The microstructure of crystal defects, e.g., dislocation patterns, are not arbitrary, and it is possible that some of them may be related to the microstructure of crystals itself, i.e., the lattice structure. We call those dislocation patterns or substructures that are related to the corresponding crystal microstructure as the

Crystal plasticity is a very common physical phenomenon. The origin of crystal plasticity is crystal dislocation motion and evolution. However, the crystal plasticity is resulted from aggregated dislocation motions, i.e., motion of dislocation pattern, rather than the motion of a single dislocation. It is because not only dislocation itself but also mutual interaction and evolution of assemble dislocations influence plastic behavior in crystalline materials. Therefore, the massive number of dislocations assembling in some specific regions in crystalline materials so that studying statistical variables from the aggregation is the only feasible way.

Based on various topological grouping of dislocations, some dislocation patterns have specific names. For example, we usually name the “wall-shape” dislocations as dislocation cells, the “strip-shape” dislocations as veins and mutually crossed “grid-shape” as dislocation labyrinth. According to past studies on crystal plasticity, many plastic phenomena in crystalline materials are strongly influenced by those micro-defects.

Recently, Li et al. [^{k}

Tantalum, as lustrous transition metal, is a highly corrosion-resistant material, and it is widely used as minor components in many alloys, as part of the refractory metals group. In current industry applications, tantalum is mainly used in following four fields: capacitors, chemical industry, alloy and hard metals [

In present work, we employed the multiscale crystal defect dynamics, or the geometrically-compatible dislocation pattern dynamics, to investigate the crystal plasticity of a special BCC crystal–

The rest of the paper is organized as follows: based on algebraic topology, construction of crystal lattice-dependent mesh and resemblance between the physics-informed mesh and real dislocation patterns will be discussed in

In this section, we first introduce the dual lattice tessellation to partition a BCC crystal into a finite element mesh of lattice process zones, which was proposed in [

In discrete dislocation dynamics (DD), the smallest unit of dislocation dynamics is the dislocation segment [

To identify the generic dislocation pattern for a given lattice structure, we need first to construct a Voronoi cell for an representative atom: placing planes normal to line segments formed by two neighboring atoms of lattice complex at midpoints. These truncation planes will form a convex polyhedron around the representative atom, and the convex polyhedron is called the Wigner-Seitz cell in crystallography for Bravais lattices. Using the terminology of crystal chemistry, we may call it as the Voronoi-Dirichelt Polyhedron (VDP). However, VDP is a bigger set of unit cells that includes the Wigner-Seitz cell. This is because different choice of representative atoms will lead to different shape of VDP. For our current BCC model, we choose the nearest 8 atoms and the second nearest 6 atoms as representative atoms. Therefore, the corresponding VDP is a truncated octahedron or the standard Wigner-Seitz cell for the BCC lattice as shown in

The complete steps to have dual lattice process zone tessellation:

(1) Based on crystal lattices of whole atoms in 2D or 3D space, we can have correspondingdual lattice;

(2) Then we shall scale the VDPs down, put them at atom positions and term it as the highestorder process zone element;

(3) It is found that the whole two-dimensional or three-dimensional space cannot be coveredby the scaled VDP. Therefore, corresponding different shape of elements are used to fill inthe gaps among scaled VDP. Therefore, for BCC crystal lattice,

Therefore, in a BCC dual-lattice unit, we have (1) one 3^{rd}^{nd}^{st}

Inspired from Ariza et al. [

Different from Ariza and Ortiz's work, the present work is about dislocation pattern dynamics, and we cannot use simplicial complex p-cell to describe discrete dislocation pattern units. However, we still found that the mathematical theory of algebraic topology and exterior differential calculus can also applied to describe dislocation pattern dynamics in terms of p-cell of CW complexes. In this work, we introduce following cell index system to label points, edges, faces and volumes in different discrete dislocation pattern units, i.e., truncated octahedron elements, prism elements, wedge elements and tetrahedron elements. We use a unified symbol,

In the next we shall take

This labeling process is important in finite element method (FEM) formulation and implementation, because we need to use the exterior calculus symbol to construct the connectivity array of FEM code that maps the local indices into global indices.

By using boundary operator of exterior calculus, we can represent lines and faces. For example, as shown in

Similarly, we can use co-boundary operator

Furthermore, we can also apply the cell complex notation defined above to represent other types of elements. For example, the prism element _{i}

In Lyu et al.’s work [

Since we know that body-centered cubic (BCC) crystal structure is NOT the closed-packed structure, the most likely slip plane should be

From the projection from 3D to 2D, we understand that 2D triangle bulk element is actually the projection of the first-order process zone element in 3D DLPZ tilling, rather than the projection of 3D tetrahedron bulk crystal element; the projection of the second-order process zone element (prism element) in

Since we have the two-dimensional geometrically-compatible dislocation pattern as in

The embedded-atom model (EAM) is widely used atomistic potential model for metals in molecular dynamics, which takes into account interactions between atom nucleus as well as electron density contribution. The EAM method has been extensively used in molecular dynamics for metallic materials,

Based on general formulation of EAM [_{ij}_{ij}_{j}_{i}

Based on EAM formulation and stress-work conjugate relation, we can have a fourth-order elastic tensor formulation:
^{u}_{0} is the volume of crystal unit cell in the referential configuration, ^{u}_{b}

To find the initial equilibrium state for many-body potential problem, we use the general or generic analytical form of EAM potential reported in literature. To find the initial equilibrium state, we first consider the

To solve this nonlinear equation, we use the Newton-Raphson method to linearize the above equation,
^{i}^{i }^{ }tol

The analytical expression of

Then the bond vectors in corresponding dislocation pattern segments may be defined by the higher order Cauchy-Born model as follows:
_{w}_{p}_{t}

The constitutive relations in wedge, prism and truncated octahedron (tetradecahedron) elements can be further explicated as follows:

The higher order stress tensor are expressed in a general form as:

To formulate MCDD finite element formulation, we first consider the Hamilton principle in terms of displacement variation for a fixed time interval as:

The variation of the kinetic energy of the crystal is:

The internal virtual work of the crystal is:

and the external virtual work of the crystal is:

Substituting these formulas above into the Hamilton principle, we can have Galerkin weak form of Multiscale Crystal Defects Dynamics for a crystal as shown as follows:
_{i}(X) is the finite element shape function, and u_{i} are the element nodal displacements, we can rewrite the form of considering different types of elements as:
_{i}_{i}_{i}_{0} is crystalline density in referential configuration. We use the notation

Note that based on internal virtual work,

In our current MCDD FLEM model, we assume that on the boundary of the crystalline solid the higher order stress effects are negligible, i.e.,

In this section, we first present MCDD numerical results in uniaxial tension and pure shear tests. To model and simulate crystal plasticity in

The total energy _{ij}_{ij}_{i}_{j}_{ij}_{ij}

We use the following pair potential _{e}

The embedding energy function is expressed as follows:

The MCDD model with the size of ^{–1}. The boundary conditions for MCDD simulations are as follows: The upper 10% portion of the specimen is applied with the prescribed displacement/velocity, and the unloading is initiated at the three different strain marks set at 10%, 15% and 20% with the same strain rate. The lower 10% portion of the specimen is fixed. The rest of the boundaries are traction and higher order traction free (see [

Meanwhile, molecular dynamics (MD) simulation is conducted by using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), which is a popular open-source molecular dynamics simulation software package. The MD model or specimen has 59582 atoms. The atomic pair potential for

By choosing the unloading position at three different strain marks mentioned above, we have obtained the corresponding stress-strain relations in MCDD simulations. The obtained stress-strain relations are compared with those of MD simulations as shown in

By considering higher order Cauchy-Born rule based higher order strain gradient formulations on wedge shaped dislocation patterns, prism shaped dislocation patterns, and truncated octahedron dislocation patterns, the MCDD method shows its ability to capture inelastic behavior and the path-dependent constitutive relation of

Similar to the pure tension case, we also conducted pure shear test on an

By prescribing three different unloading strains, we can observe overall stress-strain relation in shear in comparable with that of MD simulations with the same loading and unloading conditions as shown in

Based on the results presented above, we may conclude that under pure shear loading MCDD method also offers consistent results with that obtained from MD simulations. It may be noted that in

Conventionally, size effect is not considered in constitutive relation therefore the corresponding stress-strain curve is smooth and continuous. However, with sample size decreasing to micron or sub-micron scale, material strength shows increased compared to macro-scale sample size. Some scholars give some statistical relations to describe size effect on material strength reduce. But until now, people are not clear what mechanism leads to size-effect of micron-scale materials.

Cross-slip is atomistic level phenomenon observed commonly in crystal plasticity especially when dislocation density attains to a certain value. Capturing cross-slip in micro-scale is not trivial, not to mention that cross-slip is observed in sub-micron scale.

MCDD, as a multiscale method, has advantages in simulating crystal in micron-scale. Therefore, in this numerical example, we can find that cross-slip can be captured in numerical sample with different size. Furthermore, we give a preliminary conclusion that the appearance of cross-slip may lead to size effect in crystal plasticity.

Based on [^{7} ^{–1}, 5.55 s × 10^{7} ^{–1} and 2.77 s× 10^{8} ^{–1}. The MCDD results are compared with MD simulation as

From the figure above, we can find that the yielding stress of MCDD model is much lower than perfect crystal microstructure of MD simulation results and it is approximate to results of MD model with voids inserted. Based on the model above, we construct different size models to study relation between size effect and cross-slip.

From the

Based on such simulation result, we try to have the connection between size effect and cross-slip whose mechanism is following: initially when number of dislocation is few, crystal only has the primary slip plane. With strain increases, the number of dislocation in unit area (i.e., dislocation density) will increase. When dislocation density attains at a specific value, the second slip plane will be activated therefore the prerequisite of dislocation jumping (i.e., cross-slip) forms. Under this condition, dislocation will “jump” from one plane to another. Therefore, the cross-slip will be formed. Until now, MD simulation is hard to capture cross-slip but MCDD method can capure it. The reason may be that single or very few dislocations cannot activate second slip plane which is the prerequisite of cross-slip. But the MCDD method, as a multiscale method, simulates dislocation pattern which represents the ensemble or sub-network of dislocations. Therefore, MCDD method can capture that formation of secondary slip plane and “jumping” of dislocation.

Based on analysis above, we know that dislocation density will activate cross-slip, the appearance sequence of cross-slip may determines the appearance sequence of yielding stress of materials and the difference among appearance sequence of yielding stress is the size effect. By this conclusion, we establish the connection between dislocation density and size effect and offer a preliminary explanation to size effect in micron-scale crystal sample.

At nanometer or sub-micron scale, a very strong size effect of crystal plasticity has been reported in experiments, e.g., [

We constructed three different MCDD numerical specimens at micron scale, and they are square columns with fixed length-to-cross section width ratio of 2:1. The four specimens have the sizes: _{x}_{y}_{z}

In this micron-scale MCDD specimen, it has 5952 tetrahedron elements (bulk crystal elements), 4096 wedge elements (1st order process zone or wedge dislocation pattern elements), 2752 prism elements (2nd order process zone or prim dislocation pattern elements), and 367 truncated octahedron elements (3rd process zone or truncated octahedron dislocation pattern elements). We then conducted uniaxial compression tests with prescribed displacement boundary condition with the loading strain rate: 5× 10 − ^{6} ^{−1}.

The stress-strain curves of

In this work, we have further developed the recently proposed multiscale dislocation pattern dynamics by carrying out meso-scale discrete dislocation pattern dynamics computations and simulating crystal plasticity in ^{3} above, which is not reachable for MD simulations under current computer technology.

This work is highlighted by following two advances: (1) The simulated model is able to show accurate inelastic stress-strain curve comparable to MD results, and we can even capture dislocation nucleation and growth process with obvious crystal plasticity. There is a significant development in computational efficiency, which allows us to imagine further step: simulation of super micron-scale model and comparing with experiment results; (2) MCDD method shows the potential to be used to simulate the macroscale inelastic deformation process and fatigue damage, (3) MCDD successfully capture cross-slip which is hard to be observed in MD simulations. And having a preliminary relation between cross-slip and size effect lead us to have a preliminary explanation to mechanism of size effect and (4) MCDD model demonstrates that it can successfully capture size effect of up to 3.0 ^{3} micropillar crystal model, which shows validity of MCDD method in simulating crystal plasticity.

However, the method still has some unresolved issues. For example, the thermal effect that is strongly influenced dislocation nucleation and growth has not been taken into account in the present dislocation pattern dynamics. Moreover for macro-scale dislocation pattern dynamics simulations, it will need massive MPI parallel computation to realize. For the above issues and more, we hope to deal with them in the subsequent work.

In

Elastic | EXP | EAM (_{m} |
||||||
---|---|---|---|---|---|---|---|---|

Constant | Ref. [ |
_{2 }=_{ }14 |
_{3 }=_{ }26 |
_{4 }=_{ }50 |
_{5 }=_{ }58 |
_{6 }=_{ }64 |
_{7 }=_{ }88 |
_{8 }=_{ }112 |

_{11}( |
260.91 | 264.15 | 261.01 | 260.96 | 260.96 | 260.96 | 260.96 | 260.96 |

_{12}( |
157.43 | 155.51 | 154.56 | 154.55 | 154.55 | 154.55 | 154.55 | 154.55 |

_{44}( |
81.82 | 83.83 | 82.407 | 82.4 | 82.398 | 82.398 | 82.398 | 82.398 |