Impact damage of laminated composite with energy dissipation: Part II — Damage evolution

Having developed the methodology for analyzing the failure of a ceramic/rubber/steel composite laminate impacted by a tungsten rod in Part I, Part II of the work is concerned with the progressive damage process where material continuity would be interrupted at different locations and time intervals. Depending on the time rate dependent threshold values of the surface and volume energy density, the degree and extent of damage by fragmentation, mass loss, etc. are determined by finite element calculations for time steps of 0.15, 5.0, 7.5, 10, 20, 21 and 21.5 μs. Stresses and strains possess an oscillatory character in time; they alternate in sign as the impact waves bounce back and forth in the three-layered dissimilar materials.Local strain rates of approximately 10⁵, 10³ and 10⁴ s⁻¹ are formed in the ceramic, rubber and steel layer respectively at locations underneath the tungsten rod after 16 μs of impact. A more wide range of strain ratio would have prevailed for a homogeneous layer of the same thickness. The tungsten rod is now badly fragmented while cracking near the surface of the ceramic is also predicted. Local temperature and dissipation energy density rise rapidly as time approached 20 μs. The maximum surface and volume.energy density in the ceramic near the impact region reached 260 MPa · m and 6.39 MPa, respectively. Complete disintegration of the tungsten rods occurred at 21.5 μs. At this time, the ceramic layer is perforated and the rubber layer is partially cracked. The back-up steel plate, however, remained in tack. These predictions agree qualitatively with past observations.     

Irreversibility and damage of SAFC-40R steel specimen in uniaxial tension

Macroscopic material damage is detected and assessed for the SAFC-40R steel specimen in uniaxial tension even when the stress responded linearly with strain. As the loading increased monotonically at a rate of 0.2 cm/min, the specimen first absorbed heat from the surrounding and then released heat when the strain is almost five times beyond the so-called “elastic limit”. In other words, the specimen undergoes cooling and heating with reference to the ambient temperature. This phenomenon is predicted theoretically for the first time by application of the energy density theory and the results agreed well with experimental data. Obtained is the H-function that possesses a distinct threshold at time between 21 and 22 seconds after loading. This transition is defined as the onset of disorder at which point the energy dissipation density D increases suddenly by one order of magnitude. The corresponding uniaxial stress and strain are 194.4 MPa and 0.9764·10⁻³ cm/cm, respectively. These values are lower than those normally referred to at the yield point.     

Fatigue damage in polycrystals – Part 2: Intrinsic scatter of fatigue life

Part 2 deals with the evolution of plastic flow resistance with crack growth from its minimum value (fatigue limit) towards its saturated bulk value (cyclic yield stress). The far-field stress level, the geometry of the crack and the grain size distribution of the material are those parameters that control the area of crack tip plasticity and hence the rate towards saturation. The implication of the far-field stress is held responsible for the violation of the similitude concept and the failure of the stress intensity factor to describe conditions of short cracking. However, an engineering tool based on the stress intensity factor and being able to predict the fatigue life of short cracks can be constructed, considering that the distribution of crack growth rates is intrinsically defined by the material itself. The above allows the development of a set of equations able to construct the fatigue life scatter of the material.     

Rigid-plastic and damage behavior in metal forming: Part II—model prediction

Indeterminacy of the Lévy-Mises relations in plane strain rigid-plasticity is overcome by considering change in plastic strain rate due to triaxiality. Better agreement is achieved on the coincidence of uniaxial data with the effective stress and effective strain in addition to the removal of ill conditions associated with necking stresses derived from other theories of rigid plasticity. The formulation accounts for material compressibility and makes use of the variational principle when applied to discrete locations of the continuum. These physical refinements provide the necessary accuracy for predicting potential damage sites and establishing design limits on metal forming products. Developed are damage thresholds for defining the integrity of rigid-plastic materials with nonlinear behavior. Distortion and dilatation of the material elements are mutually interactive and their proportion changes with the local strain rates in the metal forming process. Only the stationary values of the volume energy density could automatically account for the nonlinear relationship between distortion and dilation. Completed in Part I is the development of a modified theory of rigid-plasticity that better describes the yield and fracture behavior of metals. Numerical results for the permanent deformation of sheet metals are provided in Part II; they are compared with those obtained from other theories of rigid-plasticity.     

Fatigue damage in polycrystals – Part 1: The numbers two and three

The fundamental difference between the cyclic yield stress and the fatigue limit is scale. The basic rate of slip mismatch towards a saturated condition, representing polycrystalline behaviour, is a material property embedded into the grain size distribution. Deviations from its basic rate will depend on the shape of the crack and the selected crack path. This work concludes that during the first loading cycle, and for stresses equal and above the fatigue limit, the surface will always been deformed plastically.     

Evolution of fabric tensors in damage mechanics of solids with micro-cracks: Part II – Evolution of length and orientation of micro-cracks with an application to uniaxial tension

In Part II of this work, the equations of thermodynamics are employed in order to derive the exact evolution equations of the fabric tensors defined in Part I (companion paper). In this regard, a thermodynamic force that is associated with the fabric tensor is defined and utilized in the derivation of the evolution equations. A special case of uniaxial tension is solved in order to illustrate the theory.We also derive specific uncoupled equations for the evolution of the length and orientation of micro-cracks. In this regard, some interesting results are obtained. It is concluded that the micro-crack length and orientation cannot evolve simultaneously for the same set of micro-cracks. However, two different sets of micro-cracks may be considered in the same RVE where in one set the micro-crack length evolves, while in the second set the micro-crack orientation evolves.     

Prediction of failure in weldments — Part II: Joint with initial notch and crack

This investigation is concerned with predicting failure initiation sites ina butt weld joint under bending. The nonuniform load transmission characteristics through the weld metal, heat-affected zone and base material resulting from alteration in their microstructure are reflected through the macroscopic yield strength parameter. Elastic-plastic stress and strain redistribution is obtained for each increment of load increase. Analyzed in detail are the contours of constant strain energy density for determining the local and global stationary values which are assumed to be related to failure and stability of the system. Failure is predicted to initiate in the heat affected zone at the site of maximum of the minimum local strain energy density function. This corresponds to the experimental observation where cracking starts from the side of the butt joint where local stretching is maximum.     

An elastoplastic constitutive relation of whisker-reinforced composite for meso damage: Part I – formulation

An elastoplastic constitutive relation is developed for meso damage of whisker-reinforced composites. A model is constructed that includes orientation distribution of whiskers and slip systems as well as interface and crystal sliding. Evolution of damage will be addressed. Given in Part I is the formulation while examples will be illustrated in Part II.     

Stress intensity factors for two offset parallel circular cracks: Part II — Semi-infinite solid

With the aid of the formulation in [1] (R. Muki, Progress in Solid Mechanics (North-Holland, 1961)) for general three-dimensional asymmetric problems and the superposition principle, Part II of this work makes use of the method in Part I (G.A.C. Graham and Q. Lan, J. Theor. Appl. Fract. Mech. 20, 207–225 (1994) [2]) to examine the interaction of arbitrarily located penny-shaped cracks in an infinite elastic solid to the case of a semi-infinite solid. As in Part I for the infinite body, the problem of a semi-infinite solid containing two penny-shaped cracks is reduced to a system of Fredholm integral equations of the second kind. These integral equations are then solved for some special cases when cracks are far apart and far away from the boundary. Some asymptotic solutions are presented and comparisons are made with the results for the special case where there is only one crack under axisymmetric loading.     

Stress intensity factors for two offset parallel circular cracks: Part III — Elastic layer

Part III of this work is concerned with the interaction of two penny-shaped cracks in the mid-plane of an elastic layer. Two cases, namely the stress free boundary case and the fixed boundary case, are considered. It is shown that these two cases are mathematically equivalent. As in Part I (G.A.C. Graham and Q. Lan, J. Theor. Appl. Fract. Mech. 20, 207–225 (1994) [1])_for the problems of an infinite solid and Part II (G.A.C. Graham and Q. Lan, J. Theor. Appl. Fract. Mech. 20, 227–237 (1994) [2]) of a semi-infinite solid, the problem is reduced to a system of Fredholm integral equations of the second kind. These integral equations are then solved when the crack size is small compared to the distance between them and the cracks are far away from the boundaries. It is also shown that the problem decouples when the cracks are subjected to normal loading and shear loading. Asymptotic solutions are presented for these two loadings.     

Stress intensity factors for two offset parallel circular cracks: Part I — Infinite elastic solid

The method (W.D. Collins, Proc. R. Soc. London A274, 507–528 (1963); W.S. Fu and L.M. Keer, Int. J. Eng. Sci. 7, 361–372 (1969) [1,2]) used to solve co-planar penny-shaped cracks is generalized to investigate interaction of arbitrarily located penny-shaped cracks. The solution (M.K. Kassir and G.C. Sih, Three-dimensional Crack Problems (Noordhoff International, 1975) [3]) for the problem of an isolated crack in an infinite solid is applied together with the superposition principle to reduce the problem to a system of Fredholm integral equations of the second kind. These integral equations are then solved iteratively when the cracks are far apart. Some asymptotic solutions for the stress intensity factors are presented and comparisons are made whenever possible. Numerical solutions reveal some interesting phenomena.     

Theoretical and practical aspects of the ride dynamics of off-road vehicles — Part 1

Recent work on the ride vibration behaviour of off-road vehicles is reviewed. The contributions to this subject presented at the 8th ISTVS Conference in Cambridge, 1984, form the main part of the review, but these are discussed in the context of other developments which have been presented recently. It is concluded that ride vibration studies have entered a period in which refinement and optimisation are the main goals. The basic techniques of mathematical modelling and measurement procedures are well understood at most of the major research and manufacturing centres of off-road vvehicles. In the military industry, further improvements in suspension design and ride quality are evolving gradually as a result of detailed refinements. In the agricultural industry, some major decisions still remain. There is little further scope for improvement of the unsprung tractor with passive seat suspension, and so the next improvements will come from active seat suspensions, cab suspensions or axle suspensions. Although work continues in all these areas, axle suspensions currently offer the most potential.     

高温对砂岩宏细观损伤及BP神经网络单轴强度预测研究

为了研究高温后砂岩的力学特性和宏细观损伤变化,对高温作用后的砂岩进行单轴压缩试验、声波损伤检测、X射线衍射试验、扫描电镜试验,分析应力-应变曲线、峰值应力、峰值应变、弹性模量、质量损失率、X射线衍射成像和电镜扫描图像,得到砂岩的细观损伤变化对其单轴抗压强度的影响.利用BP神经网络模型对不同物理量进行训练,预测不同高温作...     

Evolution of fabric tensors in damage mechanics of solids with micro-cracks: Part I – Theory and fundamental concepts

The evolution of fabric tensors based on micro-crack distributions is formulated based on sound thermodynamic principles. In Part I of this work, the exact definition of fabric tensors based on micro-crack distributions is presented. This definition is seen to incorporate both the orientation and length of a micro-crack. In this regard, the micro-crack distribution is assumed to be radially symmetric, i.e. symmetric about a line through the origin.The equations of thermodynamics are employed in order to derive the exact evolution equations of the fabric tensors defined in the first part. In this regard, a thermodynamic force that is associated with the fabric tensor is defined and utilized in the derivation of the evolution equations. The application of the theory to the case of uniaxial tension is derived in Part II (companion paper) of this work.     

The steady Navier–Stokes/energy system with temperature‐dependent viscosity—Part 2: The discrete problem and numerical experiments

In this second part, we analyse the associated discrete problem arising from a conforming finite element method formulation of the mathematical model presented in the first part. Thus, existence and uniqueness of the discrete solution when using small enough data are stated following the same strategy used in the continuous case, with a Cea's type error estimate established as the main result. Some numerical experiments, steady and unsteady, are performed, which allow us to validate the previous mathematical model and its discrete approximation. Copyright © 2007 John Wiley & Sons, Ltd.     

A pseudo-elastic material damage model, part II: Application to the scaling of specimen size, material behavior and loading rate

A failure criterion is presented which relates the strain energy density of the material to both yielding and fracture. Cumulative material damage throughout a structural component may be monitored and the relative influence of yielding and stable crack growth assessed. The criterion is demonstrated, using finite element analysis, for center cracked panel specimens differing by material toughness values. From crack growth increment predictions using the uniaxial stress-strain behavior of the material, the criterion predicts the critical value of the strain energy density factor S_c governing crack instability.     

Consideration of the yarn–yarn interactions in meso/macro discrete model of fabric: Part II: Woven fabric under uniaxial and biaxial extension

A mesoscopic discrete model of fabric has been developed, accounting for the yarn–yarn interactions occurring at the yarn crossing points. The fabric yarns, described in their initial state by a Fourier series development, are discretized into elastic straight bars represented by stretching springs, and connected at frictionless hinges by rotational springs. In the first part of the paper, the behavior under uniaxial tension of a single yarn has been investigated, and the impact of the interactions of the transverse yarns has been quantitatively assessed. The consideration of the yarn interactions is extended in this second part at the scale of the whole network of interwoven yarns, under uniaxial and biaxial loading conditions. The effect of the transverse yarns properties under uniaxial tension is evidenced, as well as the impact of the biaxial loading ratio.     

A ‘poor man's Navier–Stokes equation’: derivation and numerical experiments—the 2‐D case

We present a systematic derivation of a discrete dynamical system directly from the two‐dimensional incompressible Navier–Stokes equations via a Galerkin procedure and provide a detailed numerical investigation (covering more than 107 cases) of the characteristic behaviours exhibited by the discrete mapping for specified combinations of the four bifurcation parameters. We show that this simple 2‐D algebraic map, which consists of a bilinearly coupled pair of logistic maps, can produce essentially any (temporal) behaviour observed either experimentally or computationally in incompressible Navier–Stokes flows as the bifurcation parameters are varied in pairs over their ranges of stable behaviours. We conclude from this that such discrete dynamical systems deserve consideration as sources of temporal fluctuations in synthetic‐velocity forms of subgrid‐scale models for large‐eddy simulation. Copyright © 2004 John Wiley & Sons, Ltd.     

High‐order 𝒞1 finite‐element interpolating schemes—Part I: Semi‐Lagrangian linear advection

This paper is devoted to the development of accurate high‐order interpolating schemes for semi‐Lagrangian advection. The characteristic‐Galerkin formulation is obtained by using a semi‐Lagrangian temporal discretization of the total derivative. The semi‐Lagrangian method requires high‐order interpolators for accuracy. A class of ??¹ finite‐element interpolating schemes is developed and two semi‐Lagrangian methods are considered by tracking the feet of the characteristic lines either from the interpolation or from the integration nodes. Numerical stability and analytical results quantifying the amount of artificial viscosity induced by the two methods are presented in the case of the one‐dimensional linear advection equation, based on the modified equation approach. Results of test problems to simulate the linear advection of a cosine hill illustrate the performance of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.