1 Introduction

Remarks on the energy release rate for an antiplane moving crack in couple stress elasticity

Abstract

This paper is concerned with the steady-state propagation of an antiplane semi-infinite crack in couple stress elastic materials. A distributed loading applied at the crack faces and moving with the same velocity of the crack tip is considered, and the influence of the loading profile variations and microstructural effects on the dynamic energy release rate is investigated. The behaviour of both energy release rate and maximum total shear stress when the crack tip speed approaches the critical speed (either that of the shear waves or that of the localised surface waves) is studied. The limit case corresponding to vanishing characteristic scale lengths is addressed both numerically and analytically by means of a comparison with classical elasticity results.

Keywords: Couple stress elasticity, Energy release rate, Couple stress surface waves, Shielding effects, Weakening effects.

Remarks on the energy release rate for an antiplane

moving crack in couple stress elasticity

L. Morini, A. Piccolroaz and G. Mishuris

Department of Civil, Environmental and Mechanical Engineering, University of Trento,

Via Mesiano 77, 38123, Trento, Italy.

Institute of Mathematical and Physical Sciences, Aberystwyth University,

Ceredigion SY23 3BZ, Wales, U.K.

1 Introduction

Influence of the microstructure on the mechanical behaviour of brittle materials such as ceramics, composites, cellular materials, foams, masonry, bones tissues, glassy and semicrystalline polymers, has been detected in many experimental analyses (Park and Lakes, 1986; Lakes, 1993; Waseem et al., 2013; Beverige et al., 2013). In particular, relevant size effects have been found when the representative scale of the deformation field becomes comparable to the length scale of the microstructure (Lakes, 1986, 1995). These size effects influence strongly the macroscopic fracture toughness of the materials (Rice et al., 1980, 1981), and cannot be predicted by classical elasticity theory. In order to describe accurately these phenomena, generalized theories of continuum mechanics involving characteristic lengths, such as micropolar elasticity (Cosserat and Cosserat, 1909), indeterminate couple stress elasticity (Koiter, 1964) and strain gradient theories (Mindlin and Eshel, 1968; Fleck and Hutchinson, 2001; Dal Corso and Willis, 2011), have been developed and used in many experimental and theoretical studies (Radi and Gei, 2004; Itou, 2013a, b).

Indeterminate couple stress elasticity theory developed by Koiter (1964) provides two distinct characteristic length scales for bending and torsion. Moreover, it includes the effects of the rotational inertia, which can be considered as an additional dynamic length scale. Full-field solution for steady-state propagating semi-infinite Mode III crack under distributed loading has been obtained by means of Fourier transform and Wiener-Hopf analytic continuation technique by Mishuris et al. (2013). A general expression for the dynamic energy release rate (ERR) corresponding to the same steady-state antiplane problem has been derived in Morini et al. (2013), and the stability of the propagation has been analyzed by means of both maximum total shear stress (Georgiadis, 2003; Radi, 2008) and energy-based Griffith criterion (Willis, 1971). In order to investigate how the variation of the applied loading can affect both energy release rate and maximum total shear stress, in this paper the solution derived in Mishuris et al. (2013) is extended considering different distributions for the loading acting on the crack faces and moving with the same velocity as that of the crack tip. In particular, the behaviour of the energy release rate in the limiting cases when the crack tip speed approaches the shear waves speed or alternatively the Rayleigh-type surface waves speed and when the characteristic scale lengths of the material vanish is studied assuming various amplitudes for the loading profile.

The paper starts with a short description of the problem of a semi-infinite Mode III crack steadily propagating in couple stress elastic materials in Section 2, followed by an overview of results concerning the dispersive propagation of antiplane surface waves. For both antiplane and in-plane problems, indeterminate couple stress theory predicts the existence of surface waves analogous to Rayleigh waves observed in plane classical elasticity (Ottosen et al., 2000). In the paper, these are referred to as couple stress surface waves, and it is demonstrated that the critical maximum value for the crack tip speed introduced in Mishuris et al. (2013) and Morini et al. (2013) coincides with the minimum velocity for couple stress surface waves propagation in the material. A velocity range for the crack propagation, denominated for brevity sub-Rayleigh regime, is introduced: in cases where subsonic couple stress surface waves propagation is detected, a maximum crack tip velocity smaller than shear waves speed in classical elastic materials is defined and explicitly evaluated as a function of the microstructural parameters, while in cases where the surface waves propagation can be only supersonic the limit value for the crack tip speed is given by . The analytical full-field solution of the problem is then addressed in Section 3 using Wiener-Hopf technique (Noble, 1958). The crack is assumed to propagate in the sub-Rayleigh regime under generalized distributed loading conditions of variable amplitude. In Section 4, the dynamic energy release rate is evaluated explicitly by means of the method developed by Freund (1972) and extended by Georgiadis (2003), Morini et al. (2013) and Gourgiotis and Piccolroaz (2013) to static and dynamic problems in couple stress elasticity.

The effects of the microstructure as well as the influence of the loading profile gradients on displacements, stress fields, maximum total shear stress and energy release rate are illustrated and discussed by means of several numerical examples in Section 5. A strong localization of the applied loading around a maximum near to the crack tip is not associated with to higher levels of the shear traction and to a larger crack opening. This behaviour, detected by maximum total shear stress analysis, means that in couple stress elastic materials the action of loading forces concentrated near to the crack tip is shielded by the microstructure. This shielding effect is confirmed also by the energy release rate analysis. It is shown indeed that the energy release rate decreases as the applied loading is more and more localized near the crack tip.

The behaviour of the energy release rate shows that if the distance between the position of application of the maximum loading and the crack tip grows, in presence of couple stress more energy is provided for propagating the crack at constant speed with respect to the classical elastic case, and then the fracture propagation is favored. Also this weakening effect is due to the microstructural contributions, and it is in agreement with the results detected in Gourgiotis et al. (2011) for plane strain crack problems under concentrated shear loading. Numerical results illustrate also that, when the crack tip speed approaches the shear waves speed in classical elastic materials or alternatively the couple stress surface waves speed, the energy release rate assumes a finite limit value depending on the microstructural parameters. Conversely, if the characteristic lengths vanish, for any arbitrary loading profile the value of the energy release rate becomes identical to that of the classical elastic case. This is an important proof of the fact that, if the microstructural effects are negligible, the material behaviour is identical to that of a classical elastic body for what concerns crack propagation. This result, observed in all the proposed numerical examples, is validated by means of the analytical evaluation of the limit of the energy release rate for vanishing characteristic lengths reported in Section 6. In this Section, indeed, it is demonstrated that, if the characteristic lengths vanish, for any arbitrary applied loading the energy release rate for couple stress materials tends to the energy release rate associated to an antiplane steady-state crack in classical elasticity.

2 Problem formulation

A Cartesian coordinate system centered at the crack-tip at time is assumed. The micropolar behaviour of the material is described by the indeterminate theory of couple stress elasticity (Koiter, 1964). The non-symmetric Cauchy stress tensor can be decomposed into a symmetric part and a skew-symmetric part , namely . The reduced tractions vector and couple stress tractions vector are defined as

(1)

where is the couple stress tensor, denotes the outward unit normal and . For the dynamic antiplane problem, stresses and couple stresses can be expressed in terms of the out-of plane displacement :

(2)
(3)
(4)

where denotes the Laplace operator, is the rotational inertia, is the elastic shear modulus, and the couple stress parameters, with . Both material parameters and depend on the microstructure and can be connected to the material characteristic lengths in bending and in torsion (Radi, 2008), namely and . Typical values of and for some classes of materials with microstructure can be found in Lakes (1986, 1995).

Substituting expressions (2), (3) and (2) in the dynamic equilibrium equations (Mishuris et al., 2013), the following equation of motion is derived:

(5)

2.1 Steady-state crack propagation

We assume that the crack propagates with a constant velocity straight along the -axis and is subjected to reduced force traction applied on the crack faces, moving with the same velocity , whereas reduced couple traction is assumed to be zero (Georgiadis, 2003),

(6)

We also assume that the function decays at infinity sufficiently fast and it is bounded at the crack tip. These requirements are the same requirements for tractions as in the classical theory of elasticity.
It is convenient to introduce a moving framework , . By assuming that the out of plane displacement field has the form , then the equation of motion (5) writes:

(7)

where is the crack velocity normalized to the shear waves speed , and is the normalized rotational inertia defined in Mishuris et al. (2013).
According to (1), the non-vanishing components of the reduced force traction and reduced couple traction vectors along the crack line , where , can be written as

(8)

respectively. By using (2), (2), (3), and (8), the loading conditions (6) on the upper crack surface require the following conditions for the function :

(9)

Ahead of the crack tip, the skew-symmetry of the Mode III crack problem requires

(10)

Note that the ratio enters the boundary conditions (9)-(10), but it does not appear into the governing PDE (7).

2.2 Preliminary analysis on couple stress surface waves propagation

In couple stress elastic materials the existence of surface waves has been demonstrated for both in-plane and antiplane problems (Ottosen et al., 2000). Considering a material occupying the upper half-plane under antiplane deformations, the solution of the governing equation (5) is assumed in the form:

(11)

where is the amplitude, k is the wave number and the radian frequency. Substituting (11) into (5) the following ODE is obtained:

(12)

where is the shear wave speed for classical elastic materials, and the superscript indicates the derivative with respect to variable. Equation (12) can be rewritten in the form

(13)

where , is the couple stress surface waves speed and is the normalized rotational inertia introduced in the previous section. Equation (13) admits the following bounded solution in the upper half-plane, vanishing for

(14)

where

(15)
(16)
(17)

Similarly to the procedure commonly carried out for studying Rayleigh waves in classical elasticity, traction-free boundary conditions are imposed at the free surface:

(18)

by using relations (2), (3), (2) together with expression (11), equation (18) becomes

(19)
(20)

Substituting expression (14) into equations (19) and (20), the following system of two algebraic equations for the unknown constants and is derived

(21)

where and the matrix is given by

the system (21) possesses non-trivial solutions only if

(22)

Expression (22) is the dispersion relation for antiplane couple stress surface waves, and the propagation velocity corresponding to a given value of the frequency or alternatively of the wave vector can be evaluated by solving this equation.

Figure 1: Variation of the normalized Rayleigh waves speed with the normalized frequency.
Figure 2: Variation of the normalized Rayleigh waves speed with the normalized wave vector.

The normalized wave speed is shown in Figs. 1 and 2 as a function of the normalized frequency and the normalized wave number , respectively. Different values for the characteristic parameter and for the normalized rotational inertia have been considered.

For small values of the rotational inertia, the value of the couple stress surface waves speed is always greater than the shear waves velocity in classical elastic materials, and then the couple stress surface waves propagation is supersonic for any value of the wave number and frequency. In particular, for the case of vanishing rotational inertia , the wave propagation is dispersive and supersonic with monotonically increasing speed, as it as been detected in Ottosen et al. (2000) and Askes and Aifantis (2011). As the rotational inertia increases, the phase speed behaviour changes: the values of may become smaller then , and it decreases with the frequency and the wave number until a limit value corresponding to and depending by and is reached. This means that for large values of the rotational inertia and high frequencies the couple stress surface waves propagation becomes subsonic, and a minimum value for the phase speed is individuated for .

For , the dispersion relation (22) exhibits the following asymptotic behaviour

(23)

The minimum value for the normalized surface waves speed, depending on and , is given by the value of for which the coefficient of the leading order term of (23) vanishes, and then it can be evaluated by solving the equation:

(24)

By means of simple algebra, it can be verified that equation (24) is equivalent to

(25)

The function introduced in expression (25) is the same defined in the Wiener-Hopf factorization of steady-state crack propagation problem in Mishuris et al. (2013), where the regime is studied and a critical limit value for the crack tip speed is individuated by relation (25). Consequently, the minimum couple stress surface waves propagation velocity coincides with the critical value for steady-state crack propagation, and the condition introduced in Mishuris et al. (2013) defines the transition between two different ranges of velocities, which further in the text will be called sub-Rayleigh and super-Rayleigh propagation regimes. These regimes are reported in the plane in Fig. 3A.
For the case the dispersion curves shown in Fig.1 are identical to that obtained in Mishuris et al. (2013) for the shear waves. Consequently, for the couple stress surface waves degenerate to shear waves and subsonic and sub-Rayleigh regimes are equivalent. This can be demonstrated by the fact that for the eigenvalue given by (16) vanishes, and only the term of the matrix (21) depending by is non-zero: in that case the factor is also zero and the solution coincides with the planar shear waves solution.

Figure 3: A): Sub-Rayleigh and super-Rayleigh regimes in the plane. The continuous line coincides with the transition between subsonic and supersonic ranges. B): Variation of as a function of .

In Fig. 3A it can be observed that for small values of the rotational inertia the crack propagation is both subsonic and sub-Rayleigh, and the limit value for the normalized crack tip speed is . As increases, the limit speed for sub-Rayleigh regime becomes smaller than for subsonic regime, and the critical velocity is determined by solving equation (24) or alternatively (25). The limit value such that for the maximum normalized velocity for sub-Rayleigh regime is given by is plotted in Fig. 3B as a function of the microstructural parameter .

3 Full-field solution

The following form for the loading applied on the crack faces is assumed

(26)

where is the Gamma function. It is important to note that the resultant force applied to the upper crack face is , indeed

(27)

Moreover, the maximum of the distributed traction is attained at . The normalized loading profile is reported in Fig. 4 as a function of for several values of the exponent and of the ratio . Note that for , the loading is bounded but different from zero at the crack tip, for the loading tends to zero at the crack tip. Moreover, as decreases, the loading is more and more concentrated around a peak close to the crack tip.

Sub-Rayleigh regime of propagation defined in previous Section is considered, so that

(28)

where the critical value is obtained by the solution of equation (24) or (25) for given values of and .

Figure 4: Distributed loading applied to the crack faces

3.1 Solution of the Wiener-Hopf equation

Since the Mode III crack problem is skew-symmetric, only the upper half-plane () is considered for deriving the solution. The direct and inverse Fourier transforms of the out-of-plane displacements are

(29)

respectively, where is a real variable and the line of integration will be defined later. Applying the Fourier transform (29) to equation (9) and using the general factorization procedure illustrated in details in Mishuris et al. (2013), the following functional equation of the Wiener–Hopf type can be obtained

(30)

where is analytic in the lower half complex -plane, and it is given by

(31)

where

(32)
(33)
(34)
(35)

and is defined in (25). The function has been factorized in Mishuris et al. (2013) as , where , and and are analytic in the upper and lower half-planes, respectively. Since sub-Rayleigh regime is investigated, is positive for all values of crack tip speed and microstructural parameters considered.

The Wiener-Hopf equation (30) can then be rewritten in the form:

(36)

The right-hand side of (36) can be easily split in the sum of plus and minus functions. Indeed, we use the fact that the function is analytical in the point and thus can be represented as

(37)

where

(38)

Note that the function exhibits the following asymptotic behaviour:

(39)

Taking this fact into account, the right-hand side of the equation (36) can be written in the form

(40)

where

(41)

and

(42)

The unknown constants are computed by evaluating the integrals:

(43)

where is an arbitrary contour centered at the point and lying in the analyticity domain. Substituting (40) in (36), we finally obtain:

(44)

The left and right hand sides of (44) are analytic functions in the upper and lower half-planes, respectively, and thus define an entire function on the -plane. The Fourier transform of the reduced force traction ahead of the crack tip and the crack opening gives and as . Therefore, both sides of (44) are bounded as and according to the Liouville’s theorem must be equal to a constant in the entire -plane. As a result, we obtain

(45)

The constant is determined by the condition that the displacement is zero at the crack tip , that is

(46)

which leads to

(47)

where is given by

(48)

Note here that according to (39), , that is the standard balance condition for this problem. The equivalence between the two alternative expressions for the constant reported in relation (47) can be easily demonstrated by applying the Cauchy integral theorem (Arfken and Weber, 2005).

3.2 Analytical representation of displacements, stresses and couple stresses

The reduced force traction ahead of the crack tip and the crack opening can be obtained applying the inverse Fourier transform (29) to expressions (45). Since the integrand does not have branch cuts along the real line, the path of integration coincides with the real -axis. Further, we introduce the change of variable , thus obtaining

(49)
(50)

The Fourier transform of stress (symmetric and skew-symmetric) and couple stress fields can be derived from (2), (3) and (2) namely

(51)
(52)
(53)

The inverse Fourier transform can be performed as explained above, thus obtaining for

(54)
(55)
(56)
(57)

4 Dynamic energy release rate

In this Section the dynamic energy release rate for a Mode III steady-state propagating crack in couple stress elastic materials under distributed loading conditions given by expression (26) is evaluated.

4.1 Explicit evaluation

The general expression for the dynamic J-integral in couple stress elasticity, including also the rotational inertia contribution, has been derived and proved to be path-independent in the steady-state case assuming traction free crack faces by Morini et al. (2013). Considering the moving framework with the origin at the crack tip introduced in Section 2, the J-integral for a steady state crack propagating along the axis is given by:

(58)

where is an arbitrary closed path surrounding the crack tip, and is the Cartesian component directed along the axis of the outward unit vector normal to , defined by . Since the distributed loading of profile (26) acting on the crack line is assumed, in our case the contribution of the crack faces must be taken into account, and then in principle the J-integral (58) is not path-independent. Nevertheless, in this Section the J-integral is used to determine the dynamic energy release rate evaluating the limit for in (58) (Freund, 1998). This means that the asymptotic expressions of displacement and stresses can be used for calculating the energy release rate. Remembering the asymptotics behaviour of displacement and stresses for antiplane cracks reported in Morini et al. (2013) and the loading function (26), it is easy to verify that in the limit the contribution of the crack faces to the J-integral (58) vanishes.

We assume the rectangular-shaped integration contour considered in Morini et al. (2013), and in order to evaluate the energy release rate we allow the height of the path along the direction to vanish and we make the limit . Assuming this type of contour, first introduced by Freund (1972), solely asymptotic expressions of displacements and stress fields are required for evaluating the energy release rate. Moreover, upon this choice of path, allowing the height of the rectangle along the direction to vanish, the integral becomes zero and then the energy release rate is given by

(59)

Since boundary conditions (9) together with anti-symmetry conditions (10) provide that the reduced traction is zero along the whole crack , the dynamic energy release rate for a steady-state Mode III crack becomes:

(60)

In the limit , the Fourier transform of displacements, total shear stress and couple stress fields derived in Section 3 assume the following behaviour:

(61)
(62)
(63)

further, we consider the following transformation formula (Roos, 1969):