The ab-initio aided strain gradient elasticity theory: a new concept for fracture nanomechanics

A BSTRACT . When the width of cracked nanocomponents made of brittle or quasi-brittle materials is less than approximately 10nm , the size of the K dominance zone becomes smaller than 2 and comparable to the fracture process zone ( -). The fracture process starts to be dominated by far-stress field terms and the critical stress intensity factor can no more represent the total fracture driving force. This means a breakdown of a classical linear elastic fracture mechanics suffering from the undesirable crack-tip stress singularity. The contribution presents a new concept expected to properly predict the critical crack driving force for nano-components: The ab-initio aided strain gradient elasticity theory (AI-SGET). In contrast to the Barenblatt cohesive model, the strain gradient elasticity theory does not require to prescribe a suitable field of cohesive tractions along the crack faces in order to eliminate the stress singularity and to exhibit cusp-like profiles of crack flanks close to the crack front in accordance with atomistic models. The only unknown and necessary quantity is the material length scale parameter which can be, e.g., determined by best strain gradient elasticity fits of ab-initio computed phonon-dispersions and near-dislocation displacement fields. Atomistic approaches can also be employed to determine fracture mechanical parameters (crack driving force, crack tip opening displacement) related to the moment of crack instability in a given material. Such AI-SGET codes can then be utilized to a successful prediction of fracture of cracked nanocomponents made of brittle or quasi-brittle materials.


INTRODUCTION
he basic hypothesis of linear fracture mechanics says that the size of the K -dominant region must be much higher than the fracture process zone incorporating inelastic deformations near the crack-tip. This gives rise to a question regarding the sufficiency of the stress intensity factor to predict the fracture toughness of nanostructures. Shimada et al. in [1] indeed showed that beyond a certain critical size of a nano-component (the specimen T width below approximately 10 15nm -), the classical continuum approach breaks down and the K -dominance zone smaller than 2 3nm becomes comparable to the fracture process zone of a constant size of about 0.5 nm where, according to atomistic models, a highly localized discrete motion of atoms takes place. The fracture process then starts to be dominated by far-stress field terms, the stress intensity factor becomes size dependent and can no more be used for characterizing the fracture toughness of cracked nano-structures. Similar behavior also exhibits the Griffith energy release rate (ERR). There are attempts in literature to capture the size dependency of the critical stress intensity factor c K using a two parameter model based on Williams's expansion but the size dependency of both c K and the critical ERR contradicts the constant critical ERR obtained from atomistic simulations [2]. It was proved in [3] that Mindlin's strain gradient elasticity theory (SGET, [4]) allows the continuum assumption to be extended beyond the limit of the classical fracture mechanics. In contrast to the Barenblatt cohesive model, the strain gradient elasticity theory does not require to prescribe a suitable field of cohesive tractions along the crack faces in order to eliminate the undesirable stress singularity and to produce cusp-like profiles of near-tip crack faces in accordance with atomistic models even when the simplest form of the SGET is used containing elastic constants and one length material parameter only. The crucial point is to identify this internal length scale parameter. This length parameter can be determined by fitting the ab initio solutions for (i) the displacement field near a screw dislocation, (ii) the critical crack tip opening displacement and (iii) the phonon dispersions using a finite element code based on the SGET. It is worth to note that all the three methods should give equivalent results. Atomistic approaches can also be employed to determine fracture mechanical parameters (crack driving force, crack tip opening displacement) related to the moment of crack instability in a given material. Coupling of these calibration procedures with the SGET-based finite element (FE) code then leads to a multi-scale approach we call the abinitio aided strain gradient elasticity theory (AI-SGET). This article is focused on a comparison of Barenblatt and SGET solutions of the stress-strain crack tip field to demonstrate a usefulness of the AI-SGET as well as its capability to remove the stress singularity and to exhibit the neartip cusp-like profile of crack flanks.

CLASSICAL VS. STRAIN GRADIENT ELASTICITY SOLUTION
he classical elasticity is not able to describe the process region at the crack tip by simply applying the equilibrium and boundary conditions prevailing at the crack front. A special model has to be used to describe the mechanical behavior inside the process region. Probably the most popular known is the Barenblatt model, see Fig. 1a. It is assumed that the Barenblatt region is developed in an isotropic elastic surrounding to be subjected to the classical linear elasticity laws. If the cohesive stress density ( ) yy x s¢ is given, then the displacements in the process region in the range 0 p x r < < for semi-infinite solid 0 y ³ are evaluated as the superposition of displacements ahead and behind the crack tip, respectively, The detailed derivation of (1) can be found in [5]. The exponent of the non-dimensional coordinate 3/2 e in (1) provides a smooth cusp-like opening of the crack face. Although the stress density yy s¢ is unknown, from the condition follows a decreasing character of the cohesion-decohesion curve as depicted in Fig A second way how to avoid the break-down of the classical fracture mechanics at the nanoscale is an application of the SGET model introduced by Mindlin. A detailed presentation of the Form II of the Mindlin's theory can be found in [8][9][10][11]. Here the strain energy density ( , ) The mode I crack tip dominant displacement field ( ) , r q u which determines the near crack tip opening displacement is of the form where 1 A , 2 A are the amplitude factors and r , q are polar coordinates. These constants are unspecified by the asymptotic analysis and must be determined by a complete solution of the boundary value problem. It should be noted that (6) is derived under the condition of the stress-free crack faces (Fig. 1b) contrary to the Barenblatt model in which the nonzero stress density yy s¢ along the crack faces is prescribed (Fig. 1a). The vectors 1 ( ; ) ijkl c q U and 2 ( ; ) ijkl c q U have a complicated form but, for an isotropic material, they can be found in [8]. Similarly to the Barenblatt solution (1) the 3/2 r variation in (6) shows a cusp-like profile of crack faces. The modelling of the process zone is the size-depended phenomena and it requires the theory taking a material length scale l into account. This length scale enters both the constitutive and the equilibrium equations and appends the dependency of the strain energy function on the gradients of the strains as one can see in (4). The consequence of this fact is a particular role of the total stress (5) in the fracture process. Its normal component yy t is depicted in Fig. 1b, where it appeares as the full-field and the asymptotic solution ahead of the crack tip having a strong singularity 3/2 r - . The total stress takes on negative values thus exhibiting a cohesive-traction character along the prospective fracture zone. As shown in [8], the asymptotic solution for the total stress yy t derived from (6)  ). It appreciably deviates from the full-field solution for the larger distances and quickly tends to zero. However, the full-field solution for the total stress yy t takes on positive values for 0.5 x l > and tends asymptotically to the limit of the classical elasticity [8]. The weakness of the SGET is the identification of the material internal length scale parameter l in (4) and (6). This crucial point can be solved by three different approaches. The first one employs the fitting of displacement fields near the crew dislocation as obtained by ab-initio (AI) approaches by the analytical SGET solution, the second one compares the critical crack tip opening displacement in the SGET FE model and the AI-aided molecular statics (MS) and the third one utilizes the fitting of the ab initio calculated phonon dispersion relations by the SGET dispersion solution. All the three methods give equivalent results, i.e., they provide identical values of the material length parameter l [12,13].

NUMERICAL RESULTS
he FE model of a center cracked nano-panel consisting of a tungsten with elastic coefficients of the cubic structure 11 523GPa c = , 12 205 GPa c = and 44 161 GPa c = is introduced as the numerical example to illustrate the cohesion relation at the crack tip. The scheme of the specimen is shown in Fig. 2. A mixed formulation of FE solution of the gradient elasticity 4th-order equations was used where instead of the 1 C displacement field it utilizes ordinary 0 C -continuous finite elements, see [14,15]. Except the available software [17], an original novel method for FE solution of the gradient elasticity 4th-order equations was developed in [16]. Detailed FE calculations of the cracked nanopanel subjected to mod I loading in terms of the SGET were carried out under the plane strain conditions. Only a quarter of the nanopanel under the remote loading 3 12 10 MPa s =´, corresponding to the moment of unstable fracture as T predicted by atomistic simulations [13], and the crack length 5.25 nm c a = were employed in the analysis. The material length scale parameter assessed from the AI-SGET analyses based on the screw dislocation and the phonon dispersions was 0.25 nm l = [13]. A fine mesh and opening of the upper face of the crack is depicted in Fig. 3. The scaled-up detail shows the cusp like opening of the near-tip crack faces.  Fig. 4. The crack face extends along the negative axis 0 x < . One can observe that the normal strains yy e take a finite value at the crack tip 0 x = contrary to the well-known classical solution, where the normal strain exhibits a square root singularity. It is also evident, that the effect of microstructure (stress gradients) is distinct only in the zone 5 x l < , according to [8]. Outside this zone the normal strains yy e and xy e tend to classical ones.
The shear strain xy e ahead of the crack tip ( 0 x > ) is zero due to the model symmetry and mode I loading, but it is non zero along the crack face ( 0 x < ). This is contrary to the classical linear elasticity result but in accordance with the CJP model taking the nonlinear effects near the crack tip into account [18]. One can suppose the pointwise convergence of xy e to the singular classical linear elasticity solution for the length scale parameter l tending to zero. The distribution of the monopolar normal stress yy s along the upper crack face as well as ahead of the crack tip is also interesting from the point of view of the cohesion-decohesion relation. Fig. 5a shows the normal stress yy s with finite value at the crack tip ( 0 x = ) copying the distribution of the normal strain yy e from Fig. 4b. The curve of yy s for 0 x < obeys the Barenblatt´s condition (2) and converges to the zero value and hence to the classical elasticity solution for 4 x l < -. Ahead of the crack tip ( 0 x > ) the convergence of yy s to the classical elasticity solution is slightly slower. The upper crack face opening is depicted in Fig. 5b, where the "closing" effect of the crack tip is obvious for 0 l x -< < . It should be emphasized that both the monopolar normal stress yy s and the displacement y u + depicted in Fig. 5 well correspond to the results obtained by ab-initio adjusted molecular static codes for cracked tungsten nanoplates [13]. Contrary to the classical elasticity, the monopolar normal stress yy s is only a partial component of the total stress yy t , which is responsible for the exhibiting of cohesive tractions along the crack faces. The full-filed total stress yy t distribution is displayed in Fig. 5c. One can deduce the strong singular character 3/2 x - of yy t and a fast degreasing influence of the stress gradients for the increasing distance x from the crack tip. For x l > the total stress yy t tends to the classical linear elasticity monopolar stress distribution yy s ahead of the crack tip.

CONCLUSIONS
he classical linear elastic fracture mechanics, suffering from the undesirable crack-tip stress singularity, breaks down when the size of cracked components becomes less than several nanometers. The article presents a new concept of coupling methods based on atomistic and continuum mechanics approaches: The ab-initio aided strain gradient elasticity theory (AI-SGET). This method is expected to properly predict both the critical crack driving force and the critical crack tip opening displacement also for cracked nano-components. Similarly to the Barenblatt model, the SGET removes the stress singularity at the crack tip and produces a cusp-like profile of crack faces near the crack tip in accordance with atomistic models. Even the simplest form of the AI-SGET involving only one unknown material lengthscale parameter l provides a reasonable form of the cohesion-decohesion stress distribution. The length-scale parameter can be determined by ab-initio methods applied to the displacement field near the screw dislocation, critical crack tip opening displacement or acoustic phonon dispersions. An important message is the fact that all these methods provide identical results. The capability of the AI-SGET method to remove the stress singularity and to exhibit a plausible near-tip cusp-like profile of crack flanks is demonstrated by the results obtained on cracked tungsten nanopanels.