In this work, we modeled the brittle fracture of shell structure in the framework of Peridynamics Mindlin-Reissener shell theory, in which the shell is described by material points in the mean-plane with its drilling rotation neglected in kinematic assumption. To improve the numerical accuracy, the stress-point method is utilized to eliminate the numerical instability induced by the zero-energy mode and rank-deficiency. The crack surface is represented explicitly by stress points, and a novel general crack criterion is proposed based on that. Instead of the critical stretch used in common peridynamic solid, it is convenient to describe the material failure by using the classic constitutive model in continuum mechanics. In this work, a concise crack simulation algorithm is also provided to describe the crack path and its development, in order to simulate the brittle fracture of the shell structure. Numerical examples are presented to validate and demonstrate our proposed model. Results reveal that our model has good accuracy and capability to represent crack propagation and branch spontaneously.

Dynamic damage and failure process of shell structure is very challenging in the computational community for its special geometric features which requires the numerical method to be capable of representing material characteristics and discontinuity in finite deformation and large rigid rotations [

Peridynamic theory has been proposed by Silling in 2000 and it has been successfully applied to solve problems in engineering and industry with material damage and fracture involved [

However, when it is extended to plate or shell structures, it is very hard to achieve reasonable results with high accuracy since the system have to use very fine discretizations to satisfy the precision requirement along the thickness direction. To address this issue, several peridynamics membrane approaches have been proposed based on various kinematic assumptions to improve the computational efficiency [

This paper is organized as follows. In

The Non-Ordinary State-Based Peridynamic Theory is a non-local continuum theory that was proposed by Silling [

where _{0} is the mass density at the initial configuration,

where _{X} is the divergence operator with respect to the initial configuration. In non-ordinary state-based Peridynamics, the force state

in which

where _{ij} = _{j}_{i}_{i}

According to the Mindlin-Reissner’s thick shell theory, the shell is modeled using a single layer of particles that having three degrees of freedom in translation and two additional rotational degrees of freedom _{1} and _{2} in the plane tangent to the shell. Drilling rotation is not considered in this context for simplicity.

In

where ^{+} − ^{−} is the thickness of the shell, and ^{0} is the linear mapping of the mean surface from the parametric space to the reference configuration space

At the mean plane of the shell, any point that belongs to the parametric space

where _{1} and _{2} are the basis vector in the parametric space. To better describe the motion of the shell, we define the convected basis vectors

in which _{1}, _{2}, and _{3}. In the following context, we assume that by adopting appropriate basis _{α} in the parametric space the local coordinates at

Now we consider the position of a point in the current configuration with a motion

where ^{1}, ^{2}) indicates the pseudo normal vector in current configuration that is mapped from parametric space, and ^{1}, ^{2}) defines the mid-plane of the shell in current configuration.

Similarly, we defined the convected basis vectors

where

where ^{1}, ^{2}) is the displacement of point on the mean plane of the shell in the current configuration. Given the above definition we note that ^{1}, ^{2}) and ^{1}, ^{2}) is the mapping from position of point we interested in the parametric space to the pseudo normal vector in initial or reference configuration, which is

To further simplify the computational process and clarify the constitutive updating objectivity, we introduce the rotation matrix ^{0} which transfer the position vector in the initial/reference configuration from the global coordinates ^{l}, such that

If the mean plane of the shell is flat and lies on the XOY plane in the initial/reference configuration, one can simply set ^{0} = ^{l}_{3} =

Likewise, we also define a rotation matrix ^{l}, such that

The motion ^{0}, and from the reference configuration to current configuration

Using chain rule, following relation could be reached

where ∇_{ξ}_{ξ}^{0} is the deformation gradient of motion ^{0} relative to the parametric configuration, respectively. Recall the equivalent non-local material differential operator defined as follows

where

where

Note the deformation gradient motion of ^{0} in terms of the convective coordinates are given by

The deformation gradient for three motions ^{0}, and

and,

in which _{α} = ^{α} ,

The in-plane tangent convected vector _{α} could be updated by

Also, one can readily have

which implies that the pseudo normal vector in the mean plane of the shell at current configuration could be updated by the following relation:

and,

which may not parallel to the normal to the tangent plane of the mid-surface.

In this work, we use the Euler-Rodrigues rotation formula to update the direction of fibers. The pseudo normal vector

where

in which

and,

Taking time derivative of

where

One could obtain the strain based on the deformation gradient under finite deformation and rotation. The spatial velocity gradient could be obtained by using the following relationship:

where

in which

where we have used the nonlocal convective gradient operator again,

Then the spatial velocity gradient could be decomposed into stretch and skew part

where

From the non-local deformation gradient, one can obtain the Green-Lagrangian strain tensor by

where ^{T}

The the Eulerian-Almansi finite strain is given by

where ^{−T}^{−1}

The Eulerian-Almansi strain could be transfered from the Green-Lagrangian strain by

To further facilitate the plane stress hypothesis, the constitutive equation is evaluated in the local coordinate system. The Eulerian-Almansi strain is first transformed into the local system by an orthogonal tensor

Using Voigt’s notation, the Almansi strain tensor in local system and the Cauchy stress tensor in local system could be expressed in vector form by

Utilizing the plane stress hypothesis, the component along the direction of plane thickness is zero, which yields

If the material is assumed to be elastic, reduced constitutive equation proposed by Hughes [

where _{33} should be calculated within the constitutive model to satisfy the zero normal stress condition _{33} = 0. Once the Cauchy stress being obtained in the local coordinate system, we transform it into global coordinate

where ^{l} is recovered from local Cauchy stress vector in Voigt’s notation. The first Piola-Kirchhoff stress in total Lagrangian formulation can be obtained in the global coordinate system,

In current work, the balance laws and nonlocal governing equations for Peridynamic shells proposed by Zhang et al. [

in which _{i}

and,

where

where

The angular momentum balance of equation of the shell is governed by

where

and,

where _{i} is the momentum stress resultant as expressed as

In non-ordinary state-based peridynamic theory, one may expect the so-called zero-energy mode, in which multiple deformation states could be related to the same unique deformation gradient, caused by the averaging of the kinematic information of all neighboring particles around one material point. It will introduce spurious nonphysical oscillations of stress, strain, and displacement field in the domain, which consequently may lead to system instability and inaccurate predictions, especially for transient and finite deformation problems.

Many pioneer work have been reported to try to address this problem from different aspect. In classic Finite Element Method, Hughes and Liu tried to eliminate the appearance of zero-energy in-plane rotational modes by using “Heterosis elements” [

In current work, we use the stress point method to reduce the rank-insufficient induced numerical divergence and oscillations. Stress points are added into the domain to help increasing the stability, as illustrated in

The four particles points on the middle surface(lamina surface) of the shell form an integral element. Arrange a stress point in the center of the integral element, shown in

In the calculation, the integral value corresponding to the Gauss point is obtained by integrating on the integration plane where the Gauss point is now, and then integrating through the line distribution of the Gauss point in the normal direction, and the obtained value is used as the corresponding function value of the stress point.

Each stress point have its own horizon in which it interacts with its neighboring material points within a certain range. The deformation gradient at the stress point is obtained by interpolation of the deformation gradient on its neighboring material point family. Stress is obtained from strain and constitutive relationship at the stress points, and then interpolated back to the material points. The computational scheme of the physical properties on the stress points and the material points is given as follows:

where _{j} and ^{s}_{i}

In conventional Peridynamic theory, bond-breaking is mainly driven by the stretch of the bonding between material points, that when the bond stretch limit was met, the bond will break spontaneously, which in turn accumulated to shape the crack surfaces. However, if a correspondence material model is utilized to describe the material response, such as plasticity, the bond-breaking criterion needs to be modified to represent the fracture properties of the material.

In current work, the Mohr-Coulomb failure criterion is adopted as the principle of bond-breaking, in which the principle stress state are used to determine the material failure, and the tension and compression could be treated separately. When such criterion being satisfied at the stress point in the domain, we treat it as fully damaged, thus any stress points that sharing the same fiber will be considered as fractured.

Under such consideration, the damage state is assumed to be initialized and propagate at the stress point of the shell, which means that cracking is realized by nucleating and extending cracks between damaged stress points. This approach of fracture modeling will surely introduce additional work to maneuver the initialization, propagation, and branching of the cracks, which need to book-keep the historic information of the crack surfaces. The Peridynamic bonding between material points will then be cut by the forming and propagating crack surfaces, which will then in turn represent the material failure in manner of material points.

The crack surface is composed of segments connected by stress points. Since the shell is represented by the material points at the mean plane, we only consider cracks running inside the mid-surface of the shell, and when certain spot of the shell being cracked, the crack go through the thickness direction which is denoted by the pseudo fiber oriented at the parametric space. In other words, when stress points that share the same pseudo fiber along the thickness direction get to a accumulated state of complete damage, the whole shell section where the stress point resides is considered to fail immediately. A new crack tip will then be generated at the stress point and thus forming the new crack surface, as illustrated in _{3}.

When the bond-breaking criteria are met, new crack tip will be generated. If there are multiple spots of the stress points satisfy the criterion, that have the largest magnitude will be chosen as the new crack tip with other candidates remain intact.If the newly nucleated crack tip is the only one in the shell, or it does not belongs to any existing crack surface, i.e., it is far away enough from all cracks in existence, we marked it the nucleation of a new crack surface. Otherwise, we evaluate the existing crack surfaces and pick one of them, from which the new crack tip has the nearest distance from. The new crack tip will be added to the surface of the chosen crack path, which then expands from the old crack tip to the new one. Hence a new crack surface segment is created. Each time new crack surfaces are developed, we need to do bond-breaking of the material points by apply crack surfaces to the connections of all the bondings. If by any chance, when new crack tip being created, we divide material points, if any, that are exactly located on the formed crack surfaces geometrically, into two. A tiny gap _{1} of the crack surface will be added to separate the two consequential material points. The normal direction of the crack can be obtained from the normal direction of the shell at the position of the stress point at the old crack tip and the relative coordinate vector of the stress point at the new and old crack tip, as

in which _{3} is the shell normal direction at the tip of the old crack. And _{on}

The resulting two material points bisect the mass and volume of the original material point, and inherit all other properties of the original material point, including linear velocity, linear acceleration, angular velocity, angular acceleration, translational displacement, and rotation angle. The newly generated crack surface can be projected back to the parametric space, in which the crack discontinuity plane in 3D is represented by a planar line. The bond-breaking process is handled in parametric space, which can take advantage of simplified geometric information.

In this work, the crack surfaces will break both the connection between stress points and material points, and those between material points themselves. The former one we already discussed in beginning of this section. For the bond-breaking between material points, the problem is simplified to whether two line segments intersect with each other, as illustrated in

in which _{1} and _{2} are the normal direction of the line segments _{CA}_{A}_{C}

During the calculation, it is first necessary to solve the damage situation of the stress point. According to the _{g}

Due to the geometric characteristics of the shell, our work does not consider the evolution of the crack in the thickness direction of the shell, which means the crack always penetrates the thickness direction of the shell instantaneously. Thus, if the damage of the stress point is larger than the limited damage _{limit}

As shown in

As shown in

As shown in

The formation of new crack surfaces will affect the interaction between particles. This process is called bond breaking. Similarly, the corresponding relationship between the stress point and the particle will also be separated by the crack surface, too. It can be judged by the simplified visibility condition mentioned above. When the relationship between any points A and B is interrupted by the crack surface, the displacement state of one point will no longer affect the other point. The relative position between them remains unchanged at the moment the key is broken. Therefore, as the crack grows, the particles around the crack are affected by the crack, but the neighboring particles remain unchanged. When solving the shape matrix K, there is no need to update the neighbors. The pre-crack is set on the geometric model in the initial step of the calculation, and thereafter, it will always exist in the model as the current crack. The flow chart of the crack simulation algorithm is illustrated in

In this section, several numerical examples are used to validate and demonstrate the validity of our implementation and capability of the proposed approach. The first two numerical cases are aimed to verify the validity and the accuracy of our implementation of the Peridynamic shell. The latter two cases are carried out to explore the capability of our model on representing the dynamic fracture of the shell.

In this section, the square clamping plate with side length ^{3}, elastic modulus ^{11}

Three different numerical discretizations are used to obtain the dynamic responses of the plate, which are 6 × 6, 11 × 11, and 21 × 21, respectively. The horizon

In this section, we present a numerical example of the scordelis-Lo roof, a cylindrical segment under gravity, as shown in

In this section, a dynamic fracture of the brittle plate is numerically modeled and simulated. A rectangular plate with a pre-notch under symmetric tensile loading is considered, as shown in ^{3}, and Poisson’s ratio _{c} = 32_{t} = 2.07

In this section, we use the same geometric and discretization model as in ^{3}, and the Poisson ratio _{c} = 300_{t} = 41^{∘} and soon turns to the original cracking orientation. For larger loading magnitude

The crack velocity is calculated by tracing the crack tip which is located by searching the most advanced with damage index higher than a given threshold (for current work, we choose 0.75).

We normalize velocities by the Rayleigh wave speed _{VR}_{VR}_{VR}

A flat shell with dimensions ^{∘}, 45^{∘}, 90^{∘}, respectively. The material of the flat shell has Young’s modulus of _{c} = 36_{t} = 2.7

^{∘}. As the deflection increases, the crack propagates continuously and maintains the initial orientation until it runs through the whole shell, as shown in

^{∘}. The initial crack is illustrated in ^{∘}, the newly propagated crack path is almost parallel to the free edge on which the external load is applied. Compare with the XFEM numerical simulation result in

^{∘}. As the deflection increase, the crack extends towards the loading edge first as illustrated in

We also analyze the damage evolution process of the flat shell with the initial crack notch at the free edge where the external loading is applied. As shown in

In the current work, we present a brittle fracture model of the Reissener-Mindlin shell based on the non-ordinary state-based Peridynamic shell theory. This model could be used in predicting the crack initialization and propagation. The validity of the model is validated through both quasi-static and dynamic analysis, and the accuracy and convergence of the shell model are also investigated through two numerical examples. Then, we demonstrate the capability of this model in simulating the crack growth by several numerical examples with preset crack. Comparing the simulation of current work and the experiment result and the numerical simulation result of XFEM, our model is convincing in predicting the crack path of brittle material thin shell. We believe our work provides a practical and reliable approach for thin shell brittle fracture in engineering computations. This fracture model shows the capability of the brittle fracture in terms of crack nucleation and branching for both in-plane and out-plane loads. The initiation and propagation of cracks are formed and developed naturally and spontaneously.

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.