An ABAQUS Implementation of Regularized Extended Finite Element Method (Rx-FEM) for Modeling the Interaction Between Matrix Cracking and Delamination in Composites

YU-JUI LIANG, JEFFREY S. MCQUIEN and ENDEL V. IARVE

ABSTRACT

Regularized Extended Finite Element Method (Rx-FEM) is a discrete damage

modeling (DDM) method, which represents an approach to progressive failure analysis (PFA) in composites when multiple individual damage events such as matrix cracks and delamination are introduced into the model via the displacement discontinuities. Rx-FEM is a variant of the eXtended Finite Element Method (x-FEM) where a continuous approximation of Heaviside step function is introduced unlike x-FEM. The proposed implementation to represent the enriched displacement field is based on the superposition of native ABAQUS elements with UMAT and UEL which is used to connect the superimposed elements. Both user subroutines are utilized to facilitate the step function based energy conditions required for mesh independent crack (MIC) propagation within a lamina. In this paper, an interface UEL is developed to connect different unidirectional plies of the laminate for modeling the inter-ply delamination and the interaction between matrix crack and delamination. Several benchmark examples with analytical solutions including the DCB, ENF and TCT models were performed to verify the accuracy of the interface UEL for mode I, mode II fracture, and the interaction between matrix crack and delamination, respectively. The ABAQUS implementation of Rx-FEM not only demonstrates accurate predictions, but also is able to use built-in ABAQUS capabilities such as contact between plies with mesh independent Rx-FEM cracks. INTRODUCTION

Discrete Damage Modeling (DDM) is based on direct simulation of

displacement discontinuities associated with individual damage events such as matrix cracking occurring within a composite ply, and the delamination at the interface between the plies. Some DDM methods employ techniques for crack growth without

________________ Yu-Jui Liang and Endel V. Iarve, Department of Mechanical and Aerospace Engineering, The University of Texas at Arlington, Arlington, TX 76019, USA Jeffrey S. McQuien, Institute for Predictive Performance Methodology, The University of Texas at Arlington Research Institute, Fort Worth, TX 76118, USA

remeshing called mesh independent crack (MIC), such as the eXtended Finite Element. Method (x-FEM) [1] and the Regularized Extended Finite Element Method (Rx-FEM) [2, 3] in particular, where the continuous approximation of the Heaviside step function changing from 0 to 1 over a narrow volume called gradient zone is introduced unlike x-FEM. This Rx-FEM formalism tying the volume integrals in the gradient zone to the surface integrals in the limit of mesh refinement was discussed in [2]. The simulation begins without any initial matrix cracks, which are then inserted based on a failure criterion during the simulation. Each MIC represents a cohesive zone, inserted independent of the mesh orientation by means of Rx-FEM, wich will begin to open under increasing load to form a matrix crack. To date this methodology is implemented in an in-house custom software program BSAM, and extensively applied to static and fatigue analysis in laminated composite structures [4, 5, 6, 7]. Commercial finite element software has several advantages including established markets, reputation, very powerful solver, and many built-in capabilities which are impractical to implement again in a research code. The implementation of DDM methodologies in commercial code is significantly more challenging than implementation of Continuum Damage modeling (CDM) methodologies, where the damage is represented as a change of element stiffness properties. To date several implementations of DDM methodology into the commercial FE software and ABAQUS [8] in particular have been reported [9, 10, 11]. What is common in these implementations is that a user defined element (UEL) was created encapsulating the kinematics of the x-FEM element pair, which is produced after the degrees of freedom (d.o.f.) are enriched. This type of implementation has wide freedom to allow customizing the integration scheme of the cracked elements and introducing either phantom nodal d.o.f. (Augmented-FEM [10]) or d.o.f. representing displacements at the crack surface and element edge intersection (Floating node method [11]). However, these implementations only allow to take advantages of a very small subset of capabilities in the commercial FE software. This approach can take advantage of the solver portion of the code, while even visualization represents challenges, not to speak of more essential feature such as contact, etc.

A different path of Rx-FEM implementation in ABAQUS was proposed by Liang et al [12]. The continuous approximation of the Heaviside step function offers unique possibility to implement the Rx-FEM in commercial FE software. The proposed implementation represents the enriched displacement field by superimposing two native ABAQUS elements instead of creating a fully encapsulated UEL. These element require UMAT for multiplication of stiffness matrix with the regularized step function, and a UEL for introducing the volumetric cohesive forces as described in Ref. [11] and below. This implementation capitalizes on utilization of original Gauss integration schema in Rx-FEM even after the enrichment is introduced to accommodate a MIC. In this article, an interface UEL was developed to connect different unidirectional plies of the laminate for modeling the inter-ply delamination and the interaction between matrix crack and delamination. Several benchmark examples with analytical solutions including the DCB, ENF and TCT models were performed to verify the accuracy of the interface UEL for mode I, mode II fracture, and the interaction between matrix crack and delamination, respectively. The ABAQUS implementation of Rx-FEM not only demonstrates accurate predictions, but also is able to use built-in ABAQUS capabilities such as contact between plies with mesh independent Rx-FEM cracks.

METHODOLOGY Rx-FEM methodology The DDM approach consists Rx-FEM modeling of matrix cracks in each ply of the laminate, and modeling the delamination between plies by using a cohesive zone formulation at the ply interface. The simulation begins without any initial matrix cracks built into the model. If the failure criterion LaRC04 [13], which is in the present work, is met a given integration point a level set function of the signed distance function of the crack surface is introduced, and a Heaviside step function associated with this level set represents the displacement discontinuity due to the crack. The crack orientation is defined parallel to the fiber direction and at an angle calculated by LaRC04 to the ply surface. In the Rx-FEM formulation [2, 3], this Heaviside step function is replaced by a continuous function changing from 0 to 1 over a narrow volume called gradient zone, and is approximated by the same shape functions as the displacements. An advantage of Rx-FEM is in maintaining a fixed Gauss integration schema throughout the analysis without regard to location and direction of the crack created during the analysis. The elements, as opposed to x-FEM are not partitioned since the crack surface in not inside an element, but rather spread over several neighbor elements. The approximation of the Heaviside step function is

𝐻 𝑥 ∑ 𝑁 𝑥 𝐻 (1) where 𝑁 is the total number of nodes, 𝑥 is the spatial coordinate, 𝑁 is the shape function associated with node a, summed over all elements sharing the nodes, and the coefficients 𝐻 are calculated as follows

H 1| |

(2)

where x is the global coordinate, f is the signed distance function of the crack surface. The principle of minimum potential energy for the enriched displacement field can be written as

δ H x W v 1-H x W v M 0 (3)

where 𝑊 𝑣 and 𝑊 𝑣 are strain energy on the two sides of the crack surface and M is the cohesive energy of the MIC. In this work, a cohesive zone model (CZM) in terms of opening tractions and opening displacements will be used for MIC propagation. Initially, the surface traction is related to the interfacial displacement jump by a high penalty stiffness. If the displacement jump exceeds a critical traction, then the cohesive softening will initiate. Complete separation is achieved when the displacement jump exceeds the final displacement. The area under the traction-displacement curve is equal to the fracture toughness. In the application of CZM to Rx-FEM, the displacement jump of the crack surface is evaluated in the volume of the gradient zone, where ∇𝐻 0, as

opposed to the traditional x-FEM crack surface based evaluation. An equivalent crack surface energy is calculated by using the following volume integral

M ∑ 𝑔 ∙ ∇𝐻 ∙ 𝑑𝑉 (4)

where the M is cohesive energy associated with crack, 𝑣 is the element domain in the volume of the gradient zone, g is pointwise cohesive energy corresponding to the displacement jump in the gradient zone, and 𝑁 is the number of elements. Although in the Rx-FEM methodology more elements will be enriched to model the same crack than in the x-FEM, additional computational work is more than compensated by the preservation of the same original Gaussian integration scheme in each Rx-FEM element. Rx-FEM implementation for modeling MIC in ABAQUS

Figure 1 shows the schematics of the implementation based on representing the enriched displacement field by using superposition of two native ABAQUS elements which are connected by a cohesive UEL. The UMAT capability is utilized to facilitate the step function based energy conditions required for crack propagation in the superimposed elements. Figure 2 shows the schematics of regularized Heaviside step function 𝐻 𝑥 for the original and duplicated elements at hand, and the gradient of regularized Heaviside step function ∇𝐻 within a single central crack region. In additional, a user element UEL is created to accommodate the cohesive tractions M. Reference [12] provides more detail which outlines the Rx-FEM implementation for modeling the MIC insertion in ABAQUS, and several examples of unnotched plates and open hole unidirectional coupons were considered.

Figure 1. Two native ABAQUS elements connected by a cohesive UEL.

Figure 2. The schematics of regularized Heaviside step function and its gradient within a crack region.

Rx-FEM implementation for modeling delamination in ABAQUS In this work, an interface cohesive UEL connecting two plies was developed to predict the inter-ply delamination and the interaction between matrix cracking and delamination. In each ply the Rx-FEM implementation is based on representing the enriched displacement field by using superposition of two native ABAQUS elements, which we shall call original and duplicate and note that these elements are defined in the input deck and active through the analysis. Thus, to describe the displacement jump within the Rx-FEM matrix crack region, the entire displacement field can be described as

𝐮 𝐻 ∙ 𝐮 1 𝐻 ∙ 𝐮 (5)

where 𝐻 is the regularized Heaviside step function, and 𝐮 and 𝐮 represent the displacement approximation of original element and duplicate element, respectively. Figure 3 shows the schematic of interface cohesive UEL as well as the normal (∆ , ∆ ) and shear (∆ , ∆ ) components of displacement jump in the local coordinates. Suppose a MIC exists in each ply as well as the associated regularized step function 𝐻 𝑥 and 𝐻 𝑥 . We will define the values of the step

function in the original element nodes in each ply first and denote them as 𝐻 ,

and 𝐻 , . The respective values of the step function in the nodes of the duplicate

elements are then computed as 1 𝐻. We will further omit the superscript Top and Bottom Ply since we have unique nodal indexes for each ply eliminating confusion as to which ply the step function belongs. Finally, the displacement jump between two plies can be computed by taking in to account Equation (5) and the notations described above as following ∆𝟏 𝐻 𝐮𝟒 1 𝐻 𝐮𝟖 𝐻 𝐮𝟏 1 𝐻 𝐮𝟓 ∆𝟐 𝐻 𝐯𝟒 1 𝐻 𝐯𝟖 𝐻 𝐯𝟏 1 𝐻 𝐯𝟓

∆𝟑 𝐻 𝐮𝟑 1 𝐻 𝐮𝟕 𝐻 𝐮𝟐 1 𝐻 𝐮𝟔 ∆𝟒 𝐻 𝐯𝟑 1 𝐻 𝐯𝟕 𝐻 𝐯𝟐 1 𝐻 𝐯𝟔 (6)

where ui and vi are the nodal displacements of normal and shear components. The interface UEL contains nodes 1-8, and calculates the cohesive traction as a result of separation, which can be obtained as

∆𝑢∆𝑣

𝑁0

0𝑁

𝑁0

0𝑁 ∆ ∆ ∆ ∆ , and 𝑁 , 𝑁 (7)

where the shape function matrix 𝑵 in the natural coordinate (s) is that of the original or duplicate element which are identical. A specific form of the traction-separation law of the implemented CZM is then used to calculate a pointwise traction vector 𝒕 , which prior to damage initiation is defined by high penalty coefficients 𝐾 and 𝐾 as

𝑡𝑡

𝐾 ∆𝑢𝐾 ∆𝑣 (8)

The mixed-mode cohesive zone interface fracture model proposed by Turon et al. [14] is used, and the reader is referred to the paper for full details. The formulation of the interface cohesive UEL requires a residual load vector often denoted as {RHS} in ABAQUS. In the present implementation, it will contain 16 components corresponding to 8 nodes with x and y d.o.f. each. Adopting the following sequence of the d.o.f.

𝑢 𝑢 𝑢 𝑢 𝑢 𝑢 𝑢 𝑢 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 (9) Therefore, an interface cohesive UEL is used to connect four native ABAQUS elements which include original and duplicate elements for top ply and bottom ply. When the top and bottom plies start to move apart, the displacement jump between plies will be created, which generates cohesive tractions and eventually softening. The energetics of such separation has been shown previously to closely approximate the fracture mechanics formulation.

Figure 3. Interface cohesive UEL and displacement separation in local coordinates.

Solution algorithm Figure 4 shows the Rx-FEM implementation in ABAQUS framework. The MIC related information is stored in Rx-FEM data structure, which is initiated at the first call of UEXTERNALDB before the first increment. In the beginning of the calculation, 𝐻 𝑥 =1 for all original elements, and the elastic stiffness of the duplicated elements

1 𝐻 𝑥 is respectively 0. The cohesive UEL for MIC is initially set to provide a tie

between the original and duplicate elements and making them move together. Since the stiffness of the duplicate elements is 0, there is no resistance and/or separation. On all subsequent calls of UEXTERNALDB at the end of each increment, the stress tensor calculated by UMAT is evaluated by the LaRC04 [13] failure criterion. If there is any element reaching failure, one or more MICs will be inserted and the MIC related information will be updated. It includes calculation of the step function 𝐻 𝑥 and its gradient ∇𝐻 in the affected elements. When ∇𝐻 0, the UEL formulation is invoked and the stiffness of the original and duplicate elements is different and not equal to 0. In this case they start moving independently. When the applied load is sufficient to overcome cohesive forces, the separation occurs and CZ begins to open to form a real matrix crack.

Figure 4. The framework of the Rx-FEM implementation in ABAQUS.

VERIFICATION MODELS Double cantilever beam (DCB) test DCB benchmark test [15] is used to verify the performance of the interface cohesive UEL in our ABAUQS implementation of Rx-FEM under pure mode I fracture. Our interface cohesive UELs are set to connect between top beam and bottom beam along the potential crack propagation path. The native ABAQUS plane strain elements CPE4 are used both for original and duplicate elements in this DCB model as well as the ENF model in our Rx-FEM implementation in ABAQUS. While not necessarily displaying MIC these examples provide benchmark for verification of the correct behavior of the proposed interface UEL. As shown in Figure 5, the dimensions of the DCB specimen are length 2L = 100 mm, thickness h = 3 mm, width is 20 mm, and the right end of the DCB model is a fixed boundary condition. An initial crack of length a = 30 mm is modeled and the material properties are given in Table 1. The mesh size 0.2

mm along the interface and 0.3 mm through the thickness are sufficiently fine, and geometrically linear analysis is applied in ABAQUS/Standard. For the pure mode I static failure analysis, the load-displacement curve of the DCB test is compared with the analytical solution as shown in Figure 6. The CBT and MBT are the classical beam theory and modified beam theory, respectively. It can be seen that the simulation result has a good agreement with the MBT analytical solution for this DCB analysis. The proposed implementation, which is based on utilization of the native ABAQUS elements allows to take advantage of a broad range of host software features, e.g. geometrically nonlinear analysis, which is also shown on Figure 6. In this particular case the two solutions are close as expected.

Table 1. Material properties for the DCB and ENF models.

Figure 5. The geometry of the DCB model.

Figure 6. Load-displacement curve of the DCB test.

End-notched flexure (ENF) test ENF benchmark test [15] is used to verify the performance of the interface cohesive UEL in our ABAUQS implementation of Rx-FEM under pure mode II failure. As shown in Figure 7, the dimensions of the ENF specimen are the same as the DCB specimen, and the material properties are given in Table 1 as well. The left end node and right end node of the ENF bottom surface are the pinned and roller boundary conditions, respectively. Similar to the DCB both the upper and bottom plies are modeled by using superimposed original and duplicate elements and connected by the developed interface cohesive UEL.

In the present case we are employing another capability of the host software which is the ability to model the contact interaction between plies. ABAQUS built-in capability including “frictionless” and “hard contact” is used at the interface in this ENF model. The mesh size 0.2 mm along the interface and 0.3 mm through the thickness are sufficiently fine, and geometric linear analysis is applied in ABAQUS/Standard. For the pure mode II static failure analysis, the load-displacement curve of the ENF test is compared with the analytical solution as shown in Figure 8. It can be seen that the simulation result has a good agreement with the analytical solution for this ENF analysis. The deformed shape of the ENF specimen is shown on Figure 9 displaying correct contact between plies. It is worth pointing out that another important feature of the host software which the Rx-FEM implementation allows utilize is the visualization of results.

Figure 7. The geometry of the ENF model.

Figure 8. Load-displacement curve of the ENF test.

Figure 9. Deformed shape of the ENF specimen displaying the contact between delaminated plies.

Transverse crack tension (TCT) test The TCT specimen, described in reference [16], was examined to evaluate the accuracy of the proposed implementation for predicting the interaction between a matrix crack and delamination. As shown in Figure 10, the TCT model consists of three 0 degree unidirectional plies with h, 2h and h from top to bottom, length L = 120 mm, thickness h = 1 mm, and width is 32 mm. An initial crack is cut through the width at the

middle length. The delaminations between the plies are modeled by using the interface cohesive UEL previously described, and a contact behavior of the ABAQUS built-in capability is also activated similar to the ENF test. The TCT model is subjected to axial tension loading, and the T300/914C material properties used in this test are summarized in Table 2. The ABAQUS plane strain elements (CPE4) are used. The mesh size 0.5 mm along the interface and 0.5 mm through the whole thickness are sufficiently fine, and geometric linear analysis is applied in ABAQUS/Standard. The applied stress versus applied displacement curve compared with the analytical analysis is shown in Figure 11. When the displacement loading is applied, the applied stress initially increases linearly until the delamination initiation between the middle ply and the top and bottom plies starting from the middle crack. Then these delaminations start to propagate in a stable manner while the applied stress remains constant. Finally, the load starts increasing again after the delaminations reach the grips, but with a different slope. It can be seen that the simulation result has a good agreement with the analytical solution for this TCT analysis.

Table 2. T300/914C Material properties for the TCT models.

Figure 10. The geometry of the TCT model.

Figure 11. Load-displacement curve of the TCT test.

CONCLUSIONS An ABAQUS implementation of Rx-FEM for modeling mesh independent matrix cracking and delamination propagation and the interaction between matrix cracking and delamination was performed in 2D formulation. An interface cohesive UEL was developed to connect two plies with Rx-FEM cracking. This UEL connects four native ABAQUS elements which include original and duplicate elements, required for mesh independent crack modeling, in the top ply and bottom ply. Several benchmark examples including DCB, ENF and TCT specimens were considered. The interface cohesive UEL provided accurate description of the delamination propagation in all cases. The proposed Rx-FEM implementation based on superposition of native host software elements allows to take advantage of various built-in host software capabilities. A geometrically nonlinear solution regime was demonstrated for the DCB example, utilization of the contact interaction was demonstrated for ENF example and TCT. In addition utilization of the native elements allows for straight forward visualization of the results. ACKNOWLEDGEMENTS This work was funded under NASA contact number NNL16AA02C with the University of Texas at Arlington, Arlington, TX.

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