A Nonlocal Model for Dislocations with Embedded Discontinuity Peridynamics
AA Nonlocal Model for Dislocations with EmbeddedDiscontinuity Peridynamics
Teng Zhao ∗ a and Yongxing Shen † a a University of Michigan - Shanghai Jiao Tong University Joint Institute,Shanghai Jiao Tong University, Shanghai, 200240, China
Abstract
We develop a novel nonlocal model of dislocations based on the frame-work of peridynamics. By embedding interior discontinuities into thenonlocal constitutive law, the displacement jump in the Volterra dislo-cation model is reproduced, intrinsic singularities in classical elasticityare regularized, and the surface effect in previous peridynamics modelsis avoided. The extended embedded discontinuity peridynamics over-comes unphysical dissipation in treating discontinuity and is still easyto be solved with the particle-based meshless method. The propertiesof the proposed dislocation model are compared with classical elasticitysolutions under the case of an edge dislocation, double edge disloca-tions, a screw dislocation and a circular dislocation loop. Numericalresults show a high consistency in displacement field while no singu-larity appears in the peridynamics model, the interaction force is inagreement with be the Peach-Koehler formula down to the core regionand high accuracy can be reached in 3D with limited computation cost.The proposed model provides a feasible tool for multiscale modelingof dislocations. Though dislocation is modeled as pre-defined displace-ment jump, it is straightforward to extend the method to model variousfracture conditions.
The physical mechanism of plasticity lies in the collective behaviors of mas-sively distributed dislocations. In mesoscale, dislocation-induced distortions ∗ [email protected] † Corresponding author: [email protected] a r X i v : . [ c s . C E ] M a y f the stress and displacement fields are fundamental to the prediction ofvarious nonlinear deformation. Through decades, dislocation models contin-uously feed a large amount of mesoscale physical simulations, e.g., crystalplasticity and dislocation dynamics[49, 30, 9, 45, 39, 1]. Compared withphenomenological constitutive models, direct simulation of solid deforma-tion with dislocations involves the microstructure evolution patterns andthus fills the gap across scales during bottom-up multiscale modeling.As a kind of lattice defects, dislocations represent irregularly arrangedatoms along a line in crystals. Since accurate stress and displacement fieldsare associated with local lattice structure in modeling dislocations, atom-istic simulation tools have shed light on capturing detailed dislocation mis-fit structure in recent years[44, 3, 57, 43, 67, 24, 55, 35], which only de-pend on lattice parameters but are free from predefined dislocation struc-ture. However, atomistic simulations including density functional theoryand molecular dynamics method meet the bottleneck of computational ef-ficiency in predicting the behaviors of large systems. For the purpose ofupscaling, one of the most promising solutions is to bridge atomistic toolswith continuum or mesoscale models together concurrently [18, 66, 65]. Theother method is to passing defect structure features into upscale modelshierarchically[2, 11, 54, 25]. Whereas, besides the intrinsic nature of mate-rial microstructures, a continuum description of dislocations is necessary forboth methods, which should share the physical interpretation of continuummechanics but also be consistent with the stress and displacement fields ofatomistic models.Generally, dislocations in the continuum scale are constructed by directlyincorporating the displacement discontinuities in solids. One basic model isto view the displacement jump as a constant equal to the Burgers vector.One such dislocation model is Volterra’s ”cut and glue” model [59]. Notwith-standing the mathematical convenience and tractability of classical contin-uum mechanics, analytical solutions of displacement and stress fields in thelinear elasticity framework are singular. Although several elegant numericalschemes were proposed to avoid singularities [4, 19, 22, 21, 31], the infiniteenergy and force resulting from singularities are still inconsistent with theatomistic models. Similar to numerical methods, the singularity can alsobe limited mathematically via introducing an artificial ”cut-off” parameter[17]. Within the framework of classical continuum theory, another categoryof attempts in removing the singularity is conducting a redistribution of theBurgers vector by energy minimization. In the well known Peierls-Nabarromodel [38, 36], the displacement field is obtained by minimization of thesum of elastic energy and stacking fault energy, which can be interpreted2s the existence of unique dislocation core structure. The dislocation coremodel is crucial in the Peierls-Nabarro model. According to constrains in theminimization procedure, standard core model [33] and isotropic core model[10] are developed. The latter provides a non-singular and self-consistentanalytical solution, available for state-of-the-art dislocation dynamics simu-lations [13, 5, 37]. The Peierls-Nabarro approaches heavily depends on thecore region definition, which is tricky to be investigated in experiments oratomistic simulations.The inconsistency between the atomistic and continuum dislocation mod-els can be attributed to the scale. In the view of bottom-up scaling, two keyfeatures of atomistic scale mechanics are distinct from classical continuummechanics: discreteness and its related nonlocality. As a lattice defect, dis-location forms in the presence of misfit interactions between atoms, yet itis necessary to highlight that the interaction in the atomistic scale is long-range. Therefore, the ignorance of nonlocality in the continuum descriptionof dislocations is doubtful. The application of generalized elasticity theo-ries in dislocations has provided promising results in removing singularities,including the Eringen’s nonlocal elasticity theory [15], gradient elasticity[26, 41, 34, 40, 61] and micropolar theories [12]. For example, in Eringen’snonlocal elasticity theory [15] the singularities in the stress field are re-moved though the singularities in the displacement field remain, suggestingthat the singularity is a result of classical continuum theory but not onlyof the structure of dislocations. Meanwhile, applications of the generalizedelasticity theories still suffer from a lack of robust solution techniques evennumerically and only recently isogeometric analysis made it hopeful [42].As alluded above, nonlocality is a key to avoid the singularities causedby dislocations. Amongst numerous generalized continuum theories, peri-dynamics is a nonlocal theory developed in the last two decades [53]. Inperidynamics, nonlocality is introduced via a reformulation of classical con-tinuum theory. Instead of partial differential equations, the governing equa-tion of peridynamics appears in an integral form to describe internal statevariables, which overcomes the singularity problems encountered in disconti-nuities and thus can be viewed as a coarse grain model upscaled from molec-ular dynamics [48]. Opposite to other coarse grain methods [64, 63, 14],peridynamics employs macroscale measurable material parameters so thatthe tricky choice of miscellaneous atomistic potential functions and othertemperature-related properties is avoided. The underlying relationship be-tween peridynamics and atomistic models suggests the potential applicationin multiscale modeling [56]. Up to now, peridynamics has shown great po-tential in modeling mesoscale defects [60] but little work has been done in3odeling dislocations. One of the main reasons is the insufficient treatmentof discontinuity. In previous studies of discontinuities with peridynamics,which mainly focused on the simulation of crack propagation, the disconti-nuities were simply assumed as the vanishing of some pairwise interactionspassing through. Unfortunately, the assumption has led to different materialproperties near the discontinuities or surface compared with the bulk part,which is called the surface effect or skin effect [27]. Corrections of the surfaceeffect near boundaries have been widely investigated and greatly improvedthe accuracy. However, the surface effect near new surfaces or internal dis-continuities is still lack of effective control [27]. In this paper, we introducean embedded discontinuity method to extend the state-based peridynamics[52] theory into simulating dislocations with Volterra’s dislocation geome-try. The state-based peridynamics model is free from the problem of fixedPoisson’s ratio in the original bond-based model. The proposed embeddeddiscontinuity method can handle dislocation induced discontinuities with-out triggering the surface effect and is well-suited in the meshless numericalframework [50]. To the authors’ knowledge, till now this is the only methodwhich can totally remove the surface effect for interior interfaces. Resultsindicate that in peridynamics theory both stress and displacement field areregularized. By introducing an interaction range parameter with clear phys-ical interpretation, the peridynamics provides a flexible framework bridgingthe atomistic models and classical continuum models.The paper is organized as follows. In Section 2, we introduce the repre-sentation of the dislocation in the continuum firstly, then the theory of thestate-based peridynamics is briefly reviewed, and a constitutive model withembedded discontinuities is derived in Section 2.3. The numerical discretiza-tion framework and solution process is the next in Section.2.4. The last partis the numerical examples for different types of dislocations, Section 3. In this section, the state-based peridynamics theory is briefly reviewed afterdefining a continuum description of dislocation, and then we give an expla-nation that why the discontinuity should be embedded in constitutive rule.Later, details about the construction method of dislocations in peridynamicsare described based on the modified Cauchy-Born rule.4igure 1: A solid body contains a Volterra dislocation mapped from refer-ence configuration to deformed configuration.
We consider a Volterra dislocation in this work. For brevity, an edge dis-location with the Burgers vector b is sketched in Fig.1. The dislocationis characterized with the core point position in 2D (dislocation line in 3D)and the glide plane. In the reference configuration B ⊂ R n , n = 2 ,
3, theglide plane of dislocation is modeled as an interface Γ inside the solid bodywhile the core is denoted as ∂ Γ. The interface Γ cuts into the solid bodyand introduces two new surfaces, denoted as Γ + and Γ − . At the time t , thematerial point X ⊂ B is mapped to the deformed configuration B t , x = ϕ ( X ) → u = x − X , (1)where u is the displacement. For a pair of conjugated material points X + ⊂ Γ + and X − ⊂ Γ − defined as X + = X − , the map creates a jump conditionacross the glide plane, x + − x − = ϕ ( X + ) − ϕ ( X − ) or (cid:74) u (cid:75) = u + − u − . (2)For dislocations, the displacement jump is constrained tangent to the glideplane and can be quantified as the Burgers vector b . In the Volterra’s model,we further assume that b is a constant for the displacement jump across theglide plane of a certain dislocation. Since the introduction of dislocationsdivided the whole domain of interest into the bulk part B and the internalinterfaces Γ, the deformation is not homogeneous, which breaks the Cauchy-Born rule and finally cause the state-based peridynamics insufficient forinterfaces. 5 .2 The state-based peridynamics The state-based peridynamics model is a general theoretical framework ofcontinuum mechanics. According to the assumption of interaction directionconstraints, the state-based peridynamics can be split into ordinary [51]and nonordinary [62] models. In this work, the framework of dislocations isdeveloped based on the linear peridynamic solids model [52], which is oneof the ordinary models and has distinguished numerical stability comparedwith the nonordinary models [16]. For an arbitrary material point X in thereference configuration B , the basic assumption of peridynamics is that anypoint X (cid:48) within a finite distance δ of X in B may exert a force upon X . Theinteraction distance is denoted as the horizon δ , and the set of interactionpoints is denoted as the neighbor of X , i.e. H x . Thus the balance law iswritten as ρ ( X ) ¨u ( X , t ) = (cid:90) H X f ( X , X (cid:48) , u ( X , t ) , u ( X (cid:48) , t ))d X (cid:48) + g ( X , t ) , (3)where ρ is the density, u is the displacement and g is the body force density. f ( X , X (cid:48) , u ( X , t ) , u ( X (cid:48) , t )) is the pairwise force density exerted on X from anpoint X (cid:48) within the horizon δ . In peridynamics, the constitutive modelingis established based on bond stretch measurement. In B , an undeformedbond is defined as ξ XX (cid:48) := X (cid:48) − X . (4)In bond-based peridynamics, the force density f between separate points X (cid:48) and X only depends on the behavior of the bond ξ XX (cid:48) . Unlike the bond-based model, the state-based peridynamics assumes that the force function f is determined by the collective bonds behavior of the neighbor. Herein, thestate is a mathematical object describing the mapping from a collection ofvariables of the neighbors to a scalar or vector-valued quantity of a specificpoint, similar to the usage of tensors in classical continuum mechanics. Thus,the concept of the state provides a tool to link the nonlocal model withclassical well studied constitutive laws. The pairwise force density exertedon X in state-based peridynamics is divided into two parts: the force vectorstate at X and X (cid:48) , f ( X , X (cid:48) , u ( X , t ) , u ( X (cid:48) , t )) = T [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) − T [ X (cid:48) , t ] (cid:104) ξ X (cid:48) X (cid:105) . (5)The underline notation here is denoted as a state. The bracket [ • ] showsthe material point at which it is defined. The angle bracket means thatit operates on the the bond ξ . In ordinary state-based model, it is further6ssumed that the force vector is collinear with the bond connecting neighborpairs in B t . The result is a force density vector pointing to x (cid:48) from x inthe deformed configuration B t . Using the deformed bond vector state, thedeformation of bond ξ XX (cid:48) can be written as, Y [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) = x (cid:48) − x . (6)Because of the collinear assumption in the ordinary state-based peridynam-ics, the force vector state can be further decomposed into a scalar-valuedforce state and a deformed direction vector state, T [ x , t ] (cid:104) ξ XX (cid:48) (cid:105) = T[ x , t ] (cid:104) ξ XX (cid:48) (cid:105) M [ x , t ] (cid:104) ξ XX (cid:48) (cid:105) , (7)where M is the deformed direction vector state, and the value is a unitvector pointing from x (cid:48) to x in B t , M [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) = x (cid:48) − x (cid:107) x (cid:48) − x (cid:107) = Y [ X , t ] (cid:104) ξ XX (cid:48) (cid:105)(cid:107) Y [ X , t ] (cid:104) ξ XX (cid:48) (cid:105)(cid:107) . (8)Compared with other upscaling models from molecular dynamics, an impor-tant advantage of peridynamics is to incorporate classical continuum con-stitutive models. The calibration of the scalar force vector in peridynamicsutilizes the strain energy density of classical continuum models. Since themodeling of dislocations is in the mesoscale, the material is assumed to beelastic. The deformation of a specific material point is measured by theextension scalar state, defined as e [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) = (cid:107) x (cid:48) − x (cid:107) − (cid:107) X (cid:48) − X (cid:107) , (9)or using the state notation e [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) = y [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) − x [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) . (10)Here, y [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) and x [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) are the magnitude of Y [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) and X [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) respectively. For brevity, the [ • ] and (cid:104)•(cid:105) parts are ne-glected in the following contents, and it refers to [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) by default.In peridynamics, the common way for deriving the constitutive relation isvia the definition of a strain energy density function W ( e ), and the scalar-valued force state is expressed as the Frechet derivative of strain energydensity, T = ∇ W ( e ) . (11)However, the definition of W in previous literature depends highly on anintact spherical neighbor, which leads to the surface effect when the discon-tinuities exist in the neighbor. In the next part, we directly find a nonlocalstrain energy density function for solid bodies containing interior disconti-nuities instead of explicit penalty methods, as reviewed by (author?) [27].7 .3 Constitutive modeling with embedded discontinuity method In the previous work [51, 28], the Cauchy-Born rule is used to build a connec-tion between classical local elasticity and the nonlocal system. Though theCauchy-Born rule has made a great impact in multiscale modeling, certainshortages do exist. The drawbacks of the Cauchy-Born rule root in the ba-sic hypothesis of uniform deformation field [32]. In the ordinary state-basedperidynamics, the application of the Cauchy-Born rule must be under theconstraints of homogeneous deformation in order to reproduce the strain en-ergy density of the corresponding local system. As a kind of inhomogeneousdeformation, the occurrence of interior discontinuity shall break the energyconservation. Especially in the previous practice of bond-break modelingof fracture, additional energy dissipation will be brought besides fractureenergy, finally leading to an ambiguous (author?) [27] crack pattern. Here,we directly start with the modification of the Cauchy-Born rule accountingfor interior discontinuities and later apply the modified Cauchy-Born ruleto build a nonlocal strain energy function.
By assuming a homogeneous small deformation, the standard Cauchy-Bornrule for a material point X with a spherical neighborhood is expressed as, F X ξ XX (cid:48) = x (cid:48) − x . (12)Here F denotes the deformation gradient tensor in classical continuum me-chanics, F = I + ∇ u and I is the identity tensor. It shall be noted that theCauchy-Born rule requires a smooth enough deformation gradient field inthe nonlocal theory. Particularly in the ordinary state-based peridynamics,the nonlocal interaction also requires F X (cid:48) ξ X (cid:48) X = x − x (cid:48) . (13)Given that ξ X (cid:48) X = − ξ XX (cid:48) , combining Eq.12 and Eq.13, the following musthold, ( F X − F X (cid:48) ) ξ XX (cid:48) = 0 ∀ X (cid:48) ∈ H X . (14)Therefore, the above condition would work within acceptable errors only ina small and affine deformation field, F X ≈ F X (cid:48) or F X = F X (cid:48) . (15)The ordinary state-based peridynamics is built upon the above assump-tion. In other words, the Cauchy-Born rule is based on a small homogeneous8eformation field. For inhomogeneous or finite deformation condition, Eq.12and Eq.13 break down. In the view of displacement discontinuity, the dis-location is a special form of inhomogeneous deformation. In order to modeldislocation, we may assume the Cauchy-Born rule is still valid for materialpoints whose neighbor is not cut by the glide plane. For material points nearthe glide plane, it is also assumed that the Cauchy-Born rule is workablefor bonds not intersecting the glide plane. But for bonds intersecting theglide plane, the standard Cauchy-Born rule need modifications to recoverthe deformation and strain energy defined in classical elasticity. As shownFigure 2: Incompatibility for interaction bond crossing discontinuity. Thedashed line is the glide plane.in Fig.2, consider a pair of adjacent points X ± lying on two sides of theglide plane Γ ± in B respectively and its corresponding position x ± in B t ,the jump condition induced by dislocations is described with the Burgersvector b , expressed as, b = x + − x − . (16)Apply the standard Cauchy-Born to bond ξ XX − and ξ X (cid:48) X + , (cid:40) F X (cid:48) ξ X (cid:48) X + = x + − x (cid:48) F X ξ XX − = x − − x (17)Combine Eq.17, F X ξ XX − − F X (cid:48) ξ X (cid:48) X + = x (cid:48) − x − b . (18)Given that ξ X (cid:48) X = − ξ XX (cid:48) , and apply the homogeneous deformation condi-tion Eq.15, the modified Cauchy-Born rule with discontinuity can be written9s F X ξ XX (cid:48) = x (cid:48) − x − b . (19)The modification still assumes a small affine deformation field, and by in-troducing the prescribed displacement jump, the Cauchy-Born rule is stillvalid. Eq.19 can also be interpreted that x + and x − are replace by x , b and x (cid:48) , relaxing the discontinuity to a nonlocal region. Similar with the classical mechanics, the nonlocal strain energy density W of the linear peridynamics solid consists the volume part and the distortionpart W (cid:16) θ, e d (cid:17) = k (cid:48) θ α (cid:16) ωe d (cid:17) • e d . (20)Here, k (cid:48) and α are material constants to be calibrated. The notation • isthe dot product between two states (cf. (author?) [51]). ω denotes theinfluence function which only depends on the magnitude of ξ , and in thiswork we employ the polynomial form proposed by [47], ω ( | ξ | ) = − (cid:16) | ξ | δ (cid:17) + 84 (cid:16) | ξ | δ (cid:17) − (cid:16) | ξ | δ (cid:17) + 20 (cid:16) | ξ | δ (cid:17) , | ξ | (cid:54) δ, , otherwise . (21)Corresponding to the classical mechanics, in Eq.20 the extension scalar state e is divided into two states for the deformation measurement: the nonlocalvolume dilatation θ and the deviatoric extension state e d , e d = e − θx .The modified Cauchy-Born rule gives an effective way to incorporate aprescribed discontinuity into the modeling process. Similar approach canbe found in (author?) [32, 58]. In the view of nonlocal interaction, themodified Cauchy-Born rule can be understood as interface-induced bond re-fraction. A bond intersecting the glide plane is shown in Fig.3. It indicatesthat the dislocation-induced discontinuity breaks the neighborhood into twosectors, and a relative slip exists in the interface. The modified Cauchy-Bornrule can be viewed as shifting the upper sector back to rebuild the conti-nuity of the neighbor. It suggests that the implementation of the modifiedCauchy-Born rule in peridynamics is simple, i.e. an embedded-discontinuitydeformed bond vector state, Y (cid:48) [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) = Y [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) − (cid:88) b α G α ( X , X (cid:48) ) , (22)10igure 3: Illustration of embedded-discontinuity bond. Dashed line: glideplane. The modified Cauchy-Born rule is viewed as the slip of the bondintersecting the glide plane.where G α is used for checking the intersection between line segment XX (cid:48) and the glide plane Γ of the α th dislocation, defined as, G α ( X , X (cid:48) ) = (cid:40) XX (cid:48) ∩ Γ α (cid:54) = ∅ , . (23)Similarly, the embedded-discontinuity extension scalar state is expressed as e (cid:48) = | Y (cid:48) | − | X | . (24)The extension scalar state is used for the description of bond elongations.According to the modified Cauchy-Born rule, the length change of a bond ξ is expressed as e (cid:48) = | F ξ | − | ξ | = 1 | ξ | ε ij ξ i ξ j , (25)where ε is the infinitesimal strain tensor in classical continuum mechanics.So far, the nonlocal dilatation θ is given by directly relating to classicalvolume dilatation. Since the only difference is to replace e with e (cid:48) betweenEq.25 and the original work in (author?) [51, 28], the modified nonlocal11ilatation θ (cid:48) is expressed as θ (cid:48) = m ( ωx ) • e (cid:48) in 3D , ν − ν − m ( ωx ) • e (cid:48) in plane stress , m ( ωx ) • e (cid:48) in plane strain . (26)Here, ν is the Poisson’s ratio, and m is the weighted volume defined as ωx • x . For the deviatoric part of the extension state, a similar equation canbe written, e (cid:48) d = e (cid:48) − θ (cid:48) x . (27)Till now, the only difference between the embedded discontinuity model andlinear peridynamics model is to replace e with e (cid:48) . Thus deformation energyEq.20 can be expressed with respect to e (cid:48) and θ (cid:48) , W (cid:48) (cid:16) θ (cid:48) , e (cid:48) d (cid:17) = k (cid:48) θ (cid:48) α (cid:16) ωe (cid:48) d (cid:17) • e (cid:48) d , (28)where k (cid:48) and α are material parameters, as calibrated in [51, 28], k (cid:48) = k in 3D ,k + µ ν + 1) (2 ν − in plane stress, k + µ .α = µm in 3D , µm in plane stress or plane strain . Here k is the bulk modulus. By Frechet derivate of W with respect to e (cid:48) ,the modified scalar force state T (cid:48) is written as∆ W = k (cid:48) θ (cid:48) ( ∇ e (cid:48) θ (cid:48) ) • ∆ e (cid:48) + α ( ωe (cid:48) d ) • ∆ e (cid:48) d = T (cid:48) • ∆ e (cid:48) . (29)12ubstitue Eq.27 and Eq.26 into Eq.29,T (cid:48) = k (cid:48) θ (cid:48) ωxm + αωe (cid:48) d in 3D , ν − ν − k (cid:48) θ (cid:48) − α ωe (cid:48) d ) • x ) ωxm + αωe (cid:48) d in plane stress , k (cid:48) θ (cid:48) − α ωe (cid:48) d ) • x ) ωxm + αωe (cid:48) d in plane strain . (30)The force vector state is modified as T [ x , t ] (cid:104) ξ XX (cid:48) (cid:105) = T (cid:48) [ x , t ] (cid:104) ξ XX (cid:48) (cid:105) M (cid:48) [ x , t ] (cid:104) ξ XX (cid:48) (cid:105) , (31)and the modified deformed direction vector state is, M (cid:48) [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) = Y (cid:48) [ X , t ] (cid:104) ξ XX (cid:48) (cid:105)(cid:107) Y (cid:48) [ X , t ] (cid:104) ξ XX (cid:48) (cid:105)(cid:107) . (32) Remark.
In the classical practice of fracture modeling with peridynamics,the crack type strong discontinuity is represented as the ”break” of the bond.The failure criterion is expressed with the stretch ratio of the bond, simpleand effective. However, after bond-breaking the standard Cauchy-Born ruleis unable to reproduce the strain energy of the remaining part, where theneighbor is not spherical. The surface effect [27] appears in the previousbond-based and ordinary state-based peridynamics is the result of such unde-sired energy loss. Instead, the embedded discontinuity peridynamics directlymodifies the interaction force for bonds crossing the discontinuity. Thus theembedded discontinuity method is a conceptually different approach. Themodification in energy can be explained as a superposition of the strain en-ergy density created by the perfectly smooth displacement field and the dis-sipation energy induced by the dislocation interface. In the sense of force,the model can also be understood as the mechanism that additional forcestates distributed along the glide plane are added as body force to force thedisplacement jump at the magnitude of the Burgers vector.
Due to the nonlinear nature of peridynamics, analytical solution can seldombe found. Hence in this work we simulate dislocations by the particle-basedmeshless numerical approach [50]. The domain is discretized with equal-spaced nodes, as shown in Fig.4. Thus the total force acting on a node canbe calculated with Riemann sum, written as, F ( X , t ) = (cid:88) H X f ( X , X (cid:48) , u ( X , t ) , u ( X (cid:48) , t ))V X (cid:48) + g ( X , t ) . (33)13ere V X (cid:48) is the volume of each node. For the partially covered volumeshown in Fig.4, the IPA-HHB algorithm [8, 46] is employed to improve theaccuracy. The boundary conditions in peridynamics has the same physicalFigure 4: Illustration of discretization. Red: partial volume. Blue: fullvolume.meaning as in classical elasticity. The traction boundary condition is appliedby reproducing the flux through boundaries and the displacement boundarycondition is applied by setting a constant value. The difference is thatthe boundary condition is applied to a layer but not a lower dimensionalgeometry. In this work, the boundary conditions are applied to a layerof nodes whose width equals to the horizon δ . The fictitious boundarylayer method [7, 27] is used to determine the displacement value for thefictitious nodes. The velocity Verlet time integration and the fast inertialrelaxation engine method [6] are used for solving static solution. Detailsof the implement is shown in Algorithm.1. In this paper, we select theparameters for FIRE algorithm as n min = 5, γ = 0 . f γ = 0 . f dec = 0 . f inc = 1 . (author?) [29], widely misuse and meaningless compar-isons with the Cauchy stress do exist since the peridynamic stress is corre-sponding to the Piola stress. Here, we use the mechanical part of the virialstress formula as the equivalent Cauchy stress measurement in the embeddeddiscontinuity peridynamics model, expressed as σ ( X ) = 12 (cid:90) H X f ( X , X (cid:48) , u ( X , t ) , u ( X (cid:48) , t )) ⊗ Y (cid:48) [ X , t ] (cid:104) ξ XX (cid:48) (cid:105) d x (cid:48) . (34) Algorithm 1
Static solver
Require: X : node position; ∆ t : time step size; n min , γ , f γ , f dec , f inc :control parameters; k , ν : material parameters; Ensure: optimal u initial γ = γ and ∆ t ; while not converged do apply the boundary condition update the position x ( t + ∆ t ) = x ( t ) + v ( t )∆ t + a ∆ t ; compute the internal force density F ( t + ∆ t ) = (cid:80) H X f ( t + ∆ t )V X (cid:48) ; update the acceleration a ( t + ∆ t ) = F ( t + ∆ t ) /ρ update the velocity v ( t + ∆ t ) = v ( t ) + ( a ( t ) + a ( t + ∆ t ))∆ t ; compute the P = F · v ; adjust the velocity by F N = F / (cid:107) F N (cid:107) ; v → (1 − γ ) v + γ F N | v | ; if P > n > n min then ∆ t → min(∆ tf inc , ∆ t max ); γ → γf γ ; n = n + 1; else v →
0; ∆ t → ∆ tf dec ; γ → γ ; n = 0; end if end while A single edge dislocation is considered as benchmark to validate the method.For an infinite domain, the analytical solutions with the Burgers vector15igure 5: Illustration of the simulation domain for an edge dislocation b = [ b, ,
0] in the classical elasticity are given in (author?) [20], u x = b π (cid:34) arctan yx + xy − ν ) (cid:0) x + y (cid:1) (cid:35) ,u y = − b π (cid:34) − ν − ν ) ln (cid:16) x + y (cid:17) + x − y − ν ) (cid:0) x + y (cid:1) (cid:35) . (35)The geometry of the simulation domain is a two dimensional square with L =10 − m, and the core of an edge dislocation with b = 8 . × − m is placedat the origin, as shown in Fig.5. The Young’s modulus and the Poisson’sratio are 1 . × Pa and 0 .
34, respectively. The plane strain conditionis assumed. For mimicking an infinite domain, the displacement solutionEq.35 in classical elasticity is applied to the fictitious boundary layers. Toanalyze the effectiveness of meshless discretization, N × N particles areequally distributed in the domain, and the convergence can be checked withcharacter number M = δN /L .In Fig.6 and 7, the stress and displacement obtained by setting N = 500and M = 3 .
15 are compared with the classical elasticity solution. Obviously,the surface effect common in previous fracture studies [27] disappears. Inthe proposed embedded discontinuity method, the bulk strain energy is to-tally reproduced since all bulk points have a full horizon. The displacementresults of peridynamics match well with the classical elasticity, and only aslight difference exists between the stress components. Besides negligible16 a) (b)(c) (d)
Figure 6: Displacement field induced by an edge dislocation with embeddeddiscontinuity peridynamics (a)(c) and classical elasticity(b)(d).numerical issues, it proves that the nonlocal interaction, or the horizon δ really redistributes the stress field. The differences of stress in Fig.7 aremainly due to three reasons: nonlocality, stress definition and numericalerrors. Nonlocality in peridynamics introduces a difference in the solutionof displacement field while δ >
0. The difference between peridynamics andclassical elasticity converges to zero when δ approaches zero, which is called δ convergence. Compared with the displacement results in Fig.6, it is ob-vious that the discrepancy mainly exists in stress results. In this work themechanical part of virial stress is used as an measurement of the Cauchy17 a) (b)(c) (d)(e) (f) Figure 7: Stress field induced by an edge dislocation with embedded discon-tinuity peridynamics (a)(c)(e) and classical elasticity(b)(d)(f)18tress, which is equivalent but still influenced by the introduction of δ .Figure 8: Displacement jump along the glide plane. Values on the horizontalaxis refer to the X coordinate in Fig.6.In Fig.8, the effectiveness of the embedded discontinuity peridynamicsin capturing the discontinuity jump is validated. As δ decreases, the dis-placement jump curve approaches the classical solution, which confirms the δ -convergence of peridynamics. Besides perfectly recovering the Burgersvector for most regions, in the near-core region the displacement jump orrecovered Burgers vector is gradually decreasing. It also appeared in thedislocation model with the gradient elasticity theory [41]. The phenomenonindicates a redistribution of the Burgers vector with embedded discontinuityperidynamics in the near core region, conceptually similar to the result withthe Peierls-Nabarro model and related non-singular theory [10].Fig.9 shows the influence of horizon on the distribution of the stress com-ponent σ xx . A series of δ was implemented in numerical simulations with afixed N = 500 but varying M . The stress σ xx is only plotted for the nearcore region along the line segment from (0 , − × − m ) to (0 , × − m ).Compared with classical elasticity, the singularity is avoided with the embed-ded discontinuity peridynamics. The stress curves obtained with different δ are finite but diverges around the core position. As δ →
0, the stress curve19igure 9: σ xx induced by an edge dislocation. Plotting along the line seg-ment from (0 , − × − m ) to (0 , × − m ) is shown. Values on thehorizontal axis refer to the X coordinates in Fig.7.near the core position gradually rises towards the classical elasticity solu-tion, which is usually described as δ -convergence. As an important featureof peridynamics, we characterize the δ -converge rate with the relative L norm between the displacement u num obtained with the embedded discon-tinuity peridynamics and the analytical solution u local in Eq.35, expressedas D u = (cid:107) u h − u (cid:107) (cid:107) u (cid:107) = (cid:115) (cid:82) B ( u num − u local ) T ( u num − u local ) d B (cid:82) B ( u local ) T ( u local ) d B . (36)Fig.10 shows the δ -convergence of displacement field. With the embeddeddiscontinuity model, the difference of the displacement field between classicalelasticity and peridynamics is relatively small, but a rapid decrease is stillshown as δ →
0. In the embedded discontinuity model, the reproduction ofthe bulk part of strain energy is guaranteed. Besides avoiding the surfaceeffect, the δ -convergence rate is kept constant even with the presence ofdiscontinuity. 20igure 10: Relative displacement difference between embedded discontinuityperidynamics and classical elasticity, calculated with Eq.36. In this section, the interaction between two edge dislocations is considered.To compare with Section 3.1, we choose the same set of material and dislo-cation parameters. In geometry, one edge dislocation is still placed at theorigin while the other is at ( L x , L y ). In discretization, we choose N = 500and M = 3 . L y =2 × − m but three different L x are shown. As seen from the figures, boththe stress and the displacement field agree well with the classical elasticityresult, and a further quantitive measurement of D u shows D u = 0 . ∼ .
33% for all L x = 0 ∼ × − m. The results suggest that the applicationof embedded discontinuity peridynamics in multiple dislocations is feasible.Another issue on the multiple dislocations modeling is the interaction force,or driving force in dislocation dynamics. In classical elasticity, the drivingforce F on unit dislocation line segment is defined as the negative derivativesof elastic strain energy E with respect to the coordinates x , F = − ∂E∂ x . (37)On the other side, the driving force is also consistent with the Peach-Koehler21 a) u x at L x = 0 m (b) u y at L x = 0 m(c) u x at L x = 1 × − m (d) u y at L x = 1 × − m(e) u x at L x = 2 × − m (f) u x at L x = 2 × − m Figure 11: Displacement field induced by two edge dislocations, unit: m.22 a) σ xx at L x = 0m (b) σ xx at L x = 1 × − m (c) σ xx at L x = 2 × − m(d) σ xy at L x = 0m (e) σ xy at L x = 1 × − m (f) σ xy at L x = 2 × − m(g) σ yy at L x = 0m (h) σ yy at L x = 1 × − m (i) σ yy at L x = 2 × − m Figure 12: Stress field induced by two edge dislocations, color bar unit: Pa.formula, F = ( σ · b ) × ξ , (38)where ξ is a unit vector tangent to the dislocation line, and σ is the Cauchystress.Here, we compute the driving force numerically by three methods, NLPK
The nonlocal Peach-Koehler force is calculated via substituting thenonlocal stress defined in Eq.34 to Eq.38, and the nonlocal stress isinterpolated at the corresponding position utilizing the numerical datafrom Sec.3.1. 23igure 13: Driving force on dislocation line.
LPK
The local Peach-Koehler force is calculated with the Peach-Koehlerformula and the analytical solution in classical elasticity. EG The energy gradient method is calculated with Eq.37 using secondorder accurate central differences respect to L x numerically. A series ofsimulations was performed by setting L y = 0m and L x = 2 n × − m, n = 0 , · · · L x →
0. The embedded peridynamics solutionavoids the singularity in energy, thus the interaction force is also finite. TheNLPK is inconsistent with the EG as L x →
0. Consider the δ = 31 . × − m, the inconsistency indicates the behavior of dislocation core is described in24he embedded discontinuity peridynamics model, and the near core regioninteraction is failed to be described with the Peach-Koehler formula. Itshould also be noted that the interaction between edge dislocations withinthe dislocation core distance is not fully understood in literature so far. Thusthe potential use of the embedded discontinuity peridynamics is not limitedto dislocation dynamics with the Peach-Koehler formula but can also beextended to study the core region behavior. Figure 14: Illustration of the simulation domain for a screw dislocationTo validate the 3D condition, a straight screw dislocation in an infinitedomain is considered. The domain geometry is shown in Fig.14. The domainis a cube with edge length L = 10 − m, and dislocation line coincides withthe Y axis. The Burgers vector b = [0 , , b ] is selected as b = 8 . × − m.The Young’s modulus and the Poisson’s ratio are 1 . × Pa and 0 . M = 3 .
15 but different N in discretization.The displacement and stress field caused by the screw dislocation with N = 80 is presented in Fig.15. Only nonzero components are shown here.All results are independent of the z coordinate by examining an arbitraryslice normal to the Z-axis and are in good accordance with the classical elas-ticity solution [20]. An in-depth comparison is preformed via plotting thedisplacement and stress components along the selected line segment withdifferent δ , as shown in Fig.16. Compared with the classical elasticity solu-tion, the singularity in σ xz vanishes for all δ with embedded discontinuity25 a) u z (b) σ xz (c) σ yz Figure 15: Displacement and stress field induce by a screw dislocation, (b)and (c) describes the stress field of a slice plane at z = − . × − mperidynamics, Fig.16a. Similar to the stress of edge dislocation in Fig.9, thestress curve passes through the origin and also gradually converges to theclassical elasticity solution as δ →
0. Thus the δ convergence is confirmed.The phenomenon is also observed in the nonsingular theory by (author?) [10] and gradient elasticity [26]. In all the above models, the regularizationof singularity is indeed by the redistribution of local energy into a nonlocalrange. The Peierls-Nabarro type model constrains the redistribution to theglide plane or jump condition while the nonlocal models including the peri-dynamics extend it to the whole domain. The embedded discontinuity peri-dynamics can also be proved to be effective by examining the displacementfield in Fig.16b. Although the stress field is regularized, a high degree ofconsistency is still maintained for the displacement field. The δ convergence26 a) σ xz (b) u z Figure 16: Displacement and stress induced by a screw dislocation. (a)stress, plotted from (0 , − , × − m to (0 , , × − m and (b) displace-ment, plotted from (2 , − , × − m to (2 , , × − m. Values on thehorizontal axis of both subfigures refer to the Y coordinate in Fig.14.27able 1: Relative displacement difference with different horizonHorizon ( × − m) M N D u (%)15.5 3.15 20 0.53277.8 3.15 40 0.28575.2 3.15 60 0.19803.9 3.15 80 0.1525can also be seen in local enlarged subfigures inside Fig.16b. Quantitively,the relative displacement difference keeps decreasing as δ →
0, shown inTable 1. Remarkably, the accuracy is achieved with very rough discretiza-tion N for reducing computation cost. Thus the embedded discontinuityperidynamics model is also less sensitive in discretization. Besides straight dislocations mentioned above, the last case is a curved dis-location in an infinite domain. Fig.17 shows the geometry of the simulationdomain, a cube with edge length L = 1 . × − m. A circular disloca-tion loop with radius R = 3 × − m is placed in the XY plane, and theBurgers vector is set as b = [ b, ,
0] with b = 2 . × − m. The Young’smodulus and the Poisson’s ratio are 1 . × Pa and 0 .
34, respectively. Weapply the displacement solution in the classical elasticity to the boundaries.The classical elasticity solution is given in (author?) [20] and is numeri-cally solved with adaptive integration. The horizon δ is 4 . × − m, andthe discretization parameters are N = 90 and M = 3 .
15. Fig.18 showsthe displacement induced by the circular dislocation loop. A displacementjump corresponding to the Burgers vector is clearly revealed in Fig.18a. Thestress field is presented in Fig.19. No significant difference appears in bothfields compared with literature results [23]. Since the stress and displace-ment field created by circular dislocation is complex, a quantitive analysisis performed by plotting the numerical solution together with the classicalelasticity solution along the line segment ( − L/ , L y , L z ) to ( L/ , L y , L z ), asshown in Fig.20. We sampled three line segments parallel to the X-axis byfixing L y and adjusting L z . It can be seen from Fig.20a that the displace-ment showing no difference between embedded discontinuity peridynamicsand the classical elasticity, in line with the presented results of the screwdislocation. However, even though for line segments far from the glide planethe stress curves obtained by embedded discontinuity peridynamics still fit28igure 17: Illustration of the simulation domain for a circular dislocationloopin well with the classical elasticity solution, differences exist along the linesegment L z = 0 . × − m. The result may be explained by the nonsingu-lar solution with embedded discontinuity, while another likely cause for thedifference is the discretization. Similar errors also appear in XFEM mod-eling of dislocations [19]. The discretization utilizes N = 90, a rough grid,by uniform node distribution, in which the loop curve is not considered, asshown in Fig.4, neither the volume V X is corrected. Consider the horizon is δ = 0 . × − m, the occurrence of above numerical fluctuation could beexplained as numerical errors in the modified Cauchy-Born rule. Therefore,the problem can be settled by refining the grid or decreasing the horizonsize. Apart from the above numerical drawbacks, the result still shows thepotential application value in modeling complex dislocations. In this paper, a nonlocal continuum framework of dislocations based on thestate-based peridynamics has been constructed. Contrast to the previouspractice of peridynamics in modeling of fracture-like discontinuity, the dis-location induced displacement discontinuity is embedded in the nonlocalconstitutive model utilizing a modified Cauchy-Born rule.This approach ex-tends the limits of the standard Cauchy-Born rule and avoids the surfaceeffect which hinders the application of peridynamics. More broadly, the29 a) u x (b) u y Figure 18: The displacement field induced by a circular dislocation loop,unit: Pa. Slice position:(a) x = 2 . × − m , y = − . × − m ;(b) x =1 . × − m , y = − . × − m .energy conservation between local and corresponding nonlocal continuum30 a) σ xx (b) σ xy Figure 19: 3D contour and slice plane of the stress field induced by a circulardislocation loop, unit: Pa. Slice position: z = 4 . × − m a) u z (b) σ xz Figure 20: Comparision of the stress and displacement with classical elas-ticity solution. For all lines, L y = 6 . × − m. Values on the horizontalaxis of both subfigures refer to the X coordinates in Fig.17.32heories is enhanced for both intact and damaged media with evolving dis-placement discontinuity. Compared with other dislocation models, the ap-proach in this paper is capable of describing different types of dislocationswithout introducing additional parameters without clear physical meaning.The introduction of nonlocality in peridynamics is via relaxing the strainenergy density to a finite range via the horizon and the influence function.Clear meaning makes it possible for future applications of multiscale mod-eling. Though not included in this paper, fitting the influence function inthe continuum is a promising way of modeling dislocation cores of differentcrystal lattices. For verification, we examined different types of dislocationsand the interaction between a pair of dislocations numerically. Surprisingly,singularities in the classical dislocation theories are regularized while onlysubtle distinction exists in the displacement field. We conclude the mainfindings as follows, • The concept of the Volterra dislocation can be perfectly reproducedwith the embedded discontinuity peridynamics. As a pre-describeddisplacement jump, the reproduced Burgers vector matches the definedone along the glide plane except the core region. For the near coreregion, the Burgers vector smoothly decreases to zero. • A benefit from the embedded treatment of interior discontinuities, sur-face effect is avoided for all cases. The interior surface effect disappearswithout any additional tracking or penalty. • The stress solutions are nonsingular for both the edge and screw dis-locations. The stress field near the dislocation core is in the samepattern as the nonsingular theory [10] and the strain gradient solu-tion [61]. As the decreasing of horizon, the embedded discontinuityperidynamics solution will converge to the classical elasticity solution. • The displacement field computed with the embedded discontinuityperidynamics reaches an extremely high accuracy towards the clas-sical elasticity. Though rough discretization was utilized for 3D cases,an accurate match between the classical elasticity and the nonlocalmodel is still reached. • The consistency of the driving force is guaranteed outside the corerange, but the Peach-Koehler formula is not valid for computing dislo-cation driving force inside the dislocation core region. It indicates themesh refinement is necessary for a low order discretization method and33he particle arrangement in the present meshless method need morecomprehension. • Numerical instability occurs in the near core region for 3D curveddislocations. Apart from the non-singularity nature, it also indicatesthat the grid discretization in the particle based meshless method needcareful rearrangement or refinement.The work is the first step towards a multiscale dislocation dynamicsframework. Currently, the Volterra type dislocation is modeled as a pre-described discontinuity, but the approach is opening doors for modeling un-known discontinuities such as complex fracture propagation modeling. Sincethe embedded discontinuity method is built upon the modified Cauchy-Bornrule, the extension for complex constitutive modeling is straightforward.Though the current study is limited to linear elasticity, it is also possible toincorporate nonlinear elasticity for capturing the complex material behaviornear dislocation cores.
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