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Meshfree modelling of brittle and ductile dynamic fracture
Accepted Manuscript Title: Meshfree modeling of brittle and ductile dynamic fracture Author: S. Wang PII: DOI: Reference:
S0734-743X(15)00215-8 http://dx.doi.org/doi: 10.1016/j.ijimpeng.2015.10.009 IE 2593
To appear in:
International Journal of Impact Engineering
Received date: Revised date: Accepted date:
31-12-2014 28-7-2015 18-10-2015
Please cite this article as: S. Wang, Meshfree modeling of brittle and ductile dynamic fracture, International Journal of Impact Engineering (2015), http://dx.doi.org/doi: 10.1016/j.ijimpeng.2015.10.009. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Meshfree Modeling of brittle and ductile dynamic fracture S. Wang Tongji University, Shanghai, P.R.China, e-mail: [email protected] Highlights --‐ First thermo--‐mechanical CPM for brittle--‐to--‐ductile fracture --‐ Study on the influence of thermal effects and strain rate effects on dynamic fracture Abstract This manuscripts presents studies from brittle to ductile fracture by a simplified meshfree method. The Johnson-Cook models is employed to model the behaviour in the bulk while the cracking particles methods (CPM) is used to model discrete cracks. The influence of a cohesive zone model on the crack paths and the crack speed is studied. It will be shown that it has only a minor influence on the results for ductile fracture while it effects the crack speed for brittle fracture. Furthermore, the influence of temperature is tested and it is shown that for ductile fracture, the temperature is the key parameter while it has less effect for brittle fracture. Double-notched and single-notched specimen are tested and the experimental results are compared to results of our computational method for different discretizations. These results reveal that our method is a good pathway to study the failure transition from brittle-to-ductile fracture. Key words: Cracking Particles Method, Dynamic Fracture, Johnson-Cook Model, Impact 1. Introduction Modeling the brittle-to-ductile failure transition of metals under impact loading has been the focus of studies for several decades. One of the first experiments that revealed this interesting phenomenon were carried out by Kalthoff and co-workers [1] in the 1980s. They subjected double-notched metal plates to an impact loading while varying the velocity of the cylindrical impactor. They found that brittle fracture occurred at low impact speed while the specimen fractures in a ductile manner when exceeding a certain velocity of the impactor. Since the existence of the double notch creates wave reflections, other researchers subsequently carried out similar experiments on single-notched specimen [2, 3], see also [4]. They found similar features including shear band propagation that gradually turned into a crack with a sharp change in the propagation direction. Many studies have been carried out in order to understand and model these types of phenomena. Initially, cohesive element simulations were conducted as reported in the famous manuscript by Xu et al. [5]. Similar simulations were carried out by several authors including [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. With the development of partition-of-unity methods [16, 17, 18] such as XFEM [19, 20], smoothed XFEM [21, 22, 23, 24, 25], the phantom node method [26, 27, 28, 29, 30, 31] or XIGA [32], powerful alternative methods for fracture have been developed. Partition of unity methods for fracture allow for arbitrary crack growth without remeshing. They have been successfully extended to multiscale analysis [33, 34, 35, 36, 37, 38, 39] and other phenomena such as interface modeling [40, 41, 42, 43, 44]. However, while XFEM was very successful in modeling brittle fracture, its success in modelling ductile fracture is still limited. Furthermore, large deformations and finite strains deteriorate the accuracy of finite element simulations. And finally, while XFEM can handle crack growth of a single crack or few cracks, they still remain not suitable for complex cracking phenomena such as branching cracks or crack
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coalescence. Alternatives to partition of unity methods for fracture are classical remeshing techniques. However, despite some recent advances [45, 46, 47, 48, 49, 50], remeshing techniques seem to be more suitable for static problems and have not been exploited in dynamic fracture so far. A powerful alternative to finite element analysis are meshfree methods [51, 52, 53] such as SPH [54], corrected SPH methods [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66], RKPM [67, 68], EFG [69] and others [70, 71, 72, 73, 74, 75]. Meshfree methods do not rely on a background mesh and therefore can easily handle large deformations [76, 77, 78] and adaptive h-refinement [79, 80]. They are particularly suited for fracture problems. Fracture in meshfree methods, sometimes called meshless methods, are modelled by different concepts. Early approaches employ the visibility method [81, 82, 83] or improvements such as the diffraction or transparency method [84]. Later approaches employ a partition-of-unity enrichment as in the extended finite element method [85, 86, 87]. Those methods have been successfully employed for problems involving fracture in thin shells [88] and fluid structure interaction problems [89, 90, 91] among others. All methods mentioned above require the representation of the crack surface which is achieved commonly by two approaches: 1. Piece-wise straight line segments or triangular facets in 3D and 2. The level set method. While the latter approach allows for the computation of fracture related kinematic relations, its use for propagating cracks and complex fracture patterns is limited. A very interesting method that does not require any representation of the crack surface has been proposed by Rabczuk and Belytschko [92, 93, 94]. This so-called cracking particles method (CPM) models the crack surface as a set of crack segments that pass the entire domain of influence of a particle. It has been shown that such a representation is beneficial even for mode II fracture such as shear bands with curved shear band paths [95, 96, 97]. It has been applied to many interesting problems [98, 99, 99, 100, 101, 102, 103] and seems ideally to study the failure transition for the above mentioned experiments. This manuscript presents one of the first coupled thermo-mechanical CPM studies to model the failure transition from brittle to ductile fracture. The manuscript is structured as follows: In the next section, the CPM is explained. The constitutive model used in this manuscript is briefly explained as well. The governing equations in weak form are briefly stated before studies on the brittle-to-ductile failure transition are provided. We show the influence of the cohesive zone model and the temperature on the crack speed before the manuscript is concluded. 2. The Cracking Particles Method The Cracking Particles Method (CPM) is based – as classical partition of unity methods – on the decomposition of the displacement field into a continuous part and discontinuous part: C D (1) u(X) = u (X ) u (X ) the superscripts C and D indicating the continuous and discontinuous part. For the continuous part, a 'standard' meshless approximation has been employed. In this manuscript, we take advantage of the element-free Galerkin (EFG) method. It has been shown [69] that the EFG approximation is given by C u ( X ) = N I ( X ) d I (2) I S
dI being the nodal parameters and the shape functions NI (X) are given by T 1 N ( X ) = p ( X ) A ( X ) G ( X ) (3)
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which are now represented in matrix form for convenience. The matrix A(X) and G(X) are computed by n
A (X) =
w(X X
T
I
) p(X I ) p (X I )
I =1
G ( X ) = w ( X X 1 )p ( X 1 ) w ( X X 2 )p ( x 2 )
w ( X X n )p ( X n )
(4) w(X−XI) being the kernel function of the EFG approximation that determines its continuity while pT(X) is the vector that contains the basis functions. In order to fulfil the patch-test, linear basis functions are used, i.e. p T ( x ) = 1 x y . The cubic B-spline is employed for our weighting function. It is C2 and provides a smoother stress distribution around a potential crack tip compared to FEM solutions which have jumps at the element edges. The discontinuous part of the displacement field is given by: D u ( X ) = N I ( X ) S I ( X ) d I (5) I S
c
where SI(X) is the step function that causes the discontinuous displacement field of cracked particles and S c is the set of particles that are cracked. Most importantly, only cracked particles are enriched and the crack crosses the entire domain of influence of the cracked particle that drastically facilitates the implementation. As the cracking particle method might lead to numerical fracture, we have employed the cracking rules as described in [94]. 3. Constitutive Model and Cohesive Zone Model The well-established Johnson Cook model [104, 105] has been employed in the bulk. It accounts for strain-rate hardening and thermal softening: n m f ( , T ) = A B 1 C ln 1 T 0
0 = 1s
(6)
1
denoting the reference strain rates and A, B, C, n and m are material parameters. The normalised temperature T is computed by T =
T T room T m elt Troom
(7)
where Troom is the room temperature and T m elt is the melting temperature; T is the 'true' temperature. The update from [106] has been implemented. For more details, the reader is referred to the above mentioned two contributions. When the material uses stability, a discrete crack modelled by the CPM is introduced. Details about the fracture criterion and its implementation can be found in [93]. In order to account for the energy dissipated after the material looses stability, a cohesive zone model is commonly employed. We have adopted the cohesive zone model from [95] that is suitable for both brittle and ductile fracture. It has the form t i = m ax k [[ u ]] i k =
m ax m ax
(8)
where the subscript i denotes either the normal or tangential crack opening. The type of failure is determined by analysing the associated eigenvalue problem of unstable materials as described e.g. in [93]. When the eigenvectors to the associated eigenvalue of the acoustic
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tensor at fracture are parallel to the crack normal, fracture is mode I dominated while mode II fracture is characterised by a ninety degree angle between these two vectors. The cut-off value in our simulation is 45 degrees. More details can be found in [95]. 4. Weak Form The governing equations in weak form are the linear momentum equation and the energy equation for adiabatic cracks and shear bands which is a reliable assumptions for the events studied in this manuscript. They are given with respect to the initial domain by
T 0 u : P d 0 u t 0 d 0 0t 0
u b d0 0
0 u u d 0 = 0
0c p 0
T t
0
T d 0 =
P : F T d 0
(9)
0
P indicating the first Piola Kirchhoff tensor, F the deformation gradient, ρ0 density in the initial configuration, b the body forces, T is the temperature, t is the traction, superimposed dots indicate the material time derivatives, and the domain in the initial configuration is given by Ω0. The temperature is discretised with the same approach as the displacement field, see section 2. 5. Results Two types of experiments are studied in this manuscript: The single-notch experiment as proposed by [107, 3] and the 'original' double-notched experiments from Kalthoff and co-workers [1]. The double-notched specimen has the disadvantage that wave reflections from the two crack tips occur which make the failure transition less reliable. Such effects can be avoided by using just a single notch. The experimental set-ups are shown in Figure 1. Simulations with two different impact velocities are studied: 25m/s and 35 m/s, subsequently called the low and high impact velocity, respectively. For the double-notched specimen, the experiments revealed brittle fracture for a low impact velocity. A mode I dominated crack propagated from the pre-notches in a 70 degree angle as also illustrated in Figure 2. A nearly symmetric crack pattern was observed suggesting the exploitation of symmetry conditions in our computational model. When the impact velocity exceeds a given threshold, a mode II dominated shear band evolves in the axial direction of the pre-notch before it slightly curves and is arrested inside the specimen. Again, symmetric fracture patterns were observed as well and therefore symmetry conditions are exploited here, too. For the single-notched specimen, ductile fracture appears in both cases. The specimen are discretised with an unstructured particle arrangement with approximately the same particle distance. The discretisation has been refined until convergence in the crack speed and (simulateneoulsy) the crack pattern has been achieved. The transition from brittle to ductile failure is obtained here naturally by the proposed method. No empirical criterion is needed that is commonly related to the initial yield strength, i.e. the parameter A, of the Johnson Cook model. The impactor has commonly been modelled by an initial condition. However, we opt to model the impactor explicitly as the results were slightly different compared to the suggested initial condition. Figure 3 illustrates the fracture pattern exemplarily for the double-notched specimen. The different fracture patterns for different velocities of the impactor can clearly be captured. They agree well with the experimental fracture pattern illustrated schematically in Figure 3. Note that these results were obtained with the fully coupled model accounting for strain-rate effects, thermal softening and employing a cohesive zone model. We tested particularly the effect of thermal effects and the cohesive zone model. Table 1
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shows the length of the shear band of the full model, the model without cohesive zones but with thermal effects and the model with cohesive zones but without thermal effects in comparison with the experimental observations. The cohesive zone model has only a small influence on the shear band length suggesting that most of the energy is dissipated by plastic deformations. However, the length of the shear band is significantly shortened neglecting temperature effects suggesting that the temperature is the driving force of the shear band. The picture is different for brittle fracture. The crack length is barely influenced by neither the temperature nor the cohesive zone. However, this observation is somehow intriguing as the crack propagates straight and crosses the entire specimen. Therefore, the crack speed is considered closely. Figure 4 illustrates the crack speed over time for the double-notched specimen for three different models in the case of brittle fracture, i.e. for low impact velocity. It can be seen that the temperature barely influences the crack speed. However, the cohesive zone model has a significant influence. It is interesting to observe a similar tendency for the shear band propagation. While the cohesive zone model did not influence the length of the shear band, it affects the shear band velocity. As can be seen the temperature also has a decent effect on the shear band velocity. Table 2 summarises the maximum shear band velocities in comparison to experimental observations. We note that methods that describe the crack path and shear band path finally as continuous crack are not capable of capturing the correct crack propagation or shear band propagation speed, respectively, see e.g. the results of [108] who significantly overestimate these speeds. However, as demonstrated by [100], the CPM is capable of capturing more realistic crack speeds. This observation holds also in our studies. 6. Discussions We presented studies on the brittle-to-ductile failure transition of single-notched and double-notched specimen. Therefore, we proposed a coupled thermo-mechanical CPM in combination with a cohesive zone model and the Jonhson-Cook constitutive model. The material stability criterion has been employed to trigger cracks and shear bands. It offers a natural transition from brittle to ductile failure. We compared our numerical results to the experimental results of [1] and [3, 107]. Or more precisely: After ensuring that our computational method can capture the principal failure mechanisms, we focused on the length of the crack and shear band and its associated crack/shear band propagation speed. We studied in particular the influence of the cohesive zone model and the influence of thermal effects and found that the crack length is not influence by thermal effects and the cohesive zone model while the key influence on the length of the shear band is temperature. Furthermore, we find that the cohesive zone model has a drastic effect on the maximum crack propagation speed and also shear band propagation speed. This finding is interesting as the length of the shear band was not affected by the cohesive zone model indicating that the course of the shear band velocity over time is different. However, the maximum shear band velocity is similar. References [1] Kalthoff JF and Winkler S. Failure mode transition at high rates of shear loading. International Conference on Impact Loading and Dynamic Behavior of Materials, 1:185–195, 1987. [2] Ravi-Chandar K, Lu J., Yang B., and Zhu Z. Failure modes transitions in polymers under high strain rate loading. International Journal of Fracture, 101:33–72, 2000. [3] Zhou M, Ravichandran G., and Rosakis A. Dynamically propagating shear bands in impact-loaded prenotched plates-I. Journal Mechanics Physics and Solids, 44:981–1006, 1996.
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Figure 1: Set-up of double-notched and single-notched specimen Figure 2: Different fracture pattern of the double-notched and single-notched specimen under impact loading Figure 3: Fracture pattern of the Kalthoff experiment Figure 4: Crack propagation speed
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Table 1: Shear band length of different models: Model A: Full therm-mechanical model with cohesive zones, Model B: Thermo-mechanical model without cohesive zones, Model C: Pure mechanical model with cohesive zones Impact speed Model A Model B Model C Experiment 25 m / s 12 11 7 11 mm 30 m / s 57 54 41 56 mm
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Table 2: Maximum shear band speed of different models: Model A: Full therm-mechanical model with cohesive zones, Model B: Thermo-mechanical model without cohesive zones, Model C: Pure mechanical model with cohesive zones Impact speed Model A Model B Model C Experiment 25 m / s 604 m / s 854 m / s 633 m / s 595 m / s 30 m / s 1288 m / s 2033 m / s 1332 m / s 1200 m / s
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S0734-743X(15)00215-8 http://dx.doi.org/doi: 10.1016/j.ijimpeng.2015.10.009 IE 2593
To appear in:
International Journal of Impact Engineering
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31-12-2014 28-7-2015 18-10-2015
Please cite this article as: S. Wang, Meshfree modeling of brittle and ductile dynamic fracture, International Journal of Impact Engineering (2015), http://dx.doi.org/doi: 10.1016/j.ijimpeng.2015.10.009. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Meshfree Modeling of brittle and ductile dynamic fracture S. Wang Tongji University, Shanghai, P.R.China, e-mail: [email protected] Highlights --‐ First thermo--‐mechanical CPM for brittle--‐to--‐ductile fracture --‐ Study on the influence of thermal effects and strain rate effects on dynamic fracture Abstract This manuscripts presents studies from brittle to ductile fracture by a simplified meshfree method. The Johnson-Cook models is employed to model the behaviour in the bulk while the cracking particles methods (CPM) is used to model discrete cracks. The influence of a cohesive zone model on the crack paths and the crack speed is studied. It will be shown that it has only a minor influence on the results for ductile fracture while it effects the crack speed for brittle fracture. Furthermore, the influence of temperature is tested and it is shown that for ductile fracture, the temperature is the key parameter while it has less effect for brittle fracture. Double-notched and single-notched specimen are tested and the experimental results are compared to results of our computational method for different discretizations. These results reveal that our method is a good pathway to study the failure transition from brittle-to-ductile fracture. Key words: Cracking Particles Method, Dynamic Fracture, Johnson-Cook Model, Impact 1. Introduction Modeling the brittle-to-ductile failure transition of metals under impact loading has been the focus of studies for several decades. One of the first experiments that revealed this interesting phenomenon were carried out by Kalthoff and co-workers [1] in the 1980s. They subjected double-notched metal plates to an impact loading while varying the velocity of the cylindrical impactor. They found that brittle fracture occurred at low impact speed while the specimen fractures in a ductile manner when exceeding a certain velocity of the impactor. Since the existence of the double notch creates wave reflections, other researchers subsequently carried out similar experiments on single-notched specimen [2, 3], see also [4]. They found similar features including shear band propagation that gradually turned into a crack with a sharp change in the propagation direction. Many studies have been carried out in order to understand and model these types of phenomena. Initially, cohesive element simulations were conducted as reported in the famous manuscript by Xu et al. [5]. Similar simulations were carried out by several authors including [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. With the development of partition-of-unity methods [16, 17, 18] such as XFEM [19, 20], smoothed XFEM [21, 22, 23, 24, 25], the phantom node method [26, 27, 28, 29, 30, 31] or XIGA [32], powerful alternative methods for fracture have been developed. Partition of unity methods for fracture allow for arbitrary crack growth without remeshing. They have been successfully extended to multiscale analysis [33, 34, 35, 36, 37, 38, 39] and other phenomena such as interface modeling [40, 41, 42, 43, 44]. However, while XFEM was very successful in modeling brittle fracture, its success in modelling ductile fracture is still limited. Furthermore, large deformations and finite strains deteriorate the accuracy of finite element simulations. And finally, while XFEM can handle crack growth of a single crack or few cracks, they still remain not suitable for complex cracking phenomena such as branching cracks or crack
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coalescence. Alternatives to partition of unity methods for fracture are classical remeshing techniques. However, despite some recent advances [45, 46, 47, 48, 49, 50], remeshing techniques seem to be more suitable for static problems and have not been exploited in dynamic fracture so far. A powerful alternative to finite element analysis are meshfree methods [51, 52, 53] such as SPH [54], corrected SPH methods [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66], RKPM [67, 68], EFG [69] and others [70, 71, 72, 73, 74, 75]. Meshfree methods do not rely on a background mesh and therefore can easily handle large deformations [76, 77, 78] and adaptive h-refinement [79, 80]. They are particularly suited for fracture problems. Fracture in meshfree methods, sometimes called meshless methods, are modelled by different concepts. Early approaches employ the visibility method [81, 82, 83] or improvements such as the diffraction or transparency method [84]. Later approaches employ a partition-of-unity enrichment as in the extended finite element method [85, 86, 87]. Those methods have been successfully employed for problems involving fracture in thin shells [88] and fluid structure interaction problems [89, 90, 91] among others. All methods mentioned above require the representation of the crack surface which is achieved commonly by two approaches: 1. Piece-wise straight line segments or triangular facets in 3D and 2. The level set method. While the latter approach allows for the computation of fracture related kinematic relations, its use for propagating cracks and complex fracture patterns is limited. A very interesting method that does not require any representation of the crack surface has been proposed by Rabczuk and Belytschko [92, 93, 94]. This so-called cracking particles method (CPM) models the crack surface as a set of crack segments that pass the entire domain of influence of a particle. It has been shown that such a representation is beneficial even for mode II fracture such as shear bands with curved shear band paths [95, 96, 97]. It has been applied to many interesting problems [98, 99, 99, 100, 101, 102, 103] and seems ideally to study the failure transition for the above mentioned experiments. This manuscript presents one of the first coupled thermo-mechanical CPM studies to model the failure transition from brittle to ductile fracture. The manuscript is structured as follows: In the next section, the CPM is explained. The constitutive model used in this manuscript is briefly explained as well. The governing equations in weak form are briefly stated before studies on the brittle-to-ductile failure transition are provided. We show the influence of the cohesive zone model and the temperature on the crack speed before the manuscript is concluded. 2. The Cracking Particles Method The Cracking Particles Method (CPM) is based – as classical partition of unity methods – on the decomposition of the displacement field into a continuous part and discontinuous part: C D (1) u(X) = u (X ) u (X ) the superscripts C and D indicating the continuous and discontinuous part. For the continuous part, a 'standard' meshless approximation has been employed. In this manuscript, we take advantage of the element-free Galerkin (EFG) method. It has been shown [69] that the EFG approximation is given by C u ( X ) = N I ( X ) d I (2) I S
dI being the nodal parameters and the shape functions NI (X) are given by T 1 N ( X ) = p ( X ) A ( X ) G ( X ) (3)
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which are now represented in matrix form for convenience. The matrix A(X) and G(X) are computed by n
A (X) =
w(X X
T
I
) p(X I ) p (X I )
I =1
G ( X ) = w ( X X 1 )p ( X 1 ) w ( X X 2 )p ( x 2 )
w ( X X n )p ( X n )
(4) w(X−XI) being the kernel function of the EFG approximation that determines its continuity while pT(X) is the vector that contains the basis functions. In order to fulfil the patch-test, linear basis functions are used, i.e. p T ( x ) = 1 x y . The cubic B-spline is employed for our weighting function. It is C2 and provides a smoother stress distribution around a potential crack tip compared to FEM solutions which have jumps at the element edges. The discontinuous part of the displacement field is given by: D u ( X ) = N I ( X ) S I ( X ) d I (5) I S
c
where SI(X) is the step function that causes the discontinuous displacement field of cracked particles and S c is the set of particles that are cracked. Most importantly, only cracked particles are enriched and the crack crosses the entire domain of influence of the cracked particle that drastically facilitates the implementation. As the cracking particle method might lead to numerical fracture, we have employed the cracking rules as described in [94]. 3. Constitutive Model and Cohesive Zone Model The well-established Johnson Cook model [104, 105] has been employed in the bulk. It accounts for strain-rate hardening and thermal softening: n m f ( , T ) = A B 1 C ln 1 T 0
0 = 1s
(6)
1
denoting the reference strain rates and A, B, C, n and m are material parameters. The normalised temperature T is computed by T =
T T room T m elt Troom
(7)
where Troom is the room temperature and T m elt is the melting temperature; T is the 'true' temperature. The update from [106] has been implemented. For more details, the reader is referred to the above mentioned two contributions. When the material uses stability, a discrete crack modelled by the CPM is introduced. Details about the fracture criterion and its implementation can be found in [93]. In order to account for the energy dissipated after the material looses stability, a cohesive zone model is commonly employed. We have adopted the cohesive zone model from [95] that is suitable for both brittle and ductile fracture. It has the form t i = m ax k [[ u ]] i k =
m ax m ax
(8)
where the subscript i denotes either the normal or tangential crack opening. The type of failure is determined by analysing the associated eigenvalue problem of unstable materials as described e.g. in [93]. When the eigenvectors to the associated eigenvalue of the acoustic
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tensor at fracture are parallel to the crack normal, fracture is mode I dominated while mode II fracture is characterised by a ninety degree angle between these two vectors. The cut-off value in our simulation is 45 degrees. More details can be found in [95]. 4. Weak Form The governing equations in weak form are the linear momentum equation and the energy equation for adiabatic cracks and shear bands which is a reliable assumptions for the events studied in this manuscript. They are given with respect to the initial domain by
T 0 u : P d 0 u t 0 d 0 0t 0
u b d0 0
0 u u d 0 = 0
0c p 0
T t
0
T d 0 =
P : F T d 0
(9)
0
P indicating the first Piola Kirchhoff tensor, F the deformation gradient, ρ0 density in the initial configuration, b the body forces, T is the temperature, t is the traction, superimposed dots indicate the material time derivatives, and the domain in the initial configuration is given by Ω0. The temperature is discretised with the same approach as the displacement field, see section 2. 5. Results Two types of experiments are studied in this manuscript: The single-notch experiment as proposed by [107, 3] and the 'original' double-notched experiments from Kalthoff and co-workers [1]. The double-notched specimen has the disadvantage that wave reflections from the two crack tips occur which make the failure transition less reliable. Such effects can be avoided by using just a single notch. The experimental set-ups are shown in Figure 1. Simulations with two different impact velocities are studied: 25m/s and 35 m/s, subsequently called the low and high impact velocity, respectively. For the double-notched specimen, the experiments revealed brittle fracture for a low impact velocity. A mode I dominated crack propagated from the pre-notches in a 70 degree angle as also illustrated in Figure 2. A nearly symmetric crack pattern was observed suggesting the exploitation of symmetry conditions in our computational model. When the impact velocity exceeds a given threshold, a mode II dominated shear band evolves in the axial direction of the pre-notch before it slightly curves and is arrested inside the specimen. Again, symmetric fracture patterns were observed as well and therefore symmetry conditions are exploited here, too. For the single-notched specimen, ductile fracture appears in both cases. The specimen are discretised with an unstructured particle arrangement with approximately the same particle distance. The discretisation has been refined until convergence in the crack speed and (simulateneoulsy) the crack pattern has been achieved. The transition from brittle to ductile failure is obtained here naturally by the proposed method. No empirical criterion is needed that is commonly related to the initial yield strength, i.e. the parameter A, of the Johnson Cook model. The impactor has commonly been modelled by an initial condition. However, we opt to model the impactor explicitly as the results were slightly different compared to the suggested initial condition. Figure 3 illustrates the fracture pattern exemplarily for the double-notched specimen. The different fracture patterns for different velocities of the impactor can clearly be captured. They agree well with the experimental fracture pattern illustrated schematically in Figure 3. Note that these results were obtained with the fully coupled model accounting for strain-rate effects, thermal softening and employing a cohesive zone model. We tested particularly the effect of thermal effects and the cohesive zone model. Table 1
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shows the length of the shear band of the full model, the model without cohesive zones but with thermal effects and the model with cohesive zones but without thermal effects in comparison with the experimental observations. The cohesive zone model has only a small influence on the shear band length suggesting that most of the energy is dissipated by plastic deformations. However, the length of the shear band is significantly shortened neglecting temperature effects suggesting that the temperature is the driving force of the shear band. The picture is different for brittle fracture. The crack length is barely influenced by neither the temperature nor the cohesive zone. However, this observation is somehow intriguing as the crack propagates straight and crosses the entire specimen. Therefore, the crack speed is considered closely. Figure 4 illustrates the crack speed over time for the double-notched specimen for three different models in the case of brittle fracture, i.e. for low impact velocity. It can be seen that the temperature barely influences the crack speed. However, the cohesive zone model has a significant influence. It is interesting to observe a similar tendency for the shear band propagation. While the cohesive zone model did not influence the length of the shear band, it affects the shear band velocity. As can be seen the temperature also has a decent effect on the shear band velocity. Table 2 summarises the maximum shear band velocities in comparison to experimental observations. We note that methods that describe the crack path and shear band path finally as continuous crack are not capable of capturing the correct crack propagation or shear band propagation speed, respectively, see e.g. the results of [108] who significantly overestimate these speeds. However, as demonstrated by [100], the CPM is capable of capturing more realistic crack speeds. This observation holds also in our studies. 6. Discussions We presented studies on the brittle-to-ductile failure transition of single-notched and double-notched specimen. Therefore, we proposed a coupled thermo-mechanical CPM in combination with a cohesive zone model and the Jonhson-Cook constitutive model. The material stability criterion has been employed to trigger cracks and shear bands. It offers a natural transition from brittle to ductile failure. We compared our numerical results to the experimental results of [1] and [3, 107]. Or more precisely: After ensuring that our computational method can capture the principal failure mechanisms, we focused on the length of the crack and shear band and its associated crack/shear band propagation speed. We studied in particular the influence of the cohesive zone model and the influence of thermal effects and found that the crack length is not influence by thermal effects and the cohesive zone model while the key influence on the length of the shear band is temperature. Furthermore, we find that the cohesive zone model has a drastic effect on the maximum crack propagation speed and also shear band propagation speed. This finding is interesting as the length of the shear band was not affected by the cohesive zone model indicating that the course of the shear band velocity over time is different. However, the maximum shear band velocity is similar. References [1] Kalthoff JF and Winkler S. Failure mode transition at high rates of shear loading. International Conference on Impact Loading and Dynamic Behavior of Materials, 1:185–195, 1987. [2] Ravi-Chandar K, Lu J., Yang B., and Zhu Z. Failure modes transitions in polymers under high strain rate loading. International Journal of Fracture, 101:33–72, 2000. [3] Zhou M, Ravichandran G., and Rosakis A. Dynamically propagating shear bands in impact-loaded prenotched plates-I. Journal Mechanics Physics and Solids, 44:981–1006, 1996.
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Figure 1: Set-up of double-notched and single-notched specimen Figure 2: Different fracture pattern of the double-notched and single-notched specimen under impact loading Figure 3: Fracture pattern of the Kalthoff experiment Figure 4: Crack propagation speed
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Table 1: Shear band length of different models: Model A: Full therm-mechanical model with cohesive zones, Model B: Thermo-mechanical model without cohesive zones, Model C: Pure mechanical model with cohesive zones Impact speed Model A Model B Model C Experiment 25 m / s 12 11 7 11 mm 30 m / s 57 54 41 56 mm
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Table 2: Maximum shear band speed of different models: Model A: Full therm-mechanical model with cohesive zones, Model B: Thermo-mechanical model without cohesive zones, Model C: Pure mechanical model with cohesive zones Impact speed Model A Model B Model C Experiment 25 m / s 604 m / s 854 m / s 633 m / s 595 m / s 30 m / s 1288 m / s 2033 m / s 1332 m / s 1200 m / s
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