Simple extension of Lemaitre's elastoplastic damage model to account for hydrolytic degradation

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1 884 Simle extension of Lemaie's elastolastic damage model to account for hydrolytic degradation Absact Being motivated by the technological alications of bioabsorbable olymeric materials in the fields of biomechanics and medicine, this aer resents a simle but efficient extension of Lemaie s elastolastic damage model by incororating a chemical-based (hydrolysis) degradation term. The aim is to allow the simulation of devices subjected to both mechanical and chemical environments. The model alicability is tested by a set of numerical finite-element examles. The encouraging results show exected adequate couling between the elastolastic and chemical damages. Although the model is resently resicted to linear kinematics, the basic idea can be extended to finite sains. Keywords Bioabsorbable olymers, elastolastic damage, hydrolytic degradation a, b, * Eduardo A. Fancello Leandro P. Lindenmeyer a Carlos R. M. Roesler a Gean V. Salmoria c, b a GRANTE Grou of Analysis and Mech. Design; Mech. Engineering Deartment b LEBm Lab. of Biomechanical. Eng.; University Hosital c CIMJECT Plastic Molding Lab.; Mechanical Engineering Deartment Federal Univ. of Santa Catarina, Brazil. Received in 12 May 2013 In revised form 30 Jul 2013 *Author fancello@grante.ufsc.br 1 INTRODUCTION Medical imlants made of bioabsorbable olymeric materials have been comrehensively investigated as efficient substitutes for their metallic counterarts. Costs, mechanical behavior, and their ability of being absorbed by the body once their mechanical function has ceased are the main features that motivate the use of these olymers. Among the most used bioabsorbable olymers develoed for surgical rocedures, the most common are olyglycolic acid (PGA) and olylactic acid (PLA) coolymers. The rocess of degradation of bioabsorbable imlants is reresented by the irreversible henomenon called hydrolysis that comrises the rogressive breakage of the chemical bonds of the olymeric net due to its contact with bodily fluids. During this rocess, an initial decrease in the molecular weight is observed, followed by a loss in mechanical erformance; finally, a loss of mass is observed. Due to this sequential degradation, as the mechanical roerties of the bioabsorbable devices worsen, sesses are rogressively ansferred to the healed new tissue formed around them. In this way, they are designed to abandon their mechanical functions before being fully

2 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 885 absorbed. This final absortion is the last ste in the degradation rocess, which is found in biochemical reactions. The hydrolytic degradation of the olymer deends on the ininsic roerties of the material, and it is related to the diffusion coefficient of water, degradation rate, body size, degree of crystallinity, and mechanical sess state. Degradation can occur in the following two ways (Chen et al., 2011). In the olymer medium, if the diffusion of water is slower than the degradation, rate, then surface degradation is observed, which rogresses from the surface to the core. Conversely, when the water diffusivity allows for a raid soaking of the entire device, then bulk degradation is observed, which acts on all the material oints almost simultaneously. In the context of the resent work, we assume that the latter functions as the rimary agent of degradation for the PGA and PLA olymers. Several studies have been erformed to investigate the behavior of bioabsorbable olymers considering the effects of hydrolysis and mechanical sesses. Not aiming to resent a systematic review, we cite some examles below. An early study by Miller and Williams (1984) revealed that the virgin material of PLA olymers used in surgical sutures was subjected to sain values between 25% and 50% of their ruture sain values. The sess relaxation after days was evaluated, revealing that the rate of hydrolysis clearly increased with the magnitude of the alied sain. Chu (1985) reorted a relationshi between the alied sain and the degradation rate of PLA fibers immersed in a solution. When re-sess was alied to the material in an aqueous medium, microvoids aeared, which increased the water eneation and accelerated the degradation rocess. Zhong et al. (1993) analyzed the influence of a sain value of 4% on the degradation arameters when using PGA and PLA suture fiberssoaked in a solution of water and hydrogen eroxide. After two weeks of immersion, they concluded that the degradation rate was considerably higher in the resence of sain than its absence. Smit et al. (2008) evaluated the effect of axial comressive loading on the degradation of a sine cage made of 70/30 oly(l,dl-lactic acid)(pldlla). The results showed that the mechanical resistance of the material was deendent on the load, temerature, and humidity. Soares et al.(2008, 2009, 2010) resented constitutive models for bioabsorbable olymers that underwent degradation under alied sain. They roosed a scalar field that described the local state of degradation (isooic damage). This field is related to an evolution law that deends on the state of the sain/sess. Muliana et al. (2009) and Muliana and Rajagoal (2011,2012) analyzed the different relationshis between hydrolytic degradation, viscoelastic behavior, and water diffusion. Basic hyotheses suggest that the Young s modulus of the material decreases with degradation. In a very recent aer, Khan and El-Sayed (2013) includes a damagetye internal variable within a variational viscoelastic model to account for degradation henomenon in stents. It is interesting to note that none of the above mentioned works consider the combined influence of the damages induced by mechanical and chemical sources. For examle, it is ossible to find this couling in vascular stents, because these devices are subjected to severe mechanical action (lastic diameal exansion) during the imlantation rocedure. The resent study aims to test a simle extension of the classical Lemaie s isooic elastolastic damage model to include a time-deendent term that accounts for the dynamics of a chemical degradation rocesses. On the basis of the exerimental observations, it can be shown Latin American Journal of Solids and Suctures 11 (2014)

3 886 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation that this term deends on the sess state. The model combines the effects of elastic and lastic sains, mechanical damage, and chemical (hydrolytic) damage. Although this model is resicted to infinitesimal kinematics, it can be extended to finite deformations. Section 2 resents a short but self-contained mathematical formulation of the model. The notation used in this section is obtained from the book by de Souza Neto et al. (2008) where certain details about the technical oerations can be found. Section 3 illusates the behavior of the roosed model. To this end, a set of numerical tests are conducted to evaluate the single constitutive equation as well as its inclusion in finite-element examles. It is worth mentioning, however, that the results do not reresent any articular identified material, and the intention of these results is to demonsate the caabilities of the roosed model. A final section is added to summarize the obtained results. 2 HYDROLYTIC-PLASTIC DAMAGE MODEL Let us consider an isooic damage variable that is slit into two terms: the first is the classical ductile damage of the Lemaie s model and the other is called hydrolytic damage. While the evolution of the former is directly related to the existence of lastic sains, the latter is deendent on time (chemical action) and sess state. Now, let us define the conventional external and internal state variables, namely, the total sain ε, lastic sain ε, equivalent lastic sain e, lastic damage d, and hydrolytic damage d h. The elastic sain and total damage can be defined as ε e = ε ε, d = d + d h, (1) For simlicity, we assume that both lasticity and damage henomena are isooic in nature. In this case, the free energy can be additively decomosed into two conibutions: the elastic otential y e, deending on the elastic sain and scalar damage, and the lastic one ψ, deending on the accumulated lastic sain ε only: ψ = ψ( ε, ε,ε,d,d h ) = ψ e ( ε e,d ) + ψ ( ε ) (2) The elastic otential is related to the linear isooic material and is deendent on the bulk modulus K and shear modulus G : ( ) ρψ e = 1 d 2 ε e : D : ε e = 1 d ( ) e Gε d : ε e d K ε 2 v (3) In (3), D is the isooic elasticity tensor, ε v = [ε e ], the volumeic sain, and ε d e = ε e ε v 3 I, the deviatoric elastic sain. Considering the linear constitutive relationshi it is also ossible to exress this energy in terms of the sess: Latin American Journal of Solids and Suctures 11 (2014)

4 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 887 ρψ e = 1 2( 1 d ) σ : D 1 : σ = 1 1 d ( ) q 2 6G + 2 2K = 1 d ( ) q 2 6G + 2 2K (4) where = 1 σ 3 is the hydrostatic ressure, s = σ I is the deviatoric sess, and q = 3J 2 (s) = 3 s : s is the equivalent von Mises sess. Their relationshis with the corresond- 2 ing effective values are σ = σ ( 1 d ) =, s = s 1 d ( ), = ( 1 d ), q = q ( 1 d ) (5) On the basis of thermodynamical rinciles, the conjugated forces can be obtained from the derivative of ψ with resect to the state variables: σ = ρ ψ ε e = (1 d)d : εe, χ = ρ ψ ψ = σ, k = ρ ε ε = k(ε ) (6) Y = ρ ψ d = q 2 6G 2 2K, Y h = ρ ψ d h = Y (7) Y Y h is the energy release rate due to an increment in the lastic damage. The new conjugate force results in exactly the same conjugate value of Y ; however, this value is now related to the hydrolytic damage. Desite the fact that Y h and Y share the same value by definition, both these variables are exressed searately. The von Mises yield function is given by ( ) = q ( 1 d ) σ ( y k ) 0 (8) Φ σ,k,d where σ y (k) is the hardening law. To define the evolution rules of the internal variables, a common formalism can be used to assume the existence of a dissiative otential Ψ which ensures the automatic satisfaction of the Clausius Planck inequality (non-negative dissiation rocess) on the basis of its convexity roerties. In the resent case, the otential Ψ is exactly the same as that of the Lemaie s damage model along with a new term Ψ h : Ψ = Φ(σ,k,d) + Ψ (Y,d) + Ψ h (Y h,d) (9) Latin American Journal of Solids and Suctures 11 (2014)

5 888 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation In (9), we distinguish the yield function Φ, lastic otential Ψ, and the (new) hydrolytic otential Ψ h. Since Ψ Φ, the flow model is nonassociative. In our case, we use the simlified Lemaie and Chaboche lastic otential (de Souza Neto et al., 2008): ( ) r Ψ (Y ) = H ε ε D Y (1 d)(1 + s) r s+1, H a ( ) = 1 if a > 0 0 if a 0 (10) where r and s denote the material constants. The Heaviside ste function H is used to incororate a lastic threshold ε D for the accumulated lastic sain ε below which no lastic damage rate is allowed. The evolution of the internal variables ε, ε, and d are ininsically related to the lastic flow henomenon and can be defined based on the dissiation otential Ψ as follows: ε = γn, N = Ψ σ = Φ σ = 1 (1 d) 3 2 s s (11) ε = γh, H = Ψ k = Φ k = 1 (12) d = γv, V = Ψ Y = Ψ Y = H ε ( ε D )) Y s (1 d) r (13) The N, H, and V values defined in (11), (12), and(13), resectively, are associated with the direction of evolution of the internal variables, while the Lagrange multilier rate γ accounts for their amlitudes. In the resent roosition, the hydrolytic damage is not deendent on the lastic flow, unless it is indirectly related through the sess state. Therefore, d h is indeendent of γ. Moreover, the hydrolytic damage is a time-deendent rocess driven by the kinetics of the chemical reactions. In chemical kinetics, a classical exression that accounts for the degradation rate is given below (de Paula & Atkins, 2011,.1328): d = k(1 d) n k = A ex E a RT (14) where k is a rate constant deending on the temerature T, activation energy E a, gas constant R, and a re-exonential factor A. Moreover, the exonent n is known as the order of the reaction. In the resent case, since the temerature and activation energy are assumed to be constant, Latin American Journal of Solids and Suctures 11 (2014)

6 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 889 we assume that the value of k deends on the sain energy Y h. Then, following the sucture of (14), we define the evolution of d h as follows: d h = Ψ Y = Ψh h Y = h Y h l + g m (d 1) n (15) The hydrolytic dissiative otential consistent with (15) is then exressed as Ψ h (Y h l ) = Y h (m + 1) l + g m+1 (1 d) n (16) where l,m,g are material constants. The first simle exression arises when taking n = 1, i.e., a first-order chemical reaction, which rovides an evolution that is concetually similar to that of Soares et al. (2009). An imortant value in (15) is g that guarantees the existence of the degradation rate even in the case of absence of sesses. Although similar, the exressions in (13) and (15) behave significantly differently; in the case of the former, the damage rate increases when the damage itself increases, which is the oosite behavior of the latter. In (13), the damage evolution is a time-indeendent rocess since it follows the lastic multilier γ ; in (15), the hydrolytic damage is essentially deendent on the time scale of the chemical rocess. 2.1 Incremental udating The time integration of the revious equations over the time ste t n,t defines the corresonding incremental constitutive algorithm. Here, we use the standard full imlicit elastic redictor lastic corrector technique, and the resentation is similar to the rocedures shown in de Souza Neto et al. (2008) and Simo and Hughes (1998). We consider the set of state variables ε n, ε n,ε n,d h { n,d n } at time t n (which is comletely known) as well as the total sain ε = ε n + Δε at time t. The increment of the lastic deformation and damage comes from (11 15) and Δγ = Δt γ : ε = ε n + ΔγN, N = 1 ( ) 1 d 3 2 s s (17) h d = d + d = d n + Δd + Δd h (18) Δd Δγ = (1 d ) Y r s (19) Latin American Journal of Solids and Suctures 11 (2014)

7 890 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation Δd h = Δt(1 d ) Y h + g l m (20) In (20), the evolution law of (15) was substituted after consideringn = 1. The comutation of the elastic sain and corresonding effective sesses is given by e ε d = ε d ε e ( n + ΔγN ), ε v = ε v (21) s = 2G ( ε d ( ε dn + ΔγN )), s = (1 d ) s (22) e = Kε v = Kε v, = (1 d ) (23) The consistence condition in (8) at t takes the form 2.2 Elastic redictor ( Φ 0) Φ = q σ y ( k(ε n + Δγ) ) 0 Δγ 0, Φ Δγ = 0 (24) In this ste, we assume that the sain increment is urely elastic, i.e., Δγ = 0. In this case, the so-called ial quantities can be defined as e ε = ε ε n, ε = ε n (25) s e e = 2Gε d, = Kε v, q = 3 2 s : s (26) Y ( ) 2 ( ) 2 = q 6G + 2K (27) Since Δγ = 0, the damage increment, if it exists, can be atibuted only to the hydrolysis henomenon. Therefore, d = d h n + d (28) h = d h n + Δt (1 d n d h ) Y + g l d m (29) Latin American Journal of Solids and Suctures 11 (2014)

8 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 891 h Isolating d from (29), we get h d = d h + Δt(1 d ) Y n n 1 + Y ( + g l ) m ( + g l ) m Δt (30) which allows the comutation of s,, and q as follows: s = (1 d ) s, = (1 d ), q = (1 d ) q (31) Finally, the yield function can be evaluated for the ial values as follows: Φ := q σ y k(ε ( )) (32) If the condition Φ 0 is satisfied, then the ste is urely (damage) elastic ( Δγ = 0) and the variables are udated using the corresonding ial values: ( ) := ( ) Otherwise, Φ > 0 ; then, a lastic increment Δγ > 0 exists. In this case, the new udated variables are determined to satisfy the condition Φ = 0, which corresonds to the lastic corrector ste. 2.3 Plastic corrector ( Δγ > 0) In this ste, the following set of equations is solved: e ε d e = ε d Δγ (1 d ) 3 2 s s (33) ε = ε n + Δγ (34) Δγ d = d n + (1 d ) Y r s + Δt(1 d ) Y h + g l m (35) Φ = q σ y ( k(ε ) = 0 (36) Latin American Journal of Solids and Suctures 11 (2014)

9 892 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation From (33), it is ossible to write q = q 3GΔγ (1 d ) (37) which may be substituted in (35). Similarly, relacing (34) in (36), the system is reduced to the following air of equations: Φ = q 3GΔγ (1 d ) σ y k(ε n + Δγ) ( ) = 0 (38) Δγ d = d n + (1 d ) Y r s + Δt(1 d ) Y h + g l m (39) where Y h = Y = 1 6G q 3GΔγ (1 d ) K (40) Isolating w = 1 d from (38) and inoducing it in (39), a single nonlinear equation in Δγ can be obtained: Δγ F(Δγ) = w (Δγ) w n + w (Δγ) Y r s (41) + w (Δγ) Y h + g l m Δt = 0, (42) Note that this exression differs from the classical Lemaie s model in the last term of (41). The solution of this single nonlinear equation is obtained by using the Newton Rahson method. In de Souza Neto et al. (2008), a convenient initial value Δγ (0) is roosed as follows: Δγ (0) = q σ y 3G ( ε n ) w n Once Δγ is obtained, all the remaining variables are udated: Latin American Journal of Solids and Suctures 11 (2014)

10 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 893 ε = ε n + Δγ, = w (Δγ), q = w (Δγ) q 3GΔγ s = w (Δγ) 3GΔγ s, σ = s + I q ε = ε n + Δγ ( ) 1 d 3 2 s s Finally, the above algorithm σ = ˆσ yields the derivative. ( ε, ε n,ε n,d n ) of the incremental constitutive roblem D = σ ε roviding the consistent tangent elastolastic tensor. Clearly, a closed exression of this derivative is obtainable, but not included in this aer. 3 NUMERICAL EXAMPLES In this section, we resent a set of examles aiming to demonsate the behavior and alicability of the roosed model. The material arameters used, however, do not corresond to any secific material and are taken just as examles. It is worth mentioning that the arameters related to the hydrolytic damage are clearly deendent on the unit of time, since it is a henomenon that is significantly influenced by the time scale in which the chemical henomenon occurs. Here, since no identification rocess was considered, the time unit used will be generically called a day. The values shown in Table 1 were used for all the further studied cases and are intended to resemble those of a olymeric material. 3.1 Uniaxial tensile test Sain-conolled tensile test: Case 1 In this case, we consider a single material oint subjected to an axial tensile test conolled by axial sain. Three loads are alied that are linearly varied from zero to the maximum sain values of 0.05, 0.03, and 0.01 during a time interval of [0,120] days. Figure 1 shows the curves of the von Mises sess versus time and damage versus time for all the load conditions, which can be used to comare the behavior of the roosed lastic-hydrolytic damage model (DEPH) with a urely lastic damage (DEP) model. As exected, the hydrolytic damage curves exhibit higher values in the cases with higher sess values. In the case of the maximum sain of 0.01, no lastic sain occurs and only hydrolytic damage is visible. Latin American Journal of Solids and Suctures 11 (2014)

11 894 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation Figure 1 von Mises sess versus time and damage (D: total; DH: hydrolytic; DP: lastic) versus time (Case 1). Figure 2 von Mises sess versus time and damage versus time (Case 2). Latin American Journal of Solids and Suctures 11 (2014)

12 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation Sain-conolled tensile test: Case 2 In this examle, a similar tensile test is conducted; however, we consider that the maximum sain is alied for a short interval of 1/24 day and then maintained constant for u to 120 days. Figure 2 shows the curves of the von Mises sess versus time and damage versus time for all the load conditions. Again, the higher the load, the higher is the initial damage rate. During the initial loading (1/24 day), the hydrolytic damage is imercetible and only lastic damage is visible Sain-conolled tensile test: Case 3 The same maximum sains are linearly alied during the first 60 days and then linearly unloaded during the next 60 days. Figure 3 shows the sess comonent ( σ x ) versus sain curve and damage versus time curve Sess-conolled tensile test: Case 4 Similarly to the revious cases, in this case, the tensile test is conolled by the alied axial sesses with maximum values of 20 MPa and 26 MPa that are linearly alied during a short time ste of 1/24 day and then maintained constant. Figure 4 shows the curves of sain versus time and damage versus time. Note that in both these instances, the alied loads are less than the lastic yield value of 50 MPa. However, due to hydrolytic damage, lasticity is finally achieved, thereafter driving the material oint to collase. Figure 3 Axial sess versus time and damage versus time (Case 3). Latin American Journal of Solids and Suctures 11 (2014)

13 896 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation Sess-conolled tensile test: Case 5 In this case, the sess values of 18 MPa and 20 MPa are linearly alied for 60 days and unloaded in the next 60 days. Figure 5 shows the curves of the von Mises sess versus time and damage versus time. Again, in both these instances, the alied loads are less than the lastic yield value of 50 MPa. For the load of 20 MPa, hydrolytic damage occurs before the unloading and collase. For the load of 18 MPa, desite the increasing damage, unloading is faster and the material oint is able to withstand the load. Figure 4 Axial sain versus time and damageversus time (Case 4). Figure 5 Axial sain versus time and damage versus time (Case 5). 3.2 Axisymmeic samle Dislacement-conolled test: Case 6 Consider the axisymmeic samle shown in Figure 6. An axial load is alied by conolling the dislacement of the to surface of the samle and imosing symmey boundary conditions on the lower surface. Four-node Lagrangian finite elements (with 2D solid revolution) were used to simulate this case. The mesh is also shown in Figure 6. Latin American Journal of Solids and Suctures 11 (2014)

14 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 897 The rescribed dislacement increases linearly from zero to 0.15 mm within the time interval of [0,120] days. Figures 7 and 8 show the von Mises sess disibution and damage disibution for the two instances: 40 and 94 days, resectively. From Figure7, it is evident that song relaxation occurs in the instance when hydrolysis is taken into account. Further, the sess disibution changes significantly; the maximum value emanates from the notch and aears in the interior of the domain. F F Figure 6 Axial tensile secimen and simulated mesh Force-conolled test: Case 7 The same samle as that used in Case 6 is subjected to a conolled axial force that grows linearly from zero to 1021 N within the time interval of [0,120] days. After that, rogressive unloading is undertaken at the same rate, reaching (theoretically) zero value at 120 days. This maximum force value was set u so that the virgin material could reach the maximum sess value of 50 MPa. Figures 9 and 10 show the von Mises sess disibution and damage disibution for two different times: 60 and 79 days, resectively. A behavior similar to that in the revious case is evident; due to the resence of hydrolysis, the sesses at the notch relax and the maximum von Mises sess value is found to exist in the interior of the domain. It is noteworthy that since this is a force-conolled test, the samle cannot withstand the load if hydrolysis is considered. Figure 11 shows the axial sain at the notch in the samle. The unloading at 60 days for both the instances is clearly visible; however, since hydrolysis kees damaging the samle, lastic collase is achieved at 79 days (Figure 15, loss of convergence) Plate Samle: Case 8 In this case, we simulate a late samle using the 8-node Lagrangian brick elements (Figure 12), which can be used to reresent a 3D samle discretized with solid elements. In this case, a disibuted force is rogressively alied on the to surface of the late; the maximum value of N is achieved at 60 days. After that, the unloading hase is initiated, reaching the zero Latin American Journal of Solids and Suctures 11 (2014)

15 898 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation load value at 120 days. Figures 13 and 14 show the disibution of the axial sess comonent σ y and the damage at two times: 60 and 120 days. As exected, the behavior is similar to that in the cylindrical case; the damage due to hydrolysis significantly modifies the sess disibution at the notch. During the unloading hase, it is ossible to visualize the cometition between the remaining resistance of the secimen and the unloading tensile force. At the end of the rocess, the local comressive forces remain near the notch region for the hydrolytic model. Figure 7 Equivalent von Mises sess evolution for the EPDH (left) and EPD (right) models. Latin American Journal of Solids and Suctures 11 (2014)

16 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 899 Figure 8 Damage evolution for the EPDH (left) and EPD (right) models. Latin American Journal of Solids and Suctures 11 (2014)

17 900 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation Figure 9 Equivalent von Mises sess evolution for the EPDH (left) and EPD (right) models. Latin American Journal of Solids and Suctures 11 (2014)

18 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 901 Figure 10 Damage evolution for the EPDH (left) and EPD (right) models. Latin American Journal of Solids and Suctures 11 (2014)

19 902 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation Figure 11 Axial sain versus time. Figure 12 Tensile secimen and simulated mesh. 4 FINAL REMARKS Being motivated by the technological alications of bioabsorbable olymeric materials, the objective of this aer is the roosal of a simle extension of the classical Lemaie s elastolastic damage model that includes the reresentation of damage induced by chemical henomena (such as hydrolytic degradation). In a simle and efficient manner, this model demonsates the relationshi between the damage evolution due to mechanical and chemical actions. The alicability of the model is analyzed by undertaking a set of numerical tests. Although the model is resently resicted to linear kinematics, the idea can be extended to finite sains. A set of secific technical remarks are highlighted below: Hydrolytic damage interferes with all the state variables at all the material oints subjected to chemical activity. Moreover, this damage aears even in the absence of sesses due to the material arameter g. The evolution law for the hydrolytic damage in which the term (1 - d) ; n > 0 aears is essentially different than that of the mechanical damage. Because of the former, the damage rate decreases with increasing damage, behavior that is oosite to that given by the model of mechanical damage. Latin American Journal of Solids and Suctures 11 (2014) n

20 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 903 Mechanical damage is indeendent of time since it undergoes lastic evolution. Hydrolytic damage is essentially deendent on the time scale of the chemical rocess. The results of the investigated examles are consistent with the exectations for different situations. At the beginning of the load, the evolution of the damage is driven only by the hydrolysis rocess. When the material reaches the lastic flow, the degradation rate increases significantly due to the couled action between both the hydrolytic- and lasticity-driven damages. Figure 13 Equivalent von Mises sess evolution for the EPDH (left) and EPD (right) models. Latin American Journal of Solids and Suctures 11 (2014)

21 904 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation Figure 14 Damage evolution for the EPDH (left) and EPD (right) models. Latin American Journal of Solids and Suctures 11 (2014)

22 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation 905 In the cases when the material oint is subjected to a sain-conolled rocess, the exected sess relaxation deends on the degradation rate. On the other hand, when the material is subjected to a sess-conolled rocess, the degradation forces the material to undergo lastic flow followed by collase. In all the investigated cases, hydrolytic damage increased faster than elastolastic damage. This, however, should be atibuted to the arbiary choice of the material arameters in the examles. The roosition in (2) regarding free energy decouling, although conventional, cannot be considered a unique alternative. For examle, in the studies of Brünig (2002, 2004) and the references therein, the damage follows a kinematic descrition similar to that of lastic sains and the free energy is additively decouled in elastic (reversible) and inelastic (damage and lastic) conibutions. This aroach would certainly modify the definitions of the conjugate forces Y h and Y in (7). In site of these alternatives, the resent work has aimed to maintain the same sucture of the original Lemaie s elastolastic damage roosition (Lemaie, 1985). Summarily, it is worth mentioning that the actual reresentation of a bioabsorbable material deserves attention in several asects that have not been considered here. Among others, one such asect is the modification of the material roerties due to the chemical actions: in the resent simle damage model, the yield limit and elastic modulus of the virgin material decrease as long as the damage increases. The rocess of aging due to chemical actions may roduce embrittlement and/or stiffening of the elastic behavior instead of the softening action observed in the resent case. Acknowledgments The authors would like to thank the Brazilian agencies CNPq, CAPES and FINEP for the financial suort offered for this research. References Y. Chen, S. Zhou, and Q. Li(2011). Mathematical modeling of degradation for bulk-erosive olymers: Alications in tissue engineering scaffolds and drug delivery systems. ActaBiomaterialia, 7: C.C. Chu (1985).Sain-accelerated hydrolytic degradation of synthetic absorbable sutures. Surgical research, recent develoments. Proceedings of the First Annual Scientific Session of the Academy of Surgical Research, M. Brünig (2002). Numerical analysis and elastic-lastic deformation behavior of anisooically damaged solids. International Journal of Plasticity, 18: M. Brünig (2004) An isooic continuum damage model: Theory, and numerical analyses. Latin American Journal of Solid and Suctures, 1: E. A. de Souza Neto, D. Peric, and D. R. J. Owen (2008). Comutational methods for lasticity: Theory and alications. WILEY. K.A. Khan, T. El-Sayed (2013). A Phenomenological constitutive model for the nonlinear viscoelastic resonses of biodegradable olymers. Acta Mechanica 224(2): Latin American Journal of Solids and Suctures 11 (2014)

23 906 E. Fancello et al. / A Simle extension of Lemaie s elast. last. damage model to account for hydrolytic degradation J. Lemaie (1985). Couled Elasto-lasticity and damage constitutive equations. Com. Meth. Al. Mech. Engng, 51: N.D. Miller and D.F. Williams (1984). The in vivo and in vio degradation of oly(glycolicacid) suture material as a function of alied sain. Biomaterials, 5: A. Muliana, K.R. Rajagoal, and S.C. Subramanian (2009). Degradation of an elastic comosite cylinder due to the diffusion of a fluid. Journal of Comosite Materials, 43(11): A.H. Muliana and K.R. Rajagoal (2011).Changes in the resonse of viscoelastic solids to changes in their internal sucture. Acta Mechanica, 217(3-4): A. Muliana and K.R. Rajagoal (2012). Modeling the resonse of nonlinear viscoelastic biodegradable olymeric stents. International Journal of Solids and Suctures, 49(7-8): J. V. Simo and T. J. R. Hughes (1998). Comutational Inelasticity. Sringer. T. H.Smit, T A P Engels, P I J M Wuisman, and L E Govaert (2008). Time-deendent mechanical sength of 70/30 oly(l,dl-lactide) shedding light on the remature failure of degradable sinal cages. Sine, 33: J. S. Soares, J. E. Moore, JR., and K. R. Rajagoal (2008).Constitutive framework for biodegradable olymers with alications to biodegradable stents. ASAIO Journal, J. S.Soares, K. R. Rajagoal, and J. E. Moore Jr. (2010).Deformation-induced hydrolysis of a degradable olymeric cylindrical annulus. Biomechanics and Modeling in Mechanobiology, 9: S. P. Zhong, P. J. Doherty, and D. F. Williams (1993). The effect of alied sain on the degradation of absorbable suture in vio. Clinical Materials, 14: Latin American Journal of Solids and Suctures 11 (2014)