- Email: [email protected]
Material configurational forces applied to mixed mode crack propagation
Reference:
TAFMEC 1810
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
3 November 2016 27 December 2016 14 February 2017
Please cite this article as: Y. Guo, Q. Li, Material configurational forces applied to mixed mode crack propagation, Theoretical and Applied Fracture Mechanics (2017), doi: http://dx.doi.org/10.1016/j.tafmec.2017.02.006
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Material configurational forces applied to mixed mode crack propagation Yuli Guo, Qun Li* State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China *Corresponding author. Tel.:+86-29-82667535 Email address: [email protected] (Q. Li) ABSTRACT. The concept of material configurational forces is applied to predict the mixed mode crack propagation. A new fracture criterion based on the resultant of configurational forces (termed as C-force criterion) is proposed. The basic assumption is that the onset of crack growth occurs when the resultant of configurational forces reaches a critical value and the crack growth takes place in the direction of resultant configurational forces. An implementation of the configurational forces into the finite element is presented. The newly proposed C-force criterion is further validated through a series of examples. It is concluded that the predictions of mixed-mode crack propagation by C-force criterion are in good agreement with experimental data in the open literatures. In addition, an experimental procedure on evaluation the configurational forces is proposed by using the digital image correlation. It is demonstrated that the C-force criterion provides a more convenient and accurate procedure to predict the mixed mode crack propagation. Keywords: material configurational forces, crack, criterion, digital image correlation 1. Introduction Fracture behaviors of materials have been widely investigated concerning structural integrity, reliability, and functionality. The evaluation of both onset of crack growth and the crack growth direction involved in fracturing of materials necessitates an effective fracture criterion to indicate the instance when the onset of crack growth occurs and where the crack propagates. The well-known criteria within the framework of fracture mechanics of crack instability include the energy release rate G [11], the 1
stress intensity factors K [17], the J-integral [7,29], the crack tip opening displacement CTOD [33] and so on. The fracture criterion is formulated as the crack propagation starts when the fracture parameters reach the material-specific critical values. Among them, the stress intensity factor is appropriate with small-scale yielding assumption. The path-independent J-integral can be used as an effective fracture parameter to predict the crack stability and growth in elastic-plastic materials although the limitation of path-dependence of the J-integral is revealed near a stationary crack in a fully yielding material [5]. The CTOD has advantage in three-dimensional analyses to study constraint effects, crack tunneling, and the fracture process. However, the aforementioned fracture criteria always assume that the crack propagates along the original crack surface to overcome the fracture resistance. Although they have enabled a successful application to predict the occurrence of onset of crack growth, i.e., the critical load or the critical crack size that a structure will tolerate for characterizing the fracture resistance of materials, it is inappropriate to solve the issue of predicting the crack deflection or kinking from the original crack direction under the mixed-mode loading condition. In addition, industrial application usually requires that materials are regularly exposed to the complicated load conditions. In general, the cracks generated inside the engineering components can be subjected the mode I (opening) and the mode II (in-plane sliding) in plane problems. Any combinations of these modes are known as mixed mode crack problems. A few of attempts appear to treat the mixed mode crack problems where the crack extension in a combined stress field can grow in any arbitrary direction with reference to its original crack surface. There are a plethora of fracture criteria that have been used in predict the mixed mode crack propagation, among which the good examples include the maximum tangential stress MTS [8], the strain-energy-density S-factor [30], the maximum energy release rate MERR [15] et al. The maximum tangential stress makes the hypothesis that the crack initiates in the direction along which the tangential stress σθ possesses a stationary (maximum) value while the onset of crack growth occurs when the tangential stress reaches a critical 2
value. The strain-energy-density factor represents the singularity strength of the strain energy density field W around the crack tip which can be defined by S=Wr. It assumes that the initial crack growth takes place in the direction along which the strain-energy-density factor S possesses a stationary (minimum) value while the onset of crack growth occurs when the strain-energy-density factor reaches a critical value SC. The maximum energy release rate is to determine the energy release rate as a function of crack propagation, and then determine the critical direction by maximizing the energy release rate. However, the application of these fracture criteria in the situation of mixed mode crack are restrained by several limitations. First, the calculation of tangential stress is required to be along a critical distance rc from the crack tip. Similarly, the strain-energy-density factor as the intensity of dW/dA for the interior element is valid in the small region where a core region with radius r0 surrounding the crack tip must be observed. The predicted results of crack propagation could be disturbed by the magnitude of the critical distance. The determination of critical distance or the characteristic length, or the core region is an important issue in predicting the crack growth by the criteria of MTS and S-factor. Many researchers have attempted in the past several decades to suggest the various models to calculate the critical distances for different brittle and quasi-brittle materials [1,2,10,14]. Second, the limitation of the maximum energy release rate is to require the true values of the mixed mode energy release rate obtained by solving a class of boundary value problems of a branched crack that the branch-crack ratio must tend to zero. The analytical difficulties are associated with solving the branched crack problem. This limiting process presents a formidable task by using the maximum energy release rate. For this reason, there is a need in the search of a realistic mixed mode fracture criterion being able to describing the crack propagation. Recently, it is shown that the material configurational forces [4,12,13,19,26,27,36], plays an important role in fracture and damage mechanics. The configurational force is able to provide the information from the macroscopic perspective, suggesting a method on predicting the crack propagation. Among many others, several pioneering investigations on the 3
material configurational forces in fracture mechanics are briefly listed here: Kienzler and Herrmann [20] proposed a fracture criterion based on local properties of the Eshelby tensor for a plane crack under mixed-mode loading conditions. Ma et al. [22] proposed the vector Jk-integral as a fracture criterion based on the criterion of maximum energy release rate for multi-singularity-crack interaction problems. Simha et al. [31] used the configurational forces approach to identify a ‘plasticity influence term’ that describes the crack tip shielding or anti-shielding due to plastic deformation in the body. Baxevanakis and Giannakopoulos [4] applied the configurational forces and conserved integrals into the finite element analysis of discrete edge dislocations under mixed loading. Ozenc [27] proposed a configurational force approach to model the branching phenomenon in dynamic brittle fracture. Notwithstanding the practices of the configurational forces in the aforementioned situations, the main purpose of this paper is to present a failure criterion based on the concept of configurational forces which are regarded as the crack driving forces. The basic assumption is that the onset of crack growth occurs when the resultant of configurational forces reaches a critical value and the crack growth takes place in the direction of resultant configurational force. This configurational force based fracture criterion is validated to successfully describing the crack propagation under mixed-mode loading conditions. 2. Fracture criterion by the material configurational forces 2.1 The concept of material configurational forces The concept of material configurational stress tensor (or called Eshelby tensor) was first introduced by Eshelby [9]. Considering the gradient of the Lagrangian energy density is the most straightforward approach to derive the material configurational
stress
tensor.
The
material
configurational
stress
bji
in
small-deformation problems is defined as
b ji = W δ ji − σ jk uk ,i
(1)
where W=W(xi ,ui,j) denotes the strain energy density; δji denotes the Kronecker delta;
σjk is the stress tensor; uk denotes the components of displacements; the subscript 4
prima {},i refers to the corresponding differentiation with respect to the coordinate xi; the repeated indices denote summation over the range of indices. The physical interpretation of the configurational stresses bij (i, j=1, 2) can be explained as the change of the total potential energy at a point of an elastic infinitesimal element due to a material unit translation in xi–direction of a unit surface with normal in xj –direction [20]. The material configurational force as the most significant concept is denoted as the explicit dependent of W on xi
∂W ci = − ∂xi expl.
(2)
where ( ∂W / ∂xi )expl. denotes the explicit dependence of W on xi; Here, the material configurational force is denoted as the explicit gradient of W on xi [9]. The material configurational forces ci can be explained as the energy release rates due to translation of one material point along xi-direction. The balance laws between the configurational stress in Eq. (1) and the configurational force in Eq. (2) can be established by
bji, j + ci = 0
(3)
2.2 FEM implementation of configurational forces
Numerical implementation of configurational forces in the material space will be carried out into finite element analysis. It should be pointed that the configurational forces Ci is not calculated by the direct definition in Eq. (2). Actually, the configurational force is calculated through the configurational stress bij by the equilibrium equation Eq. (3). In order to calculate the discrete configurational force at the nodes, a weak function is introduced in numerical implementation. And then the discrete configurational forces at every node (I) over each finite element of area Ω e can be given by [24]
5
Ci( I ) = ∫ N ( I ) ci dV = ∫ bij N,(jI ) dV . Ωe
(4)
Ωe
In derivative of Eq. (4), the balance law Eq. (3) of configurational stress and force is adopted. The index (I) denotes the node local number in the specific element. N(I) are the shape function. The term of bij N ,(jI ) can be easily obtained in every integration point after the standard quantities are solved, e.g. the strain energy density, the nodal displacements, the stresses, according to the definition of bij in Eq. (1). The integration of Eq. (4) can be performed by a standard Gaussian integration method. Finally, the total configurational forces Ci( K ) on each specific node K have to be assembled by the contributions of all elements adjacent to node K, nel
Ci( K ) = U Ci( I ) .
(5)
e =1
Moreover, the configurational forces at the crack tip Citip are paid more attentions as a crack driving force. It can be interpreted as the energy release rate due to the translation of crack advance along xk-direction, respectively. The configurational force at the crack tip can be deemed as the resultant of all configurational force vectors in elements over the specific domain Ω surrounding the crack tip in the numerical evaluation. It can be concluded that the configurational force at the crack tip is independence of the domain. Numerical evaluation of configurational force can be alternatively calculated based on the resultant of all the configurational force vectors in a user-defined domain Ω surrounding the crack tip as shown in Fig. 1. C1tip =
∑
C1
Ω− domain
C2tip =
∑
(6) C2
Ω− domain
It should be mentioned that the configurational forces originate from the inhomogeneity of the material. For linear elastic fracture mechanics, it is demonstrated that the selection of the user-defined domain near the crack tip has independent effect on the calculated results. For convenience of FEM computation, 6
the directions of e1 and e2 correspond to the local coordinate system at the crack tip are considered as depicted in Fig. 1. Here, e1 is the tangential component of the local coordinate system with respect to the crack surface, while e2 represents the normal component of the local coordinate system. The components of the configurational force vector will be calculated in the local coordinate system (e1, e2) in the following analysis. 2.3 The C-force fracture criterion
The mixed mode crack propagation will be deflected from its original path due to the non-symmetric material structure or mixed-mode loading with respect to the crack plane. Herein, an intriguing theory to predict the mixed-mode crack propagation can be proposed according to the physical interpretation of the crack tip configurational forces. The C1-force is identical to the crack extension forces and it has been given as the rate of total potential energy release per unit crack-tip advance along e1-direction. Analogously, the C2-force has been given a precise and clear physical significance as the rate of total energy release by postulating that the crack will skew the tip advances along e2-direction in the local coordinate system. Under these considerations, a new fracture criterion (termed as C-force criterion) is proposed where two basic stipulations on crack propagation will now be made in plane problems: i) The initial crack growth takes place in the direction ahead of the crack tip along the uv configurational resultant forces C . The initiation angle of crack can be determined by
α = arctan
C2 C1
(7)
where α is the crack-kinking angle as shown in Fig. 2. ii) The onset of crack growth occurs when the magnitude of configurational resultant forces |C| overcomes the material resistance. i.e.,
C = C12 + C22 ≥ CR
(8) 7
where CR is the material fracture resistance which is a material constant regardless of the crack configuration and loading conditions. The value of CR provides a knowledge of mixed mode crack extension in that it specifies the fracture toughness of the material. It is well known that for a pure mode I crack in homogenous material where a negligible value of C2-force prevails, the crack will advance along the crack direction. In contrast, the introduction of mixed-mode crack problem will result in the remarkable magnitude of C2–force near the crack tip. A mixed mode fracture can be characterized by the configurational forces and the crack deflection will happen. The present mode is advantage from those the traditional fracture criterion (e.g., the J-integral) where the J-integral may not see much difference among mode I and mode II crack. The present method is based on the two parameters C1 and C2 which can distinguish the mixed mode crack. The framework of C-force criterion given in this paper could provide a useful tool to deal with problems associated with the crack deflection mechanism under mixed-mode loading condition. 3. Validations of C-force criterion applied to mixed mode crack propagation
The C-force criterion will be used to predict the mixed mode crack propagation. The special treatment of C-force criterion to predicting the crack growth path and the critical load will be clarified and addressed by a series of representative crack examples. The crack propagation is simulated by using the finite element method for each propagation step, then the mesh near the crack tip is modified to take into account the new crack advance. In FEM calculations, the crack paths are calculated by an iterative approach, shifting the crack tip by an increment ∆a along the direction of material configurational resultant forces according to the C-force criterion. The re-meshing algorithm refines the mesh close to the crack tip. That is, fine mesh is carried out in the local region near the crack tip and the singularity elements are chosen near the crack tip while the mesh of outer region away from the crack tip is fixed. Moreover, the present study doesn’t consider the crack growth rate and then the static growth of crack is assumed in all cases. Five representative crack examples are carried out and addressed by the C-force 8
criterion. In the calculation, the mesh sensitivity and the result convergence are given and discussed on each example. The purpose of section 3.1-3.4 is to observe the crack trajectories and it is demonstrated that the critical value of fracture resistance doesn’t affect the crack propagation path. Therefore, the values of C ≥ CR is always prescribed to ensure the crack growth by introducing a negligible values of CR. Furthermore, the purpose of section 3.5 is to valid the accuracy of C-force criterion to prediction the critical load where the value of CR is calculated from the analytical formulation of configurational forces in terms of the stress intensity factor for an incline crack in elastic plate (see Appendix). The predicted results of crack propagation by C-force criterion will be validated by comparing with the experimental observations in the open literatures. 3.1 Single central cracked beam with a hole
The first example corresponds to a single cracked beam with a hole, loaded in upper two points and constrained in lower two points, i.e., the four-point-bending beam. As depicted in Fig. 3, the initial crack is in the center of the bottom edge and the void contained in beam has an offset from the vertical centerline. The original crack length a=2.5mm. The mesh insensitivity is validated by a coarse and a fine mesh, as shown in Fig. 4. The coarse mesh with 9,318 elements and fine mesh with 12,778 elements are considered in order to validate the mesh independence. It can be found from Fig. 4 that the calculated chart of crack trajectory is consistent for the coarse and fine mesh. Numerical results presented by C-force criterion are compared with experiment results by Miranda et al. [23], where they experimentally observed the crack propagation for such a four-point bending single-edge crack in specimen made of code rolled SAE 1020 steel. The material elastic moduli of steel E=205 GPa, v=0.3. Both numerical predictions and experimental results are shown in Fig. 5. It indicates that the crack is always attracted by the hole. The crack will curve its path and grow toward the hole until sinking in the hole. Fig. 5 shows that the C-force criterion provides a quite good agreement of crack propagation with experimental observations. 9
3.2 Cracked beam contained three holes
This example corresponds to a cracked beam subjected a load in the center of one edge and supported by two points as depicted in Fig. 6. The three holes have an offset from the centerline and arrange along the vertical line. The monotonic load P is applied at center of specimens. The experimental results for polymethylmethacrylate (PMMA) beams has been conducted by Ingraffea and Grigoriu [16] with considering two specimens of the initial crack length, b, and its location, a, as shown by the table in Fig. 6. The elastic material constants of PMMA are elastic moduli E=3GPa and Poisson’s ratio ν=0.3. Numerical simulations of the crack trajectory are performed for two specimens with a crack increment ∆a=0.05mm of each iterative step in FEM computations. The total degrees of freedom for specimen I and II are over 93, 000 and 77, 000, respectively. The element size around the crack tip are less than 0.2mm so as to ensure the mesh insensitivity and the result convergence. Comparison of crack growth trajectory between numerical results by C-force criterion and experimental results by Ingraffea and Grigoriu [16] are made to validate the accuracy of C-force criterion to predict the mixed mode crack problems. The cloud chart of crack trajectory after 65 iterative steps for specimen I and 74 steps for specimen II by FEM is shown in left figures of Fig. 7(a)-(b) compared with the experimental results in right figures of Fig. 7(a)-(b). It is found that under the influence of three holes, the crack trajectory is deviated from the track predicted without the holes. For specimen I, the crack growth trajectory passes towards the bottom hole and deflects to the opposite side when it is close to the bottom hole and finally ended at the middle hole as depicted in Fig. 7. For specimen II, the crack directly propagates toward the hole in the center for this problem. Although the size and location of initial crack lead to the very different crack trajectory, both specimens of numerical patterns capture the curved crack in good agreement with experimental results. It demonstrates the validation of the C-force criterion so that the fracture criterion based on configurational force is verified and feasible to describe the mixed mode crack growth. 3.3 Single edge cracked plate under shear stress 10
This example corresponds to a single edge cracked plate that is constrained to a far–field shear stress along the top edge of plate, as depicted in Fig. 8. The bottom of plate is constrained. The total degrees of freedom is about 87, 000 and the element size around the crack tip is 0.08mm. The cloud chart of crack trajectory after 17 iterative steps by the C-force criterion is shown in left figures of Fig. 9 compared with the results by David et al [37] in right figures of Fig. 9. Among them, David et al. [37] used a Lepp-Delaunay based mesh refinement algorithm for triangular meshes which allows both the generation of the initial mesh and the local modification of the current mesh as the crack propagates according to the K-factor criterion. It is shown from Fig. 9 that the predicted crack trajectory is downward sloping at first and tends to parallel to horizontal line gradually which resembles the results presented in the reference work by David et al. [37]. 3.4 Two interacting edge cracks
In this example, a plate specimen exhibiting two interacting edge cracks with incipient crack lengths are considered as schematically depicted in Fig. 10. The edge cracks are distributing on the opposite edge of the plate and the initial crack directions are both parallel to the horizontal line, but two cracks located at different height. The bottom and top edges are subjected to the vertical stable displacement. Judt and Ricoeur [18] numerically solved this problems by a new application of M- and L-integral for the numerical loading analysis. The total degrees of freedom in calculation is about 13, 000 and the element size around the crack tip is 0.01mm to ensure the result convergence. Comparisons between the calculated two interacting edge crack paths by C-force criterion and numerical results by Judt and Ricoeur [18] are depicted in Fig. 11. It is found that the cracks in the plate have a strong interaction. Both edge cracks initially deflect award each other and finally convergence forming local damaged region between the crack paths. It reveals obviously from Fig. 11, the results acquired by C-force criterion are in good agreement with the results obtained by Judt and Ricoeur [18]. 3.5 An incline crack in elastic plate
In order to become familiar with the C-force criterion in predicting the crack 11
propagations, further considerations will be carried out to valid the accuracy of C-force criterion to prediction the critical load. An inclined crack problem is depicted in Fig. 12 where the initial direction of crack inclined at an angle β with the loading direction in the plane subjected to an applied far-field stress σ0. For the case of incline crack problem, the crack will spread when the resultant configurational forces attains the material resistance, i.e., |C|=CR, where CR can be regarded as a crack extension force which should be independent of loading quantities and crack configuration serving an indication of the crack toughness of the material. The predicted critical loads by C-force criterion will be compared with the experimental data obtained by Pook [28] where the precracked DTD 5050 Zn aluminum alloy specimens are tested until failure and the critical load versus the incline crack angle are reported during experiments. The measured values of σcra1/2 for different crack sizes and failure loads against the crack incline angle β are plotted in Fig. 13. The solid line represents the predicted results by C-force criterion while the dotted symbol is the experimental data. The value of CRE=1.797×1012N-2m-3 is used to characterize the fracture toughness in the C-force criterion. This value is calculated from the analytical formulation of configurational forces in terms of the stress intensity factor for an incline crack in elastic plate (see Appendix) where the values of KIC on the aluminum alloy are reported by Pook [28]. It is found that the predicted results by C-force criterion is consistent with experimental data. The important conclusion to be made here is that the proposed C-force criterion is capable of predicting the critical loads σcr by the criterion of C=CR where CR is a material fracture resistance. 4. Experimental evaluation of material configurational forces by digital image correlation
The usefulness of the newly proposed C-force criterion relies on an effective and convenient experimental tool to measure it. In this section, experimental measurement of configurational forces is developed for the typically mixed-mode crack problem. The key technique to experimentally evaluation of the configurational forces involves knowledge of all the components in the x1-x2 plane such as the strain energy density, 12
the strains, the stresses, and the displacement gradient. Therefore, a full field experimental technique that enables the assessment of the overall strain and stress distribution in the specimens is necessary. Recently, the optical measuring method using digital image correlation (DIC) provides the feasibility to determine the whole strain-displacement fields for the damaged material [32,35]. With a sequence of 2D or 3D images, the facet, usually a small subset of pixels, is traced from a reference image of the undeformed state to an image of the deformed state. With the proper sub-pixel interpolation algorithms and mapping functions, the displacement, the rotation and the strain of the body surface can be calculated. In this study, the PMMA sample is fabricated as a typically brittle materials. An inclined pre-crack is introduced into the specimen where the position of the edge crack specified by the inclination angle β =45° (see Fig. 14). The specimens are tested under the uniaxial tensile loading. The tensile test is performed by electronic universal testing machine (SVL National Lab, Xi'an Jiao Tong University, MTS Biaxial Bionix, USA). The corresponding mechanical displacements are measured by ARAMIS 4M 3D optical measuring system (GOM corporation) by DIC method. In experiment, specimens with changing gray values as they occur with random pattern are more appropriate. Therefore, the specimens are pretreated with powder sprays to obtain the good high contrast stochastic pattern to facilitate the high resolution of DIC (see Fig. 14). Two cameras are arranged by inclining with a special angle 25 degree to the surfaces and in this way the optic difference on each facet of the loaded specimen could be measured, from which the displacements of the specimen surface under the tensile loading could be calculated by using the appropriate mathematical treatments. One facet has the dimension 15×15 pixels that correspond to approximately 0.5×0.5 mm2. Experimental arrangement of specimen, MTS machine, and ARAMIS are shown in Fig. 14. The configuration and size of specimen is depicted in Fig. 15. The displacement results can be directly outputted from the ARAMIS with high accuracy, as plotted in Fig. 16(a). The other quantities such as stresses, displacement gradients enclosed in the definition of the configurational forces need to be calculated 13
from the measured displacements by the constitutive equations or the algorithm equation. Among them, the stress fields are calculated by the displacement results obtained by DIC. For an apparent linear elastic material behavior, the relationship between the stress and strain obeys the generalized Hooke law:
σ ij = 2Gε ij + λθδ ij where G and λ denote Lame constants; θ=εii is bulk strain. The strain
(9) ε ij
are
calculated by the partial derivative of experimental DIC displacement fields, i.e., ε ij = ( u i , j + u j ,i ) / 2 . Numerical results of ∂ux/∂y, and ∂uy/∂x and the stresses σy, σxy are
plotted in Fig. 16(b)-(c), respectively. It is demonstrated that the experimental results are in good agreement with those by finite element calculation although the experimental accuracies of DIC results might be influenced by a few unexpected factors including illumination, stochastic pattern, material surface, reflection, air environment, facet size, correlation criterion used in pattern match, subpixel interpolation algorithms, mapping functions, and so on. The difference between their mean values is less than 1% which can be accepted in the way of tolerance as shown in Figs. 16 and 17. After all quantities are obtained at all discrete points in specimen, the configurational forces over the user-defined domain Ω is then evaluated by numerical integration. DIC process use the same post process to obtain the configurational force as done in FEM calculation as described in section 2.2. For comparison, finite element analysis is performed for the incline crack problem to validate the accuracy of DIC to evaluate the configurational forces. In FEM simulation, the material property is extracted from the uniaxial tensile test and the calculated stress results are plotted in Fig.17. The results of the configurational forces evaluated by experiment measurement and FEM computation are given in Table 1. The configurational forces are calculated by FEM post-processed operation where a set of user-defined domains are taken into account. It is shown in Table 1 that the values of the configurational forces are 14
approximately same from 4 different domains which show the independence of configurational forces on the user-defined domains. Furthermore, the experimental results of the configurational forces are well consistent with those obtained by FEM. The relative errors as compared to the value calculated by FEM simulations are merely 1.81%. The slight discrepancies among DIC and FEM are inferred to the negligible errors from the ARAMIS 4M instrument and the smoothing technique. These errors can be tolerantly accepted in both engineering viewpoint and theoretical viewpoint. In conclusion, the present study demonstrates that DIC method used by the ARAMIS 4M instrument does provide the effective tools to measure the configurational forces in predicting the mixed-mode crack propagation.
5. Concluding remarks
An intriguing fracture criterion based on the material configurational forces is employed for predicting the critical loading and the crack growth trajectory under mixed mode I/II loading. The onset of crack growth occurs when the magnitude of configurational resultant forces reaches the crack resistance while the crack propagation is stipulated to take place in the direction of configurational resultant forces. The mechanism of configurational forces in material space as crack driving forces are similar with the physical forces in physical space as rigid driving forces. That is, the motion of rigid body is initiated when the magnitude of physical resultant forces is greater than motion resistant and the body moves always along the direction of physical resultant forces. The consistent comparisons between numerical results of crack propagation by C-force criterion and experiment results have revealed the validation of the proposed criterion. Moreover, the proposed C-force criterion overcome the limitations of some classic criteria (G-energy-release-rate, K-factor, J-integral, CTOD) which cannot be conveniently used to predict the direction of crack propagation or others (MTS, S-factor) which are sensitive to the critical distance near the crack tip. Additionally, the experimental procedure by using DIC is shown to be convenient to evaluating the configurational forces. The present C-force criterion does provide an effective 15
approach for determining the mixed mode fracture behaviors especially that it is expected to have great advantage in predicting the crack growth for the large-deformation, elastic-plastic crack problems, fatigue crack prediction which will be reported in the sequent work. Acknowledgment
This work was supported by the National Natural Science Foundation of China (Nos. 11321062, 11472205) and the Fundamental Research Funds for the Central Universities in China. Appendix. The configuration forces in terms of the stress intensity factors
For a 2D elastic material under far-field loading, the stress and displacement fields in the vicinity of the crack tip for the mixed I-II mode crack extension behavior can be given by [34]
θ θ 3θ θ θ 3θ KⅠ KⅡ σ 11 = 2π r cos 2 (1 − sin 2 sin 2 ) − 2π r sin 2 (2 + cos 2 cos 2 ) KⅠ KⅡ θ θ θ θ 3θ 3θ σ 22 = 2π r cos 2 (1 + sin 2 sin 2 ) + 2π r sin 2 cos 2 cos 2 σ 12 = σ 21 = KⅠ sin θ cos θ cos 3θ + KⅡ cos θ (1 − sin θ sin 3θ ) 2 2 2 2 2 2 2π r 2π r σ = 2 K cos θ − K sin θ for plane strain state. Ⅱ Ⅰ 33 2 2 πr
(A1)
θ θ K θ θ KΙ r r cos (κ − 1 + 2sin 2 ) + ΙΙ × sin (κ + 1 + 2cos 2 ) u1 = 2G 2π 2 2 2G 2π 2 2 θ θ r K ΙΙ r u = KΙ 2θ 2θ sin ( 1 2 cos ) cos ( 1 2sin ) κ + − − × κ − − 2 2G 2π 2 2 2G 2π 2 2
(A2)
where KI and KII denotes the stress intensity factor of mode I, II crack, respectively; G denotes the shear modulus; ν is the Poisson's ratio; κ=3-4v for plane strain;
κ=(3-4v)/(1+v) for plane stress.The strain energy density can be found from W=
2 (1 + v ) (1 + cos θ )(κ − cosθ ) KΙ + 2sin θ ( 2 cosθ − κ + 1) KΙ KΙΙ 8π Er + ( κ + 1)(1 − cos θ ) + (1 + cos θ )( 3cos θ − 1) KΙΙ2
(A3)
The material configurational forces can be calculated from Eq. (2) and Eq. (A3) 16
∂W KΙ2 cos 2θ − 2 KΙ KΙΙ sin 2θ − KΙΙ2 cos 2θ c = − = 1 ∂x1 E′π r 2 2 2 c = − ∂W = KΙ sin 2θ + KΙ KΙΙ cos 2θ − KΙΙ sin 2θ 2 ∂x2 E′π r 2
(A4)
where E’=E for plane strain, and E’=E/(1-v2) for plane stress. Meanwhile, the Eshelby stress bij can be calculated from Eqs. (1) and (A1)-(A3) b11 = cos θ ( KⅠ sin θ + K II cos θ )2 / ( E ′π r ) b12 = − ( KⅠ cos θ − K II sin θ ) KⅠ sin θ cos θ + K II (1 + cos 2 θ ) / ( E ′π r ) 2 b21 = sin θ ( KⅠ sin θ + K II cos θ ) / ( E ′π r ) 2 b22 = − cos θ ( KⅠ sin θ + K II cos θ ) / ( E ′π r )
(A5)
The crack-tip configurational forces is over a user-defined domain Ω with the boundary ∂Ω and it can be calculated as Citip = lim ∫ b ji n j ds Ω→ 0
∂Ω
(A6)
Substituting Eq. (A5) into Eq. (A6), the analytical solutions of the components of configurational forces in terms of the stress intensity factors can be given as C1tip = ∫ b j1n j ds ∂Ω π
= −∫
−π
C
tip 2
K I2 sin 2 θ − K Ι K ΙΙ sin 2θ − K ΙΙ2 cos 2 θ K 2 + K ΙΙ2 dθ = Ι E ′π E′
=∫ b j 2n j ds
(A7)
∂Ω
cos θ ( K Ι2 sin θ + 2 K Ι K ΙΙ cos θ − K ΙΙ2 sin θ ) 2K K d θ = − Ι ΙΙ E ′π E′ −π π
= −∫
Reference [1] M.R.M. Aliha, M.R. Ayatollahi, Analysis of fracture initiation angle in some cracked ceramics using the generalized maximum tangential stress criterion. Int. J. Solids Struct. 49 (2012) 1877-1883. [2] M.R. Ayatollahi, M.R.M. Aliha, Fracture analysis of some ceramics under mixed mode loading. J. Am. Ceram. Soc. 94 (2011) 561-569. [3] D. Azocar, M. Elgueta, M.C. Rivara, Automatic LEFM crack propagation method based on local Lepp–Delaunay mesh refinement. Adv. Eng. Software 41 (2010) 111–119. [4]B.K. Paxevanakis, A.E. Giannakopoulos, Finite element analysis of discrete edge dislocations: Configurational forces and conserved integrals. Int. J. Solids Struct. 62 (2015) 52-65. 17
[5] D. Carka, M. Landis, On the path-dependence of the J-integral near a stationary crack in an elastic-plastic material. J. Appl. Mech. 78 (2010) 011006. [6] Y. H. Chen, Advances in conservation laws and energy release rates, Kluwer Academic Publishers, The Netherlands (2002). [7] G.P. Cherepanov, The propagation of cracks in a continuous medium. J. Appl. Math. Mech. 31 (1967) 503-512. [8] F.G. Erdogen, G.C., Sih, On the crack extension in plates under plane loading and transverse shear. J. Basic Eng. 85 (1963) 519-525. [9] J.D. Eshelby, The equilibrium of linear arrays of dislocations. Philos. Mag. 42(1951), 351-364. [10] F.J. Gomez, M. Elices, Fracture loads for ceramic samples with rounded notches. Engng. Fract. Mech. 73 (2006) 880-894. [11] A.A. Griffith, The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. 221 (1921) 163-197. [12] M.E. Gurtin, P. Podio-Guidugli, Configurational forces and the basic laws for crack propagation. J. Mech. Phys. Solids 44 (1996) 905-927. [13] M.E. Gurtin, Configurational forces as basic concepts of continuum physics. Springer, Berlin (2000). [14] A., Hillerborg, The theoretical basis of a method to determine the fracture energy GF of concrete. Mater. Struct. 18 (1985) 291-296. [15] M.A. Hussain, S.L. Pu, J. Underwood, Strain-energy-release rate for a crack under combined Mode I and Mode II. ASTM 560 (1974) 2-28. [16] R. Ingraffea, M. Grigoriu, Probabilistic fracture mechanics: A validation of predictive capability. Report (1990) 90-8. [17] G.R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate. ASME J. Appl. Mech. 24 (1957) 361-364. [18] P.O. Judt, A. Ricoeur, Crack growth simulation of multiple cracks systems applying remote contour interaction integrals. Theor. Appl. Fract. Mec. 75 (2015) 78–88. [19] R. Kienzler, G. Herrmann, Mechanics of material space: with applications to defect and fracture mechanics, Appl. Mech. Rev. 55 (2002). B23-B24. [20] R. Kienzler, G., Herrmann, Fracture criteria based on local properties of the Eshelby tensor. Mech. Res. Comm. 29 (2002) 521-527. [21] R. Kienzler, G. Herrmann, On the properties of Eshelby tensor. Acta Mech. 125 (1997) 73-91. [22] L.F. Ma, T.J. Lu, A.M. Korsunsky, Vector J-Integral Analysis of Crack Interaction With Pre-existing Singularities. J. Appl. Mech. 73 (2006) 876-883. [23] A.C.O. Miranda, M.A. Meggiolaro, J.T.P. Castro, L.F. Martha, T.N. Bittencourt, Fatigue life and crack predictions in generic 2D structural components. Eng. Fract. Mech. 70 (2003) 1259–1279. [24] R. Mueller, S. Kolling, D. Gross, On configurational forces in the context of the finite element method. Int. J. Numer. Mech. Engng. 53 (2002) 1557-1574. [25] B. Näser, M. Kaliske, R. Muller, Material forces for inelastic models at large strains: application to fracture mechanics. Comput. Mech. 40 (2007) 1005-1013. [26] T.D. Nguyen, S. Govindjee, P.A. Klein, H. Gao, A material force method for inelastic fracture mechanics. J. Mech. Phys. Solids 53 (2005) 91-121 [27] K. Ozenc, G. Chinaryan, M. Kaliske, A configurational force approach to model the 18
branching phenomenon in dynamic brittle fracture. Eng. Fract. Mech. 157 (2016) 26-42. [28] L.P. Pook, The effect of crack angle on fracture toughness, J. Eng. Frac. Mech. 3 (1966) 205-218. [29] J.R. Rice, A path independent integral and the approximate analysis of strain concentration by notch and cracks. ASME J. Appl. Mech. 35 (1968) 379-386. [30] G.C. Sih, Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fract. 10 (1974) 305-321. [31] N.K. Simha, F.D. Fischer, G.X. Shan, C.R. Chen, O. Kolednik, J-integral and crack driving force in elastic–plastic materials. J. Mech. Phys. Solids. 56 (2008) 2876-2895. [32] J. Tracy, A. Waas, S. Daly, Experimental assessment of toughness in ceramic matrix composites using the J-integral with digital image correlation part II: application to ceramic matrix composites, J. Mater. Sci. 50 (2015) 4659-4671. [33] A.A. Wells, Unstable crack propagation in metals cleavage and fast fracture. Proc. Crack Prop. Symp. 1 (1961) 210-230. [34] M.L. Williams, On the stress distribution at the base of a stationary crack. ASME J. Appl. Mech. 24 (1957) 109-114. [35] N.Y. Yu, Q. Li, Y.H. Chen, Experimental evaluation of the M-integral in an elastic-plastic material containing multiple defects. ASME J. Appl. Mech. 80 (2013) 819-833. [36] G.A. Maugin, Sixty years of configurational mechanics (1950–2010). Mech. Res. Commun. 50 (2013) 39-49. [37] A. David, E. Marcelo, R. María, Automatic LEFM crack propagation method based on local Lepp–Delaunay mesh refinement. Adv. Eng. Softw. 41 (2010) 111-119.
Table 1 The configurational forces at the crack tip by DIC and FEM for the incline crack
C-forces (J/m3) Gx Gy |G|
DIC Domain Domain Domain Domain
Ω1
Ω2
Ω3
Ω4
0.363 -0.347 0.503
0.356 -0.359 0.505
0.374 -0.342 0.507
0.362 -0.352 0.5049
FEM
Relative error
0.371 -0.354 0.509
1.81% 1.19% 0.92%
Average 0.364 -0.350 0.505
19
List of Figure Captions Fig. 1. The configurational forces at the crack tip and FEM implementation of configurational forces in the local coordinate system (e1, e2). Fig. 2. C-force criterion in the mixed mode crack problems. Fig. 3. Single central cracked beam with a hole. Fig. 4. Results of crack trajectory under (a) coarse mesh (b) fine mesh. Fig. 5. Crack growth trajectory for single central cracked beam with a hole between numerical results by C-force criterion and experimental results by Miranda et al. [23]. Fig. 6. The cracked beam with three holes subjected to three points bending (dimensions in inches). Fig. 7. Comparison of crack growth trajectory between numerical results by C-force criterion and experimental results by Ingraffea and Grigoriu [16]. (a) Specimen I; (b) Specimen II. Fig. 8. Single edge cracked plate under shear stress. Fig. 9. Crack growth trajectory of single edge cracked plate under shear stress between numerical results by C-force criterion and those by David et al. [37]. Fig. 10. Two interacting edge cracks in pane plate. Fig. 11. Comparison of crack growth trajectory for two interacting edge cracks between numerical results by C-force criterion and those by Judt and Ricoeur [18]. Fig. 12. An incline crack in elastic plate under tensile loading. Fig. 13. Prediction of critical failure load by C-force criterion compared with the experimental data [28]. Fig. 14. Experimental evaluation of configurational forces using digital image correlation. Fig. 15. The configuration and size of specimen (unit mm). Fig. 16. Digital image correlation results for the mixed-mode incline crack in plane specimen under tensile testing.(a) displacements ux and uy; (b) displacement gradients ux,y and uy,x; (c) stresses σy, σxy. 20
Fig. 17. FEM results for the mixed-mode incline crack in plane specimen under tensile testing.(a) stresses σy; (b) σxy.
21
b·n
Ω C2 ∂Ω
e2 Crack
C1 e1
Fig. 1. The configurational forces at the crack tip and FEM implementation of configurational forces in the local coordinate system (e1, e2).
uv C
C2
α
C1
x2
O
x1
Fig. 2. C-force criterion in the mixed mode crack problems.
22
37.5
50
P
P r=5.2
30 53.2
14.8
a
P
P
62.5
125 12.5
12.5
Fig. 3. Single central cracked beam with a hole.
(a) (b) Fig. 4. Results of crack trajectory under (a) coarse mesh (b) fine mesh.
23
C-force
Exper.
Fig. 5. Crack growth trajectory for single central cracked beam with a hole between numerical results by C-force criterion and experimental results by Miranda et al. [23].
P 1.25
r=0.25
2.0 8.0
2.0
4.0
b
a
1.0
Specimen I Specimen II
9.0 Crack length a 1.5 1.0
9.0
1.0
Crack offset b 5.0 4.0
Fig. 6. The cracked beam with three holes subjected to three points bending (dimensions in inches).
24
C-force
Exper.
C-force
Exper.
(a) (b) Fig. 7. Comparison of crack growth trajectory between numerical results by C-force criterion and experimental results by Ingraffea and Grigoriu [16]. (a) Specimen I; (b) Specimen II.
τ
a=3.5
7.0
Fig. 8. Single edge cracked plate under shear stress.
25
C-force
David et al (2010)
Fig. 9. Crack growth trajectory of single edge cracked plate under shear stress between numerical results by C-force criterion and those by David et al. [37].
u0
a2
a1
W =70mm
u0 . Fig. 10. Two interacting edge cracks in pane plate.
26
C-force
Judt
and
Ricoeur
Fig. 11. Comparison of crack growth trajectory for two interacting edge cracks between numerical results by C-force criterion and those by Judt and Ricoeur [18].
x σ
2
σ
0
0
β O
σ
0
x
2a
1
σ
0
Fig. 12. An incline crack in elastic plate under tensile loading.
27
1.5 1.4
C-force criterion Exper. data (Pook, 1966)
1/2
σcra /(MPam
-1/2
)
1.3 1.2 1.1 1.0 0.9 0.8 0.7
12
2
E· CR=1.797· 10 N · m
-3
0.6 20
30
40
50
60
70
80
90
β//degree
Fig. 13. Prediction of critical failure load by C-force criterion compared with the experimental data [28].
MTS
Specimen
DIC ARAMIS
Fig. 14. Experimental evaluation of configurational forces using digital image correlation.
28
Fig. 15. The configuration and size of specimen (unit mm).
29
(a)
(b)
(c)
Fig. 16. Digital image correlation results for the mixed-mode incline crack in plane specimen under tensile testing.(a) displacements ux and uy; (b) displacement gradients ux,y and uy,x; (c) stresses σy, σxy.
30
(a)
(b)
Fig. 17. FEM results for the mixed-mode incline crack in plane specimen under tensile testing.(a) stresses σy; (b) σxy.
31
Highlights 1) The material configurational forces are proposed for predicting the critical loading and the crack growth trajectory under mixed mode I/II loading. 2) The newly proposed criterion is validated through a series of examples. 3) An experimental procedure on evaluation the configurational forces is proposed using the digital image correlation.
32
TAFMEC 1810
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
3 November 2016 27 December 2016 14 February 2017
Please cite this article as: Y. Guo, Q. Li, Material configurational forces applied to mixed mode crack propagation, Theoretical and Applied Fracture Mechanics (2017), doi: http://dx.doi.org/10.1016/j.tafmec.2017.02.006
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Material configurational forces applied to mixed mode crack propagation Yuli Guo, Qun Li* State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China *Corresponding author. Tel.:+86-29-82667535 Email address: [email protected] (Q. Li) ABSTRACT. The concept of material configurational forces is applied to predict the mixed mode crack propagation. A new fracture criterion based on the resultant of configurational forces (termed as C-force criterion) is proposed. The basic assumption is that the onset of crack growth occurs when the resultant of configurational forces reaches a critical value and the crack growth takes place in the direction of resultant configurational forces. An implementation of the configurational forces into the finite element is presented. The newly proposed C-force criterion is further validated through a series of examples. It is concluded that the predictions of mixed-mode crack propagation by C-force criterion are in good agreement with experimental data in the open literatures. In addition, an experimental procedure on evaluation the configurational forces is proposed by using the digital image correlation. It is demonstrated that the C-force criterion provides a more convenient and accurate procedure to predict the mixed mode crack propagation. Keywords: material configurational forces, crack, criterion, digital image correlation 1. Introduction Fracture behaviors of materials have been widely investigated concerning structural integrity, reliability, and functionality. The evaluation of both onset of crack growth and the crack growth direction involved in fracturing of materials necessitates an effective fracture criterion to indicate the instance when the onset of crack growth occurs and where the crack propagates. The well-known criteria within the framework of fracture mechanics of crack instability include the energy release rate G [11], the 1
stress intensity factors K [17], the J-integral [7,29], the crack tip opening displacement CTOD [33] and so on. The fracture criterion is formulated as the crack propagation starts when the fracture parameters reach the material-specific critical values. Among them, the stress intensity factor is appropriate with small-scale yielding assumption. The path-independent J-integral can be used as an effective fracture parameter to predict the crack stability and growth in elastic-plastic materials although the limitation of path-dependence of the J-integral is revealed near a stationary crack in a fully yielding material [5]. The CTOD has advantage in three-dimensional analyses to study constraint effects, crack tunneling, and the fracture process. However, the aforementioned fracture criteria always assume that the crack propagates along the original crack surface to overcome the fracture resistance. Although they have enabled a successful application to predict the occurrence of onset of crack growth, i.e., the critical load or the critical crack size that a structure will tolerate for characterizing the fracture resistance of materials, it is inappropriate to solve the issue of predicting the crack deflection or kinking from the original crack direction under the mixed-mode loading condition. In addition, industrial application usually requires that materials are regularly exposed to the complicated load conditions. In general, the cracks generated inside the engineering components can be subjected the mode I (opening) and the mode II (in-plane sliding) in plane problems. Any combinations of these modes are known as mixed mode crack problems. A few of attempts appear to treat the mixed mode crack problems where the crack extension in a combined stress field can grow in any arbitrary direction with reference to its original crack surface. There are a plethora of fracture criteria that have been used in predict the mixed mode crack propagation, among which the good examples include the maximum tangential stress MTS [8], the strain-energy-density S-factor [30], the maximum energy release rate MERR [15] et al. The maximum tangential stress makes the hypothesis that the crack initiates in the direction along which the tangential stress σθ possesses a stationary (maximum) value while the onset of crack growth occurs when the tangential stress reaches a critical 2
value. The strain-energy-density factor represents the singularity strength of the strain energy density field W around the crack tip which can be defined by S=Wr. It assumes that the initial crack growth takes place in the direction along which the strain-energy-density factor S possesses a stationary (minimum) value while the onset of crack growth occurs when the strain-energy-density factor reaches a critical value SC. The maximum energy release rate is to determine the energy release rate as a function of crack propagation, and then determine the critical direction by maximizing the energy release rate. However, the application of these fracture criteria in the situation of mixed mode crack are restrained by several limitations. First, the calculation of tangential stress is required to be along a critical distance rc from the crack tip. Similarly, the strain-energy-density factor as the intensity of dW/dA for the interior element is valid in the small region where a core region with radius r0 surrounding the crack tip must be observed. The predicted results of crack propagation could be disturbed by the magnitude of the critical distance. The determination of critical distance or the characteristic length, or the core region is an important issue in predicting the crack growth by the criteria of MTS and S-factor. Many researchers have attempted in the past several decades to suggest the various models to calculate the critical distances for different brittle and quasi-brittle materials [1,2,10,14]. Second, the limitation of the maximum energy release rate is to require the true values of the mixed mode energy release rate obtained by solving a class of boundary value problems of a branched crack that the branch-crack ratio must tend to zero. The analytical difficulties are associated with solving the branched crack problem. This limiting process presents a formidable task by using the maximum energy release rate. For this reason, there is a need in the search of a realistic mixed mode fracture criterion being able to describing the crack propagation. Recently, it is shown that the material configurational forces [4,12,13,19,26,27,36], plays an important role in fracture and damage mechanics. The configurational force is able to provide the information from the macroscopic perspective, suggesting a method on predicting the crack propagation. Among many others, several pioneering investigations on the 3
material configurational forces in fracture mechanics are briefly listed here: Kienzler and Herrmann [20] proposed a fracture criterion based on local properties of the Eshelby tensor for a plane crack under mixed-mode loading conditions. Ma et al. [22] proposed the vector Jk-integral as a fracture criterion based on the criterion of maximum energy release rate for multi-singularity-crack interaction problems. Simha et al. [31] used the configurational forces approach to identify a ‘plasticity influence term’ that describes the crack tip shielding or anti-shielding due to plastic deformation in the body. Baxevanakis and Giannakopoulos [4] applied the configurational forces and conserved integrals into the finite element analysis of discrete edge dislocations under mixed loading. Ozenc [27] proposed a configurational force approach to model the branching phenomenon in dynamic brittle fracture. Notwithstanding the practices of the configurational forces in the aforementioned situations, the main purpose of this paper is to present a failure criterion based on the concept of configurational forces which are regarded as the crack driving forces. The basic assumption is that the onset of crack growth occurs when the resultant of configurational forces reaches a critical value and the crack growth takes place in the direction of resultant configurational force. This configurational force based fracture criterion is validated to successfully describing the crack propagation under mixed-mode loading conditions. 2. Fracture criterion by the material configurational forces 2.1 The concept of material configurational forces The concept of material configurational stress tensor (or called Eshelby tensor) was first introduced by Eshelby [9]. Considering the gradient of the Lagrangian energy density is the most straightforward approach to derive the material configurational
stress
tensor.
The
material
configurational
stress
bji
in
small-deformation problems is defined as
b ji = W δ ji − σ jk uk ,i
(1)
where W=W(xi ,ui,j) denotes the strain energy density; δji denotes the Kronecker delta;
σjk is the stress tensor; uk denotes the components of displacements; the subscript 4
prima {},i refers to the corresponding differentiation with respect to the coordinate xi; the repeated indices denote summation over the range of indices. The physical interpretation of the configurational stresses bij (i, j=1, 2) can be explained as the change of the total potential energy at a point of an elastic infinitesimal element due to a material unit translation in xi–direction of a unit surface with normal in xj –direction [20]. The material configurational force as the most significant concept is denoted as the explicit dependent of W on xi
∂W ci = − ∂xi expl.
(2)
where ( ∂W / ∂xi )expl. denotes the explicit dependence of W on xi; Here, the material configurational force is denoted as the explicit gradient of W on xi [9]. The material configurational forces ci can be explained as the energy release rates due to translation of one material point along xi-direction. The balance laws between the configurational stress in Eq. (1) and the configurational force in Eq. (2) can be established by
bji, j + ci = 0
(3)
2.2 FEM implementation of configurational forces
Numerical implementation of configurational forces in the material space will be carried out into finite element analysis. It should be pointed that the configurational forces Ci is not calculated by the direct definition in Eq. (2). Actually, the configurational force is calculated through the configurational stress bij by the equilibrium equation Eq. (3). In order to calculate the discrete configurational force at the nodes, a weak function is introduced in numerical implementation. And then the discrete configurational forces at every node (I) over each finite element of area Ω e can be given by [24]
5
Ci( I ) = ∫ N ( I ) ci dV = ∫ bij N,(jI ) dV . Ωe
(4)
Ωe
In derivative of Eq. (4), the balance law Eq. (3) of configurational stress and force is adopted. The index (I) denotes the node local number in the specific element. N(I) are the shape function. The term of bij N ,(jI ) can be easily obtained in every integration point after the standard quantities are solved, e.g. the strain energy density, the nodal displacements, the stresses, according to the definition of bij in Eq. (1). The integration of Eq. (4) can be performed by a standard Gaussian integration method. Finally, the total configurational forces Ci( K ) on each specific node K have to be assembled by the contributions of all elements adjacent to node K, nel
Ci( K ) = U Ci( I ) .
(5)
e =1
Moreover, the configurational forces at the crack tip Citip are paid more attentions as a crack driving force. It can be interpreted as the energy release rate due to the translation of crack advance along xk-direction, respectively. The configurational force at the crack tip can be deemed as the resultant of all configurational force vectors in elements over the specific domain Ω surrounding the crack tip in the numerical evaluation. It can be concluded that the configurational force at the crack tip is independence of the domain. Numerical evaluation of configurational force can be alternatively calculated based on the resultant of all the configurational force vectors in a user-defined domain Ω surrounding the crack tip as shown in Fig. 1. C1tip =
∑
C1
Ω− domain
C2tip =
∑
(6) C2
Ω− domain
It should be mentioned that the configurational forces originate from the inhomogeneity of the material. For linear elastic fracture mechanics, it is demonstrated that the selection of the user-defined domain near the crack tip has independent effect on the calculated results. For convenience of FEM computation, 6
the directions of e1 and e2 correspond to the local coordinate system at the crack tip are considered as depicted in Fig. 1. Here, e1 is the tangential component of the local coordinate system with respect to the crack surface, while e2 represents the normal component of the local coordinate system. The components of the configurational force vector will be calculated in the local coordinate system (e1, e2) in the following analysis. 2.3 The C-force fracture criterion
The mixed mode crack propagation will be deflected from its original path due to the non-symmetric material structure or mixed-mode loading with respect to the crack plane. Herein, an intriguing theory to predict the mixed-mode crack propagation can be proposed according to the physical interpretation of the crack tip configurational forces. The C1-force is identical to the crack extension forces and it has been given as the rate of total potential energy release per unit crack-tip advance along e1-direction. Analogously, the C2-force has been given a precise and clear physical significance as the rate of total energy release by postulating that the crack will skew the tip advances along e2-direction in the local coordinate system. Under these considerations, a new fracture criterion (termed as C-force criterion) is proposed where two basic stipulations on crack propagation will now be made in plane problems: i) The initial crack growth takes place in the direction ahead of the crack tip along the uv configurational resultant forces C . The initiation angle of crack can be determined by
α = arctan
C2 C1
(7)
where α is the crack-kinking angle as shown in Fig. 2. ii) The onset of crack growth occurs when the magnitude of configurational resultant forces |C| overcomes the material resistance. i.e.,
C = C12 + C22 ≥ CR
(8) 7
where CR is the material fracture resistance which is a material constant regardless of the crack configuration and loading conditions. The value of CR provides a knowledge of mixed mode crack extension in that it specifies the fracture toughness of the material. It is well known that for a pure mode I crack in homogenous material where a negligible value of C2-force prevails, the crack will advance along the crack direction. In contrast, the introduction of mixed-mode crack problem will result in the remarkable magnitude of C2–force near the crack tip. A mixed mode fracture can be characterized by the configurational forces and the crack deflection will happen. The present mode is advantage from those the traditional fracture criterion (e.g., the J-integral) where the J-integral may not see much difference among mode I and mode II crack. The present method is based on the two parameters C1 and C2 which can distinguish the mixed mode crack. The framework of C-force criterion given in this paper could provide a useful tool to deal with problems associated with the crack deflection mechanism under mixed-mode loading condition. 3. Validations of C-force criterion applied to mixed mode crack propagation
The C-force criterion will be used to predict the mixed mode crack propagation. The special treatment of C-force criterion to predicting the crack growth path and the critical load will be clarified and addressed by a series of representative crack examples. The crack propagation is simulated by using the finite element method for each propagation step, then the mesh near the crack tip is modified to take into account the new crack advance. In FEM calculations, the crack paths are calculated by an iterative approach, shifting the crack tip by an increment ∆a along the direction of material configurational resultant forces according to the C-force criterion. The re-meshing algorithm refines the mesh close to the crack tip. That is, fine mesh is carried out in the local region near the crack tip and the singularity elements are chosen near the crack tip while the mesh of outer region away from the crack tip is fixed. Moreover, the present study doesn’t consider the crack growth rate and then the static growth of crack is assumed in all cases. Five representative crack examples are carried out and addressed by the C-force 8
criterion. In the calculation, the mesh sensitivity and the result convergence are given and discussed on each example. The purpose of section 3.1-3.4 is to observe the crack trajectories and it is demonstrated that the critical value of fracture resistance doesn’t affect the crack propagation path. Therefore, the values of C ≥ CR is always prescribed to ensure the crack growth by introducing a negligible values of CR. Furthermore, the purpose of section 3.5 is to valid the accuracy of C-force criterion to prediction the critical load where the value of CR is calculated from the analytical formulation of configurational forces in terms of the stress intensity factor for an incline crack in elastic plate (see Appendix). The predicted results of crack propagation by C-force criterion will be validated by comparing with the experimental observations in the open literatures. 3.1 Single central cracked beam with a hole
The first example corresponds to a single cracked beam with a hole, loaded in upper two points and constrained in lower two points, i.e., the four-point-bending beam. As depicted in Fig. 3, the initial crack is in the center of the bottom edge and the void contained in beam has an offset from the vertical centerline. The original crack length a=2.5mm. The mesh insensitivity is validated by a coarse and a fine mesh, as shown in Fig. 4. The coarse mesh with 9,318 elements and fine mesh with 12,778 elements are considered in order to validate the mesh independence. It can be found from Fig. 4 that the calculated chart of crack trajectory is consistent for the coarse and fine mesh. Numerical results presented by C-force criterion are compared with experiment results by Miranda et al. [23], where they experimentally observed the crack propagation for such a four-point bending single-edge crack in specimen made of code rolled SAE 1020 steel. The material elastic moduli of steel E=205 GPa, v=0.3. Both numerical predictions and experimental results are shown in Fig. 5. It indicates that the crack is always attracted by the hole. The crack will curve its path and grow toward the hole until sinking in the hole. Fig. 5 shows that the C-force criterion provides a quite good agreement of crack propagation with experimental observations. 9
3.2 Cracked beam contained three holes
This example corresponds to a cracked beam subjected a load in the center of one edge and supported by two points as depicted in Fig. 6. The three holes have an offset from the centerline and arrange along the vertical line. The monotonic load P is applied at center of specimens. The experimental results for polymethylmethacrylate (PMMA) beams has been conducted by Ingraffea and Grigoriu [16] with considering two specimens of the initial crack length, b, and its location, a, as shown by the table in Fig. 6. The elastic material constants of PMMA are elastic moduli E=3GPa and Poisson’s ratio ν=0.3. Numerical simulations of the crack trajectory are performed for two specimens with a crack increment ∆a=0.05mm of each iterative step in FEM computations. The total degrees of freedom for specimen I and II are over 93, 000 and 77, 000, respectively. The element size around the crack tip are less than 0.2mm so as to ensure the mesh insensitivity and the result convergence. Comparison of crack growth trajectory between numerical results by C-force criterion and experimental results by Ingraffea and Grigoriu [16] are made to validate the accuracy of C-force criterion to predict the mixed mode crack problems. The cloud chart of crack trajectory after 65 iterative steps for specimen I and 74 steps for specimen II by FEM is shown in left figures of Fig. 7(a)-(b) compared with the experimental results in right figures of Fig. 7(a)-(b). It is found that under the influence of three holes, the crack trajectory is deviated from the track predicted without the holes. For specimen I, the crack growth trajectory passes towards the bottom hole and deflects to the opposite side when it is close to the bottom hole and finally ended at the middle hole as depicted in Fig. 7. For specimen II, the crack directly propagates toward the hole in the center for this problem. Although the size and location of initial crack lead to the very different crack trajectory, both specimens of numerical patterns capture the curved crack in good agreement with experimental results. It demonstrates the validation of the C-force criterion so that the fracture criterion based on configurational force is verified and feasible to describe the mixed mode crack growth. 3.3 Single edge cracked plate under shear stress 10
This example corresponds to a single edge cracked plate that is constrained to a far–field shear stress along the top edge of plate, as depicted in Fig. 8. The bottom of plate is constrained. The total degrees of freedom is about 87, 000 and the element size around the crack tip is 0.08mm. The cloud chart of crack trajectory after 17 iterative steps by the C-force criterion is shown in left figures of Fig. 9 compared with the results by David et al [37] in right figures of Fig. 9. Among them, David et al. [37] used a Lepp-Delaunay based mesh refinement algorithm for triangular meshes which allows both the generation of the initial mesh and the local modification of the current mesh as the crack propagates according to the K-factor criterion. It is shown from Fig. 9 that the predicted crack trajectory is downward sloping at first and tends to parallel to horizontal line gradually which resembles the results presented in the reference work by David et al. [37]. 3.4 Two interacting edge cracks
In this example, a plate specimen exhibiting two interacting edge cracks with incipient crack lengths are considered as schematically depicted in Fig. 10. The edge cracks are distributing on the opposite edge of the plate and the initial crack directions are both parallel to the horizontal line, but two cracks located at different height. The bottom and top edges are subjected to the vertical stable displacement. Judt and Ricoeur [18] numerically solved this problems by a new application of M- and L-integral for the numerical loading analysis. The total degrees of freedom in calculation is about 13, 000 and the element size around the crack tip is 0.01mm to ensure the result convergence. Comparisons between the calculated two interacting edge crack paths by C-force criterion and numerical results by Judt and Ricoeur [18] are depicted in Fig. 11. It is found that the cracks in the plate have a strong interaction. Both edge cracks initially deflect award each other and finally convergence forming local damaged region between the crack paths. It reveals obviously from Fig. 11, the results acquired by C-force criterion are in good agreement with the results obtained by Judt and Ricoeur [18]. 3.5 An incline crack in elastic plate
In order to become familiar with the C-force criterion in predicting the crack 11
propagations, further considerations will be carried out to valid the accuracy of C-force criterion to prediction the critical load. An inclined crack problem is depicted in Fig. 12 where the initial direction of crack inclined at an angle β with the loading direction in the plane subjected to an applied far-field stress σ0. For the case of incline crack problem, the crack will spread when the resultant configurational forces attains the material resistance, i.e., |C|=CR, where CR can be regarded as a crack extension force which should be independent of loading quantities and crack configuration serving an indication of the crack toughness of the material. The predicted critical loads by C-force criterion will be compared with the experimental data obtained by Pook [28] where the precracked DTD 5050 Zn aluminum alloy specimens are tested until failure and the critical load versus the incline crack angle are reported during experiments. The measured values of σcra1/2 for different crack sizes and failure loads against the crack incline angle β are plotted in Fig. 13. The solid line represents the predicted results by C-force criterion while the dotted symbol is the experimental data. The value of CRE=1.797×1012N-2m-3 is used to characterize the fracture toughness in the C-force criterion. This value is calculated from the analytical formulation of configurational forces in terms of the stress intensity factor for an incline crack in elastic plate (see Appendix) where the values of KIC on the aluminum alloy are reported by Pook [28]. It is found that the predicted results by C-force criterion is consistent with experimental data. The important conclusion to be made here is that the proposed C-force criterion is capable of predicting the critical loads σcr by the criterion of C=CR where CR is a material fracture resistance. 4. Experimental evaluation of material configurational forces by digital image correlation
The usefulness of the newly proposed C-force criterion relies on an effective and convenient experimental tool to measure it. In this section, experimental measurement of configurational forces is developed for the typically mixed-mode crack problem. The key technique to experimentally evaluation of the configurational forces involves knowledge of all the components in the x1-x2 plane such as the strain energy density, 12
the strains, the stresses, and the displacement gradient. Therefore, a full field experimental technique that enables the assessment of the overall strain and stress distribution in the specimens is necessary. Recently, the optical measuring method using digital image correlation (DIC) provides the feasibility to determine the whole strain-displacement fields for the damaged material [32,35]. With a sequence of 2D or 3D images, the facet, usually a small subset of pixels, is traced from a reference image of the undeformed state to an image of the deformed state. With the proper sub-pixel interpolation algorithms and mapping functions, the displacement, the rotation and the strain of the body surface can be calculated. In this study, the PMMA sample is fabricated as a typically brittle materials. An inclined pre-crack is introduced into the specimen where the position of the edge crack specified by the inclination angle β =45° (see Fig. 14). The specimens are tested under the uniaxial tensile loading. The tensile test is performed by electronic universal testing machine (SVL National Lab, Xi'an Jiao Tong University, MTS Biaxial Bionix, USA). The corresponding mechanical displacements are measured by ARAMIS 4M 3D optical measuring system (GOM corporation) by DIC method. In experiment, specimens with changing gray values as they occur with random pattern are more appropriate. Therefore, the specimens are pretreated with powder sprays to obtain the good high contrast stochastic pattern to facilitate the high resolution of DIC (see Fig. 14). Two cameras are arranged by inclining with a special angle 25 degree to the surfaces and in this way the optic difference on each facet of the loaded specimen could be measured, from which the displacements of the specimen surface under the tensile loading could be calculated by using the appropriate mathematical treatments. One facet has the dimension 15×15 pixels that correspond to approximately 0.5×0.5 mm2. Experimental arrangement of specimen, MTS machine, and ARAMIS are shown in Fig. 14. The configuration and size of specimen is depicted in Fig. 15. The displacement results can be directly outputted from the ARAMIS with high accuracy, as plotted in Fig. 16(a). The other quantities such as stresses, displacement gradients enclosed in the definition of the configurational forces need to be calculated 13
from the measured displacements by the constitutive equations or the algorithm equation. Among them, the stress fields are calculated by the displacement results obtained by DIC. For an apparent linear elastic material behavior, the relationship between the stress and strain obeys the generalized Hooke law:
σ ij = 2Gε ij + λθδ ij where G and λ denote Lame constants; θ=εii is bulk strain. The strain
(9) ε ij
are
calculated by the partial derivative of experimental DIC displacement fields, i.e., ε ij = ( u i , j + u j ,i ) / 2 . Numerical results of ∂ux/∂y, and ∂uy/∂x and the stresses σy, σxy are
plotted in Fig. 16(b)-(c), respectively. It is demonstrated that the experimental results are in good agreement with those by finite element calculation although the experimental accuracies of DIC results might be influenced by a few unexpected factors including illumination, stochastic pattern, material surface, reflection, air environment, facet size, correlation criterion used in pattern match, subpixel interpolation algorithms, mapping functions, and so on. The difference between their mean values is less than 1% which can be accepted in the way of tolerance as shown in Figs. 16 and 17. After all quantities are obtained at all discrete points in specimen, the configurational forces over the user-defined domain Ω is then evaluated by numerical integration. DIC process use the same post process to obtain the configurational force as done in FEM calculation as described in section 2.2. For comparison, finite element analysis is performed for the incline crack problem to validate the accuracy of DIC to evaluate the configurational forces. In FEM simulation, the material property is extracted from the uniaxial tensile test and the calculated stress results are plotted in Fig.17. The results of the configurational forces evaluated by experiment measurement and FEM computation are given in Table 1. The configurational forces are calculated by FEM post-processed operation where a set of user-defined domains are taken into account. It is shown in Table 1 that the values of the configurational forces are 14
approximately same from 4 different domains which show the independence of configurational forces on the user-defined domains. Furthermore, the experimental results of the configurational forces are well consistent with those obtained by FEM. The relative errors as compared to the value calculated by FEM simulations are merely 1.81%. The slight discrepancies among DIC and FEM are inferred to the negligible errors from the ARAMIS 4M instrument and the smoothing technique. These errors can be tolerantly accepted in both engineering viewpoint and theoretical viewpoint. In conclusion, the present study demonstrates that DIC method used by the ARAMIS 4M instrument does provide the effective tools to measure the configurational forces in predicting the mixed-mode crack propagation.
5. Concluding remarks
An intriguing fracture criterion based on the material configurational forces is employed for predicting the critical loading and the crack growth trajectory under mixed mode I/II loading. The onset of crack growth occurs when the magnitude of configurational resultant forces reaches the crack resistance while the crack propagation is stipulated to take place in the direction of configurational resultant forces. The mechanism of configurational forces in material space as crack driving forces are similar with the physical forces in physical space as rigid driving forces. That is, the motion of rigid body is initiated when the magnitude of physical resultant forces is greater than motion resistant and the body moves always along the direction of physical resultant forces. The consistent comparisons between numerical results of crack propagation by C-force criterion and experiment results have revealed the validation of the proposed criterion. Moreover, the proposed C-force criterion overcome the limitations of some classic criteria (G-energy-release-rate, K-factor, J-integral, CTOD) which cannot be conveniently used to predict the direction of crack propagation or others (MTS, S-factor) which are sensitive to the critical distance near the crack tip. Additionally, the experimental procedure by using DIC is shown to be convenient to evaluating the configurational forces. The present C-force criterion does provide an effective 15
approach for determining the mixed mode fracture behaviors especially that it is expected to have great advantage in predicting the crack growth for the large-deformation, elastic-plastic crack problems, fatigue crack prediction which will be reported in the sequent work. Acknowledgment
This work was supported by the National Natural Science Foundation of China (Nos. 11321062, 11472205) and the Fundamental Research Funds for the Central Universities in China. Appendix. The configuration forces in terms of the stress intensity factors
For a 2D elastic material under far-field loading, the stress and displacement fields in the vicinity of the crack tip for the mixed I-II mode crack extension behavior can be given by [34]
θ θ 3θ θ θ 3θ KⅠ KⅡ σ 11 = 2π r cos 2 (1 − sin 2 sin 2 ) − 2π r sin 2 (2 + cos 2 cos 2 ) KⅠ KⅡ θ θ θ θ 3θ 3θ σ 22 = 2π r cos 2 (1 + sin 2 sin 2 ) + 2π r sin 2 cos 2 cos 2 σ 12 = σ 21 = KⅠ sin θ cos θ cos 3θ + KⅡ cos θ (1 − sin θ sin 3θ ) 2 2 2 2 2 2 2π r 2π r σ = 2 K cos θ − K sin θ for plane strain state. Ⅱ Ⅰ 33 2 2 πr
(A1)
θ θ K θ θ KΙ r r cos (κ − 1 + 2sin 2 ) + ΙΙ × sin (κ + 1 + 2cos 2 ) u1 = 2G 2π 2 2 2G 2π 2 2 θ θ r K ΙΙ r u = KΙ 2θ 2θ sin ( 1 2 cos ) cos ( 1 2sin ) κ + − − × κ − − 2 2G 2π 2 2 2G 2π 2 2
(A2)
where KI and KII denotes the stress intensity factor of mode I, II crack, respectively; G denotes the shear modulus; ν is the Poisson's ratio; κ=3-4v for plane strain;
κ=(3-4v)/(1+v) for plane stress.The strain energy density can be found from W=
2 (1 + v ) (1 + cos θ )(κ − cosθ ) KΙ + 2sin θ ( 2 cosθ − κ + 1) KΙ KΙΙ 8π Er + ( κ + 1)(1 − cos θ ) + (1 + cos θ )( 3cos θ − 1) KΙΙ2
(A3)
The material configurational forces can be calculated from Eq. (2) and Eq. (A3) 16
∂W KΙ2 cos 2θ − 2 KΙ KΙΙ sin 2θ − KΙΙ2 cos 2θ c = − = 1 ∂x1 E′π r 2 2 2 c = − ∂W = KΙ sin 2θ + KΙ KΙΙ cos 2θ − KΙΙ sin 2θ 2 ∂x2 E′π r 2
(A4)
where E’=E for plane strain, and E’=E/(1-v2) for plane stress. Meanwhile, the Eshelby stress bij can be calculated from Eqs. (1) and (A1)-(A3) b11 = cos θ ( KⅠ sin θ + K II cos θ )2 / ( E ′π r ) b12 = − ( KⅠ cos θ − K II sin θ ) KⅠ sin θ cos θ + K II (1 + cos 2 θ ) / ( E ′π r ) 2 b21 = sin θ ( KⅠ sin θ + K II cos θ ) / ( E ′π r ) 2 b22 = − cos θ ( KⅠ sin θ + K II cos θ ) / ( E ′π r )
(A5)
The crack-tip configurational forces is over a user-defined domain Ω with the boundary ∂Ω and it can be calculated as Citip = lim ∫ b ji n j ds Ω→ 0
∂Ω
(A6)
Substituting Eq. (A5) into Eq. (A6), the analytical solutions of the components of configurational forces in terms of the stress intensity factors can be given as C1tip = ∫ b j1n j ds ∂Ω π
= −∫
−π
C
tip 2
K I2 sin 2 θ − K Ι K ΙΙ sin 2θ − K ΙΙ2 cos 2 θ K 2 + K ΙΙ2 dθ = Ι E ′π E′
=∫ b j 2n j ds
(A7)
∂Ω
cos θ ( K Ι2 sin θ + 2 K Ι K ΙΙ cos θ − K ΙΙ2 sin θ ) 2K K d θ = − Ι ΙΙ E ′π E′ −π π
= −∫
Reference [1] M.R.M. Aliha, M.R. Ayatollahi, Analysis of fracture initiation angle in some cracked ceramics using the generalized maximum tangential stress criterion. Int. J. Solids Struct. 49 (2012) 1877-1883. [2] M.R. Ayatollahi, M.R.M. Aliha, Fracture analysis of some ceramics under mixed mode loading. J. Am. Ceram. Soc. 94 (2011) 561-569. [3] D. Azocar, M. Elgueta, M.C. Rivara, Automatic LEFM crack propagation method based on local Lepp–Delaunay mesh refinement. Adv. Eng. Software 41 (2010) 111–119. [4]B.K. Paxevanakis, A.E. Giannakopoulos, Finite element analysis of discrete edge dislocations: Configurational forces and conserved integrals. Int. J. Solids Struct. 62 (2015) 52-65. 17
[5] D. Carka, M. Landis, On the path-dependence of the J-integral near a stationary crack in an elastic-plastic material. J. Appl. Mech. 78 (2010) 011006. [6] Y. H. Chen, Advances in conservation laws and energy release rates, Kluwer Academic Publishers, The Netherlands (2002). [7] G.P. Cherepanov, The propagation of cracks in a continuous medium. J. Appl. Math. Mech. 31 (1967) 503-512. [8] F.G. Erdogen, G.C., Sih, On the crack extension in plates under plane loading and transverse shear. J. Basic Eng. 85 (1963) 519-525. [9] J.D. Eshelby, The equilibrium of linear arrays of dislocations. Philos. Mag. 42(1951), 351-364. [10] F.J. Gomez, M. Elices, Fracture loads for ceramic samples with rounded notches. Engng. Fract. Mech. 73 (2006) 880-894. [11] A.A. Griffith, The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. 221 (1921) 163-197. [12] M.E. Gurtin, P. Podio-Guidugli, Configurational forces and the basic laws for crack propagation. J. Mech. Phys. Solids 44 (1996) 905-927. [13] M.E. Gurtin, Configurational forces as basic concepts of continuum physics. Springer, Berlin (2000). [14] A., Hillerborg, The theoretical basis of a method to determine the fracture energy GF of concrete. Mater. Struct. 18 (1985) 291-296. [15] M.A. Hussain, S.L. Pu, J. Underwood, Strain-energy-release rate for a crack under combined Mode I and Mode II. ASTM 560 (1974) 2-28. [16] R. Ingraffea, M. Grigoriu, Probabilistic fracture mechanics: A validation of predictive capability. Report (1990) 90-8. [17] G.R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate. ASME J. Appl. Mech. 24 (1957) 361-364. [18] P.O. Judt, A. Ricoeur, Crack growth simulation of multiple cracks systems applying remote contour interaction integrals. Theor. Appl. Fract. Mec. 75 (2015) 78–88. [19] R. Kienzler, G. Herrmann, Mechanics of material space: with applications to defect and fracture mechanics, Appl. Mech. Rev. 55 (2002). B23-B24. [20] R. Kienzler, G., Herrmann, Fracture criteria based on local properties of the Eshelby tensor. Mech. Res. Comm. 29 (2002) 521-527. [21] R. Kienzler, G. Herrmann, On the properties of Eshelby tensor. Acta Mech. 125 (1997) 73-91. [22] L.F. Ma, T.J. Lu, A.M. Korsunsky, Vector J-Integral Analysis of Crack Interaction With Pre-existing Singularities. J. Appl. Mech. 73 (2006) 876-883. [23] A.C.O. Miranda, M.A. Meggiolaro, J.T.P. Castro, L.F. Martha, T.N. Bittencourt, Fatigue life and crack predictions in generic 2D structural components. Eng. Fract. Mech. 70 (2003) 1259–1279. [24] R. Mueller, S. Kolling, D. Gross, On configurational forces in the context of the finite element method. Int. J. Numer. Mech. Engng. 53 (2002) 1557-1574. [25] B. Näser, M. Kaliske, R. Muller, Material forces for inelastic models at large strains: application to fracture mechanics. Comput. Mech. 40 (2007) 1005-1013. [26] T.D. Nguyen, S. Govindjee, P.A. Klein, H. Gao, A material force method for inelastic fracture mechanics. J. Mech. Phys. Solids 53 (2005) 91-121 [27] K. Ozenc, G. Chinaryan, M. Kaliske, A configurational force approach to model the 18
branching phenomenon in dynamic brittle fracture. Eng. Fract. Mech. 157 (2016) 26-42. [28] L.P. Pook, The effect of crack angle on fracture toughness, J. Eng. Frac. Mech. 3 (1966) 205-218. [29] J.R. Rice, A path independent integral and the approximate analysis of strain concentration by notch and cracks. ASME J. Appl. Mech. 35 (1968) 379-386. [30] G.C. Sih, Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fract. 10 (1974) 305-321. [31] N.K. Simha, F.D. Fischer, G.X. Shan, C.R. Chen, O. Kolednik, J-integral and crack driving force in elastic–plastic materials. J. Mech. Phys. Solids. 56 (2008) 2876-2895. [32] J. Tracy, A. Waas, S. Daly, Experimental assessment of toughness in ceramic matrix composites using the J-integral with digital image correlation part II: application to ceramic matrix composites, J. Mater. Sci. 50 (2015) 4659-4671. [33] A.A. Wells, Unstable crack propagation in metals cleavage and fast fracture. Proc. Crack Prop. Symp. 1 (1961) 210-230. [34] M.L. Williams, On the stress distribution at the base of a stationary crack. ASME J. Appl. Mech. 24 (1957) 109-114. [35] N.Y. Yu, Q. Li, Y.H. Chen, Experimental evaluation of the M-integral in an elastic-plastic material containing multiple defects. ASME J. Appl. Mech. 80 (2013) 819-833. [36] G.A. Maugin, Sixty years of configurational mechanics (1950–2010). Mech. Res. Commun. 50 (2013) 39-49. [37] A. David, E. Marcelo, R. María, Automatic LEFM crack propagation method based on local Lepp–Delaunay mesh refinement. Adv. Eng. Softw. 41 (2010) 111-119.
Table 1 The configurational forces at the crack tip by DIC and FEM for the incline crack
C-forces (J/m3) Gx Gy |G|
DIC Domain Domain Domain Domain
Ω1
Ω2
Ω3
Ω4
0.363 -0.347 0.503
0.356 -0.359 0.505
0.374 -0.342 0.507
0.362 -0.352 0.5049
FEM
Relative error
0.371 -0.354 0.509
1.81% 1.19% 0.92%
Average 0.364 -0.350 0.505
19
List of Figure Captions Fig. 1. The configurational forces at the crack tip and FEM implementation of configurational forces in the local coordinate system (e1, e2). Fig. 2. C-force criterion in the mixed mode crack problems. Fig. 3. Single central cracked beam with a hole. Fig. 4. Results of crack trajectory under (a) coarse mesh (b) fine mesh. Fig. 5. Crack growth trajectory for single central cracked beam with a hole between numerical results by C-force criterion and experimental results by Miranda et al. [23]. Fig. 6. The cracked beam with three holes subjected to three points bending (dimensions in inches). Fig. 7. Comparison of crack growth trajectory between numerical results by C-force criterion and experimental results by Ingraffea and Grigoriu [16]. (a) Specimen I; (b) Specimen II. Fig. 8. Single edge cracked plate under shear stress. Fig. 9. Crack growth trajectory of single edge cracked plate under shear stress between numerical results by C-force criterion and those by David et al. [37]. Fig. 10. Two interacting edge cracks in pane plate. Fig. 11. Comparison of crack growth trajectory for two interacting edge cracks between numerical results by C-force criterion and those by Judt and Ricoeur [18]. Fig. 12. An incline crack in elastic plate under tensile loading. Fig. 13. Prediction of critical failure load by C-force criterion compared with the experimental data [28]. Fig. 14. Experimental evaluation of configurational forces using digital image correlation. Fig. 15. The configuration and size of specimen (unit mm). Fig. 16. Digital image correlation results for the mixed-mode incline crack in plane specimen under tensile testing.(a) displacements ux and uy; (b) displacement gradients ux,y and uy,x; (c) stresses σy, σxy. 20
Fig. 17. FEM results for the mixed-mode incline crack in plane specimen under tensile testing.(a) stresses σy; (b) σxy.
21
b·n
Ω C2 ∂Ω
e2 Crack
C1 e1
Fig. 1. The configurational forces at the crack tip and FEM implementation of configurational forces in the local coordinate system (e1, e2).
uv C
C2
α
C1
x2
O
x1
Fig. 2. C-force criterion in the mixed mode crack problems.
22
37.5
50
P
P r=5.2
30 53.2
14.8
a
P
P
62.5
125 12.5
12.5
Fig. 3. Single central cracked beam with a hole.
(a) (b) Fig. 4. Results of crack trajectory under (a) coarse mesh (b) fine mesh.
23
C-force
Exper.
Fig. 5. Crack growth trajectory for single central cracked beam with a hole between numerical results by C-force criterion and experimental results by Miranda et al. [23].
P 1.25
r=0.25
2.0 8.0
2.0
4.0
b
a
1.0
Specimen I Specimen II
9.0 Crack length a 1.5 1.0
9.0
1.0
Crack offset b 5.0 4.0
Fig. 6. The cracked beam with three holes subjected to three points bending (dimensions in inches).
24
C-force
Exper.
C-force
Exper.
(a) (b) Fig. 7. Comparison of crack growth trajectory between numerical results by C-force criterion and experimental results by Ingraffea and Grigoriu [16]. (a) Specimen I; (b) Specimen II.
τ
a=3.5
7.0
Fig. 8. Single edge cracked plate under shear stress.
25
C-force
David et al (2010)
Fig. 9. Crack growth trajectory of single edge cracked plate under shear stress between numerical results by C-force criterion and those by David et al. [37].
u0
a2
a1
W =70mm
u0 . Fig. 10. Two interacting edge cracks in pane plate.
26
C-force
Judt
and
Ricoeur
Fig. 11. Comparison of crack growth trajectory for two interacting edge cracks between numerical results by C-force criterion and those by Judt and Ricoeur [18].
x σ
2
σ
0
0
β O
σ
0
x
2a
1
σ
0
Fig. 12. An incline crack in elastic plate under tensile loading.
27
1.5 1.4
C-force criterion Exper. data (Pook, 1966)
1/2
σcra /(MPam
-1/2
)
1.3 1.2 1.1 1.0 0.9 0.8 0.7
12
2
E· CR=1.797· 10 N · m
-3
0.6 20
30
40
50
60
70
80
90
β//degree
Fig. 13. Prediction of critical failure load by C-force criterion compared with the experimental data [28].
MTS
Specimen
DIC ARAMIS
Fig. 14. Experimental evaluation of configurational forces using digital image correlation.
28
Fig. 15. The configuration and size of specimen (unit mm).
29
(a)
(b)
(c)
Fig. 16. Digital image correlation results for the mixed-mode incline crack in plane specimen under tensile testing.(a) displacements ux and uy; (b) displacement gradients ux,y and uy,x; (c) stresses σy, σxy.
30
(a)
(b)
Fig. 17. FEM results for the mixed-mode incline crack in plane specimen under tensile testing.(a) stresses σy; (b) σxy.
31
Highlights 1) The material configurational forces are proposed for predicting the critical loading and the crack growth trajectory under mixed mode I/II loading. 2) The newly proposed criterion is validated through a series of examples. 3) An experimental procedure on evaluation the configurational forces is proposed using the digital image correlation.
32