Intrinsic material length, Theory of Critical Distances and Gradient Mechanics: analogies and differences in processing linear-elastic crack tip stress fields

ABSTRACT

The Theory of Critical Distances (TCD) is a bi-parametrical approach suitable for predicting, under both static and high-cycle fatigue loading, the non-propagation of cracks by directly post-processing the linear-elastic stress fields, calculated according to continuum mechanics, acting on the material in the vicinity of the geometrical features being assessed. In other words, the TCD estimates static and high-cycle fatigue strength of cracked bodies by making use of a critical distance and a reference strength which are assumed to be material constants whose values change as the material microstructural features vary. Similarly, Gradient Mechanics postulates that the relevant stress fields in the vicinity of crack tips have to be determined by directly incorporating into the material constitutive law an intrinsic scale length. The main advantage of such a method is that stress fields become non-singular also in the presence of cracks and sharp notches. The above idea can be formalized in different ways allowing, under both static and high-cycle fatigue loading, the static and high-cycle fatigue assessment of cracked/notched components to be performed without the need for defining the position of the failure locations a priori.

The present paper investigates the existing analogies and differences between the TCD and Gradient Mechanics, the latter formalized according to the so-called Implicit Gradient Method, when such theories are used to process linear-elastic crack tip stress fields.

NOMENCLATURE

a

Crack/notch depth

c

Characteristic length related to shape function α

Vector of the nodal displacement components

Gradient Mechanics length scale parameters

Vector of external forces

n

Normal unit vector

r, θ

Polar coordinates

r_n

Notch root radius

vector containing the nodal gradient-enriched stress components

u

Displacements

w_g

Gross width

D

Pseudo-diffusivity matrix

E

Young's modulus

F_u

Ultimate axial force

K

Elastic stiffness matrix

K_Ic

Plane strain fracture toughness

L

Material characteristic length

M

Pseudo-mass matrix

N

Matrix containing the shape functions

α

Shape Function

β

Notch opening angle

δ

Dirac function

λ,μ

Lamé constants

Relevant (scalar) stress invariant

Vector containing the components of the (classical) stress tensor

σ_eff

Effective stress

σ_ij

Classical stress components

σ₀

Inherent material strength

σ_UTS

Ultimate tensile stress

σ₁

Maximum principal stress

ν

Poisson's ratio

ΔK_th

Range of the threshold value of the stress intensity factor

Δσ₀

Plain fatigue limit stress range

Δσ₁

Range of maximum principal stress

Δσ_eff

Range of the effective stress

Δσ_n

Range of the gross nominal stress

INTRODUCTION

Research into the mechanical behaviour of materials and structures – their deformation and failure under applied forces – has traditionally been conducted within two rather different disciplines: materials science and engineering mechanics. Mechanics-based models are able to describe the complex variations of stress, energy, etc. in bodies under load, but these models usually treat the material simplistically, as a homogeneous continuum having certain mechanical properties. Materials scientists recognize that materials possess structure at various size scales, from atoms to grains, but they generally measure mechanical properties using simple test specimens, typically loaded in uniaxial tension, thus creating a stress field which rarely occurs in real structures and components.

Clearly an interdisciplinary approach is needed, bringing together the complexities of material structure with the complexities of real stress fields. Not to do so leads to errors which are only too apparent. Many current problems in this field arise when the physical dimensions, or the gradient of the stress field, become similar in scale to the material's structure. Examples include short fatigue cracks whose growth rates are anomalously fast compared with fracture mechanics predictions¹ and the increase in measured hardness with decreasing indenter size.² In addition, many phenomena which are traditionally treated with empirical models, such as notch sensitivity, can be understood more fundamentally by appreciating the role of material structure.

So how to combine materials science with engineering mechanics? One approach is to attempt to introduce material structure specifically into a mechanics model. Pioneers in this field include John Knott, who created models in which material microstructure was incorporated along with near crack tip stress field information to predict toughness, most notably in the Ritchie, Knott and Rice (RKR) model of cleavage fracture in ferritic steels.³ Such approaches have had their successes, but have proved difficult to generalize, given the huge complexity involved in predicting local conditions of stress and strain in a model which incorporates structure. Recent advances in computing power have helped, and we can at last imagine a future in which all aspects of the material, from atomic structure upwards, can be incorporated into a mechanics-based prediction. Such a model would give invaluable insight for researchers but would probably never be the solution of choice for an engineering designer.

Another approach, and the one which is the subject of this paper, starts from traditional continuum mechanics and asks how some information about material structure might be introduced in the most simple way possible. The aim is to carry on using, for example, a finite element analysis with a simple material model, but to bring in some information about the scale of those material features which affect the mechanical behaviour under consideration, be it elastic deformation or fatigue crack growth.

The history of this approach applied to predicting fatigue and other failure modes goes back to the inter-war and immediate post-war years, with the work of Neuber⁴ who proposed a solution to the stress-gradient problem which involved averaging the stress over a certain distance, this distance being a constant for a given material. Peterson⁵ modified this approach, using the stress at a fixed point, located a certain distance from the point of maximum stress in the body. Thus were created the so-called Point Method (PM) and Line Method (LM), which will be discussed in more detail below. As Pluvinage⁶ pointed out, these two methods are special cases of a general approach which involves forming an integral function of the entire stress distribution along with a weighting factor: critical length information is contained in the weighting factor and the limits of the integration.

As time went on, the same basic concepts were rediscovered and developed in parallel in several different disciplines, including monotonic fracture in metals,⁷ polymers⁸ and fibre composites.⁹ Fracture mechanics concepts, begun by the well-known work of Griffith and Irwin, developed over exactly the same time period, but rarely overlapped with these critical distance approaches. However, a crucial link between the two, allowing the critical distance to be estimated from fracture mechanics data, was made by Whitney and Nuismer in 1974 in the field of fibre composite fracture⁹ and independently, some time later, by Tanaka¹⁰ in the area of metal fatigue. This link allowed the continuum mechanics approaches of linear-elastic fracture mechanics (LEFM) and elastic plastic fracture mechanics (EPFM) to be modified by introducing the length constant which we now call the critical distance, L.

In parallel with the above, researchers studying material elasticity, damage and plasticity also realized the length scale information was needed to make sense of apparent changes in elastic modulus and yield strength with specimen size and stress gradient. Closely related to the TCD approaches described above are so-called ‘non-local’ theories, whereby spatial averages of relevant state variables are included in the governing equations (see e.g. the work of Eringen¹¹ in elasticity and that of Pijaudier-Cabot and Bažant¹² in damage). Similar to the TCD concept, weighted averages are taken, but the weight functions tend to take more general forms. The spatial integrals of these non-local theories can also be rewritten as (a series of) higher-order spatial gradients, thus resulting in gradient elasticity,^13,14 gradient damage^15,16 or gradient plasticity theories.^17,18 It has been shown on various occasions that such integral-enriched and gradient-enriched continuum material models can be used (i) to describe crack-tip fields without singularities,¹⁹ (ii) to capture the size-dependent response of real-life specimens^20,21 and (iii) to simulate strain-softening phenomena without the extreme (and unphysical) dependence on the applied finite element size that is encountered for standard damage and plasticity theories.²²

The microstructural basis of the integral-type or gradient-type enrichment terms has been subject of intensive research. Generally, the relative importance of these terms is governed by one or more internal length scales, which in turn can be related to the geometry of discrete lattices,²³ the representative volume element (RVE) size of random materials²⁴ or the dislocation properties of the material.²⁵ The practical realization of these theories usually involves simplifications essentially similar in nature to the critical distance approaches mentioned above, i.e. the incorporation of a material constant with units of length into a continuum mechanics model. However, the similarities between critical distance theories and gradient-enriched continuum theories are even stronger in elastic analysis: both sets of theories can be considered as a two-step approach, whereby the specimen is first analysed using a standard elasticity model, after which the results are post-processed (either by evaluating the stress at the critical distance or by applying the gradient-enrichment) for interpretation of the specimen's strength and integrity. These similarities between critical distance approaches and gradient-enriched elasticity warrant a more in-depth study and have, thus, inspired the present contribution.

THE INTRINSIC LENGTH SCALE CONCEPT

As noted above, many researchers have discovered that simply introducing a parameter with units of length into a continuum mechanics approach can be effective in improving predictions, whether of elastic and plastic deformation or of failure processes such as brittle fracture and fatigue. We might reasonably ask ourselves why this rather simplistic approach has been so successful, and what meaning or significance can we attach to the value of the length constant.

In these approaches, the magnitude of the constant L is generally found empirically, being the best choice to bring predictions in line with experimental data. It may be explicitly stated, as in the PM and LM of the TCD (explained in more detail below), or it may be implied, for example, in the nature of the weighting function used. Its physical interpretation may or may not be obvious. For example, in some cases L has been found to coincide exactly with a microstructural distance which is of clear relevance to the problem. The best example of this, perhaps, is the cleavage fracture of steels, where L is found to be exactly equal to the grain size,²⁶ allowing a direct link to the mechanistic theory of the RKR model, in which fracture initiates in the first grain boundary ahead of the crack tip.³ Previously, it has been argued that the process of crack extension is discontinuous in nature, occurring in a series of jumps, sometimes controlled by the spacing of physical barriers in the microstructure: the critical distance has been interpreted as the jump distance in a modified fracture mechanics approach.²⁷

The smallest conceivable length scale in a material is the spacing between the atoms, so one might expect L to reduce to this value in the default situation where there is no other structure present, such as in amorphous glassy materials, and in nanomaterials, and there is indeed some evidence for this.²⁸ In many cases, however, the exact physical interpretation of L may be less obvious; in these cases the establishment of a value for L may give insights into the physical mechanisms at work which determine the particular mechanical behaviour in question. For example the establishment of an L value for brittle fracture of bone²⁹ confirmed the role of particular microstructural features (osteons) as barriers to crack growth in that material. Quasibrittle materials such as fibre composites often display L values for fracture which are larger than any apparent microstructure, such as the sizes and spacings of fibres.⁹ In these cases L is found to be of the same order of size as the process zone which develops in front of the crack or notch prior to failure, and which strongly influences the material's toughness. This itself is controlled by the underlying structure and properties of the constituents, though in a less obvious way.

LEFM-BASED FORMALIZATIONS OF THE TCD

From a mathematical point of view, the LEFM formalizations of the TCD suitable for addressing both the static and the high-cycle fatigue problem are based on the same fundamental ingredient: to accurately predict the non-propagation of a crack, the entire stress field acting on the material in the vicinity of the crack tip itself has to be taken into account through a suitable effective stress, σ_eff, whose definition depends on the features of the adopted integration domain.³⁰

The different formalizations of the LEFM-based TCD take then as a starting point the hypothesis that, independently of the level of ductility of the assessed material, the relevant stress fields have to be determined by adopting a linear-elastic constitutive law. This should make it evident that, as long as the TCD is used to estimate either the static strength of brittle materials or the fatigue limit of ductile metals, the above assumption results in stress analyses that are always characterized by an adequate level of accuracy. As to the influence of plastic deformations, even though the reasoning summarized in the present paper will be confined solely to linear-elastic situations, it is worth recalling here that the TCD was seen to be capable of accommodating different kinds of nonlinearities, resulting in accurate estimates also when such a theory is used to estimate both static strength^31–33 and fatigue lifetime^34,35 of sharply notched ductile materials, that is, in those situations in which the physical processes leading to final breakage are characterized by evident irreversible deformations.

The second common ingredient of the different formalizations of the TCD is that the propagation of cracks is assumed to be avoided as long as the effective stress is lower than (or, at least, equal to) a specific inherent strength that is material-dependent.

According to the considerations reported above, it is important to point out that, independently of the ambit in which the TCD is used, this theory predicts the propagation (or non-propagation) of cracks through a bi-parametrical re-analysis of the linear-elastic stress fields acting on the material in the vicinity of the crack tips, the critical distance and the inherent material strength being the two adopted parameters.

Turning back now to the calculation of the effective stress, as briefly mentioned above, the TCD idea can be formalized in different ways,³⁰ provided that a convenient material characteristic length is defined. If attention is initially focussed on the problem of estimating high-cycle fatigue strength of engineering materials containing stress raisers, the critical distance defining the size of the integration domain to be used to calculate the effective stress can be derived as follows^10,36–38:

• (1)

where is the range of the threshold value of the stress intensity factor and is the plain fatigue limit (both determined under the same load ratio, R, as the one characterizing the constant amplitude load history applied to the component being assessed).

Consider now the notched sample sketched in Fig. 1 and assume that it is subjected to an external cyclic force resulting in a cyclic stress field damaging the fatigue process zone. If the TCD is used, as done by Peterson,⁵ in the form of the PM, then the range of the effective stress, Δσ_eff, takes on the following value^10,38 (Fig. 1b):

• (2)

Figure 1. Different formalizations of the LEFM-based TCD.

If the TCD is formalized instead according to Neuber's idea⁴ (i.e. the so-called Line Method), then the range of the effective stress is calculated by averaging the range of the maximum principal stress over a line having length equal to 2L,^10,37,38 i.e. (Fig. 1c):

• (3)

By taking full advantage of Sheppard's intuition,³⁹Δσ_eff can also be determined by averaging over a semicircular area centred at the notch tip and having radius equal to L. Such a form of the TCD is known as the Area Method (AM) and it can be formalized as follows³⁸ (Fig. 1d):

• (4)

Finally, it is worth recalling here that, according to Bellett et al.,⁴⁰ the range of the effective stress can alternatively be calculated by averaging the range of the first principal stress, Δσ₁, in a hemisphere centred at the apex of the stress raiser and having radius equal to 1.54L: this is known as the Volume Method (VM).

The accuracy of the different formalizations of the TCD reviewed above was extensively checked through experimental results generated by testing both standard notches^41,42 and real components⁴³: such a systematic validation exercise allowed us to prove that the TCD is highly accurate in estimating high-cycle fatigue strength in the presence of any kinds of stress concentrators (i.e. cracks as well as short, blunt and sharp notches), resulting in predictions falling within an error interval of about 20%.

Turning now to the problem of estimating the static strength of engineering materials weakened by stress concentration phenomena, similar to what has been suggested for the high-cycle fatigue case, the characteristic length to be used to define the size of the integration domain takes on the following value^9,30,44:

• (5)

where K_Ic is the LEFM plane strain material toughness and is the inherent material strength, the latter material property being treated as a defect-free tensile strength that, in the most general case, is different from the conventional material ultimate tensile strength. ³⁰ The experimental procedure to determine will be described below in great detail.

As soon as the critical distance value, L, is known, by following the same reasoning as the one followed above to derive Δσ_eff under fatigue loading, the effective stress, σ_eff, to be used to address the static problem is suggested as being calculated according to the definitions reported below^30,44 (Fig. 1):

• (6)

• (7)

• (8)

In other words, the TCD can specifically be formalized to perform the static assessment of notched components by simply replacing in Eqs (2) to (4) the range of the maximum principal stress, Δσ₁(θ, r), with the local maximum principal stress, σ₁(θ, r), resulting from the application of the in-service static loading (Fig. 1).

As briefly mentioned above, the trickiest problem to be addressed when the TCD is attempted to be used to perform the static assessment of engineering materials is the definition of the proper value for the inherent material strength: according to definition (5), it is evident that σ₀ plays a role of primary importance, because the inherent material strength is needed to calculate critical distance L. As to the expected values for σ₀, classic brittle materials, such as ceramics, as well as some other quasi-brittle materials, such as fibre composites, are characterized by an experimental value of the inherent material strength that approaches the ultimate tensile stress.^9,44 On the contrary, when final breakage is preceded by a certain amount of plastic deformation, σ₀ takes on a value which is larger than σ_UTS.^30,34 Furthermore, the inherent material strength is seen to be different from the ultimate tensile stress also in those cases in which the failure of unnotched specimens occurs by different mechanisms, for instance by the propagation of pre-existing micro-defects or by plastic instability.^44–46 Under the above circumstances, the only reliable way then to determine the inherent material strength, σ₀, correctly is by testing specimens containing stress risers whose presence results in different stress distributions in the process zone.^45,46 Such an experimental strategy suitable for estimating σ₀ is summarized by the sketch of Fig. 2: given the material, critical distance L and inherent strength σ₀ can directly be estimated through the point at which the linear-elastic stress-distance curves, determined in incipient failure condition, intersect each other.

Figure 2. In-field procedure to estimate both σ₀ and L under static loading.

The validation exercise³⁰ performed to systematically check the accuracy of the different formalizations of the LEFM-based TCD in predicting the static strength of engineering materials weakened by different types of stress raisers proved that such an approach is highly accurate, resulting, similar to the high-cycle fatigue case, in estimates falling within an error interval of ±20%.

In the work following the similarities between Gradient Mechanics and TCD will be examined with the TCD PM being used.

CONTINUUM AND FINITE ELEMENT EQUATIONS OF GRADIENT ELASTICITY

The present discussion on gradient-enriched mechanics theories is limited to so-called gradient elasticity theories, whereby the governing equations of infinitesimal elasticity are extended with additional spatial gradients of the displacements. The first systematic development of such theories was made by Mindlin in the 1960s.^13,23 He started by formulating a model whereby macro deformation, micro deformation and gradient of the latter were all included independently. In its most generic, anisotropic format it contained no less than 903 constitutive parameters, but fortunately he also used a number of simplifications (such as isotropy and the coupling of micro and macro deformation), such that in its simplest form the equilibrium equations according to Mindlin¹³ can be written as

• (9)

where standard index notation is used and subscripts following a comma denote spatial derivatives. Furthermore, and are the Lamé constants, u are the displacements, body forces are ignored and the two length scale parameters and that accompany the higher-order gradients represent the microstructure of the material.

In the 1980s, Aifantis formulated simple gradient-enriched extensions of plasticity theory by which the finite width of shear bands could be described.^17,18 Based on these results a gradient-enriched extension of elasticity theory was formulated,^14,19,47 with a view of dispensing with the crack tip singularities that plague standard elasticity models. Although the motivations and derivations of the theories of Mindlin and Aifantis follow quite distinct paths, a formal link is easily established: the Aifantis theory can be retrieved from the more general Mindlin theory by taking simply , so that Eq. (9) becomes as

• (10)

where replaces and . However, the implications of this apparent simplification for analytical and numerical solution strategies are significant, as was explored in depth by Ru and Aifantis¹⁹ and as will now be shown. The specific structure of Eq. (10) allows the various derivatives to be factorized, by which the fourth-order partial differential equations of expression (10) can be rewritten as a set of second partial differential equations that are to be solved sequentially. First, the displacements u^c from classical elasticity theory are obtained via

• (11)

which are then used in a diffusion-type equation to solve for the gradient-enriched displacements as

• (12)

It is easily verified that Eq. (10) is retrieved by substituting Eq. (12) back into Eq. (11). Thus, this particular version of gradient elasticity due to Aifantis (i.e. where in the Mindlin model) can be reformulated such that the results of classical elasticity (cf. Eq. (11)) are post-processed to introduce the gradient enrichment (cf. Eq. (12)).

With appropriate boundary conditions, the solution of Eqs (11) and (12) is the same as the solution of Eq. (10). Unfortunately, in a finite element context the format of the boundary conditions cannot be chosen freely but instead follows from the variational principles, and for this reason it has been suggested to take the derivative of Eq. (12) and multiply this with the constitutive tensor.⁴⁸ Thus, Eq. (12) is replaced by

• (13)

where are the gradient-enriched stresses and are the stresses computed from the displacements u^c of classical elasticity.

Compared with Eq. (12), the format of Eq. (13) is somewhat of a setback in that more computational effort is required: the number of stress components exceeds the number of displacement components in 2D and 3D, therefore the system of equations corresponding to Eq. (13) is larger than that of Eq. (12). However, it has also been recognized that in order to make an assessment on fracture growth, the availability of a relevant stress invariant (rather than the entire stress tensor) is usually enough. Thus, for efficiency Eq. (13) can also be replaced by

• (14)

where is a relevant (scalar) stress invariant such as the maximum principal stress or the von Mises stress.

For the finite element discretization of the relevant equations, we will assume the two-step approach discussed above. The advantage is that all of the expressions (11)–(14) are second-order, so that the usual C⁰-continuous finite element shape functions can be used (in contrast, discretization of the fourth-order Mindlin equations would require a much more cumbersome C¹-continuous interpolation; see Refs [48–50] for further discussion of this issue). As already mentioned earlier, Eq. (11) represents the equations of classical linear elasticity, and its finite element discretization is straightforward and well-known – usually indicated as a system of equations as

• (15)

where K is the usual elastic stiffness matrix, is the vector of external forces (including contributions of body forces as well as externally applied tractions) and is a vector that contains the nodal displacement components according to classical elasticity theory.

Finite element implementation of the gradient-enrichment equations (12), (13) or (14) is straightforward as well. The lower-order and higher-order terms on the LHS of these expressions lead to contributions in the discrete equations that have the structure of the usual mass matrix and of the diffusivity matrix, respectively. More specifically, use of Eq. (13) leads to a finite element system as

• (16)

where is a pseudo-mass matrix, is a pseudo-diffusivity matrix, N is a matrix that contains the finite element shape functions used for the interpolation of the gradient-enriched unknowns, is a vector with the nodal gradient-enriched stress components and the vector collects the components of the (classical) stress tensor. Using Eq. (14) instead of Eq. (13) requires only a few modifications: the relevant stress invariant must be replaced by the stress vector , and the size of the shape function matrix N must be adjusted – its size is if Eq. (13) is used in 2D, where n is the number of nodes in the finite element mesh, and the size of N is in case Eq. (14) is used.

THE IMPLICIT GRADIENT METHOD: A LINK BETWEEN THE TCD AND GRADIENT MECHANICS

A formal link between the TCD and gradient mechanics can be established by an in-depth investigation of mechanical non-locality which is known as the Implicit Gradient Method (IGM). Starting from the considerations proposed in Ref. [51] it can be argued that a possible unification of the TCD theories or non-local theories could be expressed as follows: the strength at a given point P is somehow related to a proper combination of stress components evaluated close to point P, but not necessary only at point P. Thus, the strength can be directly related to an effective value of the stress, and such an effective value is a function of the stress in the entire considered volume V:

• (17)

where f is a general function and X is a variable point in the volume V.

It was previously pointed out that including this assumption in a general approach involves forming an integral function of the entire stress distribution along with a weighting factor,^6,51 the intrinsic length and the material properties being involved in the weighting factor assessment.

According to the non-local theory as proposed by Pijaudier-Cabot and Bažant,¹² such an effective stress σ_eff can be defined by averaging its elastic local counterpart weighted by a function α:

• (18)

where, according to the former definition, is a relevant scalar stress invariant, usually an equivalent stress and the weight function α should generally be an isotropic function of the distance |PX|. It can be easily assessed that the TCD approaches are all special cases of this comprehensive assumption.

For instance, let us consider the PM. In a general domain V (Fig. 3), in the following considered two-dimensional, n denotes a unit vector of a generic direction at point P and s is the straight line parallel to n. Usually considered points P are on the free edge, but this possible condition does not affect a general definition.

Figure 3. PM re-interpreted according to the implicit gradient idea.

We consider point A inside the body, along s and at a given distance from P, i.e. A∈s and |PA|=L/2 (Fig. 3). Then the PM is retrieved from Eq. (18) when the weight function is taken as follows:

• (19)

where δ is the Dirac function, hence equal to zero in any point X different from A. Similarly, it is possible to write the weight functions corresponding to the Line, Area and Volume Method.⁵¹

Several possible considerations arise: the first one is that each definition involves an arbitrary assumption that is the direction n. For the points lying on the boundary, a possible choice is the normal to the surface, but if the boundary is not differentiable (as for instance at edges or corners) an alternative to the normal direction shall be considered. A further consideration is that the weight function is also discontinuous. Hence, few points are considered as representative independently of the stress condition very close to them. A possible means of overcoming these problems is to use a continuous weight function having the following versatile properties: (i) attaining its maximum at the actual point P, because it is realistic that stress at point P would be the most important for its strength assessment, and (ii) decaying to zero for increasing distances from P, according to a simple guess that strength at point P should not be related to stress intensity at positions too far from it.

Probably the simplest possibility is to use a Gaussian-like function:

• (20)

as employed in Ref. [12 and 52] where an ‘intrinsic length’ appears in agreement with previous considerations. In this case the problem turns out to be the evaluation of the integral in Eq. (18). To overcome the evaluation of this integral over the entire volume V for each point P of V, Peerlings et al.¹⁵ proposed an alternative method: after a Taylor expansion of Eq. (18), the evaluation of effective stress σ_eff can be replaced by the solution of a differential equation in volume V (for more details, see Refs [15, 53 and 54]):

• (21)

where c is a characteristic length related to the weight function α(P,X) defined on the whole volume V and ∇² denotes the Laplacian operator. The constant c is another formulation of a possible ‘intrinsic length’ and it is clearly related to .

Other promising and more complex choices of weight functions are possible,⁵⁵ but even at this first stage it should be noted that Eq. (21) is the same as Eq. (14), thus establishing a formal relationship between the approaches discussed in this paper. It turns out that Eq. (21), which is an inhomogeneous Helmholtz equation usually called the ‘implicit gradient’ equation, can be obtained either by deriving it from gradient elasticity or as an appropriate evolution of Area or Volume effective stress definition.

Within this differential framework, further variations can be obtained by changing applied boundary conditions. The choice of the appropriate boundary condition could be critical and is generally dependent on the failure mechanism to be modelled, by defining the significance of the surface in the considered phenomenon. In damage mechanisms arising mainly from the surface, the most appropriate boundary condition has been found to be the Neumann-type condition:

• (22)

where n is the outward normal to the boundary and the symbol ∇ indicates the gradient operator; this condition can be easily formulated in both two-dimensional and three-dimensional problems.

Going further towards applications to real components, the effective stress definition based on (21) has been applied to static and fatigue strength assessment of specimens and real components.⁵¹ Due to its simplicity, Eq. (21), it can be solved with several multiphysics or programmable FE numerical tools, and even by certain commercial FE software packages. Taking advantage of this possibility, a variety of problems in solid mechanics and structural mechanics can be tackled, and it is particularly suitable for sharply notched components, even under multiaxial loading.⁵⁶ Its applicability to sharp three-dimensional notches, irrespective of the notch opening angle or shape, has enabled the analysis of weld beads⁵⁷ and welded joints.⁵⁸

TCD VERSUS IGM UNDER BOTH STATIC AND HIGH-CYLE FATIGUE LOADING

The considerations reported in the previous sections suggest that both the TCD and the IGM predict the propagation (or non-propagation) of cracks by using two parameters, i.e. an intrinsic material length and a reference strength, this modus operandi applying to situations involving both static and fatigue loading. From a philosophical point of view, the most remarkable difference between the above two approaches lies in the fact that the TCD post-processes the relevant linear-elastic stress fields calculated, in the vicinity of the assessed geometrical features, according to classical continuum mechanics, whereas, as postulated by Gradient Mechanics, the IGM calculates an effective stress field by directly incorporating the material characteristic length into the stress analysis itself.

As shown elsewhere,^{30,31,51,53,54,56–58} both approaches have already proven to be highly accurate in performing both the static and high-cycle fatigue assessment of cracked and notched components, resulting in estimates falling within an error interval equal to ±20%. Therefore, without bringing into question their overall accuracy, the aim of the present section is then to quantitatively compare to each other the ways the TCD and the IGM predict, by post-processing the relevant stress fields acting on the process zone, the non-propagation of cracks in engineering materials subjected to static as well as to high-cycle fatigue loading. For such a purpose, a number of suitable experimental results generated by testing crack-like notches were selected from the technical literature. Table 1 summarizes the mechanical properties of those brittle materials whose behaviour was investigated under static loading, the considered samples being loaded in both tension, three- and four-point bending (see Fig. 4). This table also gives the absolute dimensions of the tested specimens together with the magnitude of the force, F_u, resulting in their static breakage. Similarly, Table 2 summarizes the mechanical properties of the materials tested under fatigue loading as well as the absolute dimensions of the investigated samples, Δσ_0n being the range of the nominal fatigue limit referred to the gross section.

Table 1.  Geometry of the crack-like notched samples tested under static loading and corresponding material fatigue properties

Material	Ref.	[MPa]	E[MPa]	v	KIc	Geometry (Figure)	Load type^1	a[mm]	^w g [mm]	β[°]	^Fu[kN]	[MPa]	L/2 [mm]	c[mm]
^1AX = axial loading, 3PB = three-point bending,4PB = four-point bending.
PMMA (at –60 °C)	[59]	128	5050	0.4	1.7	4a	3PB8	5	28	0	1.08	150.0	0.037	0.03
						4a	3PB	14	28	0	0.44
PMMA	[60]	100	3200	0.36	1.6	4c	AX	6	30	0	1.87	101.2	0.045	0.047
PVC	[60]	140	3600	0.3	2.6	4c	AX	6	30	0	2.76	148.7	0.033	0.06
Y-PSZ	[61]	425	197000	0.2	5.9	4b	4PB	1	4	0	0.16	641.8	0.011	0.033
Si3N4	[61]	658	314000	0.2	5.3	4b	4PB	1	4	0	0.14	659.1	0.009	0.011
Alumina-7% Zirconia	[62]	290	356000	0.24	4.1	4b	4PB	5	15	30	1.60	435.2	0.017	0.035

Figure 4. Geometries of the investigated samples.

Table 2.  Geometry of the crack-like notched samples tested under fatigue loading and corresponding material fatigue properties

Material	Ref.	R	[MPa]		L/2 [mm]	c [mm]	Geometry (Figure)	Load type^1	a[mm]	^wg[mm]	β[°]	[MPa]
^1AX = axial loading, 3PB = three-point bending,4PB = four-point bending.
AL-2024-T351	[63]	−1	248	4.4	0.050	0.054	4d	AX	0.5	44.45	−	121
		0	172	4.0	0.086	0.093			0.5	44.45	−	107
SAE 1045	[63]	−1	606	9.0	0.035	0.038	4d	AX	0.12	44.45	−	358
		0	448	6.9	0.038	0.041			0.12	44.45	−	325
Mild steel	[64, 65]	−1	420	12.8	0.148	0.161	4c	AX	5.08	64	60	86
		−1	326	12.4	0.230	0.251			3	45	−	104
SM41B	[66, 67]	0	274	8.4	0.150	0.163	4d	AX	3	45	−	69
		0.4	244	6.4	0.109	0.119			3	45	−	73

As to the specific geometrical features of the stress raisers used to perform this quantitative comparison, it has to be highlighted here that the crack-like notches contained in the investigated specimens were all characterized by a value of the notch root radius always very small compared to the notch depth as well as to the absolute dimensions of the samples themselves: this aspect allowed us to analyse, without much loss of accuracy, such crack-like notches by simply treating them as sharp cracks (i.e. r_n= 0 mm in Fig. 4).³⁰

In order to correctly apply the TCD to the experimental results generated under static loading, initially the procedure sketched in Fig. 2 was applied to estimate, for any investigated material, both critical distance L and inherent static strength σ₀ (see Tables 1 and 3). The geometries of the notched samples used to determine the above material properties are summarized in Table 3 and Fig. 4. The relevant stress fields were determined through FE models solved by using commercial software ANSYS®, where the density of the mesh in the vicinity of stress raisers’ tips was gradually refined until convergence occurred.

Table 3.  Experimental results used to estimate, according to the TCD, σ₀ and L for the materials tested under static loading

Material	Ref.	Geometry (Figure)	Load type^1	a[mm]	^wn[mm]	^wg[mm]	^rn[mm]	β[°]	^Fu[kN]	[MPa]	L/2 [mm]
^1AX = axial loading, 3PB = three-point bending,4PB = four-point bending.
PMMA (at –60 °C)	[59]	4a	3PB	5	23	28	1.5	0	2.51	150.0	0.037
		4a	3PB	14	14	28	2	0	1.17
PMMA	[60]	4c	AX	6	18	30	2	0	3.98	101.2	0.045
		4c	AX	6	18	30	0.5	0	2.41
PVC	[60]	4c	AX	6	18	30	2	0	5.77	148.7	0.033
		4c	AX	6	18	30	0.3	0	2.88
Y-PSZ	[61]	4b	4PB	1	3	4	0.28	0	0.25	641.8	0.011
		4b	4PB	1	3	4	0.18	0	0.22
^Si3^N4	[61]	4b	4PB	1	3	4	0.2	0	0.17	659.1	0.09
		4b	4PB	1	3	4	0.075	0	1.84
Alumina-7% Zirconia	[62]	4b	4PB	5	10	15	0.1	30, 60	1.64	435.2	0.017
		4b	4PB	5	10	15	0.06	30, 60	1.64

On the contrary, inherent material length c in the IGM's governing equations (Table 1) was directly calculated according to the following definition⁵³:

• (23)

By comparing Eq. (23) to Eq. (5), it is straightforward to observe that the definitions of the two critical lengths to be used to apply the IGM and the TCD, respectively, are very similar. In more detail, in identity (5) constant ζ is invariably equal to unity, whereas in Eq. (23) its value changes as the definition used to calculate the equivalent stress field varies: for instance, if the IGM is applied along with the maximum principal stress criterion, as it will be done below, constant ζ is equal to 0.545.⁵³ The second relevant difference is that the IGM uses the ultimate tensile stress, σ_UTS, as the reference failure stress, whereas the TCD calculates the inherent material strength according to the procedure sketched in Fig. 2, so that, as already discussed in great detail above, under particular circumstances σ₀ can be much larger than σ_UTS. Turning back to the materials considered in the present investigation, as shown in Table 1, in general, the difference between σ₀ and σ_UTS was quite small, the only exceptions being materials Y-PSZ⁶¹ and Aluminium-7% Zirconia⁶² for which σ₀ was seen to be about 1.5 times larger than σ_UTS. On the contrary, the c values calculated according to definition (23) by taking ζ invariably equal to 0.545⁵³ were seen to be, in some cases, much larger than the corresponding lengths, L/2, estimated by taking full advantage of the simplified procedure sketched in Fig. 2 (see, for instance, materials PVC,⁶⁰ Y-PSZ⁶¹ and Aluminium-7% Zirconia⁶²).

In the stress–distance diagrams reported in Fig. 5 the stress fields determined according to classical continuum mechanics are directly compared, in the incipient failure condition, to those calculated according to the IGM, both being plotted along the crack-like notch bisector (i.e. for θ= 0 in Fig. 1a). In more detail, the conventional stress fields shown in Fig. 5 were estimated through refined linear-elastic FE models solved by using FE code ANSYS®, whereas those calculated according to the IGM, applied along with the maximum principal stress criterion, were determined, under a plane stress hypothesis, by taking full advantage of software COMSOL®.

Figure 5. Bi-parametrical post-processing (in the incipient failure condition) of the static stress fields acting on the material in the vicinity of crack-like notches performed according to both the TCD and the IGM.

According to the TCD, under static loading cracks are supposed not to propagate as long as the following condition is assured:

• (24)

where effective stress σ_eff can be calculated according to either the PM, LM, or AM – see Eqs (6) to (8), whereas is the inherent material strength estimated by following the procedure sketched in Fig. 2. For the sake of simplicity, in the charts of Fig. 5 the non-propagation condition for the investigated crack-like notches was predicted by applying the TCD solely in the form of the PM.

The IGM instead assumes that the propagation of cracks is avoided as long as the maximum value of the effective stress determined according to Eq. (21), , is lower than (or, at least, equal to) the material UTS, that is⁵³:

• (25)

In spite of the simplifying assumptions on the sharpness of the crack-like notches made to perform such a quantitative comparison, the charts of Fig. 5 make it evident that the use of both the TCD and the IGM resulted in the same level of accuracy, that is, in estimates falling within an error interval of about ±15%. This aspect is very interesting especially in light of the fact that the TCD predicts the non-propagation of cracks under static loading by using finite radius notches (see Fig. 2) to estimate the necessary material properties, i.e. critical distance L and inherent material strength σ₀. On the contrary, to perform the static assessment of cracked bodies, according to identity (23) the IGM estimates material length c by combining the conventional UTS with the LEFM plain strain fracture toughness, K_Ic. The resulting difference between the above two ways of determining the necessary material constants is clearly shown by the last chart of Fig. 5 (i.e., Aluminium-7% Zirconia⁶²): both approaches are equally accurate, even though σ₀ is much larger than σ_UTS, while the values of material lengths c and L are very close to each other.

Turning now to the high-cycle fatigue issue, the IGM postulates that a cracked body is at its fatigue limit when the following condition is assured⁵⁴:

• (26)

where is the maximum value of the range of the effective stress calculated according to Eq. (21), whereas Δσ₀ is the unnotched fatigue limit range determined for the appropriate load ratio, R=σ_min/σ_max. To address the high-cycle fatigue problem, material constant c is suggested as being estimated as follows^51,54:

• (27)

where ΔK_th is the range of the threshold value of the stress intensity factor and Δσ₀ the plain fatigue limit (both determined under the same load ratio, R, as the one characterizing the load history applied to the component being assessed).

By directly comparing definition (27) to definition (1), it is easy to observe the existing similarity between the material characteristic lengths determined according to the IGM and the TCD, respectively. In more detail, whilst in the LEFM-based formalization of the TCD constant ζ is invariably equal to unity, in the IGM's governing equations ζ changes as the hypothesis formed to calculate the effective stress varies: for instance, similar to the static case, if the IGM is formalized in terms of the maximum principal stress criterion, ζ is equal to 0.545.

Turning back to the TCD, such a theory postulates that under cyclic loading cracks do not propagate as long as the following fatigue limit condition is assured:

• (28)

where the range of the effective stress, Δσ_eff, can be calculated in terms of either the PM, LM, or AM (see Eqs (2) to (4)) and Δσ₀ is again the range of the unnotched fatigue limit.

The stress–distance diagrams of Fig. 6 show the way the TCD and the IGM post-process, at the fatigue limit, the stress fields in the vicinity of the crack-like notch tips of the investigated samples (see Table 2 and Fig. 4), where the continuum mechanics stress–distance curves were determined, using software ANSYS9®, through refined FE models, whereas the IGM stress fields were estimated by solving Eq. (21) by means of FE code COMSOL®. Such charts make it evident that both approaches are equally accurate, resulting in estimates always falling within an error interval of ±20%, this holding true independently of both material and applied load ratio, R.

Figure 6. Bi-parametrical post-processing (at the fatigue limit) of the cyclic stress fields acting on the material in the vicinity of crack-like notches performed according to both the TCD and the IGM.

To conclude, it is interesting to observe that, contrary to the static case, the fact that both the TCD and the IGM use the unnotched fatigue limit, Δσ₀, as the reference fatigue strength results in values of material characteristic lengths c and L/2 that are always very close to each other (Table 2).

DISCUSSION

The two approaches, Critical Distances and Gradient Mechanics, can both be used to make predictions of the strength of cracked specimens. Below, some analogies and differences of the two approaches will be discussed in more detail, which can also be used to inspire future research to extend the use of the above methods to real structural components:

• • 
  Definitions of the effective stress and related domains. In gradient mechanics, the non-local effective stress is computed by solving a global boundary value problem, and therefore it is a quantity defined in the whole domain. In the case of the TCD PM the effective stress is obviously a locally defined quantity, but the distance between the chosen point and the location of maximum stress implies a degree of non-local character even for this method. In the LM, AM and VM some degree of averaging over the neighbouring domain takes place, which can in principle be extended to the entire structure – an idea that has been explored above to establish a connection between Gradient Mechanics, via non-local mechanics, to the TCDs (see Fig. 3).
• • 
  Where to evaluate the effective stress for strength assessment? The previous observation regarding confined or global definitions of the effective stress carries some implications on how the structural strength is assessed. If a confined definition is used, as in the PM, then obviously the exact location of the critical point must be known a priori and this is not obvious for complicated structural geometries. Conversely, if a global approach is taken, then some ambiguity may exist on where the maximum of the effective stress occurs. As the numerical examples have shown, this maximum, in crack-like specimens, is located at a finite distance away from the crack tip, although the quantitative differences between the maximum effective stress and the value of the effective stress at the crack tip are small. There seem to be two avenues for further investigations in structural applications:
  • 1
    If strength is assessed strictly based on the maximum effective stress, then the maximum should occur at the critical locations, which means that one-to-one relations must be found between the length scales of Gradient Mechanics and of Critical Distance Theories.
  • 2
    Alternatively, one could explore a static and fatigue strength assessment based on the non-local stresses evaluated at the crack tip only. This would imply that, before performing a non-local analysis of real components, a unique rule should be defined to locate the position of the critical sites independently of the shape of the investigated geometrical feature.
    Irrespective of the chosen approach, one must keep in mind that when a non-local approach is taken, it is assumed that continuum mechanics is not able to describe the actual, micro-structural stress and strain fields. The size of this indeterminate zone is indeed the critical length! At distances comparable to the critical length a continuum model can only serve as an average indication.
• • 
  Additional material parameters. Both approaches employ a length scale parameter, and the interpretation of these two length scale parameters is quite similar. In addition, the TCDs also introduce a so-called ‘critical stress’ parameter . The analogies between gradient mechanics and critical distance methods can be used to shed further light on the interpretation and evaluation of .

CONCLUSIONS

• 1
  Both the TCD and Gradient Mechanics (applied in the form of the IGM) are successful in estimating the propagation (or non-propagation) of cracks by directly post-processing, through two material constants, the relevant stress fields acting on the process zone. This holds true both under static and high-cycle fatigue loading.
• 2
  Even though the approaches investigated in the present paper are based on different initial assumptions, given the material, the values of the characteristic length as well as of the reference strength are seen to be closely related to each other. This results in an evident compatibility of the two theories.
• 3
  The main advantage of the TCD is that accurate estimates can be obtained by simply post-processing the linear-elastic stress fields calculated according to classical continuum mechanics.
• 4
  The most interesting feature of the Gradient Mechanics based theories (such as the IGM) is that they are capable of performing the static and fatigue assessment of 2D-3D cracked bodies by directly incorporating the inherent material length into the stress analysis: this suggests that specific Finite Elements could be developed to directly design cracked and notched components, with the evident advantage that, in the presence of sharp geometrical features, such a particular way of performing the stress analysis allows stress/strain singularities to be removed from the calculated solution.