Patent Publication Number: US-2023153489-A1

Title: Systems and methods for semi-discrete modeling of delamination migration in composite laminate materials

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application No. 63/279,448, filed Nov. 15, 2021, and entitled “SYSTEMS AND METHODS FOR SEMI-DISCRETE MODELING OF DELAMINATION MIGRATION IN COMPOSITE LAMINATE MATERIALS”, which is incorporated herein by reference in its entirety. 
    
    
     FIELD OF THE DISCLOSURE 
     The present disclosure relates generally to techniques for modeling composite laminate materials and, more particularly, to systems and methods for semi-discrete modeling of delamination migration in composite laminate materials. 
     BACKGROUND 
     The background description provided herein is for the purpose of generally presenting the context of the disclosure. Work of the presently named inventor, to the extent it is described in this background section, as well as aspects of the description that may not otherwise qualify as prior art at the time of filing, are neither expressly nor impliedly admitted as prior art against the present disclosure. 
     Fiber reinforced composites are the material of choice for a wide variety of industries where light-weighting is important. This trend is attributed to their specific stiffness, strength, and tailorability. Popular composite material systems are fiber-reinforced systems due to their excellent mechanical properties and high quality manufacturable products. Available as convenient prepreg tapes, they allow an efficient manufacturing process, especially in combination with the automated fiber placement (AFP) technology. However, despite the advances in material and manufacturing technologies, progressive damage and failure modeling of composite structures still remains a great challenge. 
     The problem is particularly acute because the failure mechanisms in a composite material are particularly complex due to the presence of different length scales (e.g. micro scale with the fiber and matrix constituents and macro/meso scale with the laminae and interfaces). Given the hierarchical structure, the mechanics of composites is a highly multi-scale problem, which results in a variety of damage and failure modes, including their interactions. 
     Therefore, there is a need for techniques capable of accurately and efficiently modeling the delamination migration (and generally, the full failure progression) in composite laminate materials to enhance the overall design and resulting capabilities of the composite laminate materials. 
     SUMMARY OF THE INVENTION 
     According to an aspect of the present disclosure, a computer implemented method for semi-discrete modeling of delamination migration in composite laminate materials includes: receiving, from a user, a specimen geometry and a specimen stacking sequence; creating, by one or more processors, a finite-element (FE) mesh by: generating, using a mesh generation tool, a plurality of plies each shaped according to the specimen geometry, wherein each ply includes (i) a plurality of fibrous strips along a fiber direction and (ii) a bulk element between each of the plurality of fibrous strips, and connecting, using the mesh generation tool, the plurality of plies together based on the stacking sequence by placing a plurality of cohesive elements between each adjacent pair of plies, wherein the FE mesh defines a composite laminate material; and determining, by the one or more processors, a predicted mechanical response of the composite laminate material by: generating a constitutive model corresponding to the composite laminate material based on the FE mesh, and inputting a strain value to the constitutive model to generate the predicted mechanical response. 
     In a variation of this aspect, connecting the plurality of plies further comprises: connecting, using the mesh generation tool, the plurality of plies together based on the stacking sequence by placing the plurality of cohesive elements between each adjacent pair of plies based on a set of tie constraints. Further in this variation, the set of tie constraints includes enforcing that nodes of the plurality of cohesive elements are located on boundaries of predicted intra-ply matrix cracks. 
     In another variation of this aspect, connecting the plurality of plies further comprises: partitioning an interlayer between each adjacent pair of plies by defining a plurality of interlayer partition features that surround each fibrous strip included in each respective adjacent pair of plies. Further in this variation, the plurality of interlayer partition features surround each fibrous strip included in each respective adjacent pair of plies by satisfying or exceeding a width tolerance to shift the interlayer partition features toward bulk elements included in each respective adjacent pair of plies. Yet further in this variation, the method includes: placing a cohesive element at least at an intersection of each respective pair of interlayer partition features. 
     In yet another variation of this aspect, the constitutive model includes mixed-mode conditions to model delamination failure in the composite laminate material. 
     In another aspect of the present disclosure, a system for semi-discrete modeling of delamination migration in composite laminate materials includes: a user interface; a memory storing a set of computer-readable instructions comprising at least a mesh generation tool; and a processor interfacing with the user interface and the memory, and configured to execute the set of computer-readable instructions to cause the processor to: receive, from a user, a specimen geometry and a specimen stacking sequence; create a finite-element (FE) mesh by: generating, using a mesh generation tool, a plurality of plies each shaped according to the specimen geometry, wherein each ply includes (i) a plurality of fibrous strips along a fiber direction and (ii) a bulk element between each of the plurality of fibrous strips, and connecting, using the mesh generation tool, the plurality of plies together based on the stacking sequence by placing a plurality of cohesive elements between each adjacent pair of plies, wherein the FE mesh defines a composite laminate material; and determine a predicted mechanical response of the composite laminate material by: generating a constitutive model corresponding to the composite laminate material based on the FE mesh, and inputting a strain value to the constitutive model to generate the predicted mechanical response. 
     In a variation of this aspect, the set of computer-readable instructions further cause the processor to connect the plurality of plies by: connecting, using the mesh generation tool, the plurality of plies together based on the stacking sequence by placing the plurality of cohesive elements between each adjacent pair of plies based on a set of tie constraints. Further in this variation, the set of tie constraints includes enforcing that nodes of the plurality of cohesive elements are located on boundaries of predicted intra-ply matrix cracks. 
     In another variation of this aspect, the set of computer-readable instructions further cause the processor to connect the plurality of plies by: partitioning an interlayer between each adjacent pair of plies by defining a plurality of interlayer partition features that surround each fibrous strip included in each respective adjacent pair of plies. Further in this variation, the plurality of interlayer partition features surround each fibrous strip included in each respective adjacent pair of plies by satisfying or exceeding a width tolerance to shift the interlayer partition features toward bulk elements included in each respective adjacent pair of plies. Yet further in this variation, the set of computer-readable instructions further cause the processor to connect the plurality of plies by: placing a cohesive element at least at an intersection of each respective pair of interlayer partition features. 
     In yet another variation, the constitutive model includes mixed-mode conditions to model delamination failure in the composite laminate material. 
     In yet another aspect of the present disclosure, a non-transitory computer-readable storage medium having stored thereon a set of instructions, executable by at least one processor, for semi-discrete modeling of delamination migration in composite laminate materials that include: instructions for receiving, from a user, a specimen geometry and a specimen stacking sequence; instructions for creating a finite-element (FE) mesh by: generating, using a mesh generation tool, a plurality of plies each shaped according to the specimen geometry, wherein each ply includes (i) a plurality of fibrous strips along a fiber direction and (ii) a bulk element between each of the plurality of fibrous strips, and connecting, using the mesh generation tool, the plurality of plies together based on the stacking sequence by placing a plurality of cohesive elements between each adjacent pair of plies, wherein the FE mesh defines a composite laminate material; and instructions for determining a predicted mechanical response of the composite laminate material by: generating a constitutive model corresponding to the composite laminate material based on the FE mesh, and inputting a strain value to the constitutive model to generate the predicted mechanical response. 
     In a variation of this aspect, the set of instructions further comprise: instructions for connecting, using the mesh generation tool, the plurality of plies together based on the stacking sequence by placing the plurality of cohesive elements between each adjacent pair of plies based on a set of tie constraints. In another variation of this aspect, the set of tie constraints includes enforcing that nodes of the plurality of cohesive elements are located on boundaries of predicted intra-ply matrix cracks. 
     In another variation of this aspect, the set of instructions further comprise: instructions for partitioning an interlayer between each adjacent pair of plies by defining a plurality of interlayer partition features that surround each fibrous strip included in each respective adjacent pair of plies; and instructions for placing a cohesive element at least at an intersection of each respective pair of interlayer partition features. Further in this variation, the plurality of interlayer partition features surround each fibrous strip included in each respective adjacent pair of plies by satisfying or exceeding a width tolerance to shift the interlayer partition features toward bulk elements included in each respective adjacent pair of plies. 
     In yet another variation of this aspect, the constitutive model includes mixed-mode conditions to model delamination failure in the composite laminate material. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The figures described below depict various aspects of the system and methods disclosed herein. It should be understood that each figure depicts an embodiment of a particular aspect of the disclosed system and methods, and that each of the figures is intended to accord with a possible embodiment thereof. Further, wherever possible, the following description refers to the reference numerals included in the following figures, in which features depicted in multiple figures are designated with consistent reference numerals. 
         FIG.  1 A  illustrates an example system for semi-discrete modeling of delamination migration in composite laminate materials, in accordance various aspects disclosed herein. 
         FIG.  1 B  illustrates an example workflow for semi-discrete modeling of delamination migration in composite laminate materials utilizing several components from the example system of  FIG.  1 A , and in accordance various aspects disclosed herein. 
         FIG.  2 A  illustrates an example free meshing of bulk elements included as part of a composite laminate material, as utilized by the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. 
         FIG.  2 B  illustrates an example structure aligned meshing of fibrous strips and bulk elements included as part of a composite laminate material, as utilized by the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. 
         FIG.  2 C  illustrates an example discrete meshing of cohesive elements and bulk elements included as part of a composite laminate material, as utilized by the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. 
         FIG.  2 D  is an example generation sequence of a semi-discrete meshing of fibrous strips and bulk elements included as part of a composite laminate material, as generated by the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. 
         FIG.  3 A  illustrates an example partitioning of an interlayer within a composite laminate material, in accordance with various aspects disclosed herein. 
         FIG.  3 B  is an example generation sequence of a semi-discrete meshing overlay of plies and an interlayer included as part of a composite laminate material, as generated by the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. 
         FIG.  4 A  depicts an example open-hole tensile (OHT) laminate specimen after partitioning, in accordance with various aspects disclosed herein. 
         FIG.  4 B  depicts an example meshing at an elastic region of the example OHT laminate specimen of  FIG.  4 A , in accordance with various aspects disclosed herein. 
         FIG.  4 C  depicts an example meshing at a cut-out region of the OHT laminate specimen of  FIG.  4 A , in accordance with various aspects disclosed herein. 
         FIG.  4 D  depicts an example compatible meshing included as part of the OHT laminate specimen of  FIG.  4 A , in accordance with various aspects disclosed herein. 
         FIG.  5    illustrates differences between specimen partitioning and meshing procedures for plies and interlayers of a composite laminate material, in accordance with various aspects disclosed herein. 
         FIG.  6    is an example graph illustrating a general pre-peak and post-peak stress-strain response of a composite laminate material, in accordance with various aspects disclosed herein. 
         FIGS.  7 A- 7 E  illustrate an example mixed-mode traction-separation law for cohesive behavior of a composite laminate material, in accordance with various aspects disclosed herein. 
         FIGS.  8 A- 8 D  depict example matrix crack patterns and delamination failure of the OHT laminate specimen of  FIG.  4 A , in accordance with various aspects disclosed herein. 
         FIG.  9    illustrates an example method for semi-discrete modeling of delamination migration in composite laminate materials, in accordance various aspects disclosed herein. 
     
    
    
     DETAILED DESCRIPTION 
     Numerous computational physics-based models, usually implemented using the finite-element (FE) method, have been developed in the past two decades to predict the progressive damage and failure of composite materials. However, each of these conventional techniques generally suffers from a trade-off between a lack of discreteness resolution and/or consuming a large amount of computational resources. For example, smeared methods are more efficient than discrete methods, but are limited in their capability to accurately capture sharp cracks. Discrete methods offer higher fidelity than smeared methods, but the number of degrees of freedom rises rapidly with the number of represented cracks, resulting in a significant amount of time and processing resources required for model generation. Neither accuracy nor efficiency should be sacrificed, especially because post-processing is critical and non-trivial when user-defined elements (UEL) are utilized alongside finite element analysis software. 
     By contrast, the systems and methods of the present disclosure provide a semi-discrete damage model that solves these problems experienced by conventional techniques. Namely, the systems and methods of the present disclosure can accurately and efficiently model the delamination migration and overall full failure progression in composite laminate materials to enhance the overall design and resulting capabilities of the composite laminate materials. More specifically, the techniques of the present disclosure are capable of predicting the delamination migration mechanism with an accurate peak load and the total failure due to fiber tensile rupture in a realistic manner for a composite laminate material that was previously unachievable using conventional techniques. 
     Broadly speaking, the present techniques utilize a finite element (FE) framework to capture the interaction of intra-laminar matrix cracks and inter-laminar failure (e.g., delamination migration) while retaining the capability to predict fiber failure. The plies included as part of the composite laminate material may be modeled with continuum finite elements, and the interlayers (e.g., a thin pure resin layer, adhesive, etc.) may be represented with cohesive elements. A mesh generation tool may create compatible meshes for the interlayers in order to capture a proper load transfer between the laminae and the interfaces. A solver module may then utilize a constitutive model representing the composite laminate material to determine mechanical responses of the ply materials and the interfaces. Accordingly, the ply materials may be modeled in the constitutive model using various pre-peak and post-peak criteria, and the cohesive behavior of the interfaces may be modeled within the constitutive model using a mixed-mode traction-separation law. 
     Thus, as a result of these and other features described herein, the techniques of the present disclosure generally provide advantages over conventional systems in a number of ways. The efficient meshing procedure utilized as part of the techniques of the present disclosure generates a semi-discrete compatible mesh that captures the matrix-crack-delamination interaction of the interlayers of composite laminate materials in a manner previously unachievable by conventional systems. Further, the mixed-mode formulations included in the constitutive model capture the damage evolution of matrix cracks and delamination, thereby ensuring smooth and simultaneous vanishing of all traction components. Consequently, the techniques of the present disclosure are able to predict the delamination migration and the critical load drops within a composite laminate material more accurately than conventional systems, while outputting details of the matrix cracking and delamination migration. 
     In particular, the techniques of the present disclosure utilize cohesive elements in order to model delamination of the composite laminate material. Conventional techniques may generally attempt to perfectly match nodes between the plies and the interlayers and/or generate material points that are projections of slave nodes. These conventional techniques therefore suffer from inefficiencies or biases because matching nodes using a semi-discrete mesh are virtually impossible, and delamination predictions may be biased towards the slave surface. However, the mesh generation tool of the present disclosure may connect the interlayers of the material to the plies using tie constraints, thereby improving over these conventional techniques by removing the inefficiencies and biases. 
     To provide a general understanding of the system(s)/components utilized in the techniques of the present disclosure,  FIGS.  1 A and  1 B  illustrate, respectively, an example system  100  and an example workflow  120  utilizing several components of the system  100  that are configured for semi-discrete modeling of delamination migration in composite laminate materials. Accordingly,  FIG.  1 A  provides a general overview of the components, and  FIG.  1 B  describes several of the components and their respective functions in greater detail. 
     In any event,  FIG.  1 A  illustrates an example system  100  for semi-discrete modeling of delamination migration in composite laminate materials. It should be appreciated that the example system  100  is merely an example and that alternative or additional components are envisioned. 
     As illustrated in  FIG.  1 A , the example system  100  may be a user computing device and/or a server that includes a processor  102 , a user interface  104 , and a memory  106 . The memory  106  may store an operating system  108  capable of facilitating the functionalities as discussed herein, as well as other data  112 , and a composite laminate modeling application  110  including a free hex-dominated advancing front meshing algorithm  110   a , a structured hex meshing algorithm  110   b , a constitutive model  110   c , a mesh generation tool  110   d , and a compatible interface model  110   e . Generally, the processor  102  may interface with the memory  106  to access/execute the operating system  108 , the other data  112 , the free hex-dominated advancing front meshing algorithm  110   a , the structured hex meshing algorithm  110   b , the constitutive model  110   c , the mesh generation tool  110   d , and the compatible interface model  110   e . The other data  112  may include a set of applications configured to facilitate the functionalities as discussed herein, and/or may include other relevant data, such as display formatting data, etc. For example, the processor  102  may access the operating system  108  in order to execute applications included as part of the other data  112 , such as a modeling application (not shown) configured to facilitate functionalities associated with semi-discrete modeling of delamination migration in composite laminate materials, as discussed herein. In certain aspects, the free hex-dominated advancing front meshing algorithm  110   a  and/or the structured hex meshing algorithm  110   b  may be provided and/or executed by a suitable commercial software package, such as ABAQUS/CAE, or the like. It should be appreciated that one or more other applications are envisioned. Moreover, it should be understood that any processor (e.g., processor  102 ), user interface (e.g., user interface  104 ), and/or memory (e.g., memory  106 ) referenced herein may include one or more processors, one or more user interfaces, and/or one or more memories. 
     Generally, the processor  102  may access the memory  106  to execute the mesh generation tool  110   d  in order to automatically create, partition, and mesh the various portions of a composite laminate material (e.g., plies and interlayers). As described herein, the mesh generation tool  110   d  may also define and assign material properties, tie portions of the composite laminate material together, and define surfaces, contact interactions, boundary conditions, steps, mass scaling, and output requests. 
     The mesh generation tool  110   d  may receive a strip width, a strip spacing, a stacking sequence, a specimen geometry, and/or any other suitable input(s) from a user, and may proceed to partition the material base, such that thin strips of elements (also referenced herein as “element strips”, “fibrous strips”, and “strips”) with the specified strip width are generated along a fiber direction with the specified strip spacing separating each respective element strip. The mesh generation tool  110   d  may also access or otherwise include instructions that cause the processor  102  to execute the structured hex meshing algorithm  110   b  and/or the free hex-dominated advancing front meshing algorithm  110   a  in order to create the meshing within the element strips and the bulk elements separating the element strips. As a result, the mesh generation tool  110   d  may generate multiple plies that are subsequently connected using the compatible interface model  110   e  based on the stacking sequence by placing multiple cohesive elements between each adjacent pair of plies (also referenced herein as “neighboring plies”) to generate the completed composite laminate material. The completed composite laminate material model may then be analyzed in accordance with the constitutive model  110   c  to determine a predicted mechanical response and/or a predicted failure type of the composite laminate material. 
     More specifically, the structured hex meshing algorithm  110   b  may receive a fibrous strip width and a fibrous strip spacing from a user to determine a type and distribution of fibrous strips as part of a finite element (FE) mesh. The structured hex meshing algorithm  110   b  may generate a randomized distribution of fibrous strip materials, such that adjacent fibrous strips may be comprised of different fibrous strip materials. However, the structured hex meshing algorithm  110   b  may generate each fibrous strip at the fibrous strip spacing apart from each adjacent fibrous strip, and the algorithm  110   b  may generate each fibrous strip with a uniform width, as defined by the user-provided fibrous strip width. Moreover, the structured hex meshing algorithm  110   b  may generate each of the fibrous strips in a fiber direction, such that the fibrous strip spacing may be maintained between each set of adjacent fibrous strips. Additionally, or alternatively, in certain aspects, the structured hex meshing algorithm  110   b  may generate fibrous strips in a perpendicular and/or otherwise different direction than the fiber direction. 
     The free hex-dominated advancing front meshing algorithm  110   a  may generally receive the FE mesh populated with the fibrous strips from the structured hex meshing algorithm  110   b  to determine a distribution of bulk elements between each of the fibrous strips. The free hex-dominated advancing front meshing algorithm  110   a  may implement any suitable type of meshing procedure in accordance with any suitable free mesh algorithm (e.g., sweep mesh with advancing front algorithm). In certain aspects, the free hex-dominated advancing front meshing algorithm  110   a  and/or the structured hex meshing algorithm  110   b  may generate a mesh corresponding to the strip elements and/or the bulk elements. For example, regions of the strip elements that include a complex geometry (e.g., near a hole or cut-out) may be meshed using any suitable free mesh technique from the free hex-dominated advancing front meshing algorithm  110   a , while regions of the same strip elements that include a less complex geometry may be meshed using a structured hex meshing technique from the structured hex meshing algorithm  110   b . As another example, regions of the composite laminate material (e.g., strip elements and bulk elements) that include a less complex geometry (e.g., far from holes, cut-outs, etc.) may be meshed using either a free mesh technique or a structured meshing technique. 
     The compatible interface model  110   e  may generally receive the stacking sequence from a user along with multiple plies generated as a result of the free hex-dominated advancing front meshing algorithm  110   a  and the structured hex meshing algorithm  110   b  in order to generate interlayers between each adjacent pair of plies. Generally, the compatible interface model  110   e  may enforce that nodes of the cohesive elements are located at the boundaries of potential intra-ply matrix cracks, as described further herein. The mesh generation tool  110   d  may utilize these constraints from the compatible interface model  110   e  to partition the interlayer by generating strips in the directions of the two neighboring plies. In order to maintain a robust tie between the elements of the plies and those of the interlayer, the compatible interface model  110   e  may also include instructions to widen the strips corresponding to the interlayer by a width tolerance from the original width of the strips in the neighboring plies to shift the nodes slightly towards the bulk regions. Moreover, the compatible interface model  110   e  may include instructions that the in-plane element size may be smaller than the corresponding element size of the neighboring plies to ensure a proper level of connectivity to both neighboring plies with non-matching meshes. In this manner, the compatible interface model  110   e  may create semi-discrete and compatible meshes given any arbitrary specimen shape without mesh correction measures, which is not achievable using conventional techniques. 
     Based on the completed FE mesh resulting from the outputs of the structured hex meshing algorithm  110   b , the free hex-dominated advancing front meshing algorithm  110   a , the compatible interface model  110   e , and the mesh generation tool  110   d , the constitutive model  110   c  may describe the mechanical response of the composite laminate material represented by the completed FE mesh under various stresses and/or strains. In particular, the constitutive model  110   c  may describe the mechanical response of the composite laminate material represented by the completed FE mesh using a set of pre-peak conditions (prior to mechanical failure), transition criteria (at the point of mechanical failure), and post-peak conditions (after mechanical failure). The constitutive model  110   c  may include criteria for material behavior corresponding both to the three-dimensional (3D) continuum elements (e.g., intra-laminar elements) and 3D cohesive elements (e.g., inter-laminar elements). In order to generate the constitutive model  110   c , a user may define the material constitutive behaviors according to any suitable theory/law. For example, a user may define that Schapery theory (ST) and crack band (CB) methods should represent the continuum elements, and that a mixed-mode traction-separation law should represent the cohesive elements. 
     All regions of the FE mesh may be modeled with pre-peak non-linearity, but the failure modes of the FE mesh may be separated out into two distinct domains including matrix splitting failure and fiber failure. For example, matrix splitting failure may be only active in the fibrous strips, while fiber failure may be active in every element (e.g., fibrous strips and bulk elements). In fact, by reducing the width of the matrix-splitting strips, the discreteness of the method is highly increased (approaching discrete methods), while the efficiency is maintained by using only continuum elements with smeared crack damage methods (e.g., crack band) for the post-peak failure. Nevertheless, in certain aspects, the failure modes of the FE mesh may not be separated between the fibrous strips and the bulk elements, such that matrix splitting failure and fiber failure may be active in both the fibrous strips and the bulk elements. 
     Further, the inter-laminar portions of the FE mesh may be modeled with a mixed-mode condition(s), that may require that all tractions within the FE mesh vanish simultaneously at a crack propagation. Namely, the constitutive model  110   c  may include instructions that when a crack is fully developed, a corresponding crack surface must become traction free. 
     In this manner, the combination of the structured hex meshing algorithm  110   b , the free hex-dominated advancing front meshing algorithm  110   a , the constitutive model  110   c , the mesh generation tool  110   d , and the compatible interface model  110   e  decomposes the bulk non-linearity and localization zones of the composite laminate material and provides for a proper load transfer pathway through the resulting FE mesh. Consequently, subsequent manufacturing of composite laminate materials may be improved by incorporating the distribution, sizing, spacing, material composition, etc. of the fibrous strips and bulk elements provided in the material models output by the composite laminate modeling application  110 . For example, the composite laminate modeling application  110  may be stored in a computing device of a composite laminate material manufacturing plant, and the computing device may run the composite laminate modeling application  110  prior to a run of the manufacturing equipment to determine optimal material characteristics for a particular use-case. Based on the material characteristics (e.g., fibrous strip and bulk element distribution, sizing, spacing, material composition, inter-laminar layer element spacing, adhesive points, etc.) output in a material model by the composite laminate modeling application  110 , the manufacturing equipment may manufacture a composite laminate material with those material characteristics to produce an optimally configured composite laminate material for the particular use-case. 
     Moreover, the mesh generation tool  110   d  allows a user to quickly and easily adjust the output material model by refining the mesh, adjusting material properties, analysis parameters, and the like without disrupting other aspects of the model. For example, and as discussed further herein, the material model may be split into multiple parts (e.g., files), such that the user may adjust portions of the model confined to a particular file without affecting the other files. As a result, the tool  110   d  also enables large numbers of simulations to be run for a particular mesh by randomizing the material properties of the mesh without affecting the previously generated mesh. 
     In any event, the memory  106  may include one or more forms of volatile and/or non-volatile, fixed and/or removable memory, such as read-only memory (ROM), electronic programmable read-only memory (EPROM), random access memory (RAM), erasable electronic programmable read-only memory (EEPROM), and/or other hard drives, flash memory, MicroSD cards, and others. 
     The example system  100  may further include a user interface  104  configured to present/receive information to/from a user. As shown in  FIG.  1 A , the user interface  104  may include a display screen  104   a  and I/O components 104b (e.g., ports, capacitive or resistive touch sensitive input panels, keys, buttons, lights, LEDs). According to some aspects, a user may access the example system  100  via the user interface  104  to review outputs from the free hex-dominated advancing front meshing algorithm  110   a , the structured hex meshing algorithm  110   b , the constitutive model  110   c , the mesh generation tool  110   d , the compatible interface model  110   e , make various selections, and/or otherwise interact with the example system  100 . 
     In some aspects, the example system  100  may perform the functionalities as discussed herein as part of a “cloud” network or may otherwise communicate with other hardware or software components within the cloud to send, retrieve, or otherwise analyze data. Thus, it should be appreciated that the example system  100  may be in the form of a distributed cluster of computers, servers, machines, or the like. In this implementation, a user may utilize the distributed example system  100  as part of an on-demand cloud computing platform. Accordingly, when the user interfaces with the example system  100  (e.g., by inputting a fibrous strip width, a fibrous strip spacing, a specimen geometry, and/or a stacking sequence), the example system  100  may actually interface with one or more of a number of distributed computers, servers, machines, or the like, to facilitate the described functionalities. 
     In certain aspects, the example system  100  may communicate and interface with an external server (not shown) via a network(s). The external server may be associated with, for example, an entity that fabricates and/or otherwise purchases composite laminate materials, and may receive the completed FE mesh and/or constitutive model from the example system  100 . In particular, the external server may include or support a web server configured to host a website that enables users to submit instructions to the example system  100  for generating the completed FE mesh and/or the constitutive model. For example, the external server may enable a user to submit the fibrous strip width, the fibrous strip spacing, the specimen geometry, the stacking sequence, and/or any other inputs that the example system  100  may use to generate the completed FE mesh and constitutive model. 
     Further in these aspects, the network(s) used to connect the example system  100  to the external server may support any type of data communication via any standard or technology (e.g., GSM, CDMA, TDMA, WCDMA, LTE, EDGE, OFDM, GPRS, EV-DO, UWB, Internet, IEEE 802 including Ethernet, WiMAX, Wi-Fi, Bluetooth, and others). Moreover, the external server may include a memory as well as a processor, and the memory may store an operating system capable of facilitating the functionalities as discussed herein as well as the free hex-dominated advancing front meshing algorithm  110   a , the structured hex meshing algorithm  110   b , the constitutive model  110   c , and/or the compatible interface model  110   e . 
     Additionally, it is to be appreciated that a computer program product in accordance with an aspect may include a computer usable storage medium (e.g., standard random access memory (RAM), an optical disc, a universal serial bus (USB) drive, or the like) having computer-readable program code embodied therein, wherein the computer-readable program code may be adapted to be executed by the processor(s)  102  (e.g., working in connection with the operating system  108 ) to facilitate the functions as described herein. In this regard, the program code may be implemented in any desired language, and may be implemented as machine code, assembly code, byte code, interpretable source code or the like (e.g., via Golang, Python, Scala, C, C++, Java, Actionscript, Objective-C, Javascript, CSS, XML). In some aspects, the computer program product may be part of a cloud network of resources. 
       FIG.  1 B  illustrates an example workflow  120  for semi-discrete modeling of delamination migration in composite laminate materials utilizing several components from the example system of  FIG.  1 A , and in accordance various aspects disclosed herein. Generally, the example workflow  120  depicts the flow of data from a user through a pre-processing stage  122 , a processing stage  124 , and a post-processing stage  126 , wherein a composite laminate material mesh is generated, analyzed, and the results are displayed for user interpretation/evaluation. 
     The example workflow  120  begins with the pre-processing stage  122 , where a user may input the geometric inputs  122   a  and the material property inputs  122   b . The geometric inputs  122   a  may include, for example, the specimen geometry (e.g., the shape and dimensions of the composite laminate material, including any cut-outs such as notches or holes), the fibrous strip width and spacing, the ply stacking sequence (the “layup”), and/or any other suitable values or combinations thereof. The material property inputs  122   b  may include the material properties of the composite laminate material, such as, for example, ply thickness, ply elastic properties, Schapery parameters, strength values, fracture toughness values, and/or any other suitable values or combinations thereof. Further, the material property inputs  122   b  may include strength distribution parameters corresponding to the transverse and shear strengths of the composite material, so that each strip may be may be assigned a randomized material strength during the processing stage  124 . 
     Once these inputs  122   a ,  122   b  are received, the mesh generation tool  110   d  may automatically generate the flexible meshing model  122   c . As previously mentioned, the mesh generation tool  110   d  may create any layers associated with the model  122   c , partition the layers, and mesh the parts of the layers (e.g., the plies and interlayers). Moreover, the tool  110   d  may automatically define and assign random material properties, and may define surfaces, contact interactions, boundary conditions, steps, mass scaling, and output requests corresponding to the flexible meshing model  122   c . When the flexible meshing model  122   c  is generated, the model  122   c  may write an input file  122   d  that comprises an executable file containing all definitions and defined parameters associated with the model  122   c  to be evaluated at the processing stage  124 . 
     However, prior to processing, the input file  122   d  may be split into multiple portions that include different aspects of the original input file  122   d . Generally, large models may have correspondingly large input files (e.g., input file  122   d ), so it may be more efficient and create a more user-friendly, easier access experience to separate the mesh portion of the input file  122   d  from the material definition portion of the file  122   d . Thus, the input file split tool  122   e  may split the input file  122   d  into a main input file  122   f  and an ending input file  122   g . The main input file  122   f  may include mesh definition components, such as element connectivity, nodal coordinates, set definition, surface definitions, sections assignments, and/or any other suitable components or combinations thereof. The ending input file  122   g  may include constraints, section controls, material definitions, and step definitions, such as boundary conditions, loading, mass scaling, output requests, and/or any other suitable definitions or combinations thereof. 
     Moreover, in certain aspects, the input file split tool  122   e  may modify the input files  122   f ,  122   g  to define interior surfaces of the plies that are included in the general contact interaction to prevent inter-penetration of elements. In some aspects, the input file split tool  122   e  may modify the input files  122   f ,  122   g  by adding keywords specifying that the characteristic length of the continuum elements is defined in the processing stage  124  (e.g., by characteristic length subroutine  124   b ). 
     In any event, when the main input file  122   f  and the ending input file  122   g  are generated and modified appropriately, the files  122   f ,  122   g  may be analyzed by the solver module  124   a  as part of the processing stage  124 . However, a strength of this workflow  120  is the possibility to perform an uncertainty quantification by running a large number of simulations without editing the mesh (e.g., as included in the main input file  122   f ). In order to perform such a large number of simulations, the material property distribution tool  122   h  may receive some/all of the material property inputs  122   b , and may edit the ending input file  122   g  by randomizing the material properties included therein, thereby producing the ending x input file  122   i . The material property distribution tool  122   h  may generate any suitable number of ending x input files  122   i  in order to create a plethora of randomized material property distributions featuring the same mesh (e.g., as included in the main input file  122   f ), but that have distinct matrix mode strength distributions by sampling new values from the associated probability distributions (e.g., Monte-Carlo simulations). It should be understood that the input file split tool  122   e  and the material property distribution tool  122   h  may be optional components of the example workflow  120 , and that similar functionality may be achieved using, for example, a suitable code configured to modify the material definitions within a commercial software package (e.g., ABAQUS/CAE) in which the mesh generation tool  110   d  may be executed. Regardless, when transitioning to the processing stage  124 , the main input file  122   f  and one of the ending input file  122   g  and an ending x input file  122   i  are selected as inputs to the solver module  124   a  to perform a simulation. 
     As part of the processing stage  124 , the solver module  124   a  may utilize each of the characteristic length subroutine  124   b , the laminae (e.g., individual plies) subroutine  124   c  (including the constitutive model  110   c ), and the inter-laminar subroutine  124   d  to generate the results  124   e . The solver module  124   a  may use the characteristic length subroutine  124   b  in order to determine the characteristic length of each continuum element. The solver module  124   a  may utilize the laminae subroutine  124   c  and the inter-laminar subroutine  124   d  to determine the material constitutive behavior of the intra-laminar materials and the inter-laminar materials, respectively. It should be understood that the solver module  124   a  may utilize additional files not shown in  FIG.  1 B , such as supplementary routines for calculation of internal variables, solving equations to determine the micro damage (e.g., Schapery) parameter. 
     More specifically, the laminae subroutine  124   c  may include the constitutive model  110   c , which may combine Schapery theory, failure criteria, and mixed-mode crack band in order to model the effects of intra-laminar stress and strain conditions on the composite laminate material. The laminae subroutine  124   c  may also implement element-wise global-local randomization to further randomize material properties of the composite laminate material by a scaling factor. The subroutine  124   c  may apply the scaling locally, for example, at every integration point of every element, and the scaling factor may be calculated using a global field function based on the coordinates within the model. This randomization may be applied by the laminae subroutine  124   c  particularly to randomize the strip element strengths, bulk element properties, and may also be applied to elastic properties, interlayer properties, and even properties within individual strip elements. 
     Additionally, the inter-laminar subroutine  124   d  may include an inter-laminar constitutive model  124   e  that generally models the effects of inter-laminar stress and strain conditions on the composite laminate material. The inter-laminar constitutive model  124   e  may include mixed-mode condition(s), that may require all tractions within the FE mesh to vanish simultaneously at a crack propagation. Namely, the inter-laminar constitutive model  124   e  may include instructions that when a crack is fully developed, a corresponding crack surface must become traction free. In certain aspects, and as previously mentioned, the constitutive model  110   c  may include suitable theories/models/laws sufficient to model both the intra-laminar and the inter-laminar material responses, such that the inter-laminar constitutive model  124   e  may be included as part of the constitutive model  110   c . 
     When the solver module  124   a  has applied/utilized each subroutine  124   b ,  124   c ,  124   d , the module  124   a  may output the results  124   f  for analysis/evaluation during the post-processing stage  126 . In particular, the results  124   f  as displayed to a user may include the post-processing results  126   a , including the mechanical response of the composite laminate material under the simulated stresses and strains during the execution of the solver module  124   a , and the failure progression (e.g., delamination migration) of any failures within the composite laminate material as a result of the stresses and strains simulated by the solver module  124   a  exceeding the material tolerances. For example, the post-processing results  126   a  may include plots, graphs, charts, and/or any other suitable display configured to illustrate the mechanical response, the failure progression, and/or any other suitable result or combinations thereof for a user. 
     In order to generate the plies for the FE mesh, the systems and methods of the present disclosure may utilize multiple modeling techniques (e.g., meshes). For example, each of  FIGS.  2 A- 2 C  illustrate various meshes and/or meshing techniques that may be utilized as part of the techniques of the present disclosure. Namely,  FIG.  2 A  illustrates an example free meshing  200  of bulk elements  200   a  included as part of a composite laminate material, as utilized by the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. Generally, the bulk elements  200   a  illustrated in  FIG.  2 A  may constitute a large portion (e.g., a majority) of the resulting composite laminate material, and may be disposed in between any fibrous strips. As previously mentioned, the bulk elements  200   a  between the matrix-splitting strips may not fail in matrix modes. As a result, and as illustrated in  FIG.  2 A , the bulk elements  200   a  may have arbitrary shapes, such that a free mesh can be used for most of the elements comprising the FE mesh. This significantly reduces the processing resources/time that are otherwise necessary to produce a well-defined shape for the bulk elements  200   a , and by extension, the completed FE mesh. Regardless, as previously mentioned, the arbitrary shapes comprising the bulk elements  200   a  may be generated using a structured mesh (e.g., by the structured hex meshing algorithm  110   b ) and/or a free mesh (e.g., by the free hex-dominated advancing front meshing algorithm  110   a ) based on the geometry of the region of the composite laminate material in which the bulk elements  200   a  are located. In certain aspects, the element size of the bulk elements  200   a  are allowed to exceed the Bazant limit required in the transverse direction. As a result, only the Bazant limit for the fiber-direction may be required. 
       FIG.  2 B  illustrates an example structure aligned meshing  202  of fibrous strips  202   a  and bulk elements  202   b  included as part of a composite laminate material, as utilized by the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. As illustrated in  FIG.  2 B , the fibrous strips  202   a  are each generated in a uniform direction, and at a uniform spacing. Additionally, the bulk elements  202   b  may be uniformly generated such that each bulk element  202   b  has substantially similar and/or identical dimensions and each fibrous strip  202   a  may be separated from an adjacent strip by the width of a single bulk element  202   b . 
       FIG.  2 C  illustrates an example discrete meshing  204  of cohesive elements  204   a  and bulk elements  204   b  included as part of a composite laminate material, as utilized by the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. As illustrated in  FIG.  2 C , the cohesive elements  204   a  are each generated in a uniform direction, and at a uniform spacing. Each cohesive element  204   a  may not have in-plane stiffness, such that the elements  204   a  may not capture fiber stiffness (e.g., resulting from the fibrous strips). Moreover, the cohesive elements  204   a  may be generated at a relatively small thickness (e.g., 0 thickness), and may not be included as part of the intra-ply meshing techniques described herein. However, in certain aspects, the cohesive elements  204   a  may be included as part of the intra-ply meshing techniques described herein to, for example, help model matrix cracking within the composite laminate material. Additionally, the bulk elements  204   b  may be uniformly generated such that each bulk element  204   b  has substantially similar and/or identical dimensions and each cohesive element  204   a  may be separated from adjacent cohesive elements by the width of multiple bulk elements  204   b  (illustrated in  FIG.  2 C  as two bulk elements  204   b ). 
       FIG.  2 D  is an example generation sequence  206  of a semi-discrete meshing  208  of fibrous strips and bulk elements included as part of a composite laminate material, as generated by portions of the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. For ease of discussion, it is to be understood that the composite laminate modeling application  110  and each of the elements included therein (e.g., free hex-dominated advancing front meshing algorithm  110   a , structured hex meshing algorithm  110   b , and the mesh generation tool  110   d ) may be executed by one or more processors (e.g., processor  102 ) in order to perform the actions described herein. 
     As illustrated in  FIG.  2 D , the composite laminate modeling application  110  may receive a fibrous strip width and a fibrous strip spacing from a user, and the application  110  may then output the semi-discrete meshing  208  using the mesh generation tool  110   d , which may itself access or otherwise include instructions causing the processors to execute the free hex-dominated advancing front meshing algorithm  110   a  and the structured hex meshing algorithm  110   b . The user may select a small fibrous strip width relative to the overall size of the composite laminate material and/or the bulk elements  212   a ,  212   b  in order to enhance the discreteness and sharpness of the resulting matrix cracks, as described herein. The mesh generation tool  110   d  may then partition the meshing  208 , such that the each of the fibrous strips  210   a ,  210   b ,  210   c  are generated along the fiber direction (not shown) with the fibrous strip spacing in between each strip. The mesh generation tool  110   d  may also generate the bulk elements  212   a ,  212   b  between the fibrous strips  210   a ,  210   b ,  210   c , and both the bulk elements  212   a ,  212   b  and the fibrous strips  210   a ,  210   b ,  210   c  may be generated using either structured and/or free meshing techniques, as described further herein. However, it should be appreciated that the fibrous strips (e.g., fibrous strips  202   a ,  204   a ,  210   a ,  210   b ,  210   c ) described herein may follow any suitable fiber path, such as 3D fiber paths that include a 3D curve, and the mesh generation tool  110   d  may partition the fibrous strips along the corresponding 3D curve. For example, fibrous strip  210   a  may be a steered fiber strip placed within the composite laminate by robotic automation and/or a portion of a tow (bundle of fibers) that collectively follows a 3D curve. In this example, the mesh generation tool  110   d  may define partitions along the 3D fiber path which the fibrous strip  210   a  follows. 
     Moreover, during the partitioning process, the mesh generation tool  110   d  may define each fibrous strip  210   a ,  210   b ,  210   c  as a cell, which may be further defined as a geometric set. Each fibrous strip  210   a ,  210   b ,  210   c  that is part of a particular geometric set may be assigned the same material properties. Accordingly, the mesh generation tool  110   d  may assign each fibrous strip  210   a ,  210   b ,  210   c  (and/or each unique geometric set of strips) a unique randomized strength property, such as transverse and shear strength, in order to achieve a random material strength distribution throughout the composite laminate material. 
     Generally, a realistic composite laminate material may have randomized strength properties throughout the material. Thus, for the composite laminate modeling application  110  to model the material as realistically as possible, the strength properties of the FE mesh may be randomized throughout the specimen. This may be justified by experimental observations on the progressive nature of crack accumulation (e.g. in a cross-ply tensile test), which have indicated an increase of transverse crack density with rising axial strain. 
     Crack accumulation may be progressive for several reasons, each of which may be accounted for by the composite laminate modeling application  110  during the development of the semi-discrete meshing  208 . Once a crack develops within the composite laminate material, the stresses near that crack may relax and rise away slowly according to shear lag theory, such that no other crack may occur within a certain shear lag length of the crack. These matrix cracks usually provide stress concentrations at ply-interfaces for delamination to initiate, which redistributes and relaxes the stresses around a transverse crack. Further, the inherent nonuniformity of the composite laminate material causes the strength of the material to vary spatially. In fact, strength values measured in lamina tests may be the lower bound of an actual material strength distribution. 
     Conventionally, a meso-scale model can capture only a few of these effects with limited fidelity. In order to capture the shear lag effect and its strain and stress gradients within a small length scale, a very fine mesh in the in-plane and thickness directions would be required. Some conventional techniques use micro-mechanical models with explicit modeling of fibers and matrix, to capture the onset and propagation of matrix cracks. However, using these methods lead to an unreasonable computation cost associated with coupon or structural level modeling. 
     Most of the conventional numerical models may capture transverse cracks, but are not suited for a progressive failure analysis of actual composite structures. Typically, one element per width is used, and this issue becomes more complicated in a FE mesh of a realistically-sized problem having multi-dimensionality (e.g., having elements in x- and y-directions). If applying a distribution to all elements in a generic mesh with multiple elements along the width and length direction, the same progression of transverse cracking cannot be achieved. 
     Practically speaking, the stress along the width and length direction of a composite laminate material is almost uniform. If a row of elements along the width direction is assigned with a distribution, there is a high probability that one of the elements has a low strength due to the distribution. This probability increases with the number of elements, and as a result, there is a high probability that multiple cracks will occur at the same time at an applied strain within a few increments. To overcome this issue, the composite laminate modeling application  110  employs a separation of failure modes when generating the semi-discrete meshing  208 , where the strength for matrix cracking may be distributed along the 22-direction only (e.g., perpendicular to the matrix cracks). 
     Moreover, the type of probability distribution may be carefully chosen for the FE mesh generated by the composite laminate modeling application  110 . The semi-discrete meshing  208   may include a finite number of elements, such that it is difficult to capture the tails of a probability distribution (e.g. Weibull or Gaussian). Capturing the left tail is important, as it represents the weak locations in the corresponding composite laminate material. Thus, the composite laminate modeling application  110  may utilize a uniform distribution, which may be further justified because the semi-discrete meshing  208  may include evenly spaced fibrous strips  210   a ,  210   b ,  210   c . Each of the fibrous strips  210   a ,  210   b ,  210   c  may represent a weakest material point within a band around that strip. If that band of material points within a probabilistic distribution is reduced to a strip, the overall distribution (applied to the fibrous strips) may be altered. In fact, the choice of a uniform probability density function (PDF) may provide a high degree of alignment with experimental data to an extent that is otherwise unachievable with conventional techniques. Regardless, it should be understood that any suitable probability density function may be used within the composite laminate modeling application  110 . 
     In any event, the semi-discrete meshing  208  may include a first fibrous strip  210   a , a second fibrous strip  210   b , and a third fibrous strip  210   c . Each of the first, second, and third fibrous strips  210   a ,  210   b ,  210   c  may be comprised of different materials (indicated by different patternings). Of course, it will be appreciated that each of the fibrous strips  210   a ,  210   b ,  210   c  may be comprised of similar materials, identical materials, and/or any other suitable materials or combinations thereof. Regardless, the composite laminate modeling application  110  may randomize the material distribution of each of the fibrous strips  210   a ,  210   b ,  210   c , such that the different materials used to comprise the fibrous strips may be randomly distributed throughout the resulting FE mesh. In this manner, the strength properties of the FE mesh may be randomized to more accurately reflect a realistic modeling of a tangible composite laminate material. 
     The semi-discrete meshing  208  may also include bulk elements  212   a ,  212   b  including a first bulk element  212   a  and a second bulk element  212   b . As illustrated in  FIG.  2 D , the bulk elements  212   a ,  212   b  may have non-uniform shapes, such that the first bulk element  212   a  has a different dimensionality from the second bulk element  212   b . Similarly, the composite laminate modeling application  110  (e.g., the free hex-dominated advancing front meshing algorithm  110   a ) may generate each of the bulk elements  212   a ,  212   b  included as part of the semi-discrete meshing  208  with different individual dimensions without violating the fibrous strip spacing specified by the user. Thus, each individual bulk element (e.g., first bulk element  212   a , second bulk element  212   b ) may have unique/different dimensions, but the fibrous strip spacing between the fibrous strips  210   a ,  210   b ,  210   c  may remain constant. 
       FIG.  3 A  illustrates an example partitioning  300  of an interlayer within a composite laminate material, in accordance with various aspects disclosed herein. Generally, as previously mentioned, partitioning of the interlayer includes enforcing that the nodes of the cohesive elements are at the boundaries of potential intra-ply matrix cracks (also referenced herein as “predicted” intra-ply matrix cracks). The potential intra-ply matrix cracks may be determined based on the positioning of the elements comprising the composite laminate material. For example, the fibrous strip elements are generally defined to capture the matrix cracking within the material, such that matrix cracking may be confined to occur along the fibrous strips. In this manner, the composite laminate modeling application  110  may enforce that the nodes of the cohesive elements are near the boundaries of the fibrous strip elements included within the adjacent plies. 
     More specifically, and in reference to  FIG.  3 A , the example partitioning  300  may include a set of first ply partition features  302 , a set of second ply partition features  304 , a set of interlayer partition features  306 , and a set of fixed nodes  308 . The sets of ply partition features  302 ,  304  may have a ply width d cr , and the set of interlayer partition features  306  may be displaced relative to the sets of ply partition features  302 ,  304  (e.g., wider) by a tolerance  310 . Moreover, the sets of ply partition features  302 ,  304  may each have a respective angular orientation within the respective ply (e.g., features  302  at angle θ i  and features  304  at angle θ i+1 ), and the set of interlayer partition features  306  may have identical angular orientations in order to surround and run in parallel with the sets of ply partition features  302 ,  304 . The example partitioning  300  may also include an arbitrary boundary  312 , such as a cut-out or a free edge. 
     In certain embodiments, the example partitioning  300  may not include a set of interlayer partition features  306  proximate to each set of first ply partition features  302  and/or each set of second ply partition features  304 . Namely, in these embodiments, the example partitioning  300  may include certain subgroups and/or portions of the set of first ply partition features  302  and/or each set of second ply partition features  304  where there are no interlayer partition features  306  disposed proximate to those subgroups and/or portions. For example, the example partitioning  300  may include a set of interlayer partition features  306  proximate to every other set of first ply partition features  302  and/or each set of second ply partition features  304 , such that approximately half of the set of first ply partition features  302  and/or each set of second ply partition features  304  do not have a proximate set of interlayer partition features  306 . 
     In order to generate the example partitioning  300 , the compatible interface model  110   e  in combination with the mesh generation tool  110   d  may place integration points of cohesive elements at the corners of the intersection points of the sets of ply partition features  302 ,  304 . The mesh generation tool  110   d  may then partition strips in the directions of the two neighboring plies (e.g., represented by the sets of ply partition features  302 ,  304 ), namely θ i  and θ i+1 . To maintain a robust tie between the elements of the plies and those of the interlayer, the tool 100d may widen the strips included in the set of interlayer partition features  306  by a tolerance from the ply width d cr  to shift the set of fixed nodes  308  slightly towards the bulk regions of the composite laminate material. 
       FIG.  3 B  is an example generation sequence  320  of a semi-discrete meshing overlay of plies and an interlayer included as part of a composite laminate material, as generated by the example system of  FIG.  1 A , and in accordance with various aspects disclosed herein. As illustrated in  FIG.  3 B , the composite laminate modeling application  110  may receive a specimen geometry and a stacking sequence from a user, and the application  110  may then output the semi-discrete meshing  322  using the mesh generation tool  110   d , which may itself access or otherwise include instructions causing the processors to execute the free hex-dominated advancing front meshing algorithm  110   a  and the structured hex meshing algorithm  110   b . The mesh generation tool  110   d  may partition the meshing  322  for each ply, such that the first strip  324  corresponds to a first ply, and the second strip  326  corresponds to a second ply. The mesh generation tool  110   d  may then generate the nodes  328  within the cohesive layer connecting the first ply and the second ply, while accommodating the material boundary  330  (e.g., cut-out, free edge). 
     Generally, the composite laminate modeling application  110  may generate nodes  328  at any suitable point within the cohesive layer between two neighboring plies (e.g., at any/all vertices of meshing elements within the cohesive layer). However, the nodes of the cohesive layer that will interact with matrix cracks first (e.g., node  328 , referenced herein as “compatible nodes”) are represented by dots emphasizing their placement near one or both of the first strip  324  and/or the second strip  326 , thereby increasing the chances that the node will indeed interact with a matrix crack before other nodes disposed further from the strips  324 ,  326  within the composite laminate material. The composite laminate modeling application  110  may generate elements of the cohesive layer at in-plane element sizes relatively smaller than that of the plies to ensure a proper level of connectivity to both neighboring plies (e.g., the first ply and the second ply) with non-matching meshes (e.g., first strip  324  and second strip  326 ). As a result, the composite laminate modeling application  110  may create semi-discrete and compatible meshes of cohesive layers within composite laminate materials given any arbitrary specimen geometry without implementing mesh correction measures. 
       FIG.  4 A  depicts an example open-hole tensile (OHT) laminate specimen  400  after partitioning, in accordance with various aspects disclosed herein. Generally, the OHT laminate specimen  400  may include multiple plies (e.g., top 45 ply  402 ), each meshed with a semi-discrete meshing and a random material distribution. Each ply of the specimen  400  may include an elastic region  404  that is clamped and/or otherwise supported at the edge  406 . More specifically, the OHT laminate specimen  400  may be an output of the composite laminate modeling application  110 . It should be appreciated that OHT specimen  400  may be and/or include any suitable cut-out or notch. 
     For example, the composite laminate modeling application  110  may output the OHT laminate specimen  400  as a ply-scaled, 4 millimeters (mm) thick laminate with a stacking sequence [45 4 /90 4 / - 45 4 /0 4 ] s , a hole diameter of 3.175 mm, a width of 16 mm and a gage length of 64 mm. The material system may be IM7/8552 with a cured ply thickness of 0.125 mm. As illustrated in  FIG.  4 A , the application  110  may apply a semi-discrete mesh to the entire gage length of the specimen  400 . In particular, the 0 ply  408  may be meshed with matrix-split strips extending throughout the entire specimen length, including the elastic regions  404 . As illustrated in the example meshing  420  of  FIG.  4 B , the 0 ply  408  includes matrix-split strips extending through the meshed interior 408a and the meshed elastic region  408   b . 
     More generally, and as illustrated in the example meshing  425  of  FIG.  4 C , the mesh details and the assembly structure of the OHT laminate specimen  400  may vary throughout the plies. For example, the OHT laminate specimen  400  may include the top 45 ply  402  that includes a 45° meshing, a top 45/90 interlayer  426  that includes a 45°/90° meshing, a top 90 ply  427  that includes a 90° meshing, a top 90/-45 interlayer  428  that includes a 90°/-45° meshing, a top -45 ply  429  that includes a -45° meshing, a top -45/0 interlayer  430  that includes a -45°/0° meshing, and the 0 ply  408  that may include a 0° meshing. It should be understood that the plies/interlayers illustrated in  FIG.  4 C  are referenced as “top” plies/interlayers because they are illustrated as plies/interlayers above or otherwise on top of the remaining plies/interlayers of the OHT laminate specimen  400 , as indicated by the positive z-direction of the coordinates 442 in  FIG.  4 D . As such, it will be appreciated that the specimen  400  may include additional plies/interlayers defining a bottom portion of the specimen  400  (e.g., below the 0 ply  408 ) that are stacked symmetrically to the plies/interlayers of the top portion of the specimen  400  (e.g., starting with a bottom 0/-45 interlayer and ending with a bottom 45 ply). 
     The application  110  may generally model each ply with continuum elements with reduced integration and enhanced hourglass control, while the interlayers may be represented with cohesive elements. As illustrated in by the example compatible meshing  435  of  FIG.  4 D , the cohesive interlayers are relatively thin compared to the plies. Namely, the top 45/90 interlayer  426  is relatively thin compared to the top 45 ply  402  and the top 90 ply  427 , between which the interlayer  426  is displaced. For example, the interlayer  426  may be approximately 0.002 mm thin, which may be less than the fiber diameter within the neighboring plies  402 ,  427  of approximately 5 micrometers (µm). The interface between top 45/90 interlayer  426  and the neighboring plies  402 ,  427  may include several compatible nodes where cohesive elements may be placed to join the interlayer  426  to a respective neighboring ply  402 ,  427 . The compatible nodes  438  may join the interlayer  426  to the top 45 ply  402  on both sides of the matrix-split strip  437  contained within the ply  402 . Similarly, the compatible nodes  439  may join the interlayer  426  to the top 90 ply  427  on both sides of the matrix-split strip  440  contained within the ply  427 . Further, the interlayer  426  may include several ties  441  that join the interlayer  426  to the neighboring plies  402 ,  427  without constraining the location of the ties  441  to a matrix-split strip  437 ,  440 . 
       FIG.  5    illustrates differences between specimen partitioning and meshing procedures for plies and interlayers of a composite laminate material, in accordance with various aspects disclosed herein. Generally,  FIG.  5    includes a ply represented by the partitioned ply  502  and the meshed ply  504 . As previously described, the meshed ply  504  may represent a subsequent step of the process applied by the composite laminate modeling application  110  relative to the partitioned ply  502  in order to generate a completed composite laminate material. For example, the application  110  may generate the partitioned ply  502  in accordance with the partitioning processes described herein, and may thereafter overlay a mesh onto the partitioned ply  502  to generate the meshed ply  504 . This difference between the partitioned ply  502  and the meshed ply  504  is more explicitly represented in the contrast between the partitioned ply detailed view  502   a  and the meshed ply detailed view  504   a . In other words, the partitioned ply detailed view  502   a  features the matrix-split strips and general partitioning associated with the central hole. The meshed ply detailed view  504   a  additionally includes the meshing generated by the application  110 . 
     Moreover,  FIG.  5    includes an interlayer represented by the partitioned interlayer  506  and the meshed interlayer  508 . As previously described, the meshed interlayer  508  may represent a subsequent step of the process applied by the composite laminate modeling application  110  relative to the partitioned interlayer  506  in order to generate a completed composite laminate material. For example, the application  110  may generate the partitioned interlayer  506  in accordance with the partitioning processes described herein, and may thereafter overlay a mesh onto the partitioned interlayer  506  to generate the meshed interlayer  508 . This difference between the partitioned interlayer  506  and the meshed interlayer  508  is more explicitly represented in the contrast between the partitioned interlayer detailed view  506   a  and the meshed interlayer detailed view  508   a . In other words, the partitioned interlayer detailed view  506   a  features the strips corresponding to partitioned ply  502  and a neighboring ply (not shown)). The meshed interlayer detailed view  508   a  additionally includes the meshing generated by the application  110 . As shown in the meshed interlayer detailed view  508   a , the regions near the hole may be meshed with different element sizes and meshing algorithms (e.g., a free mesh near the hole). 
     As illustrated in  FIG.  5   , the partitioned interlayer  506  has partitioning features in two directions, creating a large number of partition plane intersections. Generally speaking, as more partitioning planes intersect with one another, the pre-processing time required to generate the full mesh (e.g., in meshed interlayer  508 ) increases. Accordingly, as illustrated in  FIG.  5   , the composite laminate modeling application  110  may apply the semi-discrete compatible mesh to the interlayer with a limited and/or otherwise reduced number of partitioning features to correspondingly reduce the pre-processing time, and thereby make the entire composite laminate material generation process more efficient. For example, with a limited number of partitioning features, the pre-processing (e.g., the pre-processing stage  122 ) may take on the order of only a few minutes. As a result, the full meshing procedure, described herein, may only require approximately 10 to 20 minutes. Further, it should be understood that the partitioning features in any particular direction may extend through any suitable portion of the material. For example, partitioning features along a 0 direction may extend through the entire gage length of the material at the -45/0 interface (e.g., within the top -45/0 interlayer  430 ). Additionally, or alternatively, cohesive properties within the interlayers may be uniform and/or randomized, as specified by a user (e.g., within the inter-laminar subroutine  124   d ). 
       FIG.  6    is an example graph  600  illustrating a general pre-peak and post-peak stress-strain response of a composite laminate material, in accordance with various aspects disclosed herein. The constitutive model  110   c  may generally model the behavior of each composite laminate material generated by the composite laminate modeling application  110  under stress/strain conditions that may appear similar to the example graph  600 . The bulk elements (e.g., bulk elements  212   a ,  212   b ) and the fibrous strips (e.g., fibrous strips  210   a ,  210   b ,  210   c ) may be modeled with the same constitutive model to minimize the overall complexity particularly because the distinct failure behaviors are due to the material strength distribution. 
     Additionally, or alternatively, the user may input a relatively large fibrous strip spacing to, for example, generate a relatively large material specimen and/or to reduce overall computational load. With a relatively large fibrous strip spacing, the corresponding bulk elements (e.g., bulk elements  212   a ,  212   b ) may be enhanced to capture some degree of a matrix splitting failure mode as well as a fiber failure mode. As a result, the constitutive behavior for these enhanced bulk elements may differ slightly from the constitutive behavior of the corresponding fibrous strip elements based on their characteristic (i.e., element) size. If the enhanced bulk elements are sufficiently small (e.g., comparable to the width of the fibrous strip elements), then the constitutive model may be the same for both elements. 
     However, if the bulk element size is large, a residual stiffness/modulus may be imposed for the enhanced bulk elements. In these instances, a crack density parameter may be introduced to quantify the damage state, and each enhanced bulk element may capture multiple matrix cracks in a smeared manner by utilizing continuum damage mechanics. Namely, the fibrous strip elements may capture sharp matrix cracks, and the enhanced bulk elements may capture additional damage in a smeared manner that may correspond to the accumulation of multiple matrix cracks occurring within one or more enhanced bulk elements. Hence, this approach may be referenced as a “mixed-fidelity” or “multi-fidelity” damage model. 
     In any event, generally, the constitutive response of a composite laminate material may include a pre-peak response  602  and a post-peak softening domain, represented by the post-peak response  604 . The pre-peak shear behavior is usually non-linear in nature due to micro-cracking (shear hackling) in the matrix. However, these micro-cracks also have an effect on the transverse stiffness of the material. Thus, to capture the material non-linearity while accounting for the adverse effects of micro-cracking on material transverse stiffness, the constitutive model  110   c  may include Schapery theory (ST) to accurately capture the micro-damage in a composite laminate material. Of course, it is to be understood that any suitable modeling theory may be applied, such as, for example, a multiscale model that captures the stiffness degradation of the composite analytically by the concentric cylinder model (CCM) and the generalized self-consistent method (GSCM). 
     In any event, a failure criterion may trigger the transition (represented by the transition point  606 ) from the pre-peak response  602  into the post-peak response  604 . Physically, when a crack propagates and becomes traction-free, all tractions at the crack tip must vanish simultaneously. However, from a numerical point of view, a mixed-mode law may be critical to minimize oscillations due to sudden stress drops. As a result, the constitutive model  110   c  may also incorporate a crack band model to account for these physical properties attendant to crack propagation within a composite laminate material. 
     A processor (e.g., processor  102 ) generating the constitutive model  110   c  may calculate the degradation of the transverse and shear moduli based on two damage functions e s (S r ) and g s (S r ) in terms of the reduced Schapery parameter: 
     
       
         
           
             
               S 
               r 
             
             = 
             
               S 
               3 
             
           
         
       
     
      where S is the energy dissipated due to micro-damage. Accordingly, the two damage functions may be included as part of pristine property equations given by: 
     
       
         
           
             
               E 
               
                 22 
               
             
             = 
             
               E 
               
                 22 
               
               0 
             
               
             
               e 
               s 
             
             
               
                 
                   S 
                   r 
                 
               
             
           
         
       
     
     
       
         
           
             
               G 
               
                 12 
               
             
             = 
             
               G 
               
                 12 
               
               0 
             
               
             
               g 
               s 
             
             
               
                 
                   S 
                   r 
                 
               
             
           
         
       
     
      where  
     
       
         
           
             
               E 
               
                 22 
               
               0 
             
           
         
       
     
      and  
     
       
         
           
             
               G 
               
                 12 
               
               0 
             
           
         
       
     
      are pristine properties that the processor may utilize when determining and solving a Schapery theory equation given by: 
     
       
         
           
             
               1 
               2 
             
             
               
                 
                   E 
                   
                     22 
                   
                   0 
                 
                 
                   
                     d 
                     
                       e 
                       s 
                     
                     
                       
                         
                           S 
                           r 
                         
                       
                     
                   
                   
                     d 
                     
                       S 
                       r 
                     
                   
                 
                 
                   ε 
                   
                     22 
                   
                   2 
                 
                 + 
                 
                   G 
                   
                     12 
                   
                   0 
                 
                 
                   
                     d 
                     
                       g 
                       s 
                     
                     
                       
                         
                           S 
                           r 
                         
                       
                     
                   
                   
                     d 
                     
                       S 
                       r 
                     
                   
                 
                 
                   γ 
                   
                     12 
                   
                   2 
                 
               
             
             = 
             -3 
             
               S 
               r 
               2 
             
           
         
       
     
     Equation (4) is generally a thermodynamics model that ensures the rate of dissipated energy within the composite laminate material is always positive, and the processor may solve equation (4) to obtain the Schapery parameter in equation (1), and the damage function values E 22  and G 12  in equations (2) and (3), respectively. In certain aspects, the processor may hold other moduli constant when solving equation (4). In any event, the processor may then apply a two-parameter fit given by: 
     
       
         
           
             
               τ 
               
                 12 
               
             
             = 
             
               
                 
                   G 
                   
                     12 
                   
                   0 
                 
               
               c 
             
             
               
                 1 
                 - 
                 
                   e 
                   
                     − 
                     
                       c 
                       
                         
                           γ 
                           
                             12 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
      to generate portions of the pre-peak response  602  and/or the post-peak response  604 . 
     The processor may generate the post-peak response  604  by utilizing a crack band model. For the post-peak response, the processor may define and/or may receive damage parameters to degrade the extensional and shear moduli, as well as Poisson’s ratios. The crack band method employed by the processor utilizes a characteristic length to ensure the correct energy dissipation and mesh objectivity. The processor may calculate the characteristic lengths using the element nodal coordinates and material point coordinates. In certain aspects, based on failure models for composites, the element size may be used as the characteristic length. 
     The processor may also determine a post-peak softening response for the composite laminate material. More particularly, the processor may parameterize the softening curve for a composite laminate material by a normalized softening function, which can differ for fiber and matrix modes. The normalized softening function may generally be defined by: 
     
       
         
           
             σ 
             = 
             σ 
             * 
             
               f 
               ¯ 
             
             
               
                 
                   
                     ε 
                     − 
                     
                       
                         ε 
                         * 
                       
                     
                   
                   
                     
                       ε 
                       ^ 
                     
                     − 
                     
                       
                         ε 
                         * 
                       
                     
                   
                 
               
             
           
         
       
     
      where  =  
     
       
         
           
             
               
                 ε 
                 − 
                 
                   
                     ε 
                     * 
                   
                 
               
               
                 
                   ε 
                   f 
                 
               
             
           
         
       
     
     is the normalized post-peak strain, σ* is the stress component at the failure initiation,  is the absolute value of the ultimate failure strain, the post-peak failure strain ε f  is given by: 
     
       
         
           
             
               ε 
               ^ 
             
               
             = 
             
               
                 ε 
                 * 
               
             
             + 
             
               ε 
               f 
             
           
         
       
     
      and fis parameterized with a cubic Bezier curve. Using the parameterization of  the processor may generate a variety of shapes related to the softening of a particular composite laminate material. For example, the processor may generate a quasi-bilinear softening law, a linear law, a quasi-trapezoidal law, and/or other shapes or combinations thereof. 
     Using these equations (e.g., equations (1)-(7)) and others derived therefrom, the processor may fully describe the post-peak response  604  as: 
     
       
         
           
             
               d 
               1 
             
             = 
             
               d 
               f 
             
             = 
             1 
             - 
             
               
                 
                   
                     
                       ε 
                       
                         11 
                       
                       * 
                     
                   
                 
               
               
                 
                   ε 
                   
                     11 
                   
                 
               
             
             
               
                 f 
                 ¯ 
               
               f 
             
             
               
                 
                   
                     
                       ε 
                       
                         11 
                       
                     
                     − 
                     
                       
                         
                           ε 
                           
                             11 
                           
                           * 
                         
                       
                     
                   
                   
                     
                       
                         ε 
                         ^ 
                       
                       
                         11 
                       
                     
                     − 
                     
                       
                         
                           ε 
                           
                             11 
                           
                           * 
                         
                       
                     
                   
                 
               
             
           
         
       
     
      where the processor, when applying equation (8), may assume that the fiber damage is only governed by the strain in the fiber direction ε 11 . 
     Additionally, a mixed-mode law may be utilized to describe matrix failure that guarantees a smooth and simultaneous vanishing of all stress components. This mixed-mode law may be governed by an effective strain, given by: 
     
       
         
           
             
               ∈ 
               e 
             
             = 
             
               
                 
                   ∈ 
                   
                     22 
                   
                   2 
                 
                 + 
                 
                   
                     
                       
                         
                           
                             
                               
                                 ∈ 
                                 ^ 
                               
                               
                                 22 
                               
                             
                           
                           
                             
                               
                                 γ 
                                 ^ 
                               
                               
                                 12 
                               
                             
                           
                         
                         
                           γ 
                           
                             12 
                           
                         
                       
                     
                   
                   2 
                 
                 + 
                 
                   
                     
                       
                         
                           
                             
                               
                                 ∈ 
                                 ^ 
                               
                               
                                 22 
                               
                             
                           
                           
                             
                               
                                 γ 
                                 ^ 
                               
                               
                                 23 
                               
                             
                           
                         
                         
                           γ 
                           
                             23 
                           
                         
                       
                     
                   
                   2 
                 
               
             
           
         
       
     
      and a minor correction may be added to the transverse behavior. In particular, the constitutive model  110   c  may include a distinction that when ∈ 22  &lt;0, d 2  is set to 0 in order to capture the contact between the crack surfaces and prevent inter-penetration. 
     Further, the constitutive model  110   c  may achieve mesh objectivity when the dissipated energy due to a crack opening is not influenced by the element characteristic length used for the crack band method. To achieve this, the constitutive model  110   c  may include a correction to the input value of the fracture energy. The actual dissipated fracture energy G may be given by: 
     
       
         
           
             G 
             = 
             
               
                 
                   
                     X 
                     
                       l 
                       e 
                     
                   
                   
                     2 
                     
                       G 
                       
                         i 
                         n 
                         p 
                         u 
                         t 
                       
                     
                   
                 
                 ⋅ 
                 
                   e 
                   
                     
                       
                         2 
                         
                           G 
                           
                             i 
                             n 
                             p 
                             u 
                             t 
                           
                         
                       
                       
                         X 
                         
                           l 
                           e 
                         
                       
                     
                   
                 
                 − 
                 
                   
                     X 
                     
                       l 
                       e 
                     
                   
                   
                     2 
                     
                       G 
                       
                         i 
                         n 
                         p 
                         u 
                         t 
                       
                     
                   
                 
                 − 
                 1 
               
             
             X 
             
               l 
               e 
             
           
         
       
     
      if logarithmic strain is used. Thus, to achieve a given fracture toughness G = G c  with a tensile strength X and characteristic length l e , the solver module (e.g., solver module  124   a ) may determine the adjusted input value G input  by solving equation 10. The solver module may utilize, for example, the Newton method upon entry into the post-peak domain. 
     The constitutive model  110   c  may also include a function for element deletion when a crack has fully developed (e.g., any damage variable is close to 1). This element deletion function may include other deletion criteria to prevent excessive element distortion, and may generally be given by: 
     
       
         
           
             Delete element if 
             
               
                 
                   
                     any  
                     
                       d 
                       i 
                     
                     &gt; 
                     0.9999 
                   
                 
                 
                   
                     any  
                     
                       ∈ 
                       i 
                     
                       
                     &lt; 
                     − 
                     1.0 
                       
                     or 
                       
                     
                       ∈ 
                       i 
                     
                       
                     &gt; 
                     1.5 
                   
                 
                 
                   
                     any 
                       
                     
                       
                         
                           γ 
                           
                             i 
                             j 
                           
                         
                       
                     
                     &gt; 
                     2.5 
                   
                 
                 
                   
                     det F &lt; 0 
                     .3 or det F 
                       
                     &gt; 
                       
                     5 
                     .0 
                   
                 
               
             
           
         
       
     
     The transition point  606  may represent transition criteria corresponding to the transition between the pre-peak response  602  to the post-peak response  604 . For example, the processor may utilize Hashin criteria to govern the transition between the pre-peak response  602  and the post-peak response  604 . However, it will be appreciated that any suitable transition criteria may be used. When modeling the transition criteria, the processor may analyze multiple failure planes for the composite laminate material. For example,  FIGS.  5 A- 5 C  illustrate several example failure planes that the processor may analyze when determining the transition criteria for a particular composite laminate material. 
       FIGS.  7 A- 7 E  illustrate an example mixed-mode traction-separation law for cohesive behavior of a composite laminate material, in accordance with various aspects disclosed herein. Generally, the delamination failure in a composite structure is most likely to happen under a mixed-mode condition. As a consequence, the inter-laminar constitutive model  124   e  may include a strict physical requirement that all tractions vanish simultaneously at the crack propagation, and when the crack is fully developed, the crack surface becomes traction-free. In order to accomplish these requirements, the inter-laminar constitutive model  124   e  may incorporate the definition of an effective separation, which allows the usage of arbitrary softening laws for each mode. 
     In particular, the principles of the mixed-mode formulation are depicted in  FIGS.  7 A- 7 E . In  FIGS.  7 A- 7 C , the graphs  700 ,  710 ,  720  illustrate the pure mode and mixed-mode responses of the interlayer. The pre-peak response across each of the graphs  700 ,  710 ,  720  is modeled as linear with a penalty stiffness Ki, where i = I, II, III represents the distinct modes. For example, the graph  700  may represent an opening mode I, and graphs  710 ,  720  may represent shear modes II and III, respectively. The constitutive equation governing these graphs  700 ,  710 ,  720  may be given as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           t 
                           I 
                         
                       
                     
                   
                   
                     
                       
                         
                           t 
                           
                             I 
                             I 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           t 
                           
                             I 
                             I 
                             I 
                           
                         
                       
                     
                   
                 
               
             
             = 
             
               
                 
                   
                     
                       
                         
                           
                             1 
                             − 
                             
                               d 
                               I 
                             
                           
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         
                           
                             1 
                             − 
                             
                               d 
                               
                                 11 
                               
                             
                           
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         
                           
                             1 
                             − 
                             
                               d 
                               
                                 I 
                                 I 
                                 I 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           K 
                           1 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         
                           K 
                           
                             I 
                             I 
                           
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         
                           K 
                           
                             I 
                             I 
                             I 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           δ 
                           I 
                         
                       
                     
                   
                   
                     
                       
                         
                           δ 
                           
                             I 
                             I 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           δ 
                           
                             I 
                             I 
                             I 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
      where the pure mode strengths are σ c  and τ c . The initiation criterion, influencing the values of the initiation stresses t I , t II  and t III  may be given generally as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         max 
                         
                           
                             0 
                             , 
                             
                               t 
                               I 
                             
                           
                         
                       
                       
                         
                           σ 
                           c 
                         
                       
                     
                   
                 
               
               2 
             
             + 
             
               
                 
                   
                     
                       
                         
                           t 
                           
                             I 
                             I 
                           
                         
                       
                       
                         
                           τ 
                           c 
                         
                       
                     
                   
                 
               
               2 
             
             + 
             
               
                 
                   
                     
                       
                         
                           t 
                           
                             I 
                             I 
                             I 
                           
                         
                       
                       
                         
                           τ 
                           c 
                         
                       
                     
                   
                 
               
               2 
             
             ≥ 
             1 
           
         
       
     
     When the initiation criterion is met at the stresses t I , t II  and t III , the inter-laminar constitutive model  124   e  may govern the cohesive response by the separation components and non-dimensionalized softening functions f i . However, the model  124   e  may assume that compression does not affect the failure evolution. 
     The inter-laminar constitutive model  124   e  may utilize fundamental material properties as the pure-mode fracture toughness values for each mode G ic , which is generally related to the area under the non-dimensionalized softening curve given by: 
     
       
         
           
             
               F 
               i 
             
             
               δ 
               i 
               f 
             
             
               
                 g 
                 ¯ 
               
               
                 i 
                 c 
               
               * 
             
             = 
             
               G 
               
                 i 
                 c 
               
             
             − 
             
               
                 
                   F 
                   i 
                   2 
                 
               
               
                 2 
                 
                   K 
                   i 
                 
               
             
           
         
       
     
      from which the quantities  be calculated by a solver module (e.g., solver module  124   a ). Moreover, the graph  730  of  FIG.  7 D  may represent a separation space and failure loci, including failure separations corresponding to each of modes I, II, and III in graphs  700 ,  710 , and  720 . The solver module may determine final failure separations δ ̂ i  for each mode by solving: 
     
       
         
           
             
               
                 δ 
                 ^ 
               
               i 
             
             = 
             
               
                 
                   δ 
                   i 
                   * 
                 
               
             
             + 
             
               δ 
               i 
               f 
             
           
         
       
     
     The inter-laminar constitutive model  124   e  may also include governing rules for the softening of each mode composed of the separation components, given by: 
     
       
         
           
             
               δ 
               e 
             
             = 
             
               
                 max 
                 
                   
                     
                       
                         0 
                         , 
                         
                           δ 
                           I 
                         
                       
                     
                   
                   2 
                 
                 + 
                 
                   
                     
                       
                         
                           
                             
                               
                                 δ 
                                 ^ 
                               
                               I 
                             
                           
                           
                             
                               
                                 δ 
                                 ^ 
                               
                               
                                 I 
                                 I 
                               
                             
                           
                         
                         
                           δ 
                           
                             I 
                             I 
                           
                         
                       
                     
                   
                   2 
                 
                 + 
                 
                   
                     
                       
                         
                           
                             
                               
                                 δ 
                                 ^ 
                               
                               I 
                             
                           
                           
                             
                               
                                 δ 
                                 ^ 
                               
                               
                                 I 
                                 I 
                                 I 
                               
                             
                           
                         
                         
                           δ 
                           
                             I 
                             I 
                             I 
                           
                         
                       
                     
                   
                   2 
                 
               
             
           
         
       
     
      where mode I may be chosen as the reference mode. The model  124   e  may also include a mixed-mode damage evolution expressed as: 
     
       
         
           
             
               d 
               i 
             
             = 
             1 
             − 
             
               
                 
                   δ 
                   e 
                   * 
                 
               
               
                 
                   δ 
                   e 
                 
               
             
             
               
                 f 
                 ¯ 
               
               i 
             
             
               
                 
                   
                     
                       δ 
                       e 
                     
                     − 
                     
                       δ 
                       e 
                       * 
                     
                   
                   
                     
                       
                         δ 
                         ^ 
                       
                       e 
                     
                     − 
                     
                       δ 
                       e 
                       * 
                     
                   
                 
               
             
               
             , 
             i 
             = 
             I 
             , 
             I 
             I 
             , 
             I 
             I 
             I 
           
         
       
     
      where  is the effective separation value at the initiation point and δ ̂ e  : =  δ ̂ I  is the final effective separation at complete failure. In this manner, the inter-laminar constitutive model  124   e  ensures the simultaneous vanishing of all tractions. For example, as illustrated in  FIG.  7 E , the graph  740  represents mixed-mode responses in the effective separation domain, and includes a separation value 741 at which all tractions for each of modes I, II, and III simultaneously vanish. Moreover, the loading/unloading behavior may be governed by the effective separation increment (e.g., the damage variables remain constant if δ e  decreases), and to prevent inter-penetration, the model  124   e  may impose a pristine penalty stiffness for mode I in the case of compression: d I =0 if δ I &lt; 0. 
       FIGS.  8 A- 8 D  depict example matrix crack patterns and delamination failure of the OHT laminate specimen  400  of  FIG.  4 A , in accordance with various aspects disclosed herein. As referenced herein, crack patterns, delamination failure, and/or any other type of mechanical response of the composite laminate material in response to applied stresses or strains (e.g., a loading state) may be a “mechanical response” of the material. Moreover, each of the figures referenced herein may be generated by the solver module  124   a  as part of the post-processing results  126   a  to illustrate, for example, how a mechanical response of the composite laminate material in response to input strain values may be determined. 
     Generally, each of  FIGS.  8 A- 8 D  illustrate the development of damage patterns within a composite laminate material (e.g., OHT laminate specimen  400 ) during various loading states of the material.  FIG.  8 A  generally illustrates the matrix damage experienced by all plies of the material near the cut-out at three loading states, while  FIG.  8 B  illustrates cohesive damage the material experienced at the same position and at the same loading states.  FIG.  8 C  illustrates the matrix damage experienced throughout the entire specimen during three successive loading states beyond those illustrated in  FIGS.  8 A and  8 B , and  FIG.  8 D  illustrates cohesive damage experienced throughout the entire specimen during the same three successive loading states. 
     In reference to  FIG.  8 A , the first matrix damage image  802   a  may represent a general level of matrix damage experienced by the composite laminate material at a first loading state. More specifically, the first matrix damage image  802   a  may represent a pre-peak damage pattern, wherein major split cracks in all plies near the hole are readily visible. The first matrix damage image  802   a  may represent a pre-peak damage pattern for the specimen because, despite forming matrix cracks (e.g., wherein the matrix-split elements are in post-peak stress-strain, as described in reference to  FIG.  6   ), the general load displacement of the specimen is such that the overall structural response of the specimen remains a pre-peak damage response. Thus, it should be understood that the pre-peak and post-peak responses for individual specimen elements (e.g., matrix-split strips modeled by  FIG.  6   ) and the pre-peak and post-peak responses for the entire specimen may be modeled at different scales. 
     Correspondingly, in reference to the 45/90 first image  812   a , the 90/-45 first image 814a, and the -45/0 first image  816   a  in  FIG.  8 B , cohesive damage (e.g., delamination) has developed between the matrix cracks, forming triangular shapes. However, in certain instances, the composite laminate modeling application  110  may initiate some matrix crack and splitting near the central hole and/or the free edges of the material prior to reaching the first loading state. 
     These cracks, experienced as both matrix cracking and cohesive damage, may continue to grow until the solver module  124   a  reaches the second loading state. For example, as illustrated in the second matrix damage image  802   b , the cracks continue to grow and a number of significant matrix cracks occur at the free edges of the material. Additionally, as illustrated in the 45/90 second image  812   b , the 90/-45 second image  814   b , and the -45/0 second image  816   b , the delamination may continue to spread with the splitting due to a strong interaction at each interface in a stable manner. Moreover, in certain aspects, some significant interface damage may also occur at the free edges in every interlayer except for the -45 /0 interface (e.g., as illustrated in the -45/0 second image  816   b ). 
     The solver module  124   a  may continue until reaching the third loading state, where an abrupt growth of a major split crack is identified. As illustrated in the third matrix damage image  802   c , the matrix damage experienced between the second and third loading state may include a substantial growth of a matrix crack  802   c   1  that extends along one side of the composite laminate material from the central hole to the free edge. The major split crack  802   c   1  may be localized to one side of the composite material, and it may induce more cracking in its vicinity as well as delamination spreading at the 45/90 and 90/-45 interfaces, extending from the central hole to the free edge. For example, as illustrated in the 45/90 third image  812   c  and the 90/-45 third image  814   c , the abrupt growth of the major split crack  802   c   1  causes delamination spreading from the central hole to the free edge. However, as illustrated in the -45/0 third image  816   c , the cohesive damage experienced at the -45/0 interface may not have advanced significantly between the second and third loading state. 
     At this point, the composite laminate material may begin experiencing excessive delamination migration and spreading that may cause a load drop of the specimen. The solve module  124   a  may continue to a fourth loading state, wherein all major matrix cracks begin to rapidly extend to the material free edges. For example, in reference to the first specimen matrix damage image  822   a  of  FIG.  8 C , the 0 split cracks may grow in an asymmetric manner. Due to a ply-to-ply interaction, the solver module  124   a  may determine that subcritical matrix damage is triggered along the major split cracks, and that delamination may grow rapidly along with the intra-ply failure. Namely, in reference to first specimen cohesive damage image  832   a  of  FIG.  8 D , the delamination includes footprints from several different interlayers of the material. This delamination migration, similar to the matrix split cracks, may not develop symmetrically, such that symmetric interfaces (e.g., the two -45/0 interfaces) may leave asymmetric or otherwise different delamination migration footprints during the loading process generated by the solver module  124   a . 
     The solver module  124   a  may then apply a fifth loading state to the composite laminate material. As shown in the second specimen matrix damage image  822   b , the 0 splitting on both sides of the central hole has almost reached the external grip sections of the specimen. Further, in reference to the second specimen cohesive damage image  832   b , the delamination migration continues to expand throughout the interlayers and more through the entire width of the specimen. As a result of the loading during the fifth loading state, the solver module  124   a   may determine that, due to the loss of structural integrity by both splitting and delamination, the overall structural stiffness of the specimen was reduced. This loss of stiffness may be noted and output as part of the results  124   f  and/or as part of the post-processing results  126   a . 
     Thereafter, the solver module  124   a  may increase the load to a sixth loading state. As illustrated in the third specimen matrix damage image  822   c  and the third specimen cohesive damage image  832   c , the split cracks and delamination may continue to grow until the 0 split cracks fully extend to the grips of the specimen. At this point, the solver module  124   a  may register/identify, for example, fiber rupture triggering near the grips of the specimen. This fiber rupture may additionally induce splitting near its vicinity, and the 0 layer may eventually fail at every grip connection except for the middle strips that did not carry any load. 
       FIG.  9    illustrates an example method  900  for semi-discrete modeling of delamination migration in composite laminate materials, in accordance various aspects disclosed herein. For ease of discussion, many of the various actions included in the method  900  may be optional, and may be described herein as performed by or with the use of a modeling application (e.g., composite laminate modeling application  110 ). However, it is to be appreciated that the various actions included in the method  900  may be performed by, for example, a local processor (e.g., processor  102 ) executing the modeling application, an external server, and/or other suitable processors or combinations thereof. 
     The example method  900  includes receiving, from a user, a specimen geometry and a specimen stacking sequence (block  902 ). The example method  900  may also include creating, by one or more processors, a finite-element (FE) mesh (block  904 ). For example, a mesh generation tool (e.g., mesh generation tool  110   d ) may generate a plurality of plies each shaped according to the specimen geometry, wherein each ply includes (i) a plurality of fibrous strips along a fiber direction and (ii) a bulk element between each of the plurality of fibrous strips. The mesh generation tool may also connect the plurality of plies together based on the stacking sequence by placing a plurality of cohesive elements between each adjacent pair of plies. This FE mesh may generally define a composite laminate material. 
     In certain aspects, connecting the plurality of plies further includes connecting, using the mesh generation tool, the plurality of plies together based on the stacking sequence by placing the plurality of cohesive elements between each adjacent pair of plies based on a set of tie constraints. Further, the set of tie constraints may include enforcing that nodes of the plurality of cohesive elements are located on boundaries of predicted intra-ply matrix cracks. For example, the cohesive elements may be located near matrix-split strips embedded within the neighboring plies, such that any matrix cracking will happen first at the matrix-split strips, and the cohesive elements may be near such matrix cracking. 
     In some aspects, connecting the plurality of plies further includes partitioning an interlayer between each adjacent pair of plies by defining a plurality of interlayer partition features that surround each fibrous strip included in each respective adjacent pair of plies. Further, the plurality of interlayer partition features may surround each fibrous strip included in each respective adjacent pair of plies by satisfying or exceeding a width tolerance to shift the interlayer partition features toward bulk elements included in each respective adjacent pair of plies. Generally, though, the width tolerance may be optional in order to increase the robustness of the tie connections. Additionally, or alternatively, the mesh generation tool and/or any other suitable processor may place a cohesive element at least at an intersection of each respective pair of interlayer partition features. 
     The example method  900  may also include determining, by the one or more processors, a predicted mechanical response of the composite laminate material (block  906 ). The processors (e.g., utilizing/executing the solver module  124   a ) may generate a constitutive model (e.g., constitutive model  110   c , inter-laminar constitutive model  124   e ) corresponding to the composite laminate material based on the FE mesh, and may input a stress value and/or strain value to the constitutive model to generate the predicted mechanical response. In certain aspects, the constitutive model may include mixed-mode conditions to model delamination failure in the composite laminate material. 
     More specifically, the solver (e.g., solver module  124   a ) may provide strain values for each integration point of every element at every iteration to the constitutive model. The Laminae subroutine  124   c  (e.g., by the constitutive model  110   c ) and/or the inter-laminar subroutine  124   d  (e.g., by the inter-laminar constitutive model  124   e ) may receive the strain values and return the strain values and corresponding calculated stress values to the solver, which may output the predicted mechanical response (e.g., results  124   f ). 
     In certain aspects, a thermal analysis may be performed before determining the predicted mechanical response in order to capture the manufacturing effects. However, it should be appreciated that the method  900  may be applied for any loading types (e.g., thermal or mechanical), where damage, cracking, and failure may occur. 
     Additional Considerations 
     Throughout this specification, plural instances may implement components, operations, or structures described as a single instance. Although individual operations of one or more methods are illustrated and described as separate operations, one or more of the individual operations may be performed concurrently, and nothing requires that the operations be performed in the order illustrated. Structures and functionality presented as separate components in example configurations may be implemented as a combined structure or component. Similarly, structures and functionality presented as a single component may be implemented as separate components. These and other variations, modifications, additions, and improvements fall within the scope of the subject matter herein. 
     Additionally, certain embodiments are described herein as including logic or a number of routines, subroutines, applications, or instructions. These may constitute either software (e.g., code embodied on a machine-readable medium or in a transmission signal) or hardware. In hardware, the routines, etc., are tangible units capable of performing certain operations and may be configured or arranged in a certain manner. In example embodiments, one or more computer systems (e.g., a standalone, client or server computer system) or one or more hardware modules of a computer system (e.g., a processor or a group of processors) may be configured by software (e.g., an application or application portion) as a hardware module that operates to perform certain operations as described herein. 
     In various embodiments, a hardware module may be implemented mechanically or electronically. For example, a hardware module may comprise dedicated circuitry or logic that is permanently configured (e.g., as a special-purpose processor, such as a field programmable gate array (FPGA) or an application-specific integrated circuit (ASIC)) to perform certain operations. A hardware module may also comprise programmable logic or circuitry (e.g., as encompassed within a general-purpose processor or other programmable processor) that is temporarily configured by software to perform certain operations. It will be appreciated that the decision to implement a hardware module mechanically, in dedicated and permanently configured circuitry, or in temporarily configured circuitry (e.g., configured by software) may be driven by cost and time considerations. 
     Accordingly, the term “hardware module” should be understood to encompass a tangible entity, be that an entity that is physically constructed, permanently configured (e.g., hardwired), or temporarily configured (e.g., programmed) to operate in a certain manner or to perform certain operations described herein. Considering embodiments in which hardware modules are temporarily configured (e.g., programmed), each of the hardware modules need not be configured or instantiated at any one instance in time. For example, where the hardware modules comprise a general-purpose processor configured using software, the general-purpose processor may be configured as respective different hardware modules at different times. Software may accordingly configure a processor, for example, to constitute a particular hardware module at one instance of time and to constitute a different hardware module at a different instance of time. 
     Hardware modules can provide information to, and receive information from, other hardware modules. Accordingly, the described hardware modules may be regarded as being communicatively coupled. Where multiple of such hardware modules exist contemporaneously, communications may be achieved through signal transmission (e.g., over appropriate circuits and buses) that connects the hardware modules. In embodiments in which multiple hardware modules are configured or instantiated at different times, communications between such hardware modules may be achieved, for example, through the storage and retrieval of information in memory structures to which the multiple hardware modules have access. For example, one hardware module may perform an operation and store the output of that operation in a memory device to which it is communicatively coupled. A further hardware module may then, at a later time, access the memory device to retrieve and process the stored output. Hardware modules may also initiate communications with input or output devices, and can operate on a resource (e.g., a collection of information). 
     The various operations of the example methods described herein may be performed, at least partially, by one or more processors that are temporarily configured (e.g., by software) or permanently configured to perform the relevant operations. Whether temporarily or permanently configured, such processors may constitute processor-implemented modules that operate to perform one or more operations or functions. The modules referred to herein may, in some example embodiments, comprise processor-implemented modules. 
     Similarly, the methods or routines described herein may be at least partially processor-implemented. For example, at least some of the operations of a method may be performed by one or more processors or processor-implemented hardware modules. The performance of certain of the operations may be distributed among the one or more processors, not only residing within a single machine, but also deployed across a number of machines. In some example embodiments, the processor or processors may be located in a single location (e.g., within a home environment, an office environment or as a server farm), while in other embodiments the processors may be distributed across a number of locations. 
     The performance of certain of the operations may be distributed among the one or more processors, not only residing within a single machine, but also deployed across a number of machines. In some example embodiments, the one or more processors or processor-implemented modules may be located in a single geographic location (e.g., within a home environment, an office environment, or a server farm). In other example embodiments, the one or more processors or processor-implemented modules may be distributed across a number of geographic locations. 
     Unless specifically stated otherwise, discussions herein using words such as “processing,” “computing,” “calculating,” “determining,” “presenting,” “displaying,” or the like may refer to actions or processes of a machine (e.g., a computer) that manipulates or transforms data represented as physical (e.g., electronic, magnetic, or optical) quantities within one or more memories (e.g., volatile memory, non-volatile memory, or a combination thereof), registers, or other machine components that receive, store, transmit, or display information. 
     As used herein any reference to “one embodiment” or “an embodiment” means that a particular element, feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment. 
     Some embodiments may be described using the expression “coupled” and “connected” along with their derivatives. For example, some embodiments may be described using the term “coupled” to indicate that two or more elements are in direct physical or electrical contact. The term “coupled,” however, may also mean that two or more elements are not in direct contact with each other, but yet still co-operate or interact with each other. The embodiments are not limited in this context. 
     As used herein, the terms “comprises,” “comprising,” “includes,” “including,” “has,” “having” or any other variation thereof, are intended to cover a non-exclusive inclusion. For example, a process, method, article, or apparatus that comprises a list of elements is not necessarily limited to only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Further, unless expressly stated to the contrary, “or” refers to an inclusive or and not to an exclusive or. For example, a condition A or B is satisfied by any one of the following: A is true (or present) and B is false (or not present), A is false (or not present) and B is true (or present), and both A and B are true (or present). 
     In addition, use of the “a” or “an” are employed to describe elements and components of the embodiments herein. This is done merely for convenience and to give a general sense of the description. This description, and the claims that follow, should be read to include one or at least one and the singular also includes the plural unless it is obvious that it is meant otherwise. 
     While the present invention has been described with reference to specific examples, which are intended to be illustrative only and not to be limiting of the invention, it will be apparent to those of ordinary skill in the art that changes, additions and/or deletions may be made to the disclosed embodiments without departing from the spirit and scope of the invention. 
     The foregoing description is given for clearness of understanding; and no unnecessary limitations should be understood therefrom, as modifications within the scope of the invention may be apparent to those having ordinary skill in the art.