The effect of local inertia around the cracktip in dynamic fracture of soft materials
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Abstract
Phasefield or gradientdamage approaches offer elegant ways to model cracks. Material stiffness decreases in the cracked region with the evolution of the phasefield or damage variable. This variable and, consequently, the decreased stiffness are spatially diffused, which essentially means the loss of the internal links and the bearing capacity of the material in a finite region. Considering the loss of material stiffness without the loss of inertial mass seems to be an incomplete idea when dynamic fracture is considered. Loss of the inertial mass in the damaged material region may have significant effect on the dynamic failure processes. In the present work, dynamic fracture is analyzed using a theory, which takes into account the local loss of both material stiffness and inertia. Numerical formulation for brittle fracture at large deformations is based on the Cosserat point method, which allows suppressing the hourglass type deformation modes in simulations. Based on the developed algorithms, the effect of the material inertia around a crack tip is studied. Two different problems with single and multiple cracks are considered. Results suggest that in dynamic fracture the localized loss of mass plays an important role at the crack tip. It is found, particularly, that the loss of inertia leads to lower stresses at the crack tip and, because of that, to narrower cracks as compared to the case in which no inertia loss is considered. It is also found that the regularized problem formulation provides global convergence in energy under the mesh refinement. At the same time, the local crack pattern might still depend on the geometry of the unstructured mesh.
Keywords
Fracture Material sink Dynamic Phasefield Coupled Inertia1 Introduction
Understanding and modeling of the crack propagation is arguably the central problem in solid mechanics. Two major classes of approaches for analysis of crack propagation are surface and bulk material failure models, also known as Cohesive Surface Model (CSM) and Continuum Damage Model (CDM), respectively. CSM defines interaction between the separating surfaces using tractionseparation laws [2, 7, 9, 11, 14, 25, 27, 35, 43, 50]. They are most effective when possible crack paths are already known. If the path is not known, then defining the criteria for nucleation and growth of cracks still remains a challenge [26]. On the other hand, CDM describes the failure via damage constitutive laws [5, 12, 15, 18, 20, 22, 23, 38, 44, 45, 49]. When using CDM, important features such as damage nucleation, propagation, and branching naturally derive from the constitutive laws. Unfortunately, numerical simulations based on the CDM are mesh sensitive: the finer meshes lead to the finer localized damage zones and lower dissipated energies [29, 30, 31]. Ultimately, damage can occur without the energy dissipation, which is physically meaningless.
To regularize CDMs by suppressing the pathological mesh sensitivity, nonlocal continuum formulations emerged [10, 21, 28, 33, 36]. These formulations naturally incorporate characteristic length as a parameter to limit the size of spatial damage localization The physical assumption underlying most nonlocal theories is longrange interaction of material particles. Unfortunately, actual interaction of particles is of shortrange (on nanometers or angstrom length scale). Hence, the physical basis for nonlocal continuum models appears disputable.
A special class of gradienttype damage theories called phasefield approach is becoming increasingly popular for modeling fracture [6, 8, 13, 16, 19]. In this approach, the phase field variable (also known as the internal damage variable in the earlier literature) is introduced in order to decrease the material stiffness. Apparently, the macroscopic degradation of the material stiffness is triggered by the breakage of the atomic/molecular bonds. However, the material bond breakage should also be accompanied by the mass/inertia loss. Agrawal and Dayal [1] and Chen et al. [8] have raised such concerns when using phasefield method for modeling dynamic fracture. However, the simultaneous loss of stiffness and inertia is nothing but the mass sink. Volokh [48] sharpened the latter notion and formulated the regularized approach to fracture as a coupled hyperelasticmassdiffusion problem.
In the present work, we first develop a numerical formulation for the theory given by [48]. Then, we use the finite element (FE) method to simulate fracture in a hyperelastic material. We consider two situations: (i) when the mass diffusion does not affect inertia and density only acts as a variable controlling the failure through stiffness. For this situation, the material in cracked regions has inertia even after failure but has no stiffness; (ii) when the material in cracked regions becomes massless due to diffusion of the mass to the surroundings. These two situations represent two extremes. Analysis of these situations gives us an insight on the effect of local inertia near the cracktip on the crack growth. Numerical performance of the theory is also evaluated for different mesh size. Two different fracture problems with single and double cracks under dynamic loading are solved. The results show that the crack growth under dynamic conditions is significantly affected by the variation of inertia around the cracktip.
The rest of the paper is organized as follows. In Section 2, we briefly present the theory and the constitutive model used for the numerical simulations. Details of the FE formulation for a plane strain Cosserat Point Element (CPE) and its implementation are presented in Section 3. Numerical results from simulations of the fracture problems are presented in Section 4. Finally, the salient conclusions from the study are presented in Section 5.
2 Theoretical formulation
2.1 Balance equations
The basic assumption of the theory is that failure and, consequently, mass flow are very localized and the momentum and energy balance equations can be written in standard form without adding momentum and energy due to the change in mass.
2.2 Constitutive equations
Constitutive law (16) is very similar to the hyperelasticity with the energy limiters, except with a different evolution equation for ζ. For the sake of brevity, details regarding the theory of energy limiters are not presented here. Readers are referred to [45, 46, 47] for details.
Parameters for AAA material
c_{1} (MPa)  c_{2} (MPa)  Φ (MPa)  m 

0.617  1.215  0.1686  10 
2.2.1 Nearly incompressible form
3 Numerical formulation and implementation
In this section, numerical formulation and implementation of the theory discussed in Section 2 is presented. Numerical solution of (2) and (12) requires spatial and temporal discretization. It is well known that the conventional finite element approach exhibits locking behavior when used for nearly incompressible materials. To overcome this deficiency, various element technologies such as hybrid formulation [32], reduced integration with hourglass control [3, 4], and enhanced assumed strains [37, 39] are developed. Although each of the abovementioned approach helps overcoming some of the known deficiency, they may have drawbacks [24]. The Cosserat Point Element (CPE) has been proven to be robust when considering typical locking phenomena (for poor element aspect ratios and nearly incompressible material response) compared to other standard finite elements [17]. Hence, in the present work, we follow Cosserat point approach for spatial discretization of momentum equation (2), which is briefly discussed in this section. For spatial discretization (12), the standard Galerkin method is used. Implementation of the element formulation, the time integration scheme, the element deletion criteria, and calculation of dissipated energy are also discussed.
3.1 Basic equations for a plane strain quadrilateral CPE
We briefly present the formulation for a plane strain CPE by [24]. A slightly different notation from [24] is used. Unless explicitly mentioned, in the following description, capital letters in subscript and/or superscript are used for nodal values and they take values as (0,1,2,3). Small letters in subscript and/or superscript are used for dimensional components, which take values as (1,2) for twodimensional cases.
3.1.1 Kinematics
3.1.2 Constitutive equations
3.1.3 Weak form of the momentum equation
3.2 Weak form of the mass balance equation
3.3 Numerical implementation and time integration scheme
3.4 Element deletion criterion and calculation of energy dissipation
Initially, \(\mathcal {H} = 1\) for all elements. At each time step, densities at the Gauss points (shown in Fig. 3) of an element are compared with the critical density ρ_{cr} = 𝜖ρ, where 𝜖 defines the criteria for deletion. At some time t = t_{d}, when the condition ρ ≤ ρ_{cr} is satisfied at all the Gauss points of the element, the value of \(\mathcal {H}\) for the element becomes 0 and it remains equal to 0 for all t > t_{d} during the simulation. Such elements with \(\mathcal {H} = 0\) will be called deleted elements.

For deleted elements, the internal force vector \({\boldsymbol f}^{I}_{\text {int}} = \boldsymbol 0\) in (46) and all calculations for \({\boldsymbol f}^{I}_{\text {int}}\) are skipped.

Ideally for any node the minimum value of ρ is 0. Hence for the deleted element \(\bar \rho \rightarrow 0\). Zero elements in the global mass matrix are not desired. Hence, for a deleted element \(\bar \rho = \rho _{cr}/4\) is used for the calculation of lumped mass matrix .

For the deleted elements first term is removed from (50), as it involves gradient calculation w.r.t current coordinates.
4 Numerical results

Case I – Element mass is always calculated using a constant value (ρ_{0}) of density. Thus, elements never loose mass, even after deletion. Density is treated as a variable affecting only the stiffness and not the inertia.

Case II – Element mass is calculated using current values (ρ) of density. When the cracks are formed by diffusion of density to the surroundings, the cracktip is surrounded by a region having very low mass and deleted elements are almost massless.
For all simulations the characteristic length l = 0.1mm is used.
4.1 ModeI crack problem
Details of the meshes S_{1} to S_{3}
No. of element  Min. edge length (mm)  Avg. min. edge length (mm)  

S _{1}  105000  0.0120  0.0133 
S _{2}  54854  0.0058  0.0188 
S _{3}  119247  0.00275  0.0128 
4.1.1 Case I
For meshes coarser than S_{1} to S_{3}, stress fields of the top and bottom cracks start interacting before they reach the boundary, which causes stress to increase at a point somewhere between these cracks. This results in different crack patterns from those in Fig. 6. For the sake of completeness, we present results for the coarser meshes (both structured and unstructured) in A. Details of these meshes are given in Table 4 in Appendix 1. Contours of ρ/ρ_{0} at the instant when the interaction of cracks starts fracture and the final crack pattern at time t = 189μs are shown in Fig. 15 in Appendix 1.
4.1.2 Case II
4.2 Double crack problem
Details of the meshes D_{1} to D_{3}
No. of element  Min. edge length (mm)  Avg. min. edge length (mm)  

D _{1}  33960  0.0103  0.0240 
D _{2}  46800  0.0167  0.0192 
D _{3}  54418  0.00281  0.0190 
4.3 Case I
4.4 Case II
5 Conclusion
Numerical formulation and implementation of the theory suggesting fracture as material sink has been presented. Numerical simulations have been used to reveal the effect of inertia near the cracktip on the crack growth in dynamic failure processes including problems with single and double cracks. Conclusions of the present study can be grouped as follows:
5.1 Role or inertia
The crack growth pattern completely changes, with and without consideration of decreasing inertia near the cracktip (case II and case I, respectively). The inertia gradient around the cracktip results in a weaker stress field at the crack tip and much localized element deletion for case II as compared to case I. Effects of local inertia observed in the present work will also be applicable for phasefield methods used for dynamic fracture. We are pleased to note that the phase field modelers started tending to the same conclusion of the necessity to cancel material inertia together with the material stiffness – see [1] and [8].
5.2 Mesh dependence
First, we should distinguish between strong and weak mesh dependence.
By the strong, we mean the classical pathological mesh dependence, which leads to the decreasing fracture energy with the decreasing mesh size and can culminate in zero energy fracture. Such strong meshdependence is suppressed by the present formulation in which failure diffusion is automatically incorporated and the mesh size is always smaller than the characteristic length of diffusion. Energy dissipation during fracture converges with the mesh size refinement.
By the weak, we mean the effect of the mesh shape and size on the specific crack pattern. In our simulations, the weak meshdependence has been observed even for sufficiently fine meshes. Mesh alterations could trigger various crack patterns for the same amount of the dissipated fracture energy. This observation could probably be explained qualitatively as follows. Fracture in real materials is affected by structural inhomogeneities. The crack path can be different in each specimen made of the same material depending upon distribution of these inhomogeneities. In numerical simulations, the materials are idealized as homogeneous. However, we believe that differences in various mesh structures work as numerical inhomogeneities, which affect the crack path and this is the reason why we observed different crack paths even for different fine meshes.
We should also note that the difference between the strong and weak mesh dependencies has never been pointed to in the literature. Some researchers expect the universal effect of the regularization as a panacea of any mesh sensitivity. While this is definitely true concerning the suppression of the zero energy fracture, generally, there are no reasons to expect a complete and universal fracture pattern emerging from the regularized formulation.
5.3 Material sink versus phase field
Material sink and phase filed approaches have very similar mathematical structure in which the additional variable describing damage obeys additional partial differential equation of the reactiondiffusion type. The spatial gradient term in this equation induces the length scale, which provides the regularization of the fracture modeling. Thus, from the mathematical standpoint, both approaches belong to the same family and a similarity of the outcome of their numerical analyses would not be surprising. At the same time, it should not escape attention of the reader that the material sink formulation is much simpler than numerous and various phase field formulations.
While mathematics of both approaches is similar, the physics is not. The phase field variable and its reactiondiffusion equation do not have any direct physical interpretation. It is just a formal tool to regularize calculations avoiding the zero energy fracture. The situation is different in the mass sink approach in which the mass density is the damage variable and the regularizing reactiondiffusion equation is the classical mass balance law. The clear physical meaning of all variables in the mass sink approach is not a matter of wording; it has a strong implication—the necessity to cancel the inertia forces in the material areas with the decreased stiffness. Physics helps! This difference between the mass sink and phase field approaches is critical for the consideration of dynamic fracture.
Footnotes
 1.
It should be mentioned here that a higher value of ρ_{cr} is used in case II, compared to that in case I. This is because the current value of density is used for the calculation of the stable time increment in case II. Using 𝜖 = 0.001 results in extremely small time steps for elements close to the cracked region, unnecessarily slowing down the simulations. For a few cases, simulations have been performed for similar values of 𝜖 in cases I and II, to ensure that the differences in results between both cases are not only due to different values of ρ_{cr}. For S_{2} results for different values of 𝜖 are shown in Fig. 17 in Appendix 2.
Notes
Acknowledgments
The authors thank Prof. Mahmood Jabareen for introducing them to the CPE formulation and the valuable discussion on it.
Funding information
The authors acknowledge the support from the Israel Science Foundation, grant No. ISF198/15.
Compliance with Ethical Standards
Conflict of interest
The authors declare that they have no conflict of interest.
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