Selecting Material Parameters in Abaqus for Cohesive Elements Defined in Terms Of Traction-Separation Dassault Systemes Simulia Corp. 166 Valley Street Providence, RI 02909-2499
Introduction
Cohesive elements enable the modeling of fracture initiation and propagation in finite element analyses. In Abaqus the constitutive response of cohesive elements may be based on a continuum description of the material or a traction-separation description of the interface. The continuum description is suitable when the actual thickness of the interface is being modeled. Alternatively, the traction-separation description is suitable when the thickness of the interface can be considered to be zero. The constitutive behavior of cohesive elements with a continuum description is prescribed by any of the conventional material models available in Abaqus, with appropriate material properties chosen based on physical testing of the bulk material constituting the cohesive layer. In the case of cohesive elements with traction-separation behavior the parameters characterizing the traction-separation relationship must be specified, including the initial stiffness, damage initiation threshold, and damage evolution properties. The initial stiffness of cohesive elements defined in terms of traction-separation does not represent a physically measurable quantity and is treated as a penalty parameter. Ideally, the stiffness of the cohesive element should be infinite so that they do not affect the overall compliance of the model before the damage initiation point; however, a finite value must be used in the finite element context. The damage initiation point corresponds to the peak value of the traction-separation relationship – the strength of the interface. The strength of the interface is a physically measurable quantity, although it may be difficult to measure it experimentally. For studying delamination growth in composite material systems, the strength of the interface can be measured using established experimental techniques as described later. However, for modeling fracture in many other materials for example, ice, heat sealed polymeric films, etc., the strength may be difficult to obtain. This observation has motivated Diehl [11, 12] to treat even the interface strength as a penalty parameter. The damage evolution of cohesive elements defined in terms of tractionseparation is described by the fracture toughness of the interface. The fracture toughness (or the critical energy release rate) is usually available in literature or handbooks for many material systems or it can be determined from experiments in conjunction with appropriate analysis of standardized test configuration.
1
In the following we provide some general guidelines for selecting the material parameters for cohesive elements defined in terms of traction-separation behavior. It is assumed that the fracture toughness is known from literature or experiments. The strength of the interface is assumed to be either known or unknown. Some guidelines and references for determining fracture toughness are also discussed. In Version 6.8 of Abaqus a surface-based cohesive behavior has been developed which does not require the specification of the initial stiffness; it is treated as a penalty parameter and is determined by Abaqus automatically using the stiffness of the underlying elements. The guidelines and experimental procedure described for damage initiation and evolution would still be applicable for modeling fracture using surface-based cohesive formulation. It is natural to think of the properties defining traction-separation relation in the following order: elastic stiffness, damage initiation, and damage evolution, as this is the order in which these properties come into effect in the deformation of cohesive elements. However, for accurate overall simulation of fracture initiation and propagation the most important property is the fracture toughness defining the damage evolution followed by the interface strength defining the damage initiation and finally the elastic stiffness. Hence the discussion below is presented in this order. Damage Evolution: Fracture Toughness
Fracture toughness as measured by the critical energy release rate is the single-most important parameter that defines the cohesive traction-separation behavior. In Abaqus, damage evolution for traction-separation cohesive behavior may be specified by giving the separation at complete failure relative to separation at damage initiation, or by specifying the fracture toughness. In both cases, the quantities can be given as a function of mode-mixity; the mode-mix dependence of data can be specified in a tabular or analytical form. Furthermore, in both cases the softening part of the traction-separation relation may be specified as linear, exponential, or tabular. For many material systems the fracture toughness has been or can be measured experimentally; however, the value of separation at the final failure as well as the shape of the softening portion of the traction-separation relation may be difficult, if not impossible, to determine. Thus, it is easier to use energy-based damage evolution with a linear softening behavior; the effect of the shape of the softening response on the overall results may be studied subsequently if desired. In the following we only discuss guidelines for specifying energy based damage evolution. The area under the traction-separation curves for various modes represents the corresponding fracture energies – G Ic, GIIc, and GIIIc. In problems where the various fracture modes are likely to interact, the mixed mode behavior must be specified as well. The fracture energies in pure modes as well as in mixed mode loading conditions can be determined experimentally. More commonly the fracture energies in pure modes are measured experimentally and the mixed mode behavior is specified via certain analytical forms that fit limited mixed-mode experimental data. Whereas the elastic stiffness and damage initiation parameters of the cohesive traction-separation relation can be treated as
2
penalty parameters that can be adjusted, the fracture energies are physical quantities that must be specified accurately. Experimental techniques are well established for determining interlaminar fracture in composite material systems: ASTM standards exist for mode I (ASTM D5528 [2]) and combined mode I - mode II loading conditions (ASTM D6671 [3]). The mode I testing standard uses a double cantilever beam (DCB) specimen whereas the combined mode I - mode II standard uses a mixed mode bending (MMB) test to measure fracture toughness over a wide range of mode I and mode II loading combinations. According to Reeder [18] ASTM is also working towards standardizing end notch flexure (ENF) test for pure mode II (Russell [19], Davidson and Sun [8]) and edge crack torsion (ECT) test for pure mode III (Lee [16], Ratcliffe [17]) fracture toughness for composite delamination fracture. These and other works must be referred to in order to conduct suitable experiments to obtain fracture energies for delamination fracture simulation. Fracture energies for interface cracks in thin layered structures such as those encountered in electronic devices applications can also be obtained experimentally. A number of experimental techniques exist including double cantilever beam, peel test, four-point bending test, and Brazil sandwich test. The Brazil-nut sandwich specimen proposed by Wang and Suo [24] can be used to measure fracture toughness of an interface crack for various mode mixity (also referred to as loading phase in interface fracture mechanics literature). Wang and Suo [24] measured the interfacial fracture toughness for aluminum/epoxy, steel/epoxy, brass/epoxy, and plexiglass/epoxy systems using the Brazil-nut sandwich specimens. On the other hand Dauskardt and co-workers [6] have measured fracture toughness for SiO 2 /TiN interface in thin layered structures encountered in microelectronic interconnects using a four-point bending test. The four point bending test results in a mixed mode loading condition. The energy corresponding to pure modes I and II may be extracted using the theory of interfacial fracture mechanics (Suo and Hutchinson [21], Hutchinson and Suo [15]). Furthermore, it should be noted that if plasticity develops in layers close to or adjacent to the interface, the measured fracture toughness will have a contribution due to plastic dissipation; this contribution is referred to as extrinsic fracture energy. The intrinsic fracture toughness of the interface will have to be deduced from the total fracture toughness. One approach that has been used is to measure the fracture toughness of the interface for different ductile layer thickness. As thick layer will have larger plastic zone compared to thin layer, the intrinsic fracture toughness is then deduced by extrapolating the measured fracture toughness to correspond to a ductile layer of zero thickness (Dauskardt et al. [6], Hutchinson and Evans [6]). Damage Initiation: Interface Strength and Mesh Requirements
The peak traction in the traction-separation relation for a particular mode of fracture corresponds to the interfacial strength in that mode. Interface strength determines the damage initiation point; beyond this point damage begins to develop in the cohesive zone leading to reduction in the traction.
3
Experiments for determining interfacial strength are not well developed for many material systems: there have been attempts to determine mode I interface strength in thin film systems by passing a laser-generated compressive pulse through the substrate. The reflection of the pulse from the free surface creates a tensile pulse that fractures the interface (Gupta et al. [13]). The laser-based technique has also been used to determine tensile and mixed-mode strength by Wang et al. [23]. On the other hand, for studying delamination initiation and propagation in fiber-reinforced composites, the interfacial strength in various modes may be estimated based on lamina strength properties. Davila (private communication, 2007) has suggested that the transverse lamina strength be taken as mode I strength and mode II and mode III strength (assumed equal) to be derived from the tensile strength of a [+45/-45] laminate. These properties are usually available for many composite material systems or may be obtained using established standards (ASTM D3518/D3518M [1]). If the interface strength is known for the material system, it can be used to guide the mesh selection strategy as discussed below. However, if such experimental data for your material system is lacking, you must provide a reasonable estimate of the interfacial strength. One strategy is to treat the interfacial strength as a penalty parameter and determine its value by numerical experiments (Diehl [11, 12]); this approach is discussed below as well. Case I: Interface strength is known
A rule-of-thumb for choosing the cohesive element size is to use 3-5 cohesive elements per adjacent continuum element in the mesh (Diehl [11, 12]). However, if the interfacial strength is known from literature or from direct experimentation, it may be used to assess and/or estimate the size of the cohesive elements in the mesh. This estimate is based on embedding sufficient number of cohesive elements within the process zone that develops in front of a crack tip. The length of the process zone is a material property for nonslender bodies and is given by: l pz = ME
Gc
(1a)
(t 0 ) 2
where Gc is the interface fracture toughness, t 0 is the interface strength, E is the modulus of elasticity of the material through which crack propagation is being simulated, and M is a parameter which is dependent on various models proposed in the literature; its value ranges from 0.21 to 1.0 (Turon et al. [22]). For crack propagation through slender bodies, the cohesive length is given by (Yang and Cox [25]): l pz
⎡ G ⎤ = ⎢ E c 2 ⎥ ⎣ (t 0 ) ⎦
1 / 4
h 3 / 4
(1b)
where h is the thickness of the slender body. For orthotropic materials with transverse isotropy the value of transverse modulus should be used in the above equations.
4
In order to obtain accurate FE results using cohesive elements, a sufficient number of elements should span the process zone. If N e cohesive elements are chosen to span the process zone, the length of the cohesive element is given by: l pz le = N e
(2)
Typically Ne = 3-5 is sufficient for accurate simulation for mode I crack propagation (Turon et al. [22]). Thus equations (1) and (2) may be used to determine the size of cohesive element and/or to assess whether the cohesive elements in the mesh are sufficiently refined. Of course, the cohesive element length should not be greater than the length of the adjoining continuum element, which is determined based on the overall deformation and accuracy considerations. In many situations the requirement of embedding a sufficient number of cohesive elements within the process zone may lead to very refined mesh that may not be desirable due to other considerations. In such situations one can artificially increase the process zone length by reducing the interface strength so that sufficient elements do span the process zone. Assuming the number of elements desired in the process zone is N e and the length of the cohesive elements in the mesh is le , Eq. (1a) and Eq. (2) can be combined to obtain the modified interface strength as: t 0 =
MEGc N e l e
(3a)
Similarly Eq. (1b) and Eq. (2) can be combined to obtain modified interface strength for slender bodies as: t 0 =
EGc h
3
4 4
N e l e
(3b)
The estimated interfacial strength should be compared with the actual interface strength and a minimum of the two must be used in the simulation, thus, t 0 = min(t 0 , t 0, actual )
(4)
It should be noted that reducing the interface strength via Equations (3) and (4) will lead to less accurate crack-tip stresses. Thus, this approach may not capture the initiation behavior of the crack accurately; however, the crack propagation will be reasonably captured (Turon et al. [22]). Furthermore, a balance between the strength reduction and appropriate mesh size should be achieved for proper simulation, i.e., if the strength obtained from Eq. (3) is much smaller than the true interface strength, it is an indication that the mesh is not suitably refined and hence mesh refinement would be a better alternative than strength reduction. On the other hand if the strength reduction is a small fraction of the true strength, one may obtain a reasonably accurate solution by reducing the strength and keeping the same mesh; thus avoiding re-meshing of the model.
5
Case II: Interface strength is not known
For many material systems the interfacial strengths are difficult to determine experimentally. In such situations the interface strengths may be treated as penalty parameters as advocated by Diehl [9-12]. For a triangular traction-separation law (Fig. 1) the interface strength can be written as a function of fracture toughness and the separation at failure ( δ f ) as: t 0 =
2Gc
(5)
δ f
With the fracture toughness known,
δ f
may be varied parametrically to obtain
values of interface strength to be used in the simulation. Based on classical fracture mechanics arguments, Diehl recommends that δ f should be made as small as possible so that the traction-separation law approaches a delta function; however, it should be large enough so as not to cause numerical instabilities. This parameter is problem and mesh dependent and it must be determined by conducting a parametric study for your problem with the initial value set as some fraction of the cohesive element length. In this case the cohesive element length must be chosen arbitrarily – a rule-of-thumb is to choose the cohesive element length such that 3-5 elements span an adjoining continuum element. Of course, the size of the adjoining continuum element should be determined from accuracy consideration and/or mesh refinement studies. For elastic DCB analysis Diehl [11] found that a good match with the analytical solution is obtained when δ f is taken as 5% of the cohesive element length. On the other hand, for inelastic peeling problem Diehl [12] found that δf should be equal to the cohesive element length for accurate simulation results. Initial Elastic Stiffness
The role of the cohesive elements is to simulate fracture initiation and propagation. Ideally the cohesive elements should have infinite stiffness so that they do not affect the overall compliance of the model before the damage initiation criterion is met. However, in a finite element model these elements have finite initial stiffness and a sufficiently high value should be chosen such that the overall initial stiffness of the model is not significantly affected by their presence. It should be noted that a very high value of stiffness may lead to convergence difficulties in Abaqus/Standard, and in Abaqus/Explicit it may lead to very small time increments. Thus an optimum value of the initial stiffness should be chosen based on all these considerations. In Abaqus the initial elastic stiffness of the cohesive elements may be specified as an uncoupled or coupled relationship between tractions and strains; the strain is defined as the ratio of separation and the constitutive thickness. Various guidelines have been proposed in the literature for choosing the stiffness of the cohesive elements. All of these guidelines pertain to uncoupled traction-strain response and you should choose the uncoupled behavior unless there is suitable data for choosing coupled elastic response.
6
t
t o
K ′
0
δ o
δ f
δ
Fig. 1: Schematic of traction-separation relationship
Some researchers (cf. Daudeville et al. [5], Zou et al. [26], Camanho et al. [4]) have proposed values of elastic stiffness for interfaces in composite material systems based on experience. On the other hand Turon et al. [22] have proposed the elastic stiffness of the traction-separation relationship in terms of the elastic modulus and the thickness of the region surrounding the interface. For mode I fracture they derive the overall stiffness of a system consisting of two sublaminates bonded with a cohesive element. The overall stiffness (normal to the interface) is given by: 1 Eeff = E E (6) 1+ ' Kn h where E is the Young’s modulus of the sublaminate, h is the thickness of the sublaminate and K n′ is the initial stiffness of the interface (defined as a ratio of traction to opening displacement, Fig. 1) in the normal direction. Abaqus requires the initial stiffness K n to be defined as the ratio of traction to strain with the strain being defined as the ratio of opening displacement to the constitutive thickness T 0 . Thus, K n′ ≡
t δ
=
t ε T 0
=
K n T 0
(7)
For the cohesive element stiffness to have negligible effect on the total stiffness of the system, E /( K n′ h) in Eq. (6) should be much smaller than 1. Expressed differently, the interface stiffness can be chosen as:
7
K n′ ≡
K n T 0
K n =
E α
=
(8a)
h
α E
h
T 0
(8b)
where the parameter α should be much larger than 1.0. Furthermore, similar expressions can be derived for shear modes (mode II and III): K s =
α s G
T 0
h
(9)
again with α s >> 1 . It should be noted that large values of the interface stiffness may cause numerical problems such as spurious oscillations of the tractions (Schellekens and de Borst [20]). Thus the interface stiffness should be large enough not to alter the overall stiffness of the model but small enough to reduce the risk of numerical problems. For Mode I fracture in a composite DCB specimen, Turon et al. [22] recommend the value of α to be 50. Diehl [9-12] has proposed an alternative approach to estimate the elastic stiffness of the cohesive element based on classical fracture mechanics and numerical stability considerations. He argues that for the cohesive traction-separation law to mimic classical Griffith fracture mechanics its shape should be a delta function, i.e., infinite stiffness and strength with finite fracture energy that is released instantaneously. Then he argues that such a representation would lead to computational instability and hence one should limit the stiffness and strength to a finite value. For a triangular traction-separation relationship, the stiffness is defined in terms of fracture energy (G c), separation at final failure (δ f ), and the damage initiation ratio ( δ ratio=δ 0 / δ f ) as: K ′ ≡
K T 0
K =
2Gc
=
(10a)
2
δ ratioδ f
2Gc 2
δ ratioδ f
T 0
(10b)
where δ 0 is the separation at damage initiation (Fig. 1) and T 0 is the user-specified initial constitutive thickness of the cohesive element (typically specified as 1.0). Diehl has parametrically studied the effects of
δ ratio
on various mode I fracture
characteristics of elastic double cantilever beam (DCB) and inelastic peeling problems. The results of his parametric studies reveal that δ ratio does not significantly affect the overall load-displacement response of the DCB specimen when compared with the benchmark solution; he recommends a value of 0.5 for this parameter. Guidelines for selecting δ f were discussed earlier in conjunction with the interface strength
8
requirements. Thus we can now obtain the initial elastic stiffness of the cohesive element for various choice of δ f and δ ratio . Note that Abaqus requires the value of K and not K ′ on the *ELASTIC keyword option. In Abaqus, the elastic stiffness of cohesive elements ( K ) is defined as a ratio of traction to strain with the strain defined as the ratio of separation to the initial constitutive thickness. This is in contrast with most literature where stiffness of cohesive elements is defined as a ratio of traction to separation, i.e., K ′ . Both the definitions are identical when the initial constitutive thickness is specified as 1.0 (which is the default setting in Abaqus). However, if a non-default value is used for the constitutive thickness you may have to re-scale the stiffness K to get the same behavior as that obtained with unit thickness. Also note that Eqs. (10) are valid for all three fracture modes, with appropriate value of fracture energy and separation at failure values. Illustration: Mode I Fracture - Double Cantilever Beam Specimen
In order to illustrate the procedure for selecting suitable material properties for cohesive elements, we consider mode I fracture in a double-cantilever beam specimen (Figure 2). The arms of the beam are made of linear elastic isotropic material with a modulus of 70.0 GPa and Poisson’s ratio 0.33. The fracture energy of the interface in mode I (assumed 2 given, based on suitable experiments) is 0.352 kJ/m . The dimensions of the arms are L = 200 mm, b = 10 mm, h = 0.2 mm. The interface is modeled via zero-thickness COH2D4 cohesive elements and the beams are modeled using B21 beam elements. An initial crack of length 20 mm is assumed. We illustrate two cases: 1) the interface strength is known and 2) the interface strength is not known. P a
h b
y
L P Fig. 2: A schematic of Double Cantilever Beam (DCB) specimen Case I: The interface strength is known
We assume that the interface strength is known (from suitable experiments or handbooks) to be 60 MPa. The initial stiffness of the cohesive elements is estimated using Eq. (7b) with α=50, h=2 mm, and T0=1.0 as:
9
K n =
E α h
T 0 =1.75 E 6 MPa
Now we have all the data to define the traction-separation relationship. In order to simulate the problem we must choose a suitable mesh for the arm as well as the interface. The size of the beam elements can be estimated based on bending consideration only (i.e., without considering fracture initiation or propagation). For the cantilever beam bending a mesh with 200 beam elements (B21 elements) along the length was found to give suitable bending response. To estimate the size of the cohesive elements we can adopt two strategies. A rule-of-thumb approach is to use finer mesh for cohesive elements with 3-5 cohesive elements for each adjoining structural element. We choose an over-meshing ratio of 5 giving the length of the cohesive element as 0.2 mm. An alternative approach for choosing the size of the cohesive element is to follow the guidelines presented earlier based on embedding suitable number of elements in the process zone of the crack tip. As the DCB model is a slender structure we use Eq. (1b) to estimate the length of the process zone using the known value of fracture toughness, G Ic 2 = 0.352 kJ/m , elastic modulus of the beam 70 GPa, the thickness of the arm, h=0.5 mm, and constant M = 0.88 as:
⎡ G ⎤ = ⎢ E c 2 ⎥ ⎣ (t 0 ) ⎦
l pz
1 / 4
h
3 / 4
= 0.96mm
Assuming 5 cohesive elements in the process zone, the length of the cohesive elements is obtained as 0.19 mm (approx. 0.2 mm). Thus in this case the rule-of-thumb and the process zone-based considerations both lead to a cohesive element length of 0.2. Finite element simulation using the above obtained material parameters and mesh size was conducted in Abaqus/Standard with viscous regularization parameter of 5e-6 to aid the convergence process. The force (P) versus opening-displacement (y) result is compared with the analytical solution. The analytical solution using Timoshenko beam theory is given by the following expressions (Diehl [9]):
P=
bh
2a
Gc Eh
⎛ 1 + ν ⎛ h ⎞ 3⎜1 + ⎜ ⎟ ⎜ 5 ⎝ a ⎠ ⎝
2
⎞ ⎟ ⎟ ⎠
; y = 4a 2
⎛ 3(1 + ν ) ⎛ h ⎞ 2 ⎞ ⎜1 + ⎜ ⎟ ⎟⎟ ⎜ 5 ⎝ a ⎠ ⎠ ⎝
Gc
3 Eh
3
1+
1 + ν ⎛ h ⎞
(10)
2
⎜ ⎟
5 ⎝ a ⎠
The comparison is shown in Fig. 3a. It is seen that the predicted loaddisplacement curve is in reasonable agreement with the analytical solution. Note that the discrepancy in the initial part of the load-displacement curve arises because the analytical solution assumes that the two arms of the DCB specimen are ideally clamped at the root of the crack tip. This is not realized in the simulation because of the local compliance of
10
the surrounding materials and the non-coupling of the rotational degree of freedom of the beam elements (Diehl [11]). Even though 200 beam elements are sufficient to model the bending of the beam accurately, we further refine the mesh and consider 400 elements along the length. The length of the cohesive element (0.167mm) is chosen such that 3 elements span each beam element; this also satisfies the requirement based on the length of the process zone considerations. The result from this simulation is compared with the analytical solution in Fig. 3b. Note that the numerical result is in a very good agreement with the analytical solution. This analysis highlights the need to perform parametric studies by varying the mesh and comparing the results with analytical and/or experimental results. In this case a mesh chosen for the arms of the DCB based on bending considerations alone did not gave a very good match when the fracture simulation was conducted.
Fig. 3a: Force versus opening-displacement results: comparison of analytical and FE solutions (beam element size 1 mm, cohesive elements size 0.2 mm)
11
Fig. 3b: Force versus opening-displacement results: comparison of analytical and FE solutions (beam elements size 0.5 mm, cohesive elements size 0.167 mm) Case II: The interface strength is not known
In this case we assume that only the fracture toughness, G Ic, is known. We consider the DCB problem with 200 B21 elements modeling the two arms. Using the rule-of-thumb we consider 5 cohesive elements per beam element giving the length of the cohesive element to be 0.2mm. As suggested by Diehl [11] we choose δ ratio = 0.5 and parameterize both the interface strength and the stiffness by a single parameter δ f (see Eq. (5) and Eq. (10b)). Now we systematically choose the value of δ f as a fraction of the cohesive element size and obtain the interface strength and stiffness using Eqs. (5) and (10b) respectively. We consider three cases by taking different fractions of cohesive element length for the parameter δ f (see Table 1). A fourth case is considered which uses the same interface strength as Case 1 but uses the interface stiffness calculated using Eq. (8b). For all cases viscous regularization parameter of 5e-6 was used to aid in the convergence process. Table 1: Cases considered for simulation
Case
1 2 3 4
Cohesive element length, Lc 0.2 0.2 0.2 0.2
Fraction, f
0.05 0.005 0.1 0.05
f =
f Lc
Interface strength, t0 MPa (Eq. 5)
Interface stiffness, K MPa (Eq. 10b)
70.4 704.0 35.2 70.4
14.08E3 14.08E5 3.520E3 1.750E6 (using Eq. 8b)
0.01 0.001 0.02 0.01
12
The results are presented in Fig. 4a and 4b where the force-displacement curves from the simulations are compared against the analytical solution; in Fig 4a all the cases are compared whereas in Fig. 4b case 2 is not included. For case 2 the nonlinear procedure failed to converge at the initiation of the fracture process and the peak force predicted was very high. Thus case 2 is unlikely to yield accurate results when compared with the analytical solution even if the convergence problem is somehow avoided by increasing the stabilization or reducing the tolerances in the solution controls. Cases 1, 3, and 4 all give good results when compared with the analytical solution with case 3 giving a better match. Note from Table 1 that the interface strength used for all cases differs from the true interface strength of 60 MPa (used in the illustration for case I where the interface strength was assumed to be known), yet good results are obtained. It should also be noted from the result of case 3 that reducing the interface strength significantly from the true interface strength of 60 MPa lead to lower peak force though the propagation was captured quite accurately implying that the interface strength can be treated as penalty parameter. Of course you must study and assess the effect of variation of the interface strength on the overall response for your problem of interest.
Fig. 4a: Force versus opening-displacement results: comparison of analytical and FE solutions – all cases (beam elements size 1 mm, cohesive elements size 0.2 mm)
13
Fig. 4b: Force versus opening-displacement results: comparison of analytical and FE solutions – case 1, 3, and 4 (beam elements size 1 mm, cohesive elements size 0.2 mm)
Conclusions
The traction-separation relationship for fracture simulation using cohesive finite elements is characterized by fracture toughness, initial stiffness, and strength of the interface. Although all three parameters are material properties of the interface, only fracture energy can be reliably measured experimentally. The other two properties are difficult to determine and may be treated as penalty or fitting parameters. If the interface strength is known it can be used to determine a suitable mesh for the finite element simulation or it can be used to assess the suitability of the chosen mesh. If the chosen mesh is unsuitable based on the above mentioned criterion you may either refine the mesh or in some cases reduce the interface strength to obtain reasonably accurate results. On the other hand if the interface strength is not known it can be treated as a penalty parameter. The stiffness of the interface is treated as a penalty parameter and can be estimated based on the stiffness of the surrounding material. This study clearly shows that there is no hard-and-fast guideline for determining the parameters of the traction-separation relationship – it is dependent on the problem under consideration. We recommended that the cohesive parameters estimated with the guidelines presented in this paper be assessed by first conducting simple benchmark simulations (for which analytical and/or experimental results are available) before using them in actual models. Furthermore, as some of the parameters estimated from one problem may not be valid for other problems, you must also compare the results of your
14
actual problem with suitable experimental results and study the effect of small variations in cohesive material parameters (especially interface stiffness and strength) on the overall simulation results. In addition it is suggested that a mesh convergence study be carried out in conjunction with the study of various material parameters to gain confidence in the simulation results.
References:
[1] ASTM Standard D3518/D3518M, 2001, Standard test method for in-plane shear response of polymer matrix composite materials by tensile test of a 45 laminate, ASTM Int., W. Conshohocken, PA. [2] ASTM Standard D5528, Test method for mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites, Annual Book of ASTM Standards, v15.03, ASTM Int., W. Conshohocken, PA. [3] ASTM Standard D6671, Test method for mixed mode I – mode II interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites, Annual Book of ASTM Standards, v15.03, ASTM Int., W. Conshohocken, PA. [4] Camanho, P. P., Davila, C. G., de Moura, M. F., 2003, Numerical simulation of mixed-mode progressive delamination in composite materials, Journal of Composite Materials v37, pp. 1415-1438. [5] Daudeville, L., Allix, O., Ladeveze, P., 1995, Delamination analysis by damage mechanics – some applications, Composites Engineering v5, pp. 17-24. [6] Dauskardt, R. H., Lane, M., Ma, Q., Krishna N., 1998, Adhesion and debonding of multi-layer thin film structures, Engineering Fracture Mechanics v61, pp. 141-162. [7] Davidson, B. D., Kruger, R., Konig, M., 1995, Three-dimensional analysis of center delaminated unidirectional and multidirectional single-leg bending specimens, Composites Science and Technology v54, pp. 385-394. [8] Davidson, B. D., Sun X., 2005, Effects of friction, geometry, and fixture compliance on the perceived toughness from three- and four-point bend end-notched flexure tests, Journal of Reinforced Plastics and Composites v24, pp. 1611-1628. [9] Diehl, T., 2005, Modeling surface-bonded structures with ABAQUS cohesive elements: beam type solutions, ABAQUS Users’ Conference, Stockholm, Sweden. [10] Diehl, T., 2006, Using ABAQUS cohesive elements to model peeling of an epoxybonded aluminum strip: a benchmark study for inelastic peeling arms, ABAQUS Users’ Conference, Boston, MA.
15
[11] Diehl, T., 2007, On using a penalty-based cohesive-zone finite element approach, Part I: Elastic solution benchmarks, International Journal of Adhesion and Adhesives, doi:10.1016/j.ijadhadh.2007.06.003 [12] Diehl, T., 2007, On using a penalty-based cohesive-zone finite element approach: Part II – Inelastic peeling of an epoxy-bonded aluminum strip, International Journal of Adhesion and Adhesives, doi:10.1016/j.ijadhadh.2007.06.004 [13] Gupta, V., Argon, A. S., Parks, D. M., Cornie, J. A., 1992, Measurement of interface strength by laser spallation technique, Journal of the Mechanics and Physics of Solids v40, pp. 141-180. [14] Hutchinson, J. W., Evans, A. G., 2000, Mechanics of materials: top-down approaches to fracture, Acta Materialia v48, pp. 125-135. [15] Hutchinson, J. W., Suo, Z., Mixed mode cracking in layered materials, Advances in Applied Mechanics v29, pp. 63-191. [16] Lee, S. M., 1993, An edge crack torsion method for mode III delamination fracture testing, Journal of Composites Technology and Research v15, pp. 193-201. [17] Ratcliffe, J., 2004, Characterization of the edge crack torsion (ECT) test for mode III th fracture toughness measurement of laminated composites, Proceedings of the 19 ASC /ASTM Technical Conference, Atlanta. [18] Reeder, J. R., 2006, 3D mixed-mode delamination fracture criteria – an st experimentalist’s perspective, Proceedings of the American Society for Composites, 21 Annual Technical Conference, Dearborn, Michigan. [19] Russell, A. J., 1982, On measurement of mode II interlaminar fracture energies, Material Report 82-O. DREP. [20] Schellekens, J. C. J., de Borst, R., 1993, A nonlinear finite-element approach for the analysis of mode-I free edge delamination in composites, International Journal of Solids and Structures v30, pp. 1239-1253. [21] Suo, Z., Hutchinson, J. W., 1989, Sandwich test specimens for measuring interface crack toughness, Materials Science and Engineering vA107, pp. 135-143. [22] Turon, A., Davila, C. G., Camanho, P. P., Costa, J., 2007, An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models, Engineering Fracture Mechanics v74, pp. 1665-1682. [23] Wang, J., Sottos, N. R., Weaver, R. L., 2004, Tensile and mixed-mode strength of a thin film-substrate interface under laser induced pulse loading, Journal of the Mechanics and Physics of Solids v52, pp. 999-1022.
16
[24] Wang, J. -S., Suo, Z., 1990, Experimental determination of interfacial toughness curves using Brazil-nut-sandwiches, Acta Metall. Mater. V38, pp. 1279-1290. [25] Yang, Q., Cox, B., 2005, Cohesive models for damage evolution in laminated composites, International Journal of Fracture v133, pp. 107-137. [26] Zou, Z., Reid, S. R., Li, S., Soden, P. D., 2002, Modeling interlaminar and intralaminar damage in filament wound pipes under quasi-static indentation, Journal of Composite Materials v36, pp. 477-499.
17
