Composites are the structural material consisting of two or more constituents that are combined at a macroscopic level and are insoluble in each other. One of the constituents is knows as reinforcing phase and the other in which it is embedded is called the matrix. The reinforcing phase material could be used in the form of fibers, flakes, or particles. The matrix phase materials are generally continuous. Examples of composite systems include concrete reinforced with steel and self reinforced carbon composite, fiber glass etc.
Carbon-fiber-reinforced polymer or carbon-fiber-reinforced plastic (CFRP or CRP), is a very strong composite material. The composite material is commonly referred to by the name of its reinforcing fibers (carbon fiber) similar to fiberglass (glass reinforced polymer). Most often epoxy is used as polymer, however other polymers, such as, vinyl ester, polyester or nylon, are also used depending upon the application and the required properties. Some composites contain a combination of carbon fiber and other fibers such as Kevlar, graphite aluminum, and fiberglass as reinforcement.
Major applications of polymer matrix composites are in aerospace and automotive fields like general & military aviation fuselage, bulkhead & floor, cargo liner, wings, dashboards, bumpers, however it enjoys reasonable use in sailboats, modern bicycles and motorcycles, primarily due to its high strength-to-weight ratio [1]. Improvement in the manufacturing techniques have reduced the he costs and manufacturing time, which makes it increasingly common in small consumer goods as well, such as canoes, fishing rods, archery bows, golf clubs, ski poles & skis, surf boards, racquets, ware - baths & shower units, furniture[2].
Approximate shipments of polymer-based composites in 1995.
(Source: Data used in figure published with permission of the SPI, Inc.; http://www.socplas.org.)
The most attracting property of the composite materials is their extraordinary high strength to weight ratios. This property makes the highly useful for the application in all high speed vehicles. Fracture toughness as a function of yield strength for monolithic metals, ceramics, and metal-ceramic composites could be seen in the following picture. A comparison between the composite and monolithic materials could easily be carried out using the figure below.(Source: Eager, T.W.,Whither advanced materials? Adv. Mater.Processes, ASM International, June 1991,25-29.)
Composite production techniques:
Materials produced with the above-mentioned methodology are generically known as composites. The material of matrix has a huge effect on the properties of the finished composite. One manufacturing technique graphite-epoxy components is the layering sheets of carbon fiber cloth into a mold in the shape of the final product.
The strength and stiffness properties of the resulting material are optimized by the make a careful choice of alignment and weave of the cloth fibers. The mold is then filled with epoxy and heating or air-curing is carried out. The resulting part is strongly resistant to corrosion, has high stiffness, and higher strength for its weight. Parts finding the application in less critical areas are manufactured by draping cloth completely over a mold and then the epoxy is either preimpregnated into the fibers (also known as pre-preg) or "painted" over it. Parts required for high-performance use single molds and are often vacuum-bagged and/or autoclave-cured, due to the fact that void and air bubbles cause a reduction in strength of the material.
Composite production process :
There are many processes for the manufacturing of carbon fiber-reinforced polymers, which particularly depend on the product to be created, the requirement of surface finish (outside gloss), and the quantity of products to be produced.
For simple part with a very low production requirement like (1-2 per day) , a vacuum bag can be used. Polishing and waxing of fiberglass, carbon fiber or aluminum mold is done, and a release agent is applied before the fabric and resin are applied. The vacuum is then established and set kept for a while to allow the piece to cure (harden). There are two ways for the application of resin to the fabric in a vacuum mold. First one is known as wet layup, where the mixing and application of two-part resin is done before it is put in the mold and placed in the bag. The second one uses a resin induction system. In this method the vacuum pulls the resin through a small tube into the bag while the dry fabric and mold are placed inside the bag and then through a tube with holes for spreading the resin evenly throughout the fabric. Hand work is required for both of these methods for applying resin evenly which gives it a glossy finish with very small pin-holes. The third method of composite materials production is called dry layup. In the methods the carbon fiber material is preimpregnated with resin (prepreg) and is applied to the mold like the way adhesive film is applied. This complete assembly is kept then in a vacuum for curing process. This method has an advantage that the amount of resin waste is minimum and lighter constructions can be achieved than wet layup. As it is difficult to drain and bleed larger amounts of resin with wet layup methods, prepreg parts generally have fewer pinholes. Autoclave pressures are used to purge the residual gases out to eliminate the pinholes with minimal resin amounts
A compression mold is used for the quicker production. A two-piece (male and female) mold usually made out of fiberglass or aluminum is bolted together with the fabric and resin between the two. It offers a benefit of being relatively clean and can be moved around or stored without a vacuum until after curing as it is only bolted together.
For complex geometries or convoluted shapes, a technique involving filament winder is used for the production of complex pieces.
Structure
Various carbon fiber-reinforced polymer components are created with a single layer of carbon fabric filled up with fiberglass. A pneumatic tool known as chopper gun is used which quickly create these types of parts by cutting fiberglass from a roll and spraying resin at the same time.
First of all a thin shell is cut out of carbon fiber with the chopper gun so that the fiberglass and resin are mixed on the spot. Either the resin is externally mixed, wherein the hardener and resin are sprayed separately, or internally, where they are mixed before application and cleaned after every use.
Fiber is the primary element of CFRP. A unidirectional sheet is usually created from fiber. A layer is made up by placing one onto each other in a quasi-isotropic layup, e.g. 0, +60, -60 degrees relative to each other. Bidirectional woven sheet can be created from elementary fiber i.e. a twill with a 2/2 weave.
End of useful life/recycling
When protection is done from the sun, Carbon fiber-reinforced polymers (CFRPs) have an almost infinite service lifetime. In contrary to steels these have no endurance limit when exposed to cyclic loading. However when the decommissioning of CFRPs is required, they cannot be melted down in air like many metals. If these are free of vinyl (PVC or polyvinyl chloride) and other halogenated polymers, a thermal decomposition via thermal depolymerization of CFRPs can be carried out in an oxygen-free environment. This can be accomplished in a refinery in single-step process. It is then possible to capture and reuse the carbon and monomers. Carbon fibers can be reclaimed by milling or shredding CFRPs at low temperature; resulting in the dramatically shortening of the fibers by this process. The shortened fibers cause the recycled material to be weaker than the original material just as the downcycled paper. There are certain industrial applications that do not require the strength of full-length carbon fiber reinforcement and such shortened fibers are used extensively in such application. For example, chopped reclaimed carbon fiber can be used in consumer electronics, such as laptops. Even though it lacks the strength-to-weight ratio of an aerospace component, it provides excellent reinforcement of the polymers used. [1]
Composite failures:
Due to a vast spectrum of the applications a lot of research is being carried out to exactly define the mechanical behavior of composite and the major causes of their failure. The research is done to avoid any unpleasant incident which could also be fatal in some cases. After the years of research the researchers have framed out the most general causes of failures as the delamination, matrix cracking, fiber matrix debonding and the buckling of fibers [xyz].
In my project I have paid the attention on the debonding of fibers matrix interface which is very big cause of the delamination at a later stage.
http://www.rapra.net/composites/introduction/polymer-matrix-composite-applications.asp
Micromechanics of Polymer Matrix Composite
A representative volume element (RVE) of a material is the smallest part of the material that represents the material as a whole. It could be otherwise intractable to account for the distribution of the constituents of the material.
Then the composite can be considered to be made of repeating elements called the representative volume elements (RVEs). The RVE is considered to represent the composite and respond the same as the whole composite does.
In case of composite materials analysis the selection of RVE is very important and crucial in some specific type of analysis. In some cases the samples of composite materials are put uder the SEM or the X-ray tomographic image of the sample is generated and then a suitable region is selected for the RVE. Then depending upon the analysis the cohesive zones could be added and the effect could be analyzed. This is particularly the method of choice for the composite having randomly place and randomly oriented fibers.
Composite cylinder assemblage (CCA) model used for predicting elastic moduli of unidirectional composites.
Three-phase model of a composite.
For our case a single fiber is assumed to be sandwiched between the two layers of matrix. This infers the repeated structure would yield the composite material having equally spaced fibers aligned unidirectionally, as shown in the figure below.
RVE FIGURE:
In between the fiber and matrix on either side, a cohesive zone is included. The strength of cohesive is kept very small due to the fact that the effect of the variation of strength of cohesive zone on materials properties was to be investigated. A stronger cohesive zone would lead to the failure or matrix due to plasticity. Since the matrix cracking or matrix damage is not in focus, week strength of cohesive zone is modeled. A week interface tends to reduce the possibility of crack propagation due to the matrix cracking. When the crack reaches the interface, a comparatively week interface deflects the crack on the interface and further propagation of crack is stopped.
Different types of computer simulation have been run for the model to determine the mechanical behavior of the system under different loading cases. The model is loaded in all possible tensile, compressive and biaxial cases. With different cases of varying strengths of cohesive zones in normal and tangential directions, the graphs showing the mechanical behavior of model are plotted.
After the generation of data for the mechanical behavior of model, simulations were also performed to generate the appropriate data for plotting of the yield surface of composite material. For each loading case 4 different strains are applied to generate a reliable yield surface. For all four cases of the cohesive zone strengths, the yield surfaces are plotted and the results are discussed further in the coming chapter.
Stress-strain curve for a unidirectional composite under uniaxial tensile load along fibers.
It could be easily seen that due to the highly linear elastic behavior of the fibers the behavior of the composite is highly brittle. Even the elasto-plastic behavior of the matrix doesnot change much the influence of the reinforcing medium. Hence the regime that is consider to be most interesting is the one which gives a clear insight of the brittle behavior of the composite.
Also due the earlier research it is well established that the increase of the load causes the onset of damage in composite materials and due to its behavior once the damage is initiated the failure of the composite is catastrophic. It means that it doesn't take too long after the onset of failure to the complete failure of the composite. As shown in the above figure , the fibers are highly linear elastic with very high stiffness and the matrix has a lower stiffness comparatively. The composite material has the stiffness in between the major constituent and is also linear elastic. Also the ultimate stress of composite is higher than the matrix but is less than that of the fibers.
Damage :
Damage in composite materials occurs through different mechanisms that are complex and usually involve interaction between microconstituents. During the past two decades, a number of models have been developed to simulate damage and failure process in composite materials, among which the damage mechanics approach is particularly attractive in the sense that it provides a viable framework for the description of distributed damage including material stiffness degradation, initiation, growth and coalescence of microcracks and voids. Various damage models for brittle composites can be classified into micromechanical and macromechanical approaches. In the macromechanical damage approach, composite material is idealized (or homogenized) as an anisotropic homogeneous medium and damage is introduced via internal variable whose tensorial nature depends on assumptions about crack orientation [15], [28], [29], [42], [35], [43], [31]. The micromechanical damage approach, on the other hand, treats each microphase as a statistically homogeneous medium.
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The interaction of these microscopic damage mechanisms is dependent on a number of factors, the most important being the adhesion between the constituent phases. Experimental studies have shown that the dominant damage mechanism involved in transverse ply cracking is debonding occurring at the fiber matrix interface. As these interfacial debonds grow, small resin bridges formed between them. Under sustained or increased loading these resin bridges underwent significant plastic deformation until they ruptured, causing final fracture of the ply.
The computational framework used to model deformation in the carbon fiber/epoxy composite is similar to that pioneered by Llorca and coworkers [8-9] where, a cohesive zone model is used to predict the onset of fiber-matrix debonding while the non-linear behavior in the matrix phase is modeled using the Mohr-Coulomb plasticity theory. The recently developed Nearest Neighbor Algorithm (NNA) that can accurately reproduce a statistically equivalent fiber distribution for the high volume fraction composites is used to generate the finite element models for the analysis. Finally, the effect of damage accumulation due to cyclic loading is assessed in order to try and better understand the role that fiber-matrix debonding and matrix plasticity play in the overall macroscopic response of the composite. The fibers are assumed to be linear elastic. The matrix is assumed to behave as an elastic-plastic solid. The behavior of the matrix phase is sensitive to the hydrostatic stress and as a result the Mohr-Coulomb yield criterion is employed. The interface damage tends to occur in closely neighboring fibers, however, the interfacial cracks propagated away from adjacent fibers, towards a matrix rich region. More importantly, the effect of increasing the fracture energy has been to increase the strain to failure of the composite. This has important implications as it is the low strain to failure of transverse plies which causes damage to initiate in cross-ply laminates [1]. Through the process of damage accumulation, micro-cracks in these transverse plies cause other damage mechanisms to evolve, such as inter-laminar delamination or even fiber fracture in neighboring plies.
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Polymer matrix composites (PMC) are routinely analyzed by assembling laminae response into laminate response models [8] .The laminate-level response to external loads is then decomposed into laminae responses. That is, the point stress and strains on each homogeneous orthotropic lamina are found. The application of such modeling of damage has a major shortcoming that a large number of material constants are required for the representation of equivalent orthotropic material [4-6,22].The data are scarcer with regards to damage evolution damage evolution models and failure criteria can be formulated at the constituent-level model proposed herein accounts for different initiation, evolution, and failure of the two main constituents (fiber and matrix) and, with the addition of an interphase model, it accounts for other effects not captured by the constituent models.
Loss of transverse isotropy at the lamina-level due to damage can be predicted [12]. In laminate analysis, each lamina is considered as a homogeneous material. The characteristic length of a material element over which the stress and strains do not change rapidly is the lamina thickness. The fiber diameter, fiber spacing, and dimensions of micro-cracks are much smaller than the lamina thickness. Therefore, fiber breaks, matrix crazes and micro-cracks can be analyzed as distributed damage.
Since damage of the fiber phase contributes only to loss of stiffness and strength in the fiber direction, the characteristic length of the fiber phase is of the order of the fiber length, supporting the assumption that fiber break can be modeled as distributed damage [19,29]. In tensile loading normal to the fibers, matrix cracks grow along the fiber length and can exceed the lamina thickness, which seems to invalidate the assumption of distributed damage. But if that happens to a unidirectional lamina, such cracks lead to immediate failure.
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Cohesive Zone Modeling:
The cohesive process zone model is a general model having the capability to deal with the nonlinear zone ahead of the crack tip--arising due to plasticity or micro-cracking--present in many materials has been reviewed. It has the ability not only to adequately predict the behavior and response of uncracked structures, but also bodies with blunt notches. A review of the cohesive model is made and determination of the softening function has been emphasized, which is an essential ingredient of the cohesive model, by inverse analysis procedures. Some examples of the predictive capability of the cohesive zone model are also presented when applied to different materials; concrete, PMMA and steel. Linear elastic fracture mechanics (LEFM) could be used for solving fracture problems in case the crack-like notch exists and the nonlinear zone ahead of crack tip is negligibly small. Hillerborg used this model with the name of 'fictious crack model' [23].
Broberg [4] proposed the description of process as a decomposition of cells. This description is analogous to the FEM description of discretization of the domain. When the cells are assumed to be cubic they resemble the Bazant's crack band approach [3]. In fact the Broberg's cell approach is equivalent to Bazant's crack band approach when the cells form a band along the crack path.
The cell approach restricts the size variations while in the crack band approach the finite element size can be varied consistently. Also the band approach associates the same stress-strain behavior to all the elements in the model for the hardening branch of stress-strain curve but the cell approach assigns the properties of continuum to the material outside the process zone and is not related to the properties of cell, except for the elastic response.
The cohesive crack process zone in the limiting case can be a cell model when the cells become flat and in the limit they have zero width. It is then the same as using Interface elements in FEM.
[Ref 3, Chapter 8]. Cohesive crack behavior is defined by the relation between cohesive stress and the relative displacement between the upper and lower face of the cell also known as the cohesive crack width.
Information only on the descending part of the load-displacement curve is contained by the cohesive crack curve. In order to use cohesive zones in homogenous bodies the initial stress must be the absolute maximum, however, a monotonically decreasing function or a function with a relative maximum less than the initial tensile strength is also valid. Anomalous behavior is produced with a secondary maximum greater than the absolute tensile strength which also produces secondary crack in the immediate vicinity.
For the non-homogenous bodies like the interface between two elastic blocks joined by bridged fibers, the only condition to be checked is that the stresses inside the blocks would not produce cracks outside the joint.
The softening function is considered as a material property. Two properties of the softening function are most important: tensile strength and the cohesive fracture energy.
The characteristic length is an inverse measure of degree of brittleness. It further relates to the size of a fully developed fracture process zone.
A straight forward way to measure the softening curve is by stable tensile testing of specimens. In principle such an approach lead to complete stress-displacement curves but practically it has some drawbacks. It leads to the use of alternate methods like inverse analysis or data reduction. However the results are usually not reproducible even with the same material because the procedures involved are based on simultaneously solving a set of functional equations.
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Cohesive-zone modeling for a fiber-reinforced polymer-matrix composite has been carried out. It is shown that a two-parameter model with a characteristic toughness and a characteristic strength can be used to predict the fracture of notched or cracked specimens. A comparison between numerical predictions and experimental observations of a fracture test is made to determine two parameters. A two-parameter model describes quite well the engineering behaviors in terms of strength, deformation and energy dissipation; however some extra details may be required about the cohesive law such as the matrix-cracking strength when the characteristic dimensions of the composite structure (e.g., the initial crack length or ligament length) are very small. Further the transitions between stable and catastrophic crack growth in the composite is predicted quite well by the cohesive-zone model, therefore, a broad understanding of the energy dissipation during fracture that occurs in these different regimes is permitted.
Fracture of brittle-matrix composites is generally modeled with the fracture-mechanics approach, in such a fashion that bridging fibers are left behind the tip as a sharp crack propagates into the matrix [31]. The bridging fibers tend to reduce the energy-release rate available to propagate the matrix crack. An initially unbridged crack existing in the composite will begin to propagate into the matrix if the applied energy-release rate equals (1-cf)Гm , where Гm is the matrix toughness and cf is the area fraction of fibers on the crack plane.
An important distinguishing feature of these cohesive-zone models from bridged-crack models is the automatic introduction of a strength-based fracture criterion (cohesive strength) in conjunction with an energy-based fracture criterion (toughness) for the material ahead of the crack tip . Cohesive-zone elements are embedded along the fracture plane that deform according to a traction-separation law having the appropriate strength and toughness (area under the traction- separation curve) [33,46,47]. In such a simple two-parameter form of the cohesive zone model, matrix cracking and fiber-bridging are not distinguished [43]. If a distinction between these two phenomena is necessary, the traction-separation law is to be split into two components - one associated with matrix cracking and one with fiber bridging.
The characterization of the constitutive properties of the composite is done by performing uniaxial tensile tests and Iosipescu shear tests on specimen .The composite properties could be treated as being transversely isotropic. The determination of the cohesive parameters of the composites is carried out by means of compact-tension specimens with the specific dimensions. The load-displacement curves are obtained from these specimens at a specific displacement rate.
A characteristic of compact-tension geometry is shared with the double-cantilever beam in such a way that the decreasing portions of the load-displacement curves, which are obtained after the crack begins to grow, are very sensitive indicators of the mode-I toughness, Г (area under the traction-separation curve), and relatively insensitive to the cohesive strength. Therefore, the toughness was obtained by matching this region of the load-displacement curves to predictions from numerical analyses that incorporated a cohesive zone at the crack tip. The approach appears to predict accurately the onset of catastrophic failure in static tests i.e. a mode of fracture in which the crack makes a transition to a dynamic mode, causing rupture with no further input of energy into the system.
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http://www.mate.tue.nl/mate/pdfs/8961.pdf
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An important issue when considering failure is the observation that most engineering materials are not perfectly brittle in the Griffith sense, but display some ductility after reaching the strength limit. In fact, there exists a small zone in front of the crack tip, in which small-scale yielding, micro-cracking and void initiation, growth and coalescence take place. If this fracture process zone is sufficiently small compared to the structural dimensions, linear-elastic fracture mechanics concepts apply. However, if this is not the case, the cohesive forces that
Schematic representation of a cohesive zone.
Stress-displacement curves for a ductile solid (left) and a quasi-brittle solid (right).
exist in this fracture process zone must be taken into account, and cohesive-zone models must
be used, which were introduced by Barenblatt [3] and Dugdale [4] for elastic-plastic fracture in ductile metals, and for quasi-brittle materials by Hillerborg et al. [5] in his so-called fictitious crack model.
In the past two decades, cohesive-zone models have been recognized to be an important tool for describing fracture in engineering materials. Especially when the crack path is known in advance, either from experimental evidence, or because of the structure of the material (such as in laminated composites), cohesive-zone models have been used with great success. In those cases, the mesh can be constructed such that the crack path a priori coincides with the element boundaries. By inserting interface elements between continuum elements along the potential crack path, a cohesive crack can be modeled exactly. Figures 3 and 4 show this for mixed-mode fracture in a single-edge notched (SEN) concrete beam [11]. In this example the mesh has been designed such that the interface elements, which are equipped with a quasi-brittle cohesive zone model, are exactly located at the position of the experimentally observed crack path [12].
Another good example where the potential of cohesive-zone models can be exploited fully using traditional discrete interface elements, e.g. Reference [13], is the analysis of delamination in layered composite materials [14-17]. Since the propagation of delaminations is then restricted to the interfaces between the plies, inserting interface elements at these locations permits an exact simulation of the failure mode. To allow for a more arbitrary direction of crack propagation, Xu and Needleman [18] have inserted interface elements equipped with a cohesive-zone model between all continuum elements. A related method, using remeshing, was proposed by Camacho and Ortiz [19].
Although analyses with this approach provide much insight, see also References [20, 21], they suffer from a certain mesh bias, since the direction of crack propagation is not entirely free, but is restricted to interelement boundaries. Another drawback is that the method is not suitable for large-scale analyses. For these two reasons smeared numerical representations of cohesive-zone models have appeared, including the emergence of some, initially unforeseen mathematical difficulties, which can only be overcome in a rigorous fashion by resorting to higher-order continuum models.
What I have to do What I have selected Conclusion:
The fibers are assumed to be linear elastic and homogenous solid. However due to their anisotropic nature, the fiber materials is modeled with engineering constants. It is also assumed that the fiber fracture does not occur due to transverse tensile loading.
The matrix is assumed to behave as an elastic-plastic, homogenous solid. The matrix material is assumed to be isotropic. Since the matrix phase consists of Epoxy material which is sensitive to the hydrostatic stresses, the Mohr-Coulomb yield criterion is employed.
The modeling of interface is a done using the cohesive zone modeling. At the interface between the fiber and the matrix surface based cohesive behavior is included with a pure master-slave formulation. The nodes at the interface are made to obey the cohesive law. The initiation of damage is controlled by the maximum stress criteria. It follows that the damage would be initiated at the interface when the stress at the interface increases than the specified limit.
After the initiation of damage the damage evolves linearly. The starting point for the evolution of damage is the achievement of the maximum stress at the interface. Linear softening is assumed to occur at the interface during the evolution of damage. The criterion of complete failure is related to the fracture toughness of interface. The energy at the interface must reach the fracture toughness of the material for the damage to reach its maximum.
By the variation of this fracture toughness, we actually increase the area under traction-separation curve. Since the maximum stress, which acts as a level to start the damage initiation, is kept constant the final displacement at the fracture must increase.
Since in the current project only the damage due to delamination is considered, hence only the interface is modeled using the cohesive zones. There is no cohesive zone inclusion in the matrix hence other composite defects like matrix cracking, crack deflection are not taken into account.
A two dimensional RVE is made with a single fiber sandwiched between the two matrix components. The fiber volume fraction is kept to be 0.6.This RVE is then tested under different tensile, compressive and combined loading combinations.
The mechanical behavior of the RVE is studied under the plane strain conditions. The effect of variation of the maximum stress and the maximum energy in the normal and shear directions is observed.
Different load cases are made for a set of varying maximum stresses and maximum energies. For each load case the stress-strain curve is plotted to graphically observe the effect of variation in different parameters.
Following are the material properties used in the project.
Constituent Material Properties:
Elastic Properties
Fiber (HTA)
Matrix(6376)
E11 (GPa)
238
3.63
E22 (GPa)
28
E33 (GPa)
28
Ï…12
0.23
0.34
Ï… 23
0.33
Ï… 31
0.03
G12 (GPa)
24
G23 (GPa)
7.2
G31 (GPa)
24
Reference : Vaughan, T.J., McCarthy, C.T., Micromechanical Modeling of the Transverse Damage Behavior in Fiber Reinforced Composites, Composites Science and Technology (2010), doi: 10.1016/j.compscitech.2010.12.006
In the current project I have used the above mentioned material model. The interface between the fibers and matrix are modeled using the cohesive zone modeling technique. Further to investigate the effect of the variation of the cohesive zone strength , a comprehensive parametric study is carried out in which the strength of cohesive zone is varied in the direction normal to the fibers and in the direction parallel to the fibers. Thus 4 cases of varying cohesive zone strength are studied. As mentioned in the material model, the properties of cohesive zone are given as the maximum stress required to initiate the damage and the energy required for the complete separation. Thus , when the applied stress reaches a certain value the damage initiates in the cohesive zone and it keeps on increasing till the energy absorbed is equal to a certain amount and that is then the completion of the damage.
Following table shows the description of the cases investigated
Case No.
Properties in normal direction
Properties in shear direction
Case 1
10mJ/m2
0.2MPa
0
Case 2
10mJ/m2
0.2MPa
10mJ/m2
0.2MPa
Case 3
100mJ/m2
200MPa
10mJ/m2
0.2MPa
Case 4
10mJ/m2
0.2MPa
100mJ/m2
200MPa
Results:
Based on the results of the simulation for different cases, the mechanical behavior of the model is plotted. The stress-strain curves are plotted and also a comparison is done. Each graph shows the response of model under the same loading for all four different cases.
Uni-axial tension along positive x-axis
This is the simplest case of loading for our model in which the model is loaded under uniaxial tension along positive x-axis. As represented by the model data, we can see that the behavior of the models for Case 1, Case2 and Case4 are the same are they overlap each other. It is worth mentioning that the cohesive zones in these models have different shear strength but same normal strength so the behavior is same i.e. the stress just reaches the strength of cohesive zone and well below the yield stress of matrix, it fails. However, quite convincingly the maximum load taken by the case 3 is considerably higher than the other. It was quite expected because the strength of cohesive zone was higher than the other cases.
This case was taken as the test case for the verification of the behavior of the model. The behavior of model was compared with the results of XZY et.al [XYZ]. It was observed that the behavior of the stress stain curve is in good agreement with their work. Further it is also compared to the work of [XYZ] and the behavior of the model is compared with it.
The maximum stress for Case3 was XYZ and the maximum stress for all other cases was XYZ
Bi-axial tension along positive x-axis and positive y-axis
The maximum stress for Case 3 was and the maximum stress for all other cases was
Uni-axial tension along positive y-axis
The minimum stress for for all other cases was
Bi-axial tension along negative x-axis and positive y-axis
The maximum stress for for all other cases was
Uni-axial compression along negative y-axis
The minimum stress for cases was
Bi-axial tension along negative x-axis and negative y-axis
The maximum stress for all cases was
Uni-axial tension along negative y-axis
The minimum stress for all cases was
Bi-axial tension along positive x-axis and negative y-axis
The maximum stress for Case 3 was and the maximum stress for all other cases was
Discussion
Summary
