Numerical prediction of creep crack growth in different geometries using simplified multiaxial void growth model

Abstract

This paper considers the prediction of creep crack growth (CCG) in different fracture mechanics geometries using finite element (FE) analysis based on a material independent simplified multiaxial failure strain model at the crack tip. The comparison is first made by modelling C(T) specimen tests under plane stress and plane strain conditions using creep properties of a C–Mn steel at 360°C. In addition, in order to examine CCG due to different geometries, a single edge notch specimen (SENT), centre cracked tension specimen (CCT) and three-point bending (3PB) specimen have been modelled and analysed. In all cases, it is found, depending on the geometry, that for this steel at low creep temperatures the applied load develops a high reference stress/yield stress (σ_ref/σ_y) ratio, which helps reduce constraint at the crack tip. The predictions are analysed under plane stress/plane strain loading conditions identifying the effects of geometry on cracking rates and the implications for predicting long term test or component failure times exceeding where the applied σ_ref/σ_y<<1.     

A mechanical analysis of metallic tritide aging by helium bubble growth

A simple mechanical model is proposed for the aging of a metallic tritide. The material is assumed to be elastic–power law viscoplastic. Part of the helium atoms generated by tritium decay form spherical bubbles that weaken the elastic moduli of the overall material. By contrast, others can be stored in solid solution in the matrix and are likely to increase the moduli. Two variants of the model are compared, assuming either instantaneous or finite rate diffusion of helium. They predict globally similar evolutions of the gas pressure inside the bubbles, the geometrical parameters (bubble radius, overall swelling), as well as the matrix and overall elastic moduli. The results are in good agreement with atomistic calculations of the pressure evolution. Furthermore, recent experimental measurements of the Young modulus changes during aging are better reproduced when He diffusion rate is finite, thus supporting the second variant of the model.     

Two-parameter fracture mechanics and circumferential crack growth in surface cracked pipelines using line-spring elements

A procedure for constraint correction of crack growth resistance curves for single edge notched specimens and for pipe geometries is presented. The procedure is based on FE models with the combination of shell- and line-spring finite elements. Crack tip opening displacement and T-stress are employed, and ductile crack growth is accounted for. Experimental crack growth resistance curves are obtained for both single edge notched tension- and bending-specimens for different crack depths to cover significantly different constraint levels. To account for different constraint levels, a method to scale the resistance curve using the T-stress is implemented. The analyses include ductile crack growth in both the circumferential and thickness directions. The effect of circumferential crack growth with biaxial loading is also presented. The results from the line-spring model are compared with detailed 3D-models for verification of the implementation of circumferential crack growth. The importance of including crack growth in circumferential direction is discussed based on numerical parametric studies. A measure to quantify the importance of circumferential crack growth is proposed.     

A surface acoustic wave technique for monitoring the growth behavior of small surface fatigue cracks

The theory of Kino and Auld which relates the reflection coefficient of acoustic waves from a crack to its size is summarized. A scattering model is evaluated from this theory concerning the reflection of surface acoustic waves (SAW) from a small surface fatigue crack at a frequency such that the crack depth is much smaller than the acoustic wavelength. Acoustic predictions of crack depth are compared to postfracture measurements of depth for small surface cracks in Pyrex glass, 7075-T651 aluminum, and 4340 steel. Additionally, the minimum detectable crack depth as limited by the acoustic noise level is determined for several typical aluminum and steel alloys. The utility of SAW reflection coefficient measurements for inferring crack depth, crack growth, and crack opening behaviorin situ during fatigue cycling is discussed.     

On reaction network implementations of neural networks

This paper is concerned with the utilization of deterministically modelled chemical reaction networks for the implementation of (feed-forward) neural networks. We develop a general mathematical framework and prove that the ordinary differential equations (ODEs) associated with certain reaction network implementations of neural networks have desirable properties including (i) existence of unique positive fixed points that are smooth in the parameters of the model (necessary for gradient descent) and (ii) fast convergence to the fixed point regardless of initial condition (necessary for efficient implementation). We do so by first making a connection between neural networks and fixed points for systems of ODEs, and then by constructing reaction networks with the correct associated set of ODEs. We demonstrate the theory by constructing a reaction network that implements a neural network with a smoothed ReLU activation function, though we also demonstrate how to generalize the construction to allow for other activation functions (each with the desirable properties listed previously). As there are multiple types of ‘networks’ used in this paper, we also give a careful introduction to both reaction networks and neural networks, in order to disambiguate the overlapping vocabulary in the two settings and to clearly highlight the role of each network’s properties.     

On the Fracture Models Determined by the Continuum-strong Discontinuity Approach

The paper focuses on the Continuum Strong Discontinuity Approach (CSDA) to fracture mechanics, and the traction-separation cohesive laws induced from continuum dissipative models as their projections onto the failure interface. They are compared with the cohesive laws commonly used for the fracture simulation in quasi-brittle materials, typically concrete. Emphasis is placed in the analysis of the mechanical stress-strain states induced by the CSDA into the fracture process zone: first when the damage mechanism is initiated and, after, when the cohesive model determines the crack response. The influence of the material parameters, particularly the fracture energy and the initial continuum softening modulus, in the obtained phenomenological responses is also analyzed. Representative numerical solutions of fracture problems are finally presented.     

Controllable crystal growth of Mg(OH)2 hexagonal flakes and their surface modification using graft polymerization

Mg(OH)₂ crystals with excellent flame retardance in the application of polymer materials are always in demand. Herein, regular and well-dispersed Mg(OH)₂ hexagonal flakes were hydrothermally prepared with the existence of polyethylene glycol (PEG) and then modified by surface grafting-polymerization of methyl methacrylate (MMA) monomers. The results showed that the morphology and dispersity of Mg(OH)₂ relied on the precise control of the reaction parameters including hydrothermal conditions, the molecular weight and additive amount of PEG. PEG with a molecular weight of 8000 exhibited an enhanced directing role due to its more appropriate length of molecular chains and intensive interaction with the formed Mg(OH)₂ crystallites. The molecular chains of PEG-8000 can be preferentially adsorbed onto the (0 0 1) and (1 0 1) planes of Mg(OH)₂ crystallites and sub-micro hexagonal flakes with low-polarity were consequently assembled. The optimal conditions for preparing Mg(OH)₂ hexagonal flakes with higher crystallinity and more regular morphology were determined to be hydrothermal treatment at 120 °C for 12 h with 3 wt% PEG-8000. The dimension and decomposition temperature of the end products were 400 ~ 500 nm and 388 °C, respectively. The surface graft-modified Mg(OH)₂ hexagonal flakes exhibited high hydrophobicity with a water contact angle of 148°, indicating an excellent compatibility with polymers.     

Deep convolutional neural networks for eigenvalue problems in mechanics

We show that deep convolutional neural networks (CNNs) can massively outperform traditional densely connected neural networks (NNs) (both deep or shallow) in predicting eigenvalue problems in mechanics. In this sense, we strike out in a new direction in mechanics computations with strongly predictive NNs whose success depends not only on architectures being deep but also being fundamentally different from the widely used to date. We consider a model problem: predicting the eigenvalues of one-dimensional (1D) and two-dimensional (2D) phononic crystals. For the 1D case, the optimal CNN architecture reaches 98% accuracy level on unseen data when trained with just 20 000 samples, compared to 85% accuracy even with 100 000 samples for the typical network of choice in mechanics research. We show that, with relatively high data efficiency, CNNs have the capability to generalize well and automatically learn deep symmetry operations, easily extending to higher dimensions and our 2D case. Most importantly, we show how CNNs can naturally represent mechanical material tensors, with its convolution kernels serving as local receptive fields, which is a natural representation of mechanical response. Strategies proposed are applicable to other mechanics' problems and may, in the future, be used to sidestep cumbersome algorithms with purely data-driven approaches based upon modern deep architectures.     

Fatigue growth behavior for surface crack in welding joints under combined compressive and bending stresses

It has been demonstrated by experiments that crack can grow under cyclic compressive loading. However, it is difficult to observe and describe accurately by mathematical methods. In addition, cracks may close under compressive loading, which also increases the complexity of the problem. The fatigue growth behavior for surface cracks under biaxial loadings was studied by fatigue tests of HTS-A steel. According to experimental evidences, it is concluded that the transverse compressive stress not only changes the fracture morphology but also affect crack propagation life. Considering the influence of the compressive stress, this paper proposed an equivalent SIF and crack growth model subjected to compressive and bending stresses on the basis of McEvily formula. Finally, comparisons are made between prediction results and experimental data.     

Early fatigue crack growth as the damage accumulation process

A probabilistic approach to modeling of the initial stage of fatigue crack growth is suggested based on the concepts of continuum damage mechanics. The material is presented as a set of microstructural elements with randomly distributed properties. Both the grains and intergranular boundaries are considered as the elements of microstructure. The parameters of resistance of each element to damage accumulation are considered as random variables. These parameters are distributed among the elements independently that allows to model the damage process in polycrystalline materials. The damage measure depends on the characteristic normal and tangential stresses in order to take into account the tensile and shear fracture modes for each element of microstructure. It is assumed that a nucleus of a crack is initially present near the body surface as a single completely ruptured element. The final damage of an element is considered as the crack advancement. The crack is modelled as a sequence of ruptured grains for the transgranular fracture, and as a sequence of couples of neighboring ruptured grains when the intergranular rupture is considered. Numerical simulation is performed to illustrate feasibility of the proposed model. In particular, non-planar crack propagation, blunting, kinking and branching of cracks at the early stage is demonstrated. The non-monotonous pattern of the short crack growth process is observed. Statistical scattering of the current crack size and the crack growth rate as functions of the cycle number and the crack depth is studied.     

Fatigue crack growth thresholds-the influence of Young''s modulus and fracture surface roughness

In the present investigation it is shown that the effective fatigue threshold is uniquely correlated to the Young's modulus for a wide range of metallic and composite materials (ΔK_th,eff=1.64·10⁻²·E). It is also demonstrated that the crack closure level K_cl increases with increased roughness of the fracture surface . K_cl and are quantitatively related via the equation for steels with widely different mechanical properties and grain sizes (120 MPa<R_p<1100 MPa, 1 μm<λ<100 μm). This relation can be extended to materials other than steels (e.g. aluminium and WC-Co alloys) by normalising against Young's modulus. The roughness value represents the standard deviation of height of the fracture surface and is shown to be simply related to the length and angle distributions of the linear length elements constituting the fracture profile.     

On tandem stochastic networks with time-deteriorating product quality

We consider an n-site tandem stochastic production network where each product moves sequentially through the sites, and the product's quality deteriorates with its sojourn time in the system. At each site the product goes through two stages: the first stage is a processing operation with a generally-distributed random duration. This operation either does or does not conclude successfully; in the latter case, the operation is repeated immediately. Once the processing operation concludes successfully, the product goes through an inspection stage lasting a generally-distributed random duration. At the end of the inspection the product's state is determined as follows: either (i) it requires additional processing and moves forward to the next site; or (ii) it is found ‘good’ and exits the network with quality value depending on its total sojourn time in the system; or (iii) it is declared ‘failed’, discarded, and exits the network with zero quality value. Two scenarios are analysed: (i) a new product enters the system only after the preceding product has exited and (ii) the network is a tandem Jackson-type system. For each scenario, we construct both time-dependent and quality-dependent performance measures. In the case where the sites can be arranged in an arbitrary order, we derive easy to implement optimal index-type policies of ordering the sites so as to maximise the quality rate of the production network.     

A radial point interpolation meshless method extended with an elastic rate-independent continuum damage model for concrete materials

An advanced discretization meshless technique, the radial point interpolation method (RPIM), is applied to analyze concrete structures using an elastic continuum damage constitutive model. Here, the theoretical basis of the material model and the computational procedure are fully presented. The plane stress meshless formulation is extended to a rate-independent damage criterion, where both compressive and tensile damage evolutions are established based on a Helmholtz free energy function. Within the return-mapping damage algorithm, the required variable fields, such as the damage variables and the displacement field, are obtained. This study uses the Newton–Raphson nonlinear solution algorithm to achieve the nonlinear damage solution. The verification, where the performance is assessed, of the proposed model is demonstrated by relevant numerical examples available in the literature.     

Neural network modelling for damage behaviour of composites using full-field strain measurements

This paper presents a neural network modelling method for damage behaviour of composite materials in conjunction with full-field strain measurements. The proposed method utilises the overall structural response of a laminate composite specimen to develop the constitutive model of a single ply unidirectional laminate. Based on an energy principle, a performance function for training the neural networks is derived in terms of the applied external work and the induced strain energy. This allows the proposed method to develop the neural networks without the presence of stress information that is not necessarily obtainable in experiments with non-uniform deformation. The use of neural networks also enables the proposed method to model the damage behaviour without the constraints on the parameter space, such that a more representative model is developed for the actual material behaviour. An example of tailoring the proposed method to model the in-plane shear damage behaviour of a carbon fibre reinforced plastic (CFRP) laminate is demonstrated as well as its numerical validation. The practical application of the proposed method to multi-axial damage-related nonlinear behaviour of composite is presented using the experimental data obtained from a tensile test with an open-hole specimen.     

Combination of the material force concept and the extended finite element method for mixed mode crack growth simulations

A novel approach to simulate crack growth within an extended finite element framework is presented. The introduced approach combines the material force concept and the extended finite element method (xFEM) that is not straight forward and faces the major problem that a crack tip node, which is required for the evaluation of the material force, is not available within an xFEM framework. The introduced concept enables an efficient single step evaluation of the crack state and the crack growth direction based on a continuum mechanics approach and represents an alternative to the common procedure of using the stress intensity factor solution within a stress or energy‐based empirical formulation for the determination of the crack growth direction. Two different approaches are introduced that evaluate the crack tip material force within the xFEM based on a domain or contour approach, both providing equivalent results. After an evaluation of the method, a major focus is set on crack growth investigations with increased complexity, including mixed mode loading and crack interaction with other discontinuities. The influence of different evaluation parameters is studied by comparing the results with empirical, experimental and alternative numerical solutions and confirms the applicability and capability of the proposed combination of both concepts. Copyright © 2010 John Wiley & Sons, Ltd.     

A new curvature technique calculation for surface tension contribution in PLIC-VOF method

The volume of fluid (VOF) methods have been used for numerous numerical simulations. Among these techniques used to define the moving interface, the piecewise linear interface reconstruction (PLIC-VOF) is one of the most accurate. A study of the superficial tension impact on two-phase flow with free surface is presented. A new method based on direct staggered grid is developped to include surface tension in PLIC-VOF. The new numerical curvature calculation method doesn't need smoothed colour function and leads to less “spurious current”. This technique is applied to the calculus of surface tension force in the case of the rise of air bubble in viscous liquid and the fall of liquid drop in the same liquid on free surface. Droplets, thin layer and capillarity waves are observed after the free surface rupture for different Bond number. The influence of surface tension calculus is then obvioused and when the drop hit the free surface, wavelets propagate toward the virtual boundaries imposed.     

Self-organised growth on strained substrates: the influence of anisotropic strain, surface energy and surface diffusivity

The morphological stability of coherent thin films subjected to unequal in-plane biaxial strains is investigated to determine how non-uniform strain states can be used to influence the growth of self-organised island nanostructures. The evolution via surface diffusion is modelled analytically using a small perturbation approach and allows for anisotropies in the surface energy and the surface diffusivity. It is shown that conventional uniform biaxial epitaxy does not provide a driving force towards a particular wavelength as is popularly assumed. This reduces the potential for highly self-organised growth. It is predicted that improvements in island size, shape and spatial distributions can be obtained under certain conditions of anisotropic strain, surface energy and surface diffusivity. This increase in uniformity would be beneficial to the construction of practical devices. Enhancing surface diffusivity anisotropy via the application of an applied strain could offer the most realistic opportunity for controlling the growth of self-assembled structures this way.     

Fatigue and creep crack growth of Alloy 800 and Alloy 617 at high temperatures

A common evaluation is given for creep crack growth and fatigue crack growth experiments which have been performed at the companies ABB, Siemens-KWU and KFA. The materials under investigation were X10NiCrAlTi32 20 (Alloy 800) and NiCr22Col2Mo (Alloy 617). Several production lots and semi-finished materials as well as welded materials have been tested. Testing techniques differed at the different labs. In order to eliminate the influence of individual testing techniques, material from some production lots was investigated at different labs. The given data cover fatigue crack growth (the materials were tested between room temperature and 1050°C; the influence of temperature, R?ratio, and frequency was investigated) and creep crack growth (Alloy 800 was tested between 550°C and 900°C, Inconel 617 between 800°C and 1000°C; the evaluation was done on the basis of the fracture mechanics parameters K₁ and C*).     

人工神经网络的单调序列逼近

设Nn,Ф是以Ф为激活函数的具有n+1个神经元的前向单隐层人工神经网络的全体.主要证明了,若f∈C[0,1],则对于任意的ε>0,存在两个神经网络序列{Pn,Ф}和{Qn,Ф},使得在[0,1]上Qn,Ф(x)≤Qn+1,Ф(x)≤f(x)≤Pn+1,Ф(x)≤Pn,Ф(x),而且Pn,Ф(x)-Qn,Ф(x)≤(6+4(2~(1/2)))En,Ф(f),这里的En,Ф(f)为Nn,Ф中的元对f的最佳逼近.     

On slippage induced by surface diffusion

The mechanism of surface diffusion is taken at the basis of the phenomenon of slippage of the contact line of a liquid film. With the aid of the condition of continuity of the traction vectors at the solid-liquid interface, we obtained an evolution equation for the velocity of the fluid particles at the wall which shows a marked resemblance with Millikan's equation for the slippage coefficient of gases and reduces, in the limit of small surface diffusivity, to the classical Stokes-Einstein model. The influence of surface roughness is explicitely taken into account and, among other results, cases of absence of slip caused by the attachment of the liquid film to the solid surface and of slippage solely induced by surface roughness are found. Finally, the effect of the surface deformation upon the surface velocity of the fluid particles is examined in some detail.