Elastic Stress-strain Relations Research Articles

Stress wave effects and their mechanisms on stress-strain curves in the elastic phase of SHPB tests

In Split Hopkinson Pressure Bar (SHPB) tests, the stress wave effects during the elastic phase of the stress-strain curve and their influence mechanisms are crucial. Due to the extremely short duration of the elastic deformation phase, the stress wave effects on the specimen during this stage cannot be ignored. This leads to significant errors in the obtained elastic stress-strain curves. However, the dynamic compressive elastic stress-strain relationship forms the basis for studying the viscoelastic behavior of materials. Accurate determination of elastic yield stress and yield strain is also essential for deriving accurate plastic stress-strain relationships. Quantitative research on the stress wave effects during the elastic compression phase of SHPB tests is fundamental for decoupling and obtaining accurate material elastic curves. This paper conducts a quantitative theoretical analysis of the structural effects caused by stress wave evolution during the elastic compression phase, based on the assumption of plane waves. It studies the deviation characteristics and main factors of the phenomenological engineering stress-strain curves of the specimen compared to the actual material stress-strain curves under different conditions, revealing the influence rules and mechanisms of this deviation. The maximum stress deviation value and its corresponding dimensionless time, as well as the variation trend of the maximum stress deviation value within different fluctuation intervals, are calculated. Additionally, the study investigates the cases where the incident wave is arc-shaped or a combination of arc and linear waves. The findings provide theoretical references for the precise design and accurate data processing of SHPB tests.

Direct measurement of the effective properties of an additively manufactured titanium octet truss unit cell using high energy X-ray diffraction

Additively manufactured lattice structures offer a high specific strength-to-density ratio in comparison to conventional, fully dense parts. The small feature size and intricate geometry of lattice structures make it challenging to directly characterize their mechanical properties. This study presents direct measurement of the elastic mechanical properties of an additively manufactured titanium alloy octet truss unit cell. Strut stress and strain were measured using high-energy X-ray diffraction during quasi-static compressive loading. The crystallographic stress-strain relationship was converted into the stress-strain relationship of the octet truss lattice unit cell using the known symmetry of the struts. The sample studied herein was best fit by an orthorhombic linear elastic stress-strain relationship. This technique shows promise for studying additively manufactured lattice structures at length scales not previously reported.

Physical mechanism of the variation of longitudinal wave velocity with porosity in seafloor sediments

In order to reveal the response mechanism of longitudinal wave velocity and related parameters, through the statistical analysis of the measurement data of seafloor sediments, it is found that there is a minimum value of the change of longitudinal wave velocity with porosity of seafloor sediments. The minimum values of the empirical formula for the change of longitudinal wave velocity with porosity of sediments in different sea areas and their critical porosity values are uniformly analyzed. The calculation shows that the minimum variation range of longitudinal wave velocity with porosity of sediments in different sea areas is between 1468.59 and 1520.22 m/s, and the corresponding critical porosity value changes between 0.69 and 0.88. The analysis shows that the empirical formulas of different scholars belong to the unified mathematical expression derived from acoustic theory. The study shows that the essential reason for the formation of the minimum of longitudinal wave velocity in seafloor sediment is that the coupling of sediment grains affects the propagation of longitudinal wave. It is further revealed that the physical mechanism of the change of longitudinal wave velocity is that the change of contact or suspension state of seafloor sediment grains in seawater affects the elastic stress-strain relations of sediments.

Analysis of soil behavior under freezing and thawing conditions

Understanding of the thermal, hydraulic and mechanical behavior of soils under freezing and thawing is important in solving problems of permafrost engineering and artificial ground freezing. Accurate mathematical descriptions of heat and moisture transport and the associated mechanical deformations is critical for developing such understanding. Phase change and transition during freezing and thawing are characterized by a strong non-linearity due to the rapid variations of thermo-physical properties of the material and the release of the latent heat of solidification. Such processes cannot be easily and accurately modelled via the use of local (differential) formulations [1,2]. A non-local formulation, developed within the framework of peridynamics [3–5] is proposed here for ground freezing and thawing. It expands the domain of applicability of the first work on non-local modelling of heat transfer with water flow in saturated porous media [6] into modelling the thermo-hydro-mechanical behavior of unsaturated soils.
 The mathematical description of unsaturated soil behavior is based on three governing equations of conservations of water mass, energy and linear momentum, which are complemented by constitutive relationships that represent the change of soil properties.
 Using the local mathematical model of unsaturated soils [7,8], a peridynamics mathematical framework is formulated. Following [6], the bond-based peridynamics considers a soil body as a collection of particles, see Fig. 1 for the 1D case. The term `bond' refers to the interactions between two points, located at depths and . The point interacts with all other points within a certain finite region , called the horizon of z.
 The peridynamic form of the water mass conservation is:
 where is the water pressure head; is time; is the volumetric content of a soil component; is the density of a soil component; and is the vertical coordinate. The micro-diffusivity function can be defined following [9]. Here, the subscripts i, l, a and s refer to ice, liquid water, air and soil particles respectively.
 The peridynamic form of the energy conservation is:
 where is temperature; is the latent heat of water solidification; is the average thermal conductivity of soil; is the vector of water flux; is the average heat capacity of unsaturated soils and is the specific heat capacity of liquid water. Here, is a micro-conductivity function and is a micro-velocity function, for more details see [6,10].
 The peridynamic form of the linear momentum conservation is:
 where is the acceleration of the peridynamics particle ; is a pairwise force density vector between particles and that depends on its initial and deformed positions Z and Z’; is the horizon volume associated with the particle ; is the body force on the particle . The elastic stress-strain relation for unsaturated soils can be stated as:
 where is the stress increment tensor; is soil compressive modulus for unsaturated conditions, and is the strain increment tensor due to the elastic deformation of the soil matrix, that can be find from the relation for the total strain increment tensor of unsaturated soils . Here is the strain increment tensor corresponding to the change of soil volume due to the frost heave; and is the strain increment tensor corresponding to the plastic deformations of soils.
 The numerical implementation of the peridynamic formulation is verified by comparing the results of simulations with analytical solutions and the formulation is validated by comparing these results with experimental data under the condition of one-dimensional heat and water transfer obtained by laboratory experiments. In the experiments, soil cylinders were cooled from one surface by a source of negative temperature. As a result, the freezing front propagated along the cylinder. The temperature profile in the column was changing, and water migration was observed. The developed model is shown to predict the results of these experiments well.
 The proposed peridynamic model is the first to combine the flow of liquid water with heat transfer and mechanical deformations in the presence of a change of the water phase state in unsaturated soils. It is demonstrated that for the soils with relatively high initial water content, the developed approach describes effectively the change of liquid water content and predict soil deformation. The set of integral-differential equations are solved by a simple numerical scheme.

Incipience of Plastic Flow in Aluminum with Nanopores: Molecular Dynamics and Machine-Learning-Based Description

Incipience of plastic flow in nanoporous metals under tension is an important point for the development of mechanical models of dynamic (spall) fracture. Here we study axisymmetric deformation with tension of nanoporous aluminum with different shapes and sizes of nanopores by means of molecular dynamics (MD) simulations. Random deformation paths explore a sector of tensile loading in the deformation space. The obtained MD data are used to train an artificial neural network (ANN), which approximates both an elastic stress–strain relationship in the form of tensor equation of state and a nucleation strain distance function. This ANN allows us to describe the elastic stage of deformation and the transition to the plastic flow, while the following plastic deformation and growth of pores are described by means of a kinetic model of plasticity and fracture. The parameters of this plasticity and fracture model are identified by the statistical Bayesian approach, using MD curves as the training data set. The present research uses a machine-learning-based approximation of MD data to propose a possible framework for construction of mechanical models of spall fracture in metals.

Effective Elasticity Tensor of Fiber-Reinforced Orthorhombic Composite Materials with Fiber Distribution Parallel to Plane

An orthogonal composite material Ω with fibers consists of a matrix and orthothombic distribution fibers. In addition to the matrix properties, the fiber properties and the fiber volume fraction, the effective (macroscopic) elastic stress–strain constitutive relation of Ω is related to the fiber direction distribution. Until now, there have been few papers that give an explicit formula of the macroscopic elastic stress–strain constitutive relation of Ω with the effect of the fiber direction distribution. Taking the expanded coefficients of the Fourier series as the fiber direction distribution coefficients, we give a formula of the fiber direction distribution parallel to a plane computed through the fiber directions. By the self-consistent estimates, we derive an explicit formula of the macroscopic elastic stress–strain constitutive relation of Ω with the fiber direction distribution coefficients. Since all tensors are represented in Kelvin notation, the macroscopic elastic stress–strain constitutive relation of Ω can be derived and computed only by matrix manipulations. To check the explicit formula, we use the FEM computation to obtain the macroscopic elastic stress–strain relation of Ω for three examples. The computational results of the explicit formula for the three examples are consistent with those of the FEM simulations.

Machine-Learning-Based Model of Elastic—Plastic Deformation of Copper for Application to Shock Wave Problem

Molecular dynamics (MD) simulations explored the deformation behavior of copper single crystal under various axisymmetric loading paths. The obtained MD dataset was used for the development of a machine-learning-based model of elastic–plastic deformation of copper. Artificial neural networks (ANNs) approximated the elastic stress–strain relation in the form of tensor equation of state, as well as the thresholds of homogeneous nucleation of dislocations, phase transition and the beginning of spall fracture. The plastic part of the MD curves was used to calibrate the dislocation plasticity model by means of the probabilistic Bayesian algorithm. The developed constitutive model of elastic–plastic behavior can be applied to simulate the shock waves in thin copper samples under dynamic impact.

Nonlinear elasticity of prestressed single crystals at high pressure and various elastic moduli

A general nonlinear theory for the elasticity of pre-stressed single crystals is presented. Various types of elastic moduli are defined, their importance is determined, and relationships between them are presented. In particular, B moduli are present in the relationship between the Jaumann objective time derivative of the Cauchy stress and deformation rate and are broadly used in computational algorithms in various finite-element codes. Possible applications to simplified linear solutions for complex nonlinear elasticity problems are outlined and illustrated for a superdislocation. The effect of finite rotations is fully taken into account and analyzed. Different types of the bulk and shear moduli under different constraints are defined and connected to the effective properties of polycrystalline aggregates. Expressions for elastic energy and stress-strain relationships for small distortions with respect to a pre-stressed configuration are derived in detail. Under initial hydrostatic load, general consistency conditions for elastic moduli and compliances are derived that follow from the existence of the generalized tensorial equation of state under hydrostatic loading obtained from single crystal or polycrystal. It is shown that B moduli can be found from the expression for the Gibbs energy. However, higher-order elastic moduli defined from the Gibbs energy do not have any meaning since they do not directly participate in any known equations, like stress-strain relationships and wave propagation equation. The deviatoric projection of B can also be found from the expression for the elastic energy for isochoric small strain increments, and the missing components of B can be found from the consistency conditions. Numerous inconsistencies and errors in the known works are analyzed.

Novel Ring Compression Test Method to Determine the Stress-Strain Relations and Mechanical Properties of Metallic Materials

Although there are methods for testing the stress-strain relation and strength, which are the most fundamental and important properties of metallic materials, their application to small-volume materials and tube components is limited. In this study, based on energy density equivalence, a new dimensionless elastoplastic load-displacement model for compressed metal rings with isotropy and constitutive power law is proposed to describe the relations among the geometric dimensions, Hollomon law parameters, load, and displacement. Furthermore, a novel test method was developed to determine the elastic modulus, stress-strain relation, yield and tensile strength via ring compression test. The universality and accuracy of the method were verified within a wide range of imaginary materials using finite element analysis (FEA), and the results show that the stress-strain curves obtained by this method are consistent with those inputted in the FEA program. Additionally, a series of ring compression tests were performed for seven metallic materials. It was found that the stress-strain curves and mechanical properties predicted by the method agreed with the uniaxial tensile results. With its low material consumption, the ring compression test has the potential to be as an alternative to traditional tensile test when direct tension method is limited.

Intrinsic mechanical properties of hexagonal multiple principal element alloy TiZrHf: An ab initio prediction

The mechanical properties of novel hexagonal close-packed medium entropy alloy TiZrHf have been studied using first-principles method based on special quasi-random structure. The elastic properties and stress-strain relations of unitary Ti, Zr, Hf, binary alloys TiZr, TiHf and ZrHf have also been studied to benchmark the calculation accuracy. The derived elastic constants suggest the mechanical stability of TiZrHf alloy and all three binary alloys. The elastic constants of TiZrHf indicate a weak strengthening effect. Relatively, TiZrHf has larger strength and stiffness, and also exhibits small elastic anisotropy from several criteria. Especially, ideal strength is further studied. The ideal tensile strength (ITS) of TiZrHf along with all unitary and binary materials takes place in the [112¯0] direction. The ITS of TiZrHf is 3.87 GPa and the corresponding critical tensile strain is 0.08. The ideal shear strength (ISS) for all studied materials occurs in the (101¯0) <112¯0> shear system. The obtained ISS τ{10-10}[11-20] for TiZrHf is 2.11 GPa at critical strain of ~0.09. The initial slopes of tensile and shear stress - strain curves correspond well to the tensile Young's modulus and shear modulus computed from elastic constants. Moreover, the resolved shear stress estimated from the ITS is smaller than the ISS τ{10-10}[11-20] for each calculated alloy, indicating that the studied alloys are preference to shear slip before tensile failure. From shear stress-stain relation, the intrinsic shearability and half-width of the dislocation core are also studied for all calculated alloys. Then the inherent mechanism of mechanical properties of TiZrHf is further studied by examination of the detailed bond length distribution and the charge density distribution evolution. At critical point under tensile loading, the simultaneous breakdown of bonds results in a steep drop in tensile stress due to homogeneity. Whereas gradual breakdown of chemical bonds under shear deformation is predicted.

Assessment of Springback Behaviour of 800-1200 MPa Dual-Phase Steel Grades

Springback occurs in sheet metal forming due to elastic strain recovery after removal of process forces respectively after opening of the tool. For this reason, a precise description of springback requires the elastic stress-strain relationship described by the Young’s modulus as well as the internal stress distribution of the part before unloading. In this context, the Bauschinger effect influences the stress state before springback due to premature plastification during load reversal or load path change. As is well known, the stress-strain curve of a material during unloading is non-linear because of additional microplastic strain, which is reflected in a decrease of the Young’s modulus. The aim of this work is to characterize the aforementioned phenomena and their effect on springback for three dual-phase steels namely DH800, DH1000 and DP1200LY. For this purpose, cyclic tensile-compression tests as well as loading and unloading loops within uniaxial tensile tests are performed at different plastic strains. To evaluate the springback behavior of the investigated materials, two different hat-profiles geometries are investigated. By comparing the springback of dual-phase steels on part level, the significance of different material influences with regard to springback is evaluated. The results show that the investigated dual-phase steels exhibit a pronounced Bauschinger effect and a considerable amount of microplastic strain with increasing total strain. However, the comparison between the springback of the hat-profiles and the determined material parameters proves a significant influence of the elastic strain on springback, while microplastic strain and the Bauschinger effect have a minor influence.

Elastic constants obtained analytically from microscopic features for regularly arranged elliptical particle assembly

By the micro-structural mechanics approach, this study establishes the quantitative stress–strain relationships at the elastic stage with respect to the microscopic features of the particles for anisotropic granular materials that are composed of regularly arranged elliptical particles with the same size. Firstly, the elliptical particle assembly is equated with a lattice network described by beam elements attached to the center of particles. Then, the elastic stress–strain relationships, which exactly show the features of orthotropic micropolar continuum, are established through analyzing the triangular and hexagonal cells based on the principle of energy balance. Finally, the analytical expressions of eight independent parameters in stress–strain relationships are obtained as the functions of particle shape, size, and microscopic contact stiffnesses. Further, the relationship between microscopic parameters and macroscopic elastic constants for anisotropic granular materials is proposed. The analytical expressions are verified by comparison between theoretical and discrete element method (DEM) results for elliptical particle assembly, and the influences of microscopic parameters on the macroscopic elastic constants are investigated in detail according to the proposed analytical expressions. Our work provides some useful insights for the microscopic explanation and the influences of microscopic parameters on the macroscopic mechanical behavior of granular materials.

Stress–strain relationship in elastic stage of fractured rock mass

Discontinuity is the main factor affecting the stress–strain relationship of rock mass. Based on the statistical parameters of the fracture network of rock mass and combining them with the strain energy theory from continuum and fracture mechanics, the elastic stress–strain relationship of fractured rock mass is proposed and its elastic modulus is derived. The stress–strain relationship can truly reflect the weakening and anisotropy of the mechanical properties of rock mass. On this basis, experimental and calculated stress–strain curves are compared, both of which exhibited the same slope. Additionally, monitoring data from a tunnel excavation is compared with the numerical simulation results, indicating that the displacement of both is basically consistent. Then, we compare three constitutive models with the statistical mechanics of rock mass (SMRM) model to highlight the advantages of the latter in terms of reliability and adaptability. Finally, based on the results of a bearing plate test of a rock mass at Jindong Bridge, the anisotropy of the elastic modulus and its influencing factors are studied. The results show that the weakening and anisotropy of the elastic modulus of the rock mass are notable, and the modulus changes according to the inter-angle between the stress axis and the normal of the discontinuity, reaching the minimum at an angle of 57.5°. The modulus is positively related to the friction angle and the cohesion of the plane and is inversely correlated to the average radius and density of the discontinuities. When the average radius of the discontinuities exceeds 3 m, the weakening of the elastic modulus of the rock mass tends to be stable. The stress environment also has an effect on the elastic modulus, which rapidly increases with the normal stress at the planes.

Energy Evolution of Rock under Different Stress Paths and Establishment of A Statistical Damage Model

In consideration of the influence of energy dissipation on the rock compression process, the traditional elastic stress-strain relationship was improved by adding the energy release rate. The microintensity of the rock was assumed to satisfy the Weibull distribution. The improved stress-strain relationship was transformed into a triaxial stress-strain model in consideration of the loading damage combined with mesodamage mechanics. The characteristic points of the stress-strain curve and the experimental data were utilized to calculate the distribution parameters of the rock under different conditions. The model curve was compared with the experimental data. Results showed that the stress-strain curve of the rock has peak points in the conventional triaxial compression test, but the radial peak of the rock was not evident in the unloading test. The energy of the rock under the loading path was smaller than that under the unloading path. The model curve exhibited a high degree of fitting with the experimental data, and its correlation coefficient was relatively large (> 0.90), which indicated that the model can clearly represent the actual damage and evolution law of the surrounding rock. Furthermore, the established model can also reflect the rock’s stress-strain relationship, which can serve as a practical guide for relevant applications. Finally, the distribution parameters also reflected the evolution law of rock energy and provided a method for the assessment of rock energy and damage trend.

Novel prediction models for composite elastic modulus of circular recycled aggregate concrete-filled steel tubes

Incorporating recycled aggregate concrete (RAC) into steel tubes provides a value-added and promising solution that can reduce environmental impact of waste concrete and meanwhile increase mechanical performances and durability of RAC material. Composite elastic modulus (CEM) is one of the most essential structural properties in the design and evaluation of recycled aggregate concrete-filled steel tubes (RACFST); however, it is found that available design provisions developed for conventional concrete members fail to predict the CEM of RACFST with good accuracy.This paper proposes a set of novel models to address this both research and engineering gap. A reliable experimental database is first established containing test results of circular RACFST beam and column specimens under axial and/or flexural loading. Based on this work, a mathematical approach, i.e., the grey correlation analysis, is used to evaluate the sensitivity influence of key parameters on the CEM of circular RACFST. It is demonstrated that the replacement ratio of recycled concrete aggregate (RCA) is less decisive on CEM than some other parameters such as the steel tube yield strength and the diameter-to-thickness ratio, but the RCA replacement ratio is still significant and should be considered for better representing the elastic stiffness of circular RACFST. Derivations based on elasticity theory are then presented to obtain an elastic stress-strain relationship and hence the expression of CEM of concentrically-loaded circular RACFST. A comparison between the theoretical results and the experimental ones is made to validate the proposed model. For easy use in routine design, two empirical models in compliance with design codes based on the superposition principle are also provided to estimate respectively the composite elastic compressive stiffness and flexural stiffness of circular RACFST. A good agreement between the predictions using the empirical models and the test data is exhibited.

Growth of coherent BGaN films using BBr3 gas as a boron source in plasma assisted molecular beam epitaxy

Incorporating boron into gallium nitride to make BxGa1-xN solid solutions would create an avenue for extreme alloys due to the fact that wurtzite phase BN has a larger band gap and smaller lattice parameters compared to GaN. In this paper, the authors report the growth of high crystal quality, random alloy BxGa1-xN thin films with x up to 3.04% grown on (0001) Ga-face GaN on sapphire substrates using plasma assisted molecular beam epitaxy and BBr3 gas as a B source. High resolution x-ray diffraction was used to measure both the c plane spacing and the strain state of the films. It was determined that the films were fully coherent to the GaN substrate. Elastic stress-strain relations and Vegard's law were used to calculate the composition. Atom probe tomography was used to confirm that the BxGa1-xN films were random alloys. In addition to demonstrating a growth technique for high crystal quality BxGa1-xN thin films, this paper demonstrated the use of BBr3 as a novel B source in plasma assisted molecular beam epitaxy.

Mechanical, Dynamical and Thermodynamic Properties of Al-3wt%Mg from First Principles

Abstract The mechanical, dynamical and thermodynamic properties of Al-3wt%Mg have been investigated using the first-principles method. The calculated structural parameter is in good agreement with previous works. Results for the elastic modulus, stress-strain relationships, ideal tensile and shear strengths are presented. Al-3wt%Mg is found to have larger moduli and higher strengths than Al, which is consistent with its exploitation in Al precipitate-hardening mechanisms. The partial density of states (PDOS) show that the partly covalent-like bonding through Al p-Mg s hybridization is the origin of excellent mechanical properties of Al-3wt%Mg. The phonon dispersion curves indicate that Al-3wt%Mg is dynamically stable at ambient pressure and 0 K. Furthermore, the Helmholtz free energy ΔF, the entropy S, the constant-volume specific heat CV and the phonon contribution to the internal energy ΔE are predicted using the phonon density of states. We expect that our work can provide useful guidance to help with the performance of Al-3wt%Mg.

Flexural and Shear Behaviors of Reinforced Alkali-Activated Slag Concrete Beams

The material properties of cement-zero concrete using alkali-activators have been studied extensively as the latest response to reduce the CO2 exhaust of the cement industry. However, it is also critical to evaluate the behavior of reinforced concrete beams made of alkali-activated slag (AAS) concrete in terms of flexure and shear to promote the applicability of AAS concrete as structural material. Accordingly, nine types of beam specimens with various ratios of tensile steel and stirrup were fabricated and subject to bending and shear tests. The results show that the flexural and shear behaviors of the reinforced AAS concrete members are practically similar to those made of normal concrete and indicate the applicability of the conventional design code given that the lower density of slag is considered. In addition, a framework using the elastic modulus and stress-strain relation from earlier research is adopted to carry out nonlinear finite element analysis reflecting the material properties of AAS concrete. The numerical results exhibit good agreement with the experimental results and demonstrate the validity of the analytical model.

Finite Element Simulation of GFRP Reinforced Concrete Beam Externally Strengthened With CFRP Plates

The construction technology now has become more and more advanced allowing the development of new technologies or material to replace the previous one and also solved some of the troubles confronted by construction experts. The Glass Fibre Reinforced Polymer (GFRP) composite is an alternative to replace the current usage of steel as it is rust proof and stronger in terms of stiffness compared to steel. Furthermore, GFRP bars have a high strength-to-weight ratio, making them attractive as reinforcement for concrete structures. However, the tensile behavior of GFRP bars is characterized by a linear elastic stress–strain relationship up to failure and, therefore, concrete elements reinforced with GFRP reinforcement exhibit brittle failure without warning. Design codes encourage over-reinforced GFRP design since it is more progressive and leads to a less catastrophic failure with a higher degree of deformability. Moreover, because of GFRP low modulus of elasticity, GFRP reinforced concrete members exhibit larger deflections and wider cracks width than steel reinforced concrete. This aims of this paper is to developed 2D Finite Element (FE) models that can accurately simulate the respond on an improvement in the deflection of GFRP reinforced concrete beam externally strengthened with CFRP plates on the tension part of beam. The prediction of flexural response according to RCCSA software was also discussed. It was observed that the predicted FE results are given similar result with the experimental measured test data. Base on this good agreement, a parametric study was the performed using the validation FE model to investigate the effect of flexural reinforcement ratio and arrangement of the beams strengthened with different regions of CFRP plates.

Modelling of anelastic deformation in dual-phase steel for improved springback simulation

Abstract Classical elasto-plastic models assume linear elastic stress-strain relations for all stresses within the yield surface. Closer examination discloses a nonlinear relation in the elastic domain that is dependent on the prior plastic deformation. The ‘unloading strain’ can be decomposed in a linear elastic contribution and an anelastic contribution that is related to reversible dislocation movement in the crystal lattice. The anelastic contribution in the total recovered strain upon unloading is significant and therefore should be considered in accurate springback predictions. Modelling of this phenomenon with E-modulus degradation is fundamentally incorrect and only gives a fair strain prediction after completely unloading the material. In springback situations, inner fibres of the sheet material are partly unloaded and outer fibres are even reloaded in compression. Therefore, a model is required that includes the amount of plastic pre-loading and the amount of unloading separately. For implementation of the model in a finite element code, it needs to be formulated in the complete 6-dimensional stress space and not only for uniaxial stresses. A model is presented that can be applied for arbitrary strain paths and that is consistent with the main observations in uniaxial loading-unloading-reloading experiments.