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Micro-structural analyses are an important tool to understand material behavior on a macroscopic scale. The analysis of a microstructure is usually computationally very demanding and there are several reduced order modeling techniques available in literature to limit the computational costs of repetitive analyses of a single representative volume element. These techniques to speed up the integration at the micro-scale can be roughly divided into two classes; methods interpolating the integrand and cubature methods. The empirical interpolation method (high-performance reduced order modeling) and the empirical cubature method are assessed in terms of their accuracy in approximating the full-order result. A micro-structural volume element is therefore considered, subjected to four load-cases, including cyclic and path-dependent loading. The differences in approximating the micro- and macroscopic quantities of interest are highlighted, e.g. micro-fluctuations and stresses. Algorithmic speed-ups for both methods with respect to the full-order micro-structural model are quantified. The pros and cons of both classes are thereby clearly identified.

The need for personalised and smart products drives the development of structural electronics with mass-customisation capability. A number of challenges need to be overcome in order to address the potential of complete free form manufacturing of electronic devices. One key challenge is the integration of conductive structures and components into 3D printed devices by combining different materials and printing techniques that have nearly incompatible printing conditions. In this paper, several methods to integrate electronic circuits and components into a 3D printed structure are discussed. The functional performance of the resulting structures is described. Structural parts were manufactured with a stereolithography-based 3D printing technique, which was interrupted to pick and place electronic components, followed by either direct writing or squeegee filling of conductive material. A thermal curing step was applied to enhance the bonding and improve the electrical performance. Optical micrography, 4-point resistance measurement and cross-sectional analysis were performed to evaluate functionality.

In this paper we use an eXtended Finite Element Method based model for the simulation of shear fracture in fully saturated porous materials. The fracture is incorporated as a strong discontinuity in the displacement field by exploiting the partition of unity property of finite element shape functions. The pressure is assumed to be continuous across the fracture. However, the pressure gradient, i.e. the fluid flow, can be discontinuous. The failure process is described by the cohesive zone approach and a Tresca fracture condition without dilatancy. We investigate the propagation of a shear fracture under compression asking the question whether or not a Tresca criterion can result in stepwise propagation in a poroelastic medium. In order to evaluate possible numerical artefacts, we also look at the influence of the element size and the magnitude of a time increment. The performance of the X-FEM model and the influence of the pore pressure on the fracture propagation are addressed. Our simulations do not show evidence for step wise progression in mode II failure.

© 2016 Elsevier LtdFracture of engineering alloys in the presence of hydrogen commonly occurs by decohesion along grain boundaries via a mechanism known as hydrogen induced decohesion (HID). This mechanism is investigated here by analysing the mode-I fracture of a single crystal with plastic flow in the crystal described by discrete dislocation plasticity (DDP) and material separation (decohesion) modelled using a cohesive zone formulation. The motion of dislocations is assumed to be unaffected by hydrogen diffusion. While the cohesive strength is assumed to be reduced proportional to the local hydrogen concentration. Two limiting cases are analysed: (i) the fast diffusion limit where the hydrogen within the material is assumed to be at chemical equilibrium throughout the loading so that there is a high hydrogen concentration in the regions of high hydrostatic stress around dislocations and near the crack tip and (ii) the slow diffusion limit where we assume that there is no appreciable hydrogen diffusion over the duration of loading and thus the hydrogen concentration remains spatially uniform as in a stress-free material. The lower cohesive strength at high hydrogen concentrations results in reduced dislocation activity around the crack tip and a reduction in the material toughness. In fact, at the highest hydrogen concentrations analysed here, crack growth primarily occurs in an elastic manner. However, surprisingly the calculations predicted that the toughness in the fast diffusion case was no more than 12% lower compared to the slow diffusion case suggesting that the stress concentrations due to the dislocation structures and the crack tip fields have only a minor effect on the toughness reduction in the presence of hydrogen. The DDP calculations are finally used to investigate the sensitivity of the material toughness to the grain boundary cohesive strength. The calculations show that the toughness of materials with a small cohesive opening at the peak cohesive traction are more sensitive to hydrogen loading. We speculate that this result might be used as a guide in grain boundary engineering to design alloys that are less sensitive to hydrogen embrittlement by the HID mechanism.

In this paper, we present a general partition of unity-based cohesive zone model for fracture propagation and nucleation in saturated porous materials. We consider both two-dimensional isotropic and orthotropic media based on the general Biot theory. Fluid flow from the bulk formation into the fracture is accounted for. The fracture propagation is based on an average stress approach. This approach is adjusted to be directionally depended for orthotropic materials. The accuracy of the continuous part of the model is addressed by performing Mandel’s problem for isotropic and orthotropic materials. The performance of the model is investigated with a propagating fracture in an orthotropic material and by considering fracture nucleation and propagation in an isotropic mixed-mode fracture problem. In the latter example we also investigated the influence of the bulk permeability on the numerical results.

An approach is proposed to incorporate gradient-enhanced damage models in shell elements. The approach is elaborated for a solid-like shell element, which is advantageous because of the availability of nodes at the top and bottom shell surfaces, and the presence of a three-dimensional strain state. Some simple examples are given to demonstrate the versatility and the convergence of the method.

Interface elements are a powerful tool for modelling discontinuities. Herein, we develop an interface element that is based on the isogeometric analysis concept. Through Bezier extraction, the novel interface element can be casted in the same format as conventional interface elements. Consequently, the isogeometric interface element can be implemented in a straightforward manner in an existing finite element software by a mere redefinition of the shape functions. The interface elements share the advantages of isogeometric continuum elements in that they can exactly model the geometry. On the other hand, they inherit the simplicity of conventional interface elements, but also some deficiencies, such as the occurrence of traction oscillations when a high interface stiffness is used. The extension towards poroelasticity is rather straightforward, and in this situation, the smoother flow profiles and the ensuing preservation of local mass balance are additional advantages. These are demonstrated at the hand of some example problems.

SUMMARY We present an alternative numerical approach for predicting the behaviour of a deformable fluid-saturated porous medium. The conventional finite element technology is replaced by isogeometric analysis that uses non-uniform rational B-splines. The ability of these functions to provide higher-order continuity and to exactly represent complex geometries makes isogeometric analysis a suitable candidate for accurately modeling a poroelastic medium. After some preliminaries regarding the formulation of isogeometric finite elements using Bezier extraction and a concise outline of poroelasticity theory, we describe how isogeometric finite elements can be used for a mixed formulation that results in case of a porous medium. The manuscript concludes by one-dimensional and two-dimensional examples, which demonstrate the superiority of the results of isogeometric finite element analysis in terms of the smoothness of the results compared with conventional finite element analysis and suggest that the requirement on the minimum time step for consolidation problems can be mitigated using interpolation functions that possess a higher-order continuity. Copyright © 2013 John Wiley & Sons, Ltd.

An isogeometric solid-like shell formulation is proposed in which B-spline basis functions are used to construct the mid-surface of the shell. In combination with a linear Lagrange shape function in the thickness direction, this yields a complete three-dimensional representation of the shell. The proposed shell element is implemented in a standard finite element code using Bezier extraction. The formulation is verified using different benchmark tests.

In the cohesive surface model cohesive tractions are transmitted across a two-dimensional surface, which is embedded in a three-dimensional continuum. The relevant kinematic quantities are the local crack opening displacement and the crack sliding displacement, but there is no kinematic quantity that represents the stretching of the fracture plane. As a consequence, in-plane stresses are absent, and fracture phenomena as splitting cracks in concrete and masonry, or crazing in polymers, which are governed by stress triaxiality, cannot be represented properly. In this paper we extend the cohesive surface model to include in-plane kinematic quantities. Since the full strain tensor is now available, a three-dimensional stress state can be computed in a straightforward manner. The cohesive band model is regarded as a subgrid scale fracture model, which has a small, yet finite thickness at the subgrid scale, but can be considered as having a zero thickness in the discretisation method that is used at the macroscopic scale. The standard cohesive surface formulation is obtained when the cohesive band width goes to zero. In principle, any discretisation method that can capture a discontinuity can be used, but partition-of-unity based finite element methods and isogeometric finite element analysis seem to have an advantage since they can naturally incorporate the continuum mechanics. When using interface finite elements, traction oscillations that can occur prior to the opening of a cohesive crack, persist for the cohesive band model. Example calculations show that Poisson contraction influences the results, since there is a coupling between the crack opening and the in-plane normal strain in the cohesive band. This coupling holds promise for capturing a variety of fracture phenomena, such as delamination buckling and splitting cracks, that are difficult, if not impossible, to describe within a conventional cohesive surface model. © 2013 Springer Science+Business Media Dordrecht.

A general numerical model has been developed for fluid flow in a progressively fracturing porous medium subject to large deformations. The fluid flow away from the crack is modelled in a standard manner using Darcy’s relation. In the discontinuity a similar relation is assumed for the fluid flow, but with a different permeability to take into account the higher porosity within the crack due to progressive damage evolution. The crack is described in a discrete manner by exploiting the partition-of-unity property of finite element shape functions. The nucleation and the opening of micro-cracks are modelled by a traction-separation relation. A heuristic approach is adopted to model the orientation of the cracks at the interfaces in the deformed configuration. A two-field formulation is derived, with the solid and the fluid velocities as unknowns. The weak formulation is obtained, assuming a Total Lagrangian formulation. This naturally leads to a set of coupled equations for the continuous and for the discontinuous parts of the mixture. The resulting discrete equations are nonlinear due to the cohesive-crack model, the large-deformation kinematic relations, and the coupling terms between the fine scale and the coarse scale. The capabilities of the model are shown at the hand of some example problems.

The development of models for a priori assessment of the reliability of micro electromechanical systems is of crucial importance for the further development of such devices. In this contribution a partition of unity-based cohesive zone finite element model is employed to mimic crack nucleation and propagation in a piezoelectric continuum. A multiscale framework to appropriately represent the influence of the microstructure on the response of a miniaturized component is proposed. It is illustrated that using the proposed multiscale method a representative volume element exists. Numerical simulations are performed to demonstrate the constitutive homogenization framework. Copyright © 2009 John Wiley & Sons, Ltd.

The static failure behaviour of the fibre-metal laminate GLARE is examined using 3D finite element simulations. The configuration analysed is a centre-cracked tensile specimen composed of two aluminium layers sandwiching a cross-plied, fibre-epoxy layer. The crack and delamination growths are simulated by means of interface elements equipped with a mixed-mode damage model. The mode-mixity is derived from an energy criterion typically used in linear elastic fracture mechanics studies. The damage kinetic law is rate-dependent, in order to simulate rate effects during interfacial delamination and to avoid numerical convergence problems due to crack bifurcations. The numerical implementation of the interface damage model is based on a backward Euler approach. In the boundary value problem studied, the failure responses of GLARE specimens containing elastic aluminium layers and elasto-plastic aluminium layers are compared. The development of plastic deformations in the aluminium layers stabilizes the effective failure response, and increases the residual strength of the laminate. For a 'quasi-brittle' GLARE specimen with elastic aluminium layers, the residual strength is governed by the toughness for interfacial delamination, and is in close correspondence with the residual strength obtained from a closed-form expression derived from energy considerations. Conversely, for a 'ductile' GLARE specimen with elasto-plastic aluminium layers, the residual strength is also determined by the relation between the fracture strength and the yield strength of the aluminium. The amount of constraint by the horizontal displacements at the vertical specimen edges has a moderate to small influence on the residual strength. Furthermore, the ultimate laminate strength is lower for a larger initial crack length, and shows to be in good correspondence with experimental values. © 2008 Elsevier Ltd. All rights reserved.

A robust method to trace the equilibrium path in non-linear solid mechanics problems is proposed. A general arc-length constraint based on the energy release rate is developed. Constraints have been derived for the cases of geometrically linear damage, geometrically linear plasticity and geometrically non-linear damage. All three constraints can efficiently be applied in a finite element context. Numerical simulations demonstrate that the proposed framework gives robust results for these cases. Applicability of the proposed framework to other types of constitutive and/or kinematic behaviour is predicted.

A discrete model for analyzing the interaction between plastic flow and martensitic phase transformations is developed. The model is intended for simulating the microstructure evolution in a single crystal of austenite that transforms non-homogeneously into martensite. The plastic flow in the untransformed austenite is simulated using a plane-strain discrete dislocation model. The phase transformation is modeled via the nucleation and growth of discrete martensitic regions embedded in the austenitic single crystal. At each instant during loading, the coupled elasto-plasto-transformation problem is solved using the superposition of analytical solutions for the discrete dislocations and discrete transformation regions embedded in an infinite homogeneous medium and the numerical solution of a complementary problem used to enforce the actual boundary conditions and the heterogeneities in the medium. In order to describe the nucleation and growth of martensitic regions, a nucleation criterion and a kinetic law suitable for discrete regions are specified. The constitutive rules used in discrete dislocation simulations are supplemented with additional evolution rules to account for the phase transformation. To illustrate the basic features of the model, simulations of specimens under plane-strain uniaxial extension and contraction are analyzed. The simulations indicate that plastic flow reduces the average stress at which transformation begins, but it also reduces the transformation rate when compared with benchmark simulations without plasticity. Furthermore, due to local stress fluctuations caused by dislocations, martensitic systems can be activated even though transformation would not appear to be favorable based on the average stress. Conversely, the simulations indicate that the plastic hardening behavior is influenced by the reduction in the effective austenitic grain size due to the evolution of transformation. During cyclic simulations, the coupled plasticity-transformation model predicts plastic deformations during unloading, with a significant increase in dislocation density. This information is relevant for the development of meso- and macroscopic elasto-plasto-transformation models.

Delamination in composite structures is best modelled at a mesoscopic level. In this approach, the plies are modelled as continua, which can either be assumed to behave linearly elastically or to degrade according to a damage law. Delamination in the interfaces between them is modelled using a discrete relation between interface tractions and relative displacements. Key in this type of modelling is the presence of a work of separation or fracture energy, which governs the growth of the delamination. This cohesive-surface approach has traditionally been implemented numerically using special interface elements which connect the continuum elements that are used to model the plies. Exploiting the partition-of-unity property of finite element shape functions to model the interface separation process offers some advantages, since interfaces can be inserted at the onset of delamination and not a priori, as in the conventional approach. As a consequence, elastic compliance of the interface prior to onset of delamination, spurious traction oscillations ahead of the delamination front and spurious wave reflections because of the presence of a high stiffness value are avoided. Moreover, unlike the conventional approach, unstructured meshes can be employed.

In this contribution we present a partition of unity based model for the simulation of hydraulic fracturing processes. Bulk poroelasticity is based on the Biot theory. The pressure in the fracture is included as an additional degree of freedom. A Fracture can grow in arbitrary directions by using the Camacho Ortiz fracture criterion with a cohesive zone formulation. The performance of the numerical model is addressed by considering fracture propagation from a 2D borehole. The initial stress field is validated with Kirsch's analytical solution. The results from the numerical model indicate that preferred direction of a hydraulic fracture is in the direction of the highest confining stress. In future works this model will include the nucleation of fractures and can be applied to more complex hydraulic fracturing situations.

An isogeometric continuum shell formulation is proposed in which NURBS basis functions are used to construct the reference surface of the shell. Through-the-thickness behavior is interpolated using a higher-order B-spline which is in contrast to the standard continuum shell (solid-like shell) formulation where a linear Lagrange shape function is typically used in the thickness direction. The present formulation yields a complete isogeometric representation of the continuum shell. The shell element is implemented in a standard finite element code using Bézier extraction which facilitates numerical integration on the reference surface of the shell. Through-the-thickness integration is done using a connectivity array which determines the support of a B-spline basis function over an element. The formulation has been verified using different linear and geometrically non-linear examples. The ability of the formulation in modelling buckling of static delaminations in composite materials is also demonstrated. © 2013 Elsevier B.V.

Multi-scale methods provide a new paradigm in many branches of sciences, including applied mechanics. However, at lower scales continuum mechanics can become less applicable, and more phenomena enter which involve discon- tinuities. The two main approaches to the modelling of discontinuities are briefly reviewed, followed by an in-depth discussion of cohesive models for fracture. In this discussion emphasis is put on a novel approach to incorporate triaxi- ality into cohesive-zone models, which enables for instance the modelling of crazing in polymers, or of splitting cracks in shear-critical concrete beams. This is followed by a discussion on the representation of cohesive crack models in a continuum format, where phase-field models seem promising.

A fibre–metal laminate is a composite of metal and fibre-reinforced prepreg layers. An example of such a material is Glare. It consists of alternating layers of aluminium and glass-fibre-reinforced prepreg. The material can be sensitive to delamination buckling, which occurs when a partially delaminated panel is subjected to a compressive force. The interaction of local buckling and extension of the delaminated zone typically results in a decrease of the residual strength and, eventually, in a collapse of the structure. This phenomenon can be observed in experimental tests, but numerical analyses are needed to obtain a better understanding of the mechanisms and the critical parameters. In this paper, some experimental observations are discussed regarding delamination buckling in Glare and, on the basis of these observations, a numerical model is constructed at a meso-mechanical level. In this approach, solid-like shell elements are used to model the individual layers. They are connected by interface elements, which are capable of modelling delamination between the layers.

Shales, clays, hydrogels, and tissues swell and shrink under changing osmotic conditions, which may lead to failure. The relationship between the presence of cracks and fluid flow has had little attention. The relationship between failure and osmotic conditions has had even less attention. The aim of this research is to study the effect of osmotic conditions on propagating discontinuities under different types of loads for saturated ionized porous media using the finite element method (FEM). Discontinuous functions are introduced in the shape functions of the FEM by partition of unity method, independently of the underlying mesh. Damage ahead of the crack-tip is introduced by a cohesive zone model. Tensile loading of a crack in an osmoelastic medium results in opening of the crack and high pressure gradients between the crack and the formation. The fluid flow in the crack is approximated by Couette flow. Results show that failure behavior depends highly on the load, permeability, (osmotic) prestress and the stiffness of the material. In some cases it is seen that when the crack propagation initiates, fluid is attracted to the crack-tip from the crack rather than from the surrounding medium causing the crack to close. The results show reasonable mesh-independent crack propagation for materials with a high stiffness. Stepwise crack propagation through the medium is seen due to consolidation, i.e., crack propagation alternates with pauses in which the fluid redistributes. This physical phenomenon challenges the numerical scheme. Furthermore, propagation is shown to depend on the osmotic prestressing of the medium. This mechanism may explain the tears observed in intervertebral disks as degeneration progresses.

An extension to a finite strain framework of a two-scale numerical model for propagating crack in porous material is proposed to model the fracture in intervertebral discs. In the model, a crack is described as a propagating cohesive zone by exploiting the partition-of-unity property of finite element shape functions. At the micro-scale, the flow in the cohesive crack is modelled as viscous fluid using Stokes' equations which are averaged over the cross section of the cavity. At the macro-scale, identities are derived to couple the local momentum and the mass balance to the governing equations for a saturated porous material. The resulting discrete equations are nonlinear due to the cohesive constitutive equations and the geometrically nonlinear kinematic relations. A Newton-Raphson iterative procedure is used to consistently linearise the derived system while a Crank-Nicholson scheme takes care of the time integration of the system. The derived model is used to analyse a quasi-static crack growth in confined compression under tensile loading.

A number of self healing mechanisms for composite materials have been presented in the previous chapters of this book. These methods vary from the classical concept of micro-encapsulating of healing agents in polymer systems to the autonomous healing of concrete. The key feature of these self healing mechanisms is the transport of material to the damaged zone in order to establish the healing process. Generally, this material is a fluid and its motion is driven by capillary action which enables transportation over relatively large distances requiring little or no work. In the microencapsulated polymers as developed by White et al. [1], this liquid material is a healing agent, which is enclosed in the material by micro-encapsulation. When the capsule is ruptured by a crack, the healing agent will flow into the crack, driven by capillary action. Polymerisation of this healing agent is triggered by contact with catalysts which are inserted in the material and whose position is fixed. The new polymerised material will rebond the crack surfaces.

A numerical method for crack growth is described in which the crack is not regarded as a single discontinuity that propagates continuously. Instead, the crack is represented by a set of overlapping cohesive segments. These cohesive segments are inserted into finite elements as discontinuities in the displacement field by exploiting the partition-of-unity property of shape functions. The cohesive segments can be incorporated at arbitrary locations and orientations and are not tied to any particular mesh direction. The evolution of decohesion of the segments is governed by a cohesive law. The independent specification of bulk and cohesive constitutive relations leads to a characteristic length being introduced into the formulation. The formulation permits both crack nucleation and discontinuous crack growth to be modelled. The implementation is outlined and some numerical examples are presented.

Laminates can be sensitive to delamination buckling, which occurs when a partially delaminated panel is subjected to a compressive load. The interaction of local buckling and extension of the delaminated zone typically results in a decrease of the residual strength and, eventually, in a collapse of the structure. Fortunately, this phenomenon has never been observed in experimental tests with Glare. Although the delaminated layers buckle locally, the delamination front does not propagate within the range of compressive stresses that can be expected in typical aerospace structures.