Higher order Cauchy–Born rule based multiscale cohesive zone model and prediction of fracture toughness of silicon thin films

In this work, we extend the multiscale cohesive zone model (MCZM) (Zeng and Li in Comput Methods Appl Mech Eng 199:547–556, 2010), in which interatomic potential is embedded into constitutive relation to express cohesive law in fracture process zone, to include the hierarchical Cauchy–Born rule in the process zone and to simulate three dimensional fracture in silicon thin films. The model has been applied to simulate fracture stress and fracture toughness of single-crystal silicon thin film by using the Tersoff potential. In this study, a new approach has been developed to capture inhomogeneous deformation inside the cohesive zone. For this purpose, we introduce higher order Cauchy–Born rules to construct constitutive relations for corresponding higher order process zone elements, and we introduce a sigmoidal function supported bubble mode in finite element shape function of those higher order cohesive zone elements to capture the nonlinear inhomogeneous deformation inside the cohesive zone elements. Benchmark tests with simple 3D models have confirmed that the present method can predict the fracture toughness of silicon thin films. Interestingly, this is accomplished without increasing of computational cost, because the present model does not require quadratic elements to represent heterogeneous deformation, which is the inherent weakness of the previous MCZM model. Quantitative comparisons with experimental results are performed by computing crack propagation in non-notched and initially notched silicon thin films, and it is found that our model can reproduce essential material properties, such as Young’s modulus, fracture stress, and fracture toughness of single-crystal silicon thin films.     

Mechanical characterization of Ti–5Al–2.5Sn ELI alloy at cryogenic and room temperatures

An experimental campaign consisting of tensile and fracture tests at cryogenic and room temperatures has been conducted on a Ti–5Al–2.5Sn extra-low-interstitial (ELI) alloy. It has been assessed that, at decreasing testing temperature: Young’s modulus slightly increases; yield and failure strengths increase significantly; fracture toughness decreases. Since a ductile void growth to coalescence micromechanism always governs failure in the spanned temperature interval, crack growth is simulated by allowing for material nonlinearities in the process zone, where ductile tearing takes place. Numerical results have been obtained by modeling the response of the process zone through either a cohesive model or Gurson’s constitutive law for porous-ductile media. It is shown that the latter approach can accurately describe the failure mechanism at any test temperature and for any specimen geometry, whereas the former one is not able to account for stress triaxiality at the crack tip and therefore requires a new calibration anytime the specimen geometry is varied.     

Predicting mode-II delamination suppression in z-pinned laminates

A finite element model for predicting delamination resistance of z-pin reinforced laminates under the mode-II load condition is presented. End notched flexure specimen is simulated using a cohesive zone model. The main difference of this approach to previously published cohesive zone models is that the individual bridging force exerted by z-pin is governed by a specific traction-separation law derived from a unit-cell model of single pin failure process, which is independent of the fracture toughness of the unreinforced laminate. Therefore, two separate traction-separation laws are employed; one represents unreinforced laminate properties and the other for the enhanced delamination toughness owing to the pin bridging action. This approach can account for the so-called large scale bridging effect and avoid using concentrated pin forces in numerical models, thus removing the mesh-size dependency and permitting more accurate and reliable computational solutions.     

A phenomenological multi-axial constitutive law for switching in polycrystalline ferroelectric ceramics

A phenomenological constitutive law for ferroelectric switching due to multi-axial mechanical and electrical loading of a polycrystalline material is developed. The framework of the law is based on kinematic hardening plasticity theory and has a switching surface in the space of mechanical stress and electric field that determines when non-linear response is possible. The size and shape of the switching surface in a modified electric field space remains fixed during non-linear behavior but its center moves around and thus is controlled by a kinematical hardening process. In general, the remanent polarization and the remanent strain are used as the internal variables that control how the center of the switching surface moves. However, the form presented in this paper has a one-to-one relationship between the remanent strain and the remanent polarization, simplifying the constitutive law and allowing remanent polarization to be used as the only internal variable controlling the kinematic effects. The constitutive law successfully reproduces hysteresis and butterfly loops for ferroelectric ceramics. The hysteresis and butterfly loops respond appropriately to the application of a fixed compressive stress parallel to the electric field. In addition, the law successfully handles remanent polarization rotation due to the application of electric field at an angle to the polarization direction.     

Multi-length scale micromorphic process zone model

The prediction of fracture toughness for hierarchical materials remains a challenging research issue because it involves different physical phenomena at multiple length scales. In this work, we propose a multiscale process zone model based on linear elastic fracture mechanics and a multiscale micromorphic theory. By computing the stress intensity factor in a K-dominant region while maintaining the mechanism of failure in the process zone, this model allows the evaluation of the fracture toughness of hierarchical materials as a function of their microstructural properties. After introducing a multi-length scale finite element formulation, an application is presented for high strength alloys, whose microstructure typically contains two populations of particles at different length scales. For this material, the design parameters comprise of the strength of the matrix–particle interface, the particle volume fraction and the strain-hardening of the matrix. Using the proposed framework, trends in the fracture toughness are computed as a function of design parameters, showing potential applications in computational materials design.     

A fully non-linear axisymmetrical quasi-kirchhoff-type shell element for rubber-like materials

An axisymmetrical shell element for large deformations is developed by using Ogden's non-linear elastic material law. This constitutive equation, however, demands the neglect of transverse shear deformations in order to yield a consistent theory. Therefore, the theory can be applied to thin shells only. Eventually a ‘quasi-Kirchhoff-type theory’ emerges. Within this approach the computation of the deformed director vector d is a main assumption which is essential to describe the fully non-linear bending behaviour. Furthermore, special attention is paid to the linearization procedure in order to obtain quadratic convergence behaviour within Newton's method. Finally, the finite element formulation for a conical two-node element is given. Several examples show the applicability and performance of the proposed formulation.     

Fracture toughness of exponential layer-by-layer polyurethane/poly(acrylic acid) nanocomposite films

This paper characterizes the fracture toughness of layer-by-layer (LBL) manufactured thin films with elastic polyurethane, a tough polymer, and poly(acrylic acid) as a stiffening agent. A single-edge-notch tension (SENT) specimen is used to study mode I crack propagation as a function of applied loading. Experimental results for the full-field time histories of the strain maps in the fracturing film have been analyzed to obtain R-curve parameters for the nanocomposite. In particular, by using the strain maps, details of the traction law are measured. A validated finite strain phenomenological visco-plastic constitutive model is used to characterize the nanocomposite film while a discrete cohesive zone model (DCZM) is implemented to model the fracture behavior. The LBL manufactured nanocomposite is found to display a higher fracture toughness than the unstiffened base polymer.     

A practical approach for the separation of interfacial toughness and structural plasticity in a delamination growth experiment

Interfacial delamination is a key reliability challenge in composites and micro-electronic systems due to (high density) integration of dissimilar materials. Predictive finite element models require the input of interface properties, often determined with an interface delamination growth experiment with (nearly) constant process zone, relying on the assumption of no permanent deformation in the sample structure layers. However, much evidence in the literature exists that plasticity often does occur in the sample structure during delamination experiments, which should be adequately dealt with to obtain the real interface fracture toughness that is independent of the thickness of the two sample arms. This paper presents a practical approach for the separation of interfacial toughness and structural plasticity in a delamination growth experiment on a double-cantilever beam specimen involving only small-scale plasticity at the interface. The procedure does not require knowledge of the constitutive behavior of the adherent layers. It only deals with the separation of structural plasticity in the adherents, whereas small-scale plasticity in connection with ductile interface fracture is lumped into the interface fracture toughness. The proposed approach was numerically verified for one set of parameters. Experimental assessment of the approach on industrially-relevant copper lead frame–molding compound epoxy interface structures showed a correction of the interface fracture toughness of more than a factor of two, demonstrating the potentially significant errors induced by plastic deformation of the sample structure during delamination experiments.     

Numerical techniques for the analysis of crack propagation in cohesive materials

The aim of the paper is the development, assessment and use of suitable numerical procedures for the analysis of the crack evolution in cohesive materials. In particular, homogeneous as well as heterogeneous materials, obtained by embedding short stiff fibres in a cohesive matrix, are considered. Two‐dimensional Mode I fracture problems are investigated. The cohesive constitutive law is adopted to model the process zone occurring at the crack tip. An elasto‐plastic constitutive relationship, able to take into account the processes of fibre debonding and pull‐out, is introduced to model the mechanical response of the short fibres. Two numerical procedures, based on the stress and on the energy approach, are developed to investigate the crack propagation in cohesive as well as fibre‐reinforced materials, characterized by a periodic crack distribution. The results obtained using the stress and energy approaches are compared in order to evaluate the effectiveness of the procedures. Investigations on the size effect for microcracked periodic cohesive materials, and on the beneficial effects of the fibres in improving the composite material response, are developed. Copyright © 2003 John Wiley & Sons, Ltd.     

Fracture toughness of plain concrete from three-point bend specimens

A simple method is proposed for determining the fracture toughness of plain concrete from three-point bend specimens, based on the concept of effective notch depth to account for the non-linear pre-peak load-deflection behaviour. The fracture toughness so determined is shown not to depend on the specimen size. The method improves an earlier version of the effective crack model in several ways. First, it is no longer necessary to linearize the pre-peak non-linearity, thereby eliminating the inaccurate task of establishing the limit of elastic response. Secondly, regression expressions for determining the effective notch depth should be far more accurate because they are based on an analysis of not only the authors’ test data but that of several researchers around the world. Thirdly, these expressions do not depend on the size of the test specimen. It is shown that the predictions of the effective crack model are in good agreement with two other non-linear process zone models, as far as three-point notched beam specimens are concerned.     

Unified and mixed formulation of the 4-node quadrilateral elements by assumed strain method: Application to thermomechanical problems

In this paper, a class of ‘assumed strain’ mixed finite element methods based on the Hu–Washizu variational principle is presented. Special care is taken to avoid hourglass modes and shear locking as well as volumetric locking. An unified framework for the 4-node quadrilateral solid and thermal as well as thermomechanical coupling elements with uniform reduced integration (URI) and selective numerical integration (SI) schemes is developed. The approach is simply implemented by a small change of the standard technique and is applicable to arbitrary non-linear constitutive laws including isotropic and anisotropic material behaviours. The implementation of the proposed SI elements is straightforward, while the development of the proposed URI elements requires ‘anti-hourglass stresses’ which are evaluated by classical constitutive equations. Several numerical examples are given to demonstrate the performance of the suggested formulation, including static/dynamic mechanical problems, heat conduction and thermomechanical problems.     

Influence of loading rate on fracture behaviour of extra deep drawn steel sheets

Fracture tests of extra deep drawn steel sheet were carried out at ambient temperature to investigate the effect of loading rate on fracture toughness. To determine fracture limits, fracture tests and finite‐element cohesive zone model simulation tool are used. Fracture tests are conducted at various loading rates (0.1–2.5 mm min^?1). An alternative constant traction separation law is used to account for maximum load and large load line displacements. Experimental findings, as well as cohesive zone model, show that the loading rate has no significant effect on fracture toughness till 0.4 mm min^?1; however, there is a sharp decrease in fracture toughness beyond 0.4 mm min^?1.     

基于定长裂缝试件的脆性材料尺寸效应实验方法

由于脆性或准脆性材料内各类微缺陷的影响,材料的力学性能,如名义破坏应力,　刚度以及断裂韧性等随试件的大小而改变,具有明显的尺寸效应。通常情况下,描述材料尺寸效应的Ｂａｚａｎｔ尺寸效应律是建立在一系列相似试件的基础上通过实验方法确定的。　本文提出了一种新的用含固定长度裂缝试件测定断裂韧性和有效断裂过程区大小的实验方法和计算公式。与相似试件测定方法相比,实验结果吻合很好。根据本文提出的定长裂缝试件实验方法,在保证与相似试件相同脆性指数范围的前提下,可以用小试件进行测量。     

Integration of Tresca and Mohr–Coulomb constitutive relations in plane strain elastoplasticity

This paper considers the problem of integrating the constitutive relations for Tresca and Mohr–Coulomb materials under conditions of plane strain. In the case of a Tresca material, we show that the constitutive law may be integrated exactly by assuming the strain rates d?/dλ to be constant. We also derive a semi-analytic method for integrating both types of constitutive law which assumes that the quantities d?/dλ are constant. This approach is motivated by the fact that the exact variation of the strains during a time interval is unknown and leads to a single non-linear equation in λ which can be solved efficiently to yield the unknown stresses. Finally, we compare the results from the analytic and semi-analytic methods with those from a variety of numerical integration schemes.     

Towards the robotic characterization of the constitutive response of composite materials

A historical and technical overview of a paradigm for automating research procedures on the area of constitutive identification of composite materials is presented. Computationally controlled robotic, multiple degree-of-freedom mechatronic systems are used to accelerate the rate of performing data-collecting experiments along loading paths defined in multidimensional loading spaces. The collected data are utilized for the inexpensive data-driven determination of bulk material non-linear constitutive behavior models as a consequence of generalized loading through parameter identification/estimation methodologies based on the inverse approach. The concept of the dissipated energy density is utilized as the representative encapsulation of the non-linear part of the constitutive response that is responsible for the irreversible character of the overall behavior. Demonstrations of this paradigm are given for the cases of polymer matrix composite materials systems. Finally, this computational and mechatronic infrastructure is used to create conceptual, analytical and computational models for describing and predicting material and structural performance.     

Updated Lagrangian formulation of contact problems using variational inequalities

This article is devoted to the formulation and solution of general frictional contact problems in elasto-plastic solids undergoing large deformations using variational inequalities. An updated Lagrangian formulation is adopted to develop the incremental variational inequality representing this class of problems over the loading history. The Jaumann objective stress rate is incorporated in the formulation of the elasto-plastic constitutive equations to account for large rotations, while Coulomb's law is used to model the friction forces. The resulting variational inequality is treated using mathematical programming in association with a newly developed successive approximation scheme. This scheme, which is based upon the regularization of the frictional work, is used to impose the active contact constraints identified to calculate the incremental changes in the displacement field. The newly developed approach offers the advantages of reducing the active number of variables which is highly desirable in non-linear elasto-plastic problems. The merits of the formulations are demonstrated by application to an illustrative example and to the analysis of the deep drawing process. © 1997 John Wiley & Sons, Ltd.     

Matrix dominated time dependent failure predictions in polymer matrix composites

Various types of matrix dominated failures in polymer matrix composites (PMC) are reviewed. Current methods to evaluate the modulus degradation of PMC materials are discussed including viscoelastic/plastic and continuum damage models. It is pointed out that in each case the approach is based upon developing an analytical constitutive relation for the material in order to represent a measured stress–strain response. The suggestion is made that care must be used in the measurement of stress–strain behavior such that the modeling represents the true material behavior. New digital imaging methods are suggested as a means to determine in situ properties at a local scale commensurate with the continuum modeling procedure. A little used method to model viscoelastic/plastic (linear and non-linear) effects is discussed and modified to obtain a simple and easy to use time dependent failure law. Also, a little used energy based time dependent failure criterion is presented which can be combined with a non-linear viscoelastic integral approach to provide a prediction method for the time for creep rupture under simple stress states. Each is validated with experimental data for simple stress states but their generality is such that they could be used for complex (3-D) stress states. Advantages and limitations of both are addressed. Finally, a discussion of possible fruitful research areas are presented with the view of providing engineers in industry with an easy to use accelerated life prediction procedure.     

Simulation of dynamic fracture with the Material Point Method using a mixed J-integral and cohesive law approach

A new approach to simulating fracture, in which toughness is partitioned between the crack tip and, optionally, a process zone, is applied to dynamic fracture processes. In this approach, classical fracture mechanics determines crack tip propagation, and cohesive laws characterize process zone response and determine crack root and process zone propagation. The approach is implemented in the Material Point Method, a particle method in which the fracture path is unconstrained by a body-fitted mesh. The approach is found suitable for modeling a range of dynamic fracture processes, from brittle fracture to fracture with crack bridging. A variety of ways of partitioning toughness are explored with the aim of distinguishing model parameters via experimental measurements, particularly R curves. While no unique relationship exists, R curves, or effective R curves, on a suite of materials would provide substantial insight into model parameters. Advantages to the approach are identified, both in versatility and in regards to practical matters associated with implementing numerical fracture algorithms. It is found to perform well in dynamic fracture scenarios.     

Modelling complex cracks with finite elements: a kinematically enriched constitutive model

A continuum constitutive framework with embedded cohesive interface model is presented to describe the failure of quasi-brittle materials. Both cohesive behaviour for cracking inside the fracture process zone and elastic bulk behaviour are treated at integration points making implementation straightforward. In this sense, the proposed approach is simpler than existing ones that focus on element enrichments, such as the extended finite element method, while share similarities with smeared crack models, and offers the capability to correctly model quasi-brittle failure in post-peak regime at constitutive level. In this work, the formulation is introduced, numerical algorithms described and static and dynamic fracture simulations with complex crack patterns are conducted to demonstrate the capability and advantage of the proposed approach.