Plastic Yield Criterion Research Articles

Experimental Recognition of Plastic Domain in Contact Problem Based on Full Field Metrology and Neural Network

Contact usually results in stress concentration which can easily cause the yield of materials and structures. The classic elastic–plastic yield criterion needs to utilize stress or strain field for calculation. However, most advanced full-field measurement methods output the displacement as the original data, and the fitting from displacement to strain will induce error accumulation in applications. In this paper, a plastic domain characterization method is developed that can directly judge the elastic–plastic state of materials based on the full-field displacement and neural network. By establishing and training a three-layer-based neural network, the relationship between the displacement and the elastic/plastic stage of the sampling points is modeled. A physical model is formulated based on the yield criterion and embedded in the layer of the network, which can increase the convergence rate and accuracy. Only the displacements of the contact member are required in this method, which can be easily measured by the optical metrologies. The performances of the developed method are carefully discussed through simulated data and real-world tests. Results show that the method can accurately identify the plastic domain during the tests.

A new incremental variational micro-mechanical model for porous rocks with a pressure-dependent and compression–tension asymmetric plastic solid matrix

In this paper, we shall formulate a new micro-mechanical model for describing both local and macroscopic elastic–plastic behavior of porous rocks with a pressure-sensitive and tension–compression asymmetric solid matrix. For this purpose, a new incremental variational mean field homogenization (IVMFH) method is first established. The novelty lies in the fact that the new formulation allows considering solid matrix obeying a nonlinear plastic yield criterion rather than the linear Drucker–Prager one adopted in the previous work Zhao et al. (2019). To this end, a new optimization procedure is proposed to estimate the effective incremental potential involving a stress-dependent relation between volumetric and deviatoric parts of local plastic strain field of solid matrix. The proposed incremental variational model takes into account the effect of non-uniform distribution of local stress and strain fields in solid matrix by incorporating their first and second moments in the homogenization process. The proposed model is able to provide not only macroscopic but also local mechanical responses of porous rocks. Its efficiency is assessed by a series of comparisons with full-field finite element calculations. Application examples are also presented through comparisons with experimental data for two typical porous rocks: Vosges sandstone and Lixhe chalk.

Approximate plastic yield criteria of geomaterials with pores and grains embedded in a porous matrix

Mineral grains and porosity are two common properties of geomaterials. This study focuses on establishing a macroscopic yield criterion for geomaterials having pores and mineral grains at the mesoscale which is embedded in a porous matrix. At the microscale, the solid phase is pressure-sensitive and obeys to the Drucker–Prager criterion which considers the asymmetric property between tension and compression. Based on a two-step homogenization procedure and the modified secant method, a macroscopic yield criterion is firstly derived which takes into account the effects of microstructure on the macroscopic mechanical behaviors of the studied material, such as the meso-porosity and mineral grains content at the mesoscale, the micro-porosity and the pressure-sensitive of the solid phase at the microscale. However, it overestimates the material strength in compressive loading by comparing with the Fast Fourier Transform (FFT) based numerical results. Then, this criterion is improved by considering the exact solution in pure hydrostatic loading of a porous medium with a Drucker–Prager matrix. By comparing to the numerical results with different microstructures and to experimental data, the new macroscopic yield criterion is estimated and validated. The main features of the studied geomaterial are well captured by this criterion.

Multiscale plasticity of geomaterials predicted via constrained optimization‐based granular micromechanics

AbstractA general framework to derive nonlinear elastic and elastoplastic material models from granular micromechanics is proposed, where a constraint‐based variational structure is introduced to classical grain contact‐based homogenization methods of hyperelasticity. Like the classical hyperelastic methods, reference solutions for closed‐form hyperelastic material models are analytically derived from the grain‐scale contact mechanics. However, unlike prior methods, the proposed homogenization framework defines closed‐form hyperelastoplastic material models that extend multiscale variational methods to granular plasticity. The proposed framework is used to develop novel granular micromechanics‐based macroscopic models for a Mises type solid, Drucker–Prager type plasticity, and grain‐contact cohesive‐debonding with a deviatorically and volumetrically coupled nonlinearly elastic response. Macroscopic plastic parameters and yield criteria are explicitly related to their microscale counterparts, for example, the friction coefficient governing intergranular slip. Numerical examples and comparison to measurements from the literature, including triaxial compaction of concrete, are provided to investigate model predictions and demonstrate calibration to experimental data.

Yield criteria representable by elliptic curves and Weierstrass form

Elliptic curve terminology comes from the close association with elliptic functions, and not because of any physical resemblance to an ellipse. The curves investigated here represent various yield loci in the plane having cubic algebraic relationships between the second and third invariants of the deviatoric stress tensor. A well-known yield condition attributed to Drucker falls into this classification. In addition, the more commonly used Tresca yield condition represents a limiting case of elliptic curves. All yield criteria based on elliptic curves, including the Tresca, can be parameterized in terms of the Weierstrass elliptic &-function. The properties of elliptic curves as they pertain to the formulation of various plastic yield criteria of materials are the topic of this investigation. Various perfectly plastic solutions of mode I crack problems are discussed.

More on Plastic Yield Criteria

The two plastic yield criteria known as MISES and TRESCA were proposed long ago. The Mises criterion is the one usually preferred by the strength of materials community, but tests have shown that Tresca’s criterion is the more accurate. We discuss here the two criteria, and based on those tests we conclude in favor of the Tresca criterion. But the two criteria are not that different, so when accuracy is not important, Mises may serve as a simple approximation to Tresca.

A new anisotropic elasto-plastic-damage model for quasi-brittle materials using strain energy equivalence

An anisotropic elasto-plastic-damage model is developed for quasi-brittle materials within the thermodynamics framework. The anisotropic damage is characterized by second-order damage tensors that are different under tensile and compressive loadings. The hypothesis of strain energy equivalence is adopted with two components, including the hypothesis of elastic strain energy equivalence and the hypothesis of plastic strain energy equivalence. The hypothesis of plastic strain energy equivalence is extended from metals to quasi-brittle materials so as to derive the relations of hardening stresses, which are previously assumed between the undamaged and damaged configurations for quasi-brittle materials. A plastic yield criterion is adopted simultaneously with a damage criterion to account for the coupled elasto-plastic-damage behavior under different loadings. The damage criterion can capture larger axial damage in uniaxial tension and smaller axial damage in uniaxial compression as compared with the lateral damage. The Helmholtz free energy functions are derived for three components: elastic, plastic, and damage. The constitutive laws are derived for three scenarios: the elastoplastic behavior without further damage evolution, the damage behavior without further plasticity evolution, and the coupled elasto-plastic-damage behavior. The anisotropic elasto-plastic-damage model is implemented in ABAQUS as a user material (UMAT) subroutine. The coupled constitutive laws in the continuum domain are implemented in an uncoupled way in the discrete domain with three steps, elastic predictor, plastic corrector, and damage corrector. The UMAT is validated in different conditions, including uniaxial, biaxial, and three-point bending tests. The advantage of considering the plastic free energy in anisotropic damage is illustrated through the strength envelope in biaxial loadings. The numerical results generally agree with experimental results, including the stress-strain curves in uniaxial tests, strength envelope in biaxial tests, and load-deflection curves in three-point bending tests.

Elastic–viscoplastic constitutive theory of dense granular flow and its three-dimensional numerical realization

The problem of the movement of dense granular media is common in industrial processes. Dense granular media cannot only show solid-like properties when stacked but can also flow like a liquid, exhibiting properties of fluids. Simultaneous modeling and description of these two states remain a challenge. In this study, a new constitutive model describing the motion of dense granular media is established. A linear elastic model is used to describe the solid phase. After reaching the plastic yield criterion, a viscoplastic constitutive model based on rheology is used to describe the liquid phase. The transitional relationship between these two models is deduced in detail, and the elastic–viscoplastic constitutive theory that describes the movement of dense granular media is more in line with physical reality. Smoothed particle hydrodynamic method is used to discretely solve the new model, and the relationship between smoothed particles and actual particles is illustrated. A series of basic calculation tests is used to verify the theoretical model and numerical method. Through a comparison with experiments and other numerical results, it is shown that the theoretical model and numerical method are suitable for the analysis of the movement of dense granular media and have important practical value for the preparation and processing of similar materials, three-dimensional printing, and mineral mining.

Numerical Investigation of Reinforcement of Timber Elements in Compression Perpendicular to the Grain using Densified Wood Dowels

In recent years, the construction industry has seen a greater focus on the use of sustainable construction materials to reduce the environmental impact of buildings. Timber is one such material that has seen a revival in its use due to its environmental credentials coupled with advances in the manufacture of engineered wood products and connection technologies. While timber and engineered wood products have a high strength-to-weight ratio suitable for large scale construction, timber is an orthotropic material and demonstrates poorer strength when loaded perpendicular to the grain. As a result, special consideration must be given to the design of areas of support where stress perpendicular to the grain develops in timber structures. This paper describes a study which examines the use of densified wood dowels as a sustainable reinforcement against perpendicular to the grain stresses using experimental and numerical approaches. Glued laminated timber samples were reinforced with 2, 4 and 6 densified wood dowels. The experimental results show significant improvements in load-bearing capacity can be achieved. A full 3-dimensional solid finite element model has been implemented in ABAQUS/Explicit software. The numerical model utilises cohesive zone modelling (CZM) and Hill plastic yield criterion to predict the failure behaviour of specimens utilising densified wood dowel reinforcement. The examined numerical modelling approach has been shown to give good predictions of the performance of the dowel-timber interaction and load-bearing capacity of the composite system. The numerical model has been also used in a parametric study to examine the influence of dowel diameter and dowel length on the failure behaviour. A maximum dowel length-to-diameter ratio is recommended based on the numerical results

A multi-scale model of plasticity and damage for rock-like materials with pores and inclusions

A multi-scale model is developed for modeling of plastic deformation and induced damage in rock-like materials containing pores and mineral inclusions. An analytical macroscopic plastic yield criterion is first determined from three steps of homogenization. This criterion explicitly takes into account respective effects of two populations of pores and mineral inclusions at three different scales. The accuracy of criterion is evaluated by comparisons with numerical results issued from direct simulations on representative volume elements. Then, an induced damage evolution law is defined to describe the onset and growth of micro-cracks inside the solid matrix of materials. The induced damage affects both elastic and strength properties of the solid matrix and its evolution is coupled with the plastic deformation. The performance of the proposed model is first assessed through a series of numerical examples, clearly showing influences of pores and inclusions on macroscopic responses of heterogeneous materials. Finally, the proposed model is applied to describe mechanical behaviors of clayey rocks. Comparisons between numerical results and experimental data are presented.

Ultimate Limit State Equations and Plasticity of Tubular Conical Transitions

AbstractConical transitions have wide applications in wind turbine foundation as well as oil and gas jacket type of structures. The junctions where tubular and cone meet experience a sharp stress rise from shell edge effects. Like all structures experiencing sharp stress rises, fatigue considerations are critical. In addition to fatigue, the existing offshore structural design standards also require ultimate limit state checks. It is known from the lower bound theorem of plasticity limit analysis that the junction local edge effects do not impact the global capacity. Designing for the local junction ultimate limit state contains wide variations among existing design standards. In this paper, the design practices from API RP-2A, NORSOK N-004, and ISO 19902:2020 draft are assessed. They are compared to the shell plastic yield criteria of Hodge and Ilyushin. In addition, this paper provides a semi-analytical plasticity solution to determine junction plastic deformations. The formulation is based on cylindrical shell equations coupled with deformation plasticity theory. It is found that the growth of the junction plasticity zone is limited, which is consistent with the anticipation from the lower bound limit analysis theorem. The observations made in this paper show that the local junction plasticity is a secondary issue compared to other design considerations. Its ultimate limit state design equation can afford to be more lenient if chooses for future standards’ development.

Low-velocity impact of clamped slender rectangular sandwich tubes with metal foam core

The main purpose of this work is to investigate the yield criterion and dynamic response of fully clamped slender rectangular sandwich tubes with metal foam core struck by a low-velocity heavy mass. A plastic yield criterion is proposed for the rectangular sandwich tubes with metal foam core. Using the yield criterion, analytical solutions are derived for the dynamic response of fully clamped slender rectangular sandwich tubes with metal foam core struck by a low-velocity heavy mass, in which the foam core strength effect and the interaction of bending and stretching are considered. Numerical calculations are carried out to predict the dynamic response of the rectangular sandwich tubes under low-velocity impact. The analytical model captures the numerical results reasonably. The effects of foam core strength and wall thickness of tube on the dynamic response are discussed in detail. It is shown that the present analytical model can reasonably predict the dynamic response of the rectangular sandwich tube with metal foam core under the low-velocity impact.

A constitutive model for elastoplastic-damage coupling effect of unidirectional fiber-reinforced polymer matrix composites

In this paper, an elastoplastic-damage coupling constitutive model for unidirectional fiber-reinforced polymer matrix composites (UD FRPs) is presented, which both considers the plastic-hardening and damage-softening processes. Tension-compression asymmetry and shear strength increase due to transverse compression are observed under off-axis tensile/compressive tests of UD E-glass/YPH-200. Therefore, a four-parameter plastic yield criterion considering these two effects is proposed. Applying this model to predict the off-axis tensile/compressive responses of present tests provides good agreement with experimental curves. In addition, we give a novel definition of shear damage variable based on Puck failure theory and discuss it in detail. Further, we develop a four-parameter matrix failure criterion for UD FRPs and exactly predict the off-axis failure strength.

A gradient-extended two-surface damage-plasticity model for large deformations

This work is concerned with the theoretical development, algorithmic implementation and numerical investigation of a novel thermodynamically consistent ‘two-surface’ gradient-extended damage-plasticity model for arbitrarily large deformations. It can be considered the geometrically nonlinear version of a corresponding model for small deformations which was presented recently (Brepols et al., (2017; 2018)). The terminology ‘two-surface’ stems from the fact that damage and plasticity are treated in the model as truly distinct but coupled dissipative mechanisms, meaning that individual damage loading and plastic yield criteria as well as appropriate loading/unloading conditions are used in the formulation, respectively. Such an approach makes the model flexibly adaptable to various situations in which the material under consideration shows either a more brittle- or ductile-like damaging behavior. Nonlinear Armstrong-Frederick kinematic hardening, nonlinear Voce isotropic hardening as well as nonlinear damage hardening are accounted for in the presented formulation that relies exclusively on symmetric internal variables, being beneficial, e.g., from the computational point of view. The model is formulated in the framework of Continuum Damage Mechanics and its gradient-extension is derived on the basis of the micromorphic approach according to Forest (2009, 2016). As such, it allows for the computation of mesh-insensitive results in finite element simulations involving material softening which is practically confirmed by means of several structural example problems. Furthermore, many other interesting aspects are highlighted, as, e.g., the model's implementation into finite elements, the computation of the algorithmically consistent tangent operators necessary to retain a quadratic rate of convergence in a global Newton-Raphson scheme, or a suitable time-integration of the model's evolution equations at the integration point level.

A New 3D Empirical Plastic and Damage Model for Simulating the Failure of Concrete Structure

A new plastic–damage constitutive model based on the combination of damage mechanics and classical plastic theory was developed to simulate the failure of concrete. In order to explain different material behaviors of concrete under tensile and compressive loadings, the plastic yield criterion, the different kinematic hardening rule for tension and compressive and the isotropic flow rule were established in the effective stress space. Meanwhile, two different empirical damage evolution equations were adopted: one for compression and the other for tension. A multi-axial damage influence factor was also introduced to fully describe the anisotropic damage of concrete. Finally, the model response was compared with a wide range of experiment results. The results showed that the model could well describe the nonlinear behavior of concrete in a complex stress state.

The Current Status and Development of Semi-solid Powder Forming (SPF)

Semi-solid powder forming is a promising near-net-shaped forming technology, which has the advantages of powder metallurgy and semi-solid forming, such as fine grains, low forming pressure and short process flow. It was used to prepare wide solidification range alloys and its composites. Until now, there have been many studies on the parameters, microstructure and mechanical properties of this technology, but few on the forming principles. Because deformation, solidification and densification occur simultaneously, the forming mechanism is very complex. The liquid fraction is the key factor influencing the microstructure and mechanical properties. Semi-solid compression of porous materials was carried out to study the deformation mechanism of semi-solid powders. The combination mechanism and densification process for semi-solid powder rolling has been analyzed, and the compaction behavior of powders in the semi-solid state has been studied. Shima porous yield criterion and Doraivelu plastic yield criterion were applied in the simulation of semi-solid powder rolling. Based on the Fourier heat conduction equation and the related parameters of semi-solid powder, the rolling force and relative density were simulated by using the Marc finite element software platform. The simulation results are in agreement with the experimental results. Although some achievements have been made in the theoretical research and numerical simulation, the yield criteria and mathematical models suitable for semi-solid powder forming need to be further established. In addition, further optimization of this technology and its application in commercial applications should be the research direction.

Loading rate effect on the mechanical behavior of brittle longmaxi shale in nanoindentation

Understanding the properties of sedimentary rock shale is critical for wellbore stability and hydraulic fracturing design in unconventional shale gas extraction. The mechanical properties of shale are a function of various environmental factors, such as stress, temperature, saturation and loading rate. Thus, in this study, the loading rate effect on the mechanical behavior of shale using nanoindentation was investigated. Nanoindentation tests were performed at a peak load of 500 mN and loading rates of 5–30 mN/s. In addition, X-ray diffraction (XRD) analysis was performed to identify the composition of the selected shales, and reveal the target sample is brittle. Scanning electron microscopy (SEM) analyses were performed to determine the surface morphology and demonstrate the existence of pile up and sink-in events. The results of the nanoindentation study indicate that the material behavior of rock depends on the loading rate at the nanoscale: with the increase in the loading rate, the contact hardness, Young's modulus, yield stress and indentation shear stress gradually increase. However, final penetration depths, the contact penetration depths, the maximum penetration depths and the contact stiffness decrease with increasing loading rates. Pop-ins were imperceptible at both the low and high loading rates, reflecting that the shale is brittle. Comparison of the pressure hardening coefficient in the elastic–plastic yield criterion model and the strain hardening coefficient in the strain rate sensitivity model indicates that shale is more sensitive to pressure hardening than strain hardening. A discussion regarding the uncertainty of evaluating the mechanical properties via nanoindentation indicates that the contact surface determination and the loading rate control are both critical to obtain a meaningful value for the mechanical parameters.

A micromechanical inspired model for the coupled to damage elasto-plastic behavior of geomaterials under compression

We propose an elasto-plastic model coupled with damage for the behavior of geomaterials in compression. The model is based on the properties, shown in [S. Andrieux, et al., Un modèle de matériau microfissuré pour les bétons et les roches, J. Mécanique Théorique Appliquée 5 (1986) 471?513], of microcracked materials when the microcracks are closed with a friction between their lips. That leads to a macroscopic model coupling damage and plasticity where the plasticity yield criterion is of the Drucker–Prager type with kinematical hardening. Adopting an associative flow rule for the plasticity and a standard energetic criterion for damage, the properties of such a model are illustrated in a triaxial test with a fixed confining pressure.

Analysis of plastic stress singularities of cracks and wedges under the plane stress conditions

The aim of this paper is to analyze the plastic stress singularities of the V-shaped notches and cracks under the plane stress conditions and mixed mode. Based on the formulation of assuming an asymptotic displacement field with respect to the radial coordinate centered at the notch tip, the stress components near the notch tip are expressed as the asymptotic series expansions in term of the radial coordinate by satisfying the plastic yield criterion and the nonlinear stress-strain relation. Then, the series expansions of the stress and displacement components are substituted into the governing differential equations of the plastic theory. Hence, the nonlinear second-order system of ordinary differential equations (ODEs) with the plastic stress singularity orders and the associated eigenfunctions are deduced. The interpolating matrix method is applied to solving the nonlinear eigenvalue problems of the ODEs by an iteration procedure. Thus, the leading plastic stress singularity orders and the associated eigenvectors of the displacement and stress fields in the notch tip region are achieved simultaneously. In addition, the numerical examples are given to demonstrate the accuracy and the effectiveness of the present approach to solve the plastic stress singularities of the plane V-shaped notches and cracks. Furthermore, some mechanical behaviors related to the plastic stress singularities in the notch tip region are investigated.

Effects of inclusions and pores on plastic and viscoplastic deformation of rock-like materials

The aim of this paper is to study effects of inclusion and pores on plastic and viscoplastic deformation of rock-like materials. We shall consider a class of clayey rocks with two separate scales of microstructure. At the mesoscopic scale, the material is constituted by a continuous matrix and embedded mineral inclusion. At the microscopic scale, the continuous matrix is a porous medium composed of a solid phase and spherical pores. Macroscopic deformation behavior of the material is determined by using a two-step homogenization procedure. An analytical plastic yield criterion is used for the porous matrix in the first step. It is assumed that the porous matrix exhibits both instantaneous plastic deformation and delayed viscoplastic deformation, which are described by a unified formulation. The plastic criterion is extended to serve as the viscoplastic loading function. At the mesoscopic scale, we shall investigate influences of inclusion stiffness, shape, orientation and volume fraction on plastic and viscoplastic deformation. Due to the absence of analytical solutions, we shall propose an extended Fast Fourier Transform method (FFT) to solve the nonlinear homogenization problem of the unit cell exhibiting time-dependent plastic deformation. A series of numerical simulations are performed and the obtained results show that the proposed numerical model is able to bring a finer description of complex microstructure effect than most analytical models. Finally, the efficacy of this numerical model is checked through comparisons between numerical results and experimental data in triaxial compression creep and relaxation tests on claystone.