Chapter IV: Optimal Design of 3D Printed Soft Responsive Actuators . 60
4.6 Conclusion
Figure 4.8: Optimal design and print path of the rectangular lifting actuator (right), along with printed structure (left). The darker yellow is the responsive material, while the transparent material is passive. We see that the print lines of the responsive structure follows the desired paths quite well.
Figure 4.9: Printed structure undergoing actuation in a heated oil bath. We see that the actuation is a lifting motion as expected.
Currently, these designs are undergoing testing in a mechanical testing appa- ratus. We plan to compare the compliance ratio for the loaded structure to that of intuitive designs to verify optimality.
While the developed formulation is effective for the studied scenarios, it is not without its shortcomings. Here, we designed for quasi-static conditions with complete disregard for the kinetics driving actuation. In most applications, speed of response is tremendously important. Thus, the kinetics leading to ac- tuation (e.g., heat diffusion), must be modeled and accounted for in the design formulation. Additionally, researchers have begun to develop responsive ma- terial systems with multiple actuation modes [18]. Thus, incorporating multi- functionality into the optimization framework would be a natural progression.
On the theoretical side, there remains mathematical intricacies which must be solved. In particular, structural instabilities must be rigorously treated for a fully robust design approach. However, the mathematical formulation in such settings is quite unclear. This is in part due to instability phenomena be- ing driven by the evolution of local energy minimizers, whereas the presented treatment works with global minimization. A rigorous mathematical study into these issues is required.
Even with these limitations, this work may serve as a starting point to guide actuator design towards a myriad of applications. It is straightforward to adapt the formulation to other responsive material systems including magneto- responsive structures, hydrogels, and shape memory alloys. Additionally, an identical treatment of the print constraints may be used for other anisotropic 3D printed materials. Finally, the rigorous mathematical approach we take may be extended to include design considerations necessary for application deployment such as actuation kinetics and strucrual instabilities.
BIBLIOGRAPHY
[1] E. M. Ahmed. Hydrogel: Preparation, characterization, and applications:
A review. Journal of Advanced Research, 6(2):105–121, 2015.
[2] A. Akerson, B. Bourdin, and K. Bhattacharya. Optimal design of re- sponsive structures. Structural and Multidisciplinary Optimization, 65, 03 2022.
[3] L. T. Akerson, A. Mechanics, Modeling, and Shape Optimization of Elec- trostatic Zipper Actuators. In Preparation, 2023.
[4] C. P. Ambulo, M. J. Ford, K. Searles, C. Majidi, and T. H. Ware. 4d- printable liquid metal–liquid crystal elastomer composites. ACS Applied Materials & Interfaces, 13(11):12805–12813, 2021.
[5] J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity.Archive for Rational Mechanics and Analysis, 63:337–403, 1976.
[6] W. Bangerth, R. Hartmann, and G. Kanschat. Deal.II - -A general- purpose object-oriented finite element library. ACM Transactions on Mathematical Software, 33(4):1–27, 2007.
[7] M. Bendsøe and O. Sigmund. Topology Optimization: Theory, Methods and Applications. Springer, 2nd edition, 2003.
[8] M. P. Bendsøe. Optimal shape design as a material distribution problem.
Structural Optimization, 1(4):193–202, 1989.
[9] K. Bhattacharya. Microstructure of Martensite. University of Minnesota, 1991.
[10] K. Bhattacharya and R. D. James. The material is the machine. Science, 307(5706):53–54, 2005.
[11] P. Cesana and A. DeSimone. Quasiconvex envelopes of energies for ne- matic elastomers in the small strain regime and applications. Journal of the Mechanics and Physics of Solids, 59(4):787–803, 2011.
[12] P. G. Ciarlet. The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, 2002.
[13] B. Dacorogna. Direct Methods in the Calculus of Variations. Applied Mathematical Sciences. Springer Berlin Heidelberg, 1989.
[14] G. Dal Maso. An introduction to Γ-convergence, volume 8. Springer Science & Business Media, 2012.
[15] M. de Luca and A. D. Simone. Mathematical and numerical modeling of liquid crystal elastomer phase transition and deformation. MRS Online Proceedings Library, 1403:37–42, 2012.
[16] A. Desimone and L. Teresi. Elastic energies for nematic elastomers. The European physical journal. E, Soft matter, 29:191–204, 07 2009.
[17] X. Gao, J. Yang, J. Wu, X. Xin, Z. Li, X. Yuan, X. Shen, and S. Dong.
Piezoelectric actuators and motors: Materials, designs, and applications.
Advanced Materials Technologies, 5, 1 2020.
[18] Q. He, Z. Wang, Y. Wang, A. Minori, M. T. Tolley, and S. Cai. Electrically controlled liquid crystal elastomer–based soft tubular actuator with multimodal actuation. Science Advances, 5(10):eaax5746, 2019.
[19] G. Kocak, C. Tuncer, and V. Bütün. ph-responsive polymers. Polym.
Chem., 8:144–176, 2017.
[20] A. Kotikian, R. L. Truby, J. W. Boley, T. J. White, and J. A. Lewis. 3d printing of liquid crystal elastomeric actuators with spatially programed nematic order. Advanced Materials, 30(10):1706164, 2018.
[21] Y. Li, H. Yu, K. Yu, X. Guo, and X. Wang. Reconfigurable three- dimensional mesotructures of spatially programmed liquid crystal elas- tomers and their ferromagnetic composites. Advanced Functional Mate- rials, 31(23):2100338, 2021.
[22] S. Mohith, A. R. Upadhya, K. P. Navin, S. M. Kulkarni, and M. Rao.
Recent trends in piezoelectric actuators for precision motion and their applications: a review. Smart Materials and Structures, 30, 1 2021.
[23] N. L. Pedersen. Topology optimization of laminated plates with prestress.
Computers and Structures, 80(7-8):559–570, 2002.
[24] P. Penzler, M. Rumpf, and B. Wirth. A phase-field model for compliance shape optimization in nonlinear elasticity. ESAIM: Control, Optimisation and Calculus of Variations, 18(1):229–258, 2012.
[25] G. Rozvany and W. Prager. Structural Design Via Optimality Criteria:
The Prager Approach to Structural Optimization. Mechanics of Elastic and Inela. Springer Netherlands, 1989.
[26] A. Talhan and S. Jeon. Pneumatic actuation in haptic-enabled medical simulators: A review. IEEE Access, 6:3184–3200, 12 2017.
[27] S. W. Ula, N. A. Traugutt, R. H. Volpe, R. R. Patel, K. Yu, and C. M.
Yakacki. Liquid crystal elastomers: an introduction and review of emerg- ing technologies. Liquid Crystals Reviews, 6:78–107, 2018.
[28] J. Walker, T. Zidek, C. Harbel, S. Yoon, F. S. Strickland, S. Kumar, and M. Shin. Soft robotics: A review of recent developments of pneumatic soft actuators. Actuators, 9, 3 2020.
[29] Z. Wang, Z. Wang, Y. Zheng, Q. He, Y. Wang, and S. Cai. Three- dimensional printing of functionally graded liquid crystal elastomer. Sci- ence advances, 6(39):eabc0034, 2020.
[30] T. H. Ware, M. E. McConney, J. J. Wie, V. P. Tondiglia, and T. J. White.
Voxelated liquid crystal elastomers. Science, 347(6225):982–984, 2015.
[31] M. Warner and E. Terentjev. Liquid Crystal Elastomers. International Series of Monographs on Physics. OUP Oxford, 2007.
[32] T. White. Photomechanical Materials, Composites, and Systems: Wire- less Transduction of Light into Work. 05 2017.
[33] Z. Zhao and X. S. Zhang. Topology optimization of hard-magnetic soft materials. Journal of the Mechanics and Physics of Solids, 158:104628, 2022.
C h a p t e r 5
OPTIMAL STRUCTURES FOR FAILURE RESISTANCE UNDER IMPACT
A. Akerson, Optimal structures for failure resistance under impact, Journal of Mechanics and Physics of Solids 172 105172, 2023.
Abstract
The complex physics and numerous failure modes of structural impact cre- ates challenges when designing for impact resistance. While simple geometries of layered material are conventional, advances in 3D printing and additive manufacturing techniques have now made tailored geometries or integrated multi-material structures achievable. Here, we apply gradient-based topology optimization to the design of such structures. We start by constructing a variational model of an elastic-plastic material enriched with gradient phase- field damage, and present a novel method to efficiently compute its transient dynamic time evolution. We consider a finite element discretization with ex- plicit updates for the displacements. The damage field is solved through an augmented Lagrangian formulation, splitting the operator coupling between the nonlinearity and non-locality. Sensitivities over this trajectory are com- puted through the adjoint method, and we develop a numerical method to solve the resulting adjoint dynamical system. We demonstrate this formula- tion by studying the optimal design of 2D solid-void structures undergoing blast loading. Then, we explore the trade-offs between strength and tough- ness in the design of a spall-resistant structure composed of two materials of differing properties undergoing dynamic impact.
5.1 Introduction
The design of structures for impact or blast loading is encumbered by the complex interactions between wave propagation, plasticity, and material dam-
age. This leads to failure modes such as plugging, fracture, petaling, and spall which are highly dependent on the material parameters, loading conditions, and structural layout [4]. This is further complicated by the trade-offs be- tween properties such as strength and toughness when designing integrated structures of multiple materials. In practice, engineers typically start with industry standards and intuition, followed by sophisticated dynamical simu- lations to iterate on a design before it undergoes physical testing. Usually, these designs consist of simple geometries of layered materials [23, 25, 20].
However, with recent advances in additive manufacturing and 3D printing, we may now look to tailored designs with complex geometries and integrated materials [40, 2, 18]. Additionally, the exponential growth of computational capabilities makes algorithmic optimal design methods feasible. This may al- low us to efficiently design structures of unprecedented impact performance in scenarios where intuitive design is not sufficient.
Of the optimal structural design formulations, topology optimization has proven to be one of the most powerful methodologies. Density-based methods con- sider the density of material at each point in the domain as the unknown before the design is posed as an optimization problem over these densities.
Then, gradient-based optimization methods are used to iteratively update the design, where sensitivities are usually computed through the adjoint method.
Originally introduced to optimize the compliance of linear elastic structures [8], density-based topology optimization has since been applied to a wide range of applications including acoustic band-gaps [44], piezoelectric transducers [45], micro-electro-mechanical systems [37], energy conversion devices [14], and fluid structure interaction [51]. Another common method is level-set topology op- timization. Here, the boundary of the structure is defined as a level set of a scalar-valued function, and optimization is performed over this function using the shape-gradient [1]. Finally, phase-field approaches remain popular as their variational form yields a favorable mathematical structure [10].
For the optimal design of impact problems, it is necessary to include transient dynamics, rate-dependent plasticity, and damage mechanics when modeling the material response. Past studies have addressed optimal design for transient dynamic evolution with elastic material models [43, 33]. Additionally, plastic- ity has been considered in both quasi-static [42, 49, 15, 48, 30] and dynamic settings [32, 22]. However, a structure with damage has only been considered
in the quasi-static case. This has been studied in both the ductile [26, 27]
and quasi-brittle [16, 34, 6] regime to design damage resistant structures. A variational mechanics model, where solutions are computed through energy principles, are favored to accurately model the physics and provide mathe- matical structure. Furthermore, an efficient computational method for these fields is necessary, as the iterative design process requires repeatedly simulating the dynamics for updated designs.
To address the above mentioned requirements, we consider small-strain, rate- dependent plasticity enriched with continuum damage through a variational phase-field model in a transient dynamic setting. To efficiently simulate the dynamic response, we consider a finite element discretization where we employ an explicit update scheme for the displacement fields, and an implicit update for both the plasticity and damage. Because these irreversible damage updates are both nonlinear and non-local in nature, a direct computation would be prohibitively expensive. To this end, we use an operator-splitting augmented Lagrangian alternating direction method of multipliers. By introducing an auxiliary damage and Lagrange multiplier field, we accurately and efficiently solve for the damage updates by iterating between a nonlinear local problem, a linear global problem, and a Lagrange multiplier update.
We look to optimize the material placement of the structure over the dynamic trajectory for a given objective function. By assuming the material parameters are dependent on a continuous design variable, we derive sensitivities through the adjoint method. This results in an adjoint dynamical system that we solve in a manner which shares similarities to the forward problem numerics. We use a explicit update scheme for the adjoint displacement variable, and another augmented Lagrangian method for the adjoint damage variable. However, the adjoint problem is solved backwards in time, and the adjoint damage operator is linear rather than the nonlinear operator seen in the forward problem. With the adjoint solution, we compute the sensitivities and update the design.
We start in Section 5.2 by presenting the energy functional for system, then discuss the dynamic equilibrium relations. We apply the adjoint method, where sensitivities and adjoint relations are derived for a general objective.
In Section 5.3 we detail the solution process. First, we apply an augmented Lagrangian to operator split the damage updates. Then, using a finite ele- ment discretization, we solve the system with explicit displacement updates,
followed by implicit plasticity and damage updates. We demonstrate the ac- curacy and efficiency of the numerical scheme by considering the solution con- vergence and time-scaling for a model problem. We use a similar numerical scheme for the adjoint system and the associated dual variables. Next, in Section 5.4, we discuss material interpolation schemes through intermediate densities for both solid-void structures and multi-material designs. In Sec- tion 5.5, we demonstrate the methodology by looking at two examples. First we consider the design of 2D solid-void structures optimized for blast loading.
Next, we explore the trade-offs between strength and toughness in a two ma- terial spall-resistant structure undergoing impact. Finally, in Section 5.6, we summarize our findings and discuss further directions.
5.2 Theoretical Formulation 5.2.1 Forward Problem
We consider an elastic-plastic material capable of sustaining damage occupy- ing a bounded, open domain Ω ⊂ Rn in its reference configuration over time [0, T]. We assume prescribed loads on∂fΩ⊂∂Ω and prescribed displacements on ∂uΩ ⊂ ∂Ω. We consider small-strain, rate-dependent J-2 plasticity with isotropic hardening to model the plasticity [35, 29]. Damage is measured by the phase-field scalar quantity a : Ω×[0, T] 7→ [0,1], where values of 0 and 1 correspond to the undamaged and fully damaged states. Here, we use a phase-field fracture model which we adapt for damage by considering a finite length scale [11]. These models have been modified for ductile fracture by including small-strain plasticity [13], and we adopt a similar formulation. We assume the material parameters are dependent on a design field η: Ω7→[0,1]
which determines the species of material at each point. We consider a vari- ational structure, where minimization principles yields the internal variable evolution [36]. Thus, we consider the incremental energy
E(u, q, εp, a, η) =
Z
Ω
We(ε, εp, a, η) +d(a)
Wp(q, η) +
Z t 0
g∗( ˙q, η)dt
+Gc(η) 4cw
"
wa(a, η)
`(η) +`(η)k∇ak2
#
+
Z t 0
ψ∗( ˙a, η)dt
dΩ,
(5.1)
whereu: Ω×[0, T]7→Rnis the displacement field,εp : Ω×[0, T]7→Rn×nis the volume preserving plastic strain, and q : Ω×[0, T] 7→R+ is the accumulated plastic strain whose evolution is defined by
˙ q=
s2
3ε˙p·ε˙p. (5.2)
Weis the stored elastic energy density, which accounts for the tension-compression asymmetry in its damage dependence [3],
We(ε, εp, a, η) = K(η)
2 tr−(εe)2 +d(a)
"
K(η)
2 tr+(εe)2 +µ(η)εeD :εeD
#
, (5.3) where K and µare the bulk and shear moduli. d(a) models the weakening of the material with damage,
d(a) = (1−a)2+d1a2, (5.4) where d1 << 1. εe = ε− εp is the elastic strain, and εeD is its deviatoric component. tr+(ε) and tr−(ε) are the positive and negative parts of the strain trace,
tr+(ε) = max(tr(ε),0), tr−(ε) = min(tr(ε),0). (5.5) This decomposition of the volumetric strain allows for tension-compression asymmetry in the damage model; the tensile bulk modulus is affected by dam- age, while the compressive bulk modulus remains unaffected. Wp and wa are the plastic and damage hardening functions, respectively. The damage param- eters Gc and ` control the toughness and damage length scale, with cw as a normalization constant. Finally, the rate dependence of both the damage and plastic hardening is handled by the dissipation potentials ψ∗ and g∗, respec- tively. These functions also account for irreversibility, as they take a value of
+∞for negative rates,
g∗( ˙q, η) =
¯
g∗( ˙q, η) q˙≥0
∞ q <˙ 0
, ψ∗( ˙a, η) =
ψ¯∗( ˙a, η) a˙ ≥0
∞ a <˙ 0
. (5.6) For the plastic potentials, we consider power-law hardening and rate-sensitivity functions
Wp(q, η) =σy
q+ nεp0 n+ 1
q εp0
!(n+1)/n
,
¯
g∗( ˙q, η) = mσyε˙p0 m+ 1
˙ q ε˙p0
!(m+1)/m
.
(5.7)
εp0 and ˙εp0 are the reference plastic strain and strain rate and σy is the initial yield stress. n and m are the powers for the hardening and rate sensitivity, with the perfecty plastic and rate-indepdendent cases occurring as n → ∞+
and m → ∞+, respectively [35]. These plastic hardening and rate-hardening parameters may all depend on η. For the damage hardening, we consider a quadratic function
wa(a, η) =w1a+ (1−w1)a2, (5.8) where w1 ∈ [0,1], which ensures wa(1) = 1, and may be dependent on η.
For simplicity, we consider the damage to be rate-independent by choosing ψ¯∗( ˙a, η) = 0. Here, we scale both the plastic potential and shear modulus with the same damage function d(a). Thus, the yield strength and Mises stress have the same damage dependence, leading to damage independent plastic updates. A further discussion on the behavior of a similar material model can be found in [13]. However, this choice of constitutive is not essential, and the methodologies we present below remain general.
We consider dynamic evolution through the incremental action integral L(u, q, εp, a, η) =
Z t2
t1
E(u, q, εp, a, η)−
Z
Ω
ρ(η)
2 |u|˙ 2dΩ
−
Z
Ω
fb ·u dΩ−
Z
∂fΩ
f ·u dS
dt,
(5.9)
wherefb andf are the body force and surface tractions, and ρ is the material density. Stationarity of this action integral gives the dynamic evolution and the kinetics of the internal variables [31]
0 =
Z
Ω
"
ρ¨u·δu+ ∂We
∂ε · ∇δu−fb·δu
#
dΩ
−
Z
∂fΩ
f ·δu dΩ ∀δu∈ U, (5.10a)
0∈σ¯M − ∂Wp
∂q −∂g∗, on Ω, (5.10b)
0 = ˙εp−qM˙ on Ω, (5.10c)
0∈ ∂We
∂a +∂d
∂a
Wp+
Z t 0
g∗( ˙q)dt
− ∇ · Gc` 2cw∇a
!
+ Gc 4cw`
∂wa
∂a +∂ψ∗ on Ω, (5.10d)
a= 0 on ∂uΩ, ∇a·n = 0 on∂fΩ. (5.10e) Here, we assume quiescent initial conditions. U is the space of admissible displacement variations
U ={u∈H1(Ω), u= 0 on ∂uΩ}. (5.11) (5.10a) is the second-order dynamic evolution of the displacement field. (5.10b) and (5.10c) are the yield relation and the evolution of the plastic strain, where
¯
σM is the normalized Mises stress (divided through by d(a)), andM is the di- rection of plastic flow. (5.10d) is the irreversible evolution of the damage field, with (5.10e) being the boundary conditions fora. The differential inclusion in the yield relation and damage equilibrium enforces the irreversibility of their respective internal variables. A further discussion on the damage evolution relation (5.10d) can be found in [28].
We briefly comment on the regularity of the solution to the forward prob- lem. The plastic strains may be discontinuous in space, however, they remain continuous in time as the rate-dependence provides temporal regularity. The damage field, a ∈ L∞((0, T);H1(Ω;Rn)), is continuous in space while being possibly discontinuous in time in the rate-independent case ( ¯ψ∗( ˙a, η) = 0). Fi- nally, the displacement field,u∈H1((0, T);H1(Ω;Rn)), is continuous in both space and time as the inertia provides temporal regularity.
5.2.2 Sensitivities and Adjoint Problem
We look to find the design field η(x) such that an objective, dependent on the dynamic trajectory, is minimized. Thus, we consider a general objective of
integral form min
η(x) O(η) :=
Z T 0
Z
Ω
o(u, q, εp, a, η)dΩ dt subject to: Equillibrium relations in (5.10).
(5.12) To conduct gradient-based optimization, the variation of the objective with η must be computed. For this, we employ the adjoint method [38]. We introduce fieldsξ,γ,µ, andbas the dual variables to the displacement, plastic hardening, plastic strain, and the damage fields, respectively. We consider the necessary Kuhn-Tucker conditions for the irreversible equilibrium relations, and carry out the adjoint calculation. The full details of this can be found in 5.A. This gives the total variation of the objective as
O,ηδη =
Z T 0
Z
Ω
(∂o
∂η + ∂ρ
∂ηu¨·ξ+ ∂2We
∂ε∂η · ∇ξ +ba˙ ∂2We
∂a∂η +∂d
∂a
∂Wp
∂η + ∂d
∂a
Z t
0
∂g∗
∂η dτ
!
+ 1 2cw
∂(Gc`)
∂η ∇(ba)˙ · ∇a+ba˙ wa0 4cw
∂(Gc/`)
∂η + ∂2ψ∗
∂a∂η˙
!
+γq˙ ∂σ¯M
∂η − ∂σ0
∂η − ∂2g∗
∂q∂η˙
! )
δη dΩdt,
(5.13)
where the adjoint variables satisfy the dynamic evolution 0 =
Z
Ω
"
∇ξ· ∂2We
∂ε∂ε +ba˙∂2We
∂a∂ε +γq˙∂σ¯M
∂ε −qµ˙ · ∂M
∂ε
!
· ∇δηu +ρξ¨·δηu+ ∂o
∂u ·δηu
#
dΩ ∀δηu∈ U, (5.14a)
d dt
"
γ σ¯M −σ0− ∂g¯∗
∂q˙
!
−γq˙∂2g¯∗
∂q˙2 +∂¯g∗
∂q˙
Z T t
bad˙ 0(a)dτ
!
−µ·M
#
= ∂o
∂q +bad˙ 0(a)∂Wp
∂q −γq˙∂σ0
∂q on Ω, (5.14b)
dµ dt = ∂o
∂εp +∇ξ· ∂2We
∂ε∂εp +ba˙∂2We
∂a∂εp +γq˙∂σ¯M
∂εp −qµ˙ · ∂M
∂εp on Ω, (5.14c)
d dt
"
Dab+∂2ψ¯∗
∂a˙2 ba˙
#
= ∂o
∂a + ∂2We
∂a∂ε · ∇ξ +ba˙ ∂2We
∂a2 + Gc 4cw`
∂2wa
∂a2
!
+bad˙ 00
Wp+
Z t 0
g∗dτ
− ∇ · Gc` 2cw∇(ba)˙
!
on Ω, (5.14d) ξ|t=T = 0, ξ|˙t=T = 0,
γ|t=T = 0, µ|t=T = 0, b|t=T = 0, where
Da= ∂We
∂a +∂d
∂a
Wp+
Z t
0
g∗dτ
− ∇ · Gc` 2cw∇a
!
+ Gc
4`cw
∂wa
∂a +∂ψ¯∗
∂a˙ . (5.15) These are dependent on the forward problem solution and must be solved back- wards in time. Once the forward problem is solved in time for u(t), a(t), q(t), andεp(t), they can be used to solve the adjoint problem backwards in time for ξ(t), b(t), γ(t), and µ(t). The sensitivities can then be computed from (5.13).
Details of the numerical methods to solve the forward and adjoint problem are discussed in the proceeding section.
It should be noted that both the adjoint problem and expression for the sensi- tivities may have issues with well-posedness. As we have used an elastic energy function which remains strongly convex, the Hessian which appears above is well defined. However, for a different choice of elastic energy this may not be the case. Furthermore, the convexity of the adjoint problem with respect to the entire variable set {ξ, γ, µ, b} is not established, which may lead to an ill-posed problem. While inertia is thought to provide some temporal regu- larity, the reader is nontheless cautioned in this regard. Additionally, issues may arise from the possible temporal discontinuities of the damage field dis- cussed previously in Section 5.2.1. Thus, the ˙a found in the adjoint relations may not be well-defined. A rigorous investigation into these matter would be quite worthwhile. However, after discretization we find that the presented formulation is sufficient in practice, perhaps providing the required regularity.
5.3 Numerics
5.3.1 Forward Problem
We discuss the details for the numerical evolution of the forward dynamics.
First we introduce an augmented Lagrangian formulation to split the nonlin- ear and non-local operator coupling in the damage field equilibrium. Then, using a finite element discretization, we discuss the computational procedure for updating the displacements, plasticity and damage variables. Finally, we study the accuracy and efficiency of our formulation by studying the solution behavior for varying mesh sizes.
Augmented Lagrangian
The differential inclusion and gradient terms in the damage evolution of (5.10d) result in a nonlinear and non-local state equation for the damage updates.
While there exist methods to directly solve these non-local constrained prob- lems, they result in expensive computations that would be required at every timestep. Thus, we consider an augmented Lagrangian formulation to split this operator, and solve the system using an alternating direction method of multipliers (ADMM) [19, 17]. This method has been used to efficiently solve non-linear elasticity problems with internal variable evolution [52]. We intro- duce the auxiliary fieldα∈L2(Ω) and constraina=αweakly for all time with the Lagrange multiplier λ ∈ L2(Ω) and penalty factor r. Thus, we consider