Asymptotic analysis of an end-loaded, transversely isotropic, elastic, semi-infinite strip weak in shear

We use linear elasticity to study a transversely isotropic (or specially orthotropic), semiinfinite slab in plane strain, free of traction on its faces and at infinity and subject to edge loads or displacements that produce stresses and displacements that decay in the axial direction. The governing equations (which are identical to those for a strip in plane stress, free of traction on its long sides and at infinity, and subject to tractions or displacements on its short side) are reduced, in the standard way. to a fourth-order partial differential equation with boundary conditions for a dimensionless Airy stress function ƒ. We study the asymptotic solutions to this equation for four sets of end conditions—traction, mixed (two), displacement—as g3, the ratio of the shear modulus to the geometric mean of the axial and transverse extensional moduli, approaches zero. In all cases, the solutions for ƒ consist of a “wide” boundary layer that decays slowly in the axial direction (over a distance that is long compared to the width of the strip) plus a “narrow” boundary layer that decays rapidly in the axial direction (over a distance that is short compared to the width of the strip). Moreover, we find that the narrow boundary layer has a “sinuous” part that varies rapidly in the transverse direction, but which, to lowest order, does not enter the boundary conditions nor affect the transverse normal stress or the displacements. Because the exact biorthogonality condition for the cigenfunctions associated with ƒ can be replaced by simpler orthogonality conditions in the limit as →b 0, we are able to obtain, to lowest order, explicit formulae for the coeflicients in the eigenfunction expansions of ƒ for the four different end conditions.     

Simplified prediction of the creep buckling of cylinders under external pressure. Part 2: Experimental verification

In this paper, the simplified method, proposed in (Combescure, 1998), for the prediction of creep buckling is compared to experimental results. The model is applied to predict the buckling time of two sets of experiments on cylinders subjected to uniform external pressure. It is shown that the proposed model is satisfactory for this type of prediction: in all cases, the times up to failure predicted by the model are generally lower than the experimental failure times. The model is rather conservative for thicker cylinders. However, it appears that a very detailed geometrical imperfection survey would be necessary if a highly accurate assessement of the creep failure time were sought. It has been observed experimentally that creep buckling is a very dangerous failure mode: nothing seems to happen during a very long “incubation” period but, when the initial imperfection reaches some critical value, buckling then suddenly occurs. For thin cylinders, the level of creep strain at which the instability starts to develop is much lower than the strain at which the tertiary creep initiates; the instability is thus clearly generated from the interaction between the material and the geometrical nonlinearity.     

Velocity field equations and strain localization

The velocity field equations of a hypo-elastic material are derived using arbitrary curvilinear coordinates in the actual configuration of the body, and emphasizing “higher order” terms which would disappear in an infinitesimal strain theory. As an example, bifurcation analyses are carried out with the help of the velocity field equations, for the thin rod and the thin plate under uniaxial tension. The results agree with those from the literature, at least if the “higher order” terms are disregarded. Differences may result from the way these terms are incorporated in the hypo-elastic approach.     

DYNAMICS AND STABILITY OF PINNED–CLAMPED AND CLAMPED–PINNED CYLINDRICAL SHELLS CONVEYING FLUID

The paper examines the dynamics and stability of fluid-conveying cylindrical shells having pinned–clamped or clamped–pinned boundary conditions, where “pinned” is an abbreviation for “simply supported”. Flügge's equations are used to describe the shell motion, while the fluid-dynamic perturbation pressure is obtained utilizing the linearized potential flow theory. The solution is obtained using two methods — the travelling wave method and the Fourier-transform approach. The results obtained by both methods suggest that the negative damping of the clamped–pinned systems and positive damping of the pinned–clamped systems, observed by previous investigators for any arbitrarily small flow velocity, are simply numerical artefacts; this is reinforced by energy considerations, in which the work done by the fluid on the shell is shown to be zero. Hence, it is concluded that both systems are conservative.     

Molecular dynamics study of size, temperature and rate dependent thermomechanical properties of copper nanofilms

Molecular dynamics simulations are performed to study the thermomechanical properties of copper nanofilms at different temperatures and extremely-high loading rates. The results show a drastic temperature softening effect on the film strength and modulus. The increase of strain rate could result in a much higher strength while the modulus is relatively less affected. It is shown, based on the stress results, that the observed “smaller is softer” and “smaller is stronger” behaviors of nanofilms might be due to the surface plasticity and the volumetric dislocations, respectively. It is also found that the thinner a nanofilm, the smaller the thermal expansion coefficient. The present work reveals that the quasistatic thermomechanical properties of bulk copper at room temperature might be inadequate for the continuum-based study of thermomechanical response of copper nanofilms due to ultrafast laser heating.     

Effect of gravity on the vibration of vertical cantilevers

The free vibration of a vertically-oriented, thin, prismatic cantilever is influenced by weight. That is, the natural frequencies (and to a lesser extent, mode shapes) are affected by the application of a linearly varying axial load. A beam with an “upward” orientation, i.e., with the free end above the clamped end, will experience a de-stiffening effect, up to the point of self-weight buckling (at zero effective stiffness). A beam in a “downward” orientation will be stiffened by the weight of the beam. This technical note describes some simple experiments on very slender strips and their (vertical) orientation and shows a close correlation with theory.     

Modeling and validation of hitch loading effects on tractor yaw dynamics

This paper develops a yaw dynamic model for a farm tractor with a hitched implement, which can be used to understand the effect of tractor handling characteristics for design applications and for new automated steering control systems. Dynamic equations which use a tire-like model to capture the characteristics of the implement are found to adequately describe the tractor implement yaw dynamics. This model is termed the “3-wheeled” Bicycle Model since it uses an additional wheel (from the traditional bicycle model used to capture lateral dynamics of passenger vehicles) to account for the implement forces. The model only includes effects of lateral forces as it neglects differential longitudinal or draft forces between inner and outer sides of the vehicle. Experiments are taken to verify the hitch model using a three-dimensional force dynamometer. This data shows the implement forces are indeed proportional to lateral velocity and that differential draft forces can be neglected as derived in the “3-wheeled” Bicycle Model. Steady state and dynamic steering data are used for implements at varying depths and speeds to quantify the variation in the hitch loading. The dynamic data is used to form empirical transfer function estimates (ETFEs) of the implements and depths in order to determine the coefficients used in the “3-wheeled” Bicycle Model. Changes in a single parameter, called the hitch cornering stiffness, can capture the various implement configurations. Finally, a model that includes front wheel drive forces is derived. Experiments are taken which provide a preliminary look into the effect of four-wheel drive traction forces, and show a difference with two-wheel versus four-wheel drive, on the yaw dynamics of a tractor with the hitched implement.     

Fatigue crack growth in high and low strength steel under torsional loading

During loading of a crack in mode III the crack surfaces in contact slide against each other giving rise to friction, abrasion and mutual support, thereby reducing the effective stress at the crack tip (“sliding mode crack closure”). This phenomenon was investigated in a high strength steel (AISI 4340) and in a low strength steel (AISI C1018) in circumferentially notched specimens under pure cyclic torsion and combined loading (cyclic torsion plus static axial load). The influence of sliding mode crack closure on fatigue crack propagation is shown and “true” crack growth values (without the sliding mode crack closure influence) are determined on the basis of an extrapolation procedure. Explanations are given for causes of the various fracture modes observed, such as “factory roof” fracture, macroscopically flat mode III fracture and “lamella” fracture. Finally the scientific and technical importance of sliding mode crack closure is demonstrated.     

A “quasi-elastic” affine formulation for the homogenised behaviour of nonlinear viscoelastic polycrystals and composites

The derivation of the overall behaviour of nonlinear viscoelastic (or rate-dependent elastoplastic) heterogeneous materials requires a linearisation of the constitutive equations around uniform per phase stress (or strain) histories. The resulting Linear Comparison Material (LCM) has to be linear thermoviscoelastic to fully retain the viscoelastic nature of phase interactions. Instead of the exact treatment of this LCM (i.e., correspondence principle and inverse Laplace transforms) as proposed by the “classical” affine formulation, an approximate treatment is proposed here. First considering Maxwellian behaviour, comparisons for a single phase as well as for two-phase materials (with “parallel” and disordered morphologies) show that the “direct inversion method” of Laplace transforms, initially proposed by Schapery (1962), has to be adapted to fit correctly exact responses to creep loading while a more general method is proposed for other loading paths. When applied to nonlinear viscoelastic heterogeneous materials, this approximate inversion method gives rise to a new formulation which is consistent with the classical affine one for the steady-state regimes. In the transient regime, it leads to a significantly more efficient numerical resolution, the LCM associated to the step by step procedure being no more thermoviscoelastic but thermoelastic. Various comparisons for nonlinear viscoelastic polycrystals responses to creep as well as relaxation loadings show that this “quasi-elastic” formulation yields results very close to classical affine ones, even for high contrasts.     

Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity

In the context of single-crystal strain gradient plasticity, we focus on the simple shear of a constrained strip in order to study the effects of the material parameters possibly involved in the modelling. The model consists of a deformation theory suggested and left undeveloped by Bardella [(2007). Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved. Int. J. Plasticity 23, 296–322] in which, for each glide, three dissipative length scales are considered; they enter the model through the definition of an effective slip which brings into the isotropic hardening function the relevant plastic strain gradients, averaged by means of a p-norm. By means of the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]), the model further involves an energetic material length scale. The application suggests that two dissipative length scales may be enough to qualitatively describe the size effect of metals at the microscale, and they are chosen in such a way that the higher-order state variables of the model be the dislocation densities. Moreover, we show that, depending on the crystallography, the size effect governed by the defect energy may be different from what expected (based on the findings of [Bardella, L., 2006. A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 54, 128–160] and [Gurtin et al. 2007. Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878]), leading mostly to some strengthening. In order to investigate the model capability, we also exploit a Γ-convergence technique to find closed-form solutions in the “isotropic limit”. Finally, we analytically show that in the “perfect plasticity” case, should the dissipative length scales be set to zero, the presence of the sole energetic length scale may lead, as in standard plasticity, to non-uniqueness of solutions.     

Implicit BEM formulations for usual and sensitivity problems in elasto-plasticity using the consistent tangent operator concept

This paper presents boundary element method (BEM) formulations for usual and sensitivity problems in (small strain) elasto-plasticity using the concept of the local consistent tangent operator (CTO). “Usual” problems here refer to analysis of nonlinear problems in structural and solid continua, for which Simo and Taylor first proposed the use of the CTO within the context of the finite element method (FEM). A new implicit BEM scheme for such problems, using the CTO, is presented first. A formulation for sensitivity analysis follows. It is shown that the sensitivity of the strain increment, associated with an infinitesimal variation of some design parameter, solves a linear problem which is governed by the (converged value of the) same global CTO as the one that appears in the usual problem. Numerical results for both usual and sensitivity problems are shown for a one-dimensional example. They demonstrate the effectiveness of the present approach. In particular, accurate sensitivities with respect to material parameters (e.g., exponent of the power-type hardening law) are obtained even with few integration cells and for large load increments.     

POST-BUCKLING AND IMPERFECTION SENSITIVITY OF A COLUMN WITH AN EQUILATERAL TRIANGULAR CROSS-SECTION IN THE PLASTIC RANGE

The asymmetric initial post-buckling corresponding to the lowest bifurcation load andthe imperfection sensitivity are analysed in detail for a compressed simply supported column with anequilateral triangular cross-section in the plastic range.The effect of elastic unloading is taken into ac-count in the analysis.The asymptotic relations,including high order terms,among the load,the am-plitude of imperfection and the amplitude of the bifurcation mode are realized.The results show thatthe maximum supported load is very sensitive to imperfection.     

A damage model for concrete beams in compression

A usual class of phenomenological “1 − d” damage models is revisited: starting from experimental observations showing that the Young’s and tangent moduli evolve linearly during a compression test on concrete, an appropriate expression for the damage threshold is obtained, which differs from the usual ones. The linear variation of the moduli with respect to the strain variation allows to simplify the incremental equilibrium equations and to improve the accuracy of the simulation.     

Effect of CNT length and CNT-matrix interphase in carbon nanotube (CNT) reinforced composites

The load transfer mechanisms and effective moduli of single-walled nanotube (SWNT) reinforced composites are studied using a continuum model. A “critical” fiber length is defined for full load transfer by numerically evaluating the strain-energy-changes for different fiber lengths. The effective longitudinal Young’s modulus and bulk modulus of the composite are derived. The effect of the interphase is also discussed. The results indicate the fiber length is critical both to the load transfer efficiency and effective moduli of the composite. The SWNT-matrix interphase plays an important role in load transfer efficiency but affects the effective moduli only slightly.     

Modelling of particle-wall collisions in confined gas-particle flows

This paper demonstrates that numerical simulations of confined particulate two-phase flows require a detailed modelling of particle—wall collisions which includes the wall surface structure and the particle shape. These effects are taken into account by “irregular bouncing” models which are based on the statistical treatment of the collision process. In this study, results obtained using various “irregular bouncing” models based on the impulse equations for a particle—wall collision are considered and compared with experimental observations. The wall roughness is simulated by assuming that the particle collides with a virtual wall which has a randomly distributed inclination with respect to the plane, smooth wall. A Gaussian distribution for this random inclination showed the best agreement with experimental results. Numerical predictions of a turbulent two—phase flow in a vertical channel, where the particle phase is treated using a Lagrangian approach, showed that the different models applied for a particle-wall collision have a strong effect on the particle velocity fluctuations and the mass flux profiles in the region of fully developed flow. The numerical simulations using the irregular bouncing models yielded considerably higher values for the particle velocity fluctuations, which also agreed better with the experimental values. This effect was most pronounced for large particles, where the distance they need to respond to the fluid flow is larger than the characteristic dimension of the confinement. On the other hand, the motion of small particles is less affected by the choice of the wall-collision model. These effects of the wall roughness on the velocity fluctuations of the dispersed phase have not been considered in previous studies using irregular bouncing models.     

The shakedown load boundary of an elastic-perfectly plastic structure

In the hypothesis of small displacements and combined time-variable/steady loads, the geometrical-mechanical properties of the shakedown load boundary are investigated. It is shown that, in the load space, the shakedown load boundary plays the role of yield surface, and that a certain plastic strain accumulation vector—characterizing some impending inadaptation collapse mechanism—obeys the normality rule, whereas a specific form of the maximum plastic work theorem constitutes an effective tool for the evaluation of the shakedown limit load corresponding to a specified inadaptation collapse mode. The equations governing the state of the structure at the shakedown limit are provided and the related collapse mechanism is shown to specify the shape of the steady-state response of the structure to a periodic load enveloping the load domain with an intensity slightly above the shakedown limit.

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Sommario Nell'ipotesi di piccoli spostamenti e di plasticità perfetta, si studiano le proprietà geometrico-meccaniche della frontiera del dominio dei carichi di adattamento. Si mostra come, nello spazio dei carichi, la suddetta frontiera giuoca il ruolo di superficie di plasticizzazione ed un particolare vettore di deformazione plastica accumulata—che caratterizza il meccanismo di non adattamento—segue la legge della normalità, mentre una forma specifica del teorema del massimo lavoro plastico costituisce un valido strumento per la valutazione del carico limite corrispondente ad un dato meccanismo di non adattamento. Si forniscono altresì le equazioni che governano lo stato della struttura al limite di adattamento, mostrando la capacità del relativo meccanismo di collasso a rappresentare la risposta a lungo termine della struttura a dei carichi periodici inviluppanti il dominio dei carichi con una intensità un poco al di sopra del limite di adattamento.

     

正交铺设层合板的蠕变屈曲分析

研究了正交铺设层合板的蠕变失稳问题。为了更好地模拟实际情况，在单层板的本构关系中，材料各主方向模量的松也时间均取不同值，并在建立控制方程时考虑了横向剪切变形的影响，通过理论分析，得到了粘弹性层合板的瞬时弹性临界载荷和持久临界载荷，并在算例中首次利用时间增量方法得到了有初始度层合板在长期受地的蠕变变表，计算结果表明了持久临界载荷对于粘弹性层合板的具体含义，从而使粘弹性层合板的为屈曲问题有了较为完整的     

Constitutive equations for hot-working of metals

Elevated temperature deformation processing - “hot-working,” is an important step during the manufacturing of most metal products. Central to any successful analysis of a hot-working process is the use of appropriate rate and temperature-dependent constitutive equations for large, interrupted inelastic deformations, which can faithfully account for strain-hardening, the restoration processes of recovery and recrystallization and strain rate and temperature history effects. In this paper we develop a set of phenomenological, internal variable type constitutive equations describing the elevated temperature deformation of metals. We use a scalar and a symmetric, traceless, second-order tensor as internal variables which, in an average sense, represent an isotropic and an anisotropic resistance to plastic flow offered by the internal state of the material. In this theory, we consider small elastic stretches but large plastic deformations (within the limits of texturing) of isotropic materials. Special cases (within the constitutive framework developed here) which should be suitable for analyzing hot-working processes are indicated.