Constitutive relations for a rubber cord under three-dimensional stress states

The transverse modulus of a rubber cord is experimentally determined. It is shown that, in the framework of the effective layer model, it is problematic to find both the bending stiffness and the lateral compression stiffness with equal adequacy.     

Creep rupture under multi-axial states of stress

R are presented of metallographic examinations of specimens which have been creep tested at high temperatures and under different conditions of steady applied stress. The interaction between the growth of micro-fissures or voids and the mode of final rupture is discussed. The applied-stress versus rupture-life characteristics of a number of commercial alloys are presented together with simple expressions which are shown to describe approximately the stress-sensitive nature of their rupture behaviour.     

Constitutive equations for describing the elastoplastic deformation of elements of a body along small-curvature paths in view of the stress mode

Equations relating the components of the stress and strain tensors (constitutive equations) are formulated in terms of Euler coordinates. The equations describe the finite elastoplastic deformation of an isotropic body along paths of small curvature. It is assumed that the stress deviator is coaxial with the plastic-strain differential deviator. The relationships between the first and second invariants of the stress and strain tensors in the case of complex elastoplastic deformation of the body’s elements are determined from base tests on tubular specimens loaded along rectilinear paths for several values of the stress mode angle. Methods for specification of these relationships are proposed. The assumptions adopted to derive the constitutive equations are validated experimentally __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 62–72, April 2006.     

Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres

The isotropic compression of polydisperse packings of frictionless spheres is modeled with the Discrete Element Method (DEM). The evolution of coordination number, fraction of rattlers, isotropic fabric, and pressure (isotropic stress) is reported as function of volume fraction for different system parameters. The power law relationship, with power $\approx 1/2$, between coordination number and volume fraction is confirmed in the jammed state for a broad range of volume fractions and for different (moderate) polydispersities. The polydispersity in the packing causes a shift of the critical volume fraction, i.e., more heterogeneous packings jam at higher volume fractions. Close to jamming, the coordination number and the jamming volume fraction itself depend on both history and rate. At larger densities, neither the deformation history nor the loading rate have a significant effect on the evolution of the coordination number.Concerning the fabric tensor, comparing our DEM results to theoretical predictions, good agreement for different polydispersities is observed. An analytical expression for the pressure as function of isotropic (volumetric) strain is proposed for polydisperse packings, based on the assumption of uniform deformation. We note that, besides the implicit proportionality to contact number density (or fabric), no single power-law is evidenced in the relation between pressure and isotropic strain. However, starting from zero pressure at the jamming point, a linear term with a quadratic correction describes the stress evolution rather well for a broad range of densities and for various polydispersities. Finally, an incremental evolution equation is proposed for both fabric and stress, as function of isotropic strain, and involving the coordination number and the fraction of rattlers, as starting point for further studies involving anisotropic deformations.     

Constitutive relations for no-tension materials

Some general constitutive assumptions for elastic materials on which unilateral constraints are placed on stresses are reviewed; in particular the case of negative semi-definite stresses (no-tension materials) is analyzed. A rate-dependent, degenerating, no-tension material is proposed to model masonry structures. The corresponding material response is compared with the holonomic response of the classical normal no-tension model.

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Sommario Dopo avere riesaminato alcune restrizioni costitutive generali per materiali elastici in cui la tensione è vincolata unilateralmente, si analizza il caso speciale dei materiali non reagenti a trazione. Un particolare materialeno-tension, con superficie limite suscettibile di degradazione e con relazioni costitutive formulate in termini di incrementi di deformazione, è proposto per modellare le strutture murarie. La corrispondente risposta materiale è confrontata con la risposta olonoma del classico modellono-tension elastico normale.

     

A self-consistent relation for the time-dependent creep of polycrystals

A self-consistent relation with a weakening constraint power in the matrix is derived for the primary and steady-state creep of polycrystals. This derivation makes use of a linear viscoelastic comparison material, under which the constraint power of the creeping matrix is found to decrease exponentially with the ratio of the elastic shear modulus to the secant creep modulus, or when expressed alternately, with the ratio of the creep strain to the elastic strain. Such a dramatic decrease leads one to believe that the overall creep strain of the polycrystal as calculated by the traditional elastic-constraint model could be far too low; a direct comparison between the two, however, quickly reveals that the accuracy of the elastic-constraint model is better than what is initially thought, and certainly far better than in the rate-independent plasticity. With this new relation, the creep heterogeneity among the constituent grains are then studied in details. It is demonstrated that while the creep strains of the more favorably oriented grains tend to increase, and those of the less favorably oriented ones decrease, their creep rates become virtually uniform during the long-term, steady-state creep. This suggests that grain compatibility, instead of stress equilibrium, is the more dominant factor governing the grain-boundary condition during the steady-state process.     

Constitutive modelling of anisotropic creep deformation in single crystal blade alloys SRR99 and CMSX-4

A damage mechanics based model has been developed to model stress rupture and creep behaviour of the first and second generation single crystal superalloys SRR99 and CMSX-4. In this article the creep behaviour of CMSX-4 in several different orientations at 950°C is simulated using finite elements, these simulations are compared with the results of creep tests. In order that the effects of rotation and specimen bending can be accounted for in the analysis the entire creep specimen is modelled. The FE program ABAQUS has been used and the slip system model is written using a User MATerial subroutine (UMAT). EBSD (electron back scattered diffraction) measurements of the lattice rotations occurring during creep indicate that the active slip systems at 950°C are the <101>{111} and <112>{111} systems, our simulations show that the creep results can be explained by activating these two families of slip system. There is strong microstructural evidence that the significant components of the hardening matrix should be those causing self and latent hardening of the <101>{111} systems and latent hardening by the <101>{111} systems on the <112>{111} systems.     

Generalized stress/strain rate relations for creep of thin shells; a comparison of “exact” and approximate relations

A set of approximate generalized stress/strain rate relations which has been used for the stationary creep analysis of thin shells is compared with the corresponding ‘exact’ relations. The comparison is made by computing the functions from which the relations are derived and plotting the corresponding surfaces. Results are included for a limiting condition in which the stress/strain rate relations become those for a rigid-plastic material obeying the von Mises yield condition and associated flow rule. Although the comparison is made only for conditions valid in a cylindrical shell under rotationally symmetric loading, it indicates the errors which are likely to occur when the approximate relations are used in stationary creep analysis.     

Constitutive relations for a growing masonry with setting mortar

A model is proposed for averaging a periodic block structure, namely, a growing brick masonry body with setting interlayers of bonding mortar. The brick (block) material is assumed to be elastic, and the setting mortar is described by the model of an inhomogeneously aging viscoelastic medium. The obtained system of constitutive relations describes an anisotropic inhomogeneously aging viscoelastic medium and contains a small parameter that is the ratio of the hardening interlayer thickness to the brick thickness. Also presented is an example of solving the problem of erecting a brickwork (wall) deviating from the vertical in the gravity field.     

Constitutive modeling of unsaturated soil behavior under axisymmetric stress states using a stress/suction-controlled cubical test cell

Two elasto-plastic critical state-based formulations for modeling the constitutive behavior of an unsaturated soil are presented and briefly discussed. Computational constitutive drivers were implemented to allow for the numerical simulation of suction-controlled conventional triaxial tests. Simulations were obtained from explicit and implicit integration techniques. The algorithms support numerical analyses in a deviatoric stress plane by using a mixed control constitutive driver in conjunction with a generalized cam-clay model, within a constant-suction scheme. The results from a series of suction-controlled conventional triaxial tests conducted on several identically prepared, 10-cm side, cubical specimens of compacted silty sand, using a recently developed stress/suction-controlled cubical testing device, were used for validation of the models and the enhanced features proposed in the present work. Matric suction states in the specimens were induced and maintained constant during testing by using the axis-translation technique.     

Creep damage constitutive equations for multi-axial states of stress for 0.5Cr0.5Mo0.25V ferritic steel at 590°C

Multi-axial creep damage constitutive equations are developed for 0.5Cr0.5Mo0.25V ferritic steel at 590°C. It is aimed to overcome the deficiency of inconsistency in predicted and observed creep strains existed in previous formulations [Q. Xu, in: European Conference on Advances in Mechanical Behaviour, Plasticity and Damage, 7–9 November 2000, Tours, France; Creep Damage Mechanics Research Report, The School of Engineering, The University of Huddersfield, 2000; I.J. Perrin, D.R. Hayhurst, J. Strain Anal. 31 (4) (1996) 299]. In this paper, the lifetime and strain at failure are qualitatively validated in a wider range of states of stress by which the capability of producing consistent results is demonstrated.     

Constitutive relations for the interaction force in multicomponent particulate flows

In the mechanics of multiphase (or multicomponent) mixtures, one of the outstanding issues is the formulation of constitutive relations for the interaction force. In this paper, we give a brief review of the various relations proposed for this interaction force. The review is tilted toward presenting the works of those who have used the mixture theory (or the theory of interacting continua) to derive or to propose a relationship for the interaction (or diffusive) force. We propose a constitutive relation which is general and frame-indifferent and thus suitable for use in many flow conditions. At the end, we provide an alternative approach for finding the drag force on a particle in a particulate mixture. This approach has been used in the non-Newtonian fluid mechanics to find the drag force on surfaces.     

Constitutive relations for single crystal based on two surface description

A two surface model is used to describe single crystal slip. The evolution of the yield surface and outer yield surface are proposed to be directly related to the development of dislocation structures. The anisotropy of inner yield surface is attributed to both the long range and short range dislocation interactions, whereas anisotropy of the outer yield surface is assumed to depend solely on the short range interactions. The outer yield surface is not isotropic as assumed in most two surface models of other researchers. This surface is able to evolve even when the stress state is within the outer yield surface.     

Constitutive model for plastic deformation of nanocrystalline materials with shear band

A phase mixture model was used to study the plastic deformation behaviors in hardening stage of nanocrystalline materials. The material was considered as a composite of grain interior phase and grain boundary (GB) phase. The constitutive equations of the two phases were determined in term of their main deformation mechanisms. In softening stage, a shear band deformation mechanism was presented and the corresponding constitutive relation was established. Numerical simulations have shown that the predications fit well with experimental data. The investigation using the finite-element method (FEM) provided a direct insight into quantifying shear localization effect in nanocrystalline materials.     

Boltzmann superposition principle for a time-dependent soft material: assessment under creep flow field

Variety of time-dependent soft materials that undergo evolution of microstructure are known to follow the Boltzmann superposition principle when appropriately transformed from the real to the effective time domain. This behavior is attributed to obliteration of time dependency in the effective time domain by normalizing the real time by the time-dependent relaxation time. This work is aimed at assessing validity of the Boltzmann superposition principle in the effective time domain under application of step changes and ramps in stress for a time-dependent soft glassy material: an aqueous suspension of Laponite, whose rheological properties are known to show strong time and stress dependency. Interestingly, creep experiments started at different aging times and for different constant stresses indeed lead to a comprehensive time–aging time–stress superposition resulting in time and stress dependence of relaxation time and modulus. Subsequently, we analyze compliance response of the material to different kinds of stress step change and ramp combinations. We observe that except few, most of the compliance data plotted in the effective time domain does not overlap on the comprehensive superposition. We analyze this behavior and attribute the observed mismatch to the history dependence of the time evolution of modulus.