Dynamic crack-tip fields in rate-sensitive solids

The asymptotic stress and strain fields near the tip of a crack which propagates dynamically in a rate-sensitive solid are obtained under anti-plane shear and plane strain conditions. The problem is formulated within the context of a small-strain theory for a solid whose mechanical behavior under high strain rates is described by an elastic-viscoplastic constitutive relation. It is shown that, if the stresses are singular at the crack-tip, the viscoplastic relation is equivalent asymptotically to an elastic-non-linear viscous relation. Furthermore, for a certain range of the material parameter which characterizes the rate-sensitivity of the material, the elastic strain-rates near the propagating crack tip are shown to have the same asymptotic radial dependence near the propagating crack-tip as the inelastic strain-rates. This determines the order of the stress singularity uniquely. The governing equations for anti-plane shear and plane strain are then derived. The numerical results for the stress and strain fields are presented for anti-plane shear and plane strain. For the present model, the results suggest that under small-scale yielding conditions, there exists a minimum velocity for stable steady crack propagation. The implication that a terminal velocity for a running crack may exist is also discussed.     

The Anisotropic Hooke's Law for Cancellous Bone and Wood

A method of data analysis for a set of elastic constant measurements is applied to data bases for wood and cancellous bone. For these materials the identification of the type of elastic symmetry is complicated by the variable composition of the material. The data analysis method permits the identification of the type of elastic symmetry to be accomplished independent of the examination of the variable composition. This method of analysis may be applied to any set of elastic constant measurements, but is illustrated here by application to hardwoods and softwoods, and to an extraordinary data base of cancellous bone elastic constants. The solid volume fraction or bulk density is the compositional variable for the elastic constants of these natural materials. The final results are the solid volume fraction dependent orthotropic Hooke's law for cancellous bone and a bulk density dependent one for hardwoods and softwoods. This revised version was published online in August 2006 with corrections to the Cover Date.     

An efficient algorithm for elasto-viscoplastic vibrations of multi-layered composite beams using second-order theory

An efficient time-domain algorithm for plane non-linear flexural vibrations of multi-layered composite beams, which are driven into the inelastic range by severe transverse loadings, is presented. The influence of an axial static preload is considered in the sense of the quasi-linear second-order theory of structures. The inelastic parts of strain are treated as additional sources of selfstress in the linear elastic background-structure, driving the elastic response into the inelastic one. The efficiency of this exact formulation lies in the fact that linear solution techniques can be used in their most powerful form: Rubin's useful formulation for the quasi-static second-order transfer-matrix of linear elastic structures is applied in combination with modal analysis. Having in mind multi-metal beams, the classical lamination theory is assumed to be valid. Beams with overhang composed of ideal elastic-plastic and viscoplastic layers are studied as example structures. The fictitious sources of selfstress are calculated from the different material laws of the layers in a numerical time-stepping procedure, where a generalized midpoint-rule in combination with Crisfield's secant-Newton procedure is used.     

Numerical simulation of elongational flow behaviour in the cross cell experiments

In this work, numerical simulation of an elongational flow of polymer solutions is tried. The flow field is computed in a cell of simple geometry for newtonian, non-newtonian inelastic and non-newtonian elastic fluids. For the latter, an Oldroyd four-constant-model was adopted. The results are qualitatively in good agreement with the experimental results.     

Plastic work induced heating evaluation under dynamic conditions: Critical assessment

In many engineering problems involving adiabatic (dynamic) conditions the temperature rise induced by plastic deformation is usually evaluated using the inelastic heat fraction. The latter is still frequently considered as a crudely determined constant value. On the other hand, experimental investigations have shown that the inelastic heat fraction depends on strain, strain rate and temperature. Employing a phenomenological double-potential, elastic/thermoviscoplastic constitutive framework, the intrinsic dissipation form and heat equation considered correspond to salient features of inelastic behaviour of a large class of solids. For some typical strain hardening/softening and thermal softening combinations encountered, the evolution of inelastic heat fraction is being studied and quantified and finally shown to be highly strain and temperature dependent.     

Channel cracking in inelastic film/substrate systems

Studies on channel cracking are generally limited to elastic films on elastic or inelastic substrates. There are important applications were the cracking process involves extensive plasticity in both the film and substrate, however. In this work steady-state channel cracking in inelastic thin-film bilayers undergoing large-scale yielding from thermal or mechanical loading is studied with the aid of a plane-strain FEA. The plasticity of the film and substrate, represented by a Ramberg–Osgood constitutive law, each increases the energy release rate (ERR) relative to the linearly-elastic case. This effect is more pronounced under mechanical loading where the entire bilayer undergoes large-scale yielding. To help assess the analytic approach some fragmentation tests are performed using a well-bonding epoxy/aluminum system. The analysis reproduced well the observed dependence of crack initiation strain on film thickness.Ultra-thin films may be well represented by an elastic-perfectly plastic response. For such films on a flexible support the ERR remains fixed as the applied strain exceeds the yield strain of the film. Accordingly, a critical coating thickness exists below which no channel cracking is possible. The explicit relations and graphical data presented may be used for optimal design of such structures against premature failure as well as for determining fracture energy of ductile thin films.     

Entry flow behaviour of viscoelastic fluids in an annulus

Developing and fully developed velocity profiles in the entrance region of an abrupt 2-to-1 annular contraction were measured for a number of visco-elastic polymer solutions. Experimental results were obtained for Reynolds number and flow behaviour index in the range 9.8 ? Re ? 355 and 0.372 ? n ? 0.55 respectively. A momentum-energy integral technique was employed in the boundary layer analysis. The deviation from inelastic behaviour was indicated by the ratio of elastic to inertial forces, Ws/Re. Within the limits of confidence of the experimental results, good agreement with theoretical predictions was obtained and very little deviation from inelastic behaviour was observed for Ws/Re < 0.08. For the test fluids investigated, the entrance length was found to be longer than that predicted for the corresponding inelastic fluids of the same n.     

On the evolution of rotation of a solid under inelastic collisions with a plane

The motion of a solid in a homogeneous gravity field under inelastic collisions with an immovable absolutely smooth horizontal plane is considered. The body is a homogeneous ellipsoid of revolution. There exists a motion in which the ellipsoid symmetry axis is directed along a fixed vertical, the ellipsoid itself rotates about this axis at a constant angular velocity, and the ellipsoid bounce height over the plane decreases from impact to impact because of the collisions. We study the motion of the ellipsoid in a small neighborhood of the motion corresponding to this infinite-impact process. The main goal is to compute the angle between the ellipsoid symmetry axis and the vertical at the discrete time instants corresponding to the collisions. The problem is solved in the first (linear) approximation. The analysis is based on the canonical transformation method used earlier in [1] to solve problems with absolutely elastic collisions. There are quite a few studies dealing with infinite-impact processes (e.g., see the monographs [2, 3]). A method for continuous representation of systems with inelastic collisions was proposed in [4] and efficiently used in [3–5] when analyzing specific mechanical systems.     

Experimental evaluation of inelastic dynamic amplification factors for progressive collapse analysis under sudden support loss

A small-scale test setup is devised to investigate the inelastic dynamic amplification factors (DAFs) for structures subjected to sudden support loss. Based on three different definitions, experimental DAFs are calculated from static and dynamic support-release tests. Comparison results indicate that DAFs obtained from the neutral displacement response cannot account for the inelastic dynamic effect on either the displacement or force response. The displacement-based DAFs are apparently different from the force-based DAFs in the inelastic range. The former is larger than 2.0 and exhibits a concave downward variation with displacement ductility. On the contrary, the latter is less than 2.0 and exhibits a concave upward variation. Both of them may asymptotically return to the elastic DAF under large deformation as the specimen presents significant strain-hardening behavior. Pseudo-static response analysis is carried out for prediction of inelastic DAFs using the load–displacement curve obtained from the nonlinear static test. Also, analytical formulae with consideration of post-yield stiffness ratios are derived from the pseudo-static response analysis. They are proved to be capable of simulating the variation of inelastic DAFs with ductility demand.     

Relaxation behavior and modeling

Load relaxation tests deliver several orders of magnitude of inelastic strain rate data while elastic strains are converted into inelastic strains [see Lemaitre and Chaboche, 1994. (Mechanics of Solid Materials, Oxford University Press, Cambridge p. 264)]. Hart used this test for providing information on the inelastic deformation behavior for modeling purposes. Characteristic relaxation curves were obtained with ductile metals and alloys at room and high temperature showing a scaling relation derived from Hart's theory. Subsequent testing with servo-controlled testing machines and strain measurement on the gage length showed that an increase of prior strain rate also increased the average relaxation rate. For relaxation tests starting in the flow stress region, the relaxation curves can be independent of the stress and strain at the start of the relaxation tests. For the modeling of these newly found relaxation behaviors and other phenomena the viscoplasticity theory based on overstress (VBO) has been introduced. It is shown that VBO admits a long-term (asymptotic) solution that can be used with sufficient accuracy for the flow stress region of the stress–strain diagram. The long-term solution predicts the observed relaxation behaviors and that the relaxation curves coincide when shifted along the stress axis. This behavior is observed for the recently obtained data and is confirmed by two sets of the Hart-type data when they are plotted according to the new method.     

On the coupled thermo-mechanical process in the necking problem

The development of a neck in a uniaxial tension test with circular cross section is influenced by the heat production due to inelastic deformations. The theoretical investigation of this coupled thermomechanical process is based on a constitutive law defined within a general thermodynamical frame. For the numerical solution a finite element method is employed. Theoretical and experimental results are compared.     

A model of fracture in reinforced concrete arches based on lumped damage mechanics

A simplified model of cracking and damage in RC circular elements is proposed. The model can be used for the structural assessment of arches and rings. The constitutive equations are based on lumped damage mechanics which is an adaptation of fracture and continuum damage mechanics to the theory of frames with plastic hinges. An arch element is assumed to be the assemblage of an elastic circular component and two inelastic hinges where the main inelastic effects, plastic yielding of the longitudinal reinforcement and concrete cracking, are concentrated. Deformations in the elastic part are assumed to be small but the model may include some geometrically nonlinear effects due to large displacements or rotations of the hinges. The numerical examples presented in the paper show that the model describes correctly the global behavior of two structures including the softening phase.     

Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates

A robust physically consistent three-dimensional constitutive model is developed to describe the finite mechanical response of amorphous polymers over a wide range of temperatures and strain rates, including the rubbery region and for impact loading rates. This thermomechanical model is based on an elastic–viscoplastic rheological approach, wherein the effects of temperature, strain rate, and hydrostatic pressure are accounted for. Intramolecular, as well as intermolecular, interactions under large elastic–inelastic behavior are considered for the mechanisms of deformation and hardening. For a wide range of temperatures and strain rates, our simulated results for poly(methyl methacrylate) (PMMA) and polycarbonate (PC) are in good agreement with experimental observations.     

Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar

We propose a fundamentally new concept to the treatment of material instabilities and localization phenomena based on energy minimization principles in a strain-softening elastic–plastic bar. The basis is a recently developed incremental variational formulation of the local constitutive response for generalized standard media. It provides a quasi-hyperelastic stress potential that is obtained from a local minimization of the incremental energy density with respect to the internal variables. The existence of this variational formulation induces the definition of the material stability of inelastic solids based on convexity properties in analogy to treatments in elasticity. Furthermore, localization phenomena are understood as micro-structure development associated with a non-convex incremental stress potential in analogy to phase decomposition problems in elasticity. For the one-dimensional bar considered the two-phase micro-structure can analytically be resolved by the construction of a sequentially weakly lower semicontinuous energy functional that envelops the not well-posed original problem. This relaxation procedure requires the solution of a local energy minimization problem with two variables which define the one-dimensional micro-structure developing: the volume fraction and the intensity of the micro-bifurcation. The relaxation analysis yields a well-posed boundary-value problem for an objective post-critical localization analysis. The performance of the proposed method is demonstrated for different discretizations of the elastic–plastic bar which document on the mesh-independence of the results.     

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.     

An admissible form of an inelastic constitutive equation—I. General theory

From the viewpoint of irreversible thermodynamics an admissible form of rate-type constitutive equation of inelastic materials is given. The displacement gradient tensor F referred to the temporarily fixed reference frame which coincides with the Euler frame at the instant of the reference time is decomposed linearly into elastic and inelastic parts so that the procedure of formulation is simplified and clarified. The inelastic deformation rate is directly related to the internal production rate of entropy. The existence of an inelastic potential of the usual sense is not assumed, though the result can be understood to include the conventional flow theory based on an inelastic potential. An example of an elastoviscoplastic constitutive equation is given and some properties of yield surfaces are discussed.     

An analytical and numerical study of failure waves

Based on recent observations in shock experiments on glasses, a new failure process has been suggested for a certain type of brittle solids, in which a failure wave propagates through a solid at some distance behind the compressive stress wave near but below the Hugoniot elastic limit. Since the failure wave phenomenon is different from the usual inelastic shock waves, a combined analytical and numerical effort is made in this paper to explore the impact failure mechanisms associated with the failure wave. Based on the experimental data available, it appears that the physical picture of failure wave is related to local dilatation due to shear-induced microcracking. A mathematical argument then leads to the conclusion that the failure wave should be described by a diffusion equation instead of a wave equation, which is in line with the bifurcation analysis for localization problems. However, the occurrence of different governing equations in a single computational domain imposes both an analytical and a numerical challenge on the design of an efficient solution scheme. With the use of a partitioned-modeling approach, a simple solution procedure is proposed for failure wave problems, which is verified by the comparison with data.