Concept Of Configurational Forces Research Articles

Invariant integrals of crack interaction with an inhomogeneity

The crack growth can be shielded or anti-shielded by a specific inhomogeneity occurring in materials. This can be estimated from the change of the crack tip invariant Jk-, M-, L-integrals due to the presence of inhomogeneity. The closed-form solutions of the change of invariant integrals are derived for a crack interacting with the near crack-tip inhomogeneity by using the concept of material configurational forces. Two simple problems such as a phase transformation or an inhomogeneous inclusion interacting with a main crack in two-dimensional solids are solved in explicit form. The present formulations and methods are validated by the alternative weight functions or finite element analysis. It is concluded that the total components of configurational forces given by integration over the inhomogeneity can be used to evaluating the shielding or anti-shielding effect of various inhomogeneities e.g., inclusion, plasticity, dislocation, phase transformation, residual strain, on the crack growth in fracture mechanics.

Thermal shock resistance of magnesia spinel refractories—Investigation with the concept of configurational forces

Thermal shock resistance of magnesia spinel refractories—Investigation with the concept of configurational forces

Thermal shock resistance of magnesia spinel refractories—Investigation with the concept of configurational forces

In magnesia-spinel refractories, spinel inclusions are embedded in a magnesia matrix in order to increase the thermal shock resistance by micro-crack toughening. The paper investigates damage evolution in the material during the cooling at the end of the production process by a sophisticated numerical analysis. After a stress-strain analysis on a large representative volume element, the development of the damage zone between spinel grains is simulated with the concrete damaged plasticity model of ABAQUS. Application of the concept of configurational forces allows the physically correct determination of the driving force for cracks in the elastic quasi-plastic matrix material. The results demonstrate that inherent matrix cracks are unable to grow during the cooling, but they strongly influence the initiation and development of the damage zone. The pronounced damage initiation, in combination with the low crack driving force, is the main reason for the good thermal shock resistance of magnesia spinel refractories.

A discussion of fracture mechanisms in heterogeneous materials by means of configurational forces in a phase field fracture model

Composite materials are designed in order to combine the favorable properties of their components. For technical applications of these materials the determination of their effective material and fracture mechanical properties is of great interest. The study of fracture of heterogeneous composite materials, which is a necessary first step in order to derive effective fracture mechanical properties, is a difficult topic since macroscopic failure is usually accompanied by various fracture events on the microstructural scale. Finite element implementations of phase field fracture models enable the simulation of complex fracture scenarios as they occur in fracturing of heterogeneous materials. An adaption of the configurational force concept from linear elastic fracture mechanics to the phase field model provides more insight into the mechanisms underlying the evolution of fracture. In this work this ansatz is used to study the interplay between local crack driving forces and the evolution of fracture. Furthermore, the effects of the microstructure on macroscopically measurable quantities like the far field J-integral are analyzed in representative examples comprising continuously modulated as well as layered structures with varying stiffness and varying cracking resistance.

Calculation of crack driving forces of surface cracks subjected to rolling/sliding contact

The paper deals with finite element computations of crack driving forces for a surface crack in a rail under cyclic elastic–plastic conditions during wheel/rail contact. A straight crack normal to the surface is investigated. Examinated are influences of wheel load, slip, residual stresses, crack lengths and mesh around the crack tip on the crack driving forces. The specific role of the plastic deformation during one rolling cycle and the resulting development of the crack driving force is discussed. The numerical verification of the configurational force concept is done by comparison with classical J-integral investigations using linear-elastic and elastic–plastic material properties.

Overload effect revisited − Investigation by use of configurational forces

The configurational force concept enables the derivation of the incremental plasticity J-integral Jep, which is, in contrast to the conventional J-integral, physically appropriate to characterize the crack driving force in cyclically loaded elastic–plastic materials with growing cracks. In this paper we apply Jep, combined with an analysis of the configurational force distribution, for the investigation of fatigue crack growth retardation after a single tensile overload. The motivation for this investigation is that the main reason for the overload effect, i.e. crack flank contact behind or residual stresses around the growing crack tip, is today still an open question in fatigue. Numerical case studies are performed for two-dimensional Compact Tension specimens with long cracks that grow under cyclic Mode I loading and plane strain conditions. Variables of the numerical case studies are the overload ratio and the load ratio during constant cyclic loading. The influence of crack flank contact is examined by a comparison of two different simulations: The first simulation assumes frictionless contact between the upper and lower crack flank; in the second, fictive case, it is assumed that crack flank overlap is possible. The results show that all features of the overload effect even occur, if crack flank contact is not possible. Finally, the ability of the effective stress intensity range ΔKeff to characterize the overload effect is also discussed.

Improvement of fatigue life by compliant and soft interlayers

Retardation of fatigue crack growth rate due to the introduction of thin, compliant and/or soft interlayers is investigated. The mechanism is the reduction of the crack driving force in the interlayer. Fatigue tests are conducted on composites made of high-strength aluminum alloy as matrix and technically pure aluminum or adhesive as interlayer material. The adhesive interlayer causes an increase in fatigue life by a factor 20 or more, whereas the aluminum interlayer yields only a moderate improvement. Numerical simulations based on the configurational force concept are utilized for understanding. The results show new possibilities for the design of fatigue-resistant materials.

Physically appropriate characterization of fatigue crack propagation rate in elastic–plastic materials using the $$J$$ J -integral concept

The current paper discusses the physically correct evaluation of the driving force for fatigue crack propagation in elastic–plastic materials using the $$J$$ -integral concept. This is important for low-cycle fatigue and for short fatigue cracks, where the conventional stress intensity range ( $$\Delta K$$ ) concept cannot be applied. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008) , have derived the $$J$$ -integral for elastic–plastic materials with incremental theory of plasticity, $$J^{\mathrm{ep}}$$ , which is applicable for cyclic loading and/or for growing cracks, in contrast to the conventional $$J$$ -integral. The variation of this incremental plasticity $$J$$ -integral $$J^{\mathrm{ep}}$$ is studied in numerical investigations conducted on two-dimensional C(T)-specimens with long cracks under cyclic Mode I loading. The crack propagates by an increment after each load cycle. The maximum load is varied so that small- and large-scale yielding conditions prevail. Three different load ratios are considered, from pure tension to tension-compression loading. By theoretical considerations and comparisons with the variation of the crack tip opening displacement $$\delta _{\mathrm{t}}$$ , it is demonstrated that the cyclic, incremental plasticity $$J$$ -integral $$\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$$ , which is computed for a contour around the active plastic zone of the growing crack, is physically appropriate to characterize the growth rate of fatigue cracks. The validity of the experimental cyclic $$J$$ -integral, $$\Delta J^{\mathrm{exp}}$$ , proposed by Dowling and Begley (ASTM STP 590:82–103, 1976), is also investigated. The results show that $$\Delta J^{\mathrm{exp}}$$ is correct for the first load cycle, however, not fully appropriate for a growing fatigue crack.

Configurational Forces in Cutting Processes of Microstructured Titanium

AbstractThe micro cutting process of microstructured material is influenced by the heterogeneity of the microstructure. In case of cp‐titanium an hcp crystal structure is present. Therefore the material response is described with elastic anisotropy and crystal plasticity with the prismatic and basal slip systems of cp‐titanium. Also self and latent hardening are considered. The rate dependency is taken into account by a visco‐plastic evolution law. In order to investigate a micro cutting process the concept of configurational forces for standard dissipative media is used. Within the finite element method an example illustrates the effects of the heterogeneity and the grain boundary on the configurational force. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On problems with the determination of the fracture resistance for materials with spatial variations of the Young’s modulus

The paper considers the near-tip J-integral \(\text {J}_{\mathrm{tip}}\), the far-field J-integral \(\text {J}_{\mathrm{far}} \), and the experimental J-integral \(\text {J}^{\mathrm{exp}}\) in a material with sinusoidal variation of the Young’s modulus \(E\). The evaluations of \(\text {J}_{\mathrm{tip}} \) and \(\text {J}_{\mathrm{far}} \) are based on the concept of configurational forces; \(\text {J}^{\mathrm{exp}}\) is evaluated from the area below the load point displacement curve, as prescribed by the standard testing procedures. Analytic expressions and/or approximation formulae are derived, how \(\text {J}_{\mathrm{tip}} ,\,\text {J}_{\mathrm{far}} \) and \(\text {J}^{\mathrm{exp}}\) depend on the system parameters of the material, i.e. wavelength and amplitude of the \(E\)-variation and its phase shift with respect to the crack tip position, and the global specimen dimensions. The analyses show that \(\text {J}_{\mathrm{tip}} \) and \(\text {J}_{\mathrm{far}} \) exhibit a strong dependency on the phase shift, but not the experimental J-integral \(\text {J}^{\mathrm{exp}}\). This is the reason why the current procedures for fracture mechanics testing are not suitable to determine the true values of the fracture initiation toughness \(\text {J}_\mathrm{i} \) or the crack growth resistance \(R\) of a material, if the material properties exhibit a spatial variation in the direction of crack propagation. Relations are given to estimate the possible errors.

On numerical solution of elastic–plastic problems by using configurational force driven adaptive methods

In this paper, the concept of configurational forces is introduced in the context of finite element mesh refinement for elastic–ideally plastic problems. This paper also includes the numerical computation of configurational forces in the elastic and plastic domains. Methods are demonstrated on three plane problems where the analytical solution is available. The first example is a thick-walled tube loaded by internal pressure. This simple, one dimensional problem allows computation of configurational volume forces analytically to validate the finite element (FE) results. The second example is Galin׳s problem that involves an infinite plate with a circular hole loaded by biaxial tension at the infinity. This is a two dimensional problem for which the analytical solution is known with some restrictions for elastic–ideally plastic case when Tresca yield criterion is considered. The last example introduces another plane problem that follows Naghdi׳s solution on infinite wedges. For this, a new analytical solution is presented for plane stress state using von Mises yield criterion with a uniform shear loading along the boundary. R-, h-, and combined adaptive procedures are demonstrated on the above examples. Since exact stress fields are known, the norm of the difference between numerical and analytical solutions is used as the measure of error.

An easy-to-use estimate of the energy-release rate for crack arrays

Formation of crack arrays plays an increasing role in several fields of applied physics. The energy-release rate of the cracks controls the development of the array. Therefore, following the concept of configurational forces, a simplified analytical expression is provided for the energy-release rate, which is based both on numerical studies and on a specially adapted beam model. Comparisons of this easy-to-use estimate of the energy-release rate with established results from the literature as well as detailed numerical results are presented. The provided estimate of the energy-release rate can easily be extended to non-equidistant cracks and an anisotropic material.

A new view on J-integrals in elastic–plastic materials

It is well known that the application of the conventional $$J$$ -integral is connected with severe restrictions when it is applied for elastic–plastic materials. The first restriction is that the $$J$$ -integral can be used only, if the conditions of proportional loading are fulfilled, e.g. no unloading processes should occur in the material. The second restriction is that, even if this condition is fulfilled, the $$J$$ -integral does not describe the crack driving force, but only the intensity of the crack tip field. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008), have derived a $$J$$ -integral, $$J^{\mathrm{ep}} $$ , which overcomes these restrictions: $$J^{\mathrm{ep}} $$ is able to quantify the crack driving force in elastic–plastic materials in accordance with incremental theory of plasticity and it can be applied also in cases of non-proportionality, e.g. for a growing crack. The current paper deals with the characteristic properties of this new $$J$$ -integral, $$J^{\mathrm{ep}}$$ , and works out the main differences to the conventional $$J$$ -integral. In order to do this, numerical studies are performed to calculate the distribution of the configurational forces in a cyclically loaded tensile specimen and in fracture mechanics specimens. For the latter case contained, uncontained, and general yielding conditions are considered. The path dependence of $$J^{\mathrm{ep}} $$ is determined for both a stationary and a growing crack. Much effort is spent in the investigation of the path dependence of $$J^{\mathrm{ep}} $$ very close to the crack tip. Several numerical parameters are varied in order to separate numerical and physical effects and to deduce the magnitudes of the crack driving force for stationary and growing cracks. Interpretation of the numerical results leads to a new, completed picture of the $$J$$ -integral in elastic–plastic materials where $$J^{\mathrm{ep}} $$ and the conventional $$J$$ -integral complement each other. This new view allows us also to shed new light on a long-term problem, which has been called the “paradox of elastic–plastic fracture mechanics”.

Configurational Forces in Crystal Plasticity

AbstractThe surface morphology of micro machined surfaces depends on the heterogeneous microstructure. A crystal plasticity model is used to describe the plastic deformation in cp‐titanium with its hcp crystal structure. Therefore the basal and prismatic slip systems are taken into account. Furthermore, self and latent hardening are considered. The rate dependency is motivated by a visco plastic evolution law. The cutting process of cp‐titanium is modeled within the concept of configurational forces for a standard dissipative media. This framework is implemented into the finite element method. An example illustrates the effects of the microstructure on plastic deformation and configurational forces. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Crack Growth Tendency of Surface Shear Cracks in Rolling Sliding Contact

Surface cracks arising during rolling sliding contact of a wheel and a rail are investigated. A two-dimensional crack model is proposed which calculates the crack driving force using the configurational force concept. The numerical applicability of the configurational force concept for surface shear cracks under cyclic contact loading is discussed and compared to the J-integral concept. A single inclined crack in a rail loaded by an accelerated wheel is investigated. The material of the rail is described by a cyclic plastic kinematic hardening model. The evolution of the crack driving force during several cycles is investigated.

Failure theory via the concept of material configurational forces associated with the M-integral

A new failure theory based on the material configuration forces associated with the invariant M-integral is proposed to describe the content and evolution of the multi-defects localized in the body. The physical interpretation of the global M-integral is as the sum of the local energy release rate due to the self-similar expansion for each specific defect. It does provide an effective measure for the evaluation of damage level. It is found that the unique parameter of the M-integral cannot be used as a unified failure criterion to predict the damage evolution and the final failure due to the major obstacle that the critical value of the M-integral is not a problem-invariant constant and shows an apparent defect configuration-dependence. Consequently, a new failure parameter referred as the configurational damage parameter (abbreviated as Π-parameter) is proposed by the appropriate formulation via the M-integral, the remote uni-axial load, and the inner variable of the damaged area. A series of numerical examples are carried out to demonstrate that the critical value of Π-parameter is a material constant regardless of defect configurations. Furthermore, it is performed to validate the applicability of the Π-parameter as a failure criterion to predict the final failure of the locally damaged materials. Finally, a protocol of experimental measurement of the Π-parameter is proposed by method of digital image correlation to facilitate the wide application of the new failure criterion. It is concluded that the present failure theory via the configurational forces associated with the M-integral provides some outside variable features and has the advantage of predicting the structural integrity of damaged materials containing the locally distributed defects.

Multiscale modelling and simulation of micro machining of titanium

AbstractThe microscale morphology of micro machined component surfaces is directly connected to the heterogeneous microstructure. The deformation depends on the crystal structure, in case of the considered cp‐titanium, the hcp crystal structure. In a first approach the crystal plastic deformation is modeled with isotropic hardening. A visco‐plastic evolution law accounts for the rate dependency. The concept of configurational forces is used with the framework of crystal plasticity to model the cutting process of cp‐titanium. The setting is implemented into the finite element method. The examples show the effect of the material heterogeneity on the deforamtion behavior and on the related configurational forces. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analysis of Calving Events in Antarctic Ice Shelves Using Configurational Forces

AbstractPrevious studies on the sensitivity of cracks in ice shelves with different boundary conditions, stress states and density profiles revealed the need for further analyses. As the transfer of boundary conditions from dynamic ice flow simulations to the linear elastic fracture analyses proved to be a critical point in previous studies, a new approach to relate viscous and elastic material behaviour is proposed. The numerical simulations are conducted using Finite Elements utilizing the concept of configurational forces. To show the applicability of the approach, a 2‐dimensional plane stress geometry with volume loads due to the ice shelf flow is analyzed. The resulting crack path is compared to available crack paths from satellite images. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modelling the role of surface stress on the kinetics of tissue growth in confined geometries

In a previous paper we presented a theoretical framework to describe tissue growth in confined geometries based on the work of Ambrosi and Guillou [Ambrosi D, Guillou A. Growth and dissipation in biological tissues. Cont Mech Thermodyn 2007;19:245–51]. A thermodynamically consistent eigenstrain rate for growth was derived using the concept of configurational forces and used to investigate growth in holes of cylindrical geometries. Tissue growing from concave surfaces can be described by a model based on this theory. However, an apparently asymmetric behaviour between growth from convex and concave surfaces has been observed experimentally, but is not predicted by this model. This contradiction is likely to be due to the presence of contractile tensile stresses produced by cells near the tissue surface. In this contribution we extend the model in order to couple tissue growth to the presence of a surface stress. This refined growth model is solved for two geometries, within a cylindrical hole and on the outer surface of a cylinder, thus demonstrating how surface stress may indeed inhibit growth on convex substrates.

On the dynamics of moving singularities in solids under the use of the level-set method and the configurational force concept

In this paper we examine the motion of a moving singularity within a continuum body under the use of the level-set method and the material force concept. The level set method introduces an implicit representation of a singular surface where it transforms into a thin transition layer of non-zero thickness where all quantities take inhomogeneous expressions within the body. The existence of an inhomogeneous energy of the material predicts inhomogeneity forces which drive the singularity. The driving force is in fact a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine–Hugoniot jump conditions combined with non-equilibrium jump relations at the singularity. The effectiveness of the method is illustrated by the use of two examples: (1) propagating phase-transition front that corresponds to a stress-induced martensitic transformation in an elastic bar and (2) propagation of a straight brittle crack.