Computational modeling and simulation of spall fracture in polycrystalline solids by an atomistic-based interfacial zone model

Abstract

The focus of this work is to investigate spall fracture in polycrystalline materials under high-speed impact loading by using an atomistic-based interfacial zone model. We illustrate that for polycrystalline materials, increases in the potential energy ratio between grain boundaries and grains could cause a fracture transition from intergranular to transgranular mode. We also found out that the spall strength increases when there is a fracture transition from intergranular to transgranular. In addition, analysis of grain size, crystal lattice orientation and impact speed reveals that the spall strength increases as grain size or impact speed increases.

1. Introduction

Mechanical spall is a shock wave induced dynamic fracture process, such as armor penetration and blast loads. The macroscopic spall damage is strongly depending on the mesoscale structures of material based on some experimental results [1-3]. On the mesoscale level, most ceramics and metals are characterized by crystal grains and grain boundaries, heterogeneous impurities, and material defects and so on. Various experiments have been conducted to observe the shock wave propagation and material failure in various solids at micro-scale level or meso-scale level [4-8]. However, it is still difficult for the experimental methods to precisely record the dynamical process of spall inside solids because of the short time scales. Due to the limitations of experiments, some researchers attempted to develop numerical models and simulation tools to study the basic features of spall fracture. The spall simulation of copper by planar shock loading has been investigated by molecular dynamics simulations[9, 10]. Besides, a coarse-grained molecular dynamics with a Lennard-Jones potential is employed to simulate spall phenomena of a flyer specimen system by Krivtsov and Mescheryakov[11]. Ren et al.[12] used the reproducing kernel particle method (RKPM) to capture some features of the spall fracture including the role of inelastic wave pulses. To study the dynamic thermo-mechanical response of a tungsten heavy alloy, Clayton[13] applied a cohesive finite element method developed by Camacho and Ortiz [14] to simulate dynamic spall process in tungsten heavy alloys. Qian and Li[15] simulated the spall fracture in polycrystalline solids by a multiscale cohesive zone method, which is developed by Zeng and Li [17].

Although extensive research has been carried out to investigate the mechanisms of spall fracture, there still exist many elusive issues that need to be explored for spall fracture in polycrystalline solids. In fact, micro-structural characteristics such as grain shape, spatial arrangement of grains and local crystallographic orientation have influenced the material response to some extent [16-18]. For instance, the severe mechanical anisotropy within materials may be caused by the grain rotations[16], and the onset of shear band location and width depend on the grain distribution and volume fraction[17, 18]. To better understand the spall mechanisms in polycrystalline materials, we conduct an extensive analysis of spall strength when there is an intergranular to transgranular fracture transition by applying an atomistic-based interfacial zone model (AIZM). This model was developed by Zeng and Li[19, 20] and it links the atomistic crystal structure with the mesoscale material properties of the interfacial zone, which provides an effective approach to describe material inhomogeneities such as grain boundaries, biomaterial interfaces[20], slip lines and inclusions. The contribution of current work focuses on the fundamental studies of spall fracture. In present work, the distributions of stress contour and shock wave propagation during high-speed impact process in polycrystalline materials are obtained to better illustrate the spall process and fracture mechanism. The relations of polycrystalline fracture mode transition with interfacial potential energy and microstructure configuration are discussed. It is shown that potential energy will affect the fracture morphology, causing the transition from intergranular to transgranular. Additionally, the fracture mode, grain size, lattice orientation and impact speed will affect the spall strength in polycrystalline materials.

2. Constitutive modeling of the atomistic-based interfacial zone model

2.1 Local Cauchy-Born rule continuum method

To calculate the potential energy for a crystalline solid, the Cauchy-Born rule is employed in the proposed atomistic-based interfacial zone model. The so-called Cauchy-Born can describe the material behavior from atomistic information and is widely used as a standard method in continuum mechanics [21, 22]. It refers to the following procedure: let x denote the spatial position of a material point X at the time t, thus the local continuum deformation gradient is given by $\mathbf{F}=\frac{\partial \mathit{x}}{\partial \mathit{X}}$. Since in a given element e(e = 1, …, n_elem), the deformation gradient, F_e, is a homogeneous tensor, the relative deformed bond vector r_i in a unit cell results from the corresponding related undeformed atomic bond vector R_i can be defined as:

$${\mathit{r}}_i={\mathbf{F}}_e{\mathit{R}}_i,i=1,2,\dots ,n_b.$$ (1)

where n_b is the total number of bonds in a unit cell.

Consequently, the elastic energy density in any given element can be calculated by computing the potential energy density of an arbitrary unit cell inside the element:

$$W_e({\mathbf{F}}_e,{\mathit{R}}_i)=\frac{1}{2{\Omega }_0^b}\sum _{i=1}^{n_b}\varphi \left(r_i\right)=\frac{1}{2{\Omega }_0^b}\sum _{i=1}^{n_b}\varphi \left({\mathbf{F}}_e{\mathit{R}}_i\right),e=1,2,\dots ,n_{elem}$$ (2)

Where ${\Omega }_0^b$ denotes the volume of the unit cell in the referential configuration and b indicates bulk element, φ(r_i) is the atomistic potential, r_i, i = 1,2,…, n_b is the current bond length in a unit cell.

Based on the Cauchy-Born rule, the constitutive relations for the bulk medium can be established. For instance, the second Piola-Kirchhoff stress can be expressed in the following form:

$$\mathbf{S}\left(\mathbf{C}\right)=\frac{1}{2{\Omega }_0^b}\sum _{i=1}^{n_b}{\varphi }^{\prime }\left(r_i\right)\frac{\partial r_i}{\partial \mathbf{C}}=\frac{1}{2{\Omega }_0^b}\sum _{i=1}^{n_b}\frac{\partial \varphi \left({\mathbf{F}}_e{\mathit{R}}_i\right)}{\partial r_i}\frac{{\mathit{R}}_i\otimes {\mathit{R}}_i}{r_i}$$ (3)

where C = F^T · F is the right Cauchy-Green tensor.

2.2 Effective deformation gradient in the interfacial zone

Since we consider the interfacial zone as an actual physical zone, the Cauchy-Born rule can be applied in the interfacial zone area. Additionally, the thickness R of interfacial zone is chosen as ${10}^{-3}\mid s_0^{\mp }\mid \le R\le {10}^{-1}\mid s_0^{\mp }\mid$, and $s_0^{\mp }$ are the length of the sides of the adjacent bulk elements. The lower limit aspect ratio is a restriction limit beyond which numerical ill-conditioning may happen in computations. The upper limit aspect ratio may limit how much refinement we can do as the mesh size approaches to atomistic scale, because R has to be greater than 4-5 lattice spacings in order to apply the Cauchy-Born rule to the effective displacement field in the cohesive zone. Based on Cauchy-Born rule, the relative deformed lattice bond vector in each interfacial zone can be calculated as follows:

$${\mathit{r}}_i^{inter}={\mathbf{F}}_e^{inter}{\mathit{R}}_i,i=1,2,\dots ,n_{inter}.$$ (4)

If we consider the interfacial zone as one quadrilateral element sandwiched by two triangle bulk elements(see Fig. 1), the interfacial zone deformation gradient, ${\mathbf{F}}_e^{inter}$, can be expressed by the four finite element nodal displacement[19, 20]:

$$\left[\begin{array}{c}\hfill F_{11}^{inter}\hfill \\ \hfill F_{12}^{inter}\hfill \\ \hfill F_{21}^{inter}\hfill \\ \hfill F_{22}^{inter}\hfill \end{array}\right]=\frac{1}{\text{ad}-\text{cd}}\left[\begin{array}{cccc}\hfill \mathrm{d}\hfill & \hfill 0\hfill & \hfill -\mathrm{b}\hfill & \hfill 0\hfill \\ \hfill -\mathrm{c}\hfill & \hfill 0\hfill & \hfill \mathrm{a}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathrm{d}\hfill & \hfill 0\hfill & \hfill -\mathrm{b}\hfill \\ \hfill 0\hfill & \hfill -\mathrm{c}\hfill & \hfill 0\hfill & \hfill \mathrm{a}\hfill \end{array}\right]\left[\begin{array}{c}\hfill x_{l+1}^{+}-x_l^{-}\hfill \\ \hfill y_{l+1}^{+}-y_l^{-}\hfill \\ \hfill x_l^{+}-x_{l+1}^{-}\hfill \\ \hfill y_l^{+}-y_{l+1}^{-}\hfill \end{array}\right]$$ (5)

where $\mathrm{a}=X_{l+1}^{+}-X_l^{-},b=Y_{l+1}^{+}-Y_l^{-},c=X_l^{+}-X_{l+1}^{-},d=Y_l^{+}-Y_{l+1}^{-}$.

Fig. 1.

Deformation gradient in interfacial zone

Subsequently, the averaged first Piola-Kirchhoff stress tensor in each interfacial zone can be written as:

$${\mathbf{P}}_{inter}=\frac{1}{2{\Omega }_0^b}\sum _{i=1}^{n_b}\frac{\partial {\varphi }_{inter}}{\partial r_i}\frac{{\mathit{r}}_i\otimes {\mathit{R}}_i}{r_i}$$ (6)

Once we obtain the stress inside the interfacial zone, the interfacial cohesive tractions along the boundaries of adjacent bulk elements can be readily written as:

$${\mathit{T}}^{inter}={\mathbf{P}}_{inter}\cdot \mathit{N}$$ (7)

where N is the out normal of adjacent bulk FE elements.

In this model, the interfacial zone is assumed a relatively “soft” zone and the intermolecular interaction inside the interfacial zone is a type of the Van der Waals interaction between non-covalent bonds or quasi-covalent bonds. When the atomistic potential for a given bulk medium is available, the interfacial zone potential can be calculated by integrating the bulk potential over the bulk medium half space. For example, if the potential in the bulk element is considered as Lennard-Jones potential:

$${\varphi }_{bulk}=4∊\left[{\left(\frac{\sigma }{\mathrm{r}}\right)}^{12}-{\left(\frac{\sigma }{\mathrm{r}}\right)}^6\right]$$ (8)

the coarse graining interfacial potential can be obtained by analytical integration[23]:

$${\varphi }_{inter}\left(r\right)={\int }_{\text{half}\text{space}}\beta {\varphi }_{bulk}(r-r^{\prime }){\mathrm{dV}}^{\prime }=\pi \beta ∊r_0^3\left[\frac{1}{45}{\left(\frac{r_0}{r}\right)}^9-\frac{1}{3}{\left(\frac{r_0}{r}\right)}^3\right]$$ (9)

where ∊ is the depth of the potential well; σ is the finite distance at which the inter-atom potential is zero; r₀ = 2^1÷6σ is the equilibrium bond distance in the bulk material; β is the atomic density in the solid. In the interfacial zone, the crystal lattice structure is modeled as hexagonal structure and the arrangement of atoms is regular in the undeformed state, but its lattice constants and atomistic potential are different from those of the bulk medium.

2.3 Finite element implementation

A displacement-based finite element formulation is developed from the weak form of virtual work. The Lagrangian finite deformation formulation is used to account for finite strain. If x denotes the current spatial position of a material point X at the time t, the displacement vector are defined as u = x–X. During the deformation process, a Galerkin weak formulation including a contribution from the interfacial surfaces may be expressed as following[24]:

$${\int }_{\Omega }\mathbf{P}:\delta \mathbf{F}d\Omega +{\int }_{\Omega }\mathit{B}\delta \mathbf{u}d\Omega +{\int }_{\Omega }\rho \frac{{\partial }^2\mathbf{u}}{{\partial }^{2}t}\cdot \delta \mathbf{u}d\Omega ={\int }_{S_{inter}}{\mathbf{T}}^{inter}\cdot \delta \mathbf{u}dS+{\int }_{S_{ext}}\stackrel{‒}{\mathbf{T}}\cdot \delta \mathbf{u}dS$$ (10)

where Ω, S_inter, S_ext are the volume, interfacial surface and external surface of solid body in the reference configuration, respectively. B is the body force; ρ is the material density in the reference configuration; T^inter is the interfacial surface traction vector and $\stackrel{‒}{\mathbf{T}}$ is the external traction vector.

In addition, the explicit time integration scheme is based on the Newmark β-method with β = 0 and γ = 0.5 to obtain the nodal velocities and the nodal displacements[25].

3. Polycrystalline microstructure modeling

To generate different randomly shaped grain structures in polycrystalline materials and to study the effect of polycrystalline grain morphology, Voronoi tessellation has been widely used[26-29]. Before the appearance of Voronoi tessellation method, the simplified grain shapes, such as hexagons for 2D case[30], or truncated octahedrons for 3D case[31, 32], were utilized to deal with polycrystalline microstructure. However, these regular structures cannot precisely predict local stress-strain fields inside the grains[33, 34]. Thus, Voronoi tessellation method has been emerging as a powerful tool to construct arbitrary polycrystalline microstructure. Initially, the Voronoi domain is considered as partitioning of a plane with a set of n random distinct points, or nuclei[35, 36]. For a 2D case, the Voronoi region of every nucleus is defined as:

$$V_i=V\left(X_i\right)=\{X:d(X_i,X)<d(X_j,X)\}fori\ne j$$ (11)

where X_i represent the coordinates of kernel points i, d(X_i, X) denotes the Euclidean distance between X_i and X, X belongs to Voronoi region V_i and is closer to nucleus i than other nuclei, as shown in Fig. 2. In addition, the slope of the line perpendicular to each closest point connection line, e.g. line i – j as shown in Fig. 2, defines the polygonal boundary segment $\overline{ab}$.

Fig. 2.

A schematic of Voronoi grain generation

The impact test samples are generated by centroidal Voronoi tessellation as shown in Fig. 3(a). Each Voronoi cell represents an individual grain cell and all edges of grain cells are considered as grain boundaries. It can be seen from Fig. 3(b) that triangular finite element is used to mesh all grain cells after the generation of grain cells. All interfaces among bulk elements are governed by the interfacial zone model.

Fig. 3.

Polycrystalline grain cell and mesh generation: (a) grain cells and (b) finite element meshes over grain cells

4. Numerical simulations

4.1. Problem analyzed

Numerical simulations have been carried out for simulation of high speed impact induced spall fractures. For spall fracture, it is a very difficult problem that has been elusive to many existing numerical methods[37]. The exact problem statement is described in Fig. 4, in which the projectile is a rigid block with an impact velocity v₀ and the target is a plate with dimension (L_x × L_y = 0.4mm × 0.2mm) under free boundary condition.

Fig. 4.

The problem statement of the impact test

To characterize the varying grain morphology, each grain can be assigned an individual set of material properties such as lattice orientation α(α^g is grain orientation angle and α^b is grain boundary orientation), see Fig. 5. For simplicity, the lattice structure is chosen as the hexagonal lattice in the simulations as shown in Fig. 5, and we use the Lennard-Jones potential for the hexagonal crystal structure.

Fig. 5.

Lattice orientation in grains and interfacial zones

To prevent penetration during the impact process, the so-called impenetrability condition is enforced during the computation, which means the contact forces are required to push out the penetrated nodes. The pushed-out contact force required to prevent penetration are given as following[14, 38]:

$$F_{s,j}=\frac{2M_{s,j}{\delta }_{s,j}}{\Delta t^2}$$ (12)

where M_s,j is the mass of node j on the slave surface, δ_s,j is the penetration distances for all nodes j on the slave surface. Here, the surface on target plate is assumed as slave surface and the surface on rigid block is treated as master surface in the contact pairs.

4.2. Fracture growth in polycrystalline material

In this study, there are 128 grain cells in the target plate and the average grain size is around 25μm [39-42]. Meanwhile, each grain is assigned a random lattice orientation α^g and the lattice orientation in the grain boundary is set as ${\alpha }^b=\frac{1}{2}({\alpha }_A^g+{\alpha }_B^g)$. The impact velocity is set as 100m/s and the time step is chosen as Δt = 0.1 × 10⁻¹⁰s in time integration. For this case, the potential energy of the interfacial zone in grain boundaries is smaller than that in the grains, i.e. ${∊}_{inter}^{\mathrm{g}}=3{∊}_{inter}^{\mathrm{b}}$, which means the grain boundaries is weaker than the grains [43].

When the rigid flyer impacts the target, a pulse of compression stress wave is generated at the contact surface. The simulation results are shown in Fig. 6 and the propagation and reflection of the stress wave from the top edge to the bottom free edge has been observed. The phenomenon of spall process under high-speed impact has been captured, as shown in Fig. 6.

Fig. 6.

Snapshots of stress distribution(σ₂₂) for an intergranular failure process: (a) t=0.06μs; (b) t=0.12μs; (c) t=0.164μs; (d) t=0.208μs; (e) t=0.3μs; (f) t=0.4μs.

The dynamic history of particles inside a material at micro-scale level cannot be recorded by the current experimental approach [4]. Thus, the free surface velocity history at the opposite side of contact surface is widely utilized to observe the spall behavior as an approach for indirect measurement. Within the acoustic approximation, the spall strength is defined as[44, 45]:

$${\sigma }_{sp}=\frac{1}{2}\rho c{V}_{sp}$$ (13)

where ρ is the material density, c is the sound velocity in the material and V_sp = V_o – V_m. A compressive disturbance called a “spall signal” appears on the free surface velocity history as the tensile stress relaxes[4, 12]. It can be seen that the free surface velocity illustrates as dynamic behavior as shown in Fig. 7(a). Meanwhile, the reflected force on free bottom surface also appears as an oscillating force. With the spall crack initiation, the sign of reflected force changes from compression to tension as shown in Fig. 7(b). When t>0.3μs, the amplitude of free surface velocity and reflected force decrease significantly, which implies the layer of bottom surface is completely separated from the target plane as shown in Fig. 6(e) and Fig. 6(f), then the velocity verge to approximately a constant value and the reflected force decrease to zero as shown in Fig. 7(a) and Fig. 7(b).

Fig. 7.

(a) The velocity and (b) the reflected force on the bottom surface as a function of time

4.3. Mesh convergence test

Generally speaking, the results of the interfacial zone method will be sensitive to its element size. To test mesh size dependence of interfacial zone model, the exact same condition was applied to the problem as in previous section with different mesh sizes. For comparison purpose, we defined the mesh ratio R_d in grain cell as:

$$R_d=\frac{Averageelementsize}{Averagegrainsize}$$

Three cases with different mesh ratio R_d = 0.5, 0.33 and 0.25 are employed, as shown in Fig. 8. From the study, we found out that the spallation is along the same path although the mesh size varies as shown in Fig. 9. That is, the current mesh can accurately capture the spall crack path.

Fig. 8.

Different mesh ratio over grain cells: (a) Case 1 with R_d=0.5; (b) Case 2 with R_d=0.33; (c) Case 3 with R_d=0.25

Fig. 9.

The intergranular failure for different mesh ratio: (a) Case 1; (b) Case 2; (c) Case 3

On the other hand, the reflected forces $r_{\mathrm{b}}^i$ and velocities ${\mathrm{v}}_{\mathrm{b}}^i$ on the free bottom surface are calculated in each case for further comparison. As shown in Fig. 10, the reflected force and free surface velocity become closer as mesh size decreases. The errors of free surface velocity and reflected force in different cases are defined as:

$$\begin{array}{cc}\hfill Er{r}_{\mathrm{v}}=& {\int }_0^t\mid {\mathrm{v}}_{\mathrm{b}}^i-{\mathrm{v}}_{\mathrm{b}}^{i-1}\mid dt,i=2,3,4\dots \hfill \\ \hfill Er{r}_f=& {\int }_0^t\mid r_{\mathrm{b}}^i-r_{\mathrm{b}}^{i-1}\mid dt,i=2,3,4\dots \hfill \end{array}$$

Fig. 10.

(a) Free surface velocity and (b) Force profile at the free bottom surface for different meshes

From Table 1, it can be seen that errors from free surface velocity and reflected force both decrease as the element size decreases.

Table 1.

Comparison of different meshes (Velocity & Force)

Rd	Bulk Element No.	Velocity Error	Force Error
0.50	2520	----------------------	--------------------------
0.33	5101	4.78E-08	2.72E-10
0.25	8646	2.48E-08	1.40E-10

4.4. The effect of fracture morphology on spall strength

A given material can suffer transition from intergranular to transgranular fracture under different loading conditions[46]. The transition from intergranular to transgranular mode was observed by many researchers [47-49], which is determined by a variety of factors including loading conditions, microstructure of a material, interfacial properties etc. The fracture morphology, such as intergranular mode or transgranular mode, plays an important role on spall strength [45, 50].

In the atomistic-based interface zone model, the potential energy competition between grain and grain boundary will result in different fracture modes. Due to the hypothesis that the strength of grain boundaries are weaker than grains [43], we define the interfacial potential energy ratio between grain boundary and grain as ${\mathrm{R}}_{\mathrm{p}}={∊}_{inter}^{\mathrm{b}}∕{∊}_{inter}^{\mathrm{g}}$. Then we increase the potential energy ratio R_p from 30% to 100% gradually to observe the fracture mode transition. We found out that, when R_p reaches 75%, there is an intergranular to transgranular fracture mode transition. Before that the fracture are all intergranular. The Fig. 11 captured the process of fracture mode transition as R_p = 75% . When R_p keeps increasing , the fracture mode changes from partially transgranular to completely transgranular as shown in Fig. 12.

Fig. 11.

Snapshots of stress distribution(σ₂₂) for partially transgranular failure process with potential energy ratio R_p=75%: (a) t=0.048μs; (b) t=0.156μs; (c) t=0.204μs; (d) t=0.268μs; (e) t=0.34μs; (f) t=0.4μs

Fig. 12.

The snapshot of transgranular failure intense with the increase of potential energy ratio R_p in a polycrystalline solid: (a) R_p=75% ; (b) R_p=85%; (c) R_p=100%

The free surface velocity of the fracture process is illustrated in Fig. 13. From Fig. 13, it can be seen that the spall strength increases (or V_sp increases) during the intergranular to transgranular fracture mode transition. This result is qualitatively consistent with the experimental results observed by Brewer et al.[50].

Fig. 13.

Free surface velocity as a function of time at different fracture mode: (a) Completely intergranular to partially transgranular; (b) Partially transgranular to completely transgranular

We also calculated the J integral for different potential energy ratio. As described by Sorensen and Jacobsen [51], in Fig. 14, the locally evaluated J integral along the crack faces enclosing the interfacial zone is given as:

$$J_{loc}={\int }_0^{{\delta }^{\ast }}{T}^{inter}\left(\delta \right)d\delta +J_{tip}$$ (14)

with assumption of J_tip = J_o = 0, where J_tip is the J integral evaluated around the crack tip and J_o is the fracture energy of the crack tip, δ^* is the opening of the interfacial zone.

Fig. 14.

Integration paths locally around the interfacial surface for the J integral

The J integrals for different potential energy ratio R_p are shown in Fig.15. It can be seen that when J integral (J_loc) reaches approximately 45 (J/m²), there is an intergranular to transgranular fracture mode transition.

Fig. 15.

J integral curves for different potential energy ratio R_p

4.5. The effect of grain size on spall strength

The effects of the grain size on spall strength have not been well established in the literature. There are two conflicting conclusions about the grain size effect on spall strength in metals. Some researchers presented that spall strength decreases with increasing grain sizes[52], while other studies have observed that spall strength increases as grain size increasing[45, 53]. In this work, the atomic-based interfacial zone model is used to study the grain size effects on spall strength for three different grain sizes. As shown in Fig. 16, it is observed that the transgranular crack initiates for all three different grain sizes at potential energy ratio R_p = 75% . By comparing the free surface velocity of intergranular and transgranular fracture modes for all three grain sizes, it can be seen from Fig. 17 and Fig. 18 that the spall strength σ_sp decreases as grain size decreases. Our results support the conclusion that spall strength is decreasing as grain size decreasing from experimental observations [45, 53].

Fig. 16.

The snapshot of transgranular crack initiation for different grain sizes: (a) grain No. = 128; (b) grain No. = 200; (c) grain No. = 288

Fig. 17.

The free surface velocity for different grain sizes at intergranular fracture mode:(a) R_p=35%, (b) R_p=50%

Fig. 18.

The free surface velocities for different grain sizes at transgranular fracture mode:(a) R_p=75%, (b) R_p=85%

In addition, as the grain size decreasing, the fraction of transgranular fracture versus intergranular fracture is decreasing as illustrated in Fig. 16. The reason for this trend (fraction of transgranular fracture versus intergranular fracture decreasing with decreasing grain size) might be that at small grain size, there exists more grain boundaries and these offer a low resistance path along grain boundaries for spalling. The fracture will prefer the grain boundaries to spall instead of grains. The similar phenomenon has been investigated from experiment [54].

4.6. The effect of lattice orientation on spall strength

It is known that the lattice orientation has effects on fracture toughness and fault energy[24, 55], which determines the fracture mechanism to some extent. On the other hand, the severe mechanical anisotropy within materials may be caused by the grain rotations[16], and the onset of shear band location and width depend on the grain distribution and volume fraction[17, 18]. In present work, three different lattice orientations of grains are studied to investigate the variation of spall strength. The degrees of lattice orientation are 30°, 45° and 60°, as shown in Fig. 19.

Fig. 19.

Lattice orientation: (a) 30; (b) 45; (c) 60°

From Fig. 20(a), it can be seen that, at intergranular fracture mode, the free surface velocity V_sp (spall strength) will decrease when we change the lattice orientation from 30°, 45°, to 60°. However, under the transgranular mode, the spall strength almost keeps unchanged for three different lattice orientations, as illustrated in Fig. 20(b). The possible reason is that under transgranular fracture mode, it is more like a single-crystal structure, the effect of lattice orientation on the spall strength appears to be negligible. This result is in consistency with Dalton et al. [56] who reported that crystallographic orientation has no discernible effect on spall strength in single-crystal.

Fig. 20.

Free surface velocity for different lattice orientations: (a) Intergranular fracture(R_p=35%); (b) Transgranular fracture(R_p=85%)

4.7. The effect of impact speed on spall strength

To investigate how impact velocity affects the spall strength, we test three different impact velocities (80m/s, 100m/s and 125m/s) to observe the variation of spall strength. It can be seen that the spall strength will increase with increasing impact speed under intergranular or transgranular mode, as shown in Fig. 21.

Fig. 21.

The free surface velocity for different impact speed: (a) Intergranular fracture (R_p=35%); (b) Transgranular fracture(R_p=85%)

5. Discussion and conclusion

In this paper, we have applied a recently developed atomistic-based interfacial zone model to investigate the failure mechanism and spall fracture patterns under impact loadings in polycrystalline materials. In the model, atomistic structure information is used to extrapolate constitutive relations of both polycrystalline grains as well as grain boundaries. By using the AIZM, we have successfully described both grains and grain boundaries of a given polycrystalline material at the mesoscale, and studied the failure mechanisms that may happen during impact loading.

From the simulations, some important findings of present work can be summarized as: (1) when the potential energy of interfacial zone inside grains is greater than that along grain boundaries, i.e. ${∊}_{inter}^{\mathrm{g}}=3{∊}_{inter}^{\mathrm{b}}$, the spallation will follow the grain boundaries (c.f. Fig. 6). The intergranular/transgranular fracture patterns are sensitive to potential energy ratio R_p and the potential energy competition between grain and grain boundary will result in different fracture modes (c.f. Fig. 12). From the simulations, we also observed that the spall strength increases as R_p increases during the intergranular to transgranular fracture mode transition (c.f. Fig. 13); (2) the grain size study indicated that as the grain size decreasing, the fraction of transgranular fracture versus intergranular fracture is also decreasing (c.f. Fig. 16), and the spall strength decreases as well (c.f. Fig. 17 and Fig. 18). Besides, at intergranular fracture mode, the lattice orientation will affect the spall strength to some extent (c.f. Fig. 20(a)). However, at transgranular fracture mode, the effect of crystal lattice orientation on the spall strength appear to be negligible (c.f. Fig. 20(b)); (3) from the impact speed study, we found out that the spall strength increases as impact speed increases for both intergranular and transgranular fracture modes (c.f. Fig. 21).

The simulation results are qualitatively consistent with the experimental measurements, and the atomistic-based interfacial zone model can depict the crystal lattice orientation which is not achievable by traditional cohesive zone model. We have demonstrated the potential of the AIZM as a major modeling and simulation technique in both polycrystalline materials modeling as well as multiscale simulations in general. However, it should be pointed out that the proposed interfacial zone model is a continuum model and the current model may not be able to capture the complex atomistic physical mechanisms such as dislocation activity, cavity nucleation and growth along grain boundaries involved in the polycrystalline material damage process. In the model, the interfacial constitutive relations are derived from atomistic potential to describe the material interface interactions. We believe the atomistic enriched interface zone model may provide a useful numerical simulation tool to study the material failure process of polycrystalline materials at mesoscale.

Highlights.

• Modeling grain boundaries by interfacial zone model

• High-speed impact induced spall fracture in polycrystalline materials

• Investigation of intergranular/transgranular fracture transition under impact loading

• Investigating how grain size, crystal lattice orientation, and impact speed affect spall strength

Nomenclature

B

body force

c

sound velocity in the material

C

right Cauchy-Green tensor(F^TF)

d(X_i, X)

Euclidean distance between X_iand X

F_s,j

pushed-out contact force

F

deformation gradient

M_s,j

mass of node j on the slave surface

n_b

total number of bonds in a unit cell

n_elem

total number of elements

P

first Piola-Kirchhoff stress tensor

r_i

deformed bond vector

R_d

mesh ratio

R_p

interfacial potential energy ratio between grain boundary and grain

R_i

undeformed atomic bond vector

$s_0^{\mp }$

side length of the adjacent bulk elements

S_ext

external surface of the solid body in the reference configuration

S_inter

interfacial surface of the solid body in the reference configuration

S(C)

second Piola-Kirchhoff stress tensor

T^inter

interfacial surface traction vector

$\stackrel{‒}{\mathbf{T}}$

external traction vector

u

displacement vector

V_i

Voronoi region

W_e (F_e, R_i)

potential energy density of an arbitrary unit cell inside the element

X_i

coordinates of kernel points

α^b

lattice orientation in the grain boundary

α^g

lattice orientation in the grain

δ_s,j

penetration distance for node j on the slave surface

∊

depth of the potential well

ρ

material density

σ

finite distance at which the inter-atom potential is zero

σ_sp

spall strength

φ(r_i)

atomistic potential

${\Omega }_0^b$

unit cell volume

Sub and superscripts

elem

a given element

g

polycrystalline grain

inter

interfacial zone

s

slave surface