Crack-face Conditions Research Articles

Moving EMPS model for semipermeable two equal collinear interface cracks in dissimilar magneto-electro-elastic materials

The moving electric-magnetic polarization saturation (EMPS) model is studied for two equal collinear interface cracks in mode-III dissimilar magneto-electro-elastic (MEE) bimaterial under semipermeable crack face conditions. The closed-form solutions of all three possible cases of the proposed EMPS model are derived using the distributed dislocation technique (DDT) and singular integral method. An iterative scheme and explicit expressions of fracture parameters are used to implement the semipermeable crack-face conditions. The study of local energy release rate based on constant electric and magnetic breakdown strength-based saturated conditions concludes that increasing the mismatch of the MEE bimaterial can improve its fracture toughness.

Singular integral based closed-form solutions for modified EMPS models in semipermeable magneto-electro-elastic materials

The singular integral equation method is applied to present analytical solutions for modified electric-magnetic-polarization saturation (EMPS) models subjected to semipermeable center cracked magneto-electro-elastic (MEE) materials. A generalized methodology is presented to explicitly solve the EMPS model under any arbitrary saturated electric and magnetic conditions using the distributed dislocation method (DDM) and finding the analytical solutions for developed singular integral equations. The explicit expressions are derived for distributed dislocation densities, crack opening displacement (COD), crack opening potential (COP), crack opening induction (COI), saturated zone lengths, and local stress intensity factor (LSIF) for particular cases of the generalized EMPS model such as constant, linear, quadratic and cubic polynomial varying saturated electric and magnetic conditions. Using an iterative scheme and the explicit solutions of COD, COP, and COI, the exact semipermeable crack-face conditions are evaluated at the center of the crack, and the same applies throughout the crack-surface for the study of semipermeable modified EMPS models. A comparative analysis of the fracture parameters has also been carried out between the results of exact semipermeable crack-face conditions and the results of semipermeable crack-face conditions of the center crack problem to analyze the impact of polynomial varying saturated conditions on the crack-face conditions. Numerical studies performed under different electromagnetic-mechanical loadings, volume fractions, and crack-face conditions demonstrate the increasing effects in the fracture parameters' values with the degree of the polynomial varying saturated conditions.

Singular integral method based closed form solution for moving EMPS model in semipermeable two‐equal‐collinear cracked MEE material

AbstractThis paper presents the analytical closed‐form solutions of moving two equal collinear semipermeable cracks in magneto‐electro‐elastic material considering the strip electric‐magnetic polarization saturation (EMPS) model. To simulate the dynamic/moving cracks problem, two equal collinear Yoffe type cracks moving with a constant subsonic velocity along the plane of the cracks is considered. The problem is solved by considering semipermeable crack face conditions and under the in‐plane electro‐magnetic‐mechanical loading. Applying distributed dislocation technique and symmetry of the collinear cracks, the problem is reduced into simultaneous singular integral equations which have been solved in closed form expressions using finite Hilbert approach. The explicit expressions for dynamic distributed dislocation densities, crack opening displacement (COD), crack opening potential (COP), crack opening induction (COI), and local stress intensity factors (LSIFs) are evaluated for both the cases of the EMPS model (magnetic zone is smaller or greater than electric zone). For evaluation of inner and outer electric and magnetic zone lengths (MZLs), an iterative approach is applied to solve the developed two non‐linear equations. Also, to implement the semipermeable crack‐face conditions, a bisection iterative method has been applied. Based on the developed closed form solutions, numerical studies are presented for standard fracture parameters with respect to crack propagation velocity, electrical loading and inter‐crack space distance. This study shows that electric displacement and magnetic induction defined over the crack‐surface for semipermeable conditions decrease whereas the numerical values of saturated zone lengths, COD, COP, COI, and LSIFs increase with increasing crack propagation velocity. Moreover, for both static and dynamic cracks problems, the significant crack‐interaction effects have been observed on all the studied fracture parameters except the crack‐face conditions.

Cohesive zone phase field model for electromechanical fracture in multiphase piezoelectric composites

This paper introduces a coupled electromechanical finite deformation phase field model for crack propagation and interfacial decohesion in multiphase piezoelectric composites with interfaces. The crack phase field model is augmented with cohesive traction-separation laws at the material interfaces, derived from a cohesive potential function. A Gibbs free energy density function is proposed, allowing for the incorporation of the anisotropic elastic stiffness of the piezoelectric material. Numerical simulations exhibiting different failure mechanisms are carried out to demonstrate the efficacy of the model. Effects of external electric field on crack evolution and the competition between penetration and deflection of a crack impinging on an interface are investigated. Limited verification tests are conducted with theoretical results. Finally, the model is used to simulate fracture in nonuniform piezocomposite microstructures. The effect of crack propagation on the evolution of the electric field with different crack face conditions are analyzed. Differences in the electromechanical responses of piezocomposites due to different fiber distributions are also observed.

The Ogden and the extended tube model as backbone in describing electroactive polymers: advancements in modelling nonlinear behaviour and fracture.

Hyperelastic constitutive relations form the basis of advanced models for novel materials. Such elastic deformation potentials are the backbone for complex material formulations at elastic and inelastic deformations, especially when embedded into powerful frameworks like generalized standard materials, as well as multiphysical and multiscale formulations. With the focus on electroactive polymers, the article at hand demonstrates the derivation of a variational, rate-dependent electromechanical model for quasi-incompressible polymers and the derivation of an electromechanical model for regularized fracture mechanics by means of the phase-field method. Starting at the prominent Ogden and the extended tube model, some developments from the last decades are revisited and presented via the principle of virtual power, for instance, the established mixed element formulation, nonlinear viscoelasticity and electromechanical coupling. An electromechanically fully coupled representative crack element is used to derive a novel phase-field model for fracture. A key property of the proposed model is the ability to describe the electrical free-space behaviour inside the crack gap, which is demonstrated by adopting three common crack-face conditions. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Critical Comparison of Phase-Field, Peridynamics, and Crack Band Model M7 in Light of Gap Test and Classical Fracture Tests

Abstract The recently conceived gap test and its simulation revealed that the fracture energy Gf (or Kc, Jcr) of concrete, plastic-hardening metals, composites, and probably most materials can change by ±100%, depending on the crack-parallel stresses σxx, σzz, and their history. Therefore, one must consider not only a finite length but also a finite width of the fracture process zone, along with its tensorial damage behavior. The data from this test, along with ten other classical tests important for fracture problems (nine on concrete, one on sandstone), are optimally fitted to evaluate the performance of the state-of-art phase-field, peridynamic, and crack band models. Thanks to its realistic boundary and crack-face conditions as well as its tensorial nature, the crack band model, combined with the microplane damage constitutive law in its latest version M7, is found to fit all data well. On the contrary, the phase-field models perform poorly. Peridynamic models (both bond based and state based) perform even worse. The recent correction in the bond-associated deformation gradient helps to improve the predictions in some experiments, but not all. This confirms the previous strictly theoretical critique (JAM 2016), which showed that peridynamics of all kinds suffers from several conceptual faults: (1) It implies a lattice microstructure; (2) its particle–skipping interactions are a fiction; (4) it ignores shear-resisted particle rotations (which are what lends the lattice discrete particle model (LDPM) its superior performance); (3) its representation of the boundaries, especially the crack and fracture process zone faces, is physically unrealistic; and (5) it cannot reproduce the transitional size effect—a quintessential characteristic of quasibrittleness. The misleading practice of “verifying” a model with only one or two simple tests matchable by many different models, or showcasing an ad hoc improvement for one type of test while ignoring misfits of others, is pointed out. In closing, the ubiquity of crack-parallel stresses in practical problems of concrete, shale, fiber composites, plastic-hardening metals, and materials on submicrometer scale is emphasized.

Adaptive wavelet-enhanced cohesive zone phase-field FE model for crack evolution in piezoelectric composites

A novel finite deformation FE model is developed in this paper for coupled electro-mechanical response with crack evolution in heterogeneous piezoelectric composite microstructures. Fracture in these composites is driven by complex mechanical and electrical mechanisms leading to a set of strongly coupled, nonlinear governing equations. Crack propagation and interfacial decohesion in the presence of electro-mechanical fields are modeled by a unified phase-field model integrated with cohesive traction–separation laws, derived from a cohesive potential function at material interfaces. To incorporate different electrical crack face conditions, degradation functions are employed in terms of the crack phase-field order parameter. An adaptive wavelet-enhanced hierarchical framework is developed to improve computational efficiency of the coupled electromechanical-phase-field FE solver based on a phase-field error indicator. Numerical examples representing different failure mechanisms in heterogeneous piezoelectric materials and piezo-composite microstructures are solved to demonstrate the efficacy of the model.

Riemann–Hilbert approach-based analytical solutions for strip saturated two unequal collinear cracks in piezoelectric media

The objective of the work is to derive analytical solutions based on the Riemann–Hilbert (R–H) approach for semipermeable strip saturated two unequal collinear cracks in arbitrary polarized piezoelectric media. We particularly consider the influence of far field electromechanical loadings, poling direction and different crack-face boundary conditions. The problem is mathematically formulated into a set of non-homogeneous R–H problems in terms of complex potential functions (related to field components) using complex variable and extended Stroh formalism approach. After solving these equations, explicit solutions are obtained for the involved unknown complex potential functions and hence, the stress and electric displacement components at any point within the domain. Furthermore, after employing standard limiting conditions, explicit expressions for some conventional fracture parameters such as saturated zone lengths (in terms of nonlinear equations), local stress intensity factors and crack opening displacement are obtained. Numerical studies are presented for the PZT-4H material to analyze the effects of prescribed electromechanical loadings, inter-cracks distance, crack-face conditions and poling direction on the defined fracture parameters.

Investigation of Generalized SIFs of cracks in 3D piezoelectric media under various crack-face conditions

This paper investigates the influence of crack geometry, crack-face and loading conditions, and the permittivity of a medium inside the crack gap on intensity factors of planar and non-planar cracks in linear piezoelectric media. A weakly singular boundary integral equation method together with the near-front approximation is adopted to accurately determine the intensity factors. Obtained results indicate that the non-flat crack surface, the electric field, and the permittivity of a medium inside the crack gap play a crucial role on the behavior of intensity factors. The mode-I stress intensity factors (KI) for two representative non-planar cracks under different crack-face conditions are found significantly different and they possess both upper and lower bounds. In addition, KI for impermeable and semi-permeable non-planar cracks treated depends strongly on the electric field whereas those of impermeable, permeable, and semi-permeable penny-shaped cracks are identical and independent of the electric field. The stress/electric intensity factors predicted by permeable and energetically consistent models are, respectively, independent of and dependent on the electric field for the penny-shaped crack and the two representative non-planar cracks. Also, the permittivity of a medium inside the crack gap strongly affects the intensity factors for all crack configurations considered except for KI of the semi-permeable penny-shaped crack.

Accurate fracture analysis of electrically permeable/impermeable cracks in one-dimensional hexagonal piezoelectric quasicrystal junction

An accurate fracture analysis of a multi-material junction of one-dimensional hexagonal quasicrystals with piezoelectric effect is performed by using Hamiltonian mechanics incorporated in the finite element method. Two idealized electrical assumptions, including electrically permeable and impermeable crack-face conditions, are considered. In the Hamiltonian system, the analytical solutions to the multi-material piezoelectric quasicrystal around the crack tip (singular domain) are obtained and expressed in terms of symplectic eigensolutions. Therefore, the large number of nodal unknowns in the singular domain is reduced into a small set of undetermined coefficients of the symplectic series. The unknowns in the non-singular domain remain unchanged. Explicit expressions of phonon stresses, phason stresses, and electric displacement in the singular domain and newly defined fracture parameters are achieved simultaneously. Comparisons are presented to verify the proposed approach and very good agreement is reported. The key influencing parameters of the crack are discussed in detail. The effects of electrical assumptions and positions of the crack on the fracture parameters are discussed in detail.

The extended finite element method with novel crack-tip enrichment functions for dynamic fracture analysis of interfacial cracks in piezoelectric–piezomagnetic bi-layered structures

This paper investigates the dynamic fracture problems of interfacial cracks in piezoelectric–piezomagnetic (PE–PM) bi-layered composite structures under in-plane coupled electro-magneto-mechanical impact loadings by means of the extended finite element method (X-FEM). Considering the magnetoelectrically impermeable crack-face conditions and multi-filed coupled properties in the PE–PM composites, novel and more suitable crack-tip enrichment functions for interfacial cracks in PE–PM bi-layered composite structures are newly derived and implemented in the X-FEM, where the Newmark method is applied and proved to be effective. As the fracture parameter, the J-integral is evaluated using the domain-form of the path-independent contour integral. For dynamic analysis of interfacial cracks in infinite PE–PM bi-layered composite structures, absorbing layers based on the Sarma absorbing boundary conditions are adopted and applied to avoid the unphysical wave reflections at the artificially introduced boundaries in the X-FEM meshes. In the numerical examples, the validity of the proposed scheme is verified by comparing the numerical solutions provided by the X-FEM with either analytical results obtained by solving the corresponding singular integral equations or possible stationary values obtained by introducing the corresponding absorbing layers. Finally, by the numerical examples, the effects of the applied dynamic loadings, time variable and structural geometries on the dynamic J-integral are analyzed and discussed in detail. Some important conclusions are drawn, which should be helpful for the design and applications of the PE–PM layered composite structures.

Steady-state crack growth in polymer gels: A linear poroelastic analysis

Based on a linear poroelastic formulation, we present an asymptotic analysis of the crack tip fields for steady-state crack growth in polymer gels. A finite element method is then developed for numerical analysis. A semi-infinite crack in a long strip specimen subject to plane-strain loading is studied in detail. The crack-tip fields from the numerical analysis agree with the asymptotic analysis with a poroelastic stress intensity factor, which is generally smaller than the stress intensity factor predicted by linear elasticity due to poroelastic shielding. Similarly, the crack-tip energy release rate calculated by a modified J-integral method is smaller than the applied energy release rate. The size of the poroelastic crack-tip field (or K-field) is characterized by a diffusion length scale that is inversely proportional to the crack speed. For relatively fast crack growth, the diffusion length is small compared to the strip thickness (small-scale diffusion), and the poroelastic K-field transitions to an elastic K-field at a distance proportional to the diffusion length. In this case, the crack-tip energy release rate decreases with increasing crack speed under the same loading condition. For relatively slow crack growth, the diffusion length is greater than the strip thickness (large-scale diffusion), and the poroelastic crack-tip field is confined and transitions to a one-dimensional diffusion zone ahead of the crack tip. In this case, the energy release rate increases with increasing crack speed. Both immersed and not-immersed crack face conditions are considered. Under the same loading conditions, the poroelastic stress intensity factor and the modified J-integral are higher for the immersed case than not-immersed, but they approach the same values at the fast and slow crack limits. In general, if the crack-tip energy release rate is taken to be the intrinsic fracture toughness of the gel, the applied energy release rate as the apparent fracture toughness is greater due to energy dissipation associated with solvent diffusion, which is referred to as poroelastic toughening. It is proposed that the modified J-integral can be used to determine the intrinsic fracture toughness of the gel in experiments, which may or may not depend on crack speed. Moreover, the present results are found to be qualitatively consistent with previous experiments on the effects of solvent viscosity and crack-tip soaking.

Complex variable approach in studying modified polarization saturation model in two-dimensional semipermeable piezoelectric media

A modified polarization saturation model is proposed and addressed mathematically using a complex variable approach in two-dimensional (2D) semipermeable piezoelectric media. In this model, an existing polarization saturation (PS) model in 2D piezoelectric media is modified by considering a linearly varying saturated normal electric displacement load in place of a constant normal electric displacement load, applied on a saturated electric zone. A centre cracked infinite 2D piezoelectric domain subject to an arbitrary poling direction and in-plane electromechanical loadings is considered for the analytical and numerical studies. Here, the problem is mathematically modeled as a non-homogeneous Riemann-Hilbert problem in terms of unknown complex potential functions representing electric displacement and stress components. Having solved the Hilbert problem, the solutions to the saturated zone length, the crack opening displacement (COD), the crack opening potential (COP), and the local stress intensity factors (SIFs) are obtained in explicit forms. A numerical study is also presented for the proposed modified model, showing the effects of the saturation condition on the applied electrical loading, the saturation zone length, and the COP. The results of fracture parameters obtained from the proposed model are compared with the existing PS model subject to electrical loading, crack face conditions, and polarization angles.

Strip yield zone of a penny-shaped crack in a magnetoelectroelastic material under axisymmetric loadings

The strip yield zone ahead of a penny-shaped crack in a magnetoelectroelastic material, subjected to electric, magnetic and axisymmetric mechanical loadings, is evaluated analytically. Hankel transform is employed to reduce the mixed boundary value problem of the penny-shaped crack to dual integral equations, which are solved exactly under the assumption of electrically and magnetically permeable crack face conditions and the plastic strip yield zone crack model. An analytic solution to the mixed boundary problem has been obtained to predict the relationship between the length of the strip yield zone and the applied loadings. The distribution of mechanical, electric and magnetic fields in and outside of the strip yield zone in the cracked magnetoelectroelastic material has been derived analytically, and the crack opening displacement has been investigated. The effects of the mechanical, electric and magnetic loadings on the size of the yield zone are discussed in detail.

Dugdale plastic zone of a penny-shaped crack in a piezoelectric material under axisymmetric loading

The Dugdale plastic zone ahead of a penny-shaped crack in a piezoelectric material, subjected to electric and axisymmetric mechanical loadings, is evaluated analytically. Hankel transform is employed to reduce the mixed boundary-value problem of the penny-shaped crack to dual integral equations, which are solved exactly under the assumption of electrically permeable crack face conditions. A closed-form solution to the mixed boundary-value problem is obtained to predict the relationship between the length of the plastic zone and the applied loading. The stress distribution in and outside of the yield zone has been derived analytically, and the crack opening displacement has been investigated. The electric displacement has a constant value in the strip yield zone. The current Dugdale crack model leads to non-singular stress and electric fields near the crack front, and it is observed that the material properties affect the crack opening displacement.

A regularized time-domain BIEM for transient elastodynamic crack analysis in piezoelectric solids

A time domain non-hypersingular traction boundary integral equation method (BIEM) is proposed for dynamic crack analysis of piezoelectric solids. Using the boundary integral equation method, the time domain hypersingular integral equations for a dynamic crack in a 2D infinite piezoelectric solid subjected transient loads are derived. Considering the properties of the fundamental solutions, the hypersingular integral equations are reduced to singular integral equations by using the technique of integration by parts, in which the unknown functions are the tangential derivatives of the displacement and electrical potential discontinuities of the crack surfaces. To solve the time domain singular integral equations numerically, the quadrature formula of Lubich is applied for the temporal discretization, while the Gauss–Chebyshev quadrature method is used for the spatial discretization. Numerical examples are carried out to verify the accuracy of the present method by comparing the numerical results obtained by other scholars. Finally, several numerical results are presented and discussed to show the effects of the mechanical impact loading, crack-face conditions and piezoelectric coupling coefficient on the dynamic stress intensity factors.

Determination of stress intensity factors by the finite element discretized symplectic method

When rewriting the governing equations in Hamiltonian form, analytical solutions in the form of symplectic series can be obtained by the method of separation of variable satisfying the crack face conditions. In theory, there exists sufficient number of coefficients of the symplectic series to satisfy any outer boundary conditions. In practice, the matrix relating the coefficients to the outer boundary conditions is ill-conditioned unless the boundary is very simple, e.g., circular. In this paper, a new two-level finite element method using the symplectic series as global functions while using the conventional finite element shape functions as local functions is developed. With the available classical finite elements and symplectic series, the main unknowns are no longer the nodal displacements but are the coefficients of the symplectic series. Since the first few coefficients are the stress intensity factors, post-processing is not required. A number of numerical examples as well as convergence studies are given.

A moving crack propagating in a functionally graded magnetoelectroelastic strip under different crack face conditions

An anti-plane moving crack problem in a functionally graded magnetoelectroelastic strip under magneto-electrically impermeable or permeable conditions is investigated. Fourier cosine transforms are applied to reduce the mixed-boundary-value problem to dual integral equations and then Fredholm integral equations of the second kind. The equations can be solved numerically and field intensity factors are determined. The effects of the geometric size of the composite, the functionally graded parameter and the crack moving velocity on the field intensity factors are analyzed. Also the crack kinking phenomena under impermeable and permeable conditions are discussed. The results show that the coupled magneto-electrical field has great effects on the stress intensity factor and the probable kinking direction will be affected by the moving velocity, the thickness of the strip as well as crack surface magneto-electrical boundary condition remarkably.

Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions

We present a family of phase-field models for fracture in piezoelectric and ferroelectric materials. These models couple a variational formulation of brittle fracture with, respectively, (1) the linear theory of piezoelectricity, and (2) a Ginzburg–Landau model of the ferroelectric microstructure to address the full complexity of the fracture phenomenon in these materials. In these models, both the cracks and the ferroelectric domain walls are represented in a diffuse way by phase-fields. The main challenge addressed here is encoding various electromechanical crack models (introduced as crack-face boundary conditions in sharp models) into the phase-field framework. The proposed models are verified through comparisons with the corresponding sharp-crack models. We also perform two dimensional finite element simulations to demonstrate the effect of the different crack-face conditions, the electromechanical loading and the media filling the crack gap on the crack propagation and the microstructure evolution. Salient features of the results are compared with experiments.

Dynamic crack propagation in a magneto-electro-elastic solid subjected to mixed loads: Transient Mode-III problem

The transient response of a Mode-III crack propagating in a magneto-electro-elastic solid subjected to mixed loads is investigated through solving the corresponding boundary-initial-value problem in both the cracked solid region and the interior fluid region with treatment of electro-magnetically permeable and impermeable crack face conditions in a unified way. The closed-form results for the dynamic field intensity factors are used to evaluate the dynamic energy release rate through the crack-tip dynamic contour integral. The permeability of the interior fluid region relative to the cracked solid region significantly affects the magneto-electro-mechanical coupling coefficient in the Bleustein–Gulyaev wave function and, consequently, the horizontal shear surface wave speed, the dynamic field intensity factors and the dynamic energy release rate. It is revealed from dynamic fracture mechanics analysis that the dynamic energy release rate thus obtained has an odd dependence on the dynamic electric displacement intensity factor and the dynamic magnetic induction intensity factor. It is also found that the horizontal shear surface wave speed provides the limiting velocity for the propagation of a Mode-III crack in a magneto-electro-elastic solid when there is only applied traction loading.