Incremental Stress-strain Relations Research Articles

A fractional-order two-surface plasticity model for over-consolidated clays and its application to deep gallery excavation

This paper proposes a novel fractional-order two-surface constitutive model for over-consolidated clays within the framework of the fractional plasticity and bounding surface theories. It incorporates the stress-fractional operator of the loading surface into isotropic and progressive hardening rules, thus obtaining an incremental stress–strain relation without an explicit plastic flow rule. The proposed fractional-order model can replicate the main features of over-consolidated clays well, including stiffness non-linearity, the occurrence of plastic strains before high peak strength, non-associativity, weak/strong dilatancy, and cyclic behavior. In model simulations on a set of element tests along different loading paths on natural Boom clay and remolded Fujinomori clay, the proposed model reveals better agreement than the modified Cam-Clay model (MCC). Next, the fractional-order model, ACC-2 model and MCC model are implemented via a user-defined material subroutine in ABAQUS with an explicit integration scheme based on the forward modified Euler method with automatic sub-stepping and error control. The different models are then applied to the fully coupled two-dimensional hydromechanical modeling of the excavation of two deep underground galleries in natural Boom clay. The results ultimately verify the validity and applicability of the proposed model.

Thermodynamically Consistent Models of Elastic-Plastic Materials

A three-dimensional thermodynamic setting for the modelling of elastic-plastic materials is established. The second law of thermodynamics is assumed in the Clausius-Duhem form with the entropy production being given by a constitutive function. An incremental stress strain relation is derived. In essence the free energy is found to describe the elastic behaviour while the hysteretic properties are a joint consequence of the entropy production and the free energy.

Towards data-driven constitutive modelling for granular materials via micromechanics-informed deep learning

The analytical description of path-dependent elastic-plastic responses of a granular system is highly complicated because of continuously evolving microstructures and strain localisation within the system undergoing deformation. This study offers an alternative to the current analytical paradigm by developing micromechanics-informed machine-learning based constitutive modelling approaches for granular materials. A set of critical variables associated with the constitutive behaviour of granular materials are identified through an incremental stress-strain relationship analysis. Depending on the strategy to exploit the priori micromechanical knowledge, three different training strategies are explored. The first model uses only the measurable external variables to make stress predictions; the second model utilises a directed graph to link all the external strain sequences and internal microstructural evolution variables into a single prediction model comprised of a series of sub-mappings, and the third model explicitly integrates the physically important non-temporal properties with external strain paths into training through an enhanced Gated Recurrent Unit (GRU). These three models show satisfactory agreement with unseen test specimens based on multi-directional loading cases. The basic features and potential applications of each model are explained. Furthermore, the key factors for constitutive training and limitations of the current work are also discussed in detail.

Analysis of the nonlinearity of coefficient of earth pressure at rest and its calculation method for coarse-grained soils

The acquisition of a precise coefficient of earth pressure at rest（K0） is very important for earth pressure calculation, and the study of nonlinear characteristics of K0 has great practical significance. Nevertheless, the research on nonlinearity of K0 for coarse-grained soils has long been neglected. To fill the gap, this study presents a detailed analysis of the nonlinearity characteristics of K0 for coarse-grained soils. Based on the incremental stress-strain relation of Duncan-Chang model, we establish the mathematical formula of K0, and the sensitivity analysis of the K0 formula is presented. The proposed method is applied to predict the K0 of coarse-grained soil experimental data reported by others and by DEM method. Comparison results show that the calculated K0-value is very close to the experimental K0-value, particularly for coarse-grained soils. Furthermore, we compared the K0-value from Jaky formula and that from the proposed method, and discussed the influence of initial stress condition on K0 for coarse-grained soils. The research findings of this study are helpful for the understanding of nonlinearity and the precision of K0 in coarse-grained soils, and may provide reference for related earth pressure calculations.

A simple elastoplastic dynamic constitutive model of saturated structural soft clay

The dynamic constitutive relation of soil is widely applied to the ground design of marine structure and infrastructure in offshore geotechnical engineering. Increasing evidence from marine geotechnical design has shown that the traditional dynamic constitutive model has uncertain theories, many parameters, and complexity. Therefore, it is urgent to overcome these limitations and develop a simple and practical elastoplastic dynamic constitutive relation for the saturated structural soft clay. A simple dynamic constitutive relation with 5 parameters was proposed in this study, which can reflect the strain accumulation, hysteretic characteristics and cyclic degradation of the stress-strain for structural soft clay. First, the Drucker-Prager model was selected as the bounding surface, hardening modulus equation was established by the hardening law, interpolation function and structural strength coefficient. Then, the incremental stress-strain relationship was derived based on the incremental theory. The UMAT subroutine was compiled by FORTRAN language. Finally, the cyclic triaxial test and numerical simulation were carried out, and the stress-strain relationship was analyzed. The research results showed that the test results were in good agreement with the prediction results of the model.

On the mechanism of plastic shrinkage cracking in fresh cementitious materials

In this study, a continuum poromechanics approach is presented to model the plastic shrinkage cracking of fresh cementitious materials. The boundary conditions are according to the modified ASTM C1579–13 standard for mortars. The restrained deformations are linked to the restraint stresses according to the Cauchy-Navier equations of elasticity, assuming an incremental stress-strain relationship. The Bresler-Pister and Rankine failure criteria are utilized to model failure. The material parameters are adapted according to the Drucker-Prager and Griffith criteria. The crack initiation and propagation is verified experimentally by X-ray radiography. Eventually, the cracking mechanism is discussed and a safe capillary pressure limit is proposed. It is found that capillary pressure stiffening occurring before air entry, when deformations take place in the saturated state, is the predominant cause of plastic shrinkage cracking in the drying state.

Development of low drying shrinkage foamed concrete and hygro-mechanical finite element model for prefabricated building fasçade applications

Prefabricated lightweight concrete building fasçade can improve the energy efficiency of buildings and reduce the carbon emission of transportation. However, it is essential to maintain the dimensional stability of the full scale element. The drying shrinkage of lightweight foamed concrete was investigated in this study. The hypothesis of using the drying shrinkage of normal weight concrete to approximate that of lightweight foamed concrete of dry density about 1500 kg/m3 counterpart was verified. Three different strategies of reducing drying shrinkage were studied. The drying shrinkage of common ingredients of ordinary Portland cement (OPC) and ground granulated blast-furnace slag (GGBS) was commonly up to 2000–3000 με. The use of magnesium expansive agent with different calcination conditions could not reduce the drying shrinkage. The use of calcium sulfoaluminate (CSA) cement with OPC and GGBS could significantly reduce the drying shrinkage within 1000 με in standard testing environment. The formulation developed in laboratory was scaled up in a concrete production plant for prefabricated concrete elements. A lightweight full scale panel (the wet density was about 1700 kg/m3) was fabricated. The drying shrinkage of the developed formulation with CSA cement was only 161 με in the field test. A hygro-mechanical model was developed to model the diffusion, shrinkage and plastic strain evolution. The incremental stress-strain constitutive relationship of the hygro-mechanical model was derived for incorporating it into general finite element routine. The model parameters were calibrated by the drying shrinkage measurements in this study. The calibrated model demonstrated the cracking potential of three typical reinforced concrete panels of three different formulations studied in this study.

Comparative study of two elasto-plastic work-hardening spatial models for concrete

A concept of elasto-plastic, work-hardening constitutive models for the multiaxial behaviour of concrete under short-term loading and the comparison with test results is presented in this paper. Two failure surfaces are utilized: the criterion of Podgórski and the three-parameter surface of Willam and Warnke. Both triaxial failure criteria have been calibrated in terms of different multiaxial strength tests. A non-associated flow rule has been used. The plastic potential function has been assumed in the form of the Drucker-Prager cone with variation of the angle of the cone side surface. In order to cover the plastic hardening behaviour, the equivalent uniaxial stress-strain curve has been adopted. An incremental stress-strain relationship has been formulated. The results of the numerical analysis performed by a direct integration of the constitutive relationships for the biaxial stress regime have been compared with the test data.

A viscoactive constitutive modeling framework with variational updates for the myocardium

We present a constitutive modeling framework for contractile cardiac mechanics by formulating a single variational principle from which incremental stress–strain relations and kinetic rate equations for active contraction and relaxation can all be derived. The variational framework seamlessly incorporates the hyperelastic behavior of the relaxed and contracted tissue along with the rate–and length–dependent generation of contractile force. We describe a three-element, Hill-type model that unifies the active tension and active deformation approaches. As in the latter approach, we multiplicatively decompose the total deformation gradient into active and elastic parts, with the active deformation parametrizing the contractile Hill element. We adopt as internal variables the fiber, cross-fiber, and sheet normal stretch ratios. The kinetics of these internal variables are modeled via definition of a kinetic potential function derived from experimental force–velocity relations. Additionally, we account for dissipation during tissue deformation by adding a Newtonian viscous potential. To model the force activation, the kinetic equations are coupled with the calcium transient obtained from a cardiomyocyte electrophysiology model. We first analyze our model at the material point level using stress and strain versus time curves for different viscosity values. Subsequently, we couple our constitutive framework with the finite element method (FEM) and study the deformation of three-dimensional tissue slabs with varying cardiac myocyte orientation. Finally, we simulate the contraction and relaxation of an ellipsoidal left ventricular model and record common kinematic measures, such as ejection fraction, and myocardial tissue volume changes.

Error behaviour in explicit integration algorithms with automatic substepping

SummaryThis paper studies the behaviour of the error incurred when numerically integrating the elasto‐plastic mechanical relationships of a constitutive model for soils using an explicit substepping formulation with automatic error control. The correct update of all the variables involved in the numerical integration of the incremental stress–strain relationships is central to the computational performance of the integration scheme, and, although often missed in the literature, all variables (including specific volume) should be explicitly considered in the algorithmic formulation. This is demonstrated in the paper by studying, in the context of the Cam clay formulations for saturated soils, the influence that the updating of the specific volume has on the accuracy of the numerical solution. The fact that the variation of the local error with the size of the integrated strain depends on the order of local accuracy of the numerical method is also used in the paper to propose a simple and powerful strategy to verify the correctness of the implemented mathematical formulation. Copyright © 2016 John Wiley & Sons, Ltd.

Time-Independent Plasticity Based on Thermodynamic Equilibrium and Its Stability

Within the thermodynamic framework with internal variables by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19(6), pp. 433–455), Yang et al. (2014, “Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional,” ASME J. Eng. Mater. Technol., 136(2), p. 021001) established a model of time-independent plasticity of three states. In this model, equilibrium states are the states with vanishing thermodynamic forces conjugate to the internal variables, and correspond to critical points of the free energy or its complementary energy functions. Then, the conjugate forces play a role of yield functions and further lead to the consistency conditions. The model is further elaborated in this paper and extended to nonisothermal processes. It is shown that the incremental stress–strain relations are fully determined by the local curvature of the free energy or its complementary energy functions at the critical points, described by the Hessian matrices. It is further shown that the extended model can be well reformulated based on the intrinsic time in the sense of Valanis (1971, “A Theory of Viscoplasticity Without a Yield Surface, Part I. General Theory,” Arch. Mech., 23(4), pp. 517–533; 1975, “On the Foundations of the Endochronic Theory of Viscoplasticity,” Arch. Mech., 27(5–6), pp. 857–868), by taking the intrinsic time as the accumulated length of the variation of the internal variables during inelastic processes. It is revealed within this framework that the stability condition of equilibrium directly leads to Drucker (1951, “A More Fundamental Approach to Stress–Strain Relations,” First U.S. National Congress of Applied Mechanics, pp. 487–497) and Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech., 25(2), pp. 746–750) inequalities, by introducing the consistency condition into the work of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25(3), pp. 448–461). Generalized inequalities of Drucker (1951, “A More Fundamental Approach to Stress–Strain Relations,” First U.S. National Congress of Applied Mechanics, pp. 487–497) and Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech., 25(2), pp. 746–750) for nonisothermal processes are established straightforwardly based on the connection.

A 3D co-rotational beam element for steel and RC framed structures

A 3-node 3D co-rotational beam element using vectorial rotational variables is employed to consider the geometric nonlinearity in 3D space. To account for shape versatility and reinforced concrete cross-sections, fibre model has been derived and conducted. Numerical integration over the cross-section is performed, considering both normal and shear stresses. In addition, the derivations associated with material nonlinearity are given in terms of elasto-plastic incremental stress-strain relationship for both steel and concrete. Steel reinforcement is treated as elasto-plastic material with Von Mises yield criterion. Compressive concrete behaviour is described by Modified Kent and Park model, while tensile stiffening effect is taken into account as well. Through several numerical examples, it is shown that the proposed 3D co-rotational beam element with fibre model can be used to simulate steel and reinforced concrete framed structures with satisfactory accuracy and efficiency.

Finite element verification on constitutive law of AZ31 magnesium alloy at 400 °C

A constitutive law is offered for an AZ31B-H24 Mg alloy within a strain rate range of 10−5−10−2 s−1 at a temperature of 400 °C. The constitutive law, which is developed by curve fitting the tensile tests data, is expressed as a flow stress function of strain and strain rate. Furthermore, the constitutive law is embedded into a proper FE model to simulate the tensile experiments for the purpose of verifying reliability, where the incremental stress-strain relationships are calculated by an elastic-plastic theory in the finite element analysis (FEA). The results show that the stress-strain characteristics and the final deformed shapes in the FEA agree well with the experiments. In addition, the predicting analysis of constant-velocity stretch conditions and the verification of a free bulge forming experiment show that the proposed FE model is practicable for mechanical analysis on superplastic forming problems. A selective numerical method is offered for advanced superplastic analysis on AZ31 Mg alloys.

A new generalized Drucker–Prager flow rule for concrete under compression

In this paper, a new generalized Drucker–Prager flow rule for concrete under compression with infinitesimal deformation is proposed, where the plastic volumetric strain is taken as the hardening parameter. The flow rule is proposed based on two fundamental phenomenological deformation properties of concrete, and on the fact that the deviatoric part of the plastic strain has no contribution to plastic volumetric strain. There is only one material parameter needing calibration for each grade of concrete in the flow rule. This parameter has a clear physical meaning to reflect the brittleness of concrete, and is defined as the brittleness index of concrete. The flow rule has a concise and simple form similar to the traditional Drucker–Prager flow rule, and for the case of uniaxial compression, the proposed flow rule is identical to the traditional Drucker–Prager flow rule. The effect of confinement stress is well considered by multiplying the hydrostatic part of the flow rule with a simple piecewise function, which differentiates the proposed flow rule from the traditional Drucker–Prager flow rule. An iterative method is proposed to determine the incremental stress–strain relationship when determining the flow rule. The correctness and reliability of the suggested flow rule are is validated using uniaxial, biaxial, and triaxial experimental results.

A symmetrisation method for non-associated unified hardening model

This paper presents a simple method for symmetrising the asymmetric elastoplastic matrix arising from non-associated flow rules. The symmetrisation is based on mathematical transformation and does not alter the incremental stress–strain relationship. The resulting stress increment is identical to that obtained using the original asymmetrized elastoplastic matrix. The symmetrisation method is applied to integrate the Unified Hardening (UH) model where the elastoplastic matrix is asymmetric due to stress transformation. The performance of the method is verified through finite element analysis (FEA) of boundary value problems such as triaxial extension tests and bearing capacity of foundations. It is found that the symmetrisation method can improve the convergence of the FEA and reduce computational time significantly for non-associated elastoplastic models.

A Comparison of EFGM and FEM for Nonlinear Solid Mechanics Problems

In this paper, element free Galerkin method (EFGM) has been applied to solve nonlinear solid mechanics problems using updated Lagrangian approach. The nonlinear equations have been solved using Newton Raphson method. An associative J2 flow rule and isotropic hardening has been used for the modelling of elasto-plastic material behaviour. Elastic predictor and plastic corrector algorithm has been used for the integration of incremental stress-strain relation. Few test problems involving large deformations have been simulated and the results obtained by EFGM have been compared with those obtained by FEM.

VARIATION OF MODULUS OF ELASTICITY IN THE TANGENTIAL DIRECTION WITH MOISTURE CONTENT AND TEMPERATURE FOR NORWAY SPRUCE (PICEA ABIES)

Modulus of elasticity (MOE) in the tangential direction for Norway spruce, Picea abies (L.) H. Karst, was measured. Test samples were tested in three-point bending, and moisture content (MC) and temperature were varied between the green condition and 7% MC and between 20°C and 80°C, respectively. Using correction factors calculated from finite element simulations, an adjustment of measured MOE was made to the ideally tangential direction. The results show MOE and the gradients with respect to MC and temperature and how they vary with MC and temperature. The gradients are factors in gradient terms in the incremental stress-strain relation for linear elastic behaviour during load cycles where there are mechanical loads and at the same time, varying MC and temperature. The gradient terms add to the temperature and MC expansion coefficients and may be of significant size for cases with high stress, high temperature, and high MC.

A Simplified Micromechancial Model for Analyzing Viscoelastic–Viscoplastic Response of Unidirectional Fiber Composites

This study introduces a simplified micromechanical model for analyzing a combined viscoelastic–viscoplastic response of unidirectional fiber reinforced polymer (FRP) composites. The micromechanical model is derived based on a unit-cell model consisting of fiber and matrix subcells. In this micromechanical model, a limited spatial variation of the local field variables in the fiber and matrix subcells is considered in predicting the overall time-dependent response of composites. The constitutive model for the polymer matrix is based on Schapery’s viscoelastic and Perzyna’s viscoplastic models. An incremental stress–strain relation is considered in solving the time-dependent and inelastic response. A linearized prediction and iterative corrector scheme are formulated to minimize errors from the linearization within the incremental stress–strain relation such that both the micromechanical constraints and the nonlinear constitutive equations are satisfied. The goal is to provide the accurate effective stress–strain relations of the composites and the corresponding viscoelastic and viscoplastic deformation in the polymeric matrix. The micromechanical model is verified by comparing the time-dependent response of the glass FRP composites having several off-axis fiber orientations with experimental data available in the literature.

The Effect of Initial Sampling Moisture Content on Anisotropic Behavior of Remolding Saturated Silt

By using the soil static and dynamic universal tri-axial and torsional shear apparatus, the experiment was performed on remolding saturated samples of Shan-Dong Dong-Ying silt which was prepared respectively with 0%, 5%, 10% initial sampling water content. A series of stress-controlled undrained monotonic torsional shearing tests were conducted under the same conditions of initial mean principal stress as 100kPa, initial intermediate principal stress coefficient as 0.5, initial deviator stress ratio as 0.433, but with different initial orientation angle of the principal stress at 0o, 30o, 45o, 60o, 90o. During the test the intermediate principal stress coefficient and mean principal stress were controlled constantly. The anisotropic behavior of the silt were discussed through the effects of initial sampling water content and initial orientation of the principal stress on the incremental stress-strain relations of different orientations, and the strain path. It was shown that the initial sampling water content had a remarkable influence on initial anisotropic behavior. Under the same condition, the higher the initial sampling water content is, the less obviously the anisotropic behavior is. The difference between the strain path and the stress path was obvious.

A continuum model of jointed rock masses based on micromechanics and its integration algorithm

The paper is devoted to proposing a constitutive model based on micromechanics. The joints in rock masses are treated as penny-shaped inclusion in solid but not through structural planes by considering joint density, closure effect, joint geometry. The mechanical behavior of the joints is represented by an elasto-plastic constitutive law. Mori-Tanaka method is used to derive the relationship between the joint deformations and macroscopic strains. The incremental stress-strain relationship of rock masses is formulated by taking the volume average of the representative volume element. Meanwhile, the behavior of joints is obtained. By using implicit integration algorithms, the consistent tangent moduli are proposed and the method of updating stresses and joint displacements is presented. Some examples are calculated by ABAQUS user defined material subroutine based on this model.