Thermomechanical behaviour of a damageable beam in contact with two stops

A model for the thermomechanical behaviour of a beam which allows for the general evolution of material damage is presented and investigated. One end of the beam is fixed while the other is constrained to move between two stops. The contact of the free tip with the stops is modelled by the normal compliance condition. The thermal interaction between the stops and the free tip is described by a heat exchange condition where the heat transfer coefficient is a general function of the gaps between the tip and the stops. The effects on the mechanical properties of the material due to crack expansion are described by a damage field, which measures the decrease in the load-bearing capacity of the material. The damage evolves as a constrained diffusion process in which the microcracks that develop may grow or disappear. The mathematical model consists of a coupled system of energy--elasticity equations together with a nonlinear parabolic inclusion for the damage field. The existence of a local solution is established using truncation, penalization, and a priori estimates.     

Self-similar solution of a tensile crack problem in a coupled formulation

Using a self-similar variables, an asymptotic investigation is carried out into the stress fields and the rates of creep deformations and degree of damage close to the tip of a tensile crack under creep conditions in a coupled (creep - damage) plane formulation of the problem. It is shown that a domain of completely damaged material (DCDM) exists close to the crack tip. The geometry of this domain is determined for different values of the material parameters appearing in the constitutive relations of the Norton power law in the theory of steady-state creep and a kinetic equation which postulates a power law for the damage accumulation. It is shown that, if the boundary condition at the point at infinity is formulated as the condition of asymptotic approximation to the Hutchinson–Rice-Rosengren solution [Hutchinson JW. Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids 1968;16(1):13–31; Rice JR, Rosengren GF. Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids. 1968;16(1):1–12], then the boundaries of the DCDM, which are defined by means of binomial and trinomial expansions of the continuity parameter, are substantially different with respect to their dimension and shape. A new asymptotic of the for stress field, which determines the geometry of the DCDM and leads to close configurations of the DCDM constructed using binomial and trinomial asymptotic expansions of the continuity parameter, are established by an asymptotic analysis and a numerical solution of the non-linear eigenvalue problem obtained.     

Martensitic transformations and damage: A combined phase field approach

A combined continuum phase field model for martensitic transformations and damage is introduced. The present approach considers the eigenstrain within the martensitic phase which leads in the surrounding material to both tensile and compressive stresses. The damage model needs to account for an appropriate differentiation thereof, since compressive stresses should not promote fracture. Interactions between micro crack propagation and the formation of the martensitic phases are studied in two dimensions. In agreement with experimental observations, martensite forms at the crack tip and influences the crack formation. For the numerical implementation finite elements are used while for the transient terms an implicit time integration scheme is employed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of fracture of fibrous composites with brittle components

Conclusions A numerical analysis of the stress field at the tip of a crack showed high localization, increasing with increase of anisotropy and normal and shear stresses at the crack tip, which can cause various types of fracture. The use of the J-integral for estimates of the flow of elastic energy to the crack tip, simulated by a greatly elongated elliptical notch, is apparently possible in approximate calculations, taking into account localization of the zone of nonlinear behavior of the material. A scheme of estimating the crack resistance of a fibrous material with brittle components, taking into account the contribution of various forms of microfracture to energy dissipation, is proposed. The calculated value of G^* differs from the experimental data by about 20%, which, taking into account the simplicity of the calculation method, can be considered satisfactory.Report presented at the Fourth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, October, 1980).Translated from Mekhanika Kompozitnykh Materialov, No. 4, pp. 615–619, July–August, 1981.The author thanks S. T. Mileiko for formulating the problem, advice, and discussion of the work.     

A gradient model for finite strain elastoplasticity coupled with damage

This paper describes the formulation of an implicit gradient damage model for finite strain elastoplasticity problems including strain softening. The strain softening behavior is modeled through a variant of Lemaitre's damage evolution law. The resulting constitutive equations are intimately coupled with the finite element formulation, in contrast with standard local material models. A 3D finite element including enhanced strains is used with this material model and coupling peculiarities are fully described. The proposed formulation results in an element which possesses spatial position variables, nonlocal damage variables and also enhanced strain variables. Emphasis is put on the exact consistent linearization of the arising discretized equations.

A numerical set of examples comparing the results of local and the gradient formulations relative to the mesh size influence is presented and some examples comparing results from other authors are also presented, illustrating the capabilities of the present proposal.     

Multiscale damage analyis for steel structures

An approach to model the deterioration of steel structures is presented by transferring the results of a continuum damage mechanics analysis to an extended beam model which can account for the loss of structural integrity. Damage starts at the microscopic level by the initiation, growth and coalescence of voids with decreasing material resistance followed by the formation of microcracks at the mesoscale. Nevertheless, the material behavior can be sufficiently modelled on a phenomenological basis taking into account viscoplasticity, hardening effects and damage evolution. The associated model parameters are identified with the help of an evolutionary algorithm adapting numerical to experimental results. Using the finite element method a nonlocal formulation of the damage variable is required to obtain mesh-independent results by structural analysis. The maximum element size is limited by the small magnitude of the internal length. Therefore, numerical analyses of large scale 3D steel structures are computationally expensive. To reduce the effort a beam element is proposed to account for the plastic hinges and the loss of resistance in the course of damage evolution. The corresponding relationship of bending moment and curvature bases on the continuum damage mechanics model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational homogenization for heterogeneous cohesive layers incorporating size effects

A numerical framework is presented for the multiscale modelling of material layers that possess a both heterogeneous and microstructured mesostructure. The material layer is represented as a cohesive interface on the macro level, while on the meso level micromorphic representative volume elements are used. A computational homogenization approach for the cohesive material layers has been proposed to solve this nonlinear multiscale problem numerically. The present framework is particularly well suited for the modelling of damage and failure, as the micromorphic RVE is well known for its regularizing character. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a numerical scheme for curved crack propagation based on configurational forces and maximum dissipation

A numerical scheme is presented to predict crack trajectories in two dimensional components. First a relation between the curvature in mixed–mode crack propagation and the corresponding configurational forces is derived, based on the principle of maximum dissipation. With the help of this, a numerical scheme is presented which is based on a predictor–corrector method using the configurational forces acting on the crack together with their derivatives along real and test paths. With the help of this scheme it is possible to take bigger than usual propagation steps, represented by splines. Essential for this approach is the correct numerical determination of the configurational forces acting on the crack tip. The methods used by other authors are shortly reviewed and an approach valid for arbitrary non–homogenous and non–linear materials with mixed–mode cracks is presented. Numerical examples show, that the method is a able to predict the crack paths in components with holes, stiffeners etc. with good accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of Anisotropic Damage in Arterial Walls

In a certain range of overexpansion arterial walls are characterized by an orthotropic elastic material behavior. Due to different stabilities of the helically arranged fibers, i.e. breakage of collagen crosslinks between the fibers, damage effects are observed in experiments. Because of the fibrous composition it is assumed that damage mainly occurs in the fiber direction. The proposed damage model is extended to arterial wall applications by introducing a referential damage state. The damage approach is applied to a polyconvex model for the hyperelastic behavior of arteries in order to obtain a materially stable model, which guarantees the existence of solutions of the underlying boundary value problem. The performance of the proposed model is presented in a numerical example, where the overexpansion of an atherosclerotic artery is simulated. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of experimental and numerical results on metallic plates subjected to explosions

A comparative study of dynamic response including damage and rupture processes of thin metallic plates subjected to shock-wave impulses – explosions is presented. The results of the finite element numerical analysis are related to experiments. Due to high strain rate during explosions the elasto-viscoplastic Chaboche's constitutive law including damage effects has been applied. For the assumed model proper material parameters identification has been done. In the dynamic, geometrically non-linear analysis the MSC.Marc system has been used. A good correlation between numerical and experimental results has been observed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global approaches for accurate loading analyses and crack path predictions at single and two-cracks systems

Based on the work by Eshelby, the path-independent J_k-, M-, L- and interaction- or I_k-integrals were introduced and applied to cracks for the accurate calculation of crack tip loading quantities. Applying the FE-method to solve boundary value problems with cracks, numerically inaccurate values are observed within the crack tip region affecting the accuracy of local approaches. Simulating crack paths, local approaches face further problems as cracks are running towards interfaces, internal boundaries or other crack faces. Within global approaches, path-independent integrals are calculated along remote contours far from the crack tip, essentially exploiting numerically reliable data requiring special treatment only for the near-tip crack faces. To provide path-independence, additional integrals along interfaces, internal boundaries and crack faces are necessary. In this paper, new global approaches of path-independent integrals are presented and applied to the calculation of crack paths at two-cracks systems. A second focus is directed to the accurate loading analysis and crack path prediction considering anisotropic properties and material interfaces. The numerical model provides crack paths which are in good agreement with those obtained from crack growth experiments. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interfacial model for crash resistant adhesives of ductile-modified epoxy resins

Debonding in adhesive joints is modelled and analyzed with the concept of interfacial mechanics. Material equations are presented for the inelastic behaviour of ductile-modified epoxy resins. A two surface function for the onset of yielding is advantageously expressed in terms of the stress vector on the interface. The theory may be extended to material softening due to damage and to rate dependency. This simple constitutive model is not stable in the sense of Drucker's postulate. Therefore, the non-associated flow rule is modified with a quadratic stress-dependent plastic potential. The material parameters are identified by means of the finite-element simulation of the experimental setup for a bluntly glued double steel tube sample. The numerical performance of this modified model is tested at an adhesively bonded joint in the form of a T-intersection. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal design in axial motion for several frequency constraints

Governing equations for the optimal design of a rod with tip mass subject to several constraints on natural frequency are developed. The relation of this mass distribution to the globally optimum design for lowest frequency is discussed. A numerical example for two frequencies is presented. Some continuity results concerning the optimal mass distribution are presented.     

Investigation of an elastoplastic material model coupled to nonlocal damage in an implicit-gradient framework

In this work, an elastoplastic material model coupled to nonlocal damage is discussed which is based on an implicit gradient-enhanced approach. Combined nonlinear isotropic and kinematic hardening as well as continuum damage of Lemaitre-type are considered. The model is a direct nonlocal extension of a corresponding local model which was presented earlier (see e. g. [1], [2], [3]). Conclusions drawn from a numerical benchmark test performed in this study demonstrate that the nonlocal damage model is suitable to provide mesh-independent solutions in finite element simulations. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of large deformation frictional contact in powder compaction processes

In this paper, the computational aspects of large deformation frictional contact are presented in powder forming processes. The influence of powder–tool friction on the mechanical properties of the final product is investigated in pressing metal powders. A general formulation of continuum model is developed for frictional contact and the computational algorithm is presented for analyzing the phenomena. It is particularly concerned with the numerical modeling of frictional contact between a rigid tool and a deformable material. The finite element approach adopted is characterized by the use of penalty approach in which a plasticity theory of friction is incorporated to simulate sliding resistance at the powder–tool interface. The constitutive relations for friction are derived from a Coulomb friction law. The frictional contact formulation is performed within the framework of large FE deformation in order to predict the non-uniform relative density distribution during large deformation of powder die pressing. A double-surface cap plasticity model is employed together with the nonlinear contact friction behavior in numerical simulation of powder material. Finally, the numerical schemes are examined for efficiency and accuracy in modeling of several powder compaction processes.     

Experimental and Numerical Study of the Open-Hole Tensile Strength of Carbon/Epoxy Composites

Open-hole tension tests are a part of the qualification process for composite parts that need to be joined to other parts in aircraft structures [1]. With each new material, a new set of tests is required. To reduce costs, it is desirable to develop analysis tools for the prediction of damage and failure in such tests, so that the amount of testing can be reduced and predictions can be made about material behaviour early in the design process. In this paper, an experimental and numerical study is presented on the notched (open-hole) strength of high-strength carbon/epoxy composites (HTA/6376). Open-hole tension tests have been performed on specimens with three different lay-ups — quasi-isotropic, zero-dominated, and cross-ply — in accordance with procedures in available standards. The data observed are being used to develop several methods for predicting the notched strength, and results from one such method using a progressive damage analysis are presented with comparisons with experiments. The predictions of specimen stiffness and failure load were found to agree well with experiments. To gain insight into the failure process, damage progression maps are shown.     

Modeling crack micro-branching using finite elements with embedded strong discontinuities

The emphasis of this work lies in the development of a numerical method which is capable of representing the complex physical phenomena arising in the case of crack branching in brittle materials. In particular, the formation of crack micro-branches needs to be accounted for when it comes to the prediction of the propagation pattern of crack macro-branches which will ultimately lead to the failure of the material. This is achieved by numerically modeling the failure zones within the individual finite elements based on the concept of the embedded finite element method, where all the information with regard to the geometry of the failure zone is stored locally on the element level leading to a very efficient methodology capable of discretely resolving the failure zone. The main feature of the current work is the redundancy of the branching criterion based on crack tip velocity and that both, micro- as well as macro-branches can be modeled. Whether a micro-crack develops into a macro-crack solely depends on the local state of the material as it is outlined based on the application of the proposed numerical scheme on a rectangular block with a pre-existing notch set under tension. A comparison of the oscillatory behavior of the obtained crack tip velocity every time a micro-crack develops with experimental results from the literature is provided. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A strong discontinuity based adaptive refinement approach for the modeling of crack branching

In the current work, the physical phenomena of dynamic fracture of brittle materials involving crack growth, acceleration and consequent branching is simulated. The numerical modeling is based on the approach where the failure in the form of cracks or shear bands is modeled by a jump in the displacement field, the so called ‘strong discontinuity’. The finite element method is employed with this strong discontinuity approach where each finite element is capable of developing a strong discontinuity locally embedded into it. The focus in this work is on branching phenomena which is modeled by an adaptive refinement method by solving a new sub-boundary value problem represented by a finite element at the growing crack tip. The sub-boundary value problem is subjected to a certain kinematic constraint on the boundary in the form of a linear deformation constraint. An accurate resolution of the state of material at the branching crack tip is achieved which results in realistic dynamic fracture simulations. A comparison of resulting numerical simulations is provided with the experiment of dynamic fracture from the literature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A regularized damage-plasticity model: micromorphic formulation and numerical aspects

A gradient-extended damage-plasticity model is discussed which is based on a micromorphic approach according to Forest [1]. Damage and plasticity are treated as independent but strongly coupled dissipative phenomena by considering separate yield and damage loading functions to describe the onset of plastic flow and / or damage evolution. A numerical benchmark test conducted in the study reveals that the model is able to essentially cure the well-known mesh-dependence issue which is known from finite element simulations involving conventional (i. e. ‘local’) damage material models. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A regularization approach for damage models based on a displacement gradient

Common material models that take into account softening effects due to damage encounter the problem of ill-posed boundary value problems if no regularization is applied. This condition leads to a non-unique solution for the resulting algebraic system and a strong mesh dependence of the numerical results. A possible solution approach to prevent this problem is to apply regularization techniques that take into account the non-local behavior of the damage [1]. For this purpose a field function is used to couple the local damage parameter to a non-local level, in which differences between the local and non-local parameter as well as the gradient of the non-local parameter can be penalized. In contrast, we present a novel approach to regularization in which no field function is needed [2]. Hereto, the regularization is carried out by means of the divergence of the displacements and no additional quantity is necessary since the displacements are already defined on a non-local level. The idea is that with an increasing value of the damage the element's volume will increase as well. This is a result of the softening due to the occurring damage. The increasing volume can be measured by the divergence of the displacements which can be penalized by an additional energy part. The lack of any field function and the regularization by the use of the divergence of the displacements entails several numerical advantages: the computational effort is considerably reduced and the convergence behavior is improved as well. Naturally, the numerical results are mesh independent due to the regularization. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)