Unified Yield Criterion Research Articles

A flexible yield criterion for strength modeling from biaxial compression to biaxial tension

Accurate strength modeling from equi-biaxial tension (EBT) to equi-biaxial compression (EBC) is critical for the plastic behavior prediction covering the wide-range of stress triaxiality encountered in sheet metal forming. To date, however, few yield criteria are available that can precisely model the initial yield and hardening behavior under six typical stress states between EBC and EBT, simultaneously. Furthermore, there is still a lack of a unified yield criterion for accurate strength modeling across various stress state ranges. To address the issues, a theoretical framework for constructing yield criteria dependent on stress states is provided and a new analytically described isotropic yield criterion is presented in this study. The flexibility in terms of the yield locus and application range is thoroughly explored to make the new yield criterion general. Subsequently, the isotropic yield criterion is extended into an analytically described anisotropic-asymmetric yield criterion. Furthermore, the extended yield criterion is applied to capture the initial yield behavior of DP980, AA5754-O, and AZ31 sheets, and the strain hardening behavior of QP1180 sheets at various stress states ranging from EBC to EBT along different loading directions. The predicted results from the extended criterion agree well with the corresponding experimental findings. The applications demonstrate that the proposed anisotropic-asymmetric yield criterion can effectively model the initial yield and hardening behavior of HCP, BCC, and FCC metal sheets under EBT, EBC, uniaxial tension (UT), plane strain tension (PST), shear (SH), and uniaxial compression (UC) in an analytical way.

Limit state equation and failure pressure prediction model of pipeline with complex loading

Assessing failure pressure is critical in determining pipeline integrity. Current research primarily concerns the buckling performance of pressurized pipelines subjected to a bending load or axial compression force, with some also looking at the failure pressure of corroded pipelines. However, there is currently a lack of limit state models for pressurized pipelines with bending moments and axial forces. In this study, based on the unified yield criterion, we propose a limit state equation for steel pipes under various loads. The most common operating loads on buried pipelines are bending moment, internal pressure, and axial force. The proposed limit state equation for intact pipelines is based on a three-dimensional pipeline stress model with complex load coupling. Using failure data, we investigate the applicability of various yield criteria in assessing the failure pressure of pipelines with complex loads. We show that the evaluation model can be effectively used as a theoretical solution for assessing the failure pressure in such circumstances and for selecting appropriate yield criteria based on load condition differences.

A new model for thermal-mechanical coupled of gradient temperature rolling force based on geometrical unified yield criterion

In order to precisely describe the yield deformation behavior of various materials, a generalized yield criterion known as geometrical unified (GU) yield criterion is firstly established in this paper. Based on this yield criterion, an analytical rolling force model of gradient temperature rolling process was established, and the impact of the temperature field was taken into consideration. The GU yield criterion is a function of shear tension ratio, and the classical yield criteria can be obtained as its special cases. The gradient temperature rolling process of an ultra-heavy plate was analyzed depending on the GU yield criterion, and the corresponding energy functional was established. In the meanwhile, the temperature field expression was developed by taking temperature permeability into account. Finally, the analytical model of the force and energy parameters of the gradient temperature rolling process was constructed by fusing the temperature field with the energy functional. The model can very precisely estimate the rolling force and rolling torque when compared to the measured data. The errors of the rolling force and rolling torque are less than 3.27 %.

A yield curve for bi-material structures and its application in energy absorption analysis of FML beam

The yield curve for bi-material structures is proposed and applied to energy absorption analysis of fully clamped FML beam with CFRP and metal under transverse load. Firstly, fully clamped FML beam tests under transverse loading are performed to acquire the energy absorption response. Then, a unified yield criterion for bi-material structures, that contains metal and fiber reinforced plastic and CFRP has asymmetry properties under tensile and compressive load, is established and discussed. Based on the yield criterion, a theoretical analytical model for the large deformation to predict energy absorption of fully clamped FML beams with shear angle-ply under the transverse load is developed. Finally, comparison of results obtained by theoretical solution and experiments show that the theoretical analysis can predict the energy absorption capacities of FML beams. In addition, effect of significant parameters on yield curves and energy absorption capacity are discussed.

Theoretical analysis on carbon fiber reinforced aluminum laminate under off-center impact

Carbon fiber reinforced aluminum laminate (CARALL) composed of multi-layer CFRP and aluminum has better toughness and impact resistance than composite laminates due to the existence of metal. This work investigates the energy absorption mechanism and creatively proposes a theoretical analysis model to predict the dynamic response of CARALL under off-center impact. Firstly, based on a simplified ideal rigid-plastic material model, a unified yield criterion and its simplified method for hybrid plastic hinges are proposed. Then, the analytical solution for maximum transverse displacement and impact force of CARALL under off-center impact is established. Finally, it is verified by the simulation and discussed in detail. The results show that the established theoretical method can predict the structural response under low-velocity impact.

Novel analytical heat source model for cold rolling based on an energy method and unified yield criterion

Accurate calculation of the heat source in the roll gap during the cold rolling process is the basis for a well-designed coolant system, which is particularly important for reducing roll wear and improving the flatness quality of strip steel. Although many established numerical models have high precision when calculating the work roll temperature, complex modeling processes and large calculation times result in the inability to predict the work roll temperature quickly for the application of automatic control systems. Therefore, the authors developed a new analytical heat source model to calculate the axial temperature distribution of the work roll in the cold rolling process. First, according to the deformation characteristics of strip cold rolling, the deformation zone is divided into a plastic zone and two elastic zones, and the length of each deformation zone is calculated considering the effect of tension. Then, the friction heat generated in the elastic zone is calculated. Second, new exponential velocity and corresponding strain-rate fields satisfying kinematically admissible conditions are proposed to calculate the deformation heat and friction heat generated in the plastic zone. Finally, the work roll temperature prediction model is established by contemplating the heat source of the roll gap, emulsion heat transfer, air cooling, and contact heat transfer with the intermediate roll. By repeatedly optimizing the weighted coefficient d of intermediate principal shear stress on the yield criterion, the maximum error between the calculated results and the actual measured cold roll temperature data was reduced to 3.1%. The effects of the reduction ratio, rolling speed, and resistance to deformation on the deformation heat and friction heat are discussed, and the variation of temperature field of the work roll with time is analyzed quantitatively.

An analytical model for the prediction of rolling force of thin slab

ABSTRACTA rolling force model is established by using the unified yield criterion, in which both the changes of the slab temperature and the radius of the roller are considered. In order to solve the integral problem of the nonlinear specific plastic work rate, the unified yield criterion is used to analyse a simplified velocity field, and the internal deformation power is derived. Then, the shear power and friction power are also calculated based on the velocity field, and the total power is obtained by summarising these derived formulas. Finally, the analytic solution of rolling force in terms of the unified yield criterion is obtained by the variational method. Thereafter, the measured data from one domestic factory is used to check the analytical solution of rolling force, in which the temperature drop from the slab inlet to its outlet is considered. Meanwhile, the roller flattening is also considered. It is shown from the comparison and discussion that: the Mises yield criterion that is a special case of the unified yield criterion is more accurate in predicting the rolling force; the thinner the slab is, the more the temperature drop is; the calculation results by considering the roll flattening are improved.

Optimization solution of vertical rolling force using unified yield criterion

Vertical rolling is an important technique used to control the width of continuous casting slabs in the hot-rolling field. Accurate prediction of vertical rolling force is a core point maintaining rolling-mill equipment. Owing to the limitation of the algorithm in use, the prediction accuracy of most vertical rolling force models based on the energy method can only reach more than 10%. Therefore, it is challenging to optimize the rolling-force model to improve prediction accuracy. An innovative approach for optimizing the calculation of vertical rolling force with a unified yield criterion is presented in this paper. First, the maximal width of a dog-bone region is determined by the slip-line method, and the dog-bone shape is described using a sine-function model. Second, the velocity and corresponding strain-rate fields satisfying kinematically admissible conditions are proposed to calculate the total power of the vertical rolling process. Finally, the analytical solution of the rolling force and the dog-bone-shape model is obtained by repeatedly optimizing the weighted coefficient b of intermediate principal shear stress on the yield criterion. Moreover, the effectiveness of the proposed mechanical model is verified by measured data in the strip hot-rolling field and other models’ results. Results show that the prediction accuracy of the vertical rolling force model can be improved by optimizing the value of b. Then, the impacts of reduction rate, initial thickness, and friction factor on dog-bone shape size and vertical rolling force are discussed.

Lifetime Assessment for Multiaxial High-Cycle Fatigue Using Twin-Shear Unified Yield Criteria

In this paper, a life prediction model associated with maximum principal stress and equivalent shear amplitude based on twin-shear unified yield criterion for multiaxial high-cycle fatigue is proposed. The equivalent shear amplitude is the normalized format of the equivalent shear amplitude based on clusters of yield criteria embodying Tresca and the linearization of Huber-von Mises, extending the application to metallic materials. Simultaneously, the effect of mean stress on multiaxial high-cycle fatigue is considered in the proposed model. As an assessment of the new prediction model, the criterion is compared with experimental data of aluminum alloy LY12CZ and carbon structural steel SM45C published in the relevant literature, which shows that most of the data are located within an error range of less than two times the data and are in good agreement with the experiment. Moreover, the proposed model is also compared with other models, such as McDiarmid, Liu, and Freitas, to validate its competitiveness.

Through-wall yield ductile burst pressure of high-grade steel tube and casing with and without corroded defect

Service environment of tube and casing in deep and ultra-deep well is getting more and more complicated, and the internal pressure applied to tube and casing is higher, so that the high-grade steel tube and casing (HGTC) made of ductile material is widely used. Obviously, the calculation of ductile burst pressure for HGTC is one of the key factors in design and important to safe and efficient development of unconventional oil and gas resources. Here, based on the failure principle of through-wall yield and unified yield criterion, the unified formula of through-wall yield ductile burst pressure (TYBP) for tube and casing under practical service condition which can be used to calculate TYBP of API HGTC and non-API HGTC accurately and reasonably by using true material parameters, has been deduced in this paper with due consideration of the effect of Bauschinger effect (BE), material hardening and intermediate principal stress (IPS) on the TYBP, by which the influence rules of BE, IPS and yield criterion on TYBP of tube and casing have been investigated.（The effect of lattice effect (BE), material hardening and intermediate principal stress (IPS) on TYBP is accurately and reasonably derived from non-API HGTC by using real material parameters. The value of BE, IPS and the TYBP yield criterion of pipe and casing are studied. Eight kinds of classic through-wall yield ductile burst formulas for tube and casing made of new material have been be derived from a series of yield criterion, by which TYBP model of HGTC with and without corrosion defect has been established based on weighted twin shear stress yield criterion. The calculation results of TYBP model presented by this paper are the closest to the full-scale burst experimental data than the other burst models, which can provide an important reference for burst pressure design and estimation of HGTC in deep and ultra-deep well.

An enhanced bounding surface plasticity model for soft rock subjected to drained and undrained condition

A bounding surface model in general stress space is formulated for soft rock subjected to drained and undrained condition. Specifically, a UYC bounding surface is defined based on unified yield criterion by introducing bounding surface plasticity, and it can degenerate to a bounding surface adopting Mohr-Coulomb criterion or Twin-shear criterion; a UYC loading surface that is similar to the UYC bounding surface is defined; a mapping rule in general stress space is given to consider the plastic strain inside the UYC bounding surface; a modified hardening parameter and a modified stress-dilatancy is proposed to capture the residual strength and volumetric strain of soft rock, respectively. The comparison between model prediction and experimental results shows that the constitutive model well predicts the mechanical behaviours of soft rocks subjected to drained and undrained compression condition. Besides, the proposed model overcomes the limitation that the plastic strain of soft rock can’t be well captured by the traditional constitutive models based on unified strength theory.

A Modified Symmetric and Antisymmetric Decomposition-Based Three-Dimensional Numerical Manifold Method for Finite Elastic–Plastic Deformations

There are relatively few studies on large rotation or deformation by means of the three-dimensional (3D) numerical manifold method (NMM). A new modified symmetric and antisymmetric decomposition (MSAD) theory is developed and implemented into the 3D NMM, eliminating the false-volume expansion and false-rotation strain/stress problems. The Jaumann rate is used to measure the material rotation, and the geometric stiffness built on the Jaumann rate is deduced. The incremental formulas of the MSAD-based 3D NMM and a practical guide on the implementation of the MSAD theory are given in detail and exemplified. The new theory and formulas can be applied to analyze both large rotation and large deformation problems. Based on the hypoelasto-plasticity theory and the unified strength theory, the unified yield criterion with associated flow rule is implemented into the MSAD-based 3D NMM. Several typical examples are studied, showing the advantage and potential of the new MSAD theory and the MSAD-based 3D NMM.

Implicit Integration of the Unified Yield Criterion in the Principal Stress Space

AbstractAn implicit numerical integration algorithm is presented for the unified yield criterion, which could encompass most other yield criteria. The modification matrix, which is used to convert ...

Analytical solutions for limit loads of simply supported conical shells under internal pressure with unified yield criterion

In this paper, a two-moment limited-interaction unified yield condition (TLUYC) is proposed for shells. Based on the TLUYC, the plastic limit analysis of simply supported conical shells subjected to internal pressure is carried out. A semi-analytical method is presented to compute the plastic limit solution for conical shell in terms of a piecewise linear yield criterion. According to the lower and upper bound analyses, the obtained solutions are the exact solutions for the yield point stress. The moment fields, stress fields and velocity fields with respect to three special yield conditions are illustrated and compared. The results derived in this paper can be used to predict the limit pressure on a conical shell for a wide range of non-SD (strength differential) materials.

Phase field fracture in elasto-plastic solids: Variational formulation for multi-surface plasticity and effects of plastic yield surfaces and hardening

The phase field modelling has been extended from brittle fracture to ductile fracture by incorporating plasticity. However, the effects of plastic yield functions and hardening on the fracture behaviour have not been examined systematically to date. The phase field fracture coupled with multi-surface plasticity is formulated in the variational framework for the unified yield criterion, which is able to facilitate the study on different yield surfaces. First, the homogeneous solutions of fracture in elasto-plastic solids are derived analytically for 1D and 2D cases. The results show that a greater hardening modulus would lead to an ascending branch of the stress versus strain curve; and the yield function may significantly affect the stress state and phase field damage. Second, the finite element (FE) technique is implemented for modelling the phase field fracture in elasto-plastic solids, in which the stress update and consistent tangent modular matrix are derived for the unified yield criterion. Finally, three numerical examples are presented to explore the effects of the yield function and material hardening. It is found that the yield function and material hardening could significantly affect the crack propagation and the final fracture pattern. In particular, the Tresca yield function tends to create a straight crack path orthogonal to the first principal stress, while the other yield functions show no sizeable difference in their crack paths.

Analytical Solutions for Large Deflections of Functionally Graded Beams Based on Layer-Graded Beam Model

The objective of this paper is to investigate the large deflection of a slender functionally graded beam under the transverse loading. Firstly, by modeling the functionally graded beam as a layered structure with graded yield strength, a unified yield criterion for a functionally graded metallic beam is established. Based on the proposed yielding criteria, analytical solutions (AS) for the large deflections of fully clamped functionally graded beams subjected to transverse loading are formulated. Comparisons between the present solutions with numerical results are made and good agreements are found. The effects of gradient profile and gradient intensity factor on the large deflections of functionally graded beams are discussed in detail. The reliability of the present analytical model is demonstrated, and the larger the gradient variation ratio near the loading surface is, the more accurate the layer-graded beam model will be.

Application of a unified linear yield criterion in plane strain to study the strength theory effect in geotechnical engineering

The plane strain condition is a common, but polyaxial stress state for geotechnical structure designs, in which the selection of an appropriate yield or failure criterion is crucial to reasonably account for the intermediate principal stress. Under plane strain condition, a unified linear yield criterion for seven commonly used geotechnical yield criteria is presented in conjunction with the inductive method. These seven yield criteria considered in this study are the Mohr–Coulomb, Tresca, Drucker–Prager, Mogi–Coulomb, Extended Matsuoka–Nakai, Extended Lade–Duncan criteria, and the Unified Strength Theory. The generalized analytical solutions for earth pressure of retaining walls, critical load of strip foundations as well as stress and displacement of circular tunnels are derived on the basis of the proposed unified yield criterion, and their respective theoretical significance is analyzed. Thereafter, the critical load of strip foundations obtained herein is compared with two numerical results from the literature. Furthermore, the effect of strength theory on result differences of the three typical geotechnical problems by simply selecting constants, which conform to different yield criteria, is explored through a parametric study. It is found that the proposed unified yield criterion is convenient for investigating analytical solutions of the aforementioned geotechnical structures. The strength theory effect due to adopting different yield criteria is considerably significant, which cannot be ignored. Additionally, recommendations are provided on how to make use of these seven yield criteria for an optimum design.

Autofrettage and Shakedown Analyses of an Internally Pressurized Thick-Walled Cylinder Based on Strain Gradient Plasticity Solutions

Two closed-form solutions for an internally pressurized thick-walled cylinder of an elastic linear-hardening material and of an elastic power-law hardening material are first obtained using a strain gradient plasticity theory, a unified yield criterion, and Hencky's deformation theory. The strain gradient plasticity theory contains a microstructure-dependent length-scale parameter and can capture size effects observed at the micron scale. The unified yield criterion includes the intermediate principal stress and recovers the Tresca, von Mises, and twin shear yield criteria as special cases. An autofrettage analysis is then performed by using the two new solutions, which leads to the analytical determination of the elastic and plastic limiting pressures, the residual stress field, and the stress field induced by an operating pressure for each strain-hardening cylinder. This is followed by a shakedown analysis of the autofrettaged thick-walled cylinders, which results in analytical formulas for reverse yielding and elastic reloading shakedown limits. The newly obtained solutions and formulas include their classical plasticity-based counterparts as limiting cases. To quantitatively illustrate the new formulas derived, a parametric study is conducted. The numerical results reveal that the shakedown limit (as the upper bound of the autofrettage pressure) increases with the diameter ratio and with the strain hardening level. It is also found that the Tresca yield criterion gives the lowest value and the twin shear yield criterion leads to the highest value, while the von Mises yield criterion results in the intermediate value of the shakedown limit. In addition, it is observed that the shakedown limit based on the current strain gradient plasticity solutions increases with the decrease of the inner radius when the cylinder inner radius is sufficiently small, but it approaches that (a constant value independent of the inner radius) based on the classical plasticity solution when the inner radius becomes large. This predicted size (strengthening) effect at the micron scale agrees with the general trends observed experimentally.

Modeling of burst pressure for internal pressurized pipe elbow considering the effect of yield to tensile strength ratio

The main objective in this paper is to develop an analytical solution of burst pressure for pipe elbow containing the effect of yield to tensile strength (Y/T) ratio. By non-linear regression analysis, available experimental data between the strain hardening exponent and Y/T ratio were fitted to mathematical models with high values of correlation coefficients. Meanwhile, in order to reflect the effect of different yield criteria on the burst pressure, the unified yield criterion is used and an analytical solution of burst pressure expressed as a function of parameter b, Y/T ratio, curvature impact factor and the ratio of thickness to radius is ultimately derived. Comparison with traditional theoretical results and numerical results shows that the present Mises results considering the effect of Y/T ratio match the best with numerical results. Validation with experimental results is carried out and the present TSS criterion provides the best agreement with experimental ones. Moreover, the effects of parameter b, strain hardening exponent, Y/T ratio, curvature impact factor, as well as the ratio of thickness to radius on the burst pressure are also discussed systematically.

Unified solution of a borehole problem with size effect

Based on the strain gradient theory and the unified yield criterion, the borehole problem of an elasto-plastic plane strain body containing a traction-free hole subjected to uniform far-field stress is studied. The unified expressions for the plastic region radius and the stress concentration factor considering the size effect are derived on the basis of the unified yield criterion, respectively, which can be reduced to those based on the Tresca, von Mises and twin-shear criteria. The dependences of the plastic region radius and the stress concentration on the yield criteria, the material strain hardening level and the size effect are discussed. As a result, it is concluded that the influences of yield criteria and strain hardening level on the plastic region radius and the stress concentration are significant, and the size effect cannot be ignored when the hole size is in the order of tens of microns.