more realistic forming application. From the systematic comparative analyses, the following conclusions are made; (1) Caution should be exercised when finding an optimum time increment for explicit integration schemes. (2) The computation time and accuracy for both AFR and non-AFR are not much different when identical integration scheme is used. (3) For the simulation of large scale sheet metal forming applications, the explicit type stress integration algorithm can be a more practical choice considering that it does not deteriorate the computational accuracy and efficiency compared to the fully implicit algorithm.

Effect of crystal orientation on porosity evolution in a creeping single crystal

The effect of crystal orientation on porosity evolution in a face centered cubic (fcc) single crystal subject to tensile creep loading is analyzed. The matrix material is characterized by a rate power law viscous crystal plasticity constitutive relation. Three dimensional finite deformation finite element analyses of initially cubic unit cells containing a single initially spherical void are carried out for three crystal orientations of the main loading direction together with two other cases analyzed to study the effect of secondary orientations. The calculations are carried out for three values of the stress triaxiality and for each stress triaxiality value three values of the Lode parameter are considered. Two of the crystal orientations are significantly anisotropic orientations. Symmetry boundary conditions are imposed consistent with the crystal symmetry; the sides of the unit cell are not constrained to remain straight. For an anisotropic orientation, the Lode parameter can have a significant effect on the evolution of creep strain and porosity even at relatively high values of the stress triaxiality. An overall enhanced creep rate can occur with no void growth or even accompanying void collapse. Also, whether or not a stress concentration greater than the elastic stress concentration develops depends on crystal orientation.

Hydromechanical response of a bedded argillaceous rock formation to excavation and water injection

Opalinus clay is a candidate host formation for the geological disposal of nuclear wastes in Switzerland. The understanding of its long-term mechanical (M) and hydraulic (H) behaviour is an essential requirement for the assessment of its performance as a barrier against radionuclide transport. To study the HM response of Opalinus clay, a microtunnel, 13 m in length and 1 m in diameter, was excavated in that formation at the Mont Terri Underground Research Facility. The rock mass was equipped with sensors to measure the deformation and pore pressure in the rock mass during and after the excavation. A mathematical model that couples the equations of flow and mechanical equilibrium was developed to simulate the HM response of the rock mass. An anisotropic plastic constitutive relationship, based on a microstructure tensor approach, was incorporated in the model. Creep was also considered, as well as the anisotropy of permeability. It is shown that the model satisfactorily predicts the shape and extent of the excavation damage zone (EDZ), deformation, and pore pressure in the rock mass. It is also shown that anisotropy and creep play an important role in the HM response of the rock mass to excavation. The model was further used to simulate water injection tests performed at the test section in the microtunnel. The results show that EDZ, due to its high permeability, is a preferential groundwater flow path along the microtunnel.

비국부 이론을 이용한 입자 강화 복합재 이중후방응력 소성 구성방정식 모델 및 전단밴드 분석

2개의 상으로 구성된 입자 강화 복합재에 대한 균질화와 내부 상태 변수에 대해 2차 미분항이 포함된 비구역적 이론을 적용하여 탄소성 구성 방정식을 제안하였다. 열역학과 소성 포텐셜을 통해 내부 상태 변수에 대한 전개식 또한 본 논문에 포함되었다. 연속체 결함 모델을 이용, 결함 인자에 따른 물성 저하 현상도 감안되었으며 이중 후방응력이 조합된 전개식 또한 제시하였다. 일부 예에 대한 수치해석 결과, 비구역적 변수의 영향이 증가할수록 전단밴드는 감소하나 반면 특정 후방응력 전개가 지배적일수록 소성변형 집중이 증가함이 관찰되었다. 더욱이 두 개의 강소성 상으로 이루어진 복합재의 경우 강성이 높은 게재물의 비중이 증가함에 따라 전단밴드 형성이 용이한 것으로 나타났다. 그 밖에 제어변수들의 변화에 따른 전단밴드 형성에 대한 분석 결과는 Rice 소성 불안정성 분석결과와 잘 일치함 또한 밝혀졌다.

Numerical simulations of the Richtmyer-Meshkov instability in solid-vacuum interfaces using calibrated plasticity laws.

The Richtmyer-Meshkov instability of interfaces separating elastic-plastic materials from vacuum (heavy-light configuration) is studied by means of computational techniques. A fully Eulerian multimaterial algorithm that solves consistently the Euler equations and the time evolution of the deformations in the material is applied to three distinct materials (copper, aluminum, and stainless steel). If a perfectly plastic constitutive relation is considered, an empirical law is computed that relates the long-term perturbation amplitude of the interface, its maximum growth rate, the initial density, and the yield stress of the material. It is shown that this linear relation can be extended to materials that follow more complex plastic behavior which can account for rate dependency, hardening, and thermal softening, and to situations in which small-perturbation theory is no longer valid. In effect, the yield stress computed from measurements of the long-term amplitude and maximum growth rate closely matches the von Mises stress found at the interface of solid materials for a wide range of cases with different initial parameters.

Plastic behavior and constitutive relations of DH-36 steel over a wide spectrum of strain rates and temperatures under tension

To understand the plastic characteristics of DH-36 steel, uniaxial tensile tests have been performed on dog bone samples. The strain rate range is from 0.001 to 3000/s, and the initial specimen temperatures are 293–800K. To obtain the isothermal flow stress at high strain rates, dynamic recovery technique in Hopkinson Tension Bar has been used, and the interrupt and reloading tests have been performed. The value of strain rate sensitivity has been calculated based on the isothermal stresses at different strain rates. Similar to results from compressive tests, the dynamic strain aging has been observed under tension. Microstructure analysis of the samples after interrupt tests has been carried out by scanning electron microscopy (SEM). The results show that: (1) the strain rate sensitivity value is ∼0.0115 in terms of the isothermal flow stress (uncoupled with temperature) at a given strain, corresponding to 0.0045 coupled with temperature; (2) the 3rd dynamic strain aging (DSA) occurs at some relatively constant strain rates within certain temperature region under tension; DSA shifts to higher temperature or even disappears with increasing strain rates. Finally, in depth analysis of the data based on dislocation mechanisms, it leads to a physically based model which has taken into account the 3rd DSA effects. Good agreement between the theoretical prediction and experimental results has been obtained.

Erratum to: “Plane stress state problems for notched bodies whose plastic properties depend on the form of the stress state”

We consider one possible approach to the problem of describing the dependence of material plastic strain characteristics on the stress hydrostatic component arising in many porous, fractured, and other inhomogeneous materials. The plastic strain of the media under study is investigated under the plasticity assumption in the corresponding generalized form with the use of the form parameter of the stress state. The plasticity constitutive relations are stated on the basis of the plastic flow law associated with the accepted plasticity condition. For the conditions of plane stress state in the framework of the material rigid-plastic model, a system of partial differential equations is obtained and conditions for its hyperbolicity are determined. The relations for determining the stress fields and velocity fields in plastic domains are obtained, and their properties are investigated. The problem of tension of a strip with symmetric angular notches is solved, where the stress fields are determined and the continuous displacement rate field is constructed. The problem of uniform symmetric tension of a plane with a circular hole is considered. The stress fields in a strip with symmetric circular notches are examined. A comparison with solutions for plastically incompressible media whose properties are invariant with respect to the form of the stress state is performed.

Multidisciplinary design optimization of vehicle instrument panel based on multi-objective genetic algorithm

Typical multidisciplinary design optimization(MDO) has gradually been proposed to balance performances of lightweight, noise, vibration and harshness(NVH) and safety for instrument panel(IP) structure in the automotive development. Nevertheless, plastic constitutive relation of Polypropylene(PP) under different strain rates, has not been taken into consideration in current reliability-based and collaborative IP MDO design. In this paper, based on tensile test under different strain rates, the constitutive relation of Polypropylene material is studied. Impact simulation tests for head and knee bolster are carried out to meet the regulation of FMVSS 201 and FMVSS 208, respectively. NVH analysis is performed to obtain mainly the natural frequencies and corresponding mode shapes, while the crashworthiness analysis is employed to examine the crash behavior of IP structure. With the consideration of lightweight, NVH, head and knee bolster impact performance, design of experiment(DOE), response surface model(RSM), and collaborative optimization(CO) are applied to realize the determined and reliability-based optimizations, respectively. Furthermore, based on multi-objective genetic algorithm(MOGA), the optimal Pareto sets are completed to solve the multi-objective optimization(MOO) problem. The proposed research ensures the smoothness of Pareto set, enhances the ability of engineers to make a comprehensive decision about multi-objectives and choose the optimal design, and improves the quality and efficiency of MDO.

Void growth versus void collapse in a creeping single crystal

Three dimensional finite deformation finite element analyses of unit cells containing a single initially spherical void are carried out. The materials are characterized by a rate power law viscous crystal plasticity constitutive relation and three values of the power law creep exponent are considered. The evolution of porosity is calculated for a fixed value of tensile nominal stress. The overall true stress components on the unit cell are adjusted so that a fixed value of stress triaxiality and a fixed value of the Lode parameter are maintained. Results are presented for a stress triaxiality value of 1/3 and Lode parameter values of −1, 0 and 1. Two initial void volume fractions are considered: 1% and 10%. For each initial void volume fraction calculations are carried out for various initial void spacings. For sufficiently closely spaced voids, void coalescence occurs while for widely spaced voids void collapse can occur. Whether void coalescence, void collapse or neither occurs mainly depends on the value of the Lode parameter.

Quantifying long-term strength and rock damage properties from plots of shear strain versus volume strain

Laboratory tests on crystalline rock exhibit characteristics of yielding and post-yield dilation that are represented by an elastic–plastic constitutive model. The long-term rock strength was defined by the stress at volume strain reversal. At stresses higher than the long-term strength, damage in the rock increased unabated and failure of the specimen ultimately resulted. Plots of triaxial test data on granite, represented as shear strain versus volume strain, exhibited attributes of elastic–plastic behaviour. The onset of permanent inelastic dilation at the volume strain reversal represented the rock's yield stress, while the slope of the post-yield portion of this plot provided the plastic strain increment. The plastic strain increment was directly related to a material property often referred to as the dilation angle. Laboratory testing indicated that the dilation angle was independent of time and stress history and was only dependent on confining stress. Data from a few laboratory tests were sufficient to provide all the input parameters required for an elastic–plastic constitutive model using off-the-shelf analysis software. The extent of rock damage around simulated excavations was quantified and contoured using an elastic–plastic constitutive relationship with properties derived from shear strain versus volume strain plots.

Molecular dynamics simulation based size and rate dependent constitutive model of polystyrene thin films

Rate and size-dependent properties dominate the mechanical behavior of polymeric thin films used in a variety of applications related to generation, transmission and storage of energy. In this paper, molecular dynamics (MD) simulations of polystyrene thin films are used to develop a size and rate dependent finite deformation elastic–plastic constitutive relations. Different modes of deformation are considered in the MD model, and results of simulations are homogenized to yield parametric representations of the constitutive model. The hyperelastic behavior is represented by a modified Ogden type model for films of different thicknesses at different strain rates. The MD simulations suggesting a rate-dependent yield strength is developed for a perfect plasticity model with mitigated size effects. The softer material properties of thin films are found to be the consequence of a large surface layer, which has a low density with high local chain mobility.

Porosity evolution in a creeping single crystal

Experimental observations on tensile specimens in Srivastava et al (2012 in preparation) indicated that the growth of initially present processing induced voids in a nickel-based single crystal superalloy played a significant role in limiting creep life. Also, creep tests on single crystal superalloy specimens typically show greater creep strain rates and/or reduced creep life for thinner specimens than predicted by current theories. In order to quantify the role of void growth in single crystals in creep loading, we have carried out three-dimensional finite deformation finite element analyses of unit cells containing a single initially spherical void. The materials are characterized by a rate-dependent crystal plasticity constitutive relation accounting for primary and secondary creep. Two types of imposed loading are considered: an applied true stress (force/unit current area) that is time independent; and an applied nominal stress (force/unit initial area) that is time independent. Isothermal conditions are assumed. The evolution of porosity is calculated for various values of stress triaxiality and of the Lode parameter. The evolution of porosity with time is sensitive to whether constant true stress or constant nominal stress loading is applied. However, the evolution of porosity with the overall unit cell strain is insensitive to the mode of loading. At high values of stress triaxiality, the response is essentially independent of the value of the Lode parameter. At sufficiently low values of the stress triaxiality, the porosity evolution depends on the value of the Lode parameter and void collapse can occur. Also, rather large stress concentrations can develop which could play a role in the observed thickness dependence.

Use of the Richtmyer-Meshkov Instability to Infer Yield Stress at High-Energy Densities

We use the Richtmyer-Meshkov instability (RMI) at a metal-gas interface to infer the metal's yield stress (Y) under shock loading and release. We first model how Y stabilizes the RMI using hydrodynamics simulations with a perfectly plastic constitutive relation for copper (Cu). The model is then tested with molecular dynamics (MD) of crystalline Cu by comparing the inferred Y from RMI simulations with direct stress-strain calculations, both with MD at the same conditions. Finally, new RMI experiments with solid Cu validate our simulation-based model and infer Y~0.47 GPa for a 36 GPa shock.

Plastic Instabilities in Spherical Vessels for Static and Dynamic Loading

Significant changes were made in design limits for pressurized vessels in the 2007 version of the ASME code (Sec. VIII, Div. 3) and 2008 and 2009 Addenda, and these are now a part of the 2010 code. There is now a local damage-mechanics based strain-exhaustion limit, including the well-known global plastic collapse limit. Moreover, Code Case 2564 (Sec. VIII, Div. 3) has recently been approved to address impulsively loaded vessels. It is the purpose of this paper to investigate the plastic collapse limit as it applies to dynamically loaded spherical vessels. Plastic instabilities that could potentially develop in spherical shells under symmetric loading conditions are examined for a variety of plastic constitutive relations. First, literature survey of both static and dynamic instabilities associated with spherical shells is presented. Then, a general plastic instability condition for spherical shells subjected to displacement-controlled and short-duration dynamic pressure loading is given. This instability condition is evaluated for six plastic and viscoplastic constitutive relations. The role of strain rate sensitivity on the instability point is investigated. Conclusions of this work are that there are two fundamental types of instabilities associated with failure of spherical shells. In the case of impulsively loaded vessels, where the pulse duration is short compared with the fundamental period of the structure, one instability type is found not to occur in the absence of static internal pressure. Moreover, it is found that the specific role of strain rate sensitivity on the instability strain depends on the form of the constitutive relation assumed.

Stress-Strain Behavior and California Bearing Ratio of Artificially Cemented Sand

AbstractIn this study, effects of cement content and curing time on the compressive stress-strain relationship and California bearing ratio (CBR) value of artificially cemented sand cured up to 7 days was investigated. The CBR study was focused on investigating the sensitivity of this non-destructive test to measure the changes in the compacted cemented sand properties during construction (limited curing time). The cement content in the cemented sand was varied up to 6 % by weight. The strength, modulus, and unit weight of the artificially cemented sand varied from, 60 to 1250 kPa, 14 to 290 MPa, and 15.98 to 18.33 kN/m3, respectively. The CBR values for cemented sand, compacted using the standard proctor method, varied from 8 to 230 %. Compressive stress-strain relationship of cemented sand was represented by a non-linear relationship. Relationship between compressive properties of cemented sand and the CBR was also investigated. The variation of compressive strength, modulus, and CBR values with curing time were represented using hyperbolic relationships. Finite element method (FEM) was used to model the CBR test, based on the data obtained from the unconfined compression tests for 1.5, 3, and 6 % cemented sand. In the FEM analyses the cemented sand was modeled using linear elastic-perfectly plastic constitutive relationship with Mohr-Coulomb failure criteria. The ratio of predicted to measured CBR values varied from 0.67 to 1.31.

Prediction of Forming Limit of Sheet Metal Based on Stören-Rice Model

On the basis of Storen-Rice model, the limit strain formula is derived by introducing the plasticity constitutive relation coupling Logan-Hosford yield function. Limit strain of AL6111-T4 and AL2028 is calculated and compared with experimental data. The effect of index a of yield function, anisotropy parameter r on the calculation result is discussed. Index a has a significant effect on the right hand side(RHS) limit strain, while it has less relation with left hand side(LHS) limit strain. The RHS limit strain is smaller while a-value gets larger. Good correlation between the calculation result and experimental data is indicated when a-value equals to 4. The RHS limit strain is smaller while r gets larger when a equals to 2, while it has less relation with r when a equals to 4 or 6.

Simulated microstructure-sensitive extreme value probabilities for high cycle fatigue of duplex Ti–6Al–4V

A newly developed microstructure-sensitive extreme value probabilistic framework to characterize the performance/variability for damage evolution processes is exercised to compare the driving forces for fatigue crack formation (nucleation and early growth) at room temperature for four different microstructure variants of a duplex Ti–6Al–4V alloy. The aforementioned probabilistic framework links certain extreme value fatigue response parameters with microstructure attributes at fatigue critical sites through the use of marked correlation functions. By applying this framework to study the driving forces for fatigue crack formation in these microstructure variants of Ti–6Al–4V, these microstructures can be ranked in terms of relative high cycle fatigue (HCF) performance and the correlated microstructure attributes that have the most influence on the predicted fatigue response can be identified. Nonlocal fatigue indicator parameters (FIPs) based on the cyclic plastic strain averaged over domains on the length scale of the microstructure attributes (e.g., grains, phases) are used to estimate the driving force(s) for fatigue crack formation at the grain scale. By simulating multiple statistical volume elements (SVEs) using crystal plasticity constitutive relations, extreme value distributions of the predicted driving forces for fatigue crack formation are estimated using these FIPs. This strategy of using multiple SVEs contrasts with simulation based on a single representative volume element (RVE), which is often untenably large when considering extreme value responses. The simulations demonstrate that microstructures with smaller relative primary α grain sizes and lower volume fractions of the primary α grains tend to exhibit less variability and smaller magnitudes of the driving forces for fatigue crack formation. The extreme value FIPs are predicted to most likely occur at clusters of primary α grains oriented for easy basal slip. Additionally, surrounding grains/phases with soft orientation shed load to less favorably oriented primary α grains, producing extreme value FIPs.