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Stress transfer through fully bonded interface of layered materials
Accepted Manuscript Stress transfer through fully bonded interface of layered materials Huiming M. Yin, Pablo A. Prieto-Muñoz PII: DOI: Reference:
S0167-6636(13)00048-3 http://dx.doi.org/10.1016/j.mechmat.2013.03.007 MECMAT 2104
To appear in:
Mechanics of Materials
Received Date: Revised Date:
21 March 2012 15 March 2013
Please cite this article as: Yin, H.M., Prieto-Muñoz, P.A., Stress transfer through fully bonded interface of layered materials, Mechanics of Materials (2013), doi: http://dx.doi.org/10.1016/j.mechmat.2013.03.007
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Stress transfer through fully bonded interface of layered materials Huiming M. Yin*1, Pablo A. Prieto-Muñoz1 1
Department of Civil Engineering and Engineering Mechanics, Columbia University 610 Seeley W. Mudd 500 West 120th Street, New York, NY 10027
Plane strain elastic theory of stress transfer in multi-layered materials is formulated and is used to investigate the stress distribution of a coating system under a tensile load. When the substrate layer is subjected to a uniaxial load, the load is transferred to the coating through interfacial shearing stress. With the aid of a plane assumption, decoupled governing equations are obtained, and the general solution of the displacement field can be derived for both the coating and the substrate layers. Using the boundary conditions and the interfacial continuities, we obtain a closed-form solution for the elastic fields in both the overlay and the substrate layers, which takes the first-term of a series-form solution. Although the singularity effect of stress at the ends of the interface and loading points cannot be exactly illustrated due to the simplification and assumptions, the proposed formulation provides excellent agreement with the finite element results of the transferred stress in the thickness direction of the coating system. Comparisons with the existing models demonstrate the capability and limitation of the proposed formulation. This theory can serve as a baseline for future fracture analysis, inelastic analysis, and thermomechanical analysis of multi-layered materials. Keywords: stress transfer, fully bonded interface, multi-layered materials, elastic analysis, boundary-value problem, interfacial compliance
*
Corresponding author: Tel.: +1-212-851-1648 e-mail: [email protected]
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1. Introduction Multi-layered materials and structures are commonly observed in nature and can be used in advanced material design, such as sedimentary rocks (Bai, et al., 2000a, Li and Yang, 2007, Tang, et al., 2008), pavements (Prieto-Muñoz, et al., 2013, Timm, et al., 2003, Yin, 2010c, Yin, et al., 2007a), surface protective coatings (Colak, 2001, Yin, et al., 2007b), thin film/substrate systems (Beuth, 1992, Freund and Suresh, 2003, Shenoy, et al., 2000, Xia and Hutchinson, 2000, Yin, et al., 2008), and multi-ply laminates and structures (Fellah, et al., 2007, Hwu and Derby, 1999, Yang and Yin, 2011, Yin, et al., 2013). When a multi-layered system is subjected to a mechanical load or an environmental stress (Yang, et al., 2012, Yin, 2010c, Yin, et al., 2013), the deformation of the material in each layer tends to be different due to a mismatch in material properties. Interfacial shearing stress will be induced to make the deformation compatible for structural integrity. Stress transfer through the interface has been a challenging problem over the past century. There are three widely accepted methods to solve the problem. First, Stoney (1909) studied the deformation of thin film/substrate systems with a misfit strain. Similar to bending theory, the stress can be calculated from the curvature of the system. This theory has been extended to multi-layered system with the thicknesses much smaller than their length, and has been used to study thermal stress due to temperature change (Freund and Suresh, 2003, Hsueh, 2001, Li, et al., 2007). However, if one layer is much thicker or stiffer, or the layer configuration is symmetric in the thickness direction, the curvature of the multi-layered system can be very small or equal to zero, and thus this theory will not be applicable. In addition, if the deformation is driven by a stress in the plane, the curvature change will not be comparable to the overall deformation. The transfer of stress cannot be sufficiently explained by this theory.
2
The second method uses the shear lag theory, which can be traced back to Cox’s pioneering work on stress transfer between a fiber and a matrix under a tensile load (Waldhoff, et al., 2000). This theory has been widely used in the stress and strength analysis of fiber reinforced concrete. It assumes that the strong material will carry the tensile load which is transferred to the matrix through interfacial shearing stress (Beuth and Klingbeil, 1996). Various assumptions have been made to simplify the shearing stress distribution along the interface so that the stress distribution can be solved numerically or analytically. This theory has been successfully extended to analyze multi-layered material systems. In general, a displacement discontinuity is assumed to calculate the stress transfer through the interface (Hsueh, 2001, Laws and Dvorak, 1988, Nairn and Mendels, 2001). Those specific assumptions however, are only applicable to specific materials or interfacial conditions. The accuracy of the interfacial shearing stress is therefore questionable. The third method for studying stress transfer follows Hobbs’ work on layered rocks (1967). When a two-layer system is subjected to a tensile load, opening-mode fractures form in the weak layer. To obtain the stress distribution in the fractured weak layer, it is considered as a onedimensional (1D) bar. The normal stress is balanced by the distributed shearing stress along the interface. This theory has also been used in the stress analysis of pavements (Timm, et al., 2003) and thin films (Xia and Hutchinson, 2000). Although linear distribution of shearing stress was assumed along the thickness of the weak layer (Jain, et al., 2007), the 1D solution cannot accurately describe the stress distribution. Moreover, to establish the governing equation, a frictional interface was generally assumed between the two layers using a spring coefficient, which describes the relation between the interfacial shearing stress and the displacement at each interface point. The concept of an interfacial coefficient or compliance is not well established or understood yet, which causes contradictory formulations in the literature (Jain, et al., 2007,
3
Suhir, 1989, Suhir, 1991, Timm, et al., 2003, Xia and Hutchinson, 2000, Yin, et al., 2008, Yin, 2010a). In addition, because the stress variation in the thickness direction plays an important role in the fracture analysis of the weak layer, improved accuracy of stress transfer theory is highly needed. Recent research on stress transfer through multi-layered systems mainly focused on numerical and experimental methods with a fully bonded interface (Bai, et al., 2002, Li and Yang, 2007, Tang, et al., 2008). Bai and his colleagues conducted a comprehensive experimental and numerical investigation (Bai, et al., 2002, Bai, et al., 2000a, Bai and Pollard, 1999, Bai, et al., 2000b) and found that indeed the existing stress-transfer theory fails to satisfy the equations of equilibrium for an elastic boundary-value problem. It was shown in their numerical investigation that when the fracture spacing is small, the stress in the central region of the weak layer is compressive rather than tensile, thus leading to fracture saturation in layered materials. Yin and his colleagues have proposed a two-dimensional (2D) elastic solution of a film bonded to a strong substrate subjected to a tensile load (Yin, et al., 2008, Yin, et al., 2007a), which was successfully applied to interpret fracture initiation, infilling, and saturation caused by tension or temperature loads in the layered materials (Nairn and Mendels, 2001, Yin, 2010b). It provides a very accurate solution for the stress distribution in a compliant thin film/stiff substrate system. However, the interfacial condition still followed the disputable assumption of a frictional interface (Xia and Hutchinson, 2000). Although the equivalence of a frictional interface with the exact solution of a fully bonded interface (Beuth, 1992) was explored, the accuracy of the stress solution for the two-layer system with a compliant substrate is inadequate since the frictional interface cannot sufficiently explain the mechanics of stress transfer through a fully bonded
4
interface. Moreover, the elastic field in the substrate cannot be derived in the previous formulations using the friction interface. The purpose of this work is to develop a theory of stress transfer in layered materials with a fully bonded interface in order to solve the elastic boundary-value problem considering in-plane misfit strain and tensile load. Because the mechanical loading and mismatch strain are within the layer plane, it is assumed that all points of a layer plane remain in the same plane during deformation. This assumption has been widely used in the shear-lag theory as well as our previous work (Nairn and Mendels, 2001, Yin, et al., 2007a). The elastic governing equations are then decoupled and the general solution of the in-plane displacement is obtained. Using the boundary conditions and the interfacial continuities, a closed-form solution can be derived. Due to the plane assumption used in the derivation, the singularity at the edge of the interface (Bogy, 1971, Zhang and Suo, 2007) cannot be exactly modelled in the formulation. Although the local field in the neighbourhood of the edge of the interface cannot be precisely illustrated, the proposed formulation is still accurate for the overall stress distribution in the zone that is away from the singularity points. The remainder of this paper is organized as follows: Section 2 formulates the plane strain elastic theory of stress transfer in a multi-layered system and presents the general solution to the governing equations. Section 3 solves the boundary-value problem of a coating system with the substrate subjected to a tensile load. The elastic fields in both the coating and the substrate are obtained and are verified with the finite element method, which shows the accuracy of the proposed solution for analysing stress transfer. Parametric analyses are conducted to study the stress distribution in both layers for different material properties and geometric configurations. Section 4 discusses the limitations and advantages of the proposed theory compared with other
5
stress transfer theories in the literature, such as frictional interface models and shear lag models. Using the proposed theory, the elastic modulus reduction and interfacial compliance are calculated and are compared with those in the literature. Lastly, some conclusive remarks are given in Section 5.
2. Simplified governing equations and the general solutions Consider a multi-layered material system containing n layers of length thickness
, Young’s modulus
, and Poisson’s ratio
with
. Each layer has a . They are fully
bonded to their neighboring layers, as illustrated in Figure 1. When one layer, say layer 1, is subjected to a force P, the force will be transferred to the other layers through the fully bonded interfaces. Although the ends of the other layers are free, the normal stress will increase from the ends and reach a maximum at the center, which may produce an open-mode fracture in the middle. Because the width of the system is much larger than the thickness, our formulation will be based on the plane strain problem. The coordinate system is set up with the x-axis along the interface, and the y-axis vertical.
Figure 1. Schematic illustration of stress transfer in a multi-layered material system
6
Because the load P is along the x-direction, the displacement in the y-direction should be much smaller than that in the x-direction. Following the assumption made in the shear-lag theory, we assumed that all points in one layer normal to the -direction remain in the same plane after deformation or simply that
(Nairn and Mendels, 2001, Yin, et al., 2007a).
Therefore, we can write: (1) This is the so-called plane assumption, which will be used to simplify the shearing strain and stress. The constitutive law is written as
(2)
where the subscript
denotes the material layers. Substituting the above equations
into the equilibrium equations in the absence of body force, i.e.
(3) we obtain the governing equations as follows:
(4)
(5)
7
Notice that Eq. (4) is a decoupled partial differential equation of solve for
through Eq. (5). The solution of
. Once
is obtained, we can
can be used to evaluate the accuracy and
applicability of the assumption in Eq. (1). In the
-direction, the general solution for Eq. (4) can be obtained by the method of
separation of variables by using the following simplification
(6) where the unknown constant
is introduced as a uniform strain caused by the force P. Plugging
this general solution into the governing equation (i.e. (4)) produces
(7) Two independent equations can be set equal to an independent constant
(8) Two ordinary differential equations can therefore be extracted from Eq. (8) such that:
(9) The general solutions for the ordinary differential equations in (9) are written as
(10) where the constant
is given as
(11)
8
Therefore, the general solution for
in Eq. (6) can be rewritten as
(12) Now consider the governing equation in the -direction shown in Eq. (5). By using the general solution for the -direction shown in Eq. (12), the displacement in the -direction can be shown to be
(13) where
is a function to be determined by the boundary conditions of each layer.
The parameters
,
,
, and
and the unknown function
in the general solutions of
Eqs. (12) and (13) are different for each layer. For each layer, a series-form solution may be needed to satisfy the boundary conditions exactly. In that case, a set of parameters for the asymptotic solution may be used to solve for each layer. In this paper however, we will demonstrate that a closed-form solution with one function has already reached an acceptable accuracy for a symmetric three-layer system. For boundary conditions in terms of stresses, both
and
may be used through Eq. (2),
and thus the constants in Eqs. (12) and (13) should be solved by the boundary conditions and the interfacial continuities with combined terms of
and
. To demonstrate this theory, the
remainder of this paper will investigate the stress transfer of a coated bar subjected to a tension. Stress and fracture analyses will emphasize this methodology. Some assumptions will be used to simplify the formulation and a closed-form approximate solution will be obtained. 3. Stress analysis of a coating/substrate system
9
As a special case of multi-layered materials, Figure 2 shows two coating layers (thickness length
, Young’s modulus
layer (thickness
, Poisson’s ratio
, Young’s modulus
,
) fully bonded symmetrically to the substrate
, Poisson’s ratio
) subjected to a load
. The
coordinate system is set up with the origin O at the center of layer 1. The x-axis is along the loading direction, and the y-axis is perpendicular to the loading direction. Based on the symmetry, a quarter of the system can be used to represent the entire coating system. Using the theory in Section 2, we will solve the above boundary-value problem and derive the elastic field as follows.
Figure 2. Stress transfer in a coating system when the substrate is subjected a tensile stress
3.1. Simplified displacement functions For the substrate Layer 1 in Figure 2, the general solution for the displacement in the direction is rewritten as
(14) where, as shown in Eq. (11),
10
(15)
The symmetric conditions along the x and y-axes produce the following boundary conditions:
(16) Applying the above boundary conditions to Eq. (14) reduces the displacement to
(17) where the leading coefficient
, is a constant to be determined from the remaining boundary
conditions, and the superscript 1 represents the displacement for Layer 1. This formatting will be followed for the remainder of the paper. Once
is derived,
can be obtained through the integral of Eq. (13) as
(18)
where
is considered, and
is to be determined by the interfacial continuities in
the next Section. Similarly to the derivation in Eqs. (14) to (17), we can simplify the x-directional component of the displacement in the coating as
(19) where the constant
satisfies
11
(20)
Then
in the coating can also be obtained through the integral of Eq. (13) as
(21) At the upper surface,
, so that using Eq. (2) we can write
(22) Therefore, Eq. (21) can be rewritten as
(23)
Here
is to be determined by the interfacial continuities in the next Section. 3.2. The closed-form elastic solution
In summary, we have solved the displacements of
,
,
, and
in Eqs. (17), (18), (19)
and (23), from which we can solve for all stresses using Eq. (2). However, there are still four constants, namely,
,
,
, and
, and two functions, namely
and
, to be
determined by the interfacial continuities and the loading condition. Since a fully bonded interface is considered, four interfacial continuity conditions are required when
:
12
(24)
We will solve for the four parameters and two unknown functions in what follows. The first two equations in (24) can be rewritten as:
(25)
(26) Notice that the approximation of the shearing stress in Eq. (2) is used. Division of both sides of Eq. (26) by Eq. (25), respectively, yields
(27) Using the definitions of (15) and (20), we can calculate the constant c, and then
and
from
Eq. (27). The constant c changes with the elastic constants and thickness, while the fracture spacing has no effects on c. There are many solutions for constant c because of the periodicity of the
function, which suggests the need for a series-form solution. In general, the roots of c
in Eq. (27) are not periodic, which also implies that the basis functions in the series-form solution will not be orthonormal to each other. The derivation of the coefficient of each basis function will be complicated and the convergence of the solution will still be open. An example using the series form solution can be found in another paper (Liu and Yin, 2013), which considers the stress transfer in a compliant film bonded to a rigid substrate. For simplicity, the present paper uses the first root to demonstrate this theory. To solve for c, we define a parameter
13
(28)
Notice that the Poisson’s ratios of the coating and the substrate are assumed to be less than 0.5. Otherwise, the effective stiffness is Eq. (2) will be infinite. If incompressible materials are used in the coating, the plane strain assumption in this theory may be invalid and thus the formulation would need to be reconstructed. Based on the mechanics of the coating system, the first root of c can be obtained within a range of
, which makes the shearing stress along the
interface negative. More specifically, c can be obtained through the following three conditions. •
When
, c is the smallest value that satisfies
. Typically, a thick
coating on a thin substrate will fall within this category. When to approximately solve for
•
When
,
, we can use
.
is used to solve for
.A
representative case is when the coating and the substrate have the same elastic constants and the coating is half the thickness of the substrate. •
When
, we impose that c is the smallest solution meeting the constraint where . Typically, a thin coating deposited on a thick substrate will fall in this
category. When
, we can assume
, and c can be obtained as
.
14
1.8 E1/E2 = 100 1.6
E1/E2 = 1 E1/E2 = 1/100
1.4
Value for "c"
1.2 1 0.8 0.6 0.4 0.2 0 -2 10
-1
10
0
10
1
2
10
10
ξ
Figure 3. The calculation result of c based on varying values of with
.
Figure 3 illustrates the relation between the first root of c and . With the increase of 0.01
to 100
from
, the value of c is illustrated in the log coordinate of for the three cases of . With the increase of , the value of c decreases from a certain value to
zero. Notice that c has dimension of “length-1”, so that its value depends on the length scale of and
.
In addition, Eq. (25) provides the relation between
and
as
(29) Now we can rewrite the last two equations in (24) to determine
and
as follows:
(30)
15
(31)
From Eq. (31), we can obtain
(32)
Then, from Eq. (30), we can obtain
(33)
Therefore, if the constants
and
can be determined, the closed form solution of the
displacement field in the coating system will be obtained, and thus the strain and stress fields can be derived accordingly. We will use the principle of stationary potential energy to determine these two constants. Because the stresses within the coating system are caused by the load
applied in the
substrate and no external load is applied in the y-direction, we assume that the stress
.
Again, this assumption may not be accurate near the edge but is reasonable in the zone away from the edge. Using this assumption, the stress
can be approximated from Eq. (2) as,
(34)
16
Notice that
can be taken as an average stress of the resultant force distributed in the cross
section of the substrate. The Saint-Venant principle will guarantee the accuracy for the zone away from the edge. This approximation provides flexibility for application in periodic openingmode fracture analysis in Section 4, where the stress distribution in the Layer 1 is difficult to be obtained, but the resultant force is known. The strain energy density of a material point in the coating system is from two stress components
and
. Using the third equation in Eq. (2) and Eq. (34), we can obtain the
strain energy in the coating system written as
(35)
Substituting Eqs. (17) and (19) into the above equation, we can obtain
(36) where
(37)
(38)
17
and (39) The work done by the external tensile load is written as
(40)
Using the principle of the stationary potential energy
, we can write
(41) or (42) and (43) Solving the above two equations, we can obtain
(44) Therefore, a closed form solution for a coating system is obtained with the combination of displacements in Eqs. (17), (18), (19) and (23) along with the parameters solved by Eqs. (44), (27), (29), (32), and (33). 3.3. Verification with finite element results In order to observe the effects of this solution’s simplifications used to derive the elastic fields, a finite element (FE) model was used for comparison. The basic geometry used for this
18
model was
, where
was chosen to be 1. A load of 1000 N/m is applied as a
uniformly distributed pressure along the substrate. Moreover, both materials were chosen to have the same Poisson ratio of
. Initially, the effect of the material stiffness mismatch
was observed, where
, and the ratio of
was varied by: 10, 3, 1, 1/3, and
1/10. Therefore, any limitations that this solution would have on modelling different materials should be observed. The FE model was made using commercially available ABAQUS. For each model, a uniform mesh of approximately 14,000 nodes is assembled, using quadrilateral plane strain elements using quadratic formulation. The following figures illustrate the results. -8
2.5
x 10
E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM
Displacement ux (m)
2
1.5
10 3 1 1/3 1/10
1
0.5
0
0
0.5
1
1.5
2
2.5 x(m)
3
3.5
4
4.5
5
Figure 4. Comparison between the displacement in the x-direction for the analytical solution for ( ) and the FE model when along -8
2.5
x 10
E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM
Displacement ux (m)
2
1.5
10 3 1 1/3 1/10
1
0.5
0
0
0.5
1
1.5
2
2.5 x(m)
3
3.5
4
4.5
5
19
Figure 5. Comparison between the displacement in the x-direction for the analytical solution and the FE model when along for ( ) -8
2.5
x 10
E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM
Displacement ux (m)
2
1.5
10 3 1 1/3 1/10
1
0.5
0
0
0.5
1
1.5
2
2.5 x(m)
3
3.5
4
4.5
5
Figure 6. Comparison between the displacement in the x-direction for the analytical solution and the FE model when along for ( ) Figures 4-6 show the FE and analytical solution comparisons for the distribution of displacement
along the x-direction at different vertical locations. It is observed that despite
varying the stiffness mismatch (by varying the stiffness of the coating) from having a more compliant coating to having a far stiffer coating, the FE and the analytical results agree very well. When the coating is compliant, the coating produces very minor effects on the deformation of the substrate. Therefore, in Figure 4,
is almost linear for
and 3. Along the
surface, the displacement in Figure 6 is almost linear in the central part, but the slopes decrease in the region close to the fracture end. Figures 7 and 8 show how the shearing stress and the x-displacement, when
,
change along the interface for varying fracture separations and coating thicknesses, respectively. The shearing stress increases from zero at the center to the maximum at the fractured end. For a higher fracture spacing, the shearing stress remains near zero for a large range and increases significantly in the region close to the fractured end, which is consistent with the trend of the
20
singularity effect due to the material difference. These results show that the analytical solution is quite capable of capturing the physical effects of the loading prescribed. 50
Shear Stress τxy (Pa)
0
-50
λ/h1 = 10 FEM
λ/h1 = 5
-100
FEM
λ/h1 = 2
-150
FEM
λ/h1 = 1 FEM
-200
λ/h1 = 1/2 FEM -250
0
1
2
3
4
5 x(m)
6
7
8
9
10
Figure 7. Comparison between the shearing stress for the analytical solution and the FE model when along
at the interface
for varying values of
(
-8
2.5
x 10
h2/h1 = 1 FEM h2/h1 = 0.3
Displacement ux (m)
2
FEM h2/h1 = 0.1 FEM h2/h1 = 0.03
1.5
FEM 1
0.5
0
0
0.1
0.2
0.3
0.4
0.5 x/ λ
0.6
0.7
0.8
0.9
1
Figure 8. Comparison between FE and analytical results for the x-displacement along the interface for varying coating thicknesses when
, and
,
Notice that in Figures 4-8, the difference between the analytical solution and the FEM results in the region close to the fractured end is more considerable than that in the other regions. This is due to the plane assumption used, and the application of a weak form force boundary condition 21
at the fractured ends. If a series-form solution were to be used, the boundary condition may be exactly satisfied. This however, does not guarantee a better solution because of the assumption of Eq. (1). A rigorous investigation in mathematics and mechanics of this problem will be conducted in the future. 4. Results and discussion The proposed theory is based on the simplified governing equation and thus does not provide an exact solution. The assumptions and approximations adopted may impose some limitations on the application of the formulation. In summary, the following aspects are highlighted for the strengths and weaknesses of this theory: •
Because all the points in the same plane perpendicular to the y-axis are assumed to keep in the same plane during deformation (as Eq. (1)), the decoupled governing equation of
is
obtained. This assumption imposes a limitation that this model does not take into account the curvature of the layered materials, which can be well addressed through Stoney’s theory (Freund and Suresh, 2003, Stoney, 1909). Combination of these two theories considering both bending moment and axial load may provide some new perspectives of the stress transfer in layered materials, which will be investigated in future work. •
To obtain an explicit analytical solution for the governing equation, the boundary condition along the end surface of the coating system is not exactly satisfied. Using the principle of stationary potential energy provides a way to obtain a closed-form, simple solution without the loss of generality. However, it makes the stress field around the loading edge inaccurate; whereas the accuracy in the zone away from the edge can be guaranteed by the Saint Venant’s Principle.
22
•
The parameters c,
and
are determined by the interfacial continuities. However,
currently only the first root of Eq. (27) is considered. If a series-form solution is used with all roots, boundary conditions at the loading end can be more accurately modeled. However, the exactness cannot be guaranteed due to the plane assumption used. •
If multiple layers are considered, we will need to use a series form solution to make the interfacial continuity satisfied. Due to the above-mentioned assumptions, the singularity effect at the fracture tip cannot be
exactly modeled, and the stress and strain distributions along the fractured surface have not been accurately obtained. Therefore, when the fracture distribution is very dense, such as
, this
solution may not be accurate due to the end effect. Because the proposed model provides a closed-form solution of elastic field, it can be used in structural design and failure analysis of layered materials. We will use this theory to calculate the elastic modulus reduction and to investigate the interfacial compliance, while comparing the results with existing models.
4.1. Elastic modulus reduction due to opening-mode fractures When a layered material is subjected to an increasing tensile load, the elastic modulus may decrease when opening-mode micro-cracks are induced with a uniform pattern. For example, laminates with microcracks have been investigated by the shear lag models (Nairn and Mendels, 2001). The proposed theory can be used to predict the elastic modulus reduction due to the OMFs. Consider a coating system in Fig. 2. Using the loading
and the elongation
in
Eq. (17) for a uniaxial loading condition, we can obtain the average stress on the overall cross
23
section as
and the average strain of the section as
Young’s modulus, namely
, from which the effective
, is written as
(45)
For a long layered material (
, the effective Young’s modulus, namely
, can be written
as
(46)
So that the modulus reduction ratio due to OMFs can be written as
(47) where
and
are given in Eq. (44).
The shear lag models have been widely used to solve this problem by introducing some predefined shape functions of the shear stress in the thickness direction, and the modulus reduction ratio is written in the form of (Nairn and Mendels, 2001)
(48) where
is typically determined by the shape function of shear stress distribution or calibrated by
an FEM analysis. Nairn and Mendels (2001) showed that a linear shape function has provided
24
good agreement with FEM results where the difference of shearing moduli between two layers is not very large. In such a case,
can be written by
(49)
Notice that Eq. (48) was based on the plane stress problem; whereas Eq. (47) was based on the plane strain problem. To compare both equations, Eqs. (48) and (49) should be based on the Young’s moduli for the plane strain problem as
. Because the proposed theory does not
impose any assumption to the shear stress distribution, it should be applicable to more general situations. Fig. 9 shows the modulus reduction for the coating system with elastic constants as follows thickness of
and
;
and
. The substrate has a
and the coating has a thickness of
,
respectively, and the specimen has a unit width. For each case of the coating thickness, the results of Eqs. (47) and (48) have been compared with the FEM results.
1.45 Analytical h2 = 0.25 μm 1.4
Nairn and Mendels h2 = 0.25 μm FE h2 = 0.25 μm
1.35
Analytical h2 = 1 μm E0/E* Ratio
1.3
Nairn and Mendels h2 = 1 μm FE h2 = 1 μm
1.25
Analytical h2 = 4 μm 1.2
Nairn and Mendels h2 = 4 μm FE h2 = 4 μm
1.15 1.1 1.05 1
0
50
100
150
200
250
λ (μm)
25
Fig. 9 – Modulus reduction curves for the present analytical solution, the FE model, and the Nain and Mendels’ formulation (2001) with a log scale on the y axis 4.2. Interfacial compliance of bi-layered materials and parameter c One popular method to study the stress transfer in bi-layered materials is to assume a frictional interface, through which the shear stress is proportional to the displacement. A spring constant or an interfacial compliance, which are reciprocal to each other, has been introduced to describe the constitutive relation of the interface. For example, we can write
(50) where
is the interfacial compliance.
Since the shearing stress and displacement are continuous across the interface (without loss of generality from the results in Eq. (19) and the third equation in Eq. (2)) we can obtain
(51)
The interfacial compliance is generally not constant along the interface. The interfacial compliance however, can be written in the polynomial form as
(52)
26
When is small, such as a thin film deposited on a thick substrate with dense OMFs, we can approximately write terms of
and
. Disregarding higher order
, we can obtain
(53)
Notice that the definition of interfacial compliance in Eq. (50) depends on the load configuration. If the load is applied at the coating layer and transferred to the substrate, another type of the interfacial compliance can be derived with the same solution except that the material constants and geometry parameters will be switched. When this solution is superposed with a uniform tensile or compressive deformation, the shearing stress keeps the same, but the displacement will independently change with the uniform deformation. This makes the interfacial compliance difficult to be defined for general cases. Our previous model based on the frictional interface (Yin, et al., 2008) disregarded the deformation in the substrate and provided the equivalent interfacial compliance equal to the second part of Eq. (53). If we apply a uniform deformation with equivalent strain
in Eq. (19), we can obtain the same form of the interfacial compliance
as the previous one. In this case, the interfacial compliance can be correlated to parameter c. In our previous work (Yin, et al., 2008), c was defined in a slight different way that it is dimensionless. If we transform it to the same physical mean of c in this paper, written as , its form should be
(54)
27
where
with
(Beuth, 1992, Yin, et al.,
2008). However, in this paper, c is obtained by Eq. (27), which depends on the elastic moduli and thicknesses of both the film and the substrate. Fig. 10 illustrates the comparison of the parameter c obtained by Eqs. (27) and (54). Here using changing with
for three cases
,
, we draw c
In our previous work, the effect of the
substrate thickness has not been quantitatively considered and c changes in a large range; whereas the current formulation provides a small range of c for different ratios of
. Notice
that when the difference of the thicknesses between two layers is large, the accuracy of the current formulation using one term may decrease.
1.4
Calculated value of c
Proposed c for h1 = 1 1.2
Proposed c for h1 = 10 Proposed c for h1 = 100
1
c from (Yin, et al., 2008)
0.8
0.6
0.4
0.2
0 -2 10
-1
10
0
10 Ratio of E2/E1
1
10
2
10
Fig. 10 – Value of c calculated from Eq. (27) versus the value of calculated from Eq. (54), as the ratio of increases 5. Conclusions A general elastic solution for multi-layered materials is derived to predict stress transfer when one layer is subjected to a mismatched load from others. The closed-form solution for a coating system is derived to demonstrate the use of the formulation. A comparison between the
28
proposed formulation and finite element results shows that this theory exhibits a high accuracy for general dual layered systems when the thicknesses of the layers are not significantly different. This theory provides a very reasonable prediction of the elastic modulus reduction of a coating system caused by dense fractures. It also facilitates the understanding of the interfacial compliance of multi-layered material systems. The theory can be applied to general multilayered materials with other geometries such as circular or cylindrical layered systems. An experimental validation of the theory is underway. Acknowledgments This work is sponsored by the National Science Foundation CMMI 0954717 and the Department of Homeland Security CU09-1155, whose support is gratefully acknowledged. References 1. 2. 3. 4. 5. 6. 7. 8.
9.
10.
Bai, T., Maerten, L., Gross, M. R., and Aydin, A., 2002. Orthogonal cross joints: do they imply a regional stress rotation? Journal of Structural Geology 24, 77-88. Bai, T., Pollard, D. D., and Gao, H., 2000a. Explanation for fracture spacing in layered materials. Nature 403, 753-756. Bai, T. X., and Pollard, D. D., 1999. Spacing of fractures in a multilayer at fracture saturation. International Journal of Fracture 100, 23-28. Bai, T. X., Pollard, D. D., and Gross, M. R., 2000b. Mechanical prediction of fracture aperture in layered rocks. Journal of Geophysical Research-Solid Earth 105, 707-721. Beuth, J. L., 1992. Cracking of Thin Bonded Films in Residual Tension. International Journal of Solids and Structures 29, 1657-1675. Beuth, J. L., and Klingbeil, N. W., 1996. Cracking of thin films bonded to elastic-plastic substrates. Journal of the Mechanics and Physics of Solids 44, 1411-1428. Bogy, D. B., 1971. Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. Journal of Applied Mechanics 38, 377-386. Colak, A., 2001. Parametric study of factors affecting the pull-out strength of steel rods bonded into precast concrete panels. International Journal of Adhesion and Adhesives 21, 487-493. Fellah, M., Tounsi, A., Amara, K. H., and Bedia, E. A. A., 2007. Effect of transverse cracks on the effective thermal expansion coefficient of aged angle-ply composites laminates. Theoretical and Applied Fracture Mechanics 48, 32-40. Freund, L. B., and Suresh, S., 2003. Thin film materials: stress, defect formation, and surface evolution. Cambridge University Press, Cambridge. 29
11. 12. 13. 14. 15. 16.
17. 18. 19.
20. 21.
22. 23. 24.
25. 26. 27. 28.
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Hsueh, C. H., 2001. Analyses of multiple film cracking in film/substrate systems. Journal of the American Ceramic Society 84, 2955-2961. Hwu, K. L., and Derby, B., 1999. Fracture of metal/ceramic laminates - I. Transition from single to multiple cracking. Acta Materialia 47, 529-543. Jain, A., Guzina, B. B., and Voller, V. R., 2007. Effects of overburden on joint spacing in layered rocks. Journal of Structural Geology 29, 288-297. Laws, N., and Dvorak, G. J., 1988. Progressive Transverse Cracking in Composite Laminates. Journal of Composite Materials 22, 900-916. Li, Y., and Yang, C., 2007. On fracture saturation in layered rocks. International Journal of Rock Mechanics and Mining Sciences 44, 936-941. Li, Y. Q., Shen, Z. Y., Wang, L., Wang, Y. M., and Xu, H. W., 2007. Analysis and design reliability of axially compressed members with high-strength cold-formed thinwalled steel. Thin-Walled Structures 45, 473-492. Liu, Y. J., and Yin, H. M., 2013. Elastic thermal stress in a hollow circular overlay/substrate system. (submitted). Nairn, J. A., and Mendels, D. A., 2001. On the use of planar shear-lag methods for stresstransfer analysis of multilayered composites. Mechanics of Materials 33, 335-362. Prieto-Muñoz, P., Yin, H. M., and Buttlar, W. G., 2013. Two Dimensional Stress Analysis of Low Temperature Cracking in Asphalt Overlay/Substrate Systems. Journal of Materials in Civil Engineering (in press). Shenoy, V. B., Schwartzman, A. F., and Freund, L. B., 2000. Crack patterns in brittle thin films. International Journal of Fracture 103, 1-17. Stoney, G. G., 1909. The tension of metallic films deposited by electrolysis. Proceedings of the Royal Society of London Series a-Containing Papers of a Mathematical and Physical Character 82, 172-175. Suhir, E., 1989. Interfacial stresses in bimetal thermostats. Journal of Applied MechanicsTransactions of the Asme 56, 595-600. Suhir, E., 1991. Structural Analysis in Microelectronic and Fiber-Optic Systems. van Nostrand Reinhold, New York. Tang, C. A., Liang, Z. Z., Zhang, Y. B., Chang, X., Tao, X., Wang, D. G., Zhang, J. X., Liu, J. S., Zhu, W. C., and Elsworth, D., 2008. Fracture spacing in layered materials: A new explanation based on two-dimensional failure process modeling. American Journal of Science 308, 49-72. Timm, D. H., Guzina, B. B., and Voller, V. R., 2003. Prediction of thermal crack spacing. International Journal of Solids and Structures 40, 125-142. Waldhoff, A. S., Buttlar, W. G., and Kim, J., 2000. Investigation of thermal cracking at Mn/ROAD using the Superpave IDT. Proc. Canadian Technical Asphalt Assoc., 228-259. Xia, Z. C., and Hutchinson, J. W., 2000. Crack patterns in thin films. Journal of the Mechanics and Physics of Solids 48, 1107-1131. Yang, D. J., and Yin, H. M., 2011. Energy Conversion Efficiency of a Novel Hybrid Solar System for Photovoltaic, Thermoelectric, and Heat Utilization. Ieee Transactions on Energy Conversion 26, 662-670. Yang, D. J., Yuan, Z. F., Lee, P. N., and Yin, H. M., 2012. Simulation and experimental validation of heat transfer in a novel hybrid solar panel. International Journal of Heat and Mass Transfer 55, 1076-1082.
30
30. 31.
32. 33. 34.
35.
36.
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Yin, H., Paulino, G., and Buttlar, W., 2008. An explicit elastic solution for a brittle film with periodic cracks. International Journal of Fracture 153, 39-52. Yin, H. M., 2010a. Comments on "Influence of Interfacial Compliance on Thermomechanical Stresses in Multilayered Microelectronic Packaging". Ieee Transactions on Advanced Packaging 33, 353-355. Yin, H. M., 2010b. Fracture saturation and critical thickness in layered materials. International Journal of Solids and Structures 47, 1007-1015. Yin, H. M., 2010c. Opening-Mode Cracking in Asphalt Pavements Crack Initiation and Saturation. Road Materials and Pavement Design 11, 435-457. Yin, H. M., Buttlar, W. G., and Paulino, G. H., 2007a. Simplified solution for periodic thermal discontinuities in asphalt overlays bonded to rigid pavements. Journal of Transportation Engineering-Asce 133, 39-46. Yin, H. M., Paulino, G. H., Buttlar, W. G., and Sun, L. Z., 2007b. Micromechanics-based thermoelastic model for functionally graded particulate materials with particle interactions. Journal of the Mechanics and Physics of Solids 55, 132-160. Yin, H. M., Yang, D. J., Kelly, G., and Garant, J., 2013. Design and performance of a novel building integrated PV/thermal system for energy efficiency of buildings. Solar Energy 87, 184-195. Zhang, Z., and Suo, Z. G., 2007. Split singularities and the competition between crack penetration and debond at a bimaterial interface. International Journal of Solids and Structures 44, 4559-4573.
31
Highlights
>Decoupled governing equations are obtained by the plane assumption >General solution to the governing equations is derived for the BVP >The closed-form solution for a coating system is provided using stationary potential energy >The present approximate solution well agrees with the finite element results >The formulation interprets the elastic modulus reduction and interfacial compliance
32
S0167-6636(13)00048-3 http://dx.doi.org/10.1016/j.mechmat.2013.03.007 MECMAT 2104
To appear in:
Mechanics of Materials
Received Date: Revised Date:
21 March 2012 15 March 2013
Please cite this article as: Yin, H.M., Prieto-Muñoz, P.A., Stress transfer through fully bonded interface of layered materials, Mechanics of Materials (2013), doi: http://dx.doi.org/10.1016/j.mechmat.2013.03.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Stress transfer through fully bonded interface of layered materials Huiming M. Yin*1, Pablo A. Prieto-Muñoz1 1
Department of Civil Engineering and Engineering Mechanics, Columbia University 610 Seeley W. Mudd 500 West 120th Street, New York, NY 10027
Plane strain elastic theory of stress transfer in multi-layered materials is formulated and is used to investigate the stress distribution of a coating system under a tensile load. When the substrate layer is subjected to a uniaxial load, the load is transferred to the coating through interfacial shearing stress. With the aid of a plane assumption, decoupled governing equations are obtained, and the general solution of the displacement field can be derived for both the coating and the substrate layers. Using the boundary conditions and the interfacial continuities, we obtain a closed-form solution for the elastic fields in both the overlay and the substrate layers, which takes the first-term of a series-form solution. Although the singularity effect of stress at the ends of the interface and loading points cannot be exactly illustrated due to the simplification and assumptions, the proposed formulation provides excellent agreement with the finite element results of the transferred stress in the thickness direction of the coating system. Comparisons with the existing models demonstrate the capability and limitation of the proposed formulation. This theory can serve as a baseline for future fracture analysis, inelastic analysis, and thermomechanical analysis of multi-layered materials. Keywords: stress transfer, fully bonded interface, multi-layered materials, elastic analysis, boundary-value problem, interfacial compliance
*
Corresponding author: Tel.: +1-212-851-1648 e-mail: [email protected]
1
1. Introduction Multi-layered materials and structures are commonly observed in nature and can be used in advanced material design, such as sedimentary rocks (Bai, et al., 2000a, Li and Yang, 2007, Tang, et al., 2008), pavements (Prieto-Muñoz, et al., 2013, Timm, et al., 2003, Yin, 2010c, Yin, et al., 2007a), surface protective coatings (Colak, 2001, Yin, et al., 2007b), thin film/substrate systems (Beuth, 1992, Freund and Suresh, 2003, Shenoy, et al., 2000, Xia and Hutchinson, 2000, Yin, et al., 2008), and multi-ply laminates and structures (Fellah, et al., 2007, Hwu and Derby, 1999, Yang and Yin, 2011, Yin, et al., 2013). When a multi-layered system is subjected to a mechanical load or an environmental stress (Yang, et al., 2012, Yin, 2010c, Yin, et al., 2013), the deformation of the material in each layer tends to be different due to a mismatch in material properties. Interfacial shearing stress will be induced to make the deformation compatible for structural integrity. Stress transfer through the interface has been a challenging problem over the past century. There are three widely accepted methods to solve the problem. First, Stoney (1909) studied the deformation of thin film/substrate systems with a misfit strain. Similar to bending theory, the stress can be calculated from the curvature of the system. This theory has been extended to multi-layered system with the thicknesses much smaller than their length, and has been used to study thermal stress due to temperature change (Freund and Suresh, 2003, Hsueh, 2001, Li, et al., 2007). However, if one layer is much thicker or stiffer, or the layer configuration is symmetric in the thickness direction, the curvature of the multi-layered system can be very small or equal to zero, and thus this theory will not be applicable. In addition, if the deformation is driven by a stress in the plane, the curvature change will not be comparable to the overall deformation. The transfer of stress cannot be sufficiently explained by this theory.
2
The second method uses the shear lag theory, which can be traced back to Cox’s pioneering work on stress transfer between a fiber and a matrix under a tensile load (Waldhoff, et al., 2000). This theory has been widely used in the stress and strength analysis of fiber reinforced concrete. It assumes that the strong material will carry the tensile load which is transferred to the matrix through interfacial shearing stress (Beuth and Klingbeil, 1996). Various assumptions have been made to simplify the shearing stress distribution along the interface so that the stress distribution can be solved numerically or analytically. This theory has been successfully extended to analyze multi-layered material systems. In general, a displacement discontinuity is assumed to calculate the stress transfer through the interface (Hsueh, 2001, Laws and Dvorak, 1988, Nairn and Mendels, 2001). Those specific assumptions however, are only applicable to specific materials or interfacial conditions. The accuracy of the interfacial shearing stress is therefore questionable. The third method for studying stress transfer follows Hobbs’ work on layered rocks (1967). When a two-layer system is subjected to a tensile load, opening-mode fractures form in the weak layer. To obtain the stress distribution in the fractured weak layer, it is considered as a onedimensional (1D) bar. The normal stress is balanced by the distributed shearing stress along the interface. This theory has also been used in the stress analysis of pavements (Timm, et al., 2003) and thin films (Xia and Hutchinson, 2000). Although linear distribution of shearing stress was assumed along the thickness of the weak layer (Jain, et al., 2007), the 1D solution cannot accurately describe the stress distribution. Moreover, to establish the governing equation, a frictional interface was generally assumed between the two layers using a spring coefficient, which describes the relation between the interfacial shearing stress and the displacement at each interface point. The concept of an interfacial coefficient or compliance is not well established or understood yet, which causes contradictory formulations in the literature (Jain, et al., 2007,
3
Suhir, 1989, Suhir, 1991, Timm, et al., 2003, Xia and Hutchinson, 2000, Yin, et al., 2008, Yin, 2010a). In addition, because the stress variation in the thickness direction plays an important role in the fracture analysis of the weak layer, improved accuracy of stress transfer theory is highly needed. Recent research on stress transfer through multi-layered systems mainly focused on numerical and experimental methods with a fully bonded interface (Bai, et al., 2002, Li and Yang, 2007, Tang, et al., 2008). Bai and his colleagues conducted a comprehensive experimental and numerical investigation (Bai, et al., 2002, Bai, et al., 2000a, Bai and Pollard, 1999, Bai, et al., 2000b) and found that indeed the existing stress-transfer theory fails to satisfy the equations of equilibrium for an elastic boundary-value problem. It was shown in their numerical investigation that when the fracture spacing is small, the stress in the central region of the weak layer is compressive rather than tensile, thus leading to fracture saturation in layered materials. Yin and his colleagues have proposed a two-dimensional (2D) elastic solution of a film bonded to a strong substrate subjected to a tensile load (Yin, et al., 2008, Yin, et al., 2007a), which was successfully applied to interpret fracture initiation, infilling, and saturation caused by tension or temperature loads in the layered materials (Nairn and Mendels, 2001, Yin, 2010b). It provides a very accurate solution for the stress distribution in a compliant thin film/stiff substrate system. However, the interfacial condition still followed the disputable assumption of a frictional interface (Xia and Hutchinson, 2000). Although the equivalence of a frictional interface with the exact solution of a fully bonded interface (Beuth, 1992) was explored, the accuracy of the stress solution for the two-layer system with a compliant substrate is inadequate since the frictional interface cannot sufficiently explain the mechanics of stress transfer through a fully bonded
4
interface. Moreover, the elastic field in the substrate cannot be derived in the previous formulations using the friction interface. The purpose of this work is to develop a theory of stress transfer in layered materials with a fully bonded interface in order to solve the elastic boundary-value problem considering in-plane misfit strain and tensile load. Because the mechanical loading and mismatch strain are within the layer plane, it is assumed that all points of a layer plane remain in the same plane during deformation. This assumption has been widely used in the shear-lag theory as well as our previous work (Nairn and Mendels, 2001, Yin, et al., 2007a). The elastic governing equations are then decoupled and the general solution of the in-plane displacement is obtained. Using the boundary conditions and the interfacial continuities, a closed-form solution can be derived. Due to the plane assumption used in the derivation, the singularity at the edge of the interface (Bogy, 1971, Zhang and Suo, 2007) cannot be exactly modelled in the formulation. Although the local field in the neighbourhood of the edge of the interface cannot be precisely illustrated, the proposed formulation is still accurate for the overall stress distribution in the zone that is away from the singularity points. The remainder of this paper is organized as follows: Section 2 formulates the plane strain elastic theory of stress transfer in a multi-layered system and presents the general solution to the governing equations. Section 3 solves the boundary-value problem of a coating system with the substrate subjected to a tensile load. The elastic fields in both the coating and the substrate are obtained and are verified with the finite element method, which shows the accuracy of the proposed solution for analysing stress transfer. Parametric analyses are conducted to study the stress distribution in both layers for different material properties and geometric configurations. Section 4 discusses the limitations and advantages of the proposed theory compared with other
5
stress transfer theories in the literature, such as frictional interface models and shear lag models. Using the proposed theory, the elastic modulus reduction and interfacial compliance are calculated and are compared with those in the literature. Lastly, some conclusive remarks are given in Section 5.
2. Simplified governing equations and the general solutions Consider a multi-layered material system containing n layers of length thickness
, Young’s modulus
, and Poisson’s ratio
with
. Each layer has a . They are fully
bonded to their neighboring layers, as illustrated in Figure 1. When one layer, say layer 1, is subjected to a force P, the force will be transferred to the other layers through the fully bonded interfaces. Although the ends of the other layers are free, the normal stress will increase from the ends and reach a maximum at the center, which may produce an open-mode fracture in the middle. Because the width of the system is much larger than the thickness, our formulation will be based on the plane strain problem. The coordinate system is set up with the x-axis along the interface, and the y-axis vertical.
Figure 1. Schematic illustration of stress transfer in a multi-layered material system
6
Because the load P is along the x-direction, the displacement in the y-direction should be much smaller than that in the x-direction. Following the assumption made in the shear-lag theory, we assumed that all points in one layer normal to the -direction remain in the same plane after deformation or simply that
(Nairn and Mendels, 2001, Yin, et al., 2007a).
Therefore, we can write: (1) This is the so-called plane assumption, which will be used to simplify the shearing strain and stress. The constitutive law is written as
(2)
where the subscript
denotes the material layers. Substituting the above equations
into the equilibrium equations in the absence of body force, i.e.
(3) we obtain the governing equations as follows:
(4)
(5)
7
Notice that Eq. (4) is a decoupled partial differential equation of solve for
through Eq. (5). The solution of
. Once
is obtained, we can
can be used to evaluate the accuracy and
applicability of the assumption in Eq. (1). In the
-direction, the general solution for Eq. (4) can be obtained by the method of
separation of variables by using the following simplification
(6) where the unknown constant
is introduced as a uniform strain caused by the force P. Plugging
this general solution into the governing equation (i.e. (4)) produces
(7) Two independent equations can be set equal to an independent constant
(8) Two ordinary differential equations can therefore be extracted from Eq. (8) such that:
(9) The general solutions for the ordinary differential equations in (9) are written as
(10) where the constant
is given as
(11)
8
Therefore, the general solution for
in Eq. (6) can be rewritten as
(12) Now consider the governing equation in the -direction shown in Eq. (5). By using the general solution for the -direction shown in Eq. (12), the displacement in the -direction can be shown to be
(13) where
is a function to be determined by the boundary conditions of each layer.
The parameters
,
,
, and
and the unknown function
in the general solutions of
Eqs. (12) and (13) are different for each layer. For each layer, a series-form solution may be needed to satisfy the boundary conditions exactly. In that case, a set of parameters for the asymptotic solution may be used to solve for each layer. In this paper however, we will demonstrate that a closed-form solution with one function has already reached an acceptable accuracy for a symmetric three-layer system. For boundary conditions in terms of stresses, both
and
may be used through Eq. (2),
and thus the constants in Eqs. (12) and (13) should be solved by the boundary conditions and the interfacial continuities with combined terms of
and
. To demonstrate this theory, the
remainder of this paper will investigate the stress transfer of a coated bar subjected to a tension. Stress and fracture analyses will emphasize this methodology. Some assumptions will be used to simplify the formulation and a closed-form approximate solution will be obtained. 3. Stress analysis of a coating/substrate system
9
As a special case of multi-layered materials, Figure 2 shows two coating layers (thickness length
, Young’s modulus
layer (thickness
, Poisson’s ratio
, Young’s modulus
,
) fully bonded symmetrically to the substrate
, Poisson’s ratio
) subjected to a load
. The
coordinate system is set up with the origin O at the center of layer 1. The x-axis is along the loading direction, and the y-axis is perpendicular to the loading direction. Based on the symmetry, a quarter of the system can be used to represent the entire coating system. Using the theory in Section 2, we will solve the above boundary-value problem and derive the elastic field as follows.
Figure 2. Stress transfer in a coating system when the substrate is subjected a tensile stress
3.1. Simplified displacement functions For the substrate Layer 1 in Figure 2, the general solution for the displacement in the direction is rewritten as
(14) where, as shown in Eq. (11),
10
(15)
The symmetric conditions along the x and y-axes produce the following boundary conditions:
(16) Applying the above boundary conditions to Eq. (14) reduces the displacement to
(17) where the leading coefficient
, is a constant to be determined from the remaining boundary
conditions, and the superscript 1 represents the displacement for Layer 1. This formatting will be followed for the remainder of the paper. Once
is derived,
can be obtained through the integral of Eq. (13) as
(18)
where
is considered, and
is to be determined by the interfacial continuities in
the next Section. Similarly to the derivation in Eqs. (14) to (17), we can simplify the x-directional component of the displacement in the coating as
(19) where the constant
satisfies
11
(20)
Then
in the coating can also be obtained through the integral of Eq. (13) as
(21) At the upper surface,
, so that using Eq. (2) we can write
(22) Therefore, Eq. (21) can be rewritten as
(23)
Here
is to be determined by the interfacial continuities in the next Section. 3.2. The closed-form elastic solution
In summary, we have solved the displacements of
,
,
, and
in Eqs. (17), (18), (19)
and (23), from which we can solve for all stresses using Eq. (2). However, there are still four constants, namely,
,
,
, and
, and two functions, namely
and
, to be
determined by the interfacial continuities and the loading condition. Since a fully bonded interface is considered, four interfacial continuity conditions are required when
:
12
(24)
We will solve for the four parameters and two unknown functions in what follows. The first two equations in (24) can be rewritten as:
(25)
(26) Notice that the approximation of the shearing stress in Eq. (2) is used. Division of both sides of Eq. (26) by Eq. (25), respectively, yields
(27) Using the definitions of (15) and (20), we can calculate the constant c, and then
and
from
Eq. (27). The constant c changes with the elastic constants and thickness, while the fracture spacing has no effects on c. There are many solutions for constant c because of the periodicity of the
function, which suggests the need for a series-form solution. In general, the roots of c
in Eq. (27) are not periodic, which also implies that the basis functions in the series-form solution will not be orthonormal to each other. The derivation of the coefficient of each basis function will be complicated and the convergence of the solution will still be open. An example using the series form solution can be found in another paper (Liu and Yin, 2013), which considers the stress transfer in a compliant film bonded to a rigid substrate. For simplicity, the present paper uses the first root to demonstrate this theory. To solve for c, we define a parameter
13
(28)
Notice that the Poisson’s ratios of the coating and the substrate are assumed to be less than 0.5. Otherwise, the effective stiffness is Eq. (2) will be infinite. If incompressible materials are used in the coating, the plane strain assumption in this theory may be invalid and thus the formulation would need to be reconstructed. Based on the mechanics of the coating system, the first root of c can be obtained within a range of
, which makes the shearing stress along the
interface negative. More specifically, c can be obtained through the following three conditions. •
When
, c is the smallest value that satisfies
. Typically, a thick
coating on a thin substrate will fall within this category. When to approximately solve for
•
When
,
, we can use
.
is used to solve for
.A
representative case is when the coating and the substrate have the same elastic constants and the coating is half the thickness of the substrate. •
When
, we impose that c is the smallest solution meeting the constraint where . Typically, a thin coating deposited on a thick substrate will fall in this
category. When
, we can assume
, and c can be obtained as
.
14
1.8 E1/E2 = 100 1.6
E1/E2 = 1 E1/E2 = 1/100
1.4
Value for "c"
1.2 1 0.8 0.6 0.4 0.2 0 -2 10
-1
10
0
10
1
2
10
10
ξ
Figure 3. The calculation result of c based on varying values of with
.
Figure 3 illustrates the relation between the first root of c and . With the increase of 0.01
to 100
from
, the value of c is illustrated in the log coordinate of for the three cases of . With the increase of , the value of c decreases from a certain value to
zero. Notice that c has dimension of “length-1”, so that its value depends on the length scale of and
.
In addition, Eq. (25) provides the relation between
and
as
(29) Now we can rewrite the last two equations in (24) to determine
and
as follows:
(30)
15
(31)
From Eq. (31), we can obtain
(32)
Then, from Eq. (30), we can obtain
(33)
Therefore, if the constants
and
can be determined, the closed form solution of the
displacement field in the coating system will be obtained, and thus the strain and stress fields can be derived accordingly. We will use the principle of stationary potential energy to determine these two constants. Because the stresses within the coating system are caused by the load
applied in the
substrate and no external load is applied in the y-direction, we assume that the stress
.
Again, this assumption may not be accurate near the edge but is reasonable in the zone away from the edge. Using this assumption, the stress
can be approximated from Eq. (2) as,
(34)
16
Notice that
can be taken as an average stress of the resultant force distributed in the cross
section of the substrate. The Saint-Venant principle will guarantee the accuracy for the zone away from the edge. This approximation provides flexibility for application in periodic openingmode fracture analysis in Section 4, where the stress distribution in the Layer 1 is difficult to be obtained, but the resultant force is known. The strain energy density of a material point in the coating system is from two stress components
and
. Using the third equation in Eq. (2) and Eq. (34), we can obtain the
strain energy in the coating system written as
(35)
Substituting Eqs. (17) and (19) into the above equation, we can obtain
(36) where
(37)
(38)
17
and (39) The work done by the external tensile load is written as
(40)
Using the principle of the stationary potential energy
, we can write
(41) or (42) and (43) Solving the above two equations, we can obtain
(44) Therefore, a closed form solution for a coating system is obtained with the combination of displacements in Eqs. (17), (18), (19) and (23) along with the parameters solved by Eqs. (44), (27), (29), (32), and (33). 3.3. Verification with finite element results In order to observe the effects of this solution’s simplifications used to derive the elastic fields, a finite element (FE) model was used for comparison. The basic geometry used for this
18
model was
, where
was chosen to be 1. A load of 1000 N/m is applied as a
uniformly distributed pressure along the substrate. Moreover, both materials were chosen to have the same Poisson ratio of
. Initially, the effect of the material stiffness mismatch
was observed, where
, and the ratio of
was varied by: 10, 3, 1, 1/3, and
1/10. Therefore, any limitations that this solution would have on modelling different materials should be observed. The FE model was made using commercially available ABAQUS. For each model, a uniform mesh of approximately 14,000 nodes is assembled, using quadrilateral plane strain elements using quadratic formulation. The following figures illustrate the results. -8
2.5
x 10
E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM
Displacement ux (m)
2
1.5
10 3 1 1/3 1/10
1
0.5
0
0
0.5
1
1.5
2
2.5 x(m)
3
3.5
4
4.5
5
Figure 4. Comparison between the displacement in the x-direction for the analytical solution for ( ) and the FE model when along -8
2.5
x 10
E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM
Displacement ux (m)
2
1.5
10 3 1 1/3 1/10
1
0.5
0
0
0.5
1
1.5
2
2.5 x(m)
3
3.5
4
4.5
5
19
Figure 5. Comparison between the displacement in the x-direction for the analytical solution and the FE model when along for ( ) -8
2.5
x 10
E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM E1/E2 = FEM
Displacement ux (m)
2
1.5
10 3 1 1/3 1/10
1
0.5
0
0
0.5
1
1.5
2
2.5 x(m)
3
3.5
4
4.5
5
Figure 6. Comparison between the displacement in the x-direction for the analytical solution and the FE model when along for ( ) Figures 4-6 show the FE and analytical solution comparisons for the distribution of displacement
along the x-direction at different vertical locations. It is observed that despite
varying the stiffness mismatch (by varying the stiffness of the coating) from having a more compliant coating to having a far stiffer coating, the FE and the analytical results agree very well. When the coating is compliant, the coating produces very minor effects on the deformation of the substrate. Therefore, in Figure 4,
is almost linear for
and 3. Along the
surface, the displacement in Figure 6 is almost linear in the central part, but the slopes decrease in the region close to the fracture end. Figures 7 and 8 show how the shearing stress and the x-displacement, when
,
change along the interface for varying fracture separations and coating thicknesses, respectively. The shearing stress increases from zero at the center to the maximum at the fractured end. For a higher fracture spacing, the shearing stress remains near zero for a large range and increases significantly in the region close to the fractured end, which is consistent with the trend of the
20
singularity effect due to the material difference. These results show that the analytical solution is quite capable of capturing the physical effects of the loading prescribed. 50
Shear Stress τxy (Pa)
0
-50
λ/h1 = 10 FEM
λ/h1 = 5
-100
FEM
λ/h1 = 2
-150
FEM
λ/h1 = 1 FEM
-200
λ/h1 = 1/2 FEM -250
0
1
2
3
4
5 x(m)
6
7
8
9
10
Figure 7. Comparison between the shearing stress for the analytical solution and the FE model when along
at the interface
for varying values of
(
-8
2.5
x 10
h2/h1 = 1 FEM h2/h1 = 0.3
Displacement ux (m)
2
FEM h2/h1 = 0.1 FEM h2/h1 = 0.03
1.5
FEM 1
0.5
0
0
0.1
0.2
0.3
0.4
0.5 x/ λ
0.6
0.7
0.8
0.9
1
Figure 8. Comparison between FE and analytical results for the x-displacement along the interface for varying coating thicknesses when
, and
,
Notice that in Figures 4-8, the difference between the analytical solution and the FEM results in the region close to the fractured end is more considerable than that in the other regions. This is due to the plane assumption used, and the application of a weak form force boundary condition 21
at the fractured ends. If a series-form solution were to be used, the boundary condition may be exactly satisfied. This however, does not guarantee a better solution because of the assumption of Eq. (1). A rigorous investigation in mathematics and mechanics of this problem will be conducted in the future. 4. Results and discussion The proposed theory is based on the simplified governing equation and thus does not provide an exact solution. The assumptions and approximations adopted may impose some limitations on the application of the formulation. In summary, the following aspects are highlighted for the strengths and weaknesses of this theory: •
Because all the points in the same plane perpendicular to the y-axis are assumed to keep in the same plane during deformation (as Eq. (1)), the decoupled governing equation of
is
obtained. This assumption imposes a limitation that this model does not take into account the curvature of the layered materials, which can be well addressed through Stoney’s theory (Freund and Suresh, 2003, Stoney, 1909). Combination of these two theories considering both bending moment and axial load may provide some new perspectives of the stress transfer in layered materials, which will be investigated in future work. •
To obtain an explicit analytical solution for the governing equation, the boundary condition along the end surface of the coating system is not exactly satisfied. Using the principle of stationary potential energy provides a way to obtain a closed-form, simple solution without the loss of generality. However, it makes the stress field around the loading edge inaccurate; whereas the accuracy in the zone away from the edge can be guaranteed by the Saint Venant’s Principle.
22
•
The parameters c,
and
are determined by the interfacial continuities. However,
currently only the first root of Eq. (27) is considered. If a series-form solution is used with all roots, boundary conditions at the loading end can be more accurately modeled. However, the exactness cannot be guaranteed due to the plane assumption used. •
If multiple layers are considered, we will need to use a series form solution to make the interfacial continuity satisfied. Due to the above-mentioned assumptions, the singularity effect at the fracture tip cannot be
exactly modeled, and the stress and strain distributions along the fractured surface have not been accurately obtained. Therefore, when the fracture distribution is very dense, such as
, this
solution may not be accurate due to the end effect. Because the proposed model provides a closed-form solution of elastic field, it can be used in structural design and failure analysis of layered materials. We will use this theory to calculate the elastic modulus reduction and to investigate the interfacial compliance, while comparing the results with existing models.
4.1. Elastic modulus reduction due to opening-mode fractures When a layered material is subjected to an increasing tensile load, the elastic modulus may decrease when opening-mode micro-cracks are induced with a uniform pattern. For example, laminates with microcracks have been investigated by the shear lag models (Nairn and Mendels, 2001). The proposed theory can be used to predict the elastic modulus reduction due to the OMFs. Consider a coating system in Fig. 2. Using the loading
and the elongation
in
Eq. (17) for a uniaxial loading condition, we can obtain the average stress on the overall cross
23
section as
and the average strain of the section as
Young’s modulus, namely
, from which the effective
, is written as
(45)
For a long layered material (
, the effective Young’s modulus, namely
, can be written
as
(46)
So that the modulus reduction ratio due to OMFs can be written as
(47) where
and
are given in Eq. (44).
The shear lag models have been widely used to solve this problem by introducing some predefined shape functions of the shear stress in the thickness direction, and the modulus reduction ratio is written in the form of (Nairn and Mendels, 2001)
(48) where
is typically determined by the shape function of shear stress distribution or calibrated by
an FEM analysis. Nairn and Mendels (2001) showed that a linear shape function has provided
24
good agreement with FEM results where the difference of shearing moduli between two layers is not very large. In such a case,
can be written by
(49)
Notice that Eq. (48) was based on the plane stress problem; whereas Eq. (47) was based on the plane strain problem. To compare both equations, Eqs. (48) and (49) should be based on the Young’s moduli for the plane strain problem as
. Because the proposed theory does not
impose any assumption to the shear stress distribution, it should be applicable to more general situations. Fig. 9 shows the modulus reduction for the coating system with elastic constants as follows thickness of
and
;
and
. The substrate has a
and the coating has a thickness of
,
respectively, and the specimen has a unit width. For each case of the coating thickness, the results of Eqs. (47) and (48) have been compared with the FEM results.
1.45 Analytical h2 = 0.25 μm 1.4
Nairn and Mendels h2 = 0.25 μm FE h2 = 0.25 μm
1.35
Analytical h2 = 1 μm E0/E* Ratio
1.3
Nairn and Mendels h2 = 1 μm FE h2 = 1 μm
1.25
Analytical h2 = 4 μm 1.2
Nairn and Mendels h2 = 4 μm FE h2 = 4 μm
1.15 1.1 1.05 1
0
50
100
150
200
250
λ (μm)
25
Fig. 9 – Modulus reduction curves for the present analytical solution, the FE model, and the Nain and Mendels’ formulation (2001) with a log scale on the y axis 4.2. Interfacial compliance of bi-layered materials and parameter c One popular method to study the stress transfer in bi-layered materials is to assume a frictional interface, through which the shear stress is proportional to the displacement. A spring constant or an interfacial compliance, which are reciprocal to each other, has been introduced to describe the constitutive relation of the interface. For example, we can write
(50) where
is the interfacial compliance.
Since the shearing stress and displacement are continuous across the interface (without loss of generality from the results in Eq. (19) and the third equation in Eq. (2)) we can obtain
(51)
The interfacial compliance is generally not constant along the interface. The interfacial compliance however, can be written in the polynomial form as
(52)
26
When is small, such as a thin film deposited on a thick substrate with dense OMFs, we can approximately write terms of
and
. Disregarding higher order
, we can obtain
(53)
Notice that the definition of interfacial compliance in Eq. (50) depends on the load configuration. If the load is applied at the coating layer and transferred to the substrate, another type of the interfacial compliance can be derived with the same solution except that the material constants and geometry parameters will be switched. When this solution is superposed with a uniform tensile or compressive deformation, the shearing stress keeps the same, but the displacement will independently change with the uniform deformation. This makes the interfacial compliance difficult to be defined for general cases. Our previous model based on the frictional interface (Yin, et al., 2008) disregarded the deformation in the substrate and provided the equivalent interfacial compliance equal to the second part of Eq. (53). If we apply a uniform deformation with equivalent strain
in Eq. (19), we can obtain the same form of the interfacial compliance
as the previous one. In this case, the interfacial compliance can be correlated to parameter c. In our previous work (Yin, et al., 2008), c was defined in a slight different way that it is dimensionless. If we transform it to the same physical mean of c in this paper, written as , its form should be
(54)
27
where
with
(Beuth, 1992, Yin, et al.,
2008). However, in this paper, c is obtained by Eq. (27), which depends on the elastic moduli and thicknesses of both the film and the substrate. Fig. 10 illustrates the comparison of the parameter c obtained by Eqs. (27) and (54). Here using changing with
for three cases
,
, we draw c
In our previous work, the effect of the
substrate thickness has not been quantitatively considered and c changes in a large range; whereas the current formulation provides a small range of c for different ratios of
. Notice
that when the difference of the thicknesses between two layers is large, the accuracy of the current formulation using one term may decrease.
1.4
Calculated value of c
Proposed c for h1 = 1 1.2
Proposed c for h1 = 10 Proposed c for h1 = 100
1
c from (Yin, et al., 2008)
0.8
0.6
0.4
0.2
0 -2 10
-1
10
0
10 Ratio of E2/E1
1
10
2
10
Fig. 10 – Value of c calculated from Eq. (27) versus the value of calculated from Eq. (54), as the ratio of increases 5. Conclusions A general elastic solution for multi-layered materials is derived to predict stress transfer when one layer is subjected to a mismatched load from others. The closed-form solution for a coating system is derived to demonstrate the use of the formulation. A comparison between the
28
proposed formulation and finite element results shows that this theory exhibits a high accuracy for general dual layered systems when the thicknesses of the layers are not significantly different. This theory provides a very reasonable prediction of the elastic modulus reduction of a coating system caused by dense fractures. It also facilitates the understanding of the interfacial compliance of multi-layered material systems. The theory can be applied to general multilayered materials with other geometries such as circular or cylindrical layered systems. An experimental validation of the theory is underway. Acknowledgments This work is sponsored by the National Science Foundation CMMI 0954717 and the Department of Homeland Security CU09-1155, whose support is gratefully acknowledged. References 1. 2. 3. 4. 5. 6. 7. 8.
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Highlights
>Decoupled governing equations are obtained by the plane assumption >General solution to the governing equations is derived for the BVP >The closed-form solution for a coating system is provided using stationary potential energy >The present approximate solution well agrees with the finite element results >The formulation interprets the elastic modulus reduction and interfacial compliance
32