- Email: [email protected]
A stress-concentration-formula generating equation for arbitrary shallow surfaces
International Journal of Solids and Structures 69–70 (2015) 86–93
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
A stress-concentration-formula generating equation for arbitrary shallow surfaces Hector Medina School of Engineering and Computational Sciences, Liberty University, 1971 University Boulevard, Lynchburg, VA 24515, United States
a r t i c l e
i n f o
Article history: Received 27 January 2015 Received in revised form 8 June 2015 Available online 18 June 2015 Keywords: Stress concentration Analytical solution Arbitrary shallow surfaces
a b s t r a c t Analytical understanding of how stress concentrates is invaluable. An equation that generates stress concentration formulas is derived and shown to apply very well to a number of shallow irregularities on surfaces, for the plane stress conditions and to a first-order approximation. Under shallow conditions, for any 0 first-order Hölder-continuous surface function f ðxÞ, the derived equation is: kt ðxÞ ¼ 1 2H f ðxÞ , where 0 H is the Hilbert transform and f ðxÞ is the spatial derivative of f with respect to the independent variable. It is shown that using this generating equation, well-known traditional results can be easily derived. Also, a number of other stress concentration formulas for various cases are generated. Furthermore, a second-order approximation is introduced, which shows the dependence of kt on not only the slope but also on the concavity of the surface. The approach used herein can be extended to finding closed-form solutions to other integral equations possessing similar kernels for applications such as the variation of the stress intensity factor due to an arbitrary crack front profile (work in progress). Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Analytical understanding of how stress concentrates on surfaces possessing an arbitrary topography is of crucial importance in many applications. For various reasons, it is highly desirable to know if and how a material’s parcel would become the ’’weakest link’’ during the material’s service lifetime. Additionally, analytical solutions of how the stress concentrates provide a more effective route to approach certain multi-physics problems by coupling those solutions with, for example, the Navier–Stokes, Fick’s Law of diffusion, and free-energy Gibbs equations, just to name a few. Furthermore, stress-driven-reaction analyses provide insight about the stability of surfaces, but it is evident that one limitation is the unavailability of stress-concentration solutions for general configurations of surfaces. Much of the advancement on the foregoing topics has relied on the use of surface-stress-concentration distributions for the cases of sinusoidal surfaces (Gao, 1991a; Liang and Suo, 2001), cycloid rough surfaces (Chiu and Gao, 1993; Li et al., 1993), and some single-notch cases (Yu, 2005). In this document, a novel, 2-dimensional, general equation that generates stress concentration formulas for a wide variety of shallow-surface configurations is presented. Additional impetus for this work follows.
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijsolstr.2015.06.006 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.
It has been established that surfaces are not inert to their surroundings, and that the latter interplay with the elastic and plastic flows of material at the boundaries. At such inequilibrium conditions, surfaces undergo transformations which, in turn, could be catastrophic for films, interfaces, coatings, etc. For example, electronic and micro-electromechanical devices can be significantly impacted by the foregoing conditions, mostly due to the presence of large internal stresses introduced during the manufacturing processes. It has been reported that stresses in the 1-GPa order of magnitude can be present in the thin films that comprise integrated circuits and magnetic disks (Nix, 1989). A slight magnification of such high stress levels, due to surface deformation, would definitely increase the likelihood of failure. It has been found (Medina and Hinderliter, 2014) that even slightly random rough surfaces (with heights normally distributed) can most likely magnify the bulk stress by a factor of 1.6. For the case of slightly undulating surfaces, this magnifying factor can range from 2 to 31 (Gao, 1991b). An analytical description of the distribution ofconcentrated stress for an arbitrary surface would provide safety envelopes for surface topographies found experimentally. Furthermore, stress-driven surface evolution has been recognized to be an important process in the behavior of heteroepitaxial films (Gao, 1994). For specific surface morphologies, advancement on stress-driven reactions has been accomplished
1 Although it is shown in this document, in Section 4, via a second-order approximation that these numbers are a few percents lower.
87
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
by coupling analytical solutions of surface stress distribution to the total energy equation. Ever since the development of the first analytical solution for the stress concentration due to an elliptical hole in a plate provided by Inglis (Inglis, 1913), an immense number of solutions have been developed (Pearson, 1997; Medina et al., 2014; Neuber, 1958). Additionally, some attempts have been carried out to consolidate stress concentration formulas and factors, mostly via empirical methods (Pearson, 1997; Medina et al., 2014; Neuber, 1958; Noda and Takase, 1999). Although the foregoing referenced work is commendable, it is mostly focused on stress-concentration factors and not on the stress distribution along the surfaces of the geometries analyzed. In the present work, a stress-concentra tion-formula-generating equation is derived and shown to be applicable to a wide number of shallow surfaces under tension or pure bending, at least. Under shallow2 conditions, for any function3, f ðxÞ descriptive of a 2-dimensional surface, the stress concentration along the profile is given by: 0
kt ðxÞ ¼ 1 2Hðf ðxÞÞ
ð1Þ
where H stands for the general Hilbert transform, and prime represents the first derivative with respect to the given independent variable. It will be shown that using this generating equation, well-known traditional results can be easily obtained. This paper is organized in the following manner. First, the main result, a stress-concentration-formula-generating equation (SCFGE) is derived in a fashion similar to that used by the author in a previous work (Medina and Hinderliter, 2014; Medina and Hinderliter, 2012). Next, the robustness of the SCFGE is shown via derivation of previously developed and accepted results such as semi-elliptical (and semi-circular) single notch, parabolic single notch, sinusoidal surfaces, etc. Following, the stress concentration is found for other interesting profiles, and a table containing results is provided. Next, a second-order approximation is introduced and applied to the undulating surface case. Finally, this document is completed with a discussion and conclusion section.
shows some typical topographies described by f. Being f a perturbation from a previously flat configuration, an important constraint on f is that it must be of small magnitude. Assume that our surface f ðxÞ is being subjected to a remote tensional load r far enough from the surface features that Saint–Venant’s principle applies (Saint-Venant, 1855). This model can be applied, as well, to a thick rectangular beam with a slight rough surface subjected to pure bending5, since a sufficient number of layers of fibers on and near one of surfaces can be assumed to be subjected to a tensional load. Following (Medina and Hinderliter, 2014), we can define the stress and displacements for the rough surface (T ij and ui ) in terms of the reference state values (T rij and uri ) and the perturbed elements (dT ij and dui ), as:
T ij ¼ T rij þ dT ij
ð2Þ
ui ¼ uri þ dui
Consider a concentrated point force F p (see Fig. 1) acting at an arbitrary location ðx; yÞ. The purpose of using a point force is only to utilize the well-known solution for the surface stress Green’s function, and the magnitude of F p can be shown to be irrelevant. Using a similar reasoning as used by others previously (Banichuck, 1970; Goldstein and Salganik, 1970; Goldstein and Salganik, 1974; Rice, 1985) and developed in both (Gao, 1991b and Medina and Hinderliter, 2014), it is shown that perturbation contributions for the stress and displacements are given as:
R1 b G df ðvÞdv 1 ij R1 i r dui ðx; yÞ ¼ E0 1 Gxx df ðvÞdv dT ij ðx; yÞ ¼ r
ð3Þ
which is for the particular case when there is no concentrated force and thus the material is only exposed to the bulk stress; that is, F p ¼0 T xx ¼ T rxx ¼ r. In Eq. (3) E0 is the consolidated Young modulus of elasticity6; b ij is a tensor representing a kernel function which can be And G
derived from the Green’s function, Gixx , in the following manner:
b ij ¼ 1 G E0
"
l
j @Gixx @Gxx þ @xj @xi
!
2lmdij @Gkkk þ 1 2m @xk
#
2. Derivation For the first part of our derivation, we employ a first-order perturbation method. A similar approach was previously used by Banichuck (1970) and Goldstein and Salganik (1970, 1974), later by Cotterell and Rice (1980), and Rice (1985). In the foregoing references, the perturbation approach was applied to the problem of the stress intensity variation due to a crack front that deviates from flatness due to local irregularities in materials (e.g. grain boundaries). Later, this method has been used by Gao (1991b) and later by the author (Medina and Hinderliter, 2014) to develop, respectively, stress concentration factor formulas for undulating surfaces and random rough surfaces. Other related problems, such as the study of the elastic fields at the interfaces between dissimilar materials (Gao, 1991a; Grekov, 2011), have been studied using the perturbation method, as well. Consider the profile from Fig. 1, which is a slight perturbation from an originally flat surface. Profile f ðxÞ is a continuous real function that satisfies the Hölder condition4 within its domain. Fig. 2 2 Shallow means that the heights or depths of hillocks or hollows on the surface are small in comparison to their widths. For example, in the case of a sinusoidal surface, the amplitude is much smaller than the wavelength; or in the case of a single notch, the depth of the notch is much smaller than its width. 3 Actually, as it will be shown, the requirements to be imposed on f do not limit it from any practical applications. 4 This is a very important characteristic of our solution, and it is completely related to a property of the Hilbert transform to be discussed later. Most functions found in practical applications meet this criterion. Moreover, for the particular case of surfaces without cracks, this criterion is obviously always met.
where, in this case, dij is the Kronecker delta, and modulus.
ð4Þ
l is the shear
The Green’s function Gixx can be obtained by differentiating the strain energy density with respect to the point force F p (for details see (Medina and Hinderliter, 2014)) and found to be:
Gixx ðv; x; yÞ ¼
@T xx
ð5Þ
@F ip
where the stress Green’s function in Eq. (5) is understood as the surface stress at v due to a unit point force F p in the i-direction applied at the point source ðx; yÞ. Therefore, the Green’s function above, Eq. (5), can be used to find the kernel of Eq. (4), which in turned can be used to find the contribution of the stress due to the perturbation. Finally, this perturbation part of stress can be used to find the stress increased from the bulk stress, according to part (a) of Eq. (2). Extracting the other Green’s functions Giij from the point force solutions for a half-plane from (Green and Zerna, 1968), it can be shown that Eq. (2)(a) can be expressed as:
dT xx ðx; yÞ ¼ r
Z
1
1
5
6
2
pðv xÞ2
df ðvÞdv
Either a four-point bending or a three-point bending with negligible shear. E; for plane stress E0 ¼ E ; for plane strain;ðm ¼ Poisson0s ratioÞ 1m2
ð6Þ
88
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
Fig. 1. An arbitrary surface f(x) whose heights deviate slightly from the x axis. The remote load, r is applied far away from the features of the surface, in accordance to SaintVenant (1855). The material is assumed to be elastic, homogeneous, and isotropic. A point force F p is applied at some arbitrary location within the material (this is simply an artifact of the mathematical development and has no physical implications since F p is to be set to zero). The material is characterized by Young modulus E and Poisson’ratio m.
Fig. 2. Plots of a few examples of topographies that can be described by f(x). From top to bottom: Cosine pulse notch, Gaussian notch, parabolic notch, and undulating surface. Note that and A have been magnified for visualization purposes. Actually b and A k, for shallow conditions.
and (6) can be substituted into (2) with T rxx ¼ r, which after some manipulation leads to the following result:
2
2 T xx ðxÞ ¼ r41 þ PV
p
where,
df ðvÞ dv
Z
1
1
3 df ðvÞ dv dv 5 vx
is the spatial derivative of f,
ð7Þ
v is a dummy variable, and ‘‘PV’’ means that the integral is calculated in the Cauchy’s principal value sense. At any given point of the profile, Eq. (7) provides the tangential component of the stress, T xx , introduced in part (a) of Eqn. (2). So, for shallow condition of surfaces, the function of the stress concentration, kt ðxÞ, applicable on the surface and along x, can then be written as:
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
2 kt ðxÞ ¼ 41 þ
2
p
Z
PV
1
1
3 df ðvÞ dv dv 5 vx
3.1. Semi-elliptical notch
ð8Þ
We now apply the Hilbert transform (HT) to (8). The HT is defined as follows (Hahn, 1996; King, 2009):
^ ¼ 1 PV Hf ðtÞ ¼ fðtÞ
Z
p
1
1
f ðgÞdg 1 ¼ PV gt p
Z
1
1
f ðgÞdg tg
ð9Þ
Z
1
gðsÞdðt sÞds
Consider the semi-elliptical notch shown in Fig. 3, where a and b are the depth and half-width of the notch, respectively. Such a profile can be described by:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 P2b ðxÞ b
f ðxÞ ¼ a 1
ð10Þ
P2b ðxÞ ¼
1; j x j< b
Applying (9) to (8), one obtains the main result of this work:
ð11Þ
where mathematically, f represents a piece-wise, continuous, real-valued function, by definition. Physically, f is an arbitrary boundary profile of an 2-D elastic region. It is claimed herein that the formula given by Eq. (11) is a stress-concentration-formula-gen erating equation (SCFGE). Another useful form of Eq. (11) can be obtained by using the following property of the HT:
df ðxÞ d½Hf ðxÞ ¼ H dx dx
ð12Þ
Applying Eq. (13), the stress concentration distribution becomes:
kt ðxÞ ¼ 1 2
d½Hf ðxÞ dx
ð13Þ
Eq. (13) comes in handy in situations for which the solution of HT of 0 function f ðxÞ is more complicated to obtain than the HT of the function f ðxÞ itself. Let’s pause for a moment and discuss the general requirements to be imposed on f ðxÞ so that kt ðxÞ can be solved from either Eqs. (11) or (13). Let X be an open set in Rn ; h 2 N, and 0 < a 6 1. Then C h;a ðXÞ is the universe of functions g : X ! R with continuous derivatives in X of order less than or equal to h. Our requirement for f in either Eqs. (11) or (13) is:
8f : f 2 C 1;a ðXÞ
kt ðxÞ ¼ 1 2
ð14Þ
at least within the domain of interest, D 2 R. In the present application, D is the region where the stress concentration is to be evaluated, typically the notch or surface under consideration. For most single-notch cases, D is to be taken far enough from the corners, in accordance to Sain-Venant’s principle (Saint-Venant, 1855). By definition, Eq. (14) means that function f is first-order Hölder continuous. The numerous and powerful properties of the HT make Eqs. (11) or (13) very valuable. Additionally, tables of the HT are widely known, which facilitates the utilization of those equations for design, research and pedagogical purposes.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dfH½a 1 ðbxÞ2 P2b ðxÞg dx
ð17Þ
Applying the HT to the quantity inside squared brackets in Eq. (22), followed by differentiation, one gets:
8 a j x j< b ; > > > ba < b þ 2 paxffiffiffiffiffiffiffiffiffiffi ; x>b b ðbxÞ2 1 kt ðxÞ ¼ 1 2 > > a > ; x < b : b 2 paxffiffiffiffiffiffiffiffiffiffi x 2 b
ð18Þ
ðbÞ 1
Since the region of interest is within the notch7, the value of the stress concentration is the one corresponding to j x j< b, thus:
kt ¼ 1 þ 2
Applying (12) to (11):
ð16Þ
0; elsewhere
1
df ðxÞ 0 ¼ 1 2H½f ðxÞ kt ðxÞ ¼ 1 þ 2H dx
ð15Þ
where the rectangular pulse function, P2b is defined as:
Which is a linear operator that, contrary to Fourier transform, keeps the domain of the independent variable. In essence, the HT of a function g is a convolution of g with the distribution dðtÞ ¼ p1t, or:
^ ¼ ðg dÞt ¼ gðtÞ
89
a b
ð19Þ
for a shallow semi-elliptical notch and to a first-order approximation. Remarkably, this result, Eq. (19), coincides with the traditional stress concentration factor due to an elliptical hole derived by Inglis (1913), which can be applied also as a good approximation to shallow semi-elliptical notches (Pearson, 1997; Anderson, 1991). Eq. (19) implies that for a shallow notch the stress concentration within the boundary of the notch is constant. For a depth-to-width ratio, a=2b, equals to 0.1, the stress is magnified from the remote load by 40%. 3.2. Undulating surfaces The solution to the problem of the stress concentration due to slightly undulating surfaces was solved previously using Muskelishvilli’s complex analysis approach (Gao, 1991a,b). Let us show that our SCFGE, Eq. (13), can easily lead to the same result. Consider the sinusoidal profile shown in Fig. 4 (dotted line) for which the ratio amplitude-to-wavelength, A=k is small. This profile can be expressed as:
f ðxÞ ¼ A cosð2px=kÞ
ð20Þ
Applying Eq. (13) to Eq. (20), the stress concentration distribution is:
kt ðxÞ ¼ 1 2
dfH½A cosð2px=kÞg dx
ð21Þ
Applying the HT and differentiating in Eq. (21), one gets: 3. Use of the SCFGE Let us begin our demonstration of the robustness of this SCFGE result. (The reader is reminded that the following topographies are assumed to be shallow and the material to be elastic, homogeneous, and isotropic. The loading conditions can be either in tension or bending, and the remote load is assumed to be applied far from the features of interest.)
7 Theoretically, and as expected, there are stress singularities at j x j¼ b. At most common strain rates, plasticity effects sets off much faster than the time it takes for that singularity to become of realistic concern. However, it is widely known that corners are soften for proper design purposes. For the remainder of this paper and when dealing with such boundaries, the discontinuity at the extreme points will be omitted in the discussion.
90
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
Fig. 3. Semi-elliptical profile of notch, mathematically described by Eq. (15). Notch is assumed to be shallow, hence a 2b. Note that the notch’s depth has been magnified for the purpose of visualization.
Fig. 5. Plot of parabolic notch described by Eq. (23) and the corresponding stress concentration according to derived Eq. (25). The maximum stress concentration occurs at x = 0. For shallow conditions, 2b.
kt ðxÞ ¼ 1 2 ðA 2p=kÞ cosð2px=kÞ ¼ 1 þ 4p
A cosð2px=kÞ k
ð22Þ
which is consistent with the result obtained in Gao (1991a). And again, the stress concentration factor (SCF) is found when x ¼ nk=ð2pÞ, for n 2 N; SCF ¼ 1 þ 4p Ak . A plot of the stress concentration given by Eq. (22) is shown in Fig. 4 (solid line). Note that, as it would be expected, the stress at the inflection points on the undulating surface is equivalent to the remote applied stress. 3.3. Parabolic notch Consider the parabolic notch shown in Fig. 5 (dotted line). It can be expressed as:
f ðxÞ ¼ 1 þ ðx=bÞ
2
P2b ðxÞ
ð23Þ
where is a scaling factor to guaranty the shallow condition imposed on the SCFGE. Applying Eq. (13) to Eq. (23), the stress concentration can be expressed as:
K t ðxÞ ¼ 1 2
2 dfH½ 1 ðx=bÞ P2b ðxÞg dx
ð24Þ
kt ð0Þ ¼ SCF ¼ 1 þ
8 pb
For semi-parabolic notch of height and width b. Not surprisingly, note that Eq. (26) is identical to Eq. (24) in Grekov and Makarov (2004) (which was obtained using a different mathematical method), for the case when q ¼ 1. 3.4. Gaussian notch and other profiles Consider the notch of Fig. 6 (dotted line), mathematically expressed as: 2
f ðxÞ ¼ ex
ð27Þ
Applying HT to the derivative of the profile given by Eq. (27) and applying Eq. (13), the stress distribution results in:
pffiffiffiffi 4 2 kt ðxÞ ¼ 1 þ pffiffiffiffi ½1 pxex erfiðxÞ
p
erf ðizÞ i
erfiðzÞ ¼
0 1 2
1 xb ðb þ xÞð1 ðx=bÞ Þ bx ðbþxÞ 2 2 @2x
x þ b
kt ðxÞ ¼ 1 þ ln b þ 2A xb p b b b x
pffiffiffiffiffiffiffi where i ¼ 1 The SCF occurs when x ¼ 0 and it is:
Fig. 5 (solid line) shows the stress concentration distribution as described by Eq. (25). An interesting result is the maximum stress concentration, or SCF, occurring at the bottom of the notch, where x ¼ 0:
ð28Þ
where ‘‘erfi’’ stands for the imaginary error function, and can be expressed in terms of the real error function, ‘‘erf’’, as:
Which, for the region within the notch (j x j6 b), leads to:
ð25Þ
ð26Þ
4 SCF ¼ 1 þ pffiffiffiffi
p
ð29Þ
ð30Þ
The stress concentration distribution of the Gaussian notch as given by Eq. (28) is shown in Fig. 6 (solid line). Now consider the profile given in Fig. 7, expressed as:
Fig. 4. Plot of undulating surface described by Eq. (20) and stress concentration according to derived Eq. (22). Surface is assumed to be shallow, hence A k. Profile’s amplitude, A, has been magnified for the purpose of visualization.
91
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
Table 1 Stress Concentration Distribution as provided by Eqs. (11) or (13) for various profiles. b; wð2Þ is the polygamma function of order 2; Profile, f(x)
kt ðxÞ
P2b ðxÞ
1 þ p4b
1 2 1ðx=bÞ
i
h i 1 2 bp 3 wð2Þ ð12 þ bixpÞ þ wð2Þ ð12 bixpÞ 2 1 þ 2 b 1ðx=bÞ 2 2
2
sech ðx=bÞ
h
2
ðx=bÞ þ1
½ðx=bÞ þ1
Fig. 6. Plot of a Gaussian notch described by Eq. (27) and the corresponding stress concentration according to derived Eq. (28). The maximum stress concentration, at x ¼ 0, is: SCF ¼ 1 þ p4ffiffipffi. Note that the actual values to be considered are only those within the notch. Shallow conditions required 4.
to thin films in electronics, coatings, interfaces and surfaces. However, higher order approximations may be also desirable in some cases. Unfortunately, all important perturbation series diverge due to a zero radius of convergence. Thus, in the following it will be assumed that n, the order of approximation, corresponds to the term of optimal truncation. Consider again the situation proposed in Fig. 1. Using the complex potentials approach used by Muskhelishvili (1975) functions for stress and displacement fields for the elastic half-plane have been widely obtained. For the sake of conciseness, the algorithm will be not repeated here. It can be shown (Gao, 1991a; Green and Zerna, 1968; Grekov and Makarov, 2004) that for the perturbation problem stated in Section 2, the nth order approximation of the tangential stress, T xx ðxÞ can be expressed as function a potential function U: ðnÞ T xx ðxÞ
" # n X ¼ 4Re Uj
ð34Þ
j¼0
Fig. 7. Plot of a Cosine-pulse notch (dotted line) described by Eq. (31) and the corresponding stress concentration (solid line) according to derived Eq. (32). This solution is only applicable for a region within the notch far away from the corners at b ¼ 1. For the case of b = 1, the maximum stress concentration, at x ¼ 0, is: SCF ¼ 1 þ 2Siðp=2Þ. ’’Si’’ is the sine integral function.
f ðxÞ ¼ cos
px 2b
P2b ðxÞ
ð31Þ
For the sake of simplicity, the width of the notch is normalized to unity. The stress concentration within such notch is found to be:
2 p px pjx þ 1j pjx 1j Ci kt ðxÞ ¼ 1 þ sin Ci 2 2 2 p 2 p p p p px þ sinc ðx þ 1Þ sinc ðx 1Þ sin 2 2 2 2 2 p px p p þ Si ðx þ 1Þ Si ðx 1Þ cos 2 2 2
2
px cos p2 ðx þ 1Þ cos p2 ðx 1Þ þ ð32Þ cos xþ1 x1 2 where Ci and Si are, respectively, the cosine and sine integral func
tions, and sinc is the normalized cardinal sine function sinpxpx The SCF in Eq. (32) is found at x ¼ 0 to be:
SCF ¼ 1 þ 2Siðp=2Þ
ð33Þ
We have seen some simple examples of the usefulness of either Eq. (11) or (13) in deriving stress concentration and stress concentration factor formula. Table 1 shows a list of more involved derivations. 4. Extension to higher orders As stated before, the derived SCFGE builds on the first-order perturbation approach used by Rice (1985) in the analysis of the stress intensity due to a varying crack front. Therefore, it provides stress concentration formulas that can only be guaranteed for shallow surfaces, which is the case for many applications related
where holomorphic functions U’s are obtained from two suitable complex potentials, for details see Green and Zerna, 1968; Muskhelishvili, 1975; Grekov and Makarov, 2004. As it will be shown, the resulting U’s increase in complexity as the value of n increases. From Eq. (34), it can be observed that: for the zeroth-order approximation only U0 is needed; for the first-order approximation, both U0 and U1 are required; similarly, for the second-order all U0 ; U1 , and U2 need to be found, and so on. From (Grekov and Makarov, 2004) and after some simple but lengthy manipulation, the first three U functions can be expressed as, respectively:
U0 ðzÞ ¼ r=4
r
ð35aÞ 0
U1 ðzÞ ¼ H½f 2 1 U2 ðzÞ ¼ HG2 2i
ð35bÞ ð35cÞ
with G2 defined, in terms of a dummy variable f, as:
0 00 G2 ðfÞ ¼ 2irf ðfÞ if ðfÞ 2Hf ðfÞ 0 2 0 0 0 þ 4irf ðfÞ if ðfÞ 2H½f ðfÞ þ 2r½f ðfÞ
ð36Þ
The negative sign in Eqs. (35b) and (35c) stem from the definition of the Hilbert transform, Eq. (9). Thus, the zeroth-order approximation is given by Eqs. (34) and (35a) which, as expected, is simply the remote tensile load r, or:
hr i ð0Þ T xx ðxÞ ¼ 4Re ¼r 4
ð37Þ
The first-order approximation is obtained using Eq. (34) in combination with Eqs. (35a) and (35b), which can be written as:
hr r i 0 0 0 T ð1Þ Hf ¼ r 2r Hf ¼ rð1 2Hf Þ xx ðxÞ ¼ 4Re 4 2
ð38Þ
So the first-order approximation of the stress concentration is obtained dividing Eq. (38) by the remote load:
92
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
ð1Þ
kt ðxÞ ¼
ð1Þ T xx ðxÞ
r
¼ 1 2Hf
0
ð39Þ
Which is the exact same result obtained for our SCFGE, in Eq. (11). For the second-order approximation, we plug Eqs. (35a)– (35c) and (36) into (34), and simplify to:
0 00 0 0 ð2Þ T xx ðxÞ ¼ rð1 2Hf Þ þ 8r H½f Hf þ 2H½f Hf
ð40Þ
So the second-order approximation of the SCFGE is then: ð2Þ
kt ðxÞ ¼
ð2Þ T xx ðxÞ
r
0
00
00
0
¼ 1 2Hf þ 8H½f Hf 16H½f Hf
ð41Þ
Provided that the condition imposed in f in (14) changes to:
8f : f 2 C 2;a ðXÞ
ð42Þ
at least, within the domain of interest (non-flat portion of the elastic surface). Eq. (41) provides insightful information: (1) While the first-order approximation shows kt ’s dependence only on the slope of the surface, the second-order approximation shows its dependence on both slope and curvature. (2) By extrapolation, one can generalize that for an n-order approximation, kt would depend on all derivatives of order 6 n. However, despite the mathematical contribution of the second-order approximation given by Eq. (41), it must be kept in mind that higher-order approximations may quickly diverge depending on the smallness of the perturbing element. The foregoing may be corrected applying secular-type methods which involves the use of multiple scales. Another difficulty is the complexity in developing the third and fourth terms in Eq. (41) for various profiles f’s. Finally, for an nth-order approximation, the restriction on f becomes more strict, as it must belong to a Hölder space of nth order. Nonetheless, for some practical topographies it can provide even more insightful information than the first-order solution. For example, for the undulating surface, f ¼ A cosð2p=kÞ, and using the second-order stress concentration Eq. (41) (after a lengthy but straightforward calculation), it can be shown (work in progress) that the third term provided by the A 2 second-order approximation is proportional to and k cosð4px=kÞ, or: ð2Þ
kt ðxÞ ¼ 1 þ 4p
A cosð2px=kÞ þ F ðA=kÞ2 cosð4px=kÞ k
ð43Þ
where FðÞ simply stands for ‘‘a function of’’. That the third term in Eq. (43) contains ðA=kÞ2 coincides with the general series form of a perturbation-theory solution. It turns out that, for the particular case of the maximum stress concentration the relative contribution of that third term is around 6%, which would decrease the stress concentration factor (SCF) at the lowest point of each valley of the sinusoidal surface from around 2.26 to 2.13. This latter result is in agreement with finite element results reported by Gao (1991a).
5. Discussion and conclusions (I) Undoubtedly, the form of Eq. (13) makes it very simple to find formulas for the stress concentration and hence SCF for arbitrary profiles, under the assumptions prescribed. This simplicity stems from the presence of the HT, whose properties are numerous, and tables of transforms are widely available. A physico-mathematical explanation for the appearance of the HT in the SCFGE involves its connection to the Cauchy’s Integral Formula, and the fundamental use of the latter in the development of the theory of elasticity, as we know it. (By ’’blowing up’’ the contour of integration in a semi-circle in the upper plane, one can map a closed-path integral onto the real axis.)
(II) For applications involving stress-driven reactions, Eqs. (13) or (41) can be coupled to other laws such as Fick’s Laws of diffusion, Navier–Stokes equations, etc., to advance knowledge on local fluid–solid phenomena. For example, premature mechanical failure has been observed in single-crystal silicon microelectromechanical systems (MEMS) (Muhlstein et al., 2002; Allameh et al., 2003; Alsem et al., 2005). Consider the chemical reactions at the interface between a MEMS device and a wet environment. It has been found that the presence of tensile stresses increase: (1) the electrostatic potential for oxygen molecules from water to bond to Si, and (2) the hybridization of Si and O orbitals (Colombi et al., 2008). The latter mechanism promotes the breaking of existing Si-O, thus creating more free Si sites exposed to the further attack of the environment. Additionally, it has been established that the so called ’’reaction-layer’’ process is initiated by high stress concentrations created at the bottom of notches. Since the thickening of the native SiO2 is dependent on the stress concentration, coupling equation Eqs. (13) or (41) with diffusion equations, for example, could provide insight about the thickening region. (III) A similar method can be applied to other certain types of problems involving integral equations similar to that of Eq. (6); Examples include: the variation of the stress intensity due to an arbitrary crack front which deviates slightly from flat, the Hertzian contact mechanics due to a rigid punch of arbitrary surface on an elastic layer, and the elastic–plastic plane stress Barenblatt-Dugdale crack model.
Acknowledgements Words of acknowledgment be extended to Drs. G. Tepper from the Virginia Commonwealth University and B. Hinderliter from the University of Minnesota-Duluth for their insightful discussions. This work was developed under the partial sponsorship of grant: NRC-38–10-929. References Allameh, Seyed M., Shrotriya, Pranav, Butterwick, Alex, Brown, Stuart B., Soboyejo, Wole O., 2003. Surface topography evolution and fatigue fracture in polysilicon MEMS structures. J. Microelectromech. Syst. 12 (3), 313–324. Alsem, D., Stach, E., Muhlstein, C., Ritchie, R., 2005. Fatigue failure in thin-film polycrystalline silicon is due to subcritical cracking within the oxide layer. Appl. Phys. Lett. 86, 041914. Anderson, T.L., 1991. Fracture Mechanics, Fundamentals and Applications. CRC Press, Inc. Banichuck, N. Determination of the form of a curvilinear crack by a small parameter technique. Izv. An SSR, MTT 7, Vol 2, pp. 130–137, 1970 (in Russian). Chiu, C., Gao, H., 1993. Stress singularities along a cycloid rough-surface. Int. J. Solids Struct. 30, 2983–3012. Colombi, L., Cole, D., Payne, M., Gumbsch, P., 2008. Stress-driven oxidation chemistry of wet silicon surfaces. J. Phys. Chem. Lett. C 112, 12077–12080. Cotterell, B., Rice, J., 1980. Slightly curved or kinked cracks. Int. J. Fract. 16 (2), 155– 169. Gao, H., 1991a. A boundary Perturbation Analysis For Elastic Inclusions and Interfaces. Int. J. Solids Struct. 28 (6), 703–725. Gao, H., 1991b. Stress concentration at slightly undulating surfaces. J. Mech. Phys. Solids 39 (4), 443–458. Gao, H., 1994. Some general properties of stress-driven surface evolution in a heteroepitaxial thin film structure. J. Mech. Phys. Solids 42 (5), 741–772. Goldstein, R., Salganik, R. Plane Problem of curvilinear cracks in an elastic solid. Izv. An SSR, MTT 7, Vol. 3, pp. 69–82, 1970 (in Russian). Goldstein, R., Salganik, R., 1974. Brittle fracture of solids with arbitrary cracks. Int. J. Fract. 10, 507–523. Green, A.E., Zerna, W., 1968. Theoretical Elasticity, second ed. Oxford University Press, London. Grekov, M., 2011. Two types of interface defects. J. Appl. Math. Mech. 75 (4), 476– 488. Grekov, M., Makarov, S., 2004. Stress concentration near a slightly curved part of an elastic body surface. Mech. Solids. 39 (6), 40–46.
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93 Hahn, S.L., 1996. Hilbert transforms. In: Poularakis, A. (Ed.), The Transforms and Applications Handbook. CRC Press, Boca Raton, FL, ch. 7. Inglis, C.E., 1913. Stresses in a Plate Due to the Presence of Cracks and Sharp Corners. Trans. Instit. Naval Architects 55, 219–241. King, F., 2009. Hilbert transforms. Encyclopedia of Mathematics and its Applications, vols 1& 2. Cambridge Press, Cambridge, UK. Liang, J., Suo, Z., 2001. Stress-assisted reactions at a solid-fluid interface. Interface Sci. 9, 93–104. Li, J., Chiu, C., Gao, H., 1993. Cusp-like flaws along a rough surface. Thin Solid Films 236, 240–246. Medina, H., Hinderliter, B., 2012. Use of Poly (Methyl Methacrylate) in the study of randomly damaged surfaces: I. experimental approach. Polymer. http:// dx.doi.org/10.1016/j.polymer.2012.08.012. Medina, H., Hinderliter, B., 2014. The stress concentration factor for slightly roughened random surfaces: analytical solution. Int. J. Solids Struct. 51 (10), 2012–2018. Medina, Hector E., Pidaparti, Ramana, Hinderliter, Brian, 2014. Celebrating the 100th anniversary of inglis result: from a single notch to random surface stress concentration solutions. Appl. Mech. Rev. 67, 010802. Muhlstein, C., Stach, E., Ritchie, R., 2002. A reaction-layer mechanism for the delayed failure of micron-scale polycrystalline silicon structural
93
films subjected to high-cycle fatigue loading. Acta Mater. 50, 3579– 3595. Muskhelishvili, N., 1975. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff International Publishing, Leyden, The Netherlands (Translated from the Russian by J. Radok). Neuber, H., 1958. Kerbspannunglehre, second ed. Springer-Verlag GmbH, Berlin. Nix, W., 1989. Mechanical properties of thin films. Metall. Trans. A 20 (11), 2217– 2245. Noda, N., Takase, Y., 1999. Stress concentration formulas useful for any shape of notch in a round test specimen under tension and under bending. Fatigue Fract. Eng. Mater. Struct. 22, 1071–1082. Pearson, K., 1997. Stress Concentration Factors,, John Wiley and Sons. Rice, J.R., 1985. First-order variation in elastic fields due to variation in location of a planar crack front. J. Appl. Mech. 52, 571–579. Saint-Venant, A. Barrie De., 1855. Memoire sur la torsion des prismes, avec des considerations sur leur flexion. (Read June 13, 1853). Mem. Divers Savants 14, pp. 233–560. Yu, H.H., 2005. Crack nucleation from a single notch caused by stress-dependent surface reactions. Int. J. Solids Struct. 42, 3852–3866.
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
A stress-concentration-formula generating equation for arbitrary shallow surfaces Hector Medina School of Engineering and Computational Sciences, Liberty University, 1971 University Boulevard, Lynchburg, VA 24515, United States
a r t i c l e
i n f o
Article history: Received 27 January 2015 Received in revised form 8 June 2015 Available online 18 June 2015 Keywords: Stress concentration Analytical solution Arbitrary shallow surfaces
a b s t r a c t Analytical understanding of how stress concentrates is invaluable. An equation that generates stress concentration formulas is derived and shown to apply very well to a number of shallow irregularities on surfaces, for the plane stress conditions and to a first-order approximation. Under shallow conditions, for any 0 first-order Hölder-continuous surface function f ðxÞ, the derived equation is: kt ðxÞ ¼ 1 2H f ðxÞ , where 0 H is the Hilbert transform and f ðxÞ is the spatial derivative of f with respect to the independent variable. It is shown that using this generating equation, well-known traditional results can be easily derived. Also, a number of other stress concentration formulas for various cases are generated. Furthermore, a second-order approximation is introduced, which shows the dependence of kt on not only the slope but also on the concavity of the surface. The approach used herein can be extended to finding closed-form solutions to other integral equations possessing similar kernels for applications such as the variation of the stress intensity factor due to an arbitrary crack front profile (work in progress). Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Analytical understanding of how stress concentrates on surfaces possessing an arbitrary topography is of crucial importance in many applications. For various reasons, it is highly desirable to know if and how a material’s parcel would become the ’’weakest link’’ during the material’s service lifetime. Additionally, analytical solutions of how the stress concentrates provide a more effective route to approach certain multi-physics problems by coupling those solutions with, for example, the Navier–Stokes, Fick’s Law of diffusion, and free-energy Gibbs equations, just to name a few. Furthermore, stress-driven-reaction analyses provide insight about the stability of surfaces, but it is evident that one limitation is the unavailability of stress-concentration solutions for general configurations of surfaces. Much of the advancement on the foregoing topics has relied on the use of surface-stress-concentration distributions for the cases of sinusoidal surfaces (Gao, 1991a; Liang and Suo, 2001), cycloid rough surfaces (Chiu and Gao, 1993; Li et al., 1993), and some single-notch cases (Yu, 2005). In this document, a novel, 2-dimensional, general equation that generates stress concentration formulas for a wide variety of shallow-surface configurations is presented. Additional impetus for this work follows.
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijsolstr.2015.06.006 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.
It has been established that surfaces are not inert to their surroundings, and that the latter interplay with the elastic and plastic flows of material at the boundaries. At such inequilibrium conditions, surfaces undergo transformations which, in turn, could be catastrophic for films, interfaces, coatings, etc. For example, electronic and micro-electromechanical devices can be significantly impacted by the foregoing conditions, mostly due to the presence of large internal stresses introduced during the manufacturing processes. It has been reported that stresses in the 1-GPa order of magnitude can be present in the thin films that comprise integrated circuits and magnetic disks (Nix, 1989). A slight magnification of such high stress levels, due to surface deformation, would definitely increase the likelihood of failure. It has been found (Medina and Hinderliter, 2014) that even slightly random rough surfaces (with heights normally distributed) can most likely magnify the bulk stress by a factor of 1.6. For the case of slightly undulating surfaces, this magnifying factor can range from 2 to 31 (Gao, 1991b). An analytical description of the distribution ofconcentrated stress for an arbitrary surface would provide safety envelopes for surface topographies found experimentally. Furthermore, stress-driven surface evolution has been recognized to be an important process in the behavior of heteroepitaxial films (Gao, 1994). For specific surface morphologies, advancement on stress-driven reactions has been accomplished
1 Although it is shown in this document, in Section 4, via a second-order approximation that these numbers are a few percents lower.
87
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
by coupling analytical solutions of surface stress distribution to the total energy equation. Ever since the development of the first analytical solution for the stress concentration due to an elliptical hole in a plate provided by Inglis (Inglis, 1913), an immense number of solutions have been developed (Pearson, 1997; Medina et al., 2014; Neuber, 1958). Additionally, some attempts have been carried out to consolidate stress concentration formulas and factors, mostly via empirical methods (Pearson, 1997; Medina et al., 2014; Neuber, 1958; Noda and Takase, 1999). Although the foregoing referenced work is commendable, it is mostly focused on stress-concentration factors and not on the stress distribution along the surfaces of the geometries analyzed. In the present work, a stress-concentra tion-formula-generating equation is derived and shown to be applicable to a wide number of shallow surfaces under tension or pure bending, at least. Under shallow2 conditions, for any function3, f ðxÞ descriptive of a 2-dimensional surface, the stress concentration along the profile is given by: 0
kt ðxÞ ¼ 1 2Hðf ðxÞÞ
ð1Þ
where H stands for the general Hilbert transform, and prime represents the first derivative with respect to the given independent variable. It will be shown that using this generating equation, well-known traditional results can be easily obtained. This paper is organized in the following manner. First, the main result, a stress-concentration-formula-generating equation (SCFGE) is derived in a fashion similar to that used by the author in a previous work (Medina and Hinderliter, 2014; Medina and Hinderliter, 2012). Next, the robustness of the SCFGE is shown via derivation of previously developed and accepted results such as semi-elliptical (and semi-circular) single notch, parabolic single notch, sinusoidal surfaces, etc. Following, the stress concentration is found for other interesting profiles, and a table containing results is provided. Next, a second-order approximation is introduced and applied to the undulating surface case. Finally, this document is completed with a discussion and conclusion section.
shows some typical topographies described by f. Being f a perturbation from a previously flat configuration, an important constraint on f is that it must be of small magnitude. Assume that our surface f ðxÞ is being subjected to a remote tensional load r far enough from the surface features that Saint–Venant’s principle applies (Saint-Venant, 1855). This model can be applied, as well, to a thick rectangular beam with a slight rough surface subjected to pure bending5, since a sufficient number of layers of fibers on and near one of surfaces can be assumed to be subjected to a tensional load. Following (Medina and Hinderliter, 2014), we can define the stress and displacements for the rough surface (T ij and ui ) in terms of the reference state values (T rij and uri ) and the perturbed elements (dT ij and dui ), as:
T ij ¼ T rij þ dT ij
ð2Þ
ui ¼ uri þ dui
Consider a concentrated point force F p (see Fig. 1) acting at an arbitrary location ðx; yÞ. The purpose of using a point force is only to utilize the well-known solution for the surface stress Green’s function, and the magnitude of F p can be shown to be irrelevant. Using a similar reasoning as used by others previously (Banichuck, 1970; Goldstein and Salganik, 1970; Goldstein and Salganik, 1974; Rice, 1985) and developed in both (Gao, 1991b and Medina and Hinderliter, 2014), it is shown that perturbation contributions for the stress and displacements are given as:
R1 b G df ðvÞdv 1 ij R1 i r dui ðx; yÞ ¼ E0 1 Gxx df ðvÞdv dT ij ðx; yÞ ¼ r
ð3Þ
which is for the particular case when there is no concentrated force and thus the material is only exposed to the bulk stress; that is, F p ¼0 T xx ¼ T rxx ¼ r. In Eq. (3) E0 is the consolidated Young modulus of elasticity6; b ij is a tensor representing a kernel function which can be And G
derived from the Green’s function, Gixx , in the following manner:
b ij ¼ 1 G E0
"
l
j @Gixx @Gxx þ @xj @xi
!
2lmdij @Gkkk þ 1 2m @xk
#
2. Derivation For the first part of our derivation, we employ a first-order perturbation method. A similar approach was previously used by Banichuck (1970) and Goldstein and Salganik (1970, 1974), later by Cotterell and Rice (1980), and Rice (1985). In the foregoing references, the perturbation approach was applied to the problem of the stress intensity variation due to a crack front that deviates from flatness due to local irregularities in materials (e.g. grain boundaries). Later, this method has been used by Gao (1991b) and later by the author (Medina and Hinderliter, 2014) to develop, respectively, stress concentration factor formulas for undulating surfaces and random rough surfaces. Other related problems, such as the study of the elastic fields at the interfaces between dissimilar materials (Gao, 1991a; Grekov, 2011), have been studied using the perturbation method, as well. Consider the profile from Fig. 1, which is a slight perturbation from an originally flat surface. Profile f ðxÞ is a continuous real function that satisfies the Hölder condition4 within its domain. Fig. 2 2 Shallow means that the heights or depths of hillocks or hollows on the surface are small in comparison to their widths. For example, in the case of a sinusoidal surface, the amplitude is much smaller than the wavelength; or in the case of a single notch, the depth of the notch is much smaller than its width. 3 Actually, as it will be shown, the requirements to be imposed on f do not limit it from any practical applications. 4 This is a very important characteristic of our solution, and it is completely related to a property of the Hilbert transform to be discussed later. Most functions found in practical applications meet this criterion. Moreover, for the particular case of surfaces without cracks, this criterion is obviously always met.
where, in this case, dij is the Kronecker delta, and modulus.
ð4Þ
l is the shear
The Green’s function Gixx can be obtained by differentiating the strain energy density with respect to the point force F p (for details see (Medina and Hinderliter, 2014)) and found to be:
Gixx ðv; x; yÞ ¼
@T xx
ð5Þ
@F ip
where the stress Green’s function in Eq. (5) is understood as the surface stress at v due to a unit point force F p in the i-direction applied at the point source ðx; yÞ. Therefore, the Green’s function above, Eq. (5), can be used to find the kernel of Eq. (4), which in turned can be used to find the contribution of the stress due to the perturbation. Finally, this perturbation part of stress can be used to find the stress increased from the bulk stress, according to part (a) of Eq. (2). Extracting the other Green’s functions Giij from the point force solutions for a half-plane from (Green and Zerna, 1968), it can be shown that Eq. (2)(a) can be expressed as:
dT xx ðx; yÞ ¼ r
Z
1
1
5
6
2
pðv xÞ2
df ðvÞdv
Either a four-point bending or a three-point bending with negligible shear. E; for plane stress E0 ¼ E ; for plane strain;ðm ¼ Poisson0s ratioÞ 1m2
ð6Þ
88
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
Fig. 1. An arbitrary surface f(x) whose heights deviate slightly from the x axis. The remote load, r is applied far away from the features of the surface, in accordance to SaintVenant (1855). The material is assumed to be elastic, homogeneous, and isotropic. A point force F p is applied at some arbitrary location within the material (this is simply an artifact of the mathematical development and has no physical implications since F p is to be set to zero). The material is characterized by Young modulus E and Poisson’ratio m.
Fig. 2. Plots of a few examples of topographies that can be described by f(x). From top to bottom: Cosine pulse notch, Gaussian notch, parabolic notch, and undulating surface. Note that and A have been magnified for visualization purposes. Actually b and A k, for shallow conditions.
and (6) can be substituted into (2) with T rxx ¼ r, which after some manipulation leads to the following result:
2
2 T xx ðxÞ ¼ r41 þ PV
p
where,
df ðvÞ dv
Z
1
1
3 df ðvÞ dv dv 5 vx
is the spatial derivative of f,
ð7Þ
v is a dummy variable, and ‘‘PV’’ means that the integral is calculated in the Cauchy’s principal value sense. At any given point of the profile, Eq. (7) provides the tangential component of the stress, T xx , introduced in part (a) of Eqn. (2). So, for shallow condition of surfaces, the function of the stress concentration, kt ðxÞ, applicable on the surface and along x, can then be written as:
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
2 kt ðxÞ ¼ 41 þ
2
p
Z
PV
1
1
3 df ðvÞ dv dv 5 vx
3.1. Semi-elliptical notch
ð8Þ
We now apply the Hilbert transform (HT) to (8). The HT is defined as follows (Hahn, 1996; King, 2009):
^ ¼ 1 PV Hf ðtÞ ¼ fðtÞ
Z
p
1
1
f ðgÞdg 1 ¼ PV gt p
Z
1
1
f ðgÞdg tg
ð9Þ
Z
1
gðsÞdðt sÞds
Consider the semi-elliptical notch shown in Fig. 3, where a and b are the depth and half-width of the notch, respectively. Such a profile can be described by:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 P2b ðxÞ b
f ðxÞ ¼ a 1
ð10Þ
P2b ðxÞ ¼
1; j x j< b
Applying (9) to (8), one obtains the main result of this work:
ð11Þ
where mathematically, f represents a piece-wise, continuous, real-valued function, by definition. Physically, f is an arbitrary boundary profile of an 2-D elastic region. It is claimed herein that the formula given by Eq. (11) is a stress-concentration-formula-gen erating equation (SCFGE). Another useful form of Eq. (11) can be obtained by using the following property of the HT:
df ðxÞ d½Hf ðxÞ ¼ H dx dx
ð12Þ
Applying Eq. (13), the stress concentration distribution becomes:
kt ðxÞ ¼ 1 2
d½Hf ðxÞ dx
ð13Þ
Eq. (13) comes in handy in situations for which the solution of HT of 0 function f ðxÞ is more complicated to obtain than the HT of the function f ðxÞ itself. Let’s pause for a moment and discuss the general requirements to be imposed on f ðxÞ so that kt ðxÞ can be solved from either Eqs. (11) or (13). Let X be an open set in Rn ; h 2 N, and 0 < a 6 1. Then C h;a ðXÞ is the universe of functions g : X ! R with continuous derivatives in X of order less than or equal to h. Our requirement for f in either Eqs. (11) or (13) is:
8f : f 2 C 1;a ðXÞ
kt ðxÞ ¼ 1 2
ð14Þ
at least within the domain of interest, D 2 R. In the present application, D is the region where the stress concentration is to be evaluated, typically the notch or surface under consideration. For most single-notch cases, D is to be taken far enough from the corners, in accordance to Sain-Venant’s principle (Saint-Venant, 1855). By definition, Eq. (14) means that function f is first-order Hölder continuous. The numerous and powerful properties of the HT make Eqs. (11) or (13) very valuable. Additionally, tables of the HT are widely known, which facilitates the utilization of those equations for design, research and pedagogical purposes.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dfH½a 1 ðbxÞ2 P2b ðxÞg dx
ð17Þ
Applying the HT to the quantity inside squared brackets in Eq. (22), followed by differentiation, one gets:
8 a j x j< b ; > > > ba < b þ 2 paxffiffiffiffiffiffiffiffiffiffi ; x>b b ðbxÞ2 1 kt ðxÞ ¼ 1 2 > > a > ; x < b : b 2 paxffiffiffiffiffiffiffiffiffiffi x 2 b
ð18Þ
ðbÞ 1
Since the region of interest is within the notch7, the value of the stress concentration is the one corresponding to j x j< b, thus:
kt ¼ 1 þ 2
Applying (12) to (11):
ð16Þ
0; elsewhere
1
df ðxÞ 0 ¼ 1 2H½f ðxÞ kt ðxÞ ¼ 1 þ 2H dx
ð15Þ
where the rectangular pulse function, P2b is defined as:
Which is a linear operator that, contrary to Fourier transform, keeps the domain of the independent variable. In essence, the HT of a function g is a convolution of g with the distribution dðtÞ ¼ p1t, or:
^ ¼ ðg dÞt ¼ gðtÞ
89
a b
ð19Þ
for a shallow semi-elliptical notch and to a first-order approximation. Remarkably, this result, Eq. (19), coincides with the traditional stress concentration factor due to an elliptical hole derived by Inglis (1913), which can be applied also as a good approximation to shallow semi-elliptical notches (Pearson, 1997; Anderson, 1991). Eq. (19) implies that for a shallow notch the stress concentration within the boundary of the notch is constant. For a depth-to-width ratio, a=2b, equals to 0.1, the stress is magnified from the remote load by 40%. 3.2. Undulating surfaces The solution to the problem of the stress concentration due to slightly undulating surfaces was solved previously using Muskelishvilli’s complex analysis approach (Gao, 1991a,b). Let us show that our SCFGE, Eq. (13), can easily lead to the same result. Consider the sinusoidal profile shown in Fig. 4 (dotted line) for which the ratio amplitude-to-wavelength, A=k is small. This profile can be expressed as:
f ðxÞ ¼ A cosð2px=kÞ
ð20Þ
Applying Eq. (13) to Eq. (20), the stress concentration distribution is:
kt ðxÞ ¼ 1 2
dfH½A cosð2px=kÞg dx
ð21Þ
Applying the HT and differentiating in Eq. (21), one gets: 3. Use of the SCFGE Let us begin our demonstration of the robustness of this SCFGE result. (The reader is reminded that the following topographies are assumed to be shallow and the material to be elastic, homogeneous, and isotropic. The loading conditions can be either in tension or bending, and the remote load is assumed to be applied far from the features of interest.)
7 Theoretically, and as expected, there are stress singularities at j x j¼ b. At most common strain rates, plasticity effects sets off much faster than the time it takes for that singularity to become of realistic concern. However, it is widely known that corners are soften for proper design purposes. For the remainder of this paper and when dealing with such boundaries, the discontinuity at the extreme points will be omitted in the discussion.
90
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
Fig. 3. Semi-elliptical profile of notch, mathematically described by Eq. (15). Notch is assumed to be shallow, hence a 2b. Note that the notch’s depth has been magnified for the purpose of visualization.
Fig. 5. Plot of parabolic notch described by Eq. (23) and the corresponding stress concentration according to derived Eq. (25). The maximum stress concentration occurs at x = 0. For shallow conditions, 2b.
kt ðxÞ ¼ 1 2 ðA 2p=kÞ cosð2px=kÞ ¼ 1 þ 4p
A cosð2px=kÞ k
ð22Þ
which is consistent with the result obtained in Gao (1991a). And again, the stress concentration factor (SCF) is found when x ¼ nk=ð2pÞ, for n 2 N; SCF ¼ 1 þ 4p Ak . A plot of the stress concentration given by Eq. (22) is shown in Fig. 4 (solid line). Note that, as it would be expected, the stress at the inflection points on the undulating surface is equivalent to the remote applied stress. 3.3. Parabolic notch Consider the parabolic notch shown in Fig. 5 (dotted line). It can be expressed as:
f ðxÞ ¼ 1 þ ðx=bÞ
2
P2b ðxÞ
ð23Þ
where is a scaling factor to guaranty the shallow condition imposed on the SCFGE. Applying Eq. (13) to Eq. (23), the stress concentration can be expressed as:
K t ðxÞ ¼ 1 2
2 dfH½ 1 ðx=bÞ P2b ðxÞg dx
ð24Þ
kt ð0Þ ¼ SCF ¼ 1 þ
8 pb
For semi-parabolic notch of height and width b. Not surprisingly, note that Eq. (26) is identical to Eq. (24) in Grekov and Makarov (2004) (which was obtained using a different mathematical method), for the case when q ¼ 1. 3.4. Gaussian notch and other profiles Consider the notch of Fig. 6 (dotted line), mathematically expressed as: 2
f ðxÞ ¼ ex
ð27Þ
Applying HT to the derivative of the profile given by Eq. (27) and applying Eq. (13), the stress distribution results in:
pffiffiffiffi 4 2 kt ðxÞ ¼ 1 þ pffiffiffiffi ½1 pxex erfiðxÞ
p
erf ðizÞ i
erfiðzÞ ¼
0 1 2
1 xb ðb þ xÞð1 ðx=bÞ Þ bx ðbþxÞ 2 2 @2x
x þ b
kt ðxÞ ¼ 1 þ ln b þ 2A xb p b b b x
pffiffiffiffiffiffiffi where i ¼ 1 The SCF occurs when x ¼ 0 and it is:
Fig. 5 (solid line) shows the stress concentration distribution as described by Eq. (25). An interesting result is the maximum stress concentration, or SCF, occurring at the bottom of the notch, where x ¼ 0:
ð28Þ
where ‘‘erfi’’ stands for the imaginary error function, and can be expressed in terms of the real error function, ‘‘erf’’, as:
Which, for the region within the notch (j x j6 b), leads to:
ð25Þ
ð26Þ
4 SCF ¼ 1 þ pffiffiffiffi
p
ð29Þ
ð30Þ
The stress concentration distribution of the Gaussian notch as given by Eq. (28) is shown in Fig. 6 (solid line). Now consider the profile given in Fig. 7, expressed as:
Fig. 4. Plot of undulating surface described by Eq. (20) and stress concentration according to derived Eq. (22). Surface is assumed to be shallow, hence A k. Profile’s amplitude, A, has been magnified for the purpose of visualization.
91
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
Table 1 Stress Concentration Distribution as provided by Eqs. (11) or (13) for various profiles. b; wð2Þ is the polygamma function of order 2; Profile, f(x)
kt ðxÞ
P2b ðxÞ
1 þ p4b
1 2 1ðx=bÞ
i
h i 1 2 bp 3 wð2Þ ð12 þ bixpÞ þ wð2Þ ð12 bixpÞ 2 1 þ 2 b 1ðx=bÞ 2 2
2
sech ðx=bÞ
h
2
ðx=bÞ þ1
½ðx=bÞ þ1
Fig. 6. Plot of a Gaussian notch described by Eq. (27) and the corresponding stress concentration according to derived Eq. (28). The maximum stress concentration, at x ¼ 0, is: SCF ¼ 1 þ p4ffiffipffi. Note that the actual values to be considered are only those within the notch. Shallow conditions required 4.
to thin films in electronics, coatings, interfaces and surfaces. However, higher order approximations may be also desirable in some cases. Unfortunately, all important perturbation series diverge due to a zero radius of convergence. Thus, in the following it will be assumed that n, the order of approximation, corresponds to the term of optimal truncation. Consider again the situation proposed in Fig. 1. Using the complex potentials approach used by Muskhelishvili (1975) functions for stress and displacement fields for the elastic half-plane have been widely obtained. For the sake of conciseness, the algorithm will be not repeated here. It can be shown (Gao, 1991a; Green and Zerna, 1968; Grekov and Makarov, 2004) that for the perturbation problem stated in Section 2, the nth order approximation of the tangential stress, T xx ðxÞ can be expressed as function a potential function U: ðnÞ T xx ðxÞ
" # n X ¼ 4Re Uj
ð34Þ
j¼0
Fig. 7. Plot of a Cosine-pulse notch (dotted line) described by Eq. (31) and the corresponding stress concentration (solid line) according to derived Eq. (32). This solution is only applicable for a region within the notch far away from the corners at b ¼ 1. For the case of b = 1, the maximum stress concentration, at x ¼ 0, is: SCF ¼ 1 þ 2Siðp=2Þ. ’’Si’’ is the sine integral function.
f ðxÞ ¼ cos
px 2b
P2b ðxÞ
ð31Þ
For the sake of simplicity, the width of the notch is normalized to unity. The stress concentration within such notch is found to be:
2 p px pjx þ 1j pjx 1j Ci kt ðxÞ ¼ 1 þ sin Ci 2 2 2 p 2 p p p p px þ sinc ðx þ 1Þ sinc ðx 1Þ sin 2 2 2 2 2 p px p p þ Si ðx þ 1Þ Si ðx 1Þ cos 2 2 2
2
px cos p2 ðx þ 1Þ cos p2 ðx 1Þ þ ð32Þ cos xþ1 x1 2 where Ci and Si are, respectively, the cosine and sine integral func
tions, and sinc is the normalized cardinal sine function sinpxpx The SCF in Eq. (32) is found at x ¼ 0 to be:
SCF ¼ 1 þ 2Siðp=2Þ
ð33Þ
We have seen some simple examples of the usefulness of either Eq. (11) or (13) in deriving stress concentration and stress concentration factor formula. Table 1 shows a list of more involved derivations. 4. Extension to higher orders As stated before, the derived SCFGE builds on the first-order perturbation approach used by Rice (1985) in the analysis of the stress intensity due to a varying crack front. Therefore, it provides stress concentration formulas that can only be guaranteed for shallow surfaces, which is the case for many applications related
where holomorphic functions U’s are obtained from two suitable complex potentials, for details see Green and Zerna, 1968; Muskhelishvili, 1975; Grekov and Makarov, 2004. As it will be shown, the resulting U’s increase in complexity as the value of n increases. From Eq. (34), it can be observed that: for the zeroth-order approximation only U0 is needed; for the first-order approximation, both U0 and U1 are required; similarly, for the second-order all U0 ; U1 , and U2 need to be found, and so on. From (Grekov and Makarov, 2004) and after some simple but lengthy manipulation, the first three U functions can be expressed as, respectively:
U0 ðzÞ ¼ r=4
r
ð35aÞ 0
U1 ðzÞ ¼ H½f 2 1 U2 ðzÞ ¼ HG2 2i
ð35bÞ ð35cÞ
with G2 defined, in terms of a dummy variable f, as:
0 00 G2 ðfÞ ¼ 2irf ðfÞ if ðfÞ 2Hf ðfÞ 0 2 0 0 0 þ 4irf ðfÞ if ðfÞ 2H½f ðfÞ þ 2r½f ðfÞ
ð36Þ
The negative sign in Eqs. (35b) and (35c) stem from the definition of the Hilbert transform, Eq. (9). Thus, the zeroth-order approximation is given by Eqs. (34) and (35a) which, as expected, is simply the remote tensile load r, or:
hr i ð0Þ T xx ðxÞ ¼ 4Re ¼r 4
ð37Þ
The first-order approximation is obtained using Eq. (34) in combination with Eqs. (35a) and (35b), which can be written as:
hr r i 0 0 0 T ð1Þ Hf ¼ r 2r Hf ¼ rð1 2Hf Þ xx ðxÞ ¼ 4Re 4 2
ð38Þ
So the first-order approximation of the stress concentration is obtained dividing Eq. (38) by the remote load:
92
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93
ð1Þ
kt ðxÞ ¼
ð1Þ T xx ðxÞ
r
¼ 1 2Hf
0
ð39Þ
Which is the exact same result obtained for our SCFGE, in Eq. (11). For the second-order approximation, we plug Eqs. (35a)– (35c) and (36) into (34), and simplify to:
0 00 0 0 ð2Þ T xx ðxÞ ¼ rð1 2Hf Þ þ 8r H½f Hf þ 2H½f Hf
ð40Þ
So the second-order approximation of the SCFGE is then: ð2Þ
kt ðxÞ ¼
ð2Þ T xx ðxÞ
r
0
00
00
0
¼ 1 2Hf þ 8H½f Hf 16H½f Hf
ð41Þ
Provided that the condition imposed in f in (14) changes to:
8f : f 2 C 2;a ðXÞ
ð42Þ
at least, within the domain of interest (non-flat portion of the elastic surface). Eq. (41) provides insightful information: (1) While the first-order approximation shows kt ’s dependence only on the slope of the surface, the second-order approximation shows its dependence on both slope and curvature. (2) By extrapolation, one can generalize that for an n-order approximation, kt would depend on all derivatives of order 6 n. However, despite the mathematical contribution of the second-order approximation given by Eq. (41), it must be kept in mind that higher-order approximations may quickly diverge depending on the smallness of the perturbing element. The foregoing may be corrected applying secular-type methods which involves the use of multiple scales. Another difficulty is the complexity in developing the third and fourth terms in Eq. (41) for various profiles f’s. Finally, for an nth-order approximation, the restriction on f becomes more strict, as it must belong to a Hölder space of nth order. Nonetheless, for some practical topographies it can provide even more insightful information than the first-order solution. For example, for the undulating surface, f ¼ A cosð2p=kÞ, and using the second-order stress concentration Eq. (41) (after a lengthy but straightforward calculation), it can be shown (work in progress) that the third term provided by the A 2 second-order approximation is proportional to and k cosð4px=kÞ, or: ð2Þ
kt ðxÞ ¼ 1 þ 4p
A cosð2px=kÞ þ F ðA=kÞ2 cosð4px=kÞ k
ð43Þ
where FðÞ simply stands for ‘‘a function of’’. That the third term in Eq. (43) contains ðA=kÞ2 coincides with the general series form of a perturbation-theory solution. It turns out that, for the particular case of the maximum stress concentration the relative contribution of that third term is around 6%, which would decrease the stress concentration factor (SCF) at the lowest point of each valley of the sinusoidal surface from around 2.26 to 2.13. This latter result is in agreement with finite element results reported by Gao (1991a).
5. Discussion and conclusions (I) Undoubtedly, the form of Eq. (13) makes it very simple to find formulas for the stress concentration and hence SCF for arbitrary profiles, under the assumptions prescribed. This simplicity stems from the presence of the HT, whose properties are numerous, and tables of transforms are widely available. A physico-mathematical explanation for the appearance of the HT in the SCFGE involves its connection to the Cauchy’s Integral Formula, and the fundamental use of the latter in the development of the theory of elasticity, as we know it. (By ’’blowing up’’ the contour of integration in a semi-circle in the upper plane, one can map a closed-path integral onto the real axis.)
(II) For applications involving stress-driven reactions, Eqs. (13) or (41) can be coupled to other laws such as Fick’s Laws of diffusion, Navier–Stokes equations, etc., to advance knowledge on local fluid–solid phenomena. For example, premature mechanical failure has been observed in single-crystal silicon microelectromechanical systems (MEMS) (Muhlstein et al., 2002; Allameh et al., 2003; Alsem et al., 2005). Consider the chemical reactions at the interface between a MEMS device and a wet environment. It has been found that the presence of tensile stresses increase: (1) the electrostatic potential for oxygen molecules from water to bond to Si, and (2) the hybridization of Si and O orbitals (Colombi et al., 2008). The latter mechanism promotes the breaking of existing Si-O, thus creating more free Si sites exposed to the further attack of the environment. Additionally, it has been established that the so called ’’reaction-layer’’ process is initiated by high stress concentrations created at the bottom of notches. Since the thickening of the native SiO2 is dependent on the stress concentration, coupling equation Eqs. (13) or (41) with diffusion equations, for example, could provide insight about the thickening region. (III) A similar method can be applied to other certain types of problems involving integral equations similar to that of Eq. (6); Examples include: the variation of the stress intensity due to an arbitrary crack front which deviates slightly from flat, the Hertzian contact mechanics due to a rigid punch of arbitrary surface on an elastic layer, and the elastic–plastic plane stress Barenblatt-Dugdale crack model.
Acknowledgements Words of acknowledgment be extended to Drs. G. Tepper from the Virginia Commonwealth University and B. Hinderliter from the University of Minnesota-Duluth for their insightful discussions. This work was developed under the partial sponsorship of grant: NRC-38–10-929. References Allameh, Seyed M., Shrotriya, Pranav, Butterwick, Alex, Brown, Stuart B., Soboyejo, Wole O., 2003. Surface topography evolution and fatigue fracture in polysilicon MEMS structures. J. Microelectromech. Syst. 12 (3), 313–324. Alsem, D., Stach, E., Muhlstein, C., Ritchie, R., 2005. Fatigue failure in thin-film polycrystalline silicon is due to subcritical cracking within the oxide layer. Appl. Phys. Lett. 86, 041914. Anderson, T.L., 1991. Fracture Mechanics, Fundamentals and Applications. CRC Press, Inc. Banichuck, N. Determination of the form of a curvilinear crack by a small parameter technique. Izv. An SSR, MTT 7, Vol 2, pp. 130–137, 1970 (in Russian). Chiu, C., Gao, H., 1993. Stress singularities along a cycloid rough-surface. Int. J. Solids Struct. 30, 2983–3012. Colombi, L., Cole, D., Payne, M., Gumbsch, P., 2008. Stress-driven oxidation chemistry of wet silicon surfaces. J. Phys. Chem. Lett. C 112, 12077–12080. Cotterell, B., Rice, J., 1980. Slightly curved or kinked cracks. Int. J. Fract. 16 (2), 155– 169. Gao, H., 1991a. A boundary Perturbation Analysis For Elastic Inclusions and Interfaces. Int. J. Solids Struct. 28 (6), 703–725. Gao, H., 1991b. Stress concentration at slightly undulating surfaces. J. Mech. Phys. Solids 39 (4), 443–458. Gao, H., 1994. Some general properties of stress-driven surface evolution in a heteroepitaxial thin film structure. J. Mech. Phys. Solids 42 (5), 741–772. Goldstein, R., Salganik, R. Plane Problem of curvilinear cracks in an elastic solid. Izv. An SSR, MTT 7, Vol. 3, pp. 69–82, 1970 (in Russian). Goldstein, R., Salganik, R., 1974. Brittle fracture of solids with arbitrary cracks. Int. J. Fract. 10, 507–523. Green, A.E., Zerna, W., 1968. Theoretical Elasticity, second ed. Oxford University Press, London. Grekov, M., 2011. Two types of interface defects. J. Appl. Math. Mech. 75 (4), 476– 488. Grekov, M., Makarov, S., 2004. Stress concentration near a slightly curved part of an elastic body surface. Mech. Solids. 39 (6), 40–46.
H. Medina / International Journal of Solids and Structures 69–70 (2015) 86–93 Hahn, S.L., 1996. Hilbert transforms. In: Poularakis, A. (Ed.), The Transforms and Applications Handbook. CRC Press, Boca Raton, FL, ch. 7. Inglis, C.E., 1913. Stresses in a Plate Due to the Presence of Cracks and Sharp Corners. Trans. Instit. Naval Architects 55, 219–241. King, F., 2009. Hilbert transforms. Encyclopedia of Mathematics and its Applications, vols 1& 2. Cambridge Press, Cambridge, UK. Liang, J., Suo, Z., 2001. Stress-assisted reactions at a solid-fluid interface. Interface Sci. 9, 93–104. Li, J., Chiu, C., Gao, H., 1993. Cusp-like flaws along a rough surface. Thin Solid Films 236, 240–246. Medina, H., Hinderliter, B., 2012. Use of Poly (Methyl Methacrylate) in the study of randomly damaged surfaces: I. experimental approach. Polymer. http:// dx.doi.org/10.1016/j.polymer.2012.08.012. Medina, H., Hinderliter, B., 2014. The stress concentration factor for slightly roughened random surfaces: analytical solution. Int. J. Solids Struct. 51 (10), 2012–2018. Medina, Hector E., Pidaparti, Ramana, Hinderliter, Brian, 2014. Celebrating the 100th anniversary of inglis result: from a single notch to random surface stress concentration solutions. Appl. Mech. Rev. 67, 010802. Muhlstein, C., Stach, E., Ritchie, R., 2002. A reaction-layer mechanism for the delayed failure of micron-scale polycrystalline silicon structural
93
films subjected to high-cycle fatigue loading. Acta Mater. 50, 3579– 3595. Muskhelishvili, N., 1975. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff International Publishing, Leyden, The Netherlands (Translated from the Russian by J. Radok). Neuber, H., 1958. Kerbspannunglehre, second ed. Springer-Verlag GmbH, Berlin. Nix, W., 1989. Mechanical properties of thin films. Metall. Trans. A 20 (11), 2217– 2245. Noda, N., Takase, Y., 1999. Stress concentration formulas useful for any shape of notch in a round test specimen under tension and under bending. Fatigue Fract. Eng. Mater. Struct. 22, 1071–1082. Pearson, K., 1997. Stress Concentration Factors,, John Wiley and Sons. Rice, J.R., 1985. First-order variation in elastic fields due to variation in location of a planar crack front. J. Appl. Mech. 52, 571–579. Saint-Venant, A. Barrie De., 1855. Memoire sur la torsion des prismes, avec des considerations sur leur flexion. (Read June 13, 1853). Mem. Divers Savants 14, pp. 233–560. Yu, H.H., 2005. Crack nucleation from a single notch caused by stress-dependent surface reactions. Int. J. Solids Struct. 42, 3852–3866.