Finite amplitude, free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

The initiation and growth of adiabatic shear bands

A simple version of thermo/viscoplasticity theory is used to model the formation of adiabatic shear bands in high rate deformation of solids. The one dimensional shearing deformation of a finite slab is considered. For the constitutive assumptions made in this paper, homogeneous shearing produces a stress/strain response curve that always has a maximum when strain and rate hardening, plastic heating, and thermal softening are taken into account. Shear bands form if a perturbation is added to the homogeneous fields just before peak stress is obtained with these new fields being used as initial conditions. The resulting initial/boundary value problem is solved by the finite element method for one set of material parameters. The shear band grows slowly at first, then accelerates sharply, until finally the plastic strain rate in the center reaches a maximum, followed by a slow decline. Stress drops rapidly throughout the slab, and the central temperature increases rapidly as the peak in strain rate develops.     

Finite amplitude,free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

Low-dimensional intrinsic material functions for nonlinear viscoelasticity

Rheological material functions are used to form our conceptual understanding of a material response. For a nonlinear rheological response, the possible deformation protocols and material measures span a high-dimensional space. Here, we use asymptotic expansions to outline low-dimensional measures for describing leading-order nonlinear responses in large amplitude oscillatory shear (LAOS). This amplitude-intrinsic regime is sometimes called medium amplitude oscillatory shear (MAOS). These intrinsic nonlinear material functions are only a function of oscillatory frequency, and not amplitude. Such measures have been suggested in the past, but here, we clarify what measures exist and give physically meaningful interpretations. Both shear strain control (LAOStrain) and shear stress control (LAOStress) protocols are considered, and nomenclature is introduced to encode the physical interpretations. We report the first experimental measurement of all four intrinsic shear nonlinearities of LAOStrain. For the polymeric hydrogel (polyvinyl alcohol - Borax) we observe typical integer power function asymptotics. The magnitudes and signs of the intrinsic nonlinear fingerprints are used to conceptually model the mechanical response and to infer molecular and microscale features of the material.     

Shear waves in an elastic-plastic material due to a supersonic surface step load

A step shear load moves steadily on the surface of an elastic-plastic half space at a speed exceeding the elastic shear wave speed of the material. The orientation of the shear traction is such that the deformation is two-dimensional antiplane strain. Two different representations of the rate independent elastic-plastic material response are considered. The first material model is based on the associated flow rule and the Mises yield condition with isotropic hardening, whereas the second model is based on a particular flow theory of plasticity which represents incremental behavior at a corner of the instantaneous yield surface. Both models predict the same response under the same proportional loading. The stress history experienced by a typical material particle during passage of the load step is determined, and the variation of final strain with the magnitude of the load step is calculated. One conclusion resulting from comparison of results for the two material models for this problem is that the influence of yield surface vertex formation is not significant.     

Étude des temps de chauffage et de refroidissement des alliages à mémoire de forme (AMF)Heating and cooling time studies of shape memory alloys (SMA)

The knowledge of response times of active material is essential for their efficient use as actuators. In this paper the heating and cooling times of a shape memory alloy wire (NiTi) under a constant load are predicted by the integration of the corresponding heat equation. The comparison with a ‘fictitious’ material with the same material characteristics but without phase transformation shows that the response times are longer for the SMA (around 2.7 times for heating and 1.5 times for cooling). To cite this article: N. Chaillet et al., C. R. Mecanique 332 (2004).     

Anisotropic conduction of heat in a flowing polymeric material

In this paper a theory is presented which relates the thermal conductivity tensor of an amorphous polymeric material to the history of deformation of the material. The basis of the theory is formed by the network theory for polymeric materials. It will be shown that the results obtained here are in good agreement with experimental results on rubber. The effect of anisotropic heat conduction on the flow of a polymeric material will be demonstrated by the simple example of viscous heating in shear flow.Presented at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Estimation of stochastic nonlinear dynamic response of rock-fill dams with uncertain material parameters for non-stationary random seismic excitation

This research investigates the effect of uncertain material parameters on the stochastic, dynamic response of a rock-fill dam-foundation system subjected to non-stationary random excitation. The uncertain material parameter of particular interest is the shear modulus, developed from a lognormal distribution model. The stochastic seismic response model of the dam-foundation system, with uncertain material parameters and subjected to random loads is the result of a Monte Carlo simulation method. The nonlinear behavior model arises from an equivalent linear method, which considers the nonlinear variation of soil shear modulus and soil damping as a function of shear strain. Specification of the non-stationary stochastic process arises from a simulation method, which generates artificial earthquake accelerograms obtained from the product of a deterministic function of time and a stationary process. The artificial earthquake ground acceleration records reflect the characteristics of soft, medium and firm soil types. Comparison of the numerical results from these approaches provides stochasticity in earthquake seismic excitation and randomness in material parameter (shear modulus) cases. Further, the results indicate that both these cases generally influence the nonlinear dynamic response of rock-fill dams to a non-stationary seismic excitation.     

Mechanical properties of 6061 Al-Mg-Si alloy after very rapid heating

The effects of time, temperature and strain rate on the yield strength determined at elevated temperature have been investigated for 6061-T6 Al-Mg-Si alloy. To achieve very short times-at-temperature, nanosecond pulse heating produced by electron beam energy deposition was used along with one-dimensional stress-wave loading. When a relatively thick sample is heated in this way it cannot expand on the same time scale as the temperature increase. As a result, stress relief waves propagate in the material after energy deposition, and this deformation occurring at high strain-rates and elevated temperatures produces microstructural changes that reduce the strength of the alloy on the time scale of a few μs. This strength reduction occurs in addition to that due to the lowered shear modulus at-temperature, and is an essentially permanent change reflected in the greatly reduced room temperature strength of the material following nanosecond pulse heating.If the material is heated slowly enough so that the sample can expand as the temperature increases, and if the soaking time at temperature is less than required for microstructural changes in the alloy (e.g. approximately 3 s at 260°C), the yield strength measured at-temperature under wave propagation conditions drops in proportion to the shear modulus. In addition, the yield strength measurement is sensitive to the rate of deformation at elevated temperatures even for short times-at-temperature. The nature of this sensitivity is discussed in terms of thermally-activated dislocation motion.     

Response of 3-D and Plain-Weave S-2 Glass Composites in Out-of-Plane Tension and Shear

An extensive suite of experiments was conducted to characterize the mechanical response of an S-2 glass composite. The primary interest was the response of a 3-D composite, consisting of unidirectional (non-woven) layers of glass fibers interlaced by through-thickness Z-yarns. A plain-weave material was also characterized for comparison purposes. Additionally, epoxy-only specimens were fabricated to assist in understanding the contribution of the SC-15 epoxy resin in the response of the composite system. Two new specimen geometries (torsion and hourglass) were developed specifically for this characterization effort. The response of these specimens provides considerable insight into the failure mechanics of the plain weave and 3-D weave composites. It was shown that the matrix material has an elastic-plastic response, but with different strengths in tension and torsion. The response of the composite in tension is controlled by the epoxy until failure at the glass-resin interface. The strength falls to zero for the plain-weave composite, but the Z-yarns can support tensile stress until the yarns begin to fail. The fibers contribute to the elastic stiffness in shear for the plain-weave material, but the failure strength in shear is the same as the matrix. The 3-D weave composite also fails at the failure strength of the matrix, but retains some shear strength because of the Z-yarns.     

An analysis of the shear failure of rigid-linear hardening beams under impulsive loading

A theoretical rigid-plastic analysis for the dynamic shear failure of beams under impulsive loading is presented when using a travelling plastic shear hinge model which takes into account material strain hardening. The maximum dynamic shear strain and shear strain-rate can be predicted in addition to the permanent transverse deflections and other parameters. The conditions for the three modes of shear failure, i.e., excess deflection failure, excess shear strain failure and adiabatic shear failure are analyzed. The special case of an infinitesimally small plastic zone is discussed and compared with Nonaka's solution for a rigid, perfectly plastic material. The results can also be generalized to examine the dynamic response of fibre-reinforced beams.     

The Mullins Effect in a Pure Shear

The Mullins effect in a rubberlike material subjected to a pure shear deformation is studied in the context of a recent theory of stress-softening for incompressible materials proposed by Beatty and Krishnaswamy. Some general technical results characterizing the mechanical response are presented. These show that the theory delivers results consistent with the overall behavior expected of a Mullins material, but usually exhibited in uniaxial extension or equibiaxial stretch experiments. The extent of stress-softening in a pure shear is shown to be much less than that due to an equibiaxial deformation, and only slightly greater than the degree of stress-softening induced by an uniaxial deformation, all to the same stretch. The Mullins effect in an equivalent simple shear deformation, even one having a rather large angle of shear, is small. The simple shear is the least damaging deformation among all of those mentioned here. Some graphical results, based on a special class of stress-softening materials applied to two parent material models – the familiar Mooney–Rivlin and a certain biotype material model, illustrate the general conclusions obtained for arbitrary Mullins materials. The inflation of a biomaterial membrane preconditioned in a pure shear deformation demonstrates the familiar stress-softening phenomenon observed in the inflation of a balloon.     

On the application of the additive decomposition of generalized strain measures in large strain plasticity

In the present paper two thermodynamically consistent large strain plasticity models are examined and compared in finite simple shear. The first model (A) is based on the multiplicative decomposition of the deformation gradient, while the second one (B) on the additive decomposition of generalized strain measures. Both models are applied to a rigid-plastic material described by the von Mises-type yield criterion. Since both models include neither hardening nor softening law, a constant shear stress response even for large amounts of shear is expected. Indeed, the model A exhibits the true constant shear stress behavior independent of the elastic material law. In contrast, the model B leads to a spurious shear stress increase or drop such that its applicability under finite shear deformations may be questioned.     

粘贴压电层功能梯度材料Timoshenko梁的热过屈曲分析

研究了上下表面粘贴压电层的功能梯度材料Timoshenko梁在升温及电场作用下的过屈曲行为。在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了压电功能梯度Timoshenko层合梁在热-电-机械载荷作用下的几何非线性控制方程。其中,假设功能梯度的材料性质沿厚度方向按照幂函数连续变化,压电层为各向同性均匀材料。采用打靶法数值求解所得强非线性边值问题,获得了在均匀电场和横向非均匀升温场内两端固定Timoshenko梁的静态非线性屈曲和过屈曲数值解。并给出了梁的变形随热、电载荷及材料梯度参数变化的特性曲线。结果表明,通过施加电压在压电层产生拉应力可以有效地提高梁的热屈曲临界载荷,延缓热过屈曲发生。由于材料在横向的非均匀性,即使在均匀升温和均匀电场作用下,也会产生拉-弯耦合效应。但是对于两端固定的压电-功能梯度材料梁,在横向非均匀升温下过屈曲变形仍然是分叉形的。     

Nonlinear shear behavior and interlaminar shear strength of unidirectional polymer matrix composites: A numerical study

Detailed finite element implementation is presented for a recently developed technique (He et al., 2012) to characterize nonlinear shear stress–strain response and interlaminar shear strength based on short-beam shear test of unidirectional polymeric composites. The material characterization couples iterative three-dimensional finite element modeling for stress calculation with digital image correlation for strain evaluation. Extensive numerical experiments were conducted to examine the dependence of the measured shear behavior on specimen and test configurations. The numerical results demonstrate that consistent results can be achieved for specimens with various span-to-thickness ratios, supporting the accurate material properties for the carbon/epoxy composite under study.     

Oscillatory squeezing flow of a biological material

Large-amplitude oscillatory squeezing flow data are reported for a complex biological material, which is highly shear-thinning in oscillatory shear flow. This soft tissue has a linear viscoelastic limit at a strain of approximately 0.2%. The oscillatory squeezing flow data at large strain are analyzed using two constitutive models: a bi-viscosity Newtonian model, and a non-linear Maxwell model. It is found that although both models may have the same response in shape, the later matches with our non-linear experimental data better. It is also concluded that the non-linear response of the material in large amplitude oscillatory flow is mainly due to the shear thinning of the material. Received: 9 February 2000/Accepted: 22 February 2000     

Unsteady homothermal motions of fluids and isotropic solids

Motions of a certain type are shown to be possible in every compressible or incompressible, homogeneous, isotropic simple material, in the absence of body force. These motions have a specific spatial dependence and the nature of the time dependence, for a particular material, is determined by the material response function or functional for the shear stress in simple shear. It is shown further that motions of this type, and spatially homogeneous, time dependent temperatures, can be supported without application of body force or radiative supply of heat.     

Dynamic Experiments using Simultaneous Compression and Shear Loading

Material characterization at high strain rates under simultaneous compression and shear loading has been a challenge due to the differing normal and shear wave speeds. An experimental technique utilizing the compression Kolsky bar apparatus was developed to apply dynamic compression and shear loading on a specimen nearly simultaneously. Synchronization between the compression and shear loading was realized by generating the torsion wave near the specimen which minimizes the time difference between the arrival of the compression and torsion waves. This modified Kolsky bar makes it possible to characterize the dynamic response of a material to combined compression and shear impact loading. This method can also be applied to study dynamic friction behavior across an interface under controlled loading conditions. The feasibility of this method is demonstrated in the dynamic characterization of a simulant polymer bonded explosive material.     

Interlaminar shear strength of cloth-reinforced composites

A test specimen to define the interlaminar shear strength of cloth-reinforced composite materials is developed. The specimen is a hollow circular cylinder which is subjected to torsion. The experiment is employed for the determination of the warp-normal shear strength of a graphite-fiber carbon-matrix composite material. It is demonstrated that the proper failure mode takes place, while the problems associated with the use of strain gages on this porous material, nonlinearity, difference in tension and compression properties, and the influence of clamping effects are all discussed. The proposed experiment appears to be ideally suited to study the interlaminar shear response of cloth-reinforced composites.     

Time-resolved rheometry

Applicability and limits of time-resolved rheometry have been analyzed for polymers which undergo change during a theological measurement. Processes such as gelation, phase transition, polymerization or decomposition affect the molecular mobility in these polymers and therefore the rheological experiment. We propose to choose the well known effect of heating (or cooling) during the relaxation and analyze it as a paradigm for rheometry on samples with changing molecular mobility. The temperature change does not cause permanent changes in sample structure, but it affects the molecular mobility and it significantly interferes with the measurement if the temperature changes occur too fast. In this study, time-resolved mechanical spectroscopy (TRMS) was used to experimentally investigate the effect of heating on the relaxation behavior of a typical polycarbonate sample. Each data point in a cyclic frequency sweep (CFS) was taken at a different state of the material; the data were interpolated using an interactive computer program. In this fashion, a single TRMS experiment yielded a master curve over eight decades. A model for relaxation under non-isothermal conditions showed the limitations of TRMS. It could be demonstrated that TRMS worked well for sufficiently small mutation numbers, i.e., for sufficiently small changes during the measurement. A critical mutation number of 0.9 was determined for the non-isothermal case beyond which the material response became non-linear. This corresponds to a calculated relative change of the shear stress amplitude of about 90%.Dedicated to Prof. Ken Walters on the occasion of his 60th birthday.