a numerical solution technique for a class of integro-differential equations in elastodynamic crack...

Computer methods In applied

mechanics and engineerlng

ELSEWIER Comput. Methods Appl. Mech. Engrg. 162 (1998) 19-48

A numerical solution technique for a class of integro-differential equations in elastodynamic crack propagation problems

Francesco Costanzoa**, Jay R. Waltonb “Engineering Science & Mechanics Department, The Pennsylvania State University, 227 Hammond Bldg., University Park

PA 16SO2-1401, USA

‘Texas A&M University, College Station, TX 77843-3368. USA

Received 21 February 1997; revised 3 November 1997

Abstract

A strategy for the numerical solution of a type of integro-differential equation arising from the analysis of dynamic crack propagation

problems is presented. Specifically, the problem of interest is that of a mode III crack dynamically propagating in a homogeneous linear

elastic medium while the crack tip consists of a nonlinear rate dependent cohesive (or failure) zone. The mode of propagation is general, that is, not restricted to steady state or other special regimes. Furthermore, the presented solution technique is in no way restricted to mode III

problems. 0 1998 Elsevier Science S.A. All rights reserved.

1. Introduction

The key to understanding some of the most important problems in the dynamic failure of solids is in the characterization of the material’s microstructure in the vicinity of the crack tips and on the fracture plane as a whole

(1)

(2)

(3)

Experimental evidence [l-7] shows that in a variety of amorphous polymers cracks, at the onset of fracture, seem to jump from the rest configuration to a steady state motion without a transitory smooth acceleration phase; once the crack starts to propagate, its velocity remains essentially constant, ranging between 40% and 60% of the Rayleigh wave speed; as the crack approaches its maximum value, its velocity is believed to oscillate rapidly and post-mortem surfaces display a periodic microstructure [6,7].

All these phenomena do not find any explanation in traditional fracture theories that regard the crack tip as sharp (see e.g. ]8,9], see also the discussion by Glennie and Willis [lo]).

The present study is part of a larger research effort [ 111 whose goal is that of addressing the issues briefly mentioned above from the viewpoint of continuum mechanics. In particular, the main objective is to simulate microstructural effects on the fracture plane using so-called cohesive zone models, that is, by modeling the crack tip as a thin finite size region of damaged material, the latter being characterized via a set of convenient constitutive equations (see e.g. [ 12-171). The latter are, in general, nonlinear and rate dependent.

The boundary/initial value problems (B /IVPs) arising from the implementation of the aforementioned models are often very complex and require the use of numerical methods when an (approximate) solution is desired. Unfortunately, most of the numerical techniques traditionally employed in the solution of dynamic

* Corresponding author.

0045.7825/98/$19.00 0 1998 Elsevier Science S.A. All rights reserved. PII: SOO45-7825(97)00328-9

20 F. Costanzo, J.R. Walton I Comput. Methods Appl. Mech. Engrg. 162 (1998) 19-48

fracture problems require that the system at hand be discretized. This discretization process often appears to endow the system with a fictitious microstructure which, in turn, may render difficult the recognition of physically-based oscillatory phenomena as well as phenomena leading to the formation of microstructural patterns of the type suggested by experimental evidence (see e.g. [ 18-201). For this reason, the present authors have sought alternative solution methods capable of eliminating and/or reducing the effects due to numerical aberration. In particular, an approach based on complex variables and integral transforms [21-23,111 has been followed leading to the reformulation of dynamic crack propagation problems in elastic (as well as viscoelastic) media as a system of integral-differential equations. This reformulation has required the development of a novel numerical technique for the derivation of approximate solutions. This paper is devoted to the detailed presentation of the aforementioned numerical strategy together with an evaluation of its accuracy. In particular, Section 2 presents the problem’s formulation, whereas Sections 3 and 4 are devoted to the description of the proposed solution scheme and a discussion of some numerical results, respectively.

2. Problem formulation

2.1. Governing equations, initial and boundary conditions

With reference to Fig. 1, the problem considered is that of an infinite, homogeneous, elastic, isotropic body containing a crack initially at rest lying along the line x, < 0, x2 = 0. The crack is assumed to propagate in mode III after the application of some external loads, as specified later. The crack tip is assumed to be a cohesive zone, that is, a region of the crack across which a system of cohesive forces (per unit area) can be transferred. Except, at most, at time t = 0, the cohesive zone will occupy a crack ‘segment’ of finite size. For the geometrically simple case considered herein, one can therefore fully characterize the C.Z. location and size, via two functions a,(t) and a(t), referred to as the material and mathematical crack tips, respectively. The cohesive forces translating the action of the C.Z. are characterized by an appropriate set of constitutive and evolution equations (for an in-depth analysis of cohesive zone models see [ 15,171). The C.Z. constitutive relations express said forces as functions of the cohesive zone opening displacement (CZOD) and its rate along the c.z., where the CZOD is defined as the jump discontinuity suffered by the displacement field across the crack surface within the cohesive zone.

Due to the symmetry of the propagation mode, the underlying equation of motion for the system is a single wave equation for the out of plane component of the displacement field r+(x,, x2, t) in the half space x2 > 0:

% = c2 Au,, (2.1)

where c is the elastic wave speed and A denotes the two-dimensional Laplacian. The relevant bulk constitutive equation is given by

Crack Tip: a,(t)

1 Mathematical

Fig. 1. Mode III crack dynamically propagating in a linear elastic body. The crack tip is a cohesive zone of finite size.

(2.2)

F. Costanzo, J.R. Walton / Comput. Methods Appl. Mech. Engrg. 162 (1998) 19-48

where p is the elastic shear modulus. An additional physical assumption regards the regime

crack propagation is assumed to take place under the following condition

OCa’,(f)<C) 0 6 a’(t) < c .

21

of propagation:

(2.3)

In other words, it is assumed that the C.Z. tip propagation rates will not exceed the material speed of sound and

that crack healing will not occur. The overall initial/boundary value problem will be given the following initial conditions:

u,(x,,x,,O)= li,(x,,x*,O) =O, a,(O) = a(0) = 0. (2.4)

The boundary conditions are

U&,‘O, t) = 0

q&, , 0, t) = Fe&, 7 t) 5,(X,? t>

given by

for x, > u(t) ,

for x, <a,(t),

for u,(t) <x1 < u(t) ,

asx,+m,

(2.5)

where the function F,(x,, t) (with x, <u,(t)) represents a general traction distribution applied along the crack

faces. a;.(~, , t) is instead the response function of the cohesive zone, that is, the law associating a certain value

of the opening displacement and, possibly, its rate at a given instant in time t and a given point x, in the failure

zone, with a corresponding value of the cohesive force acting at (x,, t). Hence’,

c@i, t) = ~C(u,(x,, 0, t), U&x,, 0, t)) 1 u,(t) cx, s u(t) )

where SC(u,, ti3) is a given constitutive response function.

(2.6)

REMARK 2.1. The first of Eqs. (25) can be considered ‘classical’ in the study of crack propagation (c.f.

[9,21,24]). Also, when employing a cohesive zone model, this boundary condition, together with the crack tip

singularity cancellation condition, provide the necessary information to determine the motion of the mathemati-

cal crack tip.

REMARK 2.2. The second of Eqs. (2.5) is simply a traction boundary condition. It should be noted that the

function a,,(~,, 0, t) (with u,(t) <x, <u(t)) is unknown as long as the function u~(x,, 0, t) is unknown.

Furthermore, the function F,(x,, t) (with x, < u,(t)) is given as data and is therefore known. However, for cases characterized by moving loads where the assigned traction distribution extends up to the material crack tip, the

support of Fe should also be considered unknown.

Two additional equations are necessary to complete the problem formulation. One of the required conditions

is the traditional cancellation of the classical square root stress singularity at the mathematical crack tip:

K(t)=0 VtE[W, (2.7)

where the function K(t) is the stress intensity factor at the mathematical crack tip. The second condition is a

fracture criterion. The fracture criterion chosen herein is the so-called ‘critical crack opening displacement’

criterion, which will be enforced in the following sense:

if u3(u,(t), 0, t) Cd,, then d,(t) = 0,

if u~(u,(~), 0, t) = d,, then d,(t) 3 0, (2.8)

where d,, is a material (constant) parameter. In other words, whenever a violation of inequality ~~(a,@), 0, t) <

’ Eq. (2.6) has been written so as to maintain consistency with the fact that the problem at hand is defined for x2 > 0. Note, however, that

in general the C.Z. constitutive equations should be expressed as functions of the crack opening displacement a,(~,, t), where the latter is

defined as the jump discontinuity suffered by the displacement field across the crack surface, that is, 6,(z,, t) = u,+(z,, t) - u,-(z, t), where z, is

the position vector identifying points on the crack surface, u,‘(z,, t) = lim,,, + u,(z, t 55, t). 1: is the unit normal vector orienting the crack surface and .$ E R (c.f. [15,17]).

22 F. Costanzo, J.R. Walton I Compur. Methods Appl. Mech. Engrg. 162 (1998) 19-48

d,, is detected, the latter is turned into an equation for the determination of the motion of the material crack tip

a,(t). To the authors’ knowledge, no analytical solutions to the above problem have ever been presented. Numerical

solutions to problems closely related to the one defined herein are available from the literature (see [18-201).

However, these solutions have been obtained using methods that require space-time discretizations that seem to

induce a ‘discrete lattice’ effect that is difficult, if not impossible, to separate from the system’s response as

dictated by the underlying governing equations. In the next section the problem is reformulated as a system of

integro-differential equations. The latter will then be solved with a new technique which, despite the use of a

discretization strategy, offers a better control of the ‘lattice effects’ mentioned above.

2.2. Problem reformulation as a system of integro-differential equations

The dynamic crack propagation problem defined in the previous section can be looked upon as a modification

of the following initial-boundary value problem (IBVP):

IBVP,:

ii, = c2 Au, ,

u&x,, $9 0) = u’&,, $7 0) = 0 >

z+(x,, 0, t) = 0 for x1 > a(t),

v~~(x,, 0, t> = Se(x,, t) for x, <a(t),

u,(x,,x,,t)+O asx,+m,

d(t) < c )

(2.9)

where gC(x,, t) and a(t) are regarded as arbitrarily assigned functions. The problem defined in the previous

section will be denoted IBVP,, in order to distinguish it from that defined in (2.9).

Walton and Herrmann [21] have derived the general solution to IBVP, by reformulating it as a crack surface

Riemann-Hilbert problem. More generally, their solution method can be extended to the study of unsteady

dynamic crack propagation in viscoelastic solids [22]. Furthermore, it allows one to focus attention on the

system’s response on the fracture plane, thereby turning a problem defined over (x,, x2, t) into one that is

formulated in terms of (x,, t) only. Thus, instead of solving IBVP, or IBVP,, as a whole, one can construct a

specialized problem wherein the principal unknown is the evolution of a field variable restricted to a

conveniently selected crack segment. This section presents a reformulation of IBVP,, as a system of

integro-differential equations whose principal unknown is the C.Z. evolution. For a detailed discussion of a

Riemann-Hilbert approach in the solution of dynamic crack propagation problems (see [11,21-231).

In addition to the fixed reference frame (x,, x2), it is useful to consider a coordinate system (x, y) moving with

the crack tip a(t):

x=x1 -a(t), y=x,. (2.10)

Let (+(x, t) represent the crack face stress field with respect to the coordinate system (x, t):

c7(x, t) = c(x, -a(t), t) = a;,(~,, 0, t) . (2.11)

Also, let

1

0 4(x,, t) =

for --co<x,<Oandforx,>a(t),

+(x1, 0, t) for 0 <x, <a(t), (2.12)

and

g-(x, t) = fl(x, t)H( -x) , (2.13)

where d,(x,, t) is the current crack face displacement along the portion of the x,-axis contained between the initial and the present position of the C.Z. tip at a(t), and H(x) is the Heaviside step function.

The general solution to IBVP, (cf. [21]) states that for 0 <x, < a(t),

F. Costanzo, J.R. Walton I Comput. Methods Appl. Mech. Engrg. 162 (1998) 19-48 23

where

The stress intensity factor at u(t) takes on the form

K(t) = f(c@,(c7) - &dt)), 9) & .

(2.14)

(2.15)

(2.16)

As mentioned earlier, IBVP,, will now be re-stated in a ‘reduced form’ that is, as a problem whose principal

unknown is the evolution of the c.z., where the latter is defined by

DEFINITION 2.1 (Cohesive zone evolution). Let t E [0, T], with T > 0 arbitrarily assigned. The following set

of three functions will be referred to as the cohesive zone evolution during the time interval [0, T]:

a,(t), u(t), gJx, t) 9

a,(t) s x s a(t) ) (2.17)

where the functions a,(t) and u(t) represent the trajectories of the physical and mathematical crack tip,

respectively, and the function a,(~, t) represents the cohesive force (per unit crack surface).

Let the time interval [0, T] be given. Then, it can be shown (c.f. Appendix A) that the cohesive zone evolution

(c.f. Definition 2.1) consistent with IBVP,, satisfies the following set of integro-differential equations:

Stress singularity cuncelation

i

f

F,(c(q - b,(t)), 0, 4) & + q(c(q - h)(t)), 4) dq

0 (t)) qq = 0 3

CZOD governing integro-differential equation

(2.18)

(2.19)

CZ constitutive equations

a,(x, t) = a,(d,(x, t), &(x, t)) ,

Fracture criterion

d&,(t), t) c d,, 3

where

d&, t) = d”,(x, t) + d;(x, t) ,

(2.20)

(2.21)

(2.22)

&?Jt) = t - a,(t)

b,,(t) = t + a,(t>

c ’ c ’

F,(x + c(t - 2q + r), r) sr .

(2.23)

(2.24)

Eq. (2.18) is derived from Eq. (2.16) by decomposing the function o- into parts, namely the applied traction

distribution F, defined for x < u,(t) and the unknown function a, defined for u,(t) <x <u(t). The limits of

integration for the second integral in Eq. (2.18) have been obtained by explicitly taking into account that the support of the function gC(x, t) is [u,,,(t), u(t)] (c.f. Appendix A). Eqs. (2.22) to (2.24) are simple definitions. Eq.


(2.22) is an additive decomposition of the total C.Z. opening displacement into a part directly related to the

applied loads via Eq. (2.14) and a part related to the effects of the cohesive zone. It is important to note that the formulation of the C.Z. evolution problem as stated via Eqs. (2.18) to (2.21) is exact. Before proceeding to a

description of the method used to solve the above system of equations, a few remarks are in order.

REMARK 2.3 (Equivalent alternative formulation). Eq. (2.19) is a double singular integral equation, where the

singular integrals in question are Abel operators. An alternative formulation can be provided wherein Eq. (2.19)

is stated so as to contain only single integral operators. This restatement is obtained from Eq. (2.19) by inverting

the first of the Abel operators appearing on the right-hand side of the equation. This yields the following result

(c.f. Appendix A):

Mutatis mutandis, the approximate solution method presented in Section 3 is equally applicable to integral-

differential equations (2.19) and (2.25).

REMARK 2.4 (Type of equations). Whether dealing with Eq. (2.19) or Eq. (2.25), that is, whether dealing with

a double or a single integral equation, it is important to notice that these equations contain Abel operators

characterized by unknown limits of integration. In other words, the unknown functions of this problem appear

both as integrands and as limits of integration. Furthermore, assuming that a particular point in the (x, t) plane is

given, say (2, i), one can see (especially when considering Eq. (2.25)) that the aforementioned integro-

differential operators are path integrals along the two characteristic directions associated to the wave equation:

Path, :

Path,: (i,tl) given,qE b, [ -‘(i+$].

(2.26)

This fact reflects the intrinsic two-dimensional character of the problem. Thus, the domain over which the

equations must be solved is the two-dimensional region of the (x, t) plane spanned by the two paths of

integration as t is varied from 0 to T and as x is varied from a,,,(t) to a(t). From a ‘physical’ viewpoint, one can

see that the solution domain is the region of the (x, t) plane swept by the cohesive zone as it evolves during the

time interval [0, T].

To the authors knowledge, integro-differential equations like the ones discussed here have not been studied

and they cannot be reduced to a more traditional type of Volterra integro-differential equation for which a solution method, whether analytical or numerical, is known (c.f. [25]). The present authors were unable to solve

Eqs. (2.19) and (2.25) analytically, however they succeeded in devising a viable numerical solution method, as

described in the next section.

3. Approximate solution method

3.1. Method overview

The proposed solution method, although conceptually simple, requires the discussion of a fairly large number of technical details. The purpose of this section is therefore that of providing an outline of the numerical solution scheme essential concepts so as to make the following sections more readable.

The solution of Eqs. (2.18)-(2.21) is represented by two functions of time, namely the C.Z. tips a(t) and a,(t), and two functions of time as well as space, that is, the crack opening displacement (COD) and cohesive force (Us) distributions. The free-boundary problem overall solution will be determined in a step-wise fashion, that is,

F. Costanzo, J.R. Walton / Comput. Methods Appl. Mech. Engrg. 162 (1998) 19-48 25

a given time interval of interest will be subdivided in a selected number of time sub-intervals and the a and a,

values will be found along with the COD and Us distributions over the cohesive zone at each time increment.

The main difficulty in solving Eqs. (2.18)-(2.21) is the fact that the problem unknowns appear both in the

integrands and within the limits of integrations of these equations. However, as will be illustrated later, when the

problem solution is known up to the ith time increment, one can solve for the COD and g, at the i + lth time

increment if the values of a and a, are known at that time increment. This observation is the ‘conceptual’ key to the proposed numerical solution scheme. In fact, in going from the ith to the i + lth time step, one can start by

guessing the location of the C.Z. crack tips, then compute a corresponding candidate solution for the COD and

the cohesive force distributions. If the candidate solution, which is computed via Eqs. (2.19) and (2.20), satisfies

both Eq. (2.18) the stress singularity cancellation condition, and Eq. (2.21) the fracture criterion, then it is

‘declared’ to be the solution at the selected time step. If the stress singularity equation or the fracture criterion

are not satisfied, appropriate ‘corrections’ for the values of a and a, need to be calculated. Therefore, the overall

procedure is based on an iterative procedure which starts with guesses for a and a, and is driven by Eqs. (2.18)

and (2.21). At each iteration, that is, for every pair of a and a, values, the COD and cohesive force are obtained

by solving Eqs. (2.19) and (2.20).

In reality, the strategy for the determination of a and a, is split into two separate iterative schemes for

technical convenience. In going from the ith to the i + lth time step, and after an initial guess for both a and a,,

the value for a, is kept fixed, while a first iterative procedure determines the value of a corresponding to the

satisfaction of Eq. (2.18) without enforcing Eq. (2.21). The fracture criterion is checked only after a candidate

solution satisfying the singularity cancellation condition is obtained. If Eq. (2.21) (the fracture criterion) is not

satisfied, the guess for a, is corrected and a new candidate solution (satisfying the singularity cancellation

condition) is found. Thus, the iterative procedure for the determination of a can be considered as a ‘single step’

within the iterative scheme for the determination of a,. Finally, it should be noticed that solving Eqs. (2.19) and (2.20) may require the implementation of a third and distinct iterative scheme depending on whether or not Eq.

(2.20) is nonlinear. Sections 3.2 through 3.4 present a rigorous discussion of the method outlined above. In particular, Section 3.2

is devoted to clearly defining the partitioning, or discretization, of the space-time domain over which the

derived system of integral-differential equations must be solved. Section 3.2 also contains a precise definition of

what constitutes the ‘approximate’ solution to said system of equations. Section 3.3 is instead devoted to

showing how Eqs. (2.19) and (2.20) are used to find the COD and cohesive force distributions for given values

of the C.Z. tip position. Section 3.3 is perhaps the most important in the paper because it contains the description

of a new solution method for double integral equations such as Eq. (2.19). Finally, Section 3.4 illustrates how

the solution scheme proposed in Section 3.3 can be combined with standard root-finding methods to create an

overall implicit strategy for the solution of the integro-differential equation system derived in Section 2.

3.2. Discretizution and interpolation functions

As already discussed in Remark 2.4, the solution to the system of equations introduced in Section 2.2, is obtained over the two-dimensional region of the (x, t) plane swept by the evolving C.Z. during the time interval

[0, T]. Such a region of the (x, t) plane will be referred to as the solution domain or integration domain. Its

definition will be formalized as follows:

DEFINITION 3.2 (Integration domain). With reference to Fig. 2, the Integration Domain is defined to be the

region of the x-t plane spanned by the two paths of integration defined in Eqs. (2.26)-(2.27) as the cohesive zone evolves in time from t = 0 to t = T. The value T for the upper extreme of the time interval of interest is considered given among the data defining the problem.

As shown in Fig. 2, the integration domain (in the (x, t) plane) is a wedge-shaped region bounded by the trajectories of the physical and mathematical crack tips, and by the line t = T.

In order to construct a piecewise polynomial approximate solution a discretization strategy for the integration domain must be defined. For simplicity, the discussion presented herein will be limited to a strategy yielding a piecewise linear approximation for functions of time only and a piecewise bi-linear interpolation for functions of


Fig. 2. Domain of Integration

space and time. The generalization to higher order approximations is straightforward. With reference to Fig. 2,

the integration domain discretization is defined as follows:

DEFINITION 3.3 (Grid points and grid). Let two positive integer numbers n, and n, be given. The numbers n,

and n, define a grid over the integration domain consisting of M = n, X n, quadrilateral elements and

N = [(n, - 1) X n, + l] grid points (or nodes). For simplicity, each grid point is numbered according to its

position within the integration domain:

l Node 1 is the grid point of coordinates (0,O);

l Nodesnumberedfrom[(n,+l)X(i-1)+2]to[(n,+1)X(i-1)+2+n,]withi=1,...,n,areplaced

on the time line t = ih,, where

h,=$ I

h,, will be referred to as the time step size. Therefore, the kth

1) X (i - 1) + 2 + nx will occupy the position of coordinates

xk = a,(ih,) + {k - [(n, + 1) X (i - 1) + 2]}h,(ih,),

t, = ih, ,

where the function h,(t) is defined by

h,(t) = a(t) - a,(t)

n, .

Elements numbered [n, X (i - 1) + l] to [n, X i] with i = 1,. . 9 % are placed between the time lines t = (i - l)h, and t = ih, and are defined in terms of the grid points on said time lines. In particular, the mth

element, with [n, X (i - 1) + l] Cm C [n, X i] and i 5 2, is the quadrilateral identified by the following four grid points:

(3.1)

node,with(n,+l)X(i-1)+2SkS(n,+

(x,, tk) given below:

(3.2)

(3.3)

(3.4)

Node:: [(n, + 1) X (i - 2) + 21 + [m - (n, X (i - 1) + l)] ,

Node:: Node: + 1 ,

Node;: Node: + n, + 1 , (3.5)

Node: : Node: - 1 ,

where Node:, with k = 1, . . . , 4, represents the number of the grid point on the kth element corner. For i = 1, that is, for elements numbered 1 to n,, placed within the time lines t = 0 and t = h,, respectively, the node numbering is as follows:

F. Costanzo, J.R. Walton I Comput. Methods Appl. Mech. Engrg. 162 (1998) 19-48

Node:: 1 ,

Node’,“: 1 ,

Node:: m+2,

Node:: m + 1 ,

(3.6)

The above element node numbering is such that, by connecting the nodes defining an element (in the given

order) one travels along the boundary of the element at hand, in a counter-clockwise sense.

REMARK. The elements numbered 1 to n, are degenerate quadrilaterals.

In view of the above definition and using concepts commonly used in finite element methods, a generic function

f(r) of time only will be given the following representation

f(f) = ki, .mkk(rl) 7 t E [(i - 1 P,, w , v E to, 11 , i=l,...,n,, (3.7)

(3.8)

wheref’, andfi represent the values taken on by f(t) at the lower and upper extremes, respectively, of the ith

time interval. Analogously, a generic function g(x, t), for (x, t) in the mth element of time and space will be approximated as follows:

where g: are the mth element nodal values of g(x, t). The functions I++~( 5,~) are the element nodal shape

functions expressed in terms of a local coordinate system (5, v), where 7 is defined as in Eq. (3.8) and

(3.10)

where (xF,tF) with k=l,... ,4 are the coordinates of the four nodes that define the mth element. The

interpolation functions &( 5, q) for a bi-linear interpolation take on the well known form

$,(5,77)=(1-5)(1-rl)> tiz(&;)=&l -rl)>

&(&rl)=&?‘rl ~~(5~rl)=(l-O~. (3.11)

Before proceeding to a discussion of the solution scheme, it is convenient to introduce one last definition: that

of approximate solution.

DEFINITION 3.4 (Approximate solution). Given a grid for the integration domain, the approximate solution to

the reduced IBVP,, is defined to be the (finite) set consisting of the values taken on by the C.Z. evolution at the

grid points:

{a,@,), a@,), q(x,, r,)>,

k=l,...,n,, q=l,...,nx. (3.12)

Furthermore, given a positive integer i <n,, the subset

b,kJ a@,), a,(-+ t,)) 1

k=l,..., i, q=l,..., IZ,,

will be referred to as the C.Z. approximate solution up to the ith time interval.

(3.13)

It should be noted that this definition identifies the primary unknowns of the approximate solution scheme and


therefore does not include the nodal values of the function a, for those nodes lying along the trajectory of the

mathematical crack tip because these values are known a priori. In fact, when b(t) > 0, as a direct consequence of Eq. (2.14) it can be shown that d&(t), t) = 0, V t E [0, T], and, as a (not so direct) consequence of Eqs.

(2.14) and (2.7), it can be shown that d&z(t), t) = 0, V t E [0, T] also. Therefore, from Eq. (2.6) one finds that

a,(&), t) = &.(d,(a(t), t) 3 &Mt), t>) = ~JO,O) . (3.14)

3.3. Numerical evaluation of Eq. (2.19)

Let the approximate solution be known up to the ith time interval. Under these conditions both the limits of

integration and the integrand in Eq. (2.19) are known for t E [0, ti]. Hence, one can numerically evaluate the

right-hand side of Eq. (2.19) to determine the value of the cohesive part of the crack opening displacement

d’&, t) anywhere in the integration domain from t = 0 up to the ith time step. Since the applied loads are given,

the numerical evaluation of Eq. (2.24) will then provide the remaining part of the opening displacement.

Eq. (2.19), combined with Eq. (2.20), can also be used in the construction of a linear system of equations for

the determination of the unknown nodal values of the cohesive force a,(~, t) at the (i + 1)th time step. In

particular, under the assumption that the crack tip trajectories are known up to the (i + 1)th time step, Eq. (2.19)

can be used to construct the following map:

id;;1 = ma,) + if,) 7 (3.15)

where {dk} and {a,} are n, X 1 vectors containing the unknown nodal values di(x,, ti+,) and CT,(X~, ti+,)

(q=l,..., n,), respectively, [T] and (f,} are an n, X n, matrix and an n, X 1 vector of known coefficients. The aforementioned system of n, equations in the n, unknowns a,(~,, ti + , ) (q = 1, . . . , n,) is then obtained from Eq.

(2.20) via collocation at the points (x,, ti + , ) (q = 1, . . , n,), where the nodal values of d&, t) and d;(x) t) are inferred from Eq. (3.15) and expressed as functions of gc(x, t). Next, is presented a technique for the

construction of Eq. (3.15).

First, Eq. (2.19) is reformulated as

(3.16)

I 4

G(q; x, t> = o-,(x + c(t + r - 2q), r) &r . (3.17) h,;(zq~(r+xlo))

The function a_(~, t) is continuous and, by virtue of the constitutive assumption, bounded everywhere in the

integration domain. Hence, the function G( q; x, t) (regarded as a function of q only) is sufficiently smooth to

make Gauss quadrature applicable to the evaluation of Eq. (3.16). Clearly, due to the presence of a square root singularity at the upper limit of integration, the choice of weights and evaluation points for the Gauss integration

procedure are not the traditional ones used for polynomials but will have to be determined accordingly. Thus,

Eq. (3.16) can be rewritten as

d;(x, t) = - c pT F, G(q,; x, t)w, , (3.18)

where qk and wk are the No, evaluation points and weights of the Gauss-Jacobi quadrature rule for integrable

singular functions [26]. The quality of the above approximations depends on the parameter Nor and, more importantly, on the quality of the values G(q,; x, t). The latter, in turn, depend on the quality of the interpolation for the function ac(x, t) and, in general, on the quadrature rule used to evaluate the right-hand side of Eq. (3.17). As it turns out, for the proposed interpolation strategy it is possible to evaluate the right-hand side of Eq. (3.17) exactly, thus making the quality of the values G(qk; x, t) dependent only on the selected shape function for g= and on the fineness of the integration domain discretization.

Let a point P of coordinates (x,, ti+ 1) be given on the i + lth time line. Then, with reference to Fig. 3, the integral on the right-hand side of Eq. (3.16), as already discussed in Remark 2.4, can be interpreted as a line


q / I Seg.5

I

r ’ \II I, /

R(!7 v Seg.1

b

I X-

Fig. 3. Double integration scheme.

integral along Path;. The latter originates from the point P and terminates at the point Q of coordinates

(@,‘(t,+, + X, /c)), by ‘(tj+ , + x, /c)), lying along the trajectory of the mathematical crack tip. Hence, the kth

value of G, as required by the Gauss quadrature rule, is given by

I 4k

G(q,;x,, t;,,) = h,32qk-(1,+,+x,Id)

a,(c(r-(2qk-(fj+, ++)))~r)*.

For convenience, let

(3.19)

(3.20)

Thus, the right-hand side of Eq. (3.19) is a line integral along Path’,“.k, connecting the point S( qk), of coordinates

(x, + c(t; + , - qk), qk), and R(q,), of coordinates (a,(( lmik), &;,). As shown in Fig. 3, the path of integration RS goes across several elements and can therefore be divided into a number of segments, say Nk,,,, each of which

is contained within a corresponding element. As a result, the integral in Eq. (3.19) is evaluated as follows:

(3.21)

where (rL,, ru ) identify the lower and upper extremes of the pth segment formed by the intersection of Path:.”

with the underlying integration domain grid. Recalling that the function a,.(~, t) has been given the

representation in Eq. (3.9), Eq. (3.21) becomes

(3.22)

where g, yp’ is the jth nodal value of the function cC(x, t) within element M(p), the latter being the element that

contains the pth segment forming Pathysk. Also, from the third of Eqs. (3.8) and from Eq. (3.10)

44 - <xr(p)+,(tirN + -$‘p)+22(77(rN) ‘(‘) = (x~‘“’ - .x~‘“‘)+,(~r)) + (x~(~) - xy’p’)42(rl(r)) ’

(3.23)

r-ttE(l’P’

tir)= h , (3.24) ,

x(r) = 4r - bmo<JmJ) (3.25)

where (x,~(“), tyP’) are the jth nodal coordinates of the M(p)th element. Since the interpolation functions


$j( 5,~) are polynomials, and in view of Eqs. (3.23)-(3.25) (after some tedious algebra) it is possible to rewrite integrals of the type

in the form

(A,r+fi,)” dr ~ (a,bEN),

(A1*r + &)’ vZ7

(3.26)

(3.27)

where A,, A,, B1, &, a and b are (known) constants. Finally, setting qk - r = z, the above integral can be

reformulated in the ‘canonical’ form

9= I (A,z+B,)” dz

(A,z + B,)’ 4 ’ (3.28)

where the coefficients A ,, A,, B, and B,, are also constants. Depending on the powers a and 6, the above

integrals can often be integrated exactly. In particular, exact integration was possible for all the interpolation

functions used herein. When a chosen interpolation strategy allows one to reformulate a singular integral as an

integral of a smooth kernel, such as (A ,z + B, )” /(A,z + B,)b, multiplied by a singular function of known

properties, such as 1 / 4, the resulting integration technique is referred to as ‘product integration’ (see e.g. [25]).

Eqs. (3.21)-(3.28) show that Eq. (2.19) can indeed be reformulated as Eq. (3.15) as long as a first

‘accounting’ procedure is designed to determine the segments into which Pathy.k must be divided, together with

a second ‘accounting’ procedure which keeps track of whether the nodal values a:” are known or unknown,

and which assembles the known contributions into the vector (f,} and the coefficients of the unknown nodal

values into the proper location within the matrix [T]. The first of the aforementioned ‘accounting’ procedures

consists of finding the intersection of a given line with an underlying grid. Since, as already discussed, the grid

itself and, consequently, the integration paths are not known a priori, said procedure is not always trivial to code

in an efficient way. However, it does not present any conceptually important or interesting problems and

therefore it will not be discussed further. The other ‘accounting’ procedure essentially consists of an assemblage

algorithm of the type commonly found in finite element textbooks such as those in [27,28].

It is important to notice that the reduction of Eq. (2.19) to Eq. (3.15) is possible as long as the functions b,(t)

and b,Jt) are known, that is, as long as the crack tip trajectories are known. This remark is important in

understanding the implicit nature of the overall solution scheme, as it will be illustrated in the next section. The

present section will now be concluded by a derivation of the system of equations that allows one to determine

the nodal values of the function flC(x, t) at the time step i + 1 when the approximate solution is known for

t E [O, ‘;I. Let the approximate solution be known up to the ith time step together with the values of the crack tip

trajectories at t = t,, , . By evaluating Eq. (2.20) at the n, nodal points (x,, tic,) (q = (n, + 1) X i + 2, . . , (nx + 1) X i + 2 + n,) and using Eq. (3.15), one obtains the following system of equations:

{

mo;.) = ~~$4J~ &Jl) @;I = P-lbJ + (f,>

(3.29)

where [I] is the (n, X n,) identity matrix, {Ai} is the II, X 1 vector containing the unknown nodal values of the

function di(x, t) at the points (xy, tl+, ) (q=(n,+l)~i+2,...,(n,+l)Xi+2+n,), and the expression

{&[.I} represents an n, X 1 vector of values of the C.Z. constitutive response function as a functional of the vectors {dk} and {A;}. For the sake of discussion, let the C.Z. constitutive equations be of the form

oC(X, t) = U,, + KdR(X, t) + b’& , (3.30)

where fly, K and v are constant coefficients. Also, using the interpolation strategy in Eq. (3.9), it can be shown that the vector {ii} can be expressed as a function of the nodal values of the function C&(X, t), that is,

c&J = PIW + {fl’> 1 (3.31)


where [D] and CfT} are a ‘time-derivative’ n, X IZ, matrix and a n, X 1 vector of known coefficients,

respectively (c.f. [27]). Similarly, with reference to Eq. (2.24), the part of the opening displacement due to the

applied loads can be expressed as

(3.23)

Hence, from Eqs. (2.22) (2.24) and (3.30)-(3.31), the system in Eq. (3.29) becomes

[mg.I = u-1 > (3.33)

where

[K] = [I] - KiTI - dDl[Tl t (3.34)

u} = {gY,> + K(&> + @f;]) + @]({f,) + id;;)) + r’ti>. (3.35)

The above linear system of algebraic equations can then be solved for the nodal values of the function a,.(~, t) at

time t = t, + , , that is, for the remaining part of the approximate solution at the i + lth time step. Clearly, if the

C.Z. constitutive equations are not of the form given in Eq. (3.30) the resulting system of equations will not be

linear. However, it is not difficult to show that it can be solved using standard linearization together with the

Newton-Raphson method or other Quasi-Newton methods (c.f. 1261).

REMARK 3.5 (CZ0Dlc.z. tip trajectory map). The derivation of Eq. (3.33) is important because it shows that

its solution yields a map between the value of the crack tip trajectories at t = t, + , and the nodal values of the

function rr on the same time line, that is,

9 : (a,@, +, h a@, + , >I + {q.(+ $+ , )> . (3.36)

The uniqueness properties of such a map are reflected in the invertibility of the matrix [K] which, in turn, is

strongly dependent on the C.Z. constitutive equations. In any case, in view of what is presented in the next

section, it is important and useful to regard Eq. (2.19) and/or Eqs. (3.29) in terms of the map in Eq. (3.36).

3.4. Solution strategy

The C.Z. evolution is determined via a ‘step-by-step’ procedure marching forward in time. However, Eqs.

(2.18)-( 2.21) do not seem to yield direct information regarding the functions a’,, a and tic so to allow one to

formulate an explicit solution method, that is, a method whereby the approximate solution at time step i + 1 is

directly obtained from information available at time step i (i <n,). For this reason, the approximate solution

scheme proposed herein is of implicit type, that is, it attempts to converge to the approximate solution at time

step i + 1 through a sequence of iterations generated by an initial guess. In particular, as discussed in the

previous section and in Remark 3.5 in particular, it is possible to obtain the entire approximate solution at time

step i + 1 using information at previous time steps of knowledge of a,@, + , ) and a(t, + , ). Hence, one can regard the combination of Eqs. (2.19) and (2.20), together with a guess at time step i + 1 for the pair (a,,,, a) as a guess

at time i + 1 for the C.Z. evolution as a whole. For this reason it is convenient to introduce the following definition:

DEFINITION 3.5 (C.Z. evolution initial guess and iterations). Let an approximate solution for the cohesive

zone evolution be known up to a time step i < n,. Then, the triplet consisting of the initial guesses at time step

i + 1 for the physical and mathematical crack tips, respectively, together with the associated solution provided

by the map in Eq. (3.36), will be referred to as the cohesive zone evolution initial guess at time step i + 1 and it will be denoted by [a:@,+,), a”(ti+,), gp(tj+,)]. Furthermore, the triplet [a-‘,@,+ ,), a’@!_ ,), c~/,.(t,+ ,)] will be referred to as the jth iteration for the cohesive zone evolution at time step i + 1.

REMARK. In the triplet defined above, the term gL(t, + , ) in reality consists of the set of (approximate) values for the function g.(x, t, + , ) at the grid points on the time line t = tj+ , yielded by the jth iteration.

Having introduced the above definition, one can construct a solution strategy consisting of an iterative procedure

32 F. Costanzo, J.R. Walton I Comput. Methods Appl. h4ech. Engrg. 162 (1998) 19-48

structured in two phases or loops2. Each of the two loops consists, in turn, of a number of procedures and

algorithms which are depicted in the following flow chart (Fig. 4).

In order to better illustrate the solution process, it is convenient to regard the satisfaction of the stress

singularity cancellation condition and of the fracture criterion as separate from the rest of the equations that

define IBVP,,. Symbolically, this distinction will be indicated as follows:

IBVPcz = IBVP, + {SIF(a(t), t) = 0} + {CZOD(a,(t), t)) G a,,},

where SIF(a(t), t) denotes the stress intensity factor at a(t), CZOD(a,(t), t) denotes the C.Z. opening

displacement at the material crack tip, and IBVP, indicates ‘the rest of the equations’ that define IBVP,,. Once

this (practical) distinction is made, assuming that the solution up to the time step i (i <n,) is known, one sees

Data Acquisition

Loop 1 Counter Update and Loop 2 Initialization j=j+l;k=o

~(k+~)=%(t~~) + A%j ueas o&+1)

I Find Ask by solving Sm(++l)+hk,ti+l) = 0

Fig. 4. Overall numerical solution scheme flow chart.

* Strictly speaking, there are three loops. However, since the outermost loop is technically trivial (it simply entails the updating of the

time step counter) it will not be discussed.


that Loop 2 is an iterative procedure that yields a solution [&‘,(tj + , ), ii’(ti+ , ), &~(tj+, )] to the problem consisting

of IBVP,, + {SIF(u(t), t) = O}. This ‘sub-problem’ consists, in essence, in finding the value of the mathematical

crack tip position so to satisfy the stress singularity cancellation condition, keeping the variable a, fixed and

equal to the value assigned to it by Loop 1. Thus, within Loop 2 the variable ,um is not regarded as an unknown

but rather as belonging to the data of the problem. After [5’,(ti+, ), a’(t, + , ), c?‘,(I~+, )] is determined, the iteration cycle control is transferred to Loop 1 and the aforementioned solution is regarded as a candidate solution to the

full problem (i.e. IBVP,,). If [&“,(ti+ , ), Z’(r, + , ), c?i(tj+, )] satisfies the fracture criterion then it is accepted as a

true solution and the solution corresponding to the subsequent time step is sought. If, instead, the fracture

criterion is not satisfied, then a correction to the present value of the variable a, is determined at the end of

Loop 1, before re-entering Loop 2.

REMARK 3.6 (Guess for a,@,+ , )). The value of a;@,+ , ) is set as follows:

ifj=l,

if j > 1 , (3.37)

where j is the Loop 1 counter. The above equation simply indicates that in going from one step to the next the

crack tip a, is first assumed not to move at all. This choice is motivated by the fact that at t = ti+, one must

verify whether a solution implying crack arrest is admissible or not. If, at the end of the corresponding Loop 2,

the fracture criterion is violated then a correction Au,,,~ is determined.

REMARK 3.7 (Fracture criterion). The correction Au,, mentioned in Remark 3.6 is determined by solving the

following equation:

d,(~,,,(t,+, )v t,, ,I - 4, = 0 . (3.38)

The above equation can be explicitly evaluated thanks to Eqs. (2.19) and (2.24) which makes it possible to

determine the overall C.Z. opening displacement once a solution in terms of U&X, t) is obtained. Also, upon noting that Eq. (3.38) is used once a solution satisfying the stress intensity factor cancellation condition is

satisfied, that is, after an acceptable candidate solution in terms of a@, + , ) has been found, one can regard Eq.

(3.38) as an equation of the type

%a,(r, + , )I = 0 7 (3.39)

where the (nonlinear) function 9 represents Loop 2 as a whole. Eq. (3.39), although an abstraction, indicates

that the desired values of a,@;+ i ) can be found using standard root-finding techniques for (nonlinear) real

functions of one real variable, such as bisection or the secant method (see [26]). Clearly, since the solution to

Eq. (3.39) must be found via an iterative algorithm, say bisection, for every iteration in terms of u,(ti+ , ) one

must carry out all the calculations required in Loop 2, thus making the overall solution scheme computationally

intensive.

REMARK 3.8 (Stress singuluri~ cancellation condition). With reference to Fig. 4, the stress singularity

cancellation condition has been indicated by the equation

SIF(a(~;+,)+Aa,,~,+,)=O, (3.40)

where k is the Loop 2 counter. This equation, being evaluated at a fixed and known value of a,@,+, ), presents a situation analogous to that discussed in Remark 3.7 and can therefore be solved in the same fashion.

4. Results

This section presents several results obtained with the solution strategy proposed in this paper. In particular, a comparison between the closed form and numerical solutions for a very simple case is presented so to illustrate the advantages offered by the combination of Gauss quadrature in combination with product integration, both in terms of precision and overall computational efficiency.


As far as more general problems are concerned which include nonlinear C.Z. constitutive behavior in conjunction with time dependence, given the unavailability of closed form solutions, the results presented are simply intended to illustrate the solution strategy sensitivity to grid refinement.

4.1. A case with exact solution

A very simple problem, among those of interest in this paper, which may admit a closed form solution can be

defined by assigning the mathematical and physical crack tip trajectories, together with selecting a conveniently

simple C.Z. constitutive response function. The unknown function of this problem is therefore the C.Z. opening

displacement. Clearly, if the crack tip trajectories are assigned as data, neither the chosen fracture criterion nor

the stress intensity factor cancellation condition can be enforced without causing the problem to be overdetermined. Also, a closed form solution can be obtained in this fashion only if the functions (of time) used

to specify the crack tip trajectory are simple enough. In particular, the functions defining the crack tip

trajectories must be simple enough so that the functions b,(t) and b,,)(t), defined in Eqs. (2.15) and (2.23), are

easily inverted and, once they are substituted in Eqs. (2.19) and (2.24) the resulting double integrals can be

performed in closed form. Hence, with the intent of focusing the attention on the accuracy of the proposed

numerical integration strategy, that is, the combination of Gauss quadrature and product integration (c.f. Section

3.2), the following problem has been solved both numerically and in exact form:

( 1) Cohesive zone constitutive equations

~~(d,(x, t), d,(x, t)) = cry = const. V (d,, dR) (Dugdale model);

(2) Applied crack face forces

F,(x, t) = OV t E [0, T] and V x < a,(t) (stress free crack faces);

(3) Assigned crack tip trajectories

a,(t) = c, t and a(t) = c,t, ‘d t E [0, T] and with c, = const., c2 = const. (c, < c2 < c) (rectilinear crack tip

trajectories).

The solution of this problem is reported in Appendix A.

Since the externally applied crack face forces have been chosen so to always be null, the resulting (total)

crack opening displacement will consist only of the contribution due to the cohesive zone. This means that the

expected crack opening displacement profile will take on only negative values along the cohesive zone, that is,

values expressing the fact that the crack is experiencing closure of the C.Z. with interpenetration. Hence, from a

physical viewpoint, this particular problem presents serious difficulties. However, the fact remains that, from a

mathematical viewpoint, this is still a valid problem to test the solution strategy accuracy.

Before presenting the problem’s solution, it is important to notice that the values assigned to the various

parameters defining the problem are expressed in non-dimensional form. In particular,

(1) the crack opening displacement and the crack tip positions are non-dimensionalized with respect to the

critical opening displacement d,,;

(2) velocities are non-dimensionalized with respect to the shear wave speed c; (3) time is non-dimensionalized with respect to the quantity d,,/c, that is, the time it takes to travel across a

length equal to the critical crack opening displacement while moving at the speed of sound; (4) forces per unit area are non-dimensionalized with respect to the elastic shear modulus ,u;

(5) finally, the viscosity coefficient v will be non-dimensionalized with respect to the quantity c/p. Also, all calculations have been carried out using a computer code written in C, running on an IBM RISC

System/6000 workstation with a PowerPC 603 chip at 133 MHz. Table 1 shows the numerical and exact results for a case where 0 c t S 1000, c, = 0.4, c2 = 0.8 and ur = 0.5.

The time interval was subdivided into 100 time steps, and the cohesive zone was subdivided into 5 space-wise

intervals, that is, n, = 5 and n, = 100. Table 1 reports results relative to the first and last time steps. The first column in the table identifies the time step at hand. The second column reports the results obtained using Gauss-Jacobi integration in the handling of Eq. (3.17) and product integration in the handling of Eq. (3.22), whereas the second column shows the results obtained by using Gauss and Gauss-Jacobi integration for all integrals. As indicated in the top line, every time that Gauss integration was performed 20 quadrature points were used. The node number (N#) simply identifies the various nodes on the time line at hand. In particular, nodes identified by N = 1 lie along the trajectory of the physical crack tip. Also, the opening displacement along

F. Costanzo, J.R. Walton I Comput. Methods Appl. Mech. Engrg. 162 (1998) 19-48 3s

Table I

Comparison between the numerical and closed form solutions for a problem with assigned crack tip trajectories. The assigned trajectories are

straight lines and the C.Z. is a Dugdale failure zone. The numerical solution has been obtained using two different numerical quadrature

strategies. ‘N#’ stands for ‘node number,’ ‘Err.’ stands for ‘error,’ ‘ GQ/PI’ and ‘GQIGQ’ denote the Gauss quadrature/product integration

and the Gauss quadrature/Gauss quadrature strategies, respectively

nL = 5, n, = 100, T = 1000.0, Number of Gauss quadrature points = 20

N# Exact GQ/PI GQ/PI Err. GO/GO GO/GO Err.

Time interval: I

I - I .0805e - 01

2 -1,3639e-01

3 -1.4041e-01

4 - I .2972e ~ 0 1

5 -l.OllOe-01

Time interval: 100

1 -I.O805e+Ol

2 - 1.3639e + 01

3 ~1.4041e+Ol

4 -1.2972e +Ol

5 -1.0110e+01

- 1.0805e - 01

-- 1.3639e - 01

-1.4041e-01

-1.2972e-01

-l.OllOe-01

-l.O805e+Ol

-1.3639e+Ol

-1.4041e+Ol

- 1.2972e f 01

-1.0110e+Ol

Total CPU time: 9 s

4.6802e - 10

2.9852e - I2

2.9872e - 12

2.989Oe - 12

2.992le - 12

4.6800e - 10

2.9934e - 12

2.9867e - I2

2.9909e ~ 12

2.9933e - 12

- 1.0803e - 01

-1.3639e - 01

-1.4040e-01

~ 1.2972e - 01

-l.O109e-01

-l.O805e+Ol

- I .3639e + 01

-1.4040e+Ol

-1,2972e+Ol

-1.0110e+01

Total CPU time: 56 s

I .8253e - 04

8.486Oe - 06

1.3465e - 05

2.7288e - 05

7.849le - 05

I .423Oe - 08

2.1302e - 09

1.3634e - 05

3.5140e - I I

9.20OOe - I I

the trajectory of the mathematical crack tip is not reported since it is always equal to zero. The numbers

appearing under the columns entitled ‘Computed’ and ‘Exact’ represent the non-dimensional value of the crack

opening displacement at the indicated nodes. The columns entitled ‘Error’ simply report the absolute value of

the difference between the exact and computed opening displacement at a given node. Before commenting upon the above table any further, it is important to recall that the polynomial interpolation

used during integration is intended to approximate the cohesive force distribution o;.. In this particular case, a

bi-linear interpolation was used. Therefore, given that the function to be represented is constant and uniform

over the integration domain (g= = a;), the interpolation used is, in fact, an exact representation of the function

o;-. This, in turn, is an important element in order to better understand what the error measure given in the table

really represents: the error induced in the evaluation of the C.Z. opening displacement by the selected numerical

integration strategy. The results reported in Table 1 show that Gauss quadrature in conjunction with product integration is

extremely accurate. Moreover, the level of accuracy in essentially uniform, at least in this case, over the solution

domain, whereas the accuracy of Gauss integration throughout is more sensitive to location. An interesting

element is the fact that the accuracy of calculations regarding nodes on the physical crack tip trajectory is

inferior to that of other nodes. The reason for such a behavior can be found by carefully analyzing Eq. (3.18) in

conjunction to Fig. 5. Recall, in fact, that the values G(q,; x, t) used in Eq. (3.18) must be computed via line

integrals along Pathy.k, as indicated in Fig. 5, where the length of the integration path at hand depends on where

the point S(qk) is. The latter, in turn, is found along Path: at locations corresponding to the position (along

Path:) of the Gauss-Jacobi quadrature points. Therefore, for a node on the trajectory of the physical crack tip

and for those quadrature points that happen to be very close to that node the length of PathySk may be rather

small relative to resolution permitted by machine precision. This inconvenience can be overcome by a simple

but effective strategy, namely subdividing Path: in segments and compute the integral along Path;” as the sum

of integrals along each individual segment. The size of these segments can then be adapted to reach a certain

accuracy. Clearly, since the integral along the top segment is characterized by a singular integrand, whereas the integrals over the other segments are non-singular, the selected quadrature rule must be adjusted accordingly.

With reference to Fig. 5, the results shown in Table 1 have been obtained by subdividing Path; into two - - -- segments, AB and BC, respectively, such that AB/BC = 1O-6. For cases where Path; was subdivided into just

two segments, Table 2 illustrates sensitivity of the numerical integration strategy at points along the physical -- crack tip trajectory as a function of the ratio AB/BC (the type of data represented in Table 2 is the same as that in Table 1). The main point shown by Table 2 is that the numerical error due to the integration procedure is controllable even at ‘trouble spots’ such as points along the trajectory of the physical crack via a simple adaptive


Fig. 5. Integration paths for points lying along the physical crack tip trajectory.

Table 2 - _ Integration strategy sensitivity to the partitioning of the path integral in Eq. (3.18) into two segments AB and BC (c.f. Fig. 5) with relative

size ABIBC. ‘Err.’ stands for ‘error,’ ‘GQ/PI’ and ‘GQ/GQ’ denote the Gauss quadrature/product integration and the Gauss quadrature/

Gauss quadrature strategies, respectively

n, = 5, n” = 10, T = 10.0, Number of Gauss quadrature points = 10 -- ABIBC GQlPI GQ/PI Err. GQ/GQ GQ/GQ Err.

Time Interval: 1; Exact Value: - 1.080503e - 02

l/2 -1.081563e - 02

lOe-05 - 1.080503e - 02

10e - 06 - 1.080503e - 02

lOe-07 - 1.080503e - 02

Time Interval: 5 Exact Value: -5.402515e - 02

l/2 -5.407813e - 02

lOe-05 -5.402515e - 02

IOe-06 -5.402515e - 02

IOe-07 -5.402515e - 02

Time Interval: IO; Exact Value: - 1.080503e - 01

l/2 - 1.081563e - 01

lOe-05 - 1.080503e - 01

lOe-06 - 1.080503e - 01

IOe-07 - 1.080503e - 01

9.806501e - 04

1.96 1229e - 08

1.960593e - 09

1.954237e - 10

9.806501e - 04

1.961230e - 08

1.960596e - 09

1.954264e - 10

9.80650le - 04

1.961230e - 08

1.960586e - 09

1.954183e - 10

-1.081546e - 02

- 1.080469e - 02

- 1.080469e - 02

- 1.080469e - 02

-5.407789e - 02

-5.402515e - 02

-5.402515e - 02

-5.402515e - 02

-1.081558e - 01

- 1.080502e - 01

- 1.080502e - 01

- 1.080502e - 01

9.650231e - 04

3.1097OOe - 05

3.116186e - 05

3.116835e - 05

9.760983e - 04

1 O4804Oe - 07

1.224334e - 07

1.241964e - 07

9.760095e - 04

1.121407e - 06

1.137994e - 06

1.139652e - 06

strategy like the one described above. Furthermore, the combination of Gauss quadrature and product integration not only offers high accuracy but also good sensitivity to refinement of the integration algorithms.

In both Tables 1 and 2, whenever Gauss integration was performed along a certain integration path, the number of quadrature points used was kept fixed, namely 20 points for the former and 10 points for the latter. Clearly, one should expect the overall integration procedure to be sensitive to the number of Gauss quadrature points used. Table 3 complements the previous two tables by reporting the sensitivity of the overall numerical procedure on the number of gauss quadrature points used. For simplicity, the results reported in Table 3 concern the value of the crack opening displacement at the node on the physical crack tip trajectory at the very first time step. From Table 2 it can be observed how Gauss quadrature used throughout may sometimes have an erratic behavior, where the increase in the number of quadrature points does not necessarily yield an increase in accuracy. Instead, the combination of Gauss quadrature and product integration seems to offer a consistent and accurate behavior.


Table 3

Integration strategy sensitivity to the number of Gauss quadrature points. ‘NGQP’ stands for ‘number of Gauss quadrature points,’ ‘Err.’

stands for error, ‘GQIPI’ and ‘GQ/GQ’ denote the Gauss quadrature/product integration and the Gauss quadrature/Gauss quadrature

strategies, respectively --

n = 5, n I n = IO, T= 10.0, TI = 1, Node # = I, AB/BC = 1Oe - 07

Exact Value = - 1.080503e - 02

NGQP GQIPI GQ/PI Err. GQ/GQ GQ/GQ Err.

5 - 1.080503e - 02 7.39091 le - 10 - 1.0805OOe - 02 2.980622e - 06

10 - 1.080503e - 02 1.954237e - 10 - 1.080469e - 02 3.116835e - 05

15 - 1.080503e - 02 3.880632e - 11 - 1.080503e - 02 1.74130le - 11

20 - 1.080503e - 02 1649243e - 11 - 1.080503e - 02 1.830307e - 04

25 - 1.080503e - 02 8.688203e - 12 - 1.080503e - 02 3.055866e - 12

4.2. Accelerating crack with general cohesive zone models

The figures contained in this section display the solution to the full problem outlined in Section 2.1. In particular, the present section is devoted to an analysis of results obtained using rate independent as well as rate

dependent cohesive zone models (whether linear or nonlinear). Since this paper’s objective it to present the

numerical strategy employed to obtain the aforementioned solutions, the discussion contained herein will be

limited to the sensitivity of the computed results to mesh refinement and to the order of the interpolating

functions. For a discussion of the physical meaning of these results see [ 111.

The response of a Dugdale (c.f. [ 121) cohesive zone will be considered first. The main reason motivating this

choice rests in the fact that, for a Dugdale model, the chosen interpolation procedure is capable of representing

the cohesive force distribution exactly. Therefore, the results relative to the Dugdale model are useful to

understand how the accuracy of the whole procedure is affected by grid refinement as opposed to numerical

integration, which in this case, as shown in the previous section, is extremely precise. Clearly, once more

complex C.Z. models are analyzed it will be more difficult to precisely pinpoint the origin of numerical errors.

Figs. 6 and 7 depict the trajectories and the velocities of the material (physical) and the mathematical crack

tips, i.e. the functions a,(t) and u(t), respectively, in response to the applied force distribution F,(t) = FH(x - L,)H(-x)H(t), where F and Lr are constant. Physically, at time t = 0 a uniform force distribution of magnitude

F is suddenly applied over a segment of length L, in front of the physical crack tip and kept constant in time

thereafter. As indicated in Figs. 6 and 7, the system’s response in terms of crack tip motion, consists of a rather

Cohesive Zone Evolution Cohesive Zone Evolution Crack Tip Trqjectories - Dugdale Modal

Crack Tip Velocities - Dundale Model

Non-Dimensional Crack Tip Position - a/h7 0.0 200.0 400.0 600.0 800.0 11

Non-Dimensional Time - tc/&

Fig. 6. Typical response of a rate independent Dugdale cohesive zone: crack tip trajectories.

Fig. 7. Typical response of a rate independent Dugdale cohesive zone: crack tip velocities.


rapid acceleration followed by a deceleration which ultimately leads to crack arrest. The deceleration of a crack tip occurs when the loading information carried by the stress wave has reached and gone past the crack tip at hand. This is consistent with the fact that the deceleration of the physical crack tip takes place slightly before

that of the mathematical crack tip.

In order to keep the number of graphs at a minimum, the sensitivity to mesh refinement is now discussed only

via crack tip velocity lots, and, in particular, plots that concern the material crack tip velocity, the latter being,

perhaps, the most interesting and delicate element of this analysis, both from the physical and numerical

viewpoint.

Figs. 8 and 9 show how the solution in terms of physical crack tip velocity is affected by changing the

parameters n, and n, which define the grid size. In particular, Fig. 8 reports results relative to various values of

the time step size (T/n,) while keeping constant the number of elements along the cohesive zone. Fig. 9

concerns results with varying n, and fixed time step size. In both cases one can see that this particular set of

results is essentially independent of spacewise refinement and relatively insensitive with respect to time step size

as well. This behavior, although expected because of the exact nature of the cohesive force field interpolation, is still remarkable if compared to that of other forward marching time integration schemes. Clearly, greater

sensitivity to grid size is to be expected when using C.Z. models which include dependence on the opening

displacement and/or its rate. Before proceeding further, an additional remark is in order: in Figs. 8 and 9 the

time and space discretization parameters were controlled separately and independently of each other. This fact

does not seem to have a significant impact on these particular results. However, this should not be expected to

be the ‘natural’ response to mesh refinement when using the solution strategy proposed in this paper. Recall, in

fact, that the most delicate aspect of the solution scheme as a whole is represented by the integration procedure.

Since the latter is performed on path integrals along straight lines parallel to the characteristic directions of the

wave equation rather than along the x or t axes, an effective refinement strategy cannot simply rely on increasing

n, and n, disjointly, but rather it should increase these parameters in a proportional fashion so to effectively

control the size of the segments into which the path integrals are subdivided.

Figs. lo-13 display the solution to problems with the same loading conditions as those discussed so far, but with different C.Z. constitutive behaviors. The C.Z. constitutive equation used to obtain the results in Figs. 10 and 11 is

(4.1)

where, in the displayed results, uY was taken to given the (non-dimensional) value 0.5. Instead, the results in

Figs. 12 and 13 were obtained using the relation

Cohesive Zone Evolution Cohesive Zone Evolution Pbvsical Crack Tip Velocity - Dugdale Model Physical Crack Tip Velocity - Dugdale Model

Non-Dimensional Time - tc/d,, Non-Dimensional Time - tc/kp

Fig. 8. Rate independent Dugdale cohesive zone: physical crack tip velocity sensitivity to grid refinement: sensitivity to time step size.

Fig. 9. Rate independent Dugdale cohesive zone: physical crack tip velocity sensitivity to grid refinement: sensitivity to space-wise

discretization.


Cohesive Zone Evolution Cohesive Zone Evolution Physical Crack Tip Velocity - Q = 0.05 - 0.05dR

0.801 : ! I : I / / I ,

Mathematical Crack Tip Velocity - 0, = 0.05 - 0.05dR

100.0 200.0 300.0 400.0 Non-Dimensional Time tcf d,r

.O “.” 100.0 200.0 300.0 400.0 NOII -Dimensional Time - k/de.

Fig. IO. Crack tip velocity sensitivity to grid refinement for a linear rate independent cohesive zone model: physical crack tip velocity.

Fig. 11. Crack tip velocity sensitivity to grid refinement for a linear rate independent cohesive zone model: mathematical crack tip velocity.

Cohesive Zone Evolution Physical Crack Tip Velocity - CZ: Nonlinear Rate Independent

0.80, : : : : : : Time Interval: [0, 5001

: i . . . . . . . . . . . . .

- nz = 5; nt = 25; (hi = 20.00) ._... nz = 10; nt = 50; (ht = 10.00) ___ ne= 15; TZ~ = 75; (ht = 6.67) __ n. = 20; nt = 100; (ht = 5.00) __. nZ= 25; nt = 125; (hi = 4.00)

.O 200.0 300.0 400.0 Non-Dimensional Time - tc/d,,

500.0

Cohesive Zone Evolution Mathematical Crack Tio Velocitv - CZ: Nonlinear Rate Indeuendent

0.001 I]

0.0 100.0 200.0 300.0 400.0 500.0 Non-Dimensional Time - tcf d,,

Fig. 12. Crack tip velocity sensitivity to grid refinement for a nonlinear rate independent cohesive zone model: physical crack tip velocity.

Fig. 13. Crack tip velocity sensitivity to grid refinement for a nonlinear rate independent cohesive zone model: mathematical crack tip

velocity.

(4.21

\ ucr umax /

where, in non-dimensional units, a, = 0.01, Au,,,,, = 0.025 and d,,, = 0.025. In both cases, and contrary to the analysis conducted using the Dugdale model, the selected interpolation functions for the cohesive force distribution over the cohesive zone will not, in general, be able to provide an exact representation of the field a,. Therefore, one should expect the numerical approximation procedure to perform somewhat differently with respect to the case examined until now. As it turns out, the approximate solution, at least in terms of crack tip trajectories seems to converge rather quickly with mesh refinement. This result is reassuring in that it demonstrates the method’s capability to obtain a solution with relative coarse discretization, at least for the case of rate independent cohesive zone models. Clearly, in the absence of a closed form or an alternative numerical

solution one could offer more definite conclusions using an a priori error estimation algorithm. The latter is not available at the present and it definitely is a necessity to be taken care of in the future. Before proceeding to the


discussion of results concerning rate dependent models, it is of some interest to consider the behavior of the mathematical crack tip, depicted in Fig. 13. This graph shows that, for the nonlinear rate dependent cohesive zone model considered herein, the solutidn in terms of mathematical crack tip trajectory is somewhat more

sensitive than that of the physical crack tip. Such sensitivity can be observed in the form of some roughness, if

not irregular oscillations, in the initial part of the crack tip motion. Said roughness tends to disappear as the grid

refinement is increased.

Fig. 14 shows a comparison between some of the results displayed in Fig. 13 and the same results obtained

using higher order interpolation functions. In particular, a piece-wise bi-quadratic interpolation scheme was used

for both the C.Z. force and the opening displacement fields. As it can be observed in the above-mentioned graph,

the use of a higher-order interpolation eliminates the roughness present in the lower order solution. Also,

although not displayed in the graph at hand, higher-order interpolation offers a somewhat faster convergence

rate. Clearly, in order to argue this last point in a convincing way one would have to provide a quantitative

evaluation of convergence rates, and, as already mentioned, these considerations will have to be postponed until

a more complete error estimation procedure is available.

Figs. 15-18 display solutions obtained using rate dependent models. In particular, the C.Z. law used to obtain

the results in Figs. 15 and 16 was

UC = UY + lx& ) (4.3)

where, in non-dimensional units, gY = 0.05 and v = 1.0, whereas the C.Z. model used to obtain the data in Figs.

17 and 18 was

UC = (q + I&) 1 - 2 ( )

) CT

(4.4)

where, in non-dimensional units, (T = 0.05 and v = 5.0.

Most of the trends observed so far for the rate independent cases are also present in the rate-dependent cases,

although solutions in this latter case are rather sensitive to grid size and to time step size in particular. The

reason for this higher sensitivity is due to the fact that for rate dependent C.Z. models one must evaluate the C.Z.

opening displacement time derivative nodal values. This evaluation process is very different from that

concerning the opening displacement itself. In fact, the C.Z. opening displacement evaluation is carried out by

the double integration procedure at the core of the present solution method, that is, via Eq. (2.14).

Unfortunately, a similar equation for the evaluation of the opening displacement time derivative is not available, also because, by taking the time derivative of Eq. (2.14) one would have to face a double integral equation with

Cohesive Zone Evolution Mathematical Crack Tip Velocity - CZ: Nonlinear Rate Independent

-...... - rrr = 5; nt = 25; Bilinear I ..

..--. %= 5; nt = 25; Biquadratic

II 0.401 ’ I

0.0 JO.0 100.0 150.0 Non-Dimensional Time - te/dv

Fig. 14. Nonlinear rate independent cohesive zone (c.f. Eq. (4.2)): comparison between results obtained with two different interpolation

functions.


Cohesive Zone Evolution Cohesive Zone Evolution Physical Crack Tip Velocity - CZ: Linear Rate Dependent Mathematical Crack Tip Velocity - CZ: Linear Rate Dependent

n ?O ~--v-+

4 _ n, = 5; ,zt = 25; I---- i ; .L

r” ; --.nz= 25; nt = 125;

o.o\n ; : /

1”O.O 200.0 300.0 400.0 500.0 0.0 100.0 200.0 300.0 400.0 0.0 __..- Non-Dimensional Time - k/d,, Non-Dimensional Time - k/d,,

Fig. 15. Crack tip velocity sensitivity to grid refinement of a linear rate dependent cohesive zone model: physical crack tip velocity.

Fig. 16. Crack tip velocity sensitivity to grid refinement of a linear rate dependent cohesive zone model: mathematical crack tip velocity.

Cohesive Zone Evolution Cohesive Zone Evolution Physical Crack Tip Velocity - CZ: Nonlinear Rate Dependent Mathematical Crack Tb Velocity - CZ: Nonlinear Rate Deoendent

Time Interval: [0, 5001

: : -r&z= 5; nt= 25; rz.=lO. nt=50. ... /

/ : ---nz= 15; nt = 75; ..j -- &2= 20; nt = 100; / -__n_c 7K. n.- 17K.

100.0 200.0 soo.0 400.0 Non-Dimensional Time - tcf dcr

100.0 200.0 300.0 400.0 Non-Dimensional Time - k/d,,

Fig. 17. Crack tip velocity sensitivity to grid refinement of a nonlinear rate dependent cohesive zone model: physical crack tip velocity.

Fig. 18. Crack tip velocity sensitivity to grid refinement of a nonlinear rate dependent cohesive zone model: mathematical crack tip velocity.

non-integrable singularities. The present authors have therefore decided to circumvent the problem by using the

nodal values of the opening displacement (obtained via the double integration procedure) in conjunction with an interpolation scheme essentially identical to that in place for the C.Z. force field. Hence, time derivatives of the

opening displacement are obtained by differentiating the corresponding interpolation functions. This strategy entails that the quality of time derivative evaluations is significantly inferior to that of evaluations of the opening displacement function and, is perhaps, the single largest source of numerical error in solutions with rate

dependence. Another reason for the higher sensitivity displayed by these calculations is to be found in the physics of the problem. In fact, for rate dependent C.Z. models used herein there is no intrinsic bound to the magnitude of the C.Z. force, contrary to the rate-independent cases examined earlier, This consideration is extremely important in view of the fact that, in the vicinity of the physical crack tip, both the function a;(.~, t)

and dR(x, t) develop extremely steep gradients in the space direction that are extremely difficult to capture without a rather fine discretization. Clearly, the finer the mesh discretization is the longer it takes to carry out

these calculations. One possible strategy in order to accommodate the necessity for refinement without excessively compromising computational efficiency is that of implementing a convenient mesh size adaptive


algorithm. This option is currently under study. One should mention that both in the evaluation of time derivatives and in the modeling of the aforementioned steep gradients the use of higher-order approximation is extremely useful. Before concluding a further remark regarding opening displacement time derivative

evaluations is in okder:

REMARK 4.9. From a physical viewpoint, C.Z. models are useful because their use allows one to eliminate the

crack tip square root singularity typical of linear elastic fracture mechanics (LEFM). When the aforementioned

singularity is removed, the resulting crack opening displacement profile takes on the form of a cusp (c.f. [13]),

as opposed to a square root function as in the LEFM solution, and the C.Z. force field v,(.x, t) results bounded

and smooth. With this in mind, the present solution strategy seeks a solution in terms of both opening

displacement and C.Z. force which belongs to a space of smooth and bounded functions. It is for this reason that

the interpolation functions for the cohesive force field have been chosen to be polynomials. However, it should

be noted that, by virtue of Eq. (2.14) and by virtue of the iterative nature of the resent solution scheme, while

converging toward a final admissible solution, the intermediate candidate solutions in terms of opening

displacement are indeed affected by the classical crack tip singularity at the mathematical crack tip. In turn, this

entails that the value of time derivatives of the opening displacement along the mathematical crack tip trajectory

is unbounded. This fact adds additional complications to the analysis of numerical error in the case of rate

dependent problems,. Furthermore, it also suggests that, in order to enhance the convergence rate, it could be

advantageous to use special interpolation functions with square root singular derivatives (cf. [27]) for the

modeling of the opening displacement in those elements that lie along the trajectory of the mathematical crack

tip.

5. Conclusions

A novel numerical technique for the solution of a class of integro-differential equations arising in the study of

dynamical crack propagation problems has been presented. The solution procedure is based on a double integration technique which combines Gauss-Jacobi quadrature with product integration. As shown in Section

4, the high accuracy of this integration scheme makes the solution strategy as a whole rather promising, also in

view of the fact that there are only few established solution techniques available for the study of integro-

differential equations and double integral equations in particular. As far as its usefulness in the solution of

dynamic crack propagation problems is concerned, the present strategy, although not applicable to problems

with finite size bodies, offers not only high accuracy but also the capability of removing the ‘noise’ which is

typical of other methods such as the finite element method and finite difference schemes. This aspect is of

extreme value in the modeling of dynamic crack propagation because it is essential to distinguish oscillatory

behavior originated by the solution technique from that which is intrinsically present in the physical system at

hand. In this respect, the proposed solution strategy is a very valuable analysis tool in the characterization of

material response given its unique capability of allowing one to focus the attention on the cohesive zone behavior. Finally, another rather interesting aspect of the proposed numerical technique is the fact that the

present formulation allows one to easily study problems with moving loads and it can be extended to the study

of dynamic crack propagation in viscoelastic bodies and/or with general opening modes. Future developments will include a more systematic study concerning higher order interpolation functions and

error estimation. Also, the implementation of adaptive mesh refinement algorithms is currently under

consideration in conjunction with the modeling of problems with multiple cracks.

Acknowledgments

The authors gratefully acknowledge the support for this research provided by the Air Force Office of Scientific Research and the National Science Foundation through the AFOSR Grant No. F49620-96-l-0294 and the NSF Grant No. DMS-9106332.

F. Costanzo, J.R. Walton I Comput. Methods Appl. Mech. En,qrg. 162 (1998) 19-48 43

Appendix A

This appendix is devoted to the derivation of Eqs. (2.19) and (2.25). Also, this appendix contains the

derivation of the closed form solution to the problem discussed in Section (4.1). The starting point of these

derivations are Eqs. (2.10)-(2.15).

With reference to Eqs. (2.5) recall that Vx, E Iw

a;@,. t) = g*27(x,, 0, M&r, - a,(t))H(&) -x,) . (A.11

In order to make use of the above equation into Eq. (2.14) the function 4(x,, t) must be substituted in place of

the function Y appearing as the innermost integrand on the left-hand side of Eq. (2.14). This substitution is not

immediate because one must find the expression to use as first argument of the function a, that matches the first

argument of integrand (T-- in Eq. (2.14). Let ARG, E Iw be the first argument of the function uC once the

substitution has been made. Then, recalling that (T (ARG, - a(t), t) = a,,(ARG,, 0, t)H(a(t) - ARG,) (c.f. Eqs. (2.11) and (2.13)), one obtains

ARG, - a(r) =x, + c(-2q + t + b,(r))

=x, -a(r)+(t-q)c+(r-q)c,

where h,(t) = I - a@)/~. The equation here implies that

ARG, = X, + (t - q)c + (Y - q)c .

Using Eq. (A.3), (TV can be expressed as

(A.2)

(A.3)

u (x, - u(r) + (t - q)c + (r - q)c, r) = F<,(x, + (t - q)c + (r - q)c, r)

X f&,(r) -x, - (t - q)c - (r - q)c)

+ uC(x, + (t - q)c + (r - q)c, r)

X H(x, - a,(r) + (t - q)c + (r - q)c)

X H@(r) - x, - (t - q)c - (r - q)c) .

Now, let

LY = - s and d,(x,, t) = di(x,, t) + dS;(x,, t) .

where

xf@,(r)-x, -(I-q)c-(r-q)c)*,

(A.4)

(A.5)

(A.61

and

d&t,) t) = ff I :+,,d & I o4 uC(x, + c(-2q + t + r), r)

X H(x, - u,(r) + (t - q)c + (r - q)c)

X H@(r) - x, (t - q)c - (r - q)c) Gr . (A.7)

Notice that the only unknowns contained in Eq. (A.6) are functions of a,(t) and a(t). Recalling that the equations at hand are used in the overall solution scheme under the assumption that the material and mathematical crack tip trajectories are known, Eq. (A.6) can be regarded as an equation that provides a straightforward evaluation of the function di and will not be discussed any further. The present derivation will therefore focus on the determination of how the cut-off functions H(x, - a,,(r) + (t - q)c + (r - q)c) and

f&u(r) -xl - (t - q)c - (r - q)c) affect the limits of integration in Eq. (A.7), thus leading to the derivation of


Eq. (2.19), and on the transformation of Eq. (2.19) into a single integral equation, thus leading to the derivation of Eq. (2.25).

For simplicity, let

Arg, =x, -a,(r)+(t-q)c+(r-q)c,

Arg, = a(r) -x, - (t - q)c - (r - q)c .

Also, let

(A.8)

(A.9)

b,“(t) = t - a,(t> a,(t)

C and b,,(t)=t+ c , (A. 10)

where b,Jt) and b,,(t) can be thought of as the material crack tip retarded and advanced time functions,

respectively. It is important to observe that, due to the requirement b,(t) < c (and i(t) < c), the functions b,&t)

and b,,(t) (as well as their counterparts defined with respect to the mathematical crack tip, i.e. b,(t) and b, (t)3)

are strictly increasing functions of time that therefore have unique inverses for every t E R’. Thus, with

reference to the innermost integral in Eq. (A.7), consider the following inequality in the variable r:

Arg,>O w r- *=bmo(r)>2q-(t+:).

Hence, the first cut-off function gives a non-null contribution only for

r>b,d(?q-(t+?)),

As far as H(Arg,) is concerned, one obtains

Arg,>O e r<b, -f2q-(t+%)).

(A.1 1)

(A.12)

(A.13)

Now notice that b,&y) <b,‘(X) for all x E [WC, in fact, being strictly monotonic functions of their arguments,

one must have

b,,$x)=x- %(X) 4X>

c <b,(x)=x-,, (A. 14)

Thus, the combined effect of the aforementioned cut-off functions is that of yielding a non-null contribution to

Eq. (A.7) only for

(A.15)

where the interval (bj$2q - (t +x, /c)), bi’(2q - (t +x, /c)) is proper, since b,:(X) <b,‘(X) V ,y E R’ due

to the strict monotonicity of b,A and b, ’ and the physical requirement that u,(t) <u(t) V t E lRf. In fact,

recalling that b,(b,‘(t)) = t, V t E Iw+

u,(t) < u(t) CJ b,$t) > b,(t)

= b,&b,i ‘(t)) > bdb, ‘(t))

w b,‘(t) > b,,;(t).

The interval of integration for the second integral in Eq. (A.7) is therefore given by

(A. 16)

rE{(O,q)n(b,d(2q-(t+: )),b,f2q-(t++)))} v,E(b;ft+$). (A. 17)

In order to give a precise evaluation to the above intersection, recall that

3 b,(t) = f - (a(t))lc and b,(t) = t + (a(t))lc.


q>b,’ t+: ( > w q’b,f24-(t+3).

This latter inequality allows one to re-write inequality (A.15) as

bj$2q-(t+>))<r<q. (A.18)

Now there remains to establish whether or not the interval (bid (24 - (t + x, /c)), q) is proper V q E (by ‘(t + x, /

c), t). To this end, observe that in order for the aforementioned interval to be proper one must have

b,6(2q - (t +x, /c)) < q which can be easily shown to imply q < b,i:(t +x, /c). Hence, observing that

b,‘(X) < b,:(X) V x E lR+, the following constraint on the variable q is obtained:

b,‘( ) t+: <q<b,; (A.19)

In turn, this implies that the interval of integration for the first integral in Eq. (A.7) is given by the following

intersection:

(A.20)

In general, it is not true that, for a given value of t, b,: (t + x, /c) < t for all x, < u(t) (in fact, 6,;: (t + x, /c) <

t-x, <a,(t)). This leads to the conclusion that Eq. (A.7) must be rewritten in the following way:

G h,(p-_(r+x,Ic))

a,(x, + (t - q)c + (r - q)c, r) ~ GF

(A.21)

y

I dr

However, because of the monotonicity of the function b,:, when a,(t) <x, < u(t) one obtains

(A.22)

Hence, under the constraint u,(t) <x, <u(t), t = min(b,:(t +x,/c), t) and Eq. (A.21) can be re-written as

follows:

d;(x,) t) = a i :;,(,+x,,., & I b;;(2q_(,+.,,c)) q.cxl + 0 - qk + cr - qk 4 & . (A.23)

This concludes the derivation of Eq. (2.19).

Now that the effects of the cut-off functions have been properly accounted for, Eq. (A.23) can be transformed into a single integral equation. In order to accomplish this task, it is convenient to re-write Eq. (A.23) in

characteristic coordinates:

*=t+:, (A.24)

Thus, Eq. (A.23) becomes

where

F(x. 4) =

(A.25)

(A.26)

(A.27)

Eq. (A.25) can be shown to be a classical Abel integral equation. In fact, making the change of variable


4’=29-x,

Eq. (A.25) becomes

Eq. (A.29) can be inverted to obtain

(A.28)

(A.29)

(A.30)

Going back to the variable of integration q and using Eqs. (A.26) and (A.27), Eq. (A.30) takes on the form

a - at

CUT

4&x - 419 4) -- - 2 q(c(9 0,s) = 0. (A.31)

Before going back to space-time coordinates, it is convenient to take the differential operator V/at inside the first integral of Eq. (A.31). This operation must be preceded by integration by parts in order to avoid the presence of an integral operator characterized by a non-integrable singularity:

q&(X - 917 9)

(A.32)

The first term on the right-hand side of Eq. (A.32) vanishes identically. In fact, for q = by ’ (,y)c(,y - q) = a(b; ’ (,y)) and &(a( q), q) = 0 for all q 3 0. Finally, using Eq. (A.32), Eq. (A.3 l), going back to space-time coordinates, one obtains

(A.33)

where the relation (Y = -c/pr, has also been used. This concludes the derivation of Eq. (2.25). The problem discussed in Section 4.1 regards a case where the primary data provided consists of the crack tip

trajectories. In particular it is assumed that

u,,,(t) = act and u(t) = pet , (A.34)

where

O<a<P<l, (A.35)

and c is the elastic shear wave speed. Also, it is assumed that no tractions are applied on the crack faces, i.e. F,(x, , t) = 0, V x, < a,(t), and that the C.Z. constitutive equations are those of the Dugdale model (c.f. [ 12]), that is,

a&c,, t) = a, = const. , (%(O < XI < a(t)) . (A.36)

Hence, from Eqs. (AS), (A.6), (A.7) and (A.23), one has that the solution of this problem in terms of opening displacement is given by

(A.37)

For simplicity, let


7-(x,,t)=r+~, 7(x,, t) 7’(X,, t> = l+a,

7(x,, t) 7”(X,, t) = ~

1+p. (A.38)

Hence, from Eqs. (2.15), (A. 10) and (A.34), the lower limits of integration in Eq. (A.37) take on the form

bL’(t+:)=?(x,,t) and ~~~(2q-(t+~))=24T~~i.r). (A.39)

Thus, for a,(t) <x < u(t) Eq. (A.37) yields

Finally, recalling that

I ( Ja-xlJb-x) dr = ((a - b)/2) log@ - (a + b) + 2V(.X - x)(b -x)] -j/(0 - x)(b -x) , (A.41)

the expression in Eq. (A.40) yields the following result

7’ - t c&(x, , t) = Q/(7’ - f)(t - 7’) + c, 2 log

(

7’ - t

t + 7’ - 711 - 2 (7’ - #‘)(t - 7’) ) ’

where

(A.42)

(A.43)

References

rt1

VI

[31

[41

PI

@I

[71

PI

191

1101

1111

[I21

[I31

r141

L1.v

Cl61

Cl71

[I81

r191

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a numerical solution technique for a class of integro-differential equations in elastodynamic crack...

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