Curvilinear Mode-I/Mode-II interface fracture with a curvature-dependent surface tension on the boundary
A new model of fracture mechanics considered previously by Sendova and Walton SendovaWalton2010 (), Zemlyanova Zemlyanova2013 (), and Zemlyanova and Walton Zemlyanova2012 () is further developed on the example of a mixed mode curvilinear interface fracture located on the boundary of a partially debonded thin elastic inclusion embedded in an infinite thin elastic matrix. The effect of the nano-structure of the material is incorporated into the model in the form of a curvature-depended surface tension acting on the boundary of the fracture. It is shown that the introduction of the surface tension allows to eliminate the classical oscillating and power singularities of the order present in the linear elastic fracture mechanics. The mathematical methods used to solve the problem are based on the Muskhelishvili’s complex potentials and the Savruk’s integral representations. The mechanical problem is reduced to the system of singular integro-differential equations which is further reduced to a system of weakly-singular integral equations. The numerical computations and comparison with known results are presented.
Keywords:fracture, surface elasticity, surface tension, complex potentials, integral equations.
Msc:74B05, 74K20, 45E05, 45J05.
The study of brittle fracture in solids has been a subject of many investigations. Historically, the fracture mechanics problems were considered within the framework of linear elastic fracture mechanics (LEFM). The LEFM theory allows to reduce many mechanical problems to relatively simple mathematical equations which can be studied efficiently both theoretically and numerically while describing the behavior of many materials with sufficient accuracy. At the same time, LEFM contains important internal inconsistencies. In particular, while all of the equations of linear elasticity are obtained under assumption that the stresses and the strains are bounded everywhere in an object, LEFM predicts a power singularity of the order in the stresses and the strains at the crack tips. This contradiction becomes even more pronounced for an interface fracture on the boundary of two materials with different mechanical properties. In this case, in addition to the power singularity of the order , the stresses and the strains possess a power singularity of a pure imaginary order (oscillating singularity) which predicts wrinkling and interpenetration of the crack surfaces near the crack tips.
Multiple attempts have been made to remove these inconsistencies of LEFM. It has been observed that due to the stress concentration near the crack tips, the behavior of the material can no longer be considered linearly elastic. Consequently, cohesive and processing zone models have been introduced to account for non-linear elastic and plastic deformations as well as damage accumulation near the crack tips. The main difficulty in the application of these models is in specification of the constitutive properties in the cohesive or processing zones which are very difficult to obtain from experiment. Extensive work in this area has been done by many authors including classical papers by Barenblatt Barenblatt1962 () and Dugdale Dugdale1960 ().
Since the fracture initiation and propagation is a nano-scale process, it has been argued that continuum models cannot be used to accurately describe it. Various atomistic and lattice approaches have been developed to overcome this difficulty Abraham2001 (), FinebergGross1991 (), HollandMarder1998 (), SlepyanEtAl1999 (). This approach requires an accurate description of long-range and short-range intermolecular forces and also presents some computational challenges.
Blended atom-to-continuum models have recently generated a considerable attention. Among those it is necessary to point out the quasi-continuum method introduced by Tadmor et al TadmoretAl1996 () and the method based on the introduction of bridging domains between continuum and atomistic regions proposed by Xiao and Belytschko XiaoBelytschko2004 (). Atom-to-continuum models involve adjustable parameters, such as choosing the size and location of domains over which potentials acting at different length scales are blended. One of the main difficulties here is in constructing a coupled energy whose minimizers are free from uncontrollable errors (“ghost forces”) on the atomistic/continuum interface. One of the ways to overcome this difficulty has been proposed in Shapeev2011 ().
Oscillating power singularity of a pure imaginary order at the crack tips of an interface fracture has been first described by Williams in Williams1959 (), and later confirmed by other authors. Different approaches have been proposed to eliminate this particular type of singularity such as the contact zone model suggested by Comninou Comninou1990 () and the intermediate layer model suggested by Atkinson Atkinson1977 ().
Recently, several continuum models of fracture mechanics with surface excess properties on the boundary of the fracture have been studied as well. The physical motivation behind these models stems from the fact that the material particles on the boundary of the solid experience different force system compared to the particles in the bulk. The first comprehensive model of surface stressed solids has been developed by Gurtin and Murdoch GurtinMurdoch1975 (), GurtinMurdoch1978 (). Their approach is based on the notion of a thin two-dimensional membrane that is effectively bonded to the surface of a three-dimensional bulk substrate. The membrane is modeled as an elastic surface without accounting explicitly for its thickness in three dimensions. This idea generalizes the classical notions of a fracture energy and surface tension and furnishes a basis for study of mechanics of coated surfaces.
The Gurtin-Murdoch theory is very popular and has been widely applied to the study of inhomogeneities and nano-structures by many authors Duan2005a (), Mogilevskaya2010 (), Sharma2004 (), Tian2007 (). Recently Kim, Schiavone, and Ru applied the Gurtin-Murdoch theory to the modeling of fracture Schiavone2010a ()-Schiavone2012 (). Their results for a straight interface and a non-interface mode-III fracture show that taking into account the surface elasticity eliminates the square root singularities of the stresses and the strains. At the same time it has been shown Schiavone2012 () that the presence of the surface elasticity by itself is not sufficient to guarantee that the stresses and the strains are bounded at the crack tips. Weaker logarithmic singularities may still be present. Moreover, a mode-I fracture may still continue to exhibit strong square-root singularities even if the surface elasticity is taken into account Schiavone2012 ().
Another theory of solids with surface excess properties has been developed by Slattery et al SlatteryEtAl2004 (), Slatteryetal2007 (). It is assumed that the material surfaces are endowed with a curvature-dependent surface tension. This theory has been first applied to the study of fracture in OhWaltonSlattery2006 () and developed in more detail on the example of a straight mode-I fracture in SendovaWalton2010 (). The results of Sendova and Walton SendovaWalton2010 () show that for a mode-I straight non-interface fracture all of the stresses and strains are bounded if the surface tension on the boundary of the fracture depends linearly on the mean curvature of the deformed fracture. However, it has been shown later on the example of a mode-I/mode-II non-interface curvilinear plane fracture Zemlyanova2012 () and on the example of a straight mode-I/mode-II interface fracture Zemlyanova2013 () that this conclusion does not hold for non-symmetric configurations. The main conclusion of the papers SendovaWalton2010 (), Zemlyanova2013 (), Zemlyanova2012 () is that incorporation of the curvature-dependent surface tension allows to eliminate the classical singularity of the order and also oscillating singularity in the case of a straight interface crack. However, some components of the stresses and strains may still retain a weaker logarithmic singularity. It has been shown recently Walton2013 () that the modification of the curvature-dependent surface tension model, which includes a surface stretch in addition to the mean curvature, removes the logarithmic singularities in the case of a straight mode-I/mode-II fracture.
The determination of the surface energy parameters presents an interesting practical problem. One of a few studies dedicated to this problem is the paper by Mohammadi and Sharma Mohammadi2012 (). The surface energy parameters have been obtained within the framework of the Steigmann-Ogden theory by comparing theoretical results with atomistic computations for nanowires with flat and corrugated surfaces. It has been shown in Mohammadi2012 () that surface roughness significantly influences the values of the surface energy parameters. Additional discussion of these issues is available in Ergincanetal () and WeisDuan2008 ().
The focus of the current study is to continue the investigation started in Zemlyanova2013 (), Zemlyanova2012 () and to generalize the curvature-depended surface tension fracture model proposed in OhWaltonSlattery2006 (), SendovaWalton2010 () to the case of an interface curvilinear fracture of an arbitrary shape on the boundary of a partially debonded thin elastic inclusion embedded into a thin elastic plate (matrix). It is assumed that in-plane stresses act on the boundary of the fracture and at infinity of the plate. The nano-structure of the material of the inclusion and the matrix is modeled with the help of a curvature-depended surface tension which acts on the interface between two different materials or between a material and a void. It is assumed that the fracture can be of arbitrary smooth shape. This is a significant advantage of the present study since most of the fracture literature deals with straight or circular-arc-shaped cracks. The mechanical problem is reduced to a system of singular integro-differential equations using methods of complex analysis. This system is further reduced to a system of weakly-singular integral equations. The numerical solution to the system of singular integro-differential equations is obtained using approximations of the unknown functions by Taylor polynomials with unknown coefficients. This allows to reduce the approximate solution of the system of integro-differential equations to the system of linear algebraic equations. Convergence of the proposed numerical method is studied on examples. Comparison of the current results with known results obtained in CrouchMogilevskaya2006 (), PrasadSimha2003 (), SilZem2001 (), Toya1974 (), Zemlyanova2012 () is given as well. The main conclusion of the paper is that the introduction of the curvature-dependent surface tension allows to eliminate integrable power singularities of the order and oscillating singularities of the stresses and the strains at the crack-tips, and hence presents a significant improvement of the classical LEFM theory.
2 Model with a curvature-dependent surface tension
Consider an infinite thin elastic plate (matrix) which has a hole with a smooth boundary . A thin elastic inclusion of the same size as the hole is inserted into the hole and partially attached to the matrix along the line (fig. 1). The plate and the inclusion are homogeneous and isotropic. Their shear moduli and Poisson ratios are given by , and , correspondingly. The principal in-plane stresses and are applied at infinity of the plate and act in the directions constituting the angles and with the positive direction of the real axis.
Assume that the inclusion and the matrix are perfectly attached along the junction line :
where is the vector of the displacements at a point of the plate or the inclusion . Parameters without a subscript are related to the plate ; parameters with a subscript “” are related to the inclusion . Here and henceforth, the superscripts “” and “” denote the limit values of the stresses, the displacements and other parameters from the left-hand and the right-hand sides correspondingly of the lines or . The default direction of the curve is chosen to be counterclockwise (fig. 1).
Assume that the surface tension acts on the dividing lines and which separate materials with different properties (two different elastic materials on the line and an elastic material and a void/gas on the left- or the right-hand side of the line ). The surface tension allows to take into the account the effects of long-range intermolecular forces on the dividing lines , . Then the differential and the jump momentum conditions in the deformed configuration in the absence of inertial and gravitational effects become Slatteryetal2007 ():
where is the Cauchy stress tensor, is the unit normal to the fracture surface pointing into the bulk of the material, is the mean curvature, , denote the surface gradient and the surface divergence correspondingly, and the double brackets denote the jump of the quantity enclosed across the boundary of the line . The equation (2.2) is valid in the bulk of the material of the inclusion or the matrix , and the equation (2.3) is valid on the boundary lines and .
Assume that the surface tension depends linearly on the difference between the curvature of the deformed line and the curvature of the line in the unloaded configuration Zemlyanova2012 ():
where the subindex “” denotes the parameters in the initial undeformed configuration of the line , and the superindex “” should be replaced by “” to denote the surface tension parameter on the crack from the side of the inclusion , by “” to denote the surface tension parameter on the crack from the side of the plate , and by “” to denote the surface tension parameter between the inclusion and the matrix on the junction line .
Let be the parametric equation of the line (fig. 1), where the parameter is an arc length, and the values correspond to the curve , while the values correspond to the curve . Since the contour is closed, it is necessary that . Assume that the function has continuous derivatives up to the fourth order. This assumption is made for simplicity and can be somewhat relaxed. It is possible to rewrite the condition (2.3) in terms of the normal and shear stresses in the following form:
where and are the tensile and the shear components of the stress vector acting on the tangent line to the curves and .
and are given external in-plane stresses applied to the banks of the crack, the superindices “”, “”, and “” have the same meaning as in (2.4), denotes the mean curvature of the line or in the undeformed initial configuration at the point described by the arc length . The equations (2.6) and (2.7) correspond to the boundary conditions on the crack from the side of the inclusion and the matrix correspondingly, the equation (2.8) describes the boundary condition on the junction line .
3 Complex potentials and their integral representations
The presented method of the solution is based on the use of the Muskhelishvili’s complex potentials Mus1963 () and the Savruk’s integral representations Savruk1981 (). It is well-known that the stresses and and the derivatives of the displacements in the inclusion and the matrix can be expressed through two complex functions , (complex potentials) analytic in or correspondingly using the following formulas Mus1963 ():
Here for the case of plane stress and for the plane strain, and all of the parameters and functions corresponding to the inclusion should be taken with “” subscript, for example, , , , , , and so on.
The analytic functions , and , can be expressed through the jumps of the stresses and the derivatives of the displacements on the contour by using the following integral representations which are due to Savruk Savruk1981 ():
where the jumps of the stresses , and the derivatives of the displacements , on the contour in the inclusion and in the plate can be found from the formulas:
Hence, the stressed state of the inclusion and the matrix is described by the complex potentials (3.3), (3.4) which contain four unknown functions , and , defined on the contour . We will look for these functions in the class of functions satisfying the Hölder condition on the curves , . This choice guarantees the existence of all principal and limit values of the integrals of the Cauchy type in the formulas (3.3), (3.4), except for, maybe, at the end-points of the curves , .
The matrix occupies the exterior of the line , and the inclusion occupies the interior of the line . Hence, the values , , , in the formulas (3.5)-(3.8) do not have any physical meaning and can be defined formally in multiple ways. The choice of the definition depends on the ease of treatment of the resulting system of singular integro-differential equations. In this case, it is convenient to formally extend the inclusion and the matrix to a full complex plane by assuming that the stresses and the derivatives of the displacements are equal to zero outside of the line for the inclusion or inside of the line for the matrix correspondingly:
In particular, this leads to the boundary conditions:
Observe, that if we assume the boundary conditions (3.10), then from the uniqueness of the solution of the second fundamental problem of elasticity Mus1963 (), it follows that the conditions (3.11) are true as well and that the equations (3.9) are satisfied.
If the equations (3.12)-(3.13) are further substituted into the equations (2.6)-(2.8) and the equation (2.1) differentiated by , the conditions on the boundaries of the inclusion and the matrix become:
Observe, that formally extending the inclusion and the plate using the formulas (3.9) allows us to write the boundary conditions (2.1), (2.6)-(2.8) in the form (3.14)-(3.20), and, hence, to avoid the differentiation of the singular integrals in the formulas (3.21), (3.22) which would be necessary otherwise. To make sure that the equations (3.9) hold, it is sufficient to satisfy the conditions (3.12) which lead to a relatively simple system of singular integral equations:
where and are the regular kernels defined by the formulas:
Observe, that integrating the right-hand side of the equation (3.24) along the contour produces an identical zero. Thus, the general solution of the system (3.23), (3.24) contains two real constants which can be fixed by stating two additional real conditions. It is required from a physical viewpoint that the displacements are single-valued along the contour which means that the relative displacements of the crack tips should be the same traced along both left- and right-hand-side banks of the crack. This condition is not satisfied automatically and needs to be stated additionally as a part of the solution:
4 System of singular integro-differential equations
Observe, that the solutions , , , to the system (3.23), (3.27) are sought in the class of Hölder continuous functions on the curves , and can have at most integrable power singularities at the end points of the lines , . Hence, since the contour is closed it is possible to use the inversion formula for a singular integral of Cauchy type MikhPros1986 ():
where denotes the Cauchy singular integral operator on the curve , and is the identity operator.