A stochastic damage model

The theoretical defect size distribution has been derived on the basis of the kinetic formation and growth parameters. Formulas are derived for the mathematical expectation and variance for the damage in a uniformly strained material. The damage nonuniformity occurring with a Poisson distribution for the defect size in the bulk can have a substantial effect on the forecast working life. Phenomenelogical results are obtained for the working life as a function of mean defect size and generation rate, together with ones for the failure probability and the deformed material volume.Translated from Problemy Prochnosti, No. 6, pp. 3–9, June, 1990.     

A nonlocal damage theory

A nonlocal damage field theory is proposed and a response model of nonlocal elastic damage is given. A simulating experiment using aluminum alloy plates with arrays of holes shows that the load bearing capacity of a plate varies with angle between array and loading direction. The characteristics and attenuation lengths appreciably affect the constitutive equations.

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Résumé On propose une théorie du champ d'endommagement non local et on donne un modèle de réponse à un dommage non local élastique. En utilisant des tôle en alliage d'aluminium pourvue de rangées de trous pour une expérience de simulation, on montre que la capacité de résistance d'une tôle varie avec l'angle formé par une rangée de trous avec la direction de mise en charge.

     

A coupled interface-body nonlocal damage model for FRP strengthening detachment

Aim of the paper is to propose a new coupled interface-body damage model for the study of the detachment process in concrete or masonry structures strengthened with fiber reinforced polymers (FRP). In particular, a model of the FRP-concrete or -masonry interface, accounting for the coupling between the degradation of the cohesive material and the FRP detachment, is presented. To this end, a nonlocal damage model is considered for the quasi-brittle material. Regarding the interface, a model which accounts for damage, unilateral contact and friction is developed. The novelty of the proposed model consists in taking into account the coupling between the body and the interface damage, ensuring that the interface damage is not lower than the body one. Some numerical examples and a comparison with experimental data are presented in order to verify the efficiency of the proposed model in reproducing the FRP decohesion from the support cohesive material.     

A stochastic damage accumulation model for creep crack propagation

Previously a stochastic model for damage accumulation was proposed to obtain an expression for the fatigue crack propagation rate. In this paper, the model is used in order to derive the creep crack propagation law. The theoretical results show a good agreement with the experimental data. This suggests an extensive applicability of the present model. On these theoretical bases, it is concluded that the creep crack propagation rate is proportional to the process region size.     

Boundary effect on weight function in nonlocal damage model

Some insights on boundary effects in nonlocal damage modelling are addressed. Interaction stresses that are at the origin of nonlocality are expected to vanish at the boundary of a solid, in the normal direction to this boundary. Existing models do not account for such an effect. We introduce tentative modifications of the classical nonlocal damage model aimed at accounting for this boundary layer effect in a continuum modelling setting. Computations show that some nonnegligible differences may be observed between the classical and modified formulations. In a one dimensional spalling test, only the modified formulation provides a spall of finite nonzero thickness, whereas spalls smaller than the internal length cannot be obtained according to the original formulation. For the same set of model parameters, including the internal length, the fracture energy derived from the size effect test method is also very different according to both approaches. Parameters in the size effect laws for notched and unnotched specimens, obtained from computation of geometrically similar bending beams, are more consistent with the modified nonlocal model compared to the original nonlocal formulation.     

A stochastic damage accumulation model for crack initiation in high-cycle fatigue

A stochastic damage accumulation model for crack initiation in high-cycle fatigue is proposed. It is assumed that the fatigue damage is accumulated in the form of dislocations under the repeated stress and that the slip band crack is initiated when the strain energy due to a local pile-up of dislocations exceeds a critical value. The size of an initiating crack is the cell size, derived from a probabilistic argument and its depth is determined in relation to the stored dislocation energy. Our theoretical results are compared with the experimental data from a low-carbon steel S20C in order to examine the consistency of our model.     

A stochastic damage model for the rupture prediction of a multi-phase solid

The stochastic damage model, which has been proposed in Part I of this paper, is utilized to analyze quantitatively the effects of uncertainties in locations, orientations and numbers of microcracks at the macro-tip. This is accomplished by introducing a computer simulation model which incorporates a statistical characterization of geometrical parameters of a random microcrack array. The counteracting effects of microcracking on the fracture toughness, namely, toughness enhancement and toughness degradation, are explored statistically through the change of the location and size of the near tip damage zone. The effects of changes in the geometric configuration and density of microcracks at the macro-tip are also examined through the present statistical approach. The validity of the present model is verified by comparing the obtained statistical distribution with the analytic model based on the Neville distribution function. A very good fit achieved by the use of the Neville function demonstrates the potential of the present damage model, in predicting the inherent statistical distribution of the fracture toughness from the intrinsic random microdefects.     

A stochastic theory of cumulative fatigue damage

A Stochastic theory for the cumulative fatigue damage of structural component with random fatigue strength under random loading is proposed on the basis of the Stratonovich-Khasminskii theorem. The analytical solutions for the probability densities of the cumulative fatigue damage and fatigue life and for the reliability function are given for steel and reinforced concrete components, with constant fatigue strength subject to narrow band stationary Gaussian stress process with zero mean. The results agree very well with those of digital simulation. It is noted that the theory can be applied, in principle, to metallic and some nonmetallic materials, to both narrow band and wide band stress process, and can be adapted to a sequence of n stationary stress process or quasi-stationary stress process. The scatter and degradation of fatigue strength can also be incorporated into the theory.     

A nonlocal model for plug formation in plates

By means of a nonlocal viscous fluid model, an investigation is carried out of the problem of penetration of a cylindrical projectile into a plate leading to a failure of the plate by a plug formation. The effect of impact is represented by a uniform initial velocity distribution over a circular region on the surface of the plate. The behavior of this plate material is assumed to be viscous and spatially nonlocal, and only the effects of vertical shearing stress are considered. The expression of stress, velocity and displacement are obtained and the calculated displacement profiles are compared with some existing experimental profiles.     

A stochastic equipment capacity model

This paper presents a model for the determination of the optimal equipment capacity for situations where a stochastic demand of Bervicea can be met by both owned equipment operating on a regular and overtime basis and hired-in equipment from an external source.     

Numerical simulations of void linkage in model materials using a nonlocal ductile damage approximation

Experiments on the growth and linkage of 10 μm diameter holes laser drilled in high precision patterns into Al-plates were modelled with finite elements. The simulations used geometries identical to those of the experiments and incorporated ductile damage by element removal under the control of a ductile damage indicator based on the micromechanical studies of Rice and Tracey. A regularization of the problem was achieved through an integral-type nonlocal model based on the smoothing of the rate of a damage indicator D over a characteristic length L. The simulation does not predict the experimentally observed damage acceleration either in the case where no damage is included or when only a local damage model is used. However, the full three-dimensional simulations based on the nonlocal damage methodology do predict both the failure path and the failure strain at void linkage for almost all configurations studied. For the cases considered the critical parameter controlling the local deformations at void linkage was found to be the ratio between hole diameter and hole spacing.     

梯度依赖的混凝土弹塑性非局部损伤的本构模型

以连续介质力学和不可逆热力学为基础,提出了新的混凝土弹塑性损伤本构模型.在该模型中采用了塑性应变εp、各向同性损伤标量D以及应变梯度(△)ε作为内变量,其中应变梯度反应了损伤的非局部性质,本模型是梯度依赖的非局部损伤模型,严格满足热力学的基本方程.应用梯度依赖弹塑性非局部损伤本构模型得出的结果比已有的混凝土损伤塑性模型更为合理.     

A nonlocal model of fracture by crazing in polymers

We derive and numerically verify scaling laws for the macroscopic fracture energy of polymers undergoing crazing from a micromechanical model of damage. The model posits a local energy density that generalizes the classical network theory of polymers so as to account for chain failure and a nonlocal regularization based on strain-gradient elasticity. We specifically consider periodic deformations of a slab subject to prescribed opening displacements on its surfaces. Based on the growth properties of the energy densities, scaling relations for the local and nonlocal energies and for the specific fracture energy are derived. We present finite-element calculations that bear out the heuristic scaling relations.     

A stochastic model of disperse systems

We suggest a model of the structure of a disperse system that reflects the random character of the topology. We evaluate the effect of the stochastic nature of the topology on the effective conductivity. An asymptotic decrease in the conductivity with increase in the scale of the heat and mass transfer is established and explained.     

Strength distributions of warm frozen clay and its stochastic damage constitutive model

There are many defects, such as fissures and cavities, in warm frozen clay and in warm ice-rich frozen clay. These defects are distributed randomly, which make some mechanical properties of these clays exhibit great uncertainty. Thus it is unreasonable to take some deterministic values as the mechanics parameters of these frozen clays. Furthermore, warm frozen clay and warm ice-rich frozen clay are less stable than clays at colder temperatures. Therefore, it is necessary to determine the strength and constitutive relationship of these clays using probabilistic methods. In this paper, based on the characteristics of the random distribution of defects in the warm frozen clay and warm ice-rich frozen clay, the strength distribution laws for both clays are investigated at − 0.5 °C, − 1.0 °C and − 2.0 °C through a series of experimental data, respectively. The investigated results show that the Weibull distribution can more closely reflect the strength distribution law than other probability distributions. Based on the results mentioned above, a stochastic damage model for the warm frozen clay and warm ice-rich frozen clay was developed using the continuous damage theory and probability, as well as statistic theory. Comparing theoretical results of the model with experimental data at these three temperatures, respectively, it is found that the stress-strain relationships of the two kinds of frozen clays, especially their deformation characteristics after failures, could be described by the model very well. Since the range of strength variation for both kinds of frozen clays is very large, it is unsafe and unreasonable for engineering design to use the average strengths of frozen clay soils. The reliability of the strength of these two clays was investigated and discussed on the basis of the fact that the strength of warm frozen clay and warm ice-rich frozen clay satisfy the Weibull distribution.     

A contamination model for the stochastic order

Stochastic ordering among distributions has been considered in a variety of scenarios. However, it is often a restrictive model, not supported by the data even in cases in which the researcher tends to believe that a certain variable is somehow smaller than other. Alternatively, we propose to look at a more flexible version in which two distributions satisfy an approximate stochastic order relation if they are slightly contaminated versions of distributions for which stochastic order holds. The minimal level of contamination required for stochastic order to hold is used as a measure of deviation from exact stochastic order model. Our approach is based on the use of trimmings of probabilities. We discuss their connection to approximate stochastic order and provide theoretical support for its use in data analysis, proving uniform consistency and giving non-asymptotic bounds for the error probabilities of our tests. We provide simulation results and a case study for illustration.     

A stochastic model for propagation through tissue

Attenuation of ultrasonic waves is often assumed linear with respect to frequency in biological applications whereas it is considered quadratic when the propagation occurs in the atmosphere or the water. In the latter case, other studies show that a Gaussian propagation duration can explain this attenuation behavior and provide a model for the energy loss in the stationary limit. The present paper defines an equivalent random propagation duration with Cauchy distribution, which is appropriate for the propagation of ultrasound through tissue. The model adds an unobserved noise that represents the signal deterioration. In addition, the model agrees with the mode downshift in the case of a narrowband signal.     

A stochastic model for early placental development

In the human, placental structure is closely related to placental function and consequent pregnancy outcome. Studies have noted abnormal placental shape in small-for-gestational-age infants which extends to increased lifetime risk of cardiovascular disease. The origins and determinants of placental shape are incompletely understood and are difficult to study in vivo. In this paper, we model the early development of the human placenta, based on the hypothesis that this is driven by a chemoattractant effect emanating from proximal spiral arteries in the decidua. We derive and explore a two-dimensional stochastic model, and investigate the effects of loss of spiral arteries in regions near to the cord insertion on the shape of the placenta. This model demonstrates that disruption of spiral arteries can exert profound effects on placental shape, particularly if this is close to the cord insertion. Thus, placental shape reflects the underlying maternal vascular bed. Abnormal placental shape may reflect an abnormal uterine environment, predisposing to pregnancy complications. Through statistical analysis of model placentas, we are able to characterize the probability that a given placenta grew in a disrupted environment, and even able to distinguish between different disruptions.     

A new stochastic model for crack propagation

In the following paper we present a new model for crack propagation. The new model is able to involve sudden growth of crack length (jumps) and is a generalization of the well-known Paris–Erdogen law. We obtain the lifetime distribution and the residual lifetime distributions.