Elastic Constitutive Relation Research Articles

A method for calculating damage evolution in adiabatic shear band of titanium alloy

A method for calculating the evolution of the local damage variable at the adiabatic shear band (ASB) center was proposed. In the present method, the JOHNSON-COOK model and the nonlocal theory were adopted, and the damage variable formula applicable for the bilinear (linearly elastic and strain-softening) constitutive relation was further generalized to consider the plastic deformation occurring in the strain-hardening stage. Aiming at Ti-6Al-4V, the effect of strain rate on the evolution of the local damage variable at the ASB center was investigated. In addition, a parametric study was carried out, including the effects of strain-hardening exponent, strain rate sensitive coefficient, thermal-softening exponent, static shear strength, strain-hardening modulus, shear elastic modulus, work to heat conversion factor, melting temperature and initial temperature. The damage extent at the ASB center in the radial collapse experiment was assessed. It is found that at higher strain rates the damage in the ASB becomes more serious at the same average plastic shear strain of the ASB.

The effects of surface tension on the elastic properties of nano structures

In the absence of external loading, surface tension will induce a residual stress field in the bulk of nano structures. However, in the prediction of mechanical properties of nano structures, the elastic response of the bulk is usually described by classical Hooke’s law, in which the aforementioned residual stress was neglected in the existing literatures. The present paper investigates the influences of surface tension and the residual stress in the bulk induced by the surface tension on the elastic properties of nano structures. We firstly present the surface elasticity in the Lagrangian and the Eulerian descriptions and point out that even in the case of infinitesimal deformations the reference and the current configurations should be discriminated; otherwise the out-plane terms of surface displacement gradient, associated with the surface tension, may sometimes be overlooked in the Eulerian descriptions, particularly for curved and rotated surfaces. Then, the residual stress in the bulk is studied through the non-classical boundary conditions and used to construct the linear elastic constitutive relations for the bulk material. Finally, these relations are adopted to analyze the size-dependent properties of pure bending of Al nanowires. The present results show that surface tension will considerably affect the effective Young’s modulus of Al nanowires, which decrease with either the decrease of nanowires thickness or the increase of the aspect ratio.

Cavitation in elastic and hyperelastic sheets

Plane stress, axially symmetric, cavitation patterns are examined for large sheets with an embedded central circular hole. Both (radially-uniform) remote tension and internal pressure loads are considered. Material behavior is modeled by finite strain Hookean-type elastic and hyperelastic constitutive relations with logarithmic strains. Governing field equations are reduced to a single ordinary differential equation with the principal stress difference as the independent variable. Calculations reveal that under internal pressure the usual definition of cavitation state does not apply as load increases monotonously along the deformation path. Introducing the specific cavitation energy, as a practical measure in cavitation analysis, provides a unified framework for assessment of cavitation phenomena. Comparison with related patterns of spherical and cylindrical cavitation fields supports the main findings of this study.

Macroscopic Elastic Constitutive Relationship of Cast-in-Place Hollow-Core Slabs

The macroscopic Poisson ratio and elastic moduli of the cast-in-place hollow-core slab are researched in this paper. At first, the formulas for computing the Poisson’s ratio of cast-in-place hollowed slabs with bilateral ribs are deduced by theoretical means. Then, based on the characteristics of stress distribution, an effective sectional area method is proposed for computing both the bending and axial compressive elastic moduli of cylindrical hollow-core slabs at the transverse section, in which the boundary of the effective area is suggested to be at a radial distance with its vertical slope angle α0 =22.5° . Relevant formulas for computational purposes are also proposed. At last, the shear modulus of cylindrical hollow-core slab at the transverse section is proposed to be computed considering the bending-shear deformation of the varying rib and face sheets at a periodic segment, and relevant formulas for curved sides beam are also deduced. In the course of the study, the so-called transverse-rib-effec...

Determination of large deformation and fracture behaviour of starch gels from conventional and wire cutting experiments

Large deformation and fracture properties of two types of starch gels were investigated through uniaxial compression, single edge-notched bend (SENB) and wire cutting experiments. Tests were performed at various loading rates and for various starch/powder concentrations (%w/w). It was found that starch gels exhibit rate independent stress–strain behaviour but show rate-dependent fracture behaviour, i.e. stress–strain curves at three loading rates are similar but fracture stress and fracture strain increase with increasing strain rate. This is rather unusual and interesting behaviour. SENB and wire cutting experiments also revealed rate-dependent fracture behaviour and that the true fracture toughness (G c) values increase with loading/cutting speeds and starch powder concentration. In addition, the G c values from wire cutting and SENB tests were in reasonable agreement. The wire cutting process was also studied numerically using finite element techniques. A non-linear elastic constitutive relationship based on Ogden was used to model the starch gels and a frictionless condition was assumed at the wire–starch gel contact interface. A fracture criterion based on maximum principal strain was assumed for the prediction of the steady state cutting force.

Viscous constitutive relations of solid‐liquid composites in terms of grain boundary contiguity: 1. Grain boundary diffusion control model

Viscous constitutive relations of partially molten rocks deforming in the regime of grain boundary (GB) diffusion creep are derived theoretically on the basis of microstructural processes at the grain scale. The viscous constitutive relation developed in this study is based on contiguity as an internal state variable, which enables us to take into account the detailed effects of grain‐scale melt distribution observed in experiments. Compared to the elasticities derived previously for the same microstructural model, the viscosities are much more sensitive to the presence of melt and variations in contiguity. As explored in this series of three companion papers, this “contiguity” model predicts that a very small amount of melt (ϕ < 0.01) significantly reduces the bulk and shear viscosities. Furthermore, a large anisotropy in viscosity is produced by anisotropy in contiguity, which occurs in deforming partially molten rocks. These results have important implications for deformation and melt extraction at small melt fractions, as well as for shear‐induced melt segregation. The viscous and elastic constitutive relations derived in terms of contiguity bridge microscopic grain‐scale and macroscopic continuum properties. These constitutive relations are essential for investigating melt migration dynamics in a forward sense on the basis of the basic equations of two‐phase dynamics and in an inverse sense on the basis of seismological observations.

Continuum coupled cohesive zone elements for analysis of fracture in solid bodies

Embedding cohesive surfaces into finite element models is a widely used technique for the numerical simulation of material separation (i.e. crack propagation). Typically, a traction–separation law is specified that relates the magnitude of the cohesive traction to the distance between the separating surfaces. Thus the characterization of fracture in such models is not directly coupled to the bulk constitutive response, in the sense that the cohesive traction does not explicitly depend on material stretching in the plane of the fracture surface. In this work, an initially-rigid cohesive-traction formulation that is coupled to the surrounding continuum is introduced as a further development of the cohesive zone idea. In this model, the traction–separation law – and therefore the fracture phenomenology – derives directly from the bulk constitutive law. The immediate goal is an improved cohesive zone framework that naturally and logically initiates cohesive separation behavior, and couples its evolution to the material state in the region of the crack tip. A cohesive element based on this model is implemented in an explicit three-dimensional finite element code. Proof-of-concept analyses using both linear elastic and Gurson void growth constitutive relations are presented. A three-point bend simulation is found to give good agreement with experimental results.

Deformation of PDMS membrane and microcantilever by a water droplet: Comparison between Mooney–Rivlin and linear elastic constitutive models

In this paper, we studied the role of vertical component of surface tension of a water droplet on the deformation of membranes and microcantilevers (MCLs) widely used in lab-on-a-chip and micro- and nano-electromechanical system (MEMS/NEMS). Firstly, a membrane made of a rubber-like material, poly(dimethylsiloxane) (PDMS), was considered. The deformation was investigated using the Mooney–Rivlin (MR) model and the linear elastic constitutive relation, respectively. By comparison between the numerical solutions with two different models, we found that the simple linear elastic model is accurate enough to describe such kind of problem, which would be quite convenient for engineering applications. Furthermore, based on small-deflection beam theory, the effect of a liquid droplet on the deflection of a MCL was also studied. The free-end deflection of the MCL was investigated by considering different cases like a cylindrical droplet, a spherical droplet centered on the MCL and a spherical droplet arbitrarily positioned on the MCL. Numerical simulations demonstrated that the deflection might not be neglected, and showed good agreement with our theoretical analyses.

Micromechanical definition of an entropy for quasi-static deformation of granular materials

A micromechanical theory is formulated for quasi-static deformation of granular materials, which is based on information theory. A reasoning is presented that leads to the definition of an information entropy that is appropriate for quasi-static deformation of granular materials. This definition is based on the hypothesis that relative displacements at contacts with similar orientations are independent realisations of a random variable. This hypothesis is made plausible based on the results of Discrete Element simulations. The developed theory is then used to predict the elastic behaviour of granular materials in terms of micromechanical quantities. The case considered is that of two-dimensional assemblies consisting of non-rotating particles with an elastic contact constitutive relation. Applications of this case are the initial elastic (small-strain) deformation of granular materials. Theoretical results for the elastic moduli, relative displacements, energy distribution and probability density functions are compared with results obtained from the Discrete Element simulations for isotropic assemblies with various average numbers of contacts per particle and various ratios of tangential to normal contact stiffness. This comparison shows that the developed information theory is valid for loose systems, while a theory based on the uniform-strain assumption is appropriate for dense systems.

Constitutive relations and failure criterion of planar lattice composites

In this paper, analytical elastic constitutive relations and failure criterion of planar lattice composites are established based on the assessment of static equilibrium and deformation relations of the representative unit cell. Finite element simulations are carried out to validate the analytical solutions. Furthermore, optimizations are performed to minimize the weight of a cantilever sandwich beam for a given flexibility for four different categories of cellular core materials, including metal foam, metal hexagonal honeycomb, metal lattice and lattice composite. It is found that the out-of-plane constraint plays an important role in the in-plane strength of lattice materials made of unidirectional fiber reinforced composites. The present analytical approach is capable of identifying this effect. The sandwich beam of laminate lattice composite possesses the best bending performance among all the sandwich beams considered, which suggests its superiority for application as a bending bearing structure.

A nonlinear homogenization procedure for periodic masonry

Abstract The present paper deals with the problem of the determination of the in-plane behavior of masonry material. The masonry is considered as a composite material composed by a regular distribution of blocks connected by horizontal and vertical mortar joints. The overall constitutive relationships of the regular masonry are derived by a rational micromechanical and homogenization procedure. Linear elastic constitutive relationship is considered for the blocks, while a new special nonlinear constitutive law is proposed for the mortar joints. In particular, a mortar constitutive law, which accounts for the coupling of the damage and friction phenomena occurring during the loading history, is proposed; the developed model is based on an original micromechanical analysis of the damage process of the mortar joint. Then, an effective nonlinear homogenization procedure, representing the main novelty of the paper, is proposed; it is based on the transformation field analysis, using the technique of the superposition of the effects and the finite element method. The presented methodology is implemented in a numerical code. Finally, numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of masonry material.

Nonlinear transverse free vibrations of piles

The nonlinear partial differential equation governing the nonlinear transverse vibration of pile was derived under the assumption of that both the materials of pile and soil obey nonlinear elastic and linear viscoelastic constitutive relations. The approximate expressions of the nth-order main frequency and the response of the nonlinear vibration of pile with ends hinged have been obtained by the complex mode method and multiple time scales method. Results point out that the main frequency of the nonlinear system is related to not only the natural frequency of linear vibration system, but also the amplitude, damping and nonlinearity of materials. There are high-order harmonic waves with twice the main frequency, and the main frequency of three times as well as the sum and/or difference of 2 or 3 main frequencies besides the harmonic wave with the main frequency in the response of the nonlinear system. Numerical examples were given and the effect of parameters was considered in detail.

Variation in the length of perfectly elastic rods under torsion

On the basis of elastic constitutive relations that reflect geometrically nonlinear second-order effects, we refine the theory of torsion of rectilinear rods of an arbitrary transverse cross-section. In particular, we obtain a universal formula, independent of the material properties, that determines the longitudinal strain arising as the rod undergoes free torsion. According to this formula, the length of a rod made of an isotropic perfectly elastic material can, in contrast to the traditional concepts, either increase or decrease as the rod undergoes torsion. Moreover, the variation in the length depends only on the geometry of the transverse cross-section.

A general constitutive relation for linear elastic foams

A three-dimensional linear elastic constitutive relation is formulated based on a representative unit cell of foam using elasticity theory and micromechanics homogenization scheme. The displacement and strain fields of the unit cell are obtained from elasticity theory and used to derive the macroscopic strain field defined on the outer surface of unit cell through homogenization scheme. By assuming a uniform macroscopic stress on the unit cell surface and the existence of strain energy potential, the constitutive relation of linear elastic foams is obtained. The newly derived constitutive relation is a function of mechanical property of solid constituent, the geometry of cell struts, and the porosity of foams and is able to characterize the anisotropic behavior of foams due to non-uniform strut geometry. The linear elastic response of open-celled foams with both low- and medium-relative densities can be studied using the derived constitutive relation. The effective elastic modulus for uniform strut geometry is reduced from the constitutive relation and agrees well with Gibson and Ashby's semi-empirical equation, Warren and Kraynik's, and Zhu's analytical models within relative density ranging from 0 to 0.35. For non-uniform strut geometry, the calculated effective elastic moduli in three axial directions are different and the material displays anisotropic behavior. The bulk modulus shows less dependence on the variation of the strut geometry. Poisson's ratios are also reduced from the compliance matrix.

The application of a meshless method to consolidation analysis of saturated soils with anisotropic damage

This paper explores the applicability of a novel meshless method, namely the radial point interpolation method (RPIM), to the numerical simulation of consolidation of saturated soils with anisotropic damage. Firstly, an elastic constitutive relationship for the anisotropically damaged soil is derived. The RPIM is subsequently presented. Then, the damage constitutive equation is implemented into the RPIM code, in which the displacement and excess pore water pressure are spatially approximated by the same shape functions and the time domain is discretized by fully implicit integration scheme. At last, the effect of anisotropic damage on soil consolidation process is explored numerically. The results reveal that the established numerical scheme incorporating the constitutive model with anisotropic damage and the RPIM meshless method is very effective in simulating the soil–water coupling problem with good accuracy. Besides, the effects of the magnitude and direction of anisotropic damage are found to exert significant influences on the evolution of settlement and the excess pore water pressure.

ГЕОДИНАМІКА

We investigate a role of the contact friction in thrusting within the framework of the critical taper theory and according to geological settings for orogenic belts including the Ukrainian Carpathians (in Early Cretaceous time). Finite element models are used to simulate tectonic compression of sedimentary rocks by submerged stage and take into account frictional slipping on the detachment horizon. We assume a simple wedge geometry (rectangular layer 60 km long, 1.5 km thick and 2.5 km deep), plane strain state, quasistatic process and use elastic constitutive relation. Mechanical loads include gravity, water pressure on top and lateral displacement (up to 0.5 km) from the left, whereas the right side is fixed. Numerical results show specific features of the inhomogeneous stress fields for small (0.01-0.5), middle (0.5-0.64), large (0.64—0.8) and overlarge (0.8-1.15) friction coefficients. The magnitude of the tangential contact stress controls the front between sliding and sticking zones. Stress trajectories enable to predict thrust structures using Mohr-Coulomb failure criterion.

Nonlinear dynamic characteristics of piles embedded in rock

The nonlinear dynamic characteristics of a pile embedded in a rock were investigated. Suppose that both the materials of the pile and the soil around the pile obey nonlinear elastic and linear viscoelastic constitutive relations. The nonlinear partial differential equation governing the dynamic characteristics of the pile was first derived. The Galerkin method was used to simplify the equation and to obtain a nonlinear ordinary differential equation. The methods in nonlinear dynamics were employed to solve the simplified dynamical system, and the time-path curves, phase-trajectory diagrams, power spectrum, Poincare sections and bifurcation and chaos diagrams of the motion of the pile were obtained. The effects of parameters on the dynamic characteristics of the system were also considered in detail.

On the Rank 1 Convexity of Stored Energy Functions of Physically Linear Stress-Strain Relations

The rank 1 convexity of stored energy functions corresponding to isotropic and physically linear elastic constitutive relations formulated in terms of generalized stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229–242 (1968)] is analyzed. This class of elastic materials contains as special cases the stress-strain relationships based on Seth strain measures [Seth, B.: Generalized strain measure with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162–172. Pergamon, Oxford, New York (1964)] such as the St.Venant–Kirchhoff law or the Hencky law. The stored energy function of such materials has the form $$\widetilde{W}{\left( {\user2{F}} \right)} = W{\left( \alpha \right)}: = \frac{1}{2}{\sum\limits_{i = 1}^3 {f{\left( {\alpha _{i} } \right)} + \beta } }{\sum\limits_{1 \leqslant i < j \leqslant 3} {f{\left( {\alpha _{i} } \right)}f{\left( {\alpha _{j} } \right)}} },$$ where \(f:{\left( {0,\infty } \right)} \to \mathbb{R}\) is a function satisfying \(f{\left( 1 \right)} = 0,\;f'{\left( 1 \right)} = 1,\;\beta \in \mathbb{R}\), and α 1, α 2, α 3 are the singular values of the deformation gradient \({\user2{F}}\). Two general situations are determined under which \(\widetilde{W}\) is not rank 1 convex: (a) if (simultaneously) the Hessian of W at α is positive definite, \(\beta \ne 0\), and f is strictly monotonic, and/or (b) if f is a Seth strain measure corresponding to any \(m \in \mathbb{R}\). No hypotheses about the range of f are necessary.

The geometric stiffness of thick shell triangular finite elements for large rotations

AbstractThis paper is concerned with the development of the geometric stiffness matrix of thick shell finite elements for geometrically nonlinear analysis of the Newton type. A linear shell element that is comprised of the constant stress triangular membrane element and the triangular discrete Kirchhoff Mindlin theory (DKMT) plate element is ‘upgraded’ to become a geometrically nonlinear thick shell finite element. Perturbation methods are used to derive the geometric stiffness matrix from the gradient, in global coordinates, of the nodal force vector when stresses are kept fixed. The present approach follows earlier works associated with trusses, space frames and thin shells. It has the advantage of explicitness and clear physical insight. A special procedure, tailored to triangular elements is used to isolate pure rotations to enable stress recovery via linear elastic constitutive relations. Several examples are solved. The results compare well with those available in the literature. Copyright © 2005 John Wiley &amp; Sons, Ltd.

Energetics and Conserved Functional of Axially Moving Materials Undergoing Transverse Nonlinear Vibration

Axially moving materials can represent many engineering devices such as power transmission belts, elevator cables, plastic films, magnetic tapes, paper sheets, textile fibers, band saws, aerial cable tramways, and crane hoist cables @1–3#. Energetics of axially moving materials is of considerable interest in the study of axially moving materials. The total mechanical energy associated with axially moving materials is not constant when the materials travel between two supports. It is a fundamental feature of free transverse vibration of axially moving materials, while the total energy is constant for an undamped non-translating string or beam. Chubachi @4# first discussed periodicity of the energy transfer in an axially moving string. Miranker @5# analyzed energetics of an axially moving string, and derived an expression for the time rate of change of the string energy. Barakat @6# considered the energetics of an axially moving beam and found that energy flux through the supports can invalidate the linear theories of both the axially moving string and beam at sufficiently high transporting speed. Tabarrok, Leech and Kim @7# showed that the total energy of a travelling beam without tension is periodic in time. Wickert and Mote @8# pointed out that Miranker’s expression represents the local rate of change only because it neglected the energy flux at the supports, and they presented the temporal variation of the total energy related to the local rate of change through the application of the onedimensional transport theorem. They also calculated the temporal variation of energy associated with modes of moving strings and beams. Renshaw @9# examined the change of the total mechanical energy of two prototypical winching problems, which provided strikingly different examples of energy flux at a fixed orifice of an axially moving system. Lee and Mote @10,11# presented a generalized treatment of energetics of translating continua, including axially moving strings and beams. They considered the case that there were nonconservative forces acting on two boundaries. Renshaw, Rahn, Wickert and Mote @12# examined the energy of axially moving strings and beams from both Lagrangian and Eulerian views. Their studies indicated that Lagrangian and Eulerian energy functionals are not conserved for axially moving continua. Zhu and Ni @13# investigated energetics of axially moving strings and beams with arbitrarily varying lengths. Although both the Eulerian and Lagrangian functionals for the total mechanical energy of axially moving materials are generally not constant, there do exist alternative functionals that are con-