Abstract
Cracks, the major vehicle for material failure^{1}, undergo a microbranching instability at ∼40% of their sonic limiting velocity in three dimensions^{2,3,4,5,6}. Recent experiments showed that in thin systems cracks accelerate to nearly their limiting velocity without microbranching, until undergoing an oscillatory instability^{7,8}. Despite their fundamental importance, these dynamic instabilities are not explained by the classical theory of cracks^{1}, which is based on linear elasticity and an extraneous local symmetry criterion to predict crack paths^{9}. We develop a twodimensional theory for predicting arbitrary paths of ultrahighspeed cracks, which incorporates elastic nonlinearity without extraneous criteria. We show that cracks undergo an oscillatory instability controlled by smallscale, near cracktip, elastic nonlinearity. This instability occurs above an ultrahigh critical velocity and features an intrinsic wavelength proportional to the ratio of the fracture energy to the elastic modulus, in quantitative agreement with experiments. This ratio emerges as a fundamental scaling length assumed to play no role in the classical theory of cracks, but shown here to strongly influence crack dynamics.
Main
Crack propagation is the main mode of materials failure. It has been a topic of intense research for decades because of its enormous practical importance and fundamental theoretical interest. Despite considerable progress to date^{10,11,12,13,14,15}, the classical theory of brittle crack propagation^{1} still falls short of explaining the rich dynamical behaviour of highspeed cracks in brittle solids such as glass, ceramics and other engineering, geological and biological materials that break abruptly and catastrophically.
This theory, termed linear elastic fracture mechanics (LEFM)^{1}, assumes that linear elastodynamics—a continuum version of Newton’s second law together with a linear relation between stress (force) and strain (deformation)—applies everywhere inside a stressed material except for a negligibly small region near the crack tip. It predicts the instantaneous crack velocity v by equating the elastic energy release rate G, controlled by the intensity of the stress divergence near the crack tip, with the fracture energy Γ(v). The scalar equation G = Γ(v) must be supplemented with an extraneous criterion to select the crack path; the most widely used one is the principle of local symmetry^{9}, which assumes that cracks propagating along arbitrary paths feature a symmetric stress distribution near their tips.
A central prediction of this theory is that straight cracks smoothly accelerate to the Rayleigh wave speed c_{R} (the velocity of surface acoustic waves) in large enough systems^{1}. However, cracks universally undergo symmetrybreaking instabilities before reaching their theoretical limiting velocity^{2,3,4,5,6,7}. In threedimensional (3D) systems such as thick plates, instability is manifested by shortlived microcracks that branch out sideways from the main crack. This socalled microbranching instability^{2,3,4,5,6} typically occurs when v exceeds a threshold v_{c} of about 40% of c_{R}. Recent experiments in brittle gels have further shown that on reducing the thickness of the system, microbranching is suppressed and instability is manifested at a much higher speed (v_{c} ∼ 90% of the shear wave speed c_{s}) by oscillatory cracks with a welldefined intrinsic wavelength λ (refs 7,8,15). Such behaviour cannot be explained by LEFM, even qualitatively, as it contains no length scales other than the external dimensions of the system.
To investigate dynamic fracture instabilities, we use the phasefield approach^{16,17,18,19,20}. By making the neartip degradation zone spatially diffuse, this approach avoids the difficulty of tracking the evolution of sharp fracture surfaces inherent in traditional cohesive zone models^{1}. It is therefore capable of describing complex crack paths while treating both shortscale material failure and largescale elasticity, without adopting the common assumption that elasticity remains linear at arbitrary large strains near the crack tip (Fig. 1a). This generalized approach features two intrinsic length scales missing in LEFM (Fig. 1b): the size ξ of the microscopic dissipation zone around the tip, where elastic energy is dissipated while creating new fracture surfaces, and the size ℓ of the neartip nonlinear zone, where linear elasticity breaks down when strains become large. We stress that while ξ and ℓ are missing in LEFM, they are consistent with it as they remain much smaller than the system size. Moreover, ℓ scales with the ratio Γ/μ of the fracture energy to the shear modulus, but is generally much larger than Γ/μ and also depends on crack velocity. Experiments and theory suggest that this nonlinear scale may be related to the oscillatory instability^{8,15,21,22,23,24,25}, but this relationship is not fundamentally established. Here we develop a new phasefield formulation (see Methods and Supplementary Information) that maintains the wave speeds constant inside the dissipation zone, thereby avoiding spurious tipsplitting occurring in previous phasefield models at relatively low crack velocities^{26}. This new formulation allows us to model for the first time the ultrahighspeed cracks observed experimentally.
The nonlinear strain energy density e_{strain} is chosen to correspond to an incompressible neoHookean solid (see Supplementary Information) (Fig. 1a), representing generic elastic nonlinearities and quantitatively describing the experiments of refs 6,7,8,15,21,25. We consider modeI (tensile) cracks in strips of height H (in the ydirection) and length W (in the xdirection). Fixed tensile displacements u_{y}(y = ±H/2) = ±δ_{y} are imposed at the top and bottom boundaries with δ_{y} ≪ H such that strains are small and linear elasticity is valid everywhere in the sample except within a small region of size ℓ ≪ H near the crack tip, where elastic nonlinearity is important. The applied load is quantified by the stored elastic energy per unit length along x in the prestretched intact strip, G_{0} = e_{strain}H, where e_{strain} is uniquely determined by δ_{y}.
Figure 2 unprecedentedly demonstrates the existence of a rapid crack oscillatory instability in our simulations. Figure 2a shows a close up on the crack at the onset of oscillations in the material (undeformed) coordinates (see also Supplementary Fig. 1 and Supplementary Movie 1) and a corresponding sequence of crack snapshots in the spatial (deformed) coordinates, along with the strain energy density field. The results bear striking resemblance to the corresponding experimental observations in brittle gels^{7}, reproduced here in Fig. 2b (see also Supplementary Movie 2). Figure 2c shows the time evolution of the Cartesian components, (v_{x}, v_{y}), and magnitude of the crack velocity, , demonstrating that the instability appears when v exceeds a threshold v_{c} ≈0.92c_{s}. Figure 2d shows the time evolution of the oscillation amplitude A and wavelength λ, which both grow before saturating. The saturated amplitude is an order of magnitude smaller than the wavelength, in good agreement with experiments^{7}.
Importantly, we verified that the wavelength is determined by an intrinsic length scale by carrying out simulations for different system sizes, yielding negligible variations in λ (Supplementary Fig. 2). Moreover, we verified that the instability is caused by neartip elastic nonlinearity by repeating the simulations using the smallstrain (linear elastic) quadratic approximation of the nonlinear e_{strain}, corresponding to conventional LEFM. These simulations yielded straight cracks that tipsplit on surpassing a velocity of ≈0.9c_{s}, without oscillations. Since both forms of e_{strain}—nonlinear neoHookean and its smallstrain linear elastic approximation—are nearly identical everywhere in the system outside the neartip nonlinear zone, we conclude that nonlinearity within this zone is at the heart of the oscillatory instability.
To investigate the dependence of the oscillatory instability on the external load and material properties, we varied G_{0}/Γ_{0}, where Γ_{0} ≡ Γ(v = 0) is the fracture energy at onset of crack propagation, and the materialdependent ratio Γ_{0}/(μξ) ≡ ℓ_{0}/ξ controls the relative strength of neartip elastic nonlinearity and dissipation. Results of extensive simulations are shown in Fig. 3, presenting the crack velocity versus propagation distance for several G_{0}/Γ_{0} values, and ℓ_{0}/ξ = 0.29 in Fig. 3a (also used in Fig. 2) and ℓ_{0}/ξ = 1.45 in Fig. 3b. The plots clearly show that the onset of instability occurs when v exceeds a threshold value v_{c} independently of the external load. Decreasing the load simply reduces the crack acceleration and hence v exceeds v_{c} after a larger propagation distance; instability is not observed for the lowest loads in Fig. 3a, b because v has not yet reached v_{c} by the end of the simulations. Comparing Fig. 3a and Fig. 3b, v_{c} is seen to increase by only a few per cent when Γ_{0}/(μξ) is increased fivefold. This result is consistent with the experimental finding that v_{c} remains nearly constant when the ratio of the fracture energy to the shear modulus is varied several fold by altering the composition of the brittle gels, which affects both quantities and hence their ratio^{7}. Furthermore, the onset velocity v_{c} ≈ 0.92c_{s} in Figs 2 and 3a is in remarkably good quantitative agreement with experiments^{7}.
Unlike the critical velocity of instability v_{c}, the wavelength of oscillations λ has been experimentally observed to significantly vary when the fracture energy Γ(v_{c}) and the shear modulus μ were changed by varying the material composition^{8}. The phasefield framework allows one to independently vary Γ(v_{c}) and μ, and also to assess the role of the energy dissipation scale ξ. The size of the nonlinear zone ℓ is theoretically expected to be proportional to (and much larger than) Γ(v_{c})/μ, but as the prefactor is not sharply defined, we plot in Fig. 4a λ versus Γ(v_{c})/μ, both scaled by ξ, where Γ(v_{c})/μ ≈ 1.2Γ_{0}/μ is obtained from accelerating crack simulations (see Methods). We superimpose in Fig. 4a experimental measurements in brittle gels, where a value ξ ≃ 153 μm was chosen to match the yintercepts of linear best fits of both the theoretical and experimental results. The slopes dλ/d(Γ(v_{c})/μ), which are independent of the choice of ξ, are in remarkably good quantitative agreement. This agreement demonstrates that smallscale elastic nonlinearity, which is quantitatively captured by the phasefield approach, is a major determinant of the oscillation wavelength λ that increases linearly with Γ(v_{c})/μ, when Γ(v_{c})/(μξ) is sufficiently large. The existence of finite yintercepts further suggests that the dynamics on the dissipation scale also affects λ. However, it should be emphasized that oscillations exist only above a minimum value of Γ(v_{c})/(μξ) ∼ ℓ_{0}/ξ ≪ 1 (indistinguishable from the origin on the scale of Fig. 4a). This minimum reflects the fact that nonlinear effects become negligible when ℓ < ξ (or ℓ_{0} ≪ ξ, since ℓ ≫ ℓ_{0}, see Fig. 1). Below this minimum, we observe the same behaviour as for linear elasticity: stable straight crack propagation and then tipsplitting with increasing load G_{0}/Γ_{0}. Above this minimum, oscillations with a wavelength following the linear fit in Fig. 4a exist over a finite range of loads that increases with increasing Γ(v_{c})/(μξ). All in all, these results demonstrate the failure of the classical theory of fracture, which assumes that elastic nonlinearity plays no role. The singular role of elastic nonlinearity is further highlighted by our finding that the incorporation of Kelvin–Voigt viscosity^{17} in the linear elastic phasefield model does not suffice to produce an oscillatory instability. This result suggests that reversible nonlinear elastic deformation, as opposed to irreversible viscoelastic or viscoplastic dissipative processes, provides a general mechanism to destabilize highspeed cracks in conditions where nonlinear effects remain important outside the dissipation zone.
Elastic nonlinearity has been found experimentally to also affect the cracktip shape^{21,23,25}, which strongly departs at high velocities from the parabolic shape predicted by LEFM. This departure is quantified by the deviation δ of the actual tip location from its predicted location based on the parabolic shape. Figure 4b shows that δ indeed grows dramatically in a narrow range of ultrahigh velocities approaching v_{c} and can be larger than ℓ_{0} in agreement with experiments^{8,21,25}. As δ is related to ℓ, this result indicates a strong dependence of the latter on velocity. During oscillations, the cracktip shape and neartip nonlinear elastic fields become asymmetrical about the instantaneous crack propagation axis on a scale comparable to δ, as illustrated in Fig. 4c, which shows snapshots of the cracktip shape and strain energy density during one complete steadystate oscillation cycle. This illustrative sequence reveals that the asymmetry in the neartip strain fields is temporally outofphase with the instantaneous crack propagation direction, signalling a breakdown of the principle of local symmetry under dynamic conditions^{24}. How asymmetry on the scale of the nonlinear zone provides an instability mechanism, for example, the one proposed in ref. 24, remains to be further elucidated. Our newly developed nonlinear phasefield model of highspeed cracks provides a unique framework to address this and other fundamental issues, such as the basic relationship of crack oscillations and 3D microbranching suggested by recent experiments^{27}.
Methods
We use the phasefield framework^{16,17} that couples the evolution of the material displacement vector field u(x, y, t) to a scalar field φ(x, y, t) that varies smoothly in space between the fully broken (φ = 0) and pristine (φ = 1) states of the material. The present model is formulated in terms of the Lagrangian L = T − U, where
represent the potential and kinetic energy, respectively, ρ is the mass density inside the pristine material, and dV is a volume element. The form of U implies that the broken state (φ = 0) becomes energetically favoured when the strain energy density e_{strain} exceeds a threshold e_{c}, and the function g(φ) = 4φ^{3} − 3φ^{4} is a monotonously increasing function of φ that controls the softening of elastic energy at large strains. The parameters κ and e_{c} determine the size of the dissipation zone and the fracture energy in the quasistatic limit^{17,18}. The evolution equations for φ and u are derived from Lagrange’s equations
for ψ = (φ, u_{x}, u_{y}), where the functional
controls the rate of energy dissipation. As shown in the Supplementary Information, it follows from equations (1)–(3) that d(T + U)/dt = −2D ≤ 0. This gradient flow condition implies that the total energy (kinetic + potential) decreases monotonously in time due to energy dissipation near the crack tip where φ varies. In addition, we impose the standard irreversibility condition ∂_{t}φ ≤ 0.
The above model distinguishes itself from previous phasefield models^{26,28} by the formulation of the kinetic energy in equation (1). Previous models exhibit a tipsplitting instability in a velocity range (40% to 55% of c_{s} depending on the mode of fracture and Poisson’s ratio^{26,28}) much lower than the velocity in which the oscillatory instability is experimentally observed^{7}. Furthermore, tipsplitting generates two symmetrically branched cracks that are qualitatively distinct from both the 3D microbranching and 2D oscillatory instabilities. Lowvelocity tipsplitting can be suppressed by choosing f(φ) in equation (1) to be a monotonously increasing function of φ, similarly to g(φ). In particular, f(φ) = g(φ) (used here) ensures that the wave speeds remain constant inside the dissipation zone, which is physically consistent with the fact that dissipation and structural changes near crack tips in real materials do not involve large modifications of the wave speeds. As the wave speeds control the rate of transport of energy in the dissipation zone, cracks in this model can accelerate to unprecedented velocities approaching c_{s}, as observed experimentally in quasi2D geometries^{7}.
In addition, unlike conventional phasefield models, we focus on a nonlinear strain energy density given by
where F_{ij} = δ_{ij} + ∂_{j}u_{i} are the components of the deformation gradient tensor and i, j = {x, y}. It corresponds to a 2D incompressible neoHookean constitutive law, exhibiting generic elastic nonlinearities and quantitatively describing the brittle gels in experiments^{6,25}. In the smallstrain limit, neoHookean elasticity reduces to standard linear elasticity with a shear modulus μ and a 2D Poisson’s ratio ν = 1/3.
The equations are nondimensionalized by measuring length in units of ξ and time in units of τ = 1/(2χe_{c}), characterizing the timescale of energy dissipation. Crack dynamics is then controlled by only two dimensionless parameters: e_{c}/μ and β ≡ τc_{s}/ξ. The first controls the ratio ℓ_{0}/ξ = 2.9e_{c}/μ, where ℓ_{0} = Γ_{0}/μ sets the size of the nonlinear zone where elasticity breaks down (Fig. 1). The second controls the velocity dependence of the fracture energy. In the ideal brittle limit, β ≪ 1, Γ(v) ≈ Γ_{0} is independent of v. In the opposite limit, β ≫ 1, dissipation is sluggish and Γ(v) is a strongly increasing function of v. We vary e_{c}/μ between 0.05 and 1.0 to control the importance of elastic nonlinearity and choose a value β = 0.28 so that Γ(v)/Γ_{0} increases by about 20% when v varies from zero to v_{c} (see the inset of Supplementary Fig. 4), in qualitative similarity to experiments^{25}. The equations are discretized in space on a uniform square mesh with a grid spacing Δ = 0.21ξ and finitedifference approximations of spatial derivatives, and integrated in time using a Beeman’s scheme (see Supplementary Information) with a timestep size Δt = 5 × 10^{−4} τ. Largescale simulations of 10^{6}–10^{7} grid points are performed using graphics processing units with the CUDA parallel programming language.
LEFM has been experimentally validated for accelerating cracks that follow straight trajectories, prior to the onset of instabilities^{15,29}. Therefore, as a quantitative test of our highspeed crack model (see Supplementary Information), we verified the predictions of LEFM by showing that an accelerating crack centred inside a strip (as illustrated in Supplementary Fig. 3) satisfies the scalar equation of motion G = Γ(v), as long as their trajectory remains straight (v < v_{c}). We performed this test for the nonlinear form of e_{strain} defined by equation (4) by monitoring the instantaneous total crack length a (distinct from the cracktip propagation distance d plotted in Figs 2 and 3), crack velocity v, and energy release rate G calculated directly by contour integration using the Jintegral (see Supplementary Information). The results show that cracks accelerated under different loads G_{0}/Γ_{0} exhibit dramatically different v versus a curves (Supplementary Fig. 4), but the same G versus v curves (inset of Supplementary Fig. 4), which define a unique fracture energy Γ(v) (independent of the external load and crack acceleration history). Those results are consistent with the theoretical expectation that the relation G = Γ(v), which simply accounts for energy balance near the crack tip, should remain valid even in the presence of elastic nonlinearity as long as G is calculated through the Jintegral evaluated in a nondissipative region.
To investigate crack instabilities, we carried out simulations using a treadmill method^{19,26} that maintains the crack tip in the centre of the strip by periodically adding a strained layer at the right vertical boundary ahead of the crack tip and removing a layer at the opposite left boundary. This method allows us to study large crack propagation distances by mimicking an infinite strip with negligible influence of boundary effects. To further support the results presented in the main text, we carried out additional simulations to verify that the oscillation wavelength is independent of the choice of the degradation function g(φ) (Supplementary Fig. 5) and the grid spacing Δ (Supplementary Fig. 6).
Code availability.
The CUDA implementation of the phasefield simulation code is available upon request.
Data availability.
The data that support the plots within this paper and other findings of this study are available from one of the corresponding authors (A. Karma) upon reasonable request.
Additional Information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1
Freund, L. B. Dynamic Fracture Mechanics (Cambridge Univ. Press, 1990).
 2
RaviChandar, K. & Knauss, W. G. An experimental investigation into dynamic fracture: III. On steadystate crack propagation and crack branching. Int. J. Fract. 26, 141–154 (1984).
 3
Fineberg, J., Gross, S. P., Marder, M. & Swinney, H. L. Instability in dynamic fracture. Phys. Rev. Lett. 67, 457–460 (1991).
 4
Sharon, E. & Fineberg, J. Microbranching instability and the dynamic fracture of brittle materials. Phys. Rev. B 54, 7128–7139 (1996).
 5
Fineberg, J. & Marder, M. Instability in dynamic fracture. Phys. Rep. 313, 1–108 (1999).
 6
Livne, A., Cohen, G. & Fineberg, J. Universality and hysteretic dynamics in rapid fracture. Phys. Rev. Lett. 94, 224301 (2005).
 7
Livne, A., BenDavid, O. & Fineberg, J. Oscillations in rapid fracture. Phys. Rev. Lett. 98, 124301 (2007).
 8
Goldman, T., Harpaz, R., Bouchbinder, E. & Fineberg, J. Intrinsic nonlinear scale governs oscillations in rapid fracture. Phys. Rev. Lett. 108, 104303 (2012).
 9
Gol’dstein, R. V. & Salganik, R. L. Brittle fracture of solids with arbitrary cracks. Int. J. Fract. 10, 507–523 (1974).
 10
Marder, M. & Gross, S. Origin of crack tip instabilities. J. Mech. Phys. Solids 43, 1–48 (1995).
 11
AddaBedia, M., Arias, R., Ben Amar, M. & Lund, F. Dynamic instability of brittle fracture. Phys. Rev. Lett. 82, 2314–2317 (1999).
 12
Buehler, M. J., Abraham, F. F. & Gao, H. Hyperelasticity governs dynamic fracture at a critical length scale. Nature 426, 141–146 (2003).
 13
Buehler, M. J. & Gao, H. Dynamic fracture instabilities due to local hyperelasticity at crack tips. Nature 439, 307–310 (2006).
 14
Bouchbinder, E., Fineberg, J. & Marder, M. Dynamics of simple cracks. Ann. Rev. Condens. Matter Phys. 1, 371–395 (2010).
 15
Bouchbinder, E., Goldman, T. & Fineberg, J. The dynamics of rapid fracture: instabilities, nonlinearities and length scales. Rep. Progr. Phys. 77, 046501 (2014).
 16
Bourdin, B., Francfort, G. A. & Marigo, J.J. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797–826 (2000).
 17
Karma, A., Kessler, D. A. & Levine, H. Phasefield model of mode III dynamic fracture. Phys. Rev. Lett. 87, 045501 (2001).
 18
Hakim, V. & Karma, A. Laws of crack motion and phasefield models of fracture. J. Mech. Phys. Solids 57, 342–368 (2009).
 19
Pons, A. J. & Karma, A. Helical crackfront instability in mixedmode fracture. Nature 464, 85–89 (2010).
 20
Bourdin, B., Marigo, J.J., Maurini, C. & Sicsic, P. Morphogenesis and propagation of complex cracks induced by thermal shocks. Phys. Rev. Lett. 112, 014301 (2014).
 21
Livne, A., Bouchbinder, E. & Fineberg, J. Breakdown of linear elastic fracture mechanics near the tip of a rapid crack. Phys. Rev. Lett. 101, 264301 (2008).
 22
Bouchbinder, E., Livne, A. & Fineberg, J. Weakly nonlinear theory of dynamic fracture. Phys. Rev. Lett. 101, 264302 (2008).
 23
Bouchbinder, E., Livne, A. & Fineberg, J. The 1/r singularity in weakly nonlinear fracture mechanics. J. Mech. Phys. Solids 57, 1568–1577 (2009).
 24
Bouchbinder, E. Dynamic crack tip equation of motion: highspeed oscillatory instability. Phys. Rev. Lett. 103, 164301 (2009).
 25
Livne, A., Bouchbinder, E., Svetlizky, I. & Fineberg, J. The neartip fields of fast cracks. Science 327, 1359–1363 (2010).
 26
Karma, A. & Lobkovsky, A. E. Unsteady crack motion and branching in a phasefield model of brittle fracture. Phys. Rev. Lett. 92, 245510 (2004).
 27
Goldman Boué, T., Cohen, G. & Fineberg, J. Origin of the microbranching instability in rapid cracks. Phys. Rev. Lett. 114, 054301 (2015).
 28
Henry, H. Study of the branching instability using a phase field model of inplane crack propagation. Europhys. Lett. 83, 16004 (2008).
 29
Sharon, E. & Fineberg, J. Confirming the continuum theory of dynamic brittle fracture for fast cracks. Nature 397, 333–335 (1999).
Acknowledgements
This research was supported by the USIsrael Binational Science Foundation (BSF), grant no. 2012061, which provided partial support for C.H.C. E.B. acknowledges support from the William Z. and Eda Bess Novick Young Scientist Fund and the Harold Perlman Family. A.K. acknowledges support of grant number DEFG0207ER46400 from the U.S. Department of Energy, Office of Basic Energy Sciences. The authors thank M. Nicoli for his contribution to the initial development of the phasefield simulation code.
Author information
Affiliations
Contributions
All authors contributed equally to this work.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary information
Supplementary information (PDF 533 kb)
Supplementary movie
Supplementary movie 1 (MP4 2015 kb)
Supplementary movie
Supplementary movie 2 (MP4 3276 kb)
Rights and permissions
About this article
Cite this article
Chen, C., Bouchbinder, E. & Karma, A. Instability in dynamic fracture and the failure of the classical theory of cracks. Nature Phys 13, 1186–1190 (2017). https://doi.org/10.1038/nphys4237
Received:
Accepted:
Published:
Issue Date:
Further reading

Brittleductile transitions in a metallic glass
Physical Review E (2020)

Morphological transformation of the process zone at the tip of a propagating crack. I. Simulation
Physical Review E (2020)

A length scale insensitive phase field model for brittle fracture of hyperelastic solids
Engineering Fracture Mechanics (2020)

Selfemitted surface corrugations in dynamic fracture of silicon single crystal
Proceedings of the National Academy of Sciences (2020)

A phasefield model for mixedmode fracture based on a unified tensile fracture criterion
Computer Methods in Applied Mechanics and Engineering (2020)