Yield Function Research Articles

Characterization and modeling of biaxial plastic anisotropy in metallic sheets

The anisotropic behavior of cold-rolled sheet metals has been extensively studied, typically characterized by uniaxial loading tests in different directions to determine yield strengths and plastic flow (Lankford r-values). However, biaxial principal stress states often focus solely on yield loci in the RD/TD (rolling/transverse) directions, with limited studies on other loading directions. This study analyzes the relationship between the principal stress yield loci and the material principal anisotropic stress directions based on the Hill48 yield function. Biaxial tensile tests are conducted on DP590 high-strength steel using cruciform specimens cut in various directions different from the sheet RD/TD directions. The evolution of the anisotropic yield locus and plastic flow of DP590 under biaxial tensile directions beyond the traditional RD/TD anisotropic directions is revealed. Furthermore, a modified Hill48 model that considers any principal stress direction of loading is proposed, accurately describing the continuous evolution of equi-biaxial tension stress, near-plane strain stress, and plastic flow direction for different principal stress directions of loading. The issues of failing to predict plastic flow under biaxial loadings in the original Hill48 model, and the degraded representation of existing constitutive models when accounting for out-of-plane anisotropy are addressed by decoupling the relationship between anisotropic parameters and stress state. In addition, the proposed calibration strategy of anisotropic parameters, using experimental data from various principal stress directions of loading, effectively captures the anisotropic evolution caused by changes in principal stress directions. Comprehensive evaluation and verification of the plasticity model should consider yield locus for any principal stress directions of loading, and also the normal and shear planes.

Maize-Soybean Rotation and Intercropping Increase Maize Yield by Influencing the Structure and Function of Rhizosphere Soil Fungal Communities.

Soil-borne diseases are exacerbated by continuous cropping and negatively impact maize health and yields. We conducted a long-term (11-year) field experiment in the black soil region of Northeast China to analyze the effects of different cropping systems on maize yield and rhizosphere soil fungal community structure and function. The experiment included three cropping systems: continuous maize cropping (CMC), maize-soybean rotation (MSR), and maize-soybean intercropping (MSI). MSI and MSR resulted in a 3.30-16.26% lower ear height coefficient and a 7.43-12.37% higher maize yield compared to CMC. The richness and diversity of rhizosphere soil fungi were 7.75-20.26% lower in MSI and MSR than in CMC. The relative abundances of Tausonia and Mortierella were associated with increased maize yield, whereas the relative abundance of Solicoccozyma was associated with decreased maize yield. MSI and MSR had higher proportions of wood saprotrophs and lower proportions of plant pathogens than CMC. Furthermore, our findings indicate that crop rotation is more effective than intercropping for enhancing maize yield and mitigating soil-borne diseases in the black soil zone of Northeast China. This study offers valuable insights for the development of sustainable agroecosystems.

Earing Prediction of AA5754-H111 (Al-Alloy) with Linear Transformation-Based Anisotropic Drucker Yield Function under Non-Associated Flow Rule (Non-AFR).

The yield behavior of aluminum alloy 5754-H111 under different stress conditions for three kinds of plastic work is studied using an anisotropic Drucker model. It is found that when the plastic work is 30 MPa, the anisotropic Drucker model has the most accurate prediction. Comparing the Hill48 and Yld91 models with the Drucker model, the results show that both the anisotropic Drucker and Yld91 models can accurately predict the yield behavior of the alloy. Cylinder drawing finite element analysis is performed under the AFR, but it is not possible to accurately predict the position and height of earing appearance. The anisotropic Drucker model is used to predict the earing behavior under the non-AFR, which can accurately predict the earing phenomenon. Numerical simulation is conducted using three different combinations of yield functions: the anisotropic yield function and the anisotropic plastic potential function (AYAPP), the anisotropic yield function and the isotropic plastic potential function (AYIPP), and the isotropic yield function and the anisotropic plastic potential function (IYAPP). It is concluded that the influence of the plastic potential function on predicting earing behavior is more critical than that of the yield function.

Hop-to-Hug algorithm: Novel strategy to stable cutting-plane algorithm based on convexification of yield functions

Numerical challenges, incorporating non-uniqueness, non-convexity, undefined gradients, and high curvature, of the positive level sets of yield function are encountered in stress integration when utilizing the return-mapping algorithm family. These phenomena are illustrated by an assessment of four typical yield functions: modified spatially mobilized plane criterion, Lade criterion, Bigoni-Piccolroaz criterion, and micromechanics-based upscaled Drucker-Prager criterion. One remedy to these issues, named the "Hop-to-Hug" (H2H) algorithm, is proposed via a convexification enhancement upon the classical cutting-plane algorithm (CPA). The improved robustness of the H2H algorithm is demonstrated through a series of integration tests in one single material point. Furthermore, a constitutive model is implemented with the H2H algorithm into the Abaqus/Standard finite-element platform. Element-level and structure-level analyses are carried out to validate the effectiveness of the H2H algorithm in convergence. All validation analyses manifest that the proposed H2H algorithm can offer enhanced stability over the classical CPA method while maintaining the ease of implementation, in which evaluations of the second-order derivatives of yield function and plastic potential function are circumvented.

Data-driven continuum damage mechanics with built-in physics

Soft materials such as rubbers and soft tissues often undergo large deformations and experience damage degradation that impairs their function. This energy dissipation mechanism can be described in a thermodynamically consistent framework known as continuum damage mechanics. Recently, data-driven methods have been developed to capture complex material behaviors with unmatched accuracy due to the high flexibility of deep learning architectures. Initial efforts focused on hyperelastic materials, and recent advances now offer the ability to satisfy physics constraints such as polyconvexity of the strain energy density function by default. However, modeling inelastic behavior with deep learning architectures and built-in physics has remained challenging. Here we show that neural ordinary differential equations (NODEs), which we used previously to model arbitrary hyperelastic materials with automatic polyconvexity, can be extended to model energy dissipation in a thermodynamically consistent way by introducing an inelastic potential: a monotonic yield function. We demonstrate the inherent flexibility of our network architecture in terms of different damage models proposed in the literature. Our results suggest that our NODEs re-discover the true damage function from synthetic stress-deformation history data. In addition, they can accurately characterize experimental skin and subcutaneous tissue data.

Linking main ecological clusters of soil bacterial–fungal networks and nitrogen cycling genes to crop yields under diverse cropping systems in the North China Plain

BackgroundCrop rotation changes crop species and the associated management strategies, significantly influencing soil fertility and soil microbial communities. Interactions among the species in microbial communities are important for soil nutrient cycling. Yet, the contribution of soil microbial interactions to crop yield and soil nitrogen-cycle function under wheat–maize and wheat–soybean rotation conversion remains unclear. An 8-year field experiment was conducted to investigate the impact of simple [8-year wheat–maize rotation (8WM) and 8-year wheat–soybean rotation (8WS)] and diverse cropping systems [4-year wheat–soybean followed by 4-year wheat–maize rotation (4WS4WM) and 4-year wheat–maize followed by 4-year wheat–soybean rotation (4WM4WS)] on crop yield, soil properties, bacterial–fungal co-occurrence networks and nitrogen functional potentials. The abundances of genes with nitrogen fixation (nifH), nitrification (AOB and nxrA) and denitrification (narG, nirK, norB and nosZ) potentials were quantified and bacterial and fungal communities were characterized.Results4WS4WM led to higher succeeding maize yields and lower bacterial–fungal network complexity, nitrogen fixation potentials and denitrifying potentials than 8WM. Meanwhile, 4WM4WS exhibited higher succeeding wheat and soybean yields, network complexity and lower nitrifying potentials than 8WS. The ecological cluster with the most nitrifying and denitrifying bacterial species (Module#5) and that with the least species (Module#3) dominated the potentials of nitrogen fixation, nitrification and denitrification and succeeding maize yields in 4WS4WM and 8WM. Module#4 with the highest abundances of nitrifying bacteria (Nitrosomonadaceae) and Module#2 with the most species dominated the nitrifying potentials and succeeding wheat and soybean yields in 4WM4WS and 8WS. Soil water content, organic carbon, dissolved organic carbon, NO3− and pH were key drivers influencing Module#3 and Module#5, while only NH4+ significantly affected Module#2 and Module#4.ConclusionsThese findings demonstrate the importance of ecological clusters within soil microbial network in regulating crop yield and soil nitrogen cycling, and identify specific ecological clusters dominating nitrogen functional potentials in wheat–maize and wheat–soybean rotations, offering science-based recommendations for sustainable crop rotation practices.Graphical

Diverse bacterial consortia: key drivers of rhizosoil fertility modulating microbiome functions, plant physiology, nutrition, and soybean grain yield

Soybean cultivation in tropical regions relies on symbioses with nitrogen-fixing Bradyrhizobium and plant growth-promoting bacteria (PGPBs), reducing environmental impacts of N fertilizers and pesticides. We evaluate the effects of soybean inoculation with different bacterial consortia combined with PGPBs or microbial secondary metabolites (MSMs) on rhizosoil chemistry, plant physiology, plant nutrition, grain yield, and rhizosphere microbial functions under field conditions over three growing seasons with four treatments: standard inoculation of Bradyrhizobium japonicum and Bradyrhizobium diazoefficiens consortium (SI); SI plus foliar spraying with Bacillus subtilis (SI + Bs); SI plus foliar spraying with Azospirillum brasilense (SI + Az); and SI plus seed application of MSMs enriched in lipo-chitooligosaccharides extracted from B. diazoefficiens and Rhizobium tropici (SI + MSM). Rhizosphere microbial composition, diversity, and function was assessed by metagenomics. The relationships between rhizosoil chemistry, plant nutrition, grain yield, and the abundance of microbial taxa and functions were determined by generalized joint attribute modeling. The bacterial consortia had the most significant impact on rhizosphere soil fertility, which in turn affected the bacterial community, plant physiology, nutrient availability, and production. Cluster analysis identified microbial groups and functions correlated with shifts in rhizosoil chemistry and plant nutrition. Bacterial consortia positively modulated specific genera and functional pathways involved in biosynthesis of plant secondary metabolites, amino acids, lipopolysaccharides, photosynthesis, bacterial secretion systems, and sulfur metabolism. The effects of the bacterial consortia on the soybean holobiont, particularly the rhizomicrobiome and rhizosoil fertility, highlight the importance of selecting appropriate consortia for desired outcomes. These findings have implications for microbial-based agricultural practices that enhance crop productivity, quality, and sustainability.Graphical

Multi‐material topology optimization considering arbitrary strength and yield criteria constraints with single‐variable interpolation

AbstractMaterial heterogeneity gives composite constructions unique mechanical and physical qualities. Combining multiple materials takes full use of these features in stress‐constrained topology optimization. Traditional research in this field often assumes a consistent yield criterion for all possible materials but adapts their stiffness and strengths accordingly. To cope with this challenge, an innovative single‐variable interpolation approach is proposed to enable the simultaneous inclusion of distinct yield criteria and material strengths. A stress‐constrained topology optimization formulation is presented based on this yield function interpolation method, which can independently support various materials with different elastic characteristics, material strengths, and yield criteria. Then, the large‐scale problem of local stress constraints can be effectively solved by the Augmented Lagrangian (AL) method. Several two‐dimensional (2D) and three‐dimensional (3D) design scenarios are investigated to reduce the overall mass of the structure while considering stress constraints. The optimal composite designs exhibit several crucial benefits resulting from material heterogeneity, including the enlargement of the design possibilities, the dispersion of stress, and the utilization of asymmetry in tension‐compression strength.

A modular, high dynamic range passive neutron dosimeter and imaging diagnostic.

The multi-decade neutron dosimeter and imaging diagnostic (MDND) is a passive diagnostic that utilizes the polyethylene (n, p) nuclear reaction to enhance the diagnostic's sensitivity for time and energy integrated neutron measurements in the range of 2.45-14.1MeV. The MDND utilizes a combination of radiochromic film, phosphor image plates, and solid-state nuclear track detectors, with the goal of providing several orders of magnitude of dynamic range in terms of measured neutron fluence. The diagnostic design was guided by simulations in the Monte Carlo N-Particle (MCNP) transport code to determine the optimum thickness of the polyethylene convertor for maximum proton fluence incident on the detection medium as a function of incident neutron energy. In addition, the simulation results of complete diagnostic assemblies, or "stacks," were used to determine the total dynamic range of an MDND in terms of measured neutron source yield, which was found to be between around 107 and 1015 emitted into 4π with the detector located 1m away from the source. Complimentary to these simulations, individual detectors within a stack were simulated and analyzed to determine response as a function of neutron energy and yield. This work presents the diagnostic design, MCNP simulation results, and analysis of expected signals for varying neutron sources.

Correlating mechanical properties of Al/Al bimetal composite and its constituents using combined experiment and microstructure-based multiscale modeling approach

A combined experiment and multiscale modeling approach was presented to correlate the mechanical properties of a roll-bonded AA1050/AA3004 bimetal clad sheet and constituents. The mechanical responses of the 2-ply AA1050/AA3004 and constituents were measured under various loading conditions. For the measurements, each constituent was separated from the clad sheet using electrical discharge machining and uniaxial tensile tests. Continuum-scale yield functions such as isotropic von Mises and anisotropic/non-quadratic Yld2000-2d were developed for the two constituents and the bimetal clad sheet. A grain-scale crystal plasticity finite element (CPFE) simulations were also performed incorporated with newly developed representative volume element considering measured microstructural attributes. These CPFE predictions were used to identify the anisotropy coefficients for Yld2000-2d. Using the identified yield functions, finite element modeling was followed in which the mechanical properties of each constituent sheet were addressed separately. This is a new approach developed in the current work and it enables a fundamental correlation between multilayered material and its constituents. The mechanical properties of the bimetal clad sheet under the uniaxial tensions were predicted using those of the constituents and compared with calculations based on known rule of mixture (ROM) approach. The two approaches well reproduced stress–strain curves and r–value evolutions. The two FE modeling approaches were also applied to an independent test (i.e., the limiting dome height). The predictions were in good agreement with the experimental data. Finally, the mechanical properties of the bimetal clad sheet and constituents were correlated, and proper modeling scheme for multilayered materials is proposed.

Artificial neural network enhanced plasticity modeling and ductile fracture characterization of grade-1 commercial pure titanium

This study primarily aims to develop a robust modeling approach to capture the complex material behavior of CP-Ti, appeared by high anisotropy, differential hardening due to anisotropy evolution, and flow behavior sensitive to strain rate and temperature, using artificial neural networks (ANNs). Plasticity is characterized by uniaxial tension and in-plane biaxial tension tests at temperatures of 0 °C and 20 °C with strain rates of 0.001 /s and 0.01 /s, and the results are used to calibrate the non-quadratic anisotropic Yld2000–3d yield function with respect to the plastic work. In order to predict the intricate plastic deformation with the temperature and strain rate effects, two distinct ANN models are developed; one to capture the strain hardening behavior and the other to predict the anisotropic parameters in the chosen yield function. The developed ANN models predict an unseen dataset well, which is intermediate testing conditions at a temperature of 10 °C and strain rate of 0.005 /s. The ANN models, being computationally stable and adhering to conventional constitutive equations, are implemented into a user material subroutine for the ductile fracture characterization of CP-Ti sheet using the hybrid experimental-numerical analysis. The favorable agreement between experimental data and numerical predictions, particularly using the ANN models with evolving anisotropic material parameters for the Yld2000–3d yield function, underscores the significance of the differential hardening effect on the ductile fracture behavior and highlights the capabilities of ANN models to capture the complex plastic behavior of CP-Ti. The key parameters including stress triaxiality, Lode angle parameter, and equivalent plastic strain at the fracture location are extracted from the simulations, enabling the calibration of ductile fracture models, namely Johnson-Cook, Hosford-Coulomb, and Lou-2014, and construction of fracture envelopes.

Mechanical mechanism of specimen size effect on impact toughness of Al 6061: Experiment and simulation

Impact toughness has a strong size effect, which restricts the accurate evaluation of material toughness and safety of structural design. However, our understanding of the internal mechanical mechanism that influences the size effect remains incomplete. To fill this gap, this paper adopts the method of combining experiment, theory, and simulation. From the point of view of the dependence of plastic and fracture behavior of material on stress state, the internal mechanical mechanism of the size effect of impact toughness (αk) was studied. Firstly, the Charpy impact tests of different sizes were carried out on Al 6061. αk was divided into cracking initiation toughness (αini) and cracking growth toughness (αgro). The variation of αini and αgro with specimen size was analyzed. Then, through a series of tests, the influence of stress state on the plastic and fracture behavior of Al 6061 was studied. To simulate the whole process of the Charpy impact test, an uncoupled 3D fracture model was proposed. Subsequently, a stress state dependent yield function, modified Voce hardening law, and the proposed fracture model were used to simulate the Charpy impact tests. The predicted results were in good agreement with the tests. Finally, the internal mechanical mechanisms of the size effects of αini and αgro were analyzed. The results show that: (1) The αini and αgro have significant size effects, and there are significant differences between them. (2) The internal mechanical mechanism of the size effect of αini was revealed; that is, the dependence of the plastic behavior of the material on the stress state is the internal root cause of the dependence of αini on the specimen size. (3) The internal mechanical mechanism of the size effect of αgro was revealed; that is, the dependence of the plastic behavior of the material on the stress state and the fracture behavior of the material under ultra-high stress triaxiality is the internal fundamental reason affecting the size effect of αgro. (4) This study reveals the internal complex and intimate relationships between plastic behavior, fracture behavior, toughness behavior, stress state, and the size effect, providing a valuable reference for accurate and comprehensive understanding, evaluation, and selection of metal materials.

Influence of raster orientation on fracture behavior of Ti6Al4V alloy manufactured by Laser powder bed fusion

This study analyzes the fracture characteristics of Ti6Al4V alloy manufacturedby the Laser Powder Bed Fusion (LPBF) technique, specifically investigating the effect of raster orientation. A modified yield function is presented to determine the decrease in load-carrying ability attributed to shear effects. The damage variable includes both the void volume fraction and shear damage variables, taking into consideration the consequences of void interaction. The research utilizes Finite Element Analysis (FEA) together with calibrated modified GTN fracture parameters derived from Hill’s 48 parameters to accurately predict material behavior. Uniaxial tension tests are carried out on samples in different raster orientations:0°, 45°, and 90°. The Scanning Electron Microscope (SEM) is used to conduct a more thorough examination of the fracture’s characteristics. The specimens with a 90° raster orientation exhibit considerable plastic deformation with deepdimples on the fractured surface, along with irregularly shaped voids. In contrast, the sections with 45° and 0° orientations display smaller dimples and smoother cup and cone features without significant plastic deformation. The results demonstrate that the direction of the raster has a considerable impact on the plastic modulus and fracture properties of the material.

A neural network approach for the reliability analysis on failure of shallow foundations on cohesive soils

A collection of feed forward neural networks (FNN) for estimating the limit pressure load and the according displacements at limit state of a footing settlement is presented. The training procedure is through supervised learning with error loss function the mean squared error norm. The input dataset is originated from Monte Carlo simulations for a variety of loadings and stochastic uncertainty of the material of the clayey soil domain. The material yield function is the Modified Cam Clay model. The accuracy of the FNN’s is in terms of relative error no more than 10-5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-5}$$\\end{document} and this applies to all output variables. Furthermore, the epochs of the training of the FNN’s required for construction are found to be small in amount, in the order of magnitude of 90,000, leading to an alleviated data cost and computational expense. The input uncertainty of Karhunen Loeve random field sum appears to provide the most detrimental values for the displacement field of the soil domain. The most unfavorable situation for the displacement field result to limit displacements in the order of magnitude of 0.05 m, that may result to structural collapse if they appear to the founded structure. These series can provide an easy and reliable estimation for the failure of shallow foundation and therefore it can be a useful implement for geotechnical engineering analysis and design.

Establishing the relationship between generalized crystallographic texture and macroscopic yield surfaces using partial input convex neural networks

In this study, we present a methodology to predict the macroscopic yield surface of metals and metallic alloys with general crystallographic textures. In previous work, we have established the use of partially input convex neural networks (pICNN) as macroscopic yield functions of crystal plasticity simulations. However, this work was performed with an over-abundance of data, and on limited crystallographic textures. Here, we extend this study to approach more realistic material states (i.e., complex crystallographic textures), and consider data-availability as a major driver for our approach. We present our modified framework capable of handling generalized material states and demonstrate its effectiveness on samples with multi-modal textures deformed under plane stress conditions. We further describe an adaptive algorithm for the generation of training data as informed by the shape of yield surfaces to reduce the time for both the generation of training data as well as pICNN training. Finally, we will discuss errors in both training and test datasets, limitations, and future extensibility.

Investigation of the influence of plane strain constrained anisotropic plastic flow on the localized necking prediction

PurposeThe purpose of this manuscript is to investigate the influence of the satisfying of plane strain conditions on the prediction of the localized necking and to evaluate this effect by the different hardening models.Design/methodology/approachHigh-order anisotropic Drucker yield function, two types of hardening models, namely Hollomon power and Voce saturated, and the modified maximum force criterion (MMFC) are employed to predict the forming limit strains of AA3104-H19 alloy. Two identification methods, namely conventional and plane strain constrained, are applied and forming limit diagrams of the material are predicted by the incorporation of the anisotropic Drucker criterion and the hardening models into the MMFC for both calibration methods.FindingsThe enhancement in the prediction accuracy of the forming limits provided by the implementation of the plane strain constrained method is strongly dependent upon the selected hardening model type in the MMFC. About 18% improvement in the prediction of the biaxial limit strains is provided via Hollomon power law hardening, whereas the opposite result is occurred in case that the Voce saturated hardening law is used, and the predictions match with the experimental data only in the plane strain point and its surroundings.Originality/valueThe plane strain constrained identification method precisely predicts the location of the plane strain point on the yield locus and also any yield stress or anisotropy coefficient in the plane strain state is not required during the calibration, therefore the method can reduce the number of the required experiments for the constitutive characterization, and it will be attractive for both academy and industry. Additionally, the improvement in the prediction of the biaxial forming limits can be provided by applying of this method along with the appropriate hardening law.

Strict convexity of yield surfaces of some weakly-textured materials

Let Sym0 be the space of traceless symmetric second-order tensors. We say that a polycrystalline elastic–plastic material is weakly-textured if its yield function f:Sym0→R is the sum of a texture-independent isotropic part fiso and an anisotropic part which is linear in the relevant texture coefficients. Let c>0 and S≔f−1(c)⊂Sym0 be the yield surface of the weakly-textured material in question. We present a sufficient condition (*), namely that ∇2f(S) be positive definite for each S∈S, for a smooth yield surface S to be strictly convex in Sym0. We apply this sufficient condition to weakly-textured materials with yield functions that satisfy the following conditions: (i) the yield functions f and fiso are smooth; (ii) ∇2fiso(S) is positive definite for each S in Siso≔fiso−1(c)⊂Sym0. We prove that the yield surface S⊂Sym0 of such weakly-textured material is strictly convex if the texture coefficients in f are sufficiently small. As illustration for practical applications, by appealing to condition (*) we study the strict convexity of the yield surface pertaining to a weakly-textured orthorhombic aggregate of cubic crystallites which has a quadratic yield function of the type proposed by Hill in 1948. Moreover, we show that all 35 samples of cold-rolled and annealed low-carbon steel sheets studied by Stickels and Mould have their quadratic yield functions and corresponding yield surfaces strictly convex.

Numerical study of ship hydrodynamics on ice resistance during ice sheet breaking

For icebreakers to navigate polar routes, the ability to break through ice sheet is extremely important. To quantify this, technology that can estimate the mechanical behaviors of ice and ships is essential. The purpose of this study is to analyze the response of the icebreaker Araon breaking through ice sheet. Drucker-Prager yield function and fracture energy based on continuum damage mechanics were applied to the single prism ice compression simulation. The damage in the single ice initiated and evolved as the yield function and fracture energy were assigned. The icebreaking pattern was analyzed from the cone-ice sheet interaction simulations. Patterns similar to those of previous studies were observed qualitatively. The free decay of Araon was numerically analyzed using the Hydrodynamic plug-in HydroQus that was recently developed by present authors, and very accurate ship motion responses were confirmed. The effect of hydrodynamic forces on ice resistance was analyzed through ice sheet breaking simulations. It was found that radiation forces significantly affect the heave and pitch motions of the icebreaker and increase ice resistance. HydroQus has proven to be able to efficiently and accurately simulate the ship-ice interactions compared to other approaches such as computational fluid dynamics.

Phase-field model for 2D cohesive-frictional shear fracture: An energetic formulation

This paper presents a phase-field model for cohesive-frictional shear fracture. The model is derived based on two energetic principles: the energy conservation law and a variational inequality of virtual work that serves as a stability condition. We show that all the governing equations for frictional fracture can be obtained from the above two principles, including the equilibrium condition, the phase-field evolution law and, most importantly, the yield function and flow rule for the plasticity-like frictional slip. The information of crack direction is naturally included in the flow rule. Theoretical and numerical results show that the proposed model is faithfully consistent with the Mohr–Coulomb theory of frictional failure in terms of the crack nucleation stress and direction. In addition, the presented phase-field model has an explicit frictional cohesive law.

Modelling the Discount Function through the Yield Curve Trajectories of the Parsimonious Continuous De Rezende Framework

This paper obtains the term structure technique to produce yield curves from Nigerian Eurobond by using the De Rezende parametric function. There does not seem to exist any empirical proof that Nigeria has constructed a functional model to construct its discount function from the yield curves. Therefore, the De Rezende parsimoniously parametric function is deeply examined to make this framework applicable in presenting yield curves for Nigerian Euro bond constructed through the associated positive forward curve and from where the discount function is generated. The forward rate should be continuous and at the same time generate a positive yield curve trajectory. The objective of this paper is (i) construct the yield curve trajectories and (ii) to construct the discount function from the yield curve (iii) to construct the in-sample prediction to achieve results in predicting coefficients for longer maturities. The data presented involves the daily closing of the Nigerian Eurobond yield covering January to December 2022, which is fitted to the observed Nigerian Eurobond yield curve to construct the yield function. The ordinary least square is applied to estimate the parameters used in computing the in-sample yield. A test of goodness of fit was conducted showing that the model fits in well to the observed data demonstrated by the model’s high R -square adjusted through the predicted yields after obtaining the decay factors.