Elastic materials with coincident principal stress and strain axes

In this note, a representation is derived for a Cauchy stress response function that is necessary and sufficient in order that a simple material without memory be such that the principal axes of strain and stress always coincide. It is found that the material need not be isotropic. However, if the material is in addition hyperelastic, so that the work done in any cyclic motion is zero, it is shown that the material is necessarily isotropic.     

Elastic springback in plates and beams formed by bending

A plane-strain solution is presented to predict the springback of plates formed by bending. The solution assumes that the strains are small, that the material is an isotropichardening von Mises material, and either that the stress-strain diagrams for tension and compression are identical or that an average diagram can be employed. The plastic portion of each stress-strain diagram is approximated by a finite number of straight lines. Both analytical solutions and experimental evidence indicate that the much simpler plane-stress solution can be used to predict springback for both beams and plates. Experimental data confirm that the type of die has a pronounced effect on the springback for beams and plates if the material exhibits a yield point.     

Interaction of ultrasonic waves with structural damage: A diffraction analogy

This study investigates the interaction of ultrasonic waves and structural damage, i.e., cracking and corrosion. It is shown that cracking and corrosion damage produces a diffraction pattern that resembles that associated with the traditional physics of wave motion. The extension of this hypothesis implies that it may be possible to use a simple ripple tank to investigate how to best detect/sense and size a given damage state, e.g., corrosion. We also find that cracking, and corrosion damage, has a significant effect on both the amplitude and period of the waveform and also on the local (apparent) refractive index of the material and that these effects have the potential to be used as damage indicators.     

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

Consider a cylinder (not necessarily of circular cross-section) that is composed of a hyperelastic material and which is stretched parallel to its axis of symmetry. Suppose that the elastic material that constitutes the cylinder is homogeneous, transversely isotropic, and incompressible and that the deformed length of the cylinder is prescribed, the ends of the cylinder are free of shear, and the sides are left completely free. In this paper it is shown that mild additional constitutive hypotheses on the stored-energy function imply that the unique absolute minimizer of the elastic energy for this problem is a homogeneous, isoaxial deformation. This extends recent results that show the same result is valid in 2-dimensions. Prior work on this problem had been restricted to a local analysis: in particular, it was previously known that homogeneous deformations are strict (weak) relative minimizers of the elastic energy as long as the underlying linearized equations are strongly elliptic and provided that the load/displacement curve in this class of deformations does not possess a maximum.     

泡沫铝夹芯双圆管结构的准静态轴向压缩性能研究

对泡沫铝夹芯双圆管结构的准静态轴压性能进行了实验研究,发现该新型结构的比质量吸能效率远远高于传统的泡沫铝夹芯单管,并接近甚至超过相应的空管结构;其内外管变形模式均与空管不同且受内外管组合的影响.本文讨论了它的变形机理,分析了外管壁厚对其压缩行为的影响,发现增大外管壁厚有利于增大结构的行程利用率,提高结构的比质量能量吸收...     

塑性节点法的研究及其在薄壳结构中的应用

本文采用变分原理及离散塑性流动定律假设研究了弹塑性有限元的新方法—塑性节点法及其所表示的物理意义。结果表明:该方法的物理实质是所述的塑性流动定理的离散。在一般意义下,本文对其进行了修正,典型板壳例题和复杂壳体结构的弹塑性有限元分析显示,计算结果与文献资料和实验数据符合较好。证明用文中方法进行壳体结构的弹塑性分析是行之有效的和可靠的。     

Convected derivatives for differential constitutive equations

To date, no differential constitutive equation has been proposed that agrees with each of four important experimental observations in relaxation after step shear strains: that the stress is often factorable into time and strain-dependent functions, that the strain-dependent function is shear thinning, that the ratio of first normal stress difference to shear stress equals the shear strain—that is, the Lodge-Meissner relationship holds, and that there is a negative second normal stress difference. The Johnson-Segalman model satisfies three of these, but fails to satisfy the Lodge-Meissner relationship, because in step strains the principal stress and strain axes do not rotate together. Using a mathematical technique for forcing co-rotation of stress and strain axes in an arbitrary deformation, we here present an explicit differential constitutive equation that satisfies all four of the above experimental observations.     

Metric Description of Singular Defects in Isotropic Materials

Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with the configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we showthat two dimensional defectswith trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in a Euclidean space.     

Applying the calculus of variations to circular motion

We demonstrate that having found a condition for the stationary points in multivariable calculus, that condition may be substituted back into the original equation and still yield the correct stationary points. With that, we emphasise the conditions that must be met in solving multivariable stationary point problems. We further use the analogy of the stationary points problem with finding stationary paths in calculus of variations to apply the latter to circular paths in an axisymmetric potential. Surprisingly, we find that this classical problem does not meet the required conditions. We subsequently derive new conditions that must be met and suggest a possible application.     

Stresses in rotating spheres grown by accretion

Analytical expressions are obtained for the stress field in a sphere that has grown by accretion while rotating about an axis at a speed which may vary during the accretion process. It is assumed that accretion occurs by the adherence of infinitesimal particles that are stress free at the instant of attachment and that the material of the sphere behaves elastically once it has accreted. The resulting stress field differs significantly from that predicted in a sphere that was ‘manufactured’ in a stress free state and then set to rotate. The implications of these differences are discussed in the context of the mechanisms for failure in accreted planetary bodies.     

Simulation of spatial multibody systems with friction

A formulation for modeling and simulation of friction effects in spatial multibody systems is presented. Constraint reaction forces on rigid bodies that are connected by joints that support friction are derived as functions of Lagrange multipliers, using D’Alembert’s principle. Friction forces acting on bodies are calculated as a function of joint geometry, constraint reaction forces that are functions of Lagrange multipliers, and relative velocities at constraint contact points that are determined by system kinematics. Friction forces are implemented in index 0 differential-algebraic equations of motion that are solved numerically using explicit and implicit numerical integration methods. Spatial examples are presented, yielding accurate results and demonstrating that the systems are not stiff, even in the presence of friction and stiction.     

炭纤维增强铜锡锌基复合材料的摩擦磨损性能研究

采用粉末冶金法制备了炭纤维增强铜锡锌基复合材料(Cf/Cu-Sn-Zn)和ZQSn663锡青铜, 并对其力学性能和摩擦磨损性能进行了对比研究. 结果表明: 当炭纤维的体积分数φf≤12%时,Cf/Cu-Sn-Zn复合材料的力学性能高于ZQSn663锡青铜;Cf/Cu-Sn-Zn复合材料的比磨损率小于ZQSn663锡青铜,当炭纤维体积分数φf=12%时,复合材料的摩擦磨损性能最佳;在摩擦磨损过程中,Cf/Cu-Sn-Zn复合材料的磨损机制主要为粘着磨损,而锡青铜呈现出粘着磨损和磨粒磨损特征.     

On the Equivalence of the Discontinuous One- and Two-Domain Approaches for the Modeling of Transport Phenomena at a Fluid/Porous Interface

In the quest (i) to determine the form of the boundary conditions that must be applied at a fluid/porous interface and (ii) to determine the value of the jump parameters that appear in the expression for these boundary conditions, two different approaches are commonly considered: the so-called one-domain and two-domain approaches. These approaches are commonly thought to be different, and they are thus sometimes compared to each other to determine the value of jump parameters. In this article, we show that the two-domain and discontinuous one-domain approaches are actually strictly equivalent, provided that the latter is mathematically interpreted in the sense of distributions. This equivalence is shown in details for a heat conduction problem and for the more classical Darcy-Brinkman problem. We show in particular that interfacial jumps are introduced in the discontinuous one-domain approach through Dirac delta functions. Numerical issues are then discussed that show that subtle discretization truncation errors give rise to large variations that can be mis-interpreted as the sign of the existence of jump parameters.     

Study of low-order numerical effects in the two-dimensional cloud-top mixing layer

Large-eddy simulation (LES) has been extensively used as a tool to understand how various processes contribute to the dynamics of the stratocumulus layer. These studies are complicated by the fact that many processes are tied to the dynamics of the stably stratified interface that caps the stratocumulus layer, and which is inadequately resolved by LES. Recent direct numerical simulations (DNS) of isobaric mixing due to buoyancy reversal in a cloud-top mixing layer show that molecular effects are in some instances important in setting the cloud-top entrainment rate, which in turn influences the global development of the layer. This suggests that traditional LES are fundamentally incapable of representing cloud-top processes that depend on buoyancy reversal and that numerical artefacts can affect significantly the results. In this study, we investigate a central aspect of this issue by developing a test case that embodies important features of the buoyancy-reversing cloud-top layer. So doing facilitates a one-to-one comparison of the numerical algorithms typical of LES and DNS codes in a well-established case. We focus on the numerical effects only by switching off the subgrid-scale model in the LES code and using instead a molecular viscosity. We systematically refine the numerical grid and quantify numerical errors, validate convergence and assess computational efficiency of the low-order LES code compared to the high-order DNS. We show that the high-order scheme solves the cloud-top problem computationally more efficiently. On that basis, we suggest that the use of higher-order schemes might be more attractive than further increasing resolution to improve the representation of stratocumulus in LES.     

Invertibility and Weak Continuity of the Determinant for the Modelling of Cavitation and Fracture in Nonlinear Elasticity

In this paper, we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture. The main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface. Accordingly, we define a functional that measures the area of the created surface. This functional has relationships with the theory of Cartesian currents. We show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient, and that the weak limit of one-to-one almost everywhere deformations is also one-to-one almost everywhere. We then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy, taking into account orientation-preserving and non-interpenetration conditions.     

Extreme Poisson's ratios and related elastic crystal properties

With the aim of understanding anisotropic crystals that possess a negative Poisson's ratio and to lay a foundation for investigating molecular mechanisms, we discuss the definition of the ratio and establish conditions on the compliance that govern its sign. We derive results on orientation averaging that are useful in the context of anisotropy and helpful in the investigation of isotropic polycrystals. We discuss α-cristobalite, a polymorph of silicon dioxide that possesses interesting negative ratio properties in single crystals and hypothetical polycrystals. In this connection, we draw attention to the transverse compliance as an alternative and simpler metric for gaging the ratio and for orientation averaging. For α-cristobalite, we arrive at new results for the directions that yield the most negative Poisson's ratio. This result should be of value in divining the underlying molecular mechanism that explains the negative values of Poisson's ratio in α-cristobalite, a crystal of tetragonal symmetry that possesses six independent elastic constants.     

Sharp-crack limit of a phase-field model for brittle fracture

A matched asymptotic analysis is used to establish the correspondence between an appropriately scaled version of the governing equations of a phase-field model for fracture and the equations of the two-dimensional sharp-crack theory of Gurtin and Podio-Guidugli (1996) that arise on assuming that the bulk constitutive behavior is nonlinearly elastic, requiring that surface energy provides the only factor limiting crack propagation, and assuming that the fracture kinetics are isotropic. Consistent with the prominence of the configurational momentum balance at the crack tip in the latter theory, the approach capitalizes on the configurational momentum balance that arises naturally in the context of the phase-field model. The model developed and utilized here incorporates irreversibility of the phase-field evolution. This is achieved by introducing a suitable constraint and by carefully heeding the influence of that constraint on the kinetics underlying microstructural changes associated with fracture. The analysis is predicated on the assumption that the phase-field variable takes values in the closed interval between zero and unity.     

单向连续竹纤维增强聚合物的拉伸性能

描述了一种单向连续竹纤维增强聚合物试件的制作工艺过程:研究了该型竹纤维增强复合材料的拉伸力学性能。将该性能与单向连续玻璃纤维增强聚合物的拉伸力学性能进行了比较,发现竹纤维增强聚合物的拉伸模量要明显高于玻璃纤维增强聚合物的对应值,而其拉伸强度和玻璃纤维增强聚合物的相当。同时发现竹纤维增强聚合物的纵向延展性较小,呈现一次性单界面脆性断裂状况,相对地,连续玻璃纤维增强聚合物的拉伸断裂是多次多界面分段断裂。     

A numerical study of the load on cylindrical piles exerted by internal solitary waves

A continuously stratified nonlinear model is employed to simulate the generation of internal solitary waves (ISWs) over a sill by tidal flows, and it is shown that the simulated ISW-induced current field basically agrees with that observed. Then the force and torque on a supposed small-diameter vertical cylindrical pile exerted by the simulated ISW packet are calculated. According to the calculation, it is found that, no matter whether the direction of the ISW-induced current is the same as that of the tidal current or not, the force exerted by the ISW would be much larger than that by only the tidal current; if the direction of the ISW-induced current is the same as that of the tidal current, then the torque exerted by the ISW would also be much larger than that by only the tidal current; whilst if the direction of the ISW-induced current is against that of the tidal current, then the torque exerted by the ISW has the same order as that exerted by only the tidal current. It is shown that, under the same conditions, the maximum force on the cylindrical pile is 6.58×10² kN, which is larger than that by the modal separation method of Cai et al., whilst the maximum torque is 2.46×10⁵ kN m, which is less than that given by Cai et al. During the passage of the ISW, the time series of the force and torque on the cylindrical pile can also be shown. Finally, the effect of the characteristics of the Gaussian sill on the force is studied, and the resulted empirical formulas on the force with the wave amplitude and the non-dimensional variable of the sill parameters are put forward.