Stress-strain Constitutive Relation Research Articles

Analytical Model for Shear Critical Reinforced-Concrete Members

In this investigation, a nonlinear finite-element model is developed for predicting the complete load-deflection response of shear critical reinforced-concrete members. The nonlinear finite-element model employs a biaxial stress-strain constitutive relationship of concrete based on an equivalent uniaxial approach developed in this study and a simplified bilinear stress-strain relationship of reinforcing steel. The main feature of the model is its ability to predict the postpeak load-deflection response of shear critical reinforced-concrete members failing under a diagonal tension. The computational procedure developed in this study employs the secant-stiffness method and a noniterative algorithm. The predictions of the analytical model are compared with the available experimental data and the comparisons are judged to be good.

Large strain constitutive modelling and computation for isotropic, creep elastoplastic damage solids

AbstractA three‐dimensional fully coupled creep elastoplastic damage model at finite strain for isotropic non‐linear material is developed. The model is based on the thermodynamics of an irreversible process and the internal state variable theory. A hyperelastic form of stress–strain constitutive relation in conjunction with the multiplicative decomposition of the deformation gradient into elastic and inelastic parts is employed. The pressure‐dependent plasticity with strain hardening and the damage model with two damage internal variables are particularly considered. The rounding of stress–strain curves appearing in cycling loading is reproduced by introduction of the creep mechanism into the model. A numerical integration procedure for the coupled constitutive equations with three hierarchical phases is proposed. A consistent tangent matrix with consideration of the fully coupled effects at finite strain is derived. Numerical examples are tested to demonstrate the capability and performance of the present model at large strain.

Response of a circular opening in a friable low‐permeability medium to temperature and pore pressure changes

AbstractA general analysis using an incremental elastic, perfectly plastic constitutive stress–strain relationship for poroelastoplastic materials is presented to simulate an opening in a low‐permeability friable porous medium under non‐isothermal conditions. Analytical solutions are obtained for the stresses and strains around a 2‐D plane strain circular borehole. An expansion potential is introduced by combining the strains induced by temperature and pore pressure changes. Steady‐state pressures and temperatures are considered, and a non‐associated plastic flow rule is applied to calculate plastic strains. Focusing on stress distribution near a circular opening, the classic solutions for those stresses under dry and isothermal conditions are used to compare with the newly derived solution. The general poroelastoplastic effect and the thermal effect on sand production and borehole stability are addressed. We suggest that the knowledge of stress history is critical to achieve adequate solutions for displacement and stress in friable media such as clays, shales and oil sands.

Model for Shear Critical High-Strength Concrete Beams

In this paper the development of a nonlinear finite element model that employs a biaxial stress-strain constitutive relationship of concrete based on equivalent uniaxial approach is summarized. The main feature of the model is its ability to predict the post-peak load-deflection response of shear critical reinforcement concrete members failing under diagonal tension. This post-peak behavior is a relative measure of shear ductility. The predicitons of the model are compared with the available experimental data, and the comparisons are judged good. A parametric study was conducted to quantify the effects of variables influencing the failure of shear critical reinforced concrete beams.

Non-linear analysis of a seabed deposit subjected to the action of standing waves

The response of a cohesionless seabed deposit to the action of standing waves is examined in detail. The seabed deposit is assumed to be an elasto-plastic material obeying a stress-strain constitutive relation similar to the one proposed by Poorooshasb and Pietruszazck [ Soils Fdns 26(2), 1–15 (1986)], while the interaction between the solid and liquid phase is considered to follow Darcy's linear law. One important conclusion obtained from this study is to demonstrate the influence of the coefficient of permeability on the susceptibility of the sand mass to liquefy: for high values of permeability even a loose sand mass need not liquefy. This is in direct contrast to the commonly held view that the liquefaction potential is a function, only, of the sand constituency.

Large Deflections of an Elastoplastic Strain-Hardening Cantilever

This theoretical analysis of an elastoplastic cantilever examines the effects of linear strain-hardening and the development of partial elastic unloading when deflections become large. For a vertical force at the tip, unloading modifies the distribution of curvature in a substantial internal segment after the deflections become very large, but it has a negligible effect on the tip displacement. The analysis uses a bilinear stress-strain constitutive relation that results in slightly better agreement with previous experiments than a corresponding bilinear approximation for the moment-curvature relation.

Note on a random isotropic granular material with negative poisson's ratio

Poisson's ratio in generalized Hooke's law for an isotropic continuum is subject to the restriction − 1 ≤ v≤ 0.5. With the exception of recently developed low-density polymer foams, solids including granular materials have not been known to exhibit negative Poisson's ratio. This paper is concerned with the verification of constitutive stress-strain relationships proposed by Rothenburg (Micromechanics of idealized granular systems. Ph.D. thesis, Carleton University, 1980) which describe macroscopic behaviour of idealized bonded granular materials by elastic parameters ( K and v) that are formulated explicitly in terms of microstructural parameters such as interparticle stiffness, contact density and average interparticle distance. The theory includes an expression for Poisson's Ratio which is a function only of the ratio of tangential (shear) to normal contact stiffness λ A negative Poisson's ratio is predicted for both planar and three-dimensional random isotropic systems when the tangential stiffness is greater than the normal stiffness (i.e. λ > 1). The results of numerical simulation of bonded disc assemblies used to verify constitutive relationships show that systems with λ > 1 do exhibit negative Poissqn's ratio. Similar theoretical developments are summarized for three-dimensional random isotropic assemblies of bonded spheres and an analogous expression for Poisson's ratio is presented for these systems. It is noted that while a negative Poisson's ratio is theoretically possible, the micromechanieal condition for λ > 1 is physically unlikely for particles of natural materials.

Axisymmetric, Finite-Element Analysis of Stress Relaxation in Wound Magnetic Tapes

A model for the analysis of creep and stress relaxation in wound magnetic tape is presented. It is wed to predict anelastic deformation in the tape caused by initial winding stresses and geometric deformations imposed on it. The formulation assumes quasistatic, linearly viscoelastic properties and is implemented through a finite-element discretization of the governing equilibrium equation. Constitutive stress-strain relationships for an axisymmetric geometry are obtained and used in simulations that include stresses generated by the winding dynamics and geometric distortions. Results of analyses performed for various typical situations are presented, and tile results of the stress analyses are related to the anelastic deformations induced in the tape by stress relaxation. Presented as an American Society of Lubrication Engineers paper at the ASME/ASLE Lubrication Conference in San Diego, California, October 22–24, 1984.

A model for studying the stability of thoracic aorta

An analytical investigation on the mechanical instabilities of aortic walls is performed. On the basis of histological evidence and the properties of aortic tissues reported in the literature, the aorta has been considered as a nonlinearly anisotropic, viscoelastic and incompressible circular cylindrical membrane. The theory of small deformation superimposed on a state of known finite deformation has been used in carrying out the analysis. A constitutive stress-strain relation which takes into account the above-mentioned mechanical properties of aortic tissues has been considered. On the basis of the analytically derived criterion, the stability of the middle descending thoracic aorta has been studied.

Constitutive stress-strain relations for the myocardium in diastole

The importance of stress-strain myocardial constitutive relations is that they provide a criterion for behavior in vivo. Our purpose was to develop constitutive equations which are valid in diastole. The myocardium was assumed to be composed of a nonlinear viscoelastic, inhomogeneous, anisotropic (transversely isotropic) and incompressible material operating under adiabatic and isothermal conditions. The expressions contain five moduli. Two are fixed by the restriction of incompressibility, one is estimated, the remaining two refer to directions along and perpendicular to a fiber. Both possess a bimodal variation with intermodal switching occurring in late rapid filling and diastasis. They are functions of time and material constants. These constants can be observed. A dynamic test is suggested. Constitutive statements complete a set of equations sufficient for the solution of a class of boundary value problems. One type is formulated. They also permit the determination of stress from measured strain. Examples are given.

Effective elasticity of a solid suspension of spheres: I. Stress-strain constitutive relation

We consider the problem of the elastic behaviour of a random medium composed of identical spherically symmetric inclusions, each made of a linear elastic material with varying shear and bulk moduli, distributed randomly in a uniform matrix made of a linear elastic material with moduli μ 0, K 0. We consider the response of a finite sample to an externally imposed incident displacement field and by eliminating the external field obtain a linear constitutive relation between mean stress and mean strain in the material. Using the cluster expansion method of Felderhof, Ford and Cohen we show that in the thermodynamic limit of an infinite system there is a well defined effective elasticity operator for the medium expressible as a sum of terms which involve successively the problem of 1, 2,..., p,... inclusions in an infinite matrix. The elasticity operator is given in detail through second order terms in the number density of inclusions. By passing to a long wavelength limit we show how the constitutive relation reduces to a local relation expressible in terms of effective moduli μ eff, K eff.

A phenomenological theory of hysteresis damping of vibrations in ferromagnetic materials

The hysteresis damping of vibrations in magnetostrictive polycrystalline ferromagnetic materials is investigated theoretically. The constitutive relation for applied moment and twisted angle in torsion is derived directly from the stress-strain constitutive relation. Precise damping characteristics of torsional vibrations of ferromagnetic wires are obtained in small amplitudes and in large amplitudes after the saturation of magnetostriction.

A review of stress-strain paths and constitutive relations in the plastic range

Plastic strain paths produced by both radial and non-radial stress paths are reviewed for metals and alloys under uniaxial and biaxial stress. The present interpretation illustrates collective trends and offers theoretical solutions for each of several stress paths. In general the strain path is linear for a radial stress path for most materials at low to medium temperatures. The isotropic quadratic hardening rule of von Mises rarely describes the strain behaviour accurately. but a simple anisotropic quadratic form provides good agreement by the uniform hardening rule. Exceptions are found for magnesium and an associated alloy at room temperature and for the high temperature deformation in an aluminium alloy. Descriptive yield functions are presented and the anistropic constants determined for one particular function of the Bailey form. Strain paths produced by outward non radial zig-zag and stepped stress paths are approximately linear provided that they are confined to a narrow band in stress space. It is shown that an equivalent radial stress path analysis provides good agreement with experiment. The strain paths resulting from stress paths containing corners and from paths composed of partial or complete unloading followed by reloading in a different direction are non-linear. The behaviour is attributed to strain history; namely, a Bauschinger effect where unloading occurs and an interaction between direction dependent plastic strains for stress paths containing corners. For unloading-reloading paths a rotation in the strain path towards a linear direction is indicative of the transient nature of strain history. Good agreement is found with the prediction from each of three rules of anisotropic hardening. They are thus generalized to provide the plastic strain response to any stress path that is composed of linear segments. Modifications that account for the effects of non-linear work hardening and creep are discussed.

On the Response of Viscoelastic Materials to Triangle Pulse and Delta Function Excitations

The stress-strain constitutive relation for linear viscoelastic materials may be expressed on the real time axis by any of several convolution integral formulations. It has been pointed out by Tsch...

Remarks on the application of Koiter's general post-buckling theory

Application of Koiter's general buckling and post-buckling theory by using a generalized strain-displacement relationship which in fact represents a slightly non-linear stress-strain constitutive relationship is investigated. It seems that a slight non-linearity in the stress-strain curve may lead in some cases to unexpected large reduction of the classical buckling load. The non-linear constitutive law is not consistent with the basic assumptions of Koiter's theory. It is shown that a strain-displacement relationship obtained by the use of Taylor's series is preferable. For complicated structures calculation errors are inavoidable. A straight forward application of Koiter's theory for such structures would lead to incorrect interpretation. A method to avoid such a difficulty is proposed and both analytically and numerically described.

High velocity impacts against cables: nonlinear investigation of the safety conditions

This paper analyses the high velocity impact against a cable in order to define the limits of the impacting particle velocity to avoid the sudden cut of the cable. The mathematical model is based on the characteristic method in order to describe the nonlinear 3D dynamics of the cable (large deformations). The impacting object is considered as a point mass. The global system, cableimpacting mass, is described trough a quasi-linear system of hyperbolic partial differential equation with boundary condition of inertial type (the particle motion). The impact is assumed perfectly anelastic and the cable longitudinal constitutive stress-strain relationship follows the Young's law. The solution algorithm is developed on the characteristic plane trough a discrete grid of characteristic curves solved by a trial-and-error procedure. The investigations were accomplished by multiple simulations, varying the impact velocity in order to define the relation velocity-peak stress in the cable cross section. As a principal result, a parabolic relationship exists between the impact velocity and the cable stress, in particular we can state that there exists a range of velocities where the cable stress is very sensible to the impact velocity, below of such range the influence is sensibly lower. All allows to define the range of safety conditions.

Stress-strain data obtained at high rates using an expanding ring

Dynamic uniaxial tensile stress-strain data are obtained at high strain rates by measuring the kinematics of thin-ring specimens expanding symmetrically by virtue of their own inertia. Impulsively loaded to produce high initial radial velocities, expanding rings are decelerated by the radial component of the hoop stresses. Differential equations of motion are evaluated experimentally to obtain the stress-strain (constitutive) relationships which govern the magnitude of these stresses. Techniques have been developed for producing symmetric radial expansion and measuring resulting displacements precisely as a function of time. Dynamic stress-strain relationships have been obtained for 6061-T6 aluminum, 1020 cold-drawn steel, and 6Al-4V titanium. For each of these materials, displacement-time curves are observed to be parabolic within the resolution of the measurements. Results are presented as true-stress/true-strain relationships.