Bi-material interfaces are studied with cracks that end perpendicular to the interface. As is wel... more Bi-material interfaces are studied with cracks that end perpendicular to the interface. As is well-known, singularities in the stresses appear when classical elasticity is used. Moreover, the nature of the singularity depends on the difference in elastic constants of the two materials. In this paper, the gradient elasticity theory of Aifantis is used to remove these singularities. This is demonstrated for a range of ratios between the two Young's moduli.
The present paper is concerned with the use of the Modified Wöhler Curve Method to estimate fatig... more The present paper is concerned with the use of the Modified Wöhler Curve Method to estimate fatigue lifetime of thin welded joints of both steel and aluminium subjected to in-phase and out-of-phase multiaxial fatigue loading. The most important peculiarity of the above multiaxial fatigue criterion is that it can be applied by performing the stress analysis in terms of both nominal and local quantities, where in the latter case the relevant stress state at the assumed critical locations can be estimated according to either the reference radius concept or the Theory of Critical Distances. The accuracy and reliability of our multiaxial fatigue criterion was systematically checked through several experimental results taken from the literature and generated by testing, under in-phase and out-of-phase biaxial loading, welded joints of both steel and aluminium having thickness of the main tube lower than 5 mm.
The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gr... more The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourthgradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the secondgradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics.
Gradient elasticity theories are a powerful tool to describe the state of stress and strain aroun... more Gradient elasticity theories are a powerful tool to describe the state of stress and strain around sharp crack tips. With the appropriate format of field equations and boundary conditions, gradient elasticity can be used to predict non-singular stresses and strains, which can aid in simplifying engineering interpretation and the formulation of propagation criteria. The additional terms in the continuum equations are accompanied by internal length scales that represent the microstructure, and these internal length scales may be used to interpret fracture process zones or critical distances in fatigue theory. One of the difficulties of gradient elasticity theories, and the main reason why they have not been disseminated widely in the engineering communities, is that their finite element implementation is not straightforward. However, much progress has been made in recent years, and in this paper a few relatively simple implementations will be shown.
Computer Methods in Applied Mechanics and Engineering, Apr 18, 2010
The implementation of an h-adaptive Element-Free Galerkin (EFG) method in the framework of limit ... more The implementation of an h-adaptive Element-Free Galerkin (EFG) method in the framework of limit analysis is described. The naturally conforming property of meshfree approximations (with no nodal connectivity required) facilitates the implementation of h-adaptivity. Nodes may be moved, discarded or introduced without the need for complex manipulation of the data structures involved. With the use of the Taylor expansion technique, the error in the computed displacement field and its derivatives can be estimated throughout the problem domain with high accuracy. A stabilized conforming nodal integration scheme is extended to error estimators and results in an efficient and truly meshfree adaptive method. To demonstrate its effectiveness the procedure is then applied to plates with various boundary conditions.
An adaptive scheme is proposed in which the domain is split into two subdomains. One subdomain co... more An adaptive scheme is proposed in which the domain is split into two subdomains. One subdomain consists of regions where the discretization is reÿned with an h-adaptive approach, whereas in the other subdomain node relocation or r-adaptivity is used. Through this subdivision the advantageous properties of both remeshing strategies (accuracy and low computer costs, respectively) can be exploited in greater depth. The subdivision of the domain is based on the formulation of a desired element size, which renders the approach suitable for coupling with various error assessment tools. Two-dimensional linear examples where the analytical solution is known illustrate the approach. It is shown that the combined rh-adaptive approach is superior to its components r-and h-adaptivity, in that higher accuracies can be obtained compared to a purely r-adaptive approach, while the computational costs are lower than that of a purely h-adaptive approach. As such, a more exible formulation of adaptive strategies is given, in which the relative importance of attaining a pre-set accuracy and speeding-up the computational process can be set by the user.
The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gr... more The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourthgradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the secondgradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics.
A gradient viscoplasticity model has been used to analyse stationary and propagative instabilitie... more A gradient viscoplasticity model has been used to analyse stationary and propagative instabilities. It is demonstrated that the use of viscous regularisation is effective for both quasistatic and dynamic problems. Due to the influence of the length scales that are introduced in the model, the numerical simulation gives mesh-objective results with a finite width and unique orientation of the shear band. The numerical simulation of shear banding and propagative Portevin-Le Chatelier bands will be discussed. A 3D analysis of shear banding is shown to give significantly differentresults than the 2D plane strain analysis under similar conditions. Very fine meshes are needed to obtain accurate solutions for the shear band. The Arbitrary Lagrangian Eulerian remeshing method will be used to relocate elements from outside to inside the shear band to minimise computer costs.
According to Gradient Mechanics (GM), stress fields have to be determined by directly incorporati... more According to Gradient Mechanics (GM), stress fields have to be determined by directly incorporating into the stress analysis a length scale which that takes into account the material microstructural features. This peculiar modus operandi results in stress fields in the vicinity of sharp cracks which are no longer singular, even though the assessed material is assumed to obey a linear-elastic constitutive law. Given both the geometry of the cracked component being assessed and the value of the material length scale, the magnitude of the corresponding gradient enriched linear-elastic crack tip stress is then finite and it can be calculated by taking full advantage of those computational methods specifically devised to numerically implement gradient elasticity. In the present investigation, it is first shown that GM's length scale can directly be estimated from the material ultimate tensile strength and the plane strain fracture toughness through the critical distance value calculated according to the Theory of Critical Distances. Next, by post-processing a large number of experimental results taken from the literature and generated by testing cracked ceramics, it is shown that gradient enriched linearelastic crack tip stresses can successfully be used to model the transition from the short-to the long-crack regime under Mode I static loading.
The dynamic variant of Eshelby's inclusion problem plays a crucial role in many areas of mechanic... more The dynamic variant of Eshelby's inclusion problem plays a crucial role in many areas of mechanics and theoretical physics. Because of its mathematical complexity, dynamic variants of the inclusion problems so far are only little touched. In this paper we derive solutions for dynamic variants of the Eshelby inclusion problem for arbitrary scalar source densities of the eigenstrain. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method which covers both the time and frequency domain is especially useful for dynamically transforming inclusions of any shape.
Most materials exhibit rate-dependent inelastic behaviou r. Increasing strain-rate usually increa... more Most materials exhibit rate-dependent inelastic behaviou r. Increasing strain-rate usually increases the yield stress thus enlarging the elastic range. However, the ductility is gradually lost, and for some materials there exists a rather sharp transition strai n-rate after which the material behaviour is completely brittle. In this paper, a dispersion analysis of a simple phenomenological constitutive model for ductile-to- brittle
International Journal for Numerical Methods in Biomedical Engineering, 2008
Static and dynamic performance tests of a four-noded quadrilateral isoparametric membrane finite ... more Static and dynamic performance tests of a four-noded quadrilateral isoparametric membrane finite element enriched with rotational degrees of freedom (DOF) are presented. The formulation is based on displacement superposition whereby the in-plane displacements are decomposed additively into a regular, low-order contribution (governed by the nodal in-plane displacements) and a higher-order contribution (governed by the nodal in-plane rotations). The sensitivity of the element to element distortion and element aspect ratio is investigated. Comparisons with standard bilinear finite elements are made for static and dynamic loading case. It is demonstrated that the studied element exhibits an accuracy per DOF that is superior to the standard bilinear finite element.
International Journal for Numerical Methods in Engineering, 2014
In this paper, the effects of element shape on the critical time step are investigated. The commo... more In this paper, the effects of element shape on the critical time step are investigated. The common ruleof-thumb, used in practice, is that the critical time step is set by the shortest distance within an element divided by the dilatational (compressive) wave speed, with a modest safety factor. For regularly shaped elements, many analytical solutions for the critical time step are available, but this paper focusses on distorted element shapes. The main purpose is to verify whether element distortion adversely affects the critical time step or not. Two types of element distortion will be considered, namely aspect ratio distortion and angular distortion, and two particular elements will be studied: four-noded bilinear quadrilaterals (with various spatial integration schemes) and three-noded linear triangles. The maximum eigenfrequencies of the distorted elements are determined and compared to those of the corresponding undistorted elements. The critical time steps obtained from single element calculations are also compared to those from calculations based on finite element patches with multiple elements.
International Journal of Solids and Structures, 2011
In this paper, we discuss various formats of gradient elasticity and their performance in static ... more In this paper, we discuss various formats of gradient elasticity and their performance in static and dynamic applications. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order spatial derivatives of strains, stresses and/or accelerations. We focus on the versatile class of gradient elasticity theories whereby the higher-order terms are the Laplacian of the corresponding lower-order terms. One of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum. We start with discussing the general Mindlin theory, that in its most general form has 903 constitutive elastic parameters but which were reduced by Mindlin to three independent material length scales. Further simplifications are often possible. In particular, the Aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures. We also address how this can be extended to dynamics. An overview of length scale identification and quantification procedures is given. Finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions. Details are provided for particular formats of gradient elasticity that can be implemented with simple, linear finite element shape functions. New numerical results show the removal of singularities in statics and dynamics, as well as the size-dependent mechanical response predicted by gradient elasticity.
The dynamic variant of Eshelby's inclusion problem plays a crucial role in many areas of mechanic... more The dynamic variant of Eshelby's inclusion problem plays a crucial role in many areas of mechanics and theoretical physics. Because of its mathematical complexity, dynamic variants of the inclusion problems so far are only little touched. In this paper we derive solutions for dynamic variants of the Eshelby inclusion problem for arbitrary scalar source densities of the eigenstrain. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method which covers both the time and frequency domain is especially useful for dynamically transforming inclusions of any shape.
Bi-material interfaces are studied with cracks that end perpendicular to the interface. As is wel... more Bi-material interfaces are studied with cracks that end perpendicular to the interface. As is well-known, singularities in the stresses appear when classical elasticity is used. Moreover, the nature of the singularity depends on the difference in elastic constants of the two materials. In this paper, the gradient elasticity theory of Aifantis is used to remove these singularities. This is demonstrated for a range of ratios between the two Young's moduli.
The present paper is concerned with the use of the Modified Wöhler Curve Method to estimate fatig... more The present paper is concerned with the use of the Modified Wöhler Curve Method to estimate fatigue lifetime of thin welded joints of both steel and aluminium subjected to in-phase and out-of-phase multiaxial fatigue loading. The most important peculiarity of the above multiaxial fatigue criterion is that it can be applied by performing the stress analysis in terms of both nominal and local quantities, where in the latter case the relevant stress state at the assumed critical locations can be estimated according to either the reference radius concept or the Theory of Critical Distances. The accuracy and reliability of our multiaxial fatigue criterion was systematically checked through several experimental results taken from the literature and generated by testing, under in-phase and out-of-phase biaxial loading, welded joints of both steel and aluminium having thickness of the main tube lower than 5 mm.
The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gr... more The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourthgradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the secondgradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics.
Gradient elasticity theories are a powerful tool to describe the state of stress and strain aroun... more Gradient elasticity theories are a powerful tool to describe the state of stress and strain around sharp crack tips. With the appropriate format of field equations and boundary conditions, gradient elasticity can be used to predict non-singular stresses and strains, which can aid in simplifying engineering interpretation and the formulation of propagation criteria. The additional terms in the continuum equations are accompanied by internal length scales that represent the microstructure, and these internal length scales may be used to interpret fracture process zones or critical distances in fatigue theory. One of the difficulties of gradient elasticity theories, and the main reason why they have not been disseminated widely in the engineering communities, is that their finite element implementation is not straightforward. However, much progress has been made in recent years, and in this paper a few relatively simple implementations will be shown.
Computer Methods in Applied Mechanics and Engineering, Apr 18, 2010
The implementation of an h-adaptive Element-Free Galerkin (EFG) method in the framework of limit ... more The implementation of an h-adaptive Element-Free Galerkin (EFG) method in the framework of limit analysis is described. The naturally conforming property of meshfree approximations (with no nodal connectivity required) facilitates the implementation of h-adaptivity. Nodes may be moved, discarded or introduced without the need for complex manipulation of the data structures involved. With the use of the Taylor expansion technique, the error in the computed displacement field and its derivatives can be estimated throughout the problem domain with high accuracy. A stabilized conforming nodal integration scheme is extended to error estimators and results in an efficient and truly meshfree adaptive method. To demonstrate its effectiveness the procedure is then applied to plates with various boundary conditions.
An adaptive scheme is proposed in which the domain is split into two subdomains. One subdomain co... more An adaptive scheme is proposed in which the domain is split into two subdomains. One subdomain consists of regions where the discretization is reÿned with an h-adaptive approach, whereas in the other subdomain node relocation or r-adaptivity is used. Through this subdivision the advantageous properties of both remeshing strategies (accuracy and low computer costs, respectively) can be exploited in greater depth. The subdivision of the domain is based on the formulation of a desired element size, which renders the approach suitable for coupling with various error assessment tools. Two-dimensional linear examples where the analytical solution is known illustrate the approach. It is shown that the combined rh-adaptive approach is superior to its components r-and h-adaptivity, in that higher accuracies can be obtained compared to a purely r-adaptive approach, while the computational costs are lower than that of a purely h-adaptive approach. As such, a more exible formulation of adaptive strategies is given, in which the relative importance of attaining a pre-set accuracy and speeding-up the computational process can be set by the user.
The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gr... more The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourthgradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the secondgradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics.
A gradient viscoplasticity model has been used to analyse stationary and propagative instabilitie... more A gradient viscoplasticity model has been used to analyse stationary and propagative instabilities. It is demonstrated that the use of viscous regularisation is effective for both quasistatic and dynamic problems. Due to the influence of the length scales that are introduced in the model, the numerical simulation gives mesh-objective results with a finite width and unique orientation of the shear band. The numerical simulation of shear banding and propagative Portevin-Le Chatelier bands will be discussed. A 3D analysis of shear banding is shown to give significantly differentresults than the 2D plane strain analysis under similar conditions. Very fine meshes are needed to obtain accurate solutions for the shear band. The Arbitrary Lagrangian Eulerian remeshing method will be used to relocate elements from outside to inside the shear band to minimise computer costs.
According to Gradient Mechanics (GM), stress fields have to be determined by directly incorporati... more According to Gradient Mechanics (GM), stress fields have to be determined by directly incorporating into the stress analysis a length scale which that takes into account the material microstructural features. This peculiar modus operandi results in stress fields in the vicinity of sharp cracks which are no longer singular, even though the assessed material is assumed to obey a linear-elastic constitutive law. Given both the geometry of the cracked component being assessed and the value of the material length scale, the magnitude of the corresponding gradient enriched linear-elastic crack tip stress is then finite and it can be calculated by taking full advantage of those computational methods specifically devised to numerically implement gradient elasticity. In the present investigation, it is first shown that GM's length scale can directly be estimated from the material ultimate tensile strength and the plane strain fracture toughness through the critical distance value calculated according to the Theory of Critical Distances. Next, by post-processing a large number of experimental results taken from the literature and generated by testing cracked ceramics, it is shown that gradient enriched linearelastic crack tip stresses can successfully be used to model the transition from the short-to the long-crack regime under Mode I static loading.
The dynamic variant of Eshelby's inclusion problem plays a crucial role in many areas of mechanic... more The dynamic variant of Eshelby's inclusion problem plays a crucial role in many areas of mechanics and theoretical physics. Because of its mathematical complexity, dynamic variants of the inclusion problems so far are only little touched. In this paper we derive solutions for dynamic variants of the Eshelby inclusion problem for arbitrary scalar source densities of the eigenstrain. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method which covers both the time and frequency domain is especially useful for dynamically transforming inclusions of any shape.
Most materials exhibit rate-dependent inelastic behaviou r. Increasing strain-rate usually increa... more Most materials exhibit rate-dependent inelastic behaviou r. Increasing strain-rate usually increases the yield stress thus enlarging the elastic range. However, the ductility is gradually lost, and for some materials there exists a rather sharp transition strai n-rate after which the material behaviour is completely brittle. In this paper, a dispersion analysis of a simple phenomenological constitutive model for ductile-to- brittle
International Journal for Numerical Methods in Biomedical Engineering, 2008
Static and dynamic performance tests of a four-noded quadrilateral isoparametric membrane finite ... more Static and dynamic performance tests of a four-noded quadrilateral isoparametric membrane finite element enriched with rotational degrees of freedom (DOF) are presented. The formulation is based on displacement superposition whereby the in-plane displacements are decomposed additively into a regular, low-order contribution (governed by the nodal in-plane displacements) and a higher-order contribution (governed by the nodal in-plane rotations). The sensitivity of the element to element distortion and element aspect ratio is investigated. Comparisons with standard bilinear finite elements are made for static and dynamic loading case. It is demonstrated that the studied element exhibits an accuracy per DOF that is superior to the standard bilinear finite element.
International Journal for Numerical Methods in Engineering, 2014
In this paper, the effects of element shape on the critical time step are investigated. The commo... more In this paper, the effects of element shape on the critical time step are investigated. The common ruleof-thumb, used in practice, is that the critical time step is set by the shortest distance within an element divided by the dilatational (compressive) wave speed, with a modest safety factor. For regularly shaped elements, many analytical solutions for the critical time step are available, but this paper focusses on distorted element shapes. The main purpose is to verify whether element distortion adversely affects the critical time step or not. Two types of element distortion will be considered, namely aspect ratio distortion and angular distortion, and two particular elements will be studied: four-noded bilinear quadrilaterals (with various spatial integration schemes) and three-noded linear triangles. The maximum eigenfrequencies of the distorted elements are determined and compared to those of the corresponding undistorted elements. The critical time steps obtained from single element calculations are also compared to those from calculations based on finite element patches with multiple elements.
International Journal of Solids and Structures, 2011
In this paper, we discuss various formats of gradient elasticity and their performance in static ... more In this paper, we discuss various formats of gradient elasticity and their performance in static and dynamic applications. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order spatial derivatives of strains, stresses and/or accelerations. We focus on the versatile class of gradient elasticity theories whereby the higher-order terms are the Laplacian of the corresponding lower-order terms. One of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum. We start with discussing the general Mindlin theory, that in its most general form has 903 constitutive elastic parameters but which were reduced by Mindlin to three independent material length scales. Further simplifications are often possible. In particular, the Aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures. We also address how this can be extended to dynamics. An overview of length scale identification and quantification procedures is given. Finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions. Details are provided for particular formats of gradient elasticity that can be implemented with simple, linear finite element shape functions. New numerical results show the removal of singularities in statics and dynamics, as well as the size-dependent mechanical response predicted by gradient elasticity.
The dynamic variant of Eshelby's inclusion problem plays a crucial role in many areas of mechanic... more The dynamic variant of Eshelby's inclusion problem plays a crucial role in many areas of mechanics and theoretical physics. Because of its mathematical complexity, dynamic variants of the inclusion problems so far are only little touched. In this paper we derive solutions for dynamic variants of the Eshelby inclusion problem for arbitrary scalar source densities of the eigenstrain. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method which covers both the time and frequency domain is especially useful for dynamically transforming inclusions of any shape.
Uploads
Papers by Harm Askes