Crack initiation behavior of functionally graded piezoelectric material: prediction by the strain energy density criterion

Transient response of a functionally graded piezoelectric medium is considered for a through crack under the mixed-mode in-plane mechanical and electric load. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. The energy density factor criterion is applied to obtain the maximum of the minimum energy density factor. This determines the direction of crack initiation. Numerical results display the effects of material constants, loading combination parameter, mechanical loading angle and material gradient parameter on the possible fracture behavior.     

具有线性介电常数的Ogden型介电弹性体的本构关系和机电稳定性

应用多材料常数的Ogden弹性应变能函数分析了介电弹性体的力学行为,研究了介电弹性体的机电稳定性.数值结果表明,通过对材料系数(如材料常数比和电致伸缩系数等)的恰当调节可以使得介电弹性体材料或介电弹性体结构更趋稳定.这些有益于深入理解介电弹性体的机电稳定性行为,进而设计恰当的介电弹性体器件.     

A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this paper, a novel size-dependent functionally graded(FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton's principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.     

Homogenization of rectangular cross-section fibre-reinforced materials: bending–torsion effects

We study the homogenization of an elastic material in contact with periodic parallel elastic rectangular cross-section fibres of higher rigidity. The interactions between the matrix and the fibres are described by a local adhesion contact law with interfacial adhesive stiffness parameter depending on the period. Assuming that the Lamé constants in the fibres and the stiffness parameter have appropriate orders of magnitude, we derive a class of energy functionals involving extension, flexure and torsion terms.     

A note on isoenergetic stability

In this note we consider certain two-degree-of-freedom Hamiltonian systems which may be regarded as perturbations of integrable systems governed by a real parameter ε. We wish to study the stability, at fixed energy, of certain periodic solutions. Two constants are defined, computable in terms of the original Hamiltonian function and the energy. The main theorem then states that if these constants are not zero, the periodic solutions are isoenergetically stable for sufficiently small ε. The proof is an application of the Twist Theorem of Kolmogorov-Arnol'd-Moser. By way of illustration, we apply the theorem to a mechanical system consisting of coupled non-linear oscillators. The periodic solutions are the “normal modes” and ε governs the non-linearity of the system. One obtains stability criteria for arbitrary energies and small ε, or, alternatively, for arbitrary ε and small energies.     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

Performance Dependent Model for normal and high strength concretes

A new constitutive formulation, the so-called Performance Dependent Model valid for normal and high strength concretes is presented. The distinctive aspect of the proposed model is the consideration of relevant properties of concrete mix components in the evaluation of the involved material performance or quality at the macroscopic stand point. In this way, the composite features of concrete are appropriately taken into account.The model maximum strength surface is defined by means of the Performance Dependent Failure Criterion proposed by the authors in previous works. Concrete behaviors in pre and post peak regimes are modeled with a non uniform hardening law and an isotropic softening rule, respectively. To realistically reproduce the concrete ductility in pre and post peak regimes under different load scenarios, the hardening and softening laws are defined in terms of the acting confining pressure. Concrete dilatancy behavior is approached by means of a volumetric non associative flow rule. The softening law is embedded in fracture energy concepts for mode I and II types of failure. The model considers two main input material parameters: the uniaxial compressive strength and the performance parameter, a quality index defined in the context of the Performance Dependent Failure Criterion.The proposed constitutive model is able to capture the substantial differences in the failure behavior of normal and high strength concretes as well as of concretes with the same compressive strength but different mix components. The predictive capability of the model is demonstrated in the numerical analyses included in this paper where the numerical predictions are compared with experimental results related to concrete specimens of different qualities and subjected to stress histories under both compressive and tensile regimes.     

The effect of a spherical inclusion in an anisotropic solid

Summary A spherical domain within an anisotropic crystalline material is considered to have elastic constants differing from those of the remainder of the material; the particular case where the constants vanish within the sphere represents a cavity. The elastic fields inside and immediately outside the spherical domain, together with the interaction energy, are calculated for the case of a uniform stress applied at infinity. Specific examples are given for aluminum, copper, and pyrite, and numerical results are compared with those for isotropic material. The tensile stress concentration is larger for aluminum than for isotropic material while the opposite is true for pyrite. Similarly, the interaction energy of the inhomogeneity is larger for an anisotropic material than an isotropic material, but in pyrite the reverse is found.     

Optimization of the strain energy density in linear anisotropic elasticity

The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.     

On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall

In this paper we solve the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. We use a non-linear slip model relating the shear stress to the velocity at the wall and exhibiting a maximum and a minimum. We assume that the material parameters in the slip equation are such that multiple steady-state solutions do not exist. The stability of the steady-state solutions is investigated by means of a one-dimensional linear stability analysis and by numerical calculations. The instability regimes are always within or coincide with the negative-slope regime of the slip equation. As expected, the numerical results show that the instability regimes are much broader than those predicted by the linear stability analysis. Under our assumptions for the slip equation, the Newtonian solutions are stable everywhere. The interval of instability grows as one moves from the Newtonian to the upper-convected Maxwell model. Perturbing an unstable steady-state solution leads to periodic solutions. The amplitude and the period of the oscillations increase with elasticity.     

An ‘extended’ volumetric/deviatoric formulation of anisotropic damage based on a pseudo-log rate

Following a framework of elastic degradation and damage previously proposed by the authors, an ‘extended’ formulation of orthotropic damage in initially isotropic materials, based on volumetric/deviatoric decomposition, is presented. The formulation is founded on the concept of energy equivalence and makes use of second-order symmetric tensor damage variables. It is characterized by fourth-order damage-effect tensors (relating nominal to effective stresses and strains) built from the underlying second-order damage tensors and decomposed in product-form in isotropic and anisotropic parts. The formulation is developed in two steps. First, secant relations are established. In the isotropic case, the model embeds a path parameter allowing to range between pure volumetric to pure deviatoric damage. With the two undamaged material constants this makes a total of three constant parameters plus an evolving scalar damage variable, giving rise to a four-parameter model with two varying isotropic material coefficients. In the anisotropic case, the model is still characterized by the same three material constants plus three evolving variables which are the principal values of a second-order damage tensor. This leads to a six-parameter restricted form of orthotropic damage. In the second step, damage evolution rules are formulated in terms of a pseudo-logarithmic rate of damage. This allows to define meaningful conjugate forces that constitute a feasible space in which loading functions and damage evolution rules can be defined. The present ‘extended’ formulation is closed by the derivation of the tangent stiffness.     

平面碰撞与强激光加载下金属铝的层裂行为

在轻气炮和神光Ⅱ强激光装置上开展了金属铝的层裂实验。针对激光打靶层裂实验中样品自由面速度剖面后期振荡容易丢失问题,改进靶设计,获得很好效果。利用轻气炮加载和强激光加载层裂实验应变率的显著差异,并通过数值模拟,讨论了在建立具有预测能力的理论建模中需要关注的损伤成核、演化与汇合问题中的材料特性与应变率相关特性因素。结果表明,对于我们以前建立的动态损伤与断裂模型,微孔洞成核的平均半径、阈值压力、成核速率相关参数以及微孔洞长大的阈值压力等具有材料特性属性,但微孔洞的表面能以及决定材料发生完全层裂的临界损伤度等具有明显的应变率效应。另外,分析还发现,虽然层裂强度具有明显的应变率效应,但是在样品层裂当地,样品由持续拉伸向收缩转变的临界行为,取决于一个很小的临界损伤,这个临界值很可能是材料常数,与应变率无关。     

Transient anti-plane crack problem for two bonded functionally graded piezoelectric materials

In this paper, the dynamic anti-plane crack problem for two bonded functionally graded piezoelectric materials is considered. The crack is perpendicular to the interface and assumed to be electrically impermeable or permeable. Integral transforms are employed to reduce the problem to Cauchy singular equations that can be solved numerically. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor and the energy density factor are studied. It is found that for the impermeable case, the normalized dynamic stress intensity factor may increase or decrease in different time domains determined by the sign and magnitude of λ.     

Suppression or enhancement of interfacial instability in two-layer plane Couette flow of FENE-P fluids past a deformable solid layer

The linear stability of two-layer plane Couette flow of FENE-P fluids past a deformable solid layer is analyzed in order to examine the effect of solid deformability on the interfacial instability due to elasticity and viscosity stratification at the two-fluid interface. The solid layer is modeled using both linear viscoelastic and neo-Hookean constitutive equations. The limiting case of two-layer flow of upper-convected Maxwell (UCM) fluids is used as a starting point, and results for the FENE-P case are obtained by numerically continuing the UCM results for the interfacial mode to finite values of the chain extensibility parameter. For the case of two-layer plane Couette flow past a rigid solid surface, our results show that the finite extensibility of the polymer chain significantly alters the neutral stability boundaries of the interfacial instability. In particular, the two-layer Couette flow of FENE-P fluids is found to be unstable in a larger range of nondimensional parameters when compared to two-layer flow of UCM fluids. The presence of the deformable solid layer is shown to completely suppress the interfacial instability in most of the parameter regimes where the interfacial mode is unstable, while it could have a completely destabilizing effect in other parameter regimes even when the interfacial mode is stable in rigid channels. When compared with two-layer UCM flow, the two-layer FENE-P case is found in general to require solid layers with relatively lower shear modulii in order to suppress the interfacial instability. The results from the linear elastic solid model are compared with those obtained using the (more rigorous) neo-Hookean model for the solid, and good agreement is found between the two models for neutral stability curves pertaining to the two-fluid interfacial mode. The present study thus provides an important extension of the earlier analysis of two-layer UCM flow [V. Shankar, Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer: implications of fluid viscosity stratification, J. Non-Newtonian Fluid Mech. 125 (2005) 143–158] to more accurate constitutive models for the fluid and solid layers, and reaffirms the central conclusion of instability suppression in two-layer flows of viscoelastic fluids by soft elastomeric coatings in more realistic settings.     

Rate effects on toughness in elastic nonlinear viscous solids

A micromechanics-based constitutive relation for void growth in a nonlinear viscous solid is proposed to study rate effects on fracture toughness. This relation is incorporated into a microporous strip of cell elements embedded in a computational model for crack growth. The microporous strip is surrounded by an elastic nonlinear viscous solid referred to as the background material. Under steady-state crack growth, two dissipative processes contribute to the macroscopic fracture toughness—the work of separation in the strip of cell elements and energy dissipation by inelastic deformation in the background material. As the crack velocity increases, voids grow in the strain-rate strengthened microporous strip, thereby elevating the work of separation. In contrast, the energy dissipation in the background material decreases as the crack velocity increases. In the regime where the work of separation dominates energy dissipation, toughness increases with crack velocity. In the regime where energy dissipation is dominant, toughness decreases with crack velocity. Computational simulations show that the two regimes can exist in certain range of crack velocities for a given material. The existence of these regimes is greatly influenced by the rate dependence of the void growth mechanism (and the initial void size) as well as that of the bulk material. This competition between the two dissipative processes produces a U-shaped toughness-crack velocity curve. Our computational simulations predict trends that agree with fracture toughness vs. crack velocity data reported in several experimental studies for glassy polymers and rubber-modified epoxies.     

Features of Movement of a Rotational Body

Vertical motion of a rotational body in an air environment as a mechanical model of a rotochute is considered. It is assumed that, in the process of motion, the symmetry axis of the rotational body remains vertical and the rotational body itself rotates relative to this axis. The aerodynamic impact model is based on a quasistatic approach. Steady regimes of motion are identified, their stability is analyzed, and certain features of transition regimes are explored, including those related to the exchange between the energy of rotational motion and the energy of translational motion.     

梯度蜂窝面外动态压缩力学行为与吸能特性研究

蜂窝材料具有优异的抗冲击吸能特性。为进一步提高蜂窝材料的比吸能与压缩力效率,提出了一种几何参数或材料参数沿厚度方向梯度渐变的蜂窝材料模型,并针对六边形蜂窝构型研究了胞元壁厚和屈服强度梯度变化的蜂窝材料在面外动态压缩载荷下的力学行为与吸能特性。研究结果表明,通过调控梯度变化的指数,胞元壁厚或母体材料屈服强度的梯度设计均可有效降低初始峰值应力,并使蜂窝材料的比吸能和压缩力效率同时增大。研究结果可为蜂窝材料的防撞性优化设计提供新的思路。     

类桁架夹层板的等效弹性常数研究

从应变能等效出发,将具有周期性分布的夹层板的类桁架夹芯与各向异性连续材料等效,给出了相应的宏观等效弹性常数,进而用有限元方法计算了实际夹层板和等效夹层板的结构响应,用一个算例证明了该文方法的有效性.通过对类桁架夹芯的等效弹性常数的计算,结果表明该文方法可以得到较为准确的等效弹性常数,且较其它类型的均匀化方法大大降低了计算量.     

Crack-extension resistance of polycarbonate material

Crack-extension resistance for the polycarbonate material is examined by application of the strain energy density criterion and the incremental theory of plasticity. The energy state ahead of a slow moving crack in a three-point bend specimen is obtained for each load increment and used to determine the crack growth characteristics. The analytical results are displayed by plotting the strain energy density factor S as a function of crack length and compared with available experimental data on the polycarbonate material. Standard deviations and mean errors are computed for the experimentally measured and analytically determined values of S and are shown to be much lower than those based on the J-integral parameter. Modeling of the polycarbonate material by the theory of plasticity still remains much to be desired. Crack growth calculations are performed for a strain hardening parameter α = 0.85 that controls the proportion of isotropic and kinematic hardening. Nevertheless, the criterion dS/da = const. is shown to collate well with the experimental crack growth data.     

Third-order elastic,piezoelectric, and dielectric constants

The definitions of the third-order elastic, piezoelectric, and dielectric constants and the properties of the associated tensors are discussed. Based on the energy conservation and coordinate transformation, the relations among the third-order constants are obtained. Furthermore, the relations among the third-order elastic, piezoelectric, and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials. It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups, while all third-order dielectric constants are0 for 16 kinds of point groups as well as isotropic material. The reason is that some of the point groups are centrally symmetric, and the other point groups are high symmetry.These results provide the foundation to measure these constants, to choose material, and to research nonlinear problems. Moreover, these results are helpful not only for the study of nonlinear elastic and piezoelectric problems, but also for the research on flexoelectric effects and size effects.