Strain Constitutive Relations Research Articles

Correlation of constitutive response of hybrid textile reinforced concrete from tensile and flexural tests

This papers addresses the disparities that exist in measuring the constitutive properties of thin section cement composites using a combination of tensile and flexural tests. It is shown that when the test results are analyzed using a simplified linear analysis, the variability between the results of tensile and flexural strength can be as high as 200–300%. Experimental results of tension and flexural tests of laminated Textile Reinforced Concrete (TRC) composites with alkali resistant (AR) glass, carbon, aramid, polypropylene textile fabrics, and a hybrid reinforcing system with aramid and polypropylene are presented. Correlation of material properties is studied analytically using a parametric model for simulation of flexural behavior using a closed form solution based on tensile stress–strain constitutive relation. The flexural load carrying capacity of TRC composites is computed using a back-calculation approach, and parameters for a strain hardening material model are obtained using the closed form equations. While the parametric model over predicts the simulated tensile response for carbon and polypropylene TRCs, predictions are however consistent with experimental trends for aramid and glass TRCs. Detailed discussion of the differences between backcalculated and experimental tensile properties is presented. Results can be implemented as average moment–curvature relationship in the structural design and analysis of cement composites.

Comparisons between reduced order models and full 3D models for fluid–structure interaction problems in haemodynamics

When modelling the cardiovascular system, the effect of the vessel wall on the blood flow has great relevance. Arterial vessels are complex living tissues and three-dimensional specific models have been proposed to represent their behaviour. The numerical simulation of the 3D–3D Fluid–Structure Interaction (FSI) coupled problem has high computational costs in terms of required time and memory storage. Even if many possible solutions have been explored to speed up the resolution of such problem, we are far from having a 3D–3D FSI model that can be solved quickly.In 3D–3D FSI models two of the main sources of complexity are represented by the domain motion and the coupling between the fluid and the structural part. Nevertheless, in many cases, we are interested in the blood flow dynamics in compliant vessels, whereas the displacement of the domain is small and the structure dynamics is less relevant. In these situations, techniques to reduce the complexity of the problem can be used. One consists in using transpiration conditions for the fluid model as surrogate for the wall displacement, thus allowing problem’s solution on a fixed domain. Another strategy consists in modelling the arterial wall as a thin membrane under specific assumptions (Figueroa et al., 2006, Nobile and Vergara, 2008) instead of using a more realistic (but more computationally intensive) 3D elastodynamic model. Using this strategy the dynamics of the vessel motion is embedded in the equation for the blood flow. Combining the transpiration conditions with the membrane model assumption, we obtain an attractive formulation, in fact, instead of solving two different models on two moving physical domains, we solve only a Navier–Stokes system in a fixed fluid domain where the structure model is integrated as a generalized Robin condition. In this paper, we present a general formulation in the boundary conditions which is independent of the time discretization scheme choice and on the stress–strain constitutive relation adopted for the vessel wall structure.Our aim is, first, to write a formulation of a reduced order model with zero order transpiration conditions for a generic time discretization scheme, then to compare a 3D–3D FSI model and a reduced FSI one in two realistic patient-specific cases: a femoropopliteal bypass and an aorta. In particular, we are interested in comparing the wall shear stresses, in fact this quantity can be used as a risk factor for some pathologies such as atherosclerosis or thrombogenesis. More in general we want to assess the accuracy and the computational convenience to use simpler formulations based on reduced order models. In particular, we show that, in the case of small displacements, using a 3D–3D FSI linear elastic model or the correspondent reduced order one yields many similar results.

Another Solving Equation of Elastoplastic Problem Based on Stress Strain Relation

Based on relationship between elastic and plastic strain, a new basic equation system of plasticity mechanics was established. In this paper, modification is made to the new basic equation system, and another form of basic equation system is presented based on stress strain constitutive relation, which also makes it possible to apply the results from elasticity mechanics directly to plasticity mechanics and decreases the solving difficulty of plastic problem. Finally, a simple application example is given, which shows that the new form of basic equation system is correct and practicable.

Discretized virtual internal bond model for nonlinear elasticity

The virtual internal bond (VIB) is a micro–macro constitutive model. Although this model is based on a postulated discrete microstructure, it ultimately returns to a continuum constitutive relation through a homogenization process. The homogenization process can reduce the internal degrees of freedom, but it omits the effect of the individual micro bond that may play an important role in the fracture process. The present research develops a discrete system to represent the nonlinear elasticity by discretizing the continuous VIB. This discrete system is composed of unit cells, which can adopt any geometry with any number of bonds. The system is characterized by the force–displacement, not the stress–strain constitutive relationship. The nonlinear properties of this discrete system are governed by the micro-bond potential. The micro bond properties are related to Young’s modulus of the material, the volume and the bond number of the unit cell. For a given material, the unit cell has a certain topological structure and configuration. A discussion of two specific cases (the 2D triangular and 3D tetrahedral unit cells) suggests that the discrete system converges with decreasing unit cell size. In the unstructured unit cell scheme, the discrete system can almost precisely represent the initial Young’s modulus and the Poisson ratio of a nonlinear continuum. A mixed fracture example demonstrates that the present method can efficiently simulate the fracture propagation. The present paper provides a theory for developing a lattice-type mechanical model for nonlinear elasticity and provides new method for the fracture simulation of a nonlinear elastic material.

A fractional-order model on new experiments of linear viscoelastic creep of Hami Melon

This paper describes our experimental testing of linear viscoelastic creep behaviours in Hami Melon. Experimental data shows that Hami Melon has complex viscoelastic property which cannot be well described by the standard model. Consequently, this study develops a fractional derivative model to describe such complex viscoelastic creep behaviours of Hami Melon. The analytical creep function of the proposed fractional linear viscoelastic models is derived via the Boltzmann superposition principle and discrete inverse Laplace transform. And then, such analytical solutions are used to fit the experimental viscoelastic data of Hami Melon. Our study shows that the present fractional linear viscoelastic model with merely three parameters is more efficient and accurate than the generalised Kelvin viscoelastic model of six parameters to describe the stress–strain constitutive relations of Hami Melon. It is noted that the present fractional model with adjustable parameters can also be used to describe creep damage.

Investigation of the ultimate residual strengthen of a worn casing by using the arc-length algorithm

The accurate prediction of the ultimate residual strength (URS) (including collapse, burst, and tensile strength) of worn casing is important in the design and safety assessment of casings for the oil and gas industry. Based on the true stress–strain constitutive relationship of P110 steel, this paper establishes a three-dimensional (3D) finite element model (FEM) to obtain the URS of worn casing by using the arc-length algorithm. The effects of worn defect length, wear percentage, drillstring diameter, and casing thickness on the URS are investigated. Moreover, three full-scale experiments on casings with different wear percentages are conducted to verify the accuracy and reliability of the 3D FEM and of the algorithm. The validation proves that the URS of worn casing can be predicted by the 3D FEM to a satisfactory degree of accuracy. The URS has an exponential relation with defect length, but a linear relation with wear percentage, drillstring diameter, and casing thickness. The effects of numerous factors on the URS are also studied. Regression analysis is conducted on the basis of 3D FEM predictions, and the corresponding design equations for worn casing are presented.

Numerical simulation of cyclic behavior of ductile metals with a coupled damage–viscoplasticity model

In this paper, the evolution of ductile damage and the failure is investigated in the cyclic loading of ductile metals based on the rate-dependent elasto-viscoplastic damage model with the linear and nonlinear isotropic-kinematic hardening rules. The constitutive model is derived by introducing the damage variable, based on the plastic strain behavior, in both the elasto-plasticity and visco-plasticity frameworks. The cyclic behavior is modeled by using the visco-plasticity model since the rate-dependent behavior is observed experimentally for various metal alloys. In order to obtain a more realistic response in the compression zone, the crack closure effect is employed in the stress–strain constitutive relations. Finally, four different cases are studied numerically to assess the overall framework of the proposed constitutive model, including the fatigue behavior and lifetime assessment of AISI316L alloy, and the results are compared with available experimental data.

Prediction of the micro-thermo-mechanical behaviors in dispersion nuclear fuel plates with heterogeneous particle distributions

Dispersion nuclear fuel elements have promising prospects to be used in advanced nuclear reactors and disposal of nuclear wastes. They consist of fuel meat and cladding, and the fuel meat is a kind of composite fuel in which the fuel particles are embedded in the non-fissile matrix. Prediction of the micro-thermo-mechanical behaviors in dispersion nuclear plates is of importance to their irradiation safety and optimal design. In this study, the heterogeneity of the fuel particles along the thickness direction in the fuel meat is considered. The 3D finite element models have been developed respectively for two cases: (1) variation of fuel particle–particle (PP) distances for the particles near the mid-plane of the fuel meat; (2) variation of the particle-cladding (PC) distances for the fuel particles near the interface between the fuel meat and the cladding. The respective finite strain constitutive relations are developed for the fuel particle, metal matrix and cladding. The developed virtual temperature method is used to simulate irradiation swelling of the fuel particles and irradiation growth of the metal cladding. Effects of the heterogeneous distributions of the fuel particles on the micro temperature fields and the micro stress–strain fields are investigated. The obtained results indicate that: (1) as a whole, the maximum Mises stress, equivalent plastic strain and first principal stress at the matrix between the two closest particles increase with decreasing the particle–particle (PP) distance; existence of large first principal stresses there may be the main factor that induces the matrix failure; (2) variation of the particle-cladding (PC) distance has remarkable effects on the interfacial normal stress and shear stress at the interface between the fuel meat and the cladding; the first principal stress at the cladding near the interface increases dramatically when the fuel particle is closer and closer to the cladding. Thus, the proper distance between the fuel particles and the one between the fuel particle and the cladding should be considered in the optimal design; (3) the micro temperature distribution in the fuel element is hardly affected.

High Strain Rate Compression Behavior and Constitutive Relation of As-Extruded Mg-Gd-Y Magnesium Alloy

The high strain rate compression behavior of extruded Mg-Gd-Y magnesium alloy was tested by split Hopkinson pressure bar (SHPB) under the strain rates of 465s-1，2140s-1 and 3767s-1. As comparison the quasi-static compression behavior was tested in the meanwhile. The results show that the quasi-static yield stress is equivalent to that of high strain rates, but the flow stress at high strain rates are higher than that of quasi-static stain rate at the same strain. When the strain rate is increase from quasi-static to high strain rates the deformation stresses increase obviously but within the present testing high strain rates, increasing the strain rate the stress has a slight increasing, indicating that at high strain rate the stress of Mg-Gd-Y magnesium alloy is not sensitive to the strain rate. The constitutive equation between deformation stress, strain and strain rate was build based on the tested compression stress strain curves. The calculated stress strain data were compared with tested stress strain curves. The results demonstrate that when the strain rates are 0.001s-1，465s-1，2140s-1 respectively the calculated and experimental data are fit very well. The calculated stress is higher than that of tested stress if the strain rate is increase to 3767s-1 and the strain is more than 0.15. The discrepancy was explained through the physical soundness of Johnson-Cook model.

Macroscopic constitutive equations of thermo-poroelasticity derived using eigenstrain–eigenstress approaches

Macroscopic constitutive equations for thermoelastic processes in a fluid-saturated porous medium are re-derived using the notion of eigenstrain or, equivalently, eigenstress. The eigenstrain-stress approach is frequently used in micromechanics of solid multi-phase materials, such as composites. Simple derivations of the stress–strain constitutive relations and the void occupancy relationship are presented for both fully saturated and partially saturated porous media. Governing coupled equations for the displacement components and the fluid pressure are also obtained.

On thermoelastostatics of composites with nonlocal properties of constituents I. General representations for effective material and field parameters

A theory of thermoelastic composites with nonlocal properties of constituents is analyzed for multiphase elastic solids of arbitrary geometry and material symmetry. Due to their generality, one uses the nonlocal integral models because the gradient models are usually derived as approximations of corresponding integral models in the immediate (infinitely closed) vicinity of the point being considered. One explores a simplified theory for linear (macroscopically) elasticity, which differs from the classical local theory in the stress–strain constitutive relation only, whereas the equilibrium and compatibility equations remain unaltered. One obtains the new representation of the effective modulus and compliance through the mechanical influence function which does not explicitly depend (as opposed to its local counterpart) on the elastic operators of constituents. The representations for the effective eigenstrains and eigenstresses through either the mechanical influence functions or transformation influence functions are presented. The effective strain energy and potential energy are expressed in terms of only average values of the state variables and the effective properties. Representations of both the first and second statistical moments of stress and strain fields in the constituents are also performed. Many of the results were obtained as the straightforward generalizations of their local counterparts because the methods used for obtaining the mentioned results widely exploit the Hill’s (1963) condition which holds for any compatible strain field and equilibrium stress field not necessarily related to each other by a specific stress–strain relation.

Determination of Hardening Coefficient of Large Strain Constitutive Model Based on Torsion Tests

Uni-tension test and torsion test with specimens of A3 steel are completed, and deformation rate, the spin of Euler configuration to Lagrange configuration and generalized Jaumann rate of Kirchhoff stress in torsion problem are given. Large strain constitutive relationship with generalized Jaumann stress rate is studied based on torsion test. The simple shear problem is discussed by using the constitutive model given in this paper. The result indicates that the hardening coefficient of constitutive model with generalized Jaumann rate may be given by the same expression as that of constitutive model with Jaumann objective rate when determining by using torsion tests and the shear stress response curve in simple shear problem obtained by using the constitutive model in this paper is similar to the shear stress-strain curve given in solid circular shaft torsion tests.

On thermoelastostatics of composites with nonlocal properties of constituents II. Estimation of effective material and field parameters

One considers a linear thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities. It is assumed that the stress–strain constitutive relations of constituents are described by the nonlocal integral operators, whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity. The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. In a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities, one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties (local or nonlocal) of heterogeneities. In such a case, the Levin’s (1967) formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart. A solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method. The operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and “ ellipsoidal symmetry” assumption. Quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented.

Inverse Technique for Deformational Analysis of Concrete Beams with Ordinary Reinforcement and Steel Fibers

The smeared crack model, extensively applied in numerical analysis of reinforced concrete structures, is based on use of average stress-average strain constitutive relationships. This model is very effective for deformation analysis of steel fiber reinforced concrete (SFRC) members as cracking deformations of such members are restrained by fibers. One of the most critical points in the theory of SFRC is quantifying residual tension stresses for a cracked section. The purpose of the present study is to apply the inverse technique for determining residual stresses of SFRC in tension. The proposed method aims at deriving average stress-average strain relationships of cracked tensile concrete using test data of SFRC flexural members reinforced with steel bars. The paper reports results of the experimental program consisting of five beams reinforced in the tension zone with three 10mm bars in diameter. The beams had different contents of fibers, i.e. 0, 0.3, 0.5, 1.0 and 1.5% by volume. The proposed inverse technique has been applied to the test data and average stress-strain relationships in tension were derived for each beam. It has been shown that residual stresses increase with growing amount of fiber until the content reaches 1% of the element volume. Under this limit, the efficiency of fiber increases proportionally to its content, whereas, its influence becomes less significant above the limit.

Study on Spins and Deformation Rate of Solid Circular Shafts at Finite Torsion Deformation

Torsion experiments should be adopted to characterize large strain elasto-plastic behavior of material instead of traditional uni-axial tension experiments due to the plastic stability of specimen in torsion deformation. Study on spins and deformation rate in finite torsion deformation is the key to determine the material parameters by torsion experiments and understand the finite deformation characteristics of material. In this paper, five spins and deformation rate in torsion deformation with solid circular shafts are investigated in cylinder coordinates. The expressions of the deformation rate and spins, namely the material spin, the relative spin, the spin of the frame of the deformation rate, logarithmic spin and instantaneous spin considering the effect of stress, are deduced by analyzing the finite torsion deformation. The comparisons are made among all spins obtained in this paper. The results obtained in this paper are the basis of analyzing the large strain constitutive relationship based on torsion experiments with solid circular shafts.

A Test Method for Assessment of Shear Properties of Thick Composites

Transverse stress—strain constitutive relations are required for structural analysis of thick-section composites. A standard technique for assessment of transverse shear properties is the V-notched beam method. Such technique is based on strain gage measurements, which require a large specimen thickness for strain gage placement and impose tight geometry tolerances to minimize variations of strain at the gage location. A full-field strain measurement capability could enable simpler test specimen designs. A method for assessment of shear stress—strain relations using a short-beam shear (SBS) test and a digital image correlation (DIC) technique is presented in this work. The DIC technique is based on quantifying locations of a random texture on a surface to measure surface shape and deformation. V-notched-beam and SBS test results are compared for a glass/epoxy tape composite. To illustrate accuracy of SBS stress calculations, finite element results are obtained. Highly nonlinear interlaminar stress—strain relations are documented.

Failure analysis of natural gas buried X65 steel pipeline under deflection load using finite element method

A 3D parametric finite element model of the pipeline and soil is established using finite element method to perform the failure analysis of natural gas buried X65 steel pipeline under deflection load. The pipeline is assumed to be loaded in a parabolic deflection displacement along the axial direction. Based on the true stress–strain constitutive relationship of X65 steel, the elastic–plastic finite element analysis employs the arc-length algorithm and non-linear stabilization algorithm respectively to simulate the strain softening properties of pipeline after plastic collapse. Besides, effects of the soil types and model sizes on the maximum deflection displacement of pipeline are investigated. The proposed finite element method serves as a base available for the safety design and evaluation as well as engineering acceptance criterion for the failure of pipeline due to deflection.

Stress redistributions in adhesively bonded double-lap joints, with elastic–perfectly plastic adhesive behavior, subjected to axial lap-shear cyclic loading

A shear-lag model is developed in order to evaluate stress redistributions in double-lap joints under axial (tensile) lap-shear cyclic loading. The adherend materials exhibit linear elastic behavior, whereas the material of the adhesive layer satisfies the elastic–perfectly plastic shear stress–strain constitutive relation. The reference state (from which the stresses are redistributed) is based on the standard elastic–perfectly plastic shear-lag analysis for double-lap joints. The main conclusion of the current analysis is that, during unloading, shear stresses of opposite sign may develop in the plastic zones of the adhesive layer, at the ends of the overlap, without reversing the direction of the applied load. A simple model for evaluating the variation of the maximum peel stress in the adhesive layer, based on the variation of the peak shear stress, demonstrates that the sign of peel stresses may alternate, as well. Under cyclic (fatigue) loading, the range of the peak stresses in the adhesive layer is the basic parameter for the evaluation of the variation of the energy release rate and the associated crack growth rate in the overlap. In this framework, the current simplified analysis may provide a reference model for comparisons with experimental data or with results which are based on more complex numerical models. The current model can be readily extended to cover the cases of development of plastic zones in the adhesive layer with shear stresses and plastic strains of opposite sign (during unloading or during load direction change).

Nonlinear analysis of static axisymmetric deformation of the human cornea

This paper presents a simple mechanics model for analyzing the stresses, strains and displacements of the human cornea under the action of the intraocular pressure. The analysis is performed under the assumptions that when the cornea undergoes small displacements the stress–strain constitutive relation includes an isotropic, linear material of ground substance and an orthotropic, nonlinear material of collagen fibrils. The present results highlight the influence of corneal geometry and its local microstructural organisation on the corneal overall mechanical properties.

Thermodynamic-based interface model for cohesive brittle materials: Application to bond slip in RC structures

The main objectives of this work concern a new formulation for an inelastic constitutive model for bond-slip phenomena between brittle cohesive materials and reinforcement through a nondimensional zero-thickness joint finite element. The bond-slip behavior assumes the small deformation and plane strains case. It is firmly placed within the framework of thermodynamics and continuum damage mechanics accounting for frictional sliding. The finite element interface model is based on a 2D-degenerated four-node quadrilateral element which is able to account for interface normal stress as well as for the nonlinear hysteretic tangential–normal coupling of bonding. One of the most important advantages of the proposed model consists in a sufficiently rich kinematics structure for contact allowing for the implementation of refine stress–strain constitutive relations.