A generalisation to cohesive cracks evolution under effects of non-Uniform stress field

and stress fields as the unique solution, denoted (u[t, a, b],σ[t, a, b], of the following linear elastic problem posed on the cracked body with non uniform cohesive forces on the crack lips divσ[t, a, b] = 0 in Ω\([−a, a]× {0}) σ[t, a, b] = λ tr(ε(u[t, a, b]))I+ 2µε(u[t, a, b]) in Ω\([−a, a]× {0}) u[t, a, b] = tU on ∂DΩ σ[t, a, b]n = tF on ∂NΩ σ[t, a, b]e2 = 0 on (−b, b)× {0} σ[t, a, b]e2 = σce2 on ((−a,−b) ∪ (b, a))× {0} (24) Since 0 < b < a  L, the crack should perturb the elastic fields only in a neighborhood of the origin. Consequently, one can introduce in (24) the gaps of solution with elastic fields, i.e. u[t, a, b] = u[t, a, b]− tuel, σ[t, a, b] = σ[t, a, b]− tσel. A generalisation to cohesive cracks evolution under effects of non-uniform stress field 359 Using the same approximations and hypothesis as the case of a fully cohesive crack, the local problem reads as follows divσ[t, a, b] = 0 in R2\([−a, a]× {0}) σ[t, a, b] = λ tr(ε(u[t, a, b]))I+ 2µε(u[t, a, b]) in R2\([−a, a]× {0}) σ[t, a, b](x)→ 0 when |x| → ∞ σ[t, a, b] (x1, 0) e2 = T (x1) e2 on (−a, a)× {0} (25) where T (x1) =  ( − t te + t te ∞ ∑ n=1 αn x2n1 `2n ) σc if |x1| < b( 1− t te + t te ∞ ∑ n=1 αn x2n1 `2n ) σc if b < |x1| < a (26) The integral calculation give us the intensity factor KI[t, a, b] as follows KI[t, a, b] = σc √ pia [ t te − 1− t te ∞ ∑ n=1 ( a2n `2n αn n ∏ i=1 2i− 1 2i ) + 2 pi arcsin b a ] (27) The condition KI[t, a, b] = 0 gives us the first relationship between the crack lengths (a, b) and the loading parameter t t te [ 1− ∞ ∑ n=1 ( a2n `2n αn n ∏ i=1 2i− 1 2i )] = 2 pi arccos b a . (28) One can investigate the complex potential jump on cohesive zone, i.e. [[ϕ]]′(x1) pour b < |x1| < a, to compute the crack opening [[u[t, a, b]2]](x1). Using normal stress distribution (26) and the relationship (28), we obtain [[ϕ]]′ (x1) =  1 pi arctanh  b x1 √ a2 − x21 a2 − b2  + x1 √ a2 − x21 2 t te ∞ ∑ n=1 ( αn `2n n−1 ∑ i=0 (2i− 1)!! (2i)!! a2ix2(n−i−1)1 ) . (29) By a straightforward integration and using the fact that [[ϕ]](±a(t)) = 0, one gets [[ϕ]](x1). Using the relationship (21), one can deduce the crack opening at the cohesive zone tips x1 = ±b as follows [[u[a, b, t]2]](b) = piδc 2dc [ 2 b pi ln a b + t te √ 1− b 2 a2 ∞ ∑ n=1 αn `2n a2n+1 n−1 ∑ i=0 (2i− 1)!!(2(n− i− 1))!! (2i)!!(2(n− i) + 1)!! + t te √ 1− b 2 a2 ∞ ∑ n=1 αn `2n a2n+1 n−1 ∑ i=0 (2i− 1)!!(2(n− i− 1))!! (2i)!!(2(n− i) + 1)!! n−i−2 ∑ j=0 (2j + 1)!! (2j + 2)!! ( b a )2j+2 − t te √ 1− b 2 a2 ∞ ∑ n=1 αn `2n a2n+1 n−1 ∑ i=0 (2i− 1)!! (2i)!! 1 2(n− i) + 1 ( b a )2(n−i)] . (30) 360 Tuan-Hiep Pham, Je´roˆme Laverne, Jean-Jacques Marigo Using the condition given in Proposition 2.2, we obtain the second relationship between a, b and t 2 pi dc = 2 pi b ln a b + t te √ 1− b 2 a2 ∞ ∑ n=1 αn `2n a2n+1 n−1 ∑ i=0 (2i− 1)!!(2(n− i− 1))!! (2i)!!(2(n− i) + 1)!! + t te √ 1− b 2 a2 ∞ ∑ n=1 αn `2n a2n+1 n−1 ∑ i=0 (2i− 1)!! (2(n− i− 1)!!!n−i−2 (2i)!!(2(n− i) + 1)!! n−i−2 ∑ j=0 (2j + 1)!! (2j + 2)!! ( b a )2j+2 − t te √ 1− b 2 a2 ∞ ∑ n=1 αn `2n a2n+1 n−1 ∑ i=0 (2i− 1)!! (2i)!! 1 2(n− i) + 1 ( b a )2(n−i) . (31) 3.2. Representation of the three branches in particular cases We present in the present work the case of the 4th order normal stress distribution. This case corresponds to nmax = 2 in (6). In consequence, two parameters (α1, α2) and the length ` are introduced to characterize the non uniform normal stress Σ(x1). Evidently, (α1, α2) depend on the elastic solution with properties of structure, limit conditions, etc. In order to investigate analytically the problem of crack evolution, we choose the follow- ing set of values for the two parameters α1 = 2, α2 = 8 3 . (32) The normal stress distribution Σ(x1) of elastic response given by (6) becomes Σ (x1) = σc t te ( 1− 2 x 2 1 `2 − 8 3 x41 `4 ) . (33) Recall that the elastic branch begins at t = 0 and finishes when the loading parameter t the critical value te. This branch corresponds to the segment line [0, te]× {0} in the (t, a) diagram. Beyond the loading te, the crack nucleates and propagates in the structure. One considers now this evolution, which contains a fully cohesive branch and a partially non cohesive branch, corresponding to the distribution (33) of Σ(x1). 3.2.1. Fully cohesive branch At the instant t, a cohesive crack of length 2a is present in the structure. The condition for the stress intensity factor KI[t,±a] = 0 gives a relationship between the crack length and the loading parameter given by (18) which simply reads here as t te ( 1− a 2 `2 − a 4 `4 ) = 1. (34) The loading parameter t is an increasing function of the crack length. Consequently, this relationship is reversible and implies that the crack length is a continuous, monotonically increasing function of the loading. This solution is no more valid as soon as the crack opening reaches the critical value δc. Corresponding to the normal stress distribution A generalisation to cohesive cracks evolution under effects of non-uniform stress field 361 (33), the crack opening at the origin given by the general expression (22) becomes [[u[t, a]2]](0) = 8 ( 1− ν2) σc E [ 1 3 t 3 a(t)3 `2 + 2 5 t te a(t)5 `4 ] , (35) where a(t) is computed in (34). Accordingly, the crack opening at the origin x1 = 0 is also a monotonically increasing function of t for t ≥ te, and reaches the critical cohesive value δc at the instant ti given by dc pi` = ti te ( 1 3 + 2 5 a (ti) 2 `2 ) a (ti) 3 `3 . (36) 3.2.2. Partially non cohesive branch From the instant t > 0, a partially non cohesive crack exists with non cohesive part length 2b and total length 2a. The condition on the stress intensity factor KI[t, a, b] = 0 gives us the first relationship between two length (a, b) and the loading parameter t t te ( 1− a 2 `2 − a 4 `4 ) = 2 pi arccos b a . (37) Corresponding to the normal stress distribution (33), the crack opening at the tip of non cohesive part x1 = b is expressed as follows [[u[t, a, b]2]](b) = 8 ( 1− ν2) σc pi [ b pi ln a b + 1 3 t 3 ( a2 − b2)3/2 `2 + t te 1 `4 ( 2a2 3 ( a2 − b2)3/2 − 4 15 ( a2 − b2)5/2)] . (38) The condition [[u[t, a, b]2]](b) = δc provides the second relationship between (a, b, t) b ln a b + pi 3 t te ( a2 − b2)3/2 `2 + t te pi `4 ( 2a2 3 ( a2 − b2)3/2 − 4 15 ( a2 − b2)5/2) = dc. (39) Accordingly, the two lengths (a, b) are related to the loading parameter t by two rela- tionships (37) and (39). In order to study the partially non cohesive crack evolution, one introduces the following dimensionless variable α = b a ∈ (0, 1). By injecting (37) and this variable in (39), we obtain − ( 1− a 2 `2 − a 4 `4 )( a ` α ln α+ dc ` ) + 2 arccos α 3 ( 1−α2)3/2 [ a3 `3 + 2 a5 `5 − 4 5 ( 1− α2) a5 `5 ] = 0. (40) For given α ∈ (0, 1), (40) is a 5th order equation for a := a/` which depends on the parameter e := dc/`. This equation admits a unique solution, say ae(α), whose depen- dence on α is non monotonic. Specially, ae(α) starts from ai/` = √ 1− te/ti at α = 0, then is first decreasing up to am/` before to be increasing and finally tends to the limit 362 Tuan-Hiep Pham, Je´roˆme Laverne, Jean-Jacques Marigo ac := ` √√ 5− 1 2 < ` when α tends to 1, also see Fig. 4. Besides, (37) allows us to write t/te as a function of α with the parameter e t te = te(α) := 2 arccos α pi (1− ae(α)2 − ae(α)4) . (41) 14 T.-H. Pham et al. function of α with the parameter  t te = t¯(α) := 2 arccosα pi (1− a¯(α)2 − a¯(α)4) (41) The function t¯(α) starts from ti/te at α = 0, and is first monotonically decreasing up to tl/te, that minimum being reached at α = αl. Then t¯ (α) is increasing to infinity when α tends to 1, cf Figure 4. Finally, the evolution of b with α is given by b ` = b¯(α) := αa¯(α). (42) Figure 4 shows the evolution of b with α, which starts from 0 at α = 0 then increases monotonically and tends to ac = ` √√ 5− 1 2 when α tends to 1. Accordingly, the triple (t, a, b) satisfying (37) and (39) can be considered as two parametric curves (t(α), a(α)) and (t(α), b(α)) parameterized by α ∈ (0, 1) and depending on the characteristic length ` and on the ratio dc/`. In particular, the curve (t(α), a(α)) represents the partially non cohesive branch in the (t, a) diagram, cf Figure 5. Since the functions a¯(α) and t¯(α) are respectively non monotonic and monotonically decreasing for small α, the partially non cohesive branch contains a snap-back in the neighborhood of (ti, ai) and a limit point (tl, al), both points depending on ` and dc. Accordingly, the branch has the shape of a loop which can be divided into two parts: the lower part between (ti, ai) and (tl, al), the upper part after (tl, al). a or b Fig. 4. Typical dependence of a, b and t on α = b/a. Here the curves correspond to the case where dc/` = 0.1. Assuming that the critical stress σc is fixed, one can study the dependence of the Dugdale’s branches on dc at fixed `, or, on ` at fixed dc. At fixed ` and for all dc, the fully cohesive branch in the (t, a) diagram is given by (34). Only the final point (ti, ai) depends on dc, and both ti and ai are increasing functions of dc (or ), cf Figure 6. On the one hand, when dc (or ) goes to 0, then ti tends to te and ai/` tends to 0 like  1/3. On the other hand, when dc/` goes to infinity, then ai tends to ` and ti tends to infinity. Fig. 4. Typical dependence of a, b and t on α = b/a. Here the curves correspond to the case where dc/` = 0.1 The function te(α) starts from ti/te at α = 0, and is first monotonically decreasing up to tl/te, that minimum being reached at α = αl . Then t e (α) is increasing to infinity when α tends to 1, also see Fig. 4. Finally, the evolution of b with α is given by b ` = b e (α) := αae(α). (42) Fig. 4 shows the evolution of b with α, which starts from 0 at α = 0 then increases monotonically and tends to ac = ` √√ 5− 1 2 when α tends to 1. Accordingly, the triple (t, a, b) satisfying (37) and (39) can be considered as two para- metric curves (t(α), a(α)) and (t(α), b(α)) parameterized by α ∈ (0, 1) and depending on the characteristic length ` and on the ratio dc/`. In particular, the curve (t(α), a(α)) rep- resents the partially non cohesive branch in the (t, a) diagram, also see Figure 5. Since the functions ae(α) and te(α) are respectively non monotonic and monotonically decreasing for small α, the partially non cohesive branch contains a snap-back in the neighborhood of (ti, ai) and a limit point (tl , al), both points depending on ` and dc. Accordingly, the branch has the shape of a loop which can be divided into two parts: the lower part be- tween (ti, ai) and (tl , al), the upper part after (tl , al). Assuming that the critical stress σc is fixed, one can study the dependence of the Dugdale’s branches on dc at fixed `, or, on ` at fixed dc. At fixed ` and for all dc, the fully cohesive branch in the (t, a) diagram is given by (34). Only the final point (ti, ai) depends on dc, and both ti and ai are increasing functions of dc (or e), also see Fig. 5. On the one A generalisation to cohesive cracks evolution under effects of non-uniform stress field 363 hand, when dc (or e) goes to 0, then ti tends to te and ai/` tends to 0 like e1/3. On the other hand, when dc/` goes to infinity, then ai tends to ` and ti tends to infinity. A generalisation to cohesive cracks evolution under effects of non-uniform stress field 15 a or b Fig. 5. Typical graphs of the three branches in the diagram (t, a). The gray curve represents the evolution of the tip b of the non cohesive crack for the partially non cohesive branch. Here the curves correspond to the case where dc/` = 0.1. That means that the smaller the material length dc, the weaker the stabilizing effect of the stress gradient. In the same manner, for the partially non cohesive branch, the smaller the material length dc, the more accentuated the snap-back and the larger the size of the loop. At fixed dc, for a given material, one can see the influence of the intensity of the stress gradient by comparing on Figure 6 the Dugdale branches associated with different values of `. Let us recall that the higher the stress gradient, the smaller the length `, the case of a uniform stress field corresponding to ` = +∞. Accordingly, the higher the gradient, the greater the fully cohesive branch, the smaller the loop of the partially non cohesive branch and the smaller the final length of the crack. ϵ=0.01 ϵ=0.1 ϵ=0.25 ϵ=0.5 ϵ=1 ϵ=0.1 ϵ=0.25 ϵ=0.5 ϵ=1 Fig. 6. Dependence of crack evolution branches on dc for fixed `, and on ` for fixed dc. Fig. 5. Typical graphs of the three branches in the diagram (t, a). The gr y curve repr - sents the evolution of the tip b of the non co- hesive crack for the partially non cohesive branch. Here the cu ves correspond to the case where dc/` = 0.1 That means that the smaller the material length dc, the weaker the stabilizing effect of the stress gradient. In the same manner, for the partially non cohesive branch, the smaller the material length dc, the more accentuated the snap-back and the larger the size of the loop. At fixed dc, for a given material, one can see the influence of the intensity of the stress gradient by comparing on Fig. 6 the Dugdale branches associated with different values of `. Let us recall that the higher the stress gradient, the smaller the length `, the case of a uniform stress field corresponding to ` = +∞. Accord- ingly, the higher the gradient, the grea er the fully cohesive branch, the smaller the loop of the partially non cohesive branch and the small r the final le gth of the cr ck. A generalisation to cohesive cracks evolution under effects of non-uniform stress field 15 a or b Fig. 5. Typical graphs of the three branches in the diagram (t, a). The gray curve represents the evolution of the tip b of the non cohesive crack for the partially non cohesive branch. Here the curves correspond to the case where dc/` = 0.1. That means that the smaller the material length dc, the weaker the stabilizing effect of the stress gradient. In the same manner, for the partially non cohesive branch, the smaller the material length dc, the more accentuated the snap-back and the larger the size of the loop. At fixed dc, for a given material, one can see the influence of the intensity of the stress gradient by comparing on Figure 6 the Dugd le branches as ociated wi h diffe ent values of `. Let us recall that the higher the stress gradient, the smaller the length `, the case of a uniform stress field corresponding to ` = +∞. Accordingly, the higher the gradient, the greater the fully cohesive br nch, t e smaller e loop of he parti lly non cohesive branch and the smaller the final length of the crack. ϵ=0.01 ϵ=0.1 ϵ=0.25 ϵ=0.5 ϵ=1 ϵ=0.1 ϵ=0.25 ϵ=0.5 ϵ=1 Fig. 6. Dependence of crack evolution branches on dc for fixed `, and on ` for fixed dc. Fig. 6. Dependence of crack evolution branches on dc f r fixed `, and on ` for fixe dc 16 T.-H. Pham et al. On the other hand, the response under monotonically increasing loading is shown in Figure 7. Accordingly, the elastic solution est valid as long as the loading t is in the interval (0, te), then the fully cohesive crack nucleates and propagates continuously for t ∈ (te, ti). Finally, the crack length must jump at the instant ti corresponding to the apparition of non cohesive zone at the center of crack. If on neglects the inertial effects, at the instant ti the crack length jumps from the value ai to the value a ∗ i on the upper part of partially non cohesive branch. Fig. 7. Crack length evolution under monotonically increasing loading for dc/` = 0.1. Jump of crack length at t = ti. 3.2.3. Sensibility to the imperfections It would seem that the shape of the loop and the snap-back part of the partially non cohesive branch do not play any role in the crack propagation under monotonic loading. However, the loop can be observed and even that it plays a fundamental role in presence of imperfections. Indeed, we consider the case where the imperfection corresponds to a preexisting non cohesive crack along the x2 = 0, centered at O and of half-length a0 < `. Accordingly, the elastic response is no more regular, but the stress is singular at the tips ±a0 as soon as a loading is applied. There exists no more an elastic branch, but a cohesive zone must nucleate ahead the tips ±a0 as soon as t > 0 with a length a > a0 such that the singularity vanishes at the tips ±a. Consequently, the relationship between a, b and t in order that the singularity vanishes remains given by (37). Besides, the crack opening at x1 = ±b, namely Ju[t, a, b]2K(b), is always given by (38). Two relationships allows us to study the crack evolution with an initial imperfection under monotonically increasing loading. Specifically, the evolution can be divided into the two or three following parts, according to the value of a0 Fig. 7. Crack length evolution under mono- tonically increasing loading for dc/` = 0.1. Jump of crack length at t = ti On the other hand, the response under monotonically increasing loading is shown in Fig. 7. Accordingly, the elastic solution est valid as long as the loading t is in the interval (0, te), then the fully cohesive crack nucleates and propagates continuously for t ∈ (te, ti). Fi- nally, the crack length must jump at the instant ti corresponding to the apparition of non cohe- sive zone at the center of crack. If on neglects the inertial effects, at the instant ti the crack length jumps from the value ai to the value a∗i on the upper part of partially non cohesive branch. 364 Tuan-Hiep Pham, Je´roˆme Laverne, Jean-Jacques Marigo 3.2.3. Sensibility to the imperfections It would seem that the shape of the loop and the snap-back part of the partially non cohesive branch do not play any role in the crack propagation under monotonic loading. However, the loop can be observed and even that it plays a fundamental role in presence of imperfections. Indeed, we consider the case where the imperfection corresponds to a preexisting non cohesive crack along the x2 = 0, centered at O and of half-length a0 < `. Accordingly, the elastic response is no more regular, but the stress is singular at the tips ±a0 as soon as a loading is applied. There exists no more an elastic branch, but a cohesive zone must nucleate ahead the tips ±a0 as soon as t > 0 with a length a > a0 such that the singularity vanishes at the tips ±a. Consequently, the relationship between a, b and t in order that the singularity vanishes remains given by (37). Besides, the crack opening at x1 = ±b, namely [[u[t, a, b]2]](b), is always given by (38). Two relationships allows us to study the crack evolution with an initial imperfection under monotonically increasing loading. Specifically, the evolution can be divided into the two or three following parts, according to the value of a0: (1) Cohesive phase: Growth of two symmetric purely cohesive zones, the non cohesive zone tips remaining at ±a0. The relationship between a and t is given by the condition KI[t, a, a0] = 0 and hence (37) with b = a0 leads to t te = 2 pi arccos a0a( 1− a2 `2 − a4 `4 ) . (43) In consequence, a is a monotonically increasing function of t starting from a0 at t = 0. That allows us to define the cohesive branch associated with the initial crack length a0 in the diagram (t, a). In addition, for a et t satisfying the relationship (43), the crack opening with b = a0 expressed in (38) becomes [[u[t, a, a0]2]] (a0) = 8 ( 1− u2) σc E [ a0 pi ln a a0 + 1 3 t te ( a2 − a20 )3/2 `2 + t te 1 `4 ( 2a2 3 ( a2 − a20 )3/2 − 4 15 ( a2 − a20 )5/2)] . This opening reaches the critical value δc when the triple (a, a0, t) satisfies the two con- ditions (37) and (39). Consequently, the triple is the point of the partially non cohesive branch of the perfect case which corresponds to b = a0. The associated parameter α is given by the equation b e (α0) ` = a0, its uniqueness being ensured by the monotonicity of the function b e (α0). In other words the cohesive branch will finish when it intersects the loop of the perfect case. In conclu- sion, the cohesive branch starts from (0, a0) and finished at (t e (α0)te, ae(α0)`). (2) Possible jump of the crack length: Brutal propagation of the crack if the cohesive branch intersects the lower part of the loop of the perfect case. The intersection point between the purely cohesive branch and the loop of the perfect case depends on a0. If a0 is small enough, the intersection point is lower than the limit point of the loop, i.e. ae(α0) < al , A generalisation to cohesive cracks evolution under effects of non-uniform stress field 365 the crack length must jump and the crack evolution is discontinuous after the purely cohesive branch. On the other hand, if a0 is large enough, the intersection point is at or above the limit point of the loop, i.e.ae(α0) ≥ al , the evolution can continuously follow that part of the curve in the sense of increasing time since the crack length increases and no jump is necessary. (3) The continuous growth of a partially non cohesive crack. Once the upper part of the loop has been reached, which can occur after a jump, the crack evolution simply follows that upper part of the loop in the direction of increasing time and finally the crack length will tends to ac when t goes to infinity as in the perfect case. The system will finally forget its initial imperfection. All these results can be seen on Fig. 8 where are considered five cases of imperfection size. The first three, which correspond to a small initial crack length, lead to a jump whereas the last two, corresponding to a sufficiently large initial crack length, give rise to a continuous growth of the crack. Of course, the critical length of the initial crack above which the evolution is continuous depends both on ` and dc. In any case, one sees the fundamental role played by the loop of the perfect system. 18 T.-H. Pham et al. whereas the last two, corresponding to a sufficiently large initial crack length, give rise to a continuous growth of the crack. Of course, the critical length of the initial crack above which the evolution is continuous depends both on ` and dc. In any case, one sees the fundamental role played by the loop of the perfect system. 0 te ti t al a ∗i ` a dc /2 0 dc 2 dc 4.65 dc 6.5 dc dc ac Fig. 8. Evolution of the crack length a under a monotonically increasing loading for differ- ent lengths a0 of the centered initial non cohesive crack. Here, dc/` = 0.1 and a0 = 0, dc/2, dc, 2dc, 4.65 dc, 6dc. 4. Barenblatt’s cohesive crack evolution The aim of this section is to generalise the previous results by supposing that the crack evolution is governed by Barenblatt’s cohesive model. The model assumes that the normal cohesive stress σnn is no more a constant but a continuous function of displace- ment jump Ju2K. This assumption leads to integro-differential equations in the resolution of the crack evolution problem where a semi-analytical method using Chebychev polyno- mials developed in [10] is necessary. We recall some principal formulations of Barenblatt’s cohesive model, then formulate the generalised crack evolution problem in two stages: the first one of purely cohesive crack and the second one of partially non-cohesive crack. Some important results of the dimensionless resolution are presented for a special case of linear Barenblatt’s cohesive model. 4.1. Barenblatt’s model of crack opening This model is based on the principal assumption that the normal stress σnn giving the interaction between the crack lips is a continuous, monotonically decreasing function of displacement jump along the cohesive zone, i.e. JunK ≥ δc, while the Dugdale’s model Fig. 8. Evolution of the crack length a under a monotonically increasing loading for differ- ent lengths a0 of the centered initial non cohesive crack. Here, dc/` = 0.1 and a0 = 0, dc/2, dc, 2dc, 4.65dc, 6dc 4. BARENBLATT’S COHESIVE CRACK EVOLUTION The aim of this section is to generalise the previous results by supposing that the crack evolution is governed by Barenblatt’s cohesive model. The m del assumes that the normal cohesive stress σnn is no more a constant but a continuous function of displace- ment jump [[u2]]. This assumption leads to integro-differential equations in the resolu- tion of the crack evolution problem where a semi-analytical method using Chebychev polynomials developed i [15] is necessary. We recall some principal formulations of 366 Tuan-Hiep Pham, Je´roˆme Laverne, Jean-Jacques Marigo Barenblatt’s cohesive model, then formulate the generalised crack evolution problem in two stages: the first one of purely cohesive crack and the second one of partially non- cohesive crack. Some important results of the dimensionless resolution are presented for a special case of linear Barenblatt’s cohesive model. 4.1. Barenblatt’s model of crack opening This model is based on the principal assumption that the normal stress σnn giving the interaction between the crack lips is a continuous, monotonically decreasing function of displacement jump along the cohesive zone, i.e. [[un]] ≥ δc, while the Dugdale’s model assumes this physical quantity is constant in this zone (see (9)). Specifically, we have σnn  ≤ σc if [[un]] = 0= σc (1− f ([[un]])) if 0 < [[un]] < δc = 0 if [[un]] > δc (44) where f is a monotonically increasing, positive function of [[u2]] which satisfies the fol- lowing conditions f(0) = 0, f ([[un]]) = 1∀ [[un]] ≥ δc, f′ ([[un]]) ≥ 0, ∀ [[un]] ≥ 0. (45) Consequently, the surface energy density reads as Φ ([[un]]) =  ∞+ if [[un]] = 0 σc [ [[un]]− ∫ [[un]] 0 f(s)ds ] if 0 < [[un]] < δc Gc if [[un]] > δc (46) The relationship between the critical cohesive stress σc and the critical energy release rate Gc can be written as follows Gc = σc [ δc − ∫ δc 0 f(s)ds ] . Fig. 9 shows the Barenblatt’s surface energy density and the cohesive stress in function of the jump displacement. As in the Dugdale’s case, the cracks are generally divided into two zones: the cohesive zone and the non cohesive zone. A generalisation to ohesive cracks evolution under effects f n n-u iform stress field 19 assumes this physical quantit is st t in t is zone (se (9)). Specifically, we have σnn  ≤ σc if JunK = 0 = σc ( 1− f(JunK)) if 0 < JunK < δc = 0 if JunK > δc (44) where f is a monotonically increasing, positive function of Ju2K which satisfies the following conditions f(0) = 0, f(JunK) = 1 ∀JunK ≥ δc, f′(JunK) ≥ 0 ∀JunK ≥ 0 (45) Consequently, the surface ener sity reads as : Φ(JunK) =  ∞+ if JunK < 0 σc [JunK J K 0 ds ] if 0 ≤ JunK < δc Gc if JunK ≥ δc (46) The relationship between the critical cohesive stress σc and the critical energy release rate Gc can be written as follows Gc = c [ δc δc 0 (s) s ] The figure 9 shows the Barenblatt’s surface energy de sity and the cohesive stress in function of the ju p displacement. As in the Dugdal ’s case, th cracks are generally divided into two zones: the cohesive zone and the non cohesive zone. JunK Φ δc Gc JunK σnn δc σc Fig. 9. Barenblatt’s the surface energy density and the cohesive stress in function of jump dis- placement Recalling that the problem settings and the assumptions presented in Section 2 remain the same. In addition, the three types of crack state in Barenblatt’s case are always shown in the figure 3: no crack, full cohesive crack and partially non cohesive crack. One obtains the same elastic response as in Dugdale’s case where the normal stress distribution along Γ is given by (6) and the time corresponding to the nucleation of cohesive Fig. 9. Barenblatt’s the surface energy de sity and the cohesive stress in function of jump displacement A generalisation to cohesive cracks evolution under effects of non-uniform stress field 367 Recalling that the problem settings and the assumptions presented in Section 2 re- main the same. In addition, the three types of crack state in Barenblatt’s case are always shown in Fig. 3: no crack, full cohesive crack and partially non cohesive crack. One ob- tains the same elastic response as in Dugdale’s case where the normal stress distribution along Γ is given by (6) and the time corresponding to the nucleation of cohesive crack is still te. The two scale approach can always be used to study Barenblatt’s crack evolution by assuming the hierarchy of the length, i.e. d  L, ` . L. The equations are formu- lated in the Barenblatt’s general case, then a dimensionless study will be detailed for a particular case of linear cohesive law. 4.2. Formulations of two-scale approach for Barenblatt’s general cohesive model 4.2.1. Fully cohesive crack Considering a fully cohesive with length 2a > 0 at time t > 0. For given a and t, we define the associated displacement and stress fields as the unique solution, denoted (u[t, a],σ[t, a]), of the following linear elastic problem posed on the cracked body with uniform cohesive forces on the crack lips divσ[t, a] = 0 in Ω\([−a, a]× {0}) σ[t, a] = λ tr(ε(u[t, a]))I+ 2µε(u[t, a]) in Ω\([−a, a]× {0}) u[t, a] = tU on ∂DΩ σ[t, a]n = tF on ∂NΩ σ[t, a]e2 = σc (1− f ([[u(t)2]] (x1))) e2 on [−a, a]× {0} (47) In the same way as Dugdale’s case, one introduces in (47) the gaps between the solution with the elastic fields, i.e. u[t, a] = u[t, a]− tuel, σ[t, a] = σ[t, a]− tσel, where σ[t, a](x) should tend to 0 when ‖x‖ becomes large by comparison with a. In addition, the gap of the normal stress verifies σ[t, a]22(x1) = σc(1− f([[u(t)2]]))− tΣ(x1) where Σ(x1) is given by (6). Consequently, we obtain σ[t, a]22 (x1) = ( 1− t te ) σc − f([[u(t)2]](x1)) σc + 2 t te x21 `2 σc + o ( x21 ) , |x1| < a. By expanding the normal stress distribution up to the second order, one can write the problem giving the gaps (u,σ) in a neighborhood of the origin as follows divσ[t, a] = 0 in R2\([−a, a]× {0}) σ[t, a] = λ tr(ε(u[t, a]))I+ 2µε(u[t, a]) in R2\([−a, a]× {0}) σ[t, a](x)→ 0 when ‖x‖ → ∞ σ[t, a] (x1, 0) e2 = ( 1− f([[u(t)2]](x1))− t te + 2 t te x21 `2 ) σce2 when x1 ∈ (−a, a) (48) This problem can always be solved with the method of complex potentials developed by [19]. Here, the method is applied with the following normal stress distribution: T (x1) = ( 1− f ([[u(t)2]] (x1))− t te + 2 t te x21 `2 ) σc. (49) 368 Tuan-Hiep Pham, Je´roˆme Laverne, Jean-Jacques Marigo By integrating, the stress intensity factor at the crack tips reads as KI[t, a] = σc √ pia (( 1− a 2 `2 ) t te − 1 ) + σc√ pia ∫ a −a f([[u(t)2]](s)) √ a + s a− sds. The conditionKI[t, a] = 0 gives us an implicit relationship between the time t, the position of the non cohesive zone tips a and the fully cohesive crack opening: pia ( 1− ( 1− a 2 `2 ) t te ) = ∫ a −a f([[u(t)2]](s)) √ a + s a− sds. (50) This equation is valid as long as the crack opening at x1 = 0 if less than δc. The com- plex potential jump ϕ(z) through the fully cohesive crack must be considered to study the crack opening. By using the normal stress distribution (49) in the calculation, the derivative of the complex potential can be expressed as follows ϕ′(z) = σc 2 ( t te − 1 )( z√ z2 − a2 − 1 ) + σc 2 t te ( −2z √ z2 − a2 `2 + 2z2 `2 − a 2z `2 √ z2 − a2 ) − σcχ(z)2ipi ∫ Su f ([[u(t)2]](ζ)) χ (ζ+) (ζ − z)dζ. (51) In this formulation, Su = [−a, a]×{0} denotes the set of discontinuity points of displace- ment field u and χ is a complex function defined on C\Su χ :=  C\Su → Cz 7→ χ(z) = 1√ z2 − a2 (52) We deduce the complex potential jump as follows [[ϕ]](x1) = − σc ( t te − 1 ) ix1√ a2 − x21 + σc t te i x1 ( 2x21 − a2 ) `2 √ a2 − x21 + σc i pi √ a2 − x21 ∫ a −a f([[u(t)2]](s)) √ a2 − s2 s− x1 ds. By using the relationship between the jumps of the complex potentials and of the normal displacement, this expression leads to [[u(t)2]] ′(x1) =− 4 ( 1− ν2) σc E ( t te − 1 ) x1√ a2 − x21 + 4 ( 1− ν2) σc E t `e x1 ( 2x21 − a2 ) `2 √ a2 − x21 + 4 ( 1− ν2) pi √ a2 − x21 σc E ∫ a −a f ([[u(t)2]](s)) √ a2 − s2 s− x1 ds. (53) A generalisation to cohesive cracks evolution under effects of non-uniform stress field 369 The study of the fully cohesive crack evolution, i.e. the crack length a(t) and the crack opening [[u(t)2]] in function of the time t, consists in solving the two integro-differential equations (50) and (53). 4.2.2. Partially non cohesive crack Let us consider now the partially non cohesive crack evolution at time t whose non cohesive length is 2b and whose cohesive zone tips are at ±a. For given (a, b, t) with 0 0, we define the associated displacement and stress fields as the unique solution, denoted (u[t, a, b],σ[t, a, b]), of a linear elastic problem posed on the cracked body with non uniform cohesive forces on the crack lips. By using the same perturba- tion method as the fully cohesive crack calculation, one considers the following problem giving the gap fields in a neighborhood of the origin O divσ[t, a, b] = 0 in R2\([−a, a]× {0}) σ[t, a, b] = λ tr(ε[t, a, b]))I+ 2µε(u[t, a, b]) in R2\([−a, a]× {0}) σ[t, a, b](x)→ 0 when |x| → ∞ σ[t, a, b] (x1, 0) e2 = T (x1) e2 sur (−a, a)× {0} (54) where the cohesive force distribution is given by T (x1) =  ( − t te + 2 t te x21 `2 ) σc if |x1| < b( 1− f ([[u(t)2]] (x1))− t te + 2 t te x21 `2 ) σc if b < |x1| < a (55) At a given time t > 0, a and b must satisfy the two necessary conditions (see Propositions 2.2 and 2.1) { KI[t, a, b] = 0 [[u(t, a, b)2]](b) = δc By using the normal stress distribution (55) in the calculation of the stress intensity factor, we obtain K1[t, a, b] = σc √ pia (( 1− a 2 `2 ) t te − 1+ 2 pi arcsin b a ) + σc√ pia (∫ a −a f ([[u(t)2]](s)) √ a + s a− sds− ∫ b −b f ([[u(t)2]](s)) √ a + s a− sds. ) (56) The condition KI[t, a, b] = 0 gives the first implicit relationship between (a, b, t) pia ( 2 pi arccos b a − ( 1− a 2 `2 ) t te ) = a∫ −a f([[u(t)2]](x1)) √ a + s a− sds− b∫ −b f([[u(t)2]](x1)) √ a + s a− sds. (57) We consider now the normal displacement jump [[u(t, a, b)]]. Without detailing the in- tegral steps, we write directly in the following equation the implicit expression of the 370 Tuan-Hiep Pham, Je´roˆme Laverne, Jean-Jacques Marigo complex potential jump [[ϕ]]′(x1) [[ϕ]]′ (x1) =− iσcx1√ a2 − x21 ( t te ( 1− a 2 `2 ) − 2 pi arccos b a ) − 2iσc  x1 √ a2 − x21 `2 t te + 1 pi arctanh  b √ a2 − x21 x1 √ a2 − b2  + iσc pi √ a2 − x21  a∫ −a f ([[u(t)2]] (x1)) √ a2−s2 s− x1 ds− b∫ −b f ([[u(t)2]] (x1)) √ a2−s2 s− x1 ds  . (58) Finally, the relationship between thejumps of the normal displacement and of the com- plex potential gives us the following integro-differential equation for u(t)2 [[u(t)2]] ′ (x1) =− 4 ( 1− ν2) σcx1 E √ a2 − x21 ( t te ( 1− a 2 `2 ) − 2 pi arccos b a ) − 8 (1− ν2) σc E  x1 √ a2 − x21 `2 t te + 1 pi arctanh  b √ a2 − x21 x1 √ a2 − b2  + 4 ( 1−ν2) σc piE √ a2−x21  a∫ −a f([[u(t)2]](x1)) √ a2−s2 s−x1 ds− b∫ −b f([[u(t)2]](x1)) √ a2−s2 s−x1 ds  . (59) The dimensionless semi-analytical solution of the integro-differential equations are con- sidered in the simple case of a linear Barenblatt’s cohesive law. 4.3. Particular case of a linear Barenblatt’s cohesive law One assumes for this particular case that the cohesive normal stress σnn is a linear decreasing function of the normal displacement jump [[un]] as long as 0 < [[un]] < δc σnn  ≤ σc if [[un]] = 0 = σc ( 1− [[un]] δc ) if 0 < [[un]] < δc = 0 if [[un]] > δc We deduce the expression of the surface energy density in function of the normal dis- placement jump Φ ([[un]]) =  ∞+ if [[un]] < 0 σc [[un]] ( 1− [[un]] 2δc ) if 0 ≤ [[un]] < δc Gc if [[un]] > δc A generalisation to cohesive cracks evolution under effects of non-uniform stress field 371 The Barenblatt’s crack evolution in two stages (first fully cohesive, then partially non cohesive) can be studied now by using this simplified cohesive law. 4.3.1. Fully cohesive crack The relationship (50) becomes pia ( 1− ( 1− a 2 `2 ) t te ) = ∫ a −a [[u(t)2]](s) δc √ a + s a− sds. (60) The expression of the derivative of the displacement jump (53) can be rewritten as follows [[u(t)2]] ′ (x1) =− 4 ( 1− ν2) σc E t te ( 1− te t ) x1√ a2 − x21 + 4 ( 1− ν2) σc E t te x1 ( 2x21 − a2 ) `2 √ a2 − x21 + 4 ( 1− ν2) pi √ a2 − x21 σc E ∫ a −a [[u(t)2]](s) δc √ a2 − s2 s− x1 ds. (61) The crack tip position a(t) and the crack opening [[u(t)2]] are solutions of the integro- differential system of equations (60) and (61) in the case of the linear Barenblatt’s cohesive law. Let us introduce the following dimensionless variables x˜1 := x1 a ∈ [−1, 1], δt := [[u(t)2]]4λ pi2 a2 `2 t te δc , (62) where λ := pi2 4 a dc is a dimensionless variable proportional to crack length. By using these variables, Eq. (60) can be simplified as follows pi ( 1− ` 2 a2 ( 1− te t )) = 4λ pi2 ∫ 1 −1 δt(s˜) √ 1+ s˜ 1− s˜ds˜. (63) We obtain also a dimensionless form of the integro-differential equation (61) 2 pi δ ′ t (x˜1) = x˜1√ 1− x˜21 [ − ` 2 a2 ( 1− te t ) + ( 2x˜21 − 1 )] + 1 pi √ 1− x˜21 4λ pi2 ∫ 1 −1 δt(s˜) √ 1− s˜2 s˜− x˜1 ds˜. (64) This integro-differential equation can be solved semi-analytically using Chebychev polynomials developed in [15]. The expression of the dimensionless displacement jump δt of the Barenblatt’s fully cohesive crack depends on the crack length a which increases with the loading t. Without detailing the calculation, the dimensionless displacement 372 Tuan-Hiep Pham, Je´roˆme Laverne, Jean-Jacques Marigo jump δt of the Barenblatt’s fully cohesive crack corresponding to λ = 0, 0.2, 1, 2 in com- parison with Dugdale’s case is shown in Fig. 10. A generalisation to cohesive cracks evolution under effects of non-uniform stress field 25 consequence, the results obtained in this phase of Barenblatt’s fully cohesive crack can be well approximated by Dugdale’s results. This good approximation is confirmed by the figure 10. • Barenblatt’s fully cohesive crack of large length. When the crack length a is at the same order or much greater than dc, i.e. when λ & 1, the curves of the dimensionless displacement jump δ¯t corresponding to different values of λ are shown in the figure 10. In particular, δ¯t(x˜1) is a decreasing function of |x˜1|, is negligible for x˜1 = ±1 and reaches its maximal value (which is always greater than the maximum pi/3 of Dugdale’s crack) for x˜1 = 0. Besides, the loading ti corresponding to the end of the fully cohesive phase is now given by δ¯t∗i (0)a 3(t∗i )t ∗ i = dc` 2te λ=0 λ=0,2 λ=1 λ=2 Fig. 10. The dimensionless displacement jump δ¯t of the Barenblatt’s fully cohesive crack corre- sponding to λ = 0, 0.2, 1, 2 in comparison with Dugdale’s case (black curve) 4.3.2. Partially non cohesive crack By using the linear Barenblatt’s cohesive law assumptions, the implicit relationship (57) between (ab, b, t) becomes pia ( 2 pi arccos b a − ( 1− a 2 `2 ) t te ) = ∫ a −a Ju(t)2K(s) δc √ a+ s a− sds− ∫ b −b Ju(t)2K(s) δc √ a+ s a− sds (66) Fig. 10. The dimensionless displace ent j δt of the Barenblatt’s fully cohesive crack corre- sp nding to λ = 0, 0.2, 1, 2 in comparison with Dugd le’s cas (black curve) Let us now rewrite the results of Dugdale’s crack evolution in dimensionless form and compare with the Barenblatt’s crack corresponding to different crack lengths. • Dimensionless calculations of Dugdale’s fully crack evolution. The dimensionless vari- ables defined in (62) allow us to express the dimensionless normal displacement of Dug- dale’s fully cohesive crack as follows δt (x˜1) = pi 3 ( 1− x˜21 )3/2 . (65) Consequently, the formulation of the dimensionless normal displacement remains al- ways the same during the Dugdale’s fully cohesive crack evolution. In particular, δt(x˜1) is decreasing with x˜1, equals to 0 for x˜1 = ±1 and reaches its maximal values pi/3 for x˜1 = 0. Besides, the non cohesive crack appears when [[[u(t)2]](0) reaches the critical value δc. The loading ti corresponding to the end of fully cohesive phase can be calcu- lated by the following equation pi 3 a3 (ti) ti = dc`2te. • Barenblatt’s fully cohesive crack of very small length. During this phase, the crack length a is very close to the material characteristic length dc, i.e. the dimensionless vari- able λ is very close to 0. Because of a very small fully cohesive crack opening, the cohesive stress along the crack is quasi-uniform and very close to the critical value σc. In conse- quence, the results obtained in this phase of Barenblatt’s fully cohesive crack can be well approximated by Dugdale’s results. This good approximation is confirmed by Fig. 10. A generalisation to cohesive cracks evolution under effects of non-uniform stress field 373 • Barenblatt’s fully cohesive crack of large length. When the crack length a is at the same order or much greater than dc, i.e. when λ & 1, the curves of the dimensionless displacement jump δt corresponding to different values of λ are shown in Fig. 10. In particular, δt(x˜1) is a decreasing function of |x˜1|, is negligible for x˜1 = ±1 and reaches its maximal value (which is always greater than the maximum pi/3 of Dugdale’s crack) for x˜1 = 0. Besides, the loading ti corresponding to the end of the fully cohesive phase is now given by δt∗i (0)a 3 (t∗i ) t ∗ i = dc` 2te. 4.3.2. Partially non cohesive crack By using the linear Barenblatt’s cohesive law assumptions, the implicit relationship (57) between (ab, b, t) becomes pia ( 2 pi arccos b a − ( 1− a 2 `2 ) t te ) = ∫ a −a [[u(t)2]](s) δc √ a + s a− sds− ∫ b −b [[u(t)2]](s) δc √ a + s a− sds. (66) The integro-differential equation giving the normal displacement jump can be rewritten as follows [[u(t)2]] ′(x1) =− 4 ( 1− ν2) σcx1 E √ a2 − x21 ( t te ( 1− a 2 `2 ) − 2 pi arccos b a ) − 8 (1− ν2) σc E  x1 √ a2 − x21 `2 t te + 1 pi arctanh  b √ a2 − x21 x1 √ a2 − b2  + 4 ( 1− ν2) σc piE √ a2 − x21 (∫ a −a [[u(t)2]](s) δc √ a2 − s2 s− x1 ds− ∫ b −b [[u(t)2]](s) δc √ a2 − s2 s− x1 ds ) . (67) By injecting the dimensionless variables defined in (62) and setting α = b a ∈ (0, 1) in (66), we obtain pi3 4λ ( 2 pi `2 a2 te t arccos α+ 1− ` 2 a2 ) = ∫ 1 −1 δt(s˜) √ 1+ s˜ 1− s˜ds˜− ∫ α −α δt(s˜) √ 1+ s˜ 1− s˜ds˜. (68) The integro-differential equation (67) can be also rewritten under a dimensionless form 2 pi δ ′ t (x˜1) = 2 pi `2 a2 te t arccos α+ 1− ` 2 a2 − 2 x˜1√a2 − x˜21 + 1pi `2a2 tet arctanh  α √ 1− x˜21 x˜1 √ 1− α2  + 4λ pi3 √ 1− x˜21 (∫ 1 −1 δt(s˜) √ 1− s˜2 s˜− x˜1 ds˜− ∫ α −α δt(s˜) √ 1− s˜2 s˜− x˜1 ds˜ ) . (69) 374 Tuan-Hiep Pham, Je´roˆme Laverne, Jean-Jacques Marigo 5. CONCLUSION The present paper considers the problem of the nucleation and the propagation of a cohesive crack at the tip of a notch in two-dimensional elastic structures using Dugdale’s or Barenblatt’s cohesive force models where the stress field associated with a pure elastic response is assumed to be smooth and bounded, but nonuniform. Further it is supposed that the material characteristic length associated with the cohesive model is small by comparison to the dimension of the body. The crack evolution can be considered in two stages: the first one where all the crack is submitted to cohesive forces, followed by a second one where a non cohesive part appears. 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