A method for evaluating the slip failure of a composite film-substrate structure

Abstract

The application discloses a kind of composite film-base structure's sliding failure evaluation method, first, by modeling composite film as no traction model, the mechanical parameters of composite film are obtained analytically, parameter establishes new constitutive relation, subsequently using new constitutive relation gets the equilibrium equation and control equation of composite beam-soft base structure before sliding, finally by solving equation can obtain the interface sliding failure judgment criterion suitable for composite film-soft base structure model.Through analysis model, it can be concluded that the composite film-soft base structure model can be degraded to the single-layer film-soft base structure model under the plane strain assumption, and the risk of introducing error by the plane strain model is eliminated.The sliding failure evaluation method of composite film-base structure can determine whether the interface of flexible electronic device of composite film-soft base structure has sliding failure, which is closer to the actual engineering situation and eliminates the limitations of existing methods.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of flexible electronics, and particularly relates to a method for evaluating the slip failure of a composite film-substrate structure. BACKGROUND

[0002] In the working state of a flexible extensible electronic device, the soft substrate itself will bear severe and complex deformation due to the flexibility and extensibility, and the rotation and displacement transmitted to the bent hard functional film is also very complex. Due to the great difference in mechanical properties between the hard functional film and the soft substrate, the interface has extremely complex and highly concentrated load transmission, so the interface failure and damage are often more dangerous than the material itself in the actual application process, which directly affects the flexibility of the system. Therefore, it is urgent to establish a reasonable interface failure evaluation method.

[0003] In the study of the interface failure mode of multilayer materials, the commonly used method is to approximate the normal stress and shear force on the interface by using a beam model, and a slip zone model is introduced to completely obtain the stress evolution and establish the interface failure criteria by combining the widely used ideal elastic-plastic cohesive constitutive relationship with the strength theory. However, this method can only solve the interface failure problem of a single-layer film-substrate structure, which is quite different from the actual engineering situation, and the plane strain model used in this method may produce considerable errors in the mechanical analysis of flexible electronics. SUMMARY

[0004] In order to overcome the shortcomings of the prior art, the present application provides a method for evaluating the slip failure of a composite film-substrate structure. First, the composite film is modeled as a no-drag model to analytically obtain the mechanical parameters of the composite film, and a new constitutive relationship is established. Then, the equilibrium equation and control equation of the composite beam-soft substrate structure before slip are obtained using the new constitutive relationship. Finally, the interface slip failure judgment criterion applicable to the composite film-soft substrate structure model is obtained by solving the equations. Through the analysis of the model, it can be concluded that the composite film-soft substrate structure model can be degenerated into a single-layer film-soft substrate structure model under the plane strain assumption, and the risk of introducing errors by the plane strain model is eliminated. The slip failure evaluation method of the composite film-substrate structure can judge whether the interface of the flexible electronic device of the composite film-soft substrate structure has slip failure, which is closer to the actual engineering situation and eliminates the limitations of the existing method.

[0005] The technical solution adopted by the present application to solve its technical problems includes the following steps:

[0006] Step 1: Obtain the constitutive relationship of the composite film based on the no-drag model;

[0007] Step 1-1: Define the geometric and mechanical parameters involved in the composite film-substrate structure:

[0008] The thickness, plane strain modulus, and Poisson's ratio of the thin film, substrate, and adhesive layer are expressed as: h f , ν f h s , ν s and h a , ν a The subscripts f, s, and a represent the thin film, substrate, and adhesive layer, respectively; the torque, shear stress, and axial stress of the thin film and substrate are represented by M. i Q i and N i , i = f or s means that the lengths of the thin film and the substrate are l and L, respectively, l << L;

[0009] Steps 1-2: The composite film is considered as an n-layer composite beam bonded together, which is subjected to a uniformly distributed axial force f at the boundary. x and f z and bending moment m x and m z Thickness t of each layer of the composite film j With coordinate y j The relationship is t j =y j -y j-1 Under tensile and bending loads, the principal strains are along the coordinate directions, and the strains are obtained through Kirchhoff's assumptions:

[0010]

[0011] Where, ε x0 and ε z0 It is the mid-plane of the composite beam along the x and z directions. Membrane strain at κ x and κ z These are the curvatures; the normal stress, given by the linear elastic constitutive relation, is:

[0012]

[0013] Where E is Young's modulus and ν is Poisson's ratio; therefore, the axial force and bending moment per unit length are obtained by integrating over the entire thickness of the composite beam, as shown in equation (3):

[0014]

[0015] Among them, y0, y n These represent the coordinates of the lower surface of the first-layer beam and the upper surface of the nth-layer beam, respectively.

[0016] Step 1-3: The relationship between axial force / bending moment and membrane strain / curvature is given by simultaneous equations (1), (2) and (3):

[0017]

[0018] where the values of each element in the coefficient matrix are:

[0019]

[0020] where E i , v i and t i are the Young's modulus, Poisson's ratio and thickness of the i-th layer of the composite beam, respectively;

[0021] Step 1-4: For bending load, there is a no-drag condition at the boundary perpendicular to the x-axis:

[0022] f x = m x = 0 (6)

[0023] There are axial force and bending moment at the boundary perpendicular to the z-axis:

[0024]

[0025] Substituting equations (6) and (7) into equation (4), we get:

[0026]

[0027] Further, we get:

[0028]

[0029] where the elements of the coefficient matrix are the inverse of the elements of the coefficient matrix in equation (4); according to the knowledge of elasticity mechanics, we have:

[0030]

[0031] Simultaneous equations (9) and (10) give the constitutive relationship of the composite beam under bending load as follows:

[0032]

[0033] Step 2: Obtain the equilibrium equation and control equation of the composite thin film-substrate structure before slipping;

[0034] Step 2-1: The equilibrium equations of the soft substrate and the hard thin film are written as:

[0035]

[0036]

[0037] where σ and τ are the interfacial normal and shear stresses, respectively;

[0038] Step 2-2: The constitutive relations for the axial elongation and bending of the soft substrate are written as:

[0039]

[0040] where u s and w s are the axial displacement and deflection components, respectively;

[0041] Step 2-3: In the normal direction, the interfacial normal stress is expressed as:

[0042]

[0043] The shear stress is also related to the displacement components of the film and substrate:

[0044]

[0045] where u and w are the axial displacement and deflection components of the first and second parts of the interlayer, respectively, and G is the shear modulus of the adhesive layer. By combining the fourth and third order derivatives of Eqs. (15) and (16) with Eqs. (11)-(14), the coupled differential equations for the interfacial stresses are established as follows:

[0046]

[0047]

[0048] By substituting the above equations, we obtain

[0049]

[0050]

[0051] where the coefficients are given by

[0052]

[0053] By combining Eqs. (19) and (20) and eliminating the normal stress, we obtain the differential expression for the shear stress:

[0054]

[0055] where ξ = x / h a , and the coefficients are given by

[0056]

[0057] For flexible electronics of composite thin film-soft substrate structure, both satisfy k1 » k2, k3, so the last term of equation (22) is ignored;

[0058] Step 3: Solve the stress distribution before slip and the maximum interfacial stress;

[0059] Step 3-1: Since the shear stress is known to decay rapidly from the free edge x = l / 2, the approximate solution is expressed as

[0060]

[0061] where the superscript 0 represents the stress field before slip occurs, A1 and A2 are constants determined by the boundary conditions, and:

[0062]

[0063]

[0064] Neglecting the high-order infinitesimal term, the expression (24) of the shear stress differential in the approximate equation is as follows:

[0065]

[0066] where A = λ1A2 + λ2A1; substituting equation (27) into equation (17), the normal stress is:

[0067]

[0068] where B and C are constants determined by the boundary conditions, and:

[0069]

[0070]

[0071] Step 3-2: One end of the substrate in the composite thin film-soft substrate structure is subjected to a compression displacement load dL, and the other end is fixed. The system loading compression strain is defined as |ε| = dL / L; the critical buckling strain is given by the elastic stability theory: which is the minimum applied strain that causes buckling; at the same time, because the length of the thin film is negligible compared to the bending curvature, the moment and axial force of the delaminated part are considered to be constants, which are:

[0072]

[0073]

[0074] where K(·) is the first kind of complete elliptic integral, α is the maximum slope rotation angle determined by the following formula:

[0075]

[0076] The boundary conditions of the free edges of the film and substrate are:

[0077]

[0078]

[0079] The boundary conditions of the normal equilibrium are:

[0080]

[0081] The boundary conditions are:

[0082]

[0083]

[0084]

[0085] Step 3-3: Substitute the boundary conditions (36)-(39) into the expressions (24), (28) of the interfacial shear stress and normal stress, ignore the high order terms (O(h s / L), and solve the undetermined constants A1, A2, B and C:

[0086]

[0087] where

[0088]

[0089] Then the maximum stress is:

[0090]

[0091]

[0092] Step 4: Determine whether the maximum interfacial shear stress reaches the shear strength τ c , and further evaluate whether the interface fails;

[0093] When the maximum interfacial shear stress increases to the shear strength τ c with the increase of the compression strain of the system, the composite film will slide, and the corresponding compression strain is the critical strain If the composite film-substrate structure will not slide when the compression strain is loaded; if Composite film-substrate structure slip failure under compression occurs.

[0094] The beneficial effects of the present application are as follows:

[0095] (1) The present application can obtain the mechanical parameters of the composite film, establish a new constitutive relation, and further obtain a new composite film-soft substrate structure model, which can not only be degenerated into a single-layer film-soft substrate structure model under the plane strain assumption, but also eliminate the risk of introducing errors in the model under the plane strain assumption.

[0096] (2) The slip failure evaluation method of the composite film-substrate structure established by the present application can determine whether the flexible electronic device interface of the composite film-soft substrate structure slips and fails, which is closer to the actual engineering situation and eliminates the limitations of the existing method. BRIEF DESCRIPTION OF DRAWINGS

[0097] Figure 1 is a stress analysis diagram of the composite film-soft substrate structure device of the present application embodiment.

[0098] Figure 2 is a schematic diagram of the composite film of the present application embodiment.

[0099] Figure 3 is a schematic diagram of the device when degenerated into a single-layer film of the present application embodiment.

[0100] Figure 4 is a comparison of the slip failure evaluation method of the composite film-substrate structure when degenerated into a single-layer film of the present application embodiment and the existing method: Figure 4 (a) is a comparison between the maximum shear stress of the interface and the pre-compression strain of the system; Figure 4 (b) is a comparison of the typical distribution of the interface shear stress.

[0101] Figure 5 is a schematic diagram of the composite film-soft substrate structure of the present application embodiment: Figure 5 (a) is a PI / Si film; Figure 5 (b) is a PZT / Si film; Figure 5 (c) is a Pt / Si film; Figure 5 (d) is an arbitrary elastic modulus material / Si film.

[0102] Figure 6 is a mechanical property diagram of the three composite film-soft substrate structures of the present application embodiment: Figure 6 (a) is the relationship between the critical compression strain of the interface slip failure of the composite beam of different materials and the film thickness; Figure 6 (b) is the change of the interface stress component along the length direction before the slip failure of the composite beam of different materials occurs. Figure 6(c) is a graph showing the relationship between the critical compressive strain and the elastic modulus of the film material at the interface of the composite film-soft substrate structure where slip failure occurs. Detailed Implementation

[0103] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0104] To overcome the shortcomings of existing technologies, this invention provides a method for assessing slip failure in composite thin film-substrate structures. By establishing a traction-free model, the mechanical parameters of the composite thin film are analytically obtained and directly applied to the establishment of a composite thin film-soft substrate structure model. Finally, an interface slip failure judgment criterion applicable to the composite thin film-soft substrate structure model is given.

[0105] The technical solution adopted by this invention to solve its technical problem includes the following steps:

[0106] Step 1: Obtain the constitutive relation of the composite thin film based on the traction-free model:

[0107] First, the geometric and mechanical parameters of the composite beam-soft substrate structure are defined accordingly. The thickness, plane strain modulus, and Poisson's ratio of the three layers are expressed as: h f , ν f h s , ν s and h a , ν a The subscripts "f", "s", and "a" represent the thin film, substrate, and adhesive layer, respectively, and the torque, shear stress, and axial stress are represented by M. i Q i and N i (i = f or s) indicates that the lengths of the thin film and the substrate are l and L, respectively (l << L).

[0108] The composite film can be viewed as a perfectly bonded n-layer composite beam, subjected to a uniformly distributed axial force f at its boundaries. x and f z and bending moment m x and m z Thickness t of each layer i With coordinate y i The relationship is t i =y i -y i-1 Under tensile and bending loads, the principal strains are along the coordinate directions. The strain can be obtained using Kirchhoff's assumptions:

[0109]

[0110] Where, εx0 and ε z0 It is the mid-plane of the composite beam along the x and z directions. Membrane strain at κ x and κ z These are the curvatures. For laminates containing multiple materials, the intermediate plane is not necessarily a neutral plane, hence ε x0 and ε z0 It is not always zero. The normal stress given by the linear elastic constitutive relation is:

[0111]

[0112] Where E is Young's modulus and ν is Poisson's ratio. Therefore, the axial force and bending moment per unit length can be obtained by integrating over the entire thickness of the composite beam, as shown below:

[0113]

[0114] 5. Combining formulas (1), (2), and (3), the relationship between axial force / bending moment and membrane strain / curvature can be given as follows:

[0115]

[0116] The values ​​of each element in the coefficient matrix are...

[0117]

[0118] Here, E i ,ν i and t i These are Young's modulus, Poisson's ratio, and thickness of the i-th layer material in the composite beam, respectively.

[0119] For bending loads, there is no traction condition at the boundary perpendicular to the x-axis:

[0120] f x =m x =0 (7)

[0121] There is axial force and bending moment at the boundary perpendicular to the z-axis:

[0122]

[0123] Substituting equations (6) and (7) into equation (4), we get:

[0124]

[0125] Therefore, we can obtain:

[0126]

[0127] where the coefficient matrix is the inverse of the coefficient matrix in equation (4). From the knowledge of elasticity, we have

[0128]

[0129] By combining equations (9) and (10), the constitutive relation of the composite beam under typical bending load can be obtained as follows

[0130]

[0131] Step two, obtain the equilibrium equation and control equation of the composite beam-soft substrate structure before sliding:

[0132] First, consider the solution of the stress distribution under perfect bonding state (i.e. the interface does not appear sliding failure). By symmetry, only half of the system needs to be analyzed, at this time the equilibrium equation about the soft substrate and the hard film is written as:

[0133]

[0134]

[0135] where σ and τ are the interface normal stress and shear stress, respectively. The constitutive relation of the soft bottom axial elongation and bending can be written as:

[0136]

[0137] where u s and w s are the axial displacement and deflection components, respectively. Now consider the balance of the adhesive layer, which is modeled as a discrete distribution spring with given normal and shear stiffness. In the normal direction, the interface normal stress is represented as:

[0138]

[0139] The shear stress is also related to the displacement components of the film and substrate:

[0140]

[0141] where the first and second parts of the middle layer are the axial displacements of the film bottom surface and the substrate top, respectively, is the shear modulus of the adhesive layer. Combine the fourth and third order derivatives of equations (15) and (16) with equations (11)-(14), and then establish the coupled differential equations of the interface stress as follows:

[0142]

[0143]

[0144] Substitute the above equation to get

[0145]

[0146]

[0147] where each coefficient value is

[0148]

[0149] By eliminating the normal stress from equations (19) and (20), the differential expression for the shear stress is obtained:

[0150]

[0151] where ξ = x / h a and each coefficient is

[0152]

[0153] For a general composite thin film-soft substrate structure, the flexible electronic device satisfies the conclusion that k1 » k2, k3, so the last term of equation (22) can be ignored.

[0154] Step three, solving the stress distribution before slip and the maximum interfacial stress

[0155] Since the shear stress is known to decay rapidly from the free edge x = l / 2, the approximate solution can be expressed as

[0156]

[0157] where the superscript "0" represents the stress field before slip occurs, A1 and A2 are constants determined by the boundary conditions, and:

[0158]

[0159]

[0160] In practical applications, we are mainly concerned with the area near the edge of the film, which is several times the thickness of the adhesive layer. Therefore, by ignoring the high-order infinitesimal term, we can approximate the expression (24) of the shear stress differential as:

[0161]

[0162] where A = λ1A2 + λ2A1. Substituting equation (27) into equation (17), the normal stress is:

[0163]

[0164] where B and C are constants determined by the boundary conditions, and

[0165]

[0166]

[0167] The end of the substrate in the composite film-soft substrate structure is subjected to a compressive displacement load dL, while the other end is fixed. The system loading compressive strain is defined as |ε| = dL / L. The elastic stability theory gives the critical buckling strain: which is the minimum applied strain that causes buckling. At the same time, because the length of the film can be neglected compared to the bending curvature, the moment and axial force of the delaminated part can be considered as constants, which are:

[0168]

[0169]

[0170] where K(·) is the first kind of complete elliptic integral, α is the maximum slope rotation angle (absolute value) determined by

[0171]

[0172] Further, the boundary conditions of the free edges of the film and the substrate can be obtained as:

[0173]

[0174]

[0175] Combining the symmetry, the boundary conditions of the normal equilibrium are

[0176]

[0177] Using the above conditions, the boundary conditions can be obtained as

[0178]

[0179]

[0180]

[0181] The boundary conditions (36)-(39) are brought into the expressions of the interfacial shear stress and normal stress (24), (28), and the high-order terms (O(h s / L)) are ignored. The undetermined constants A1, A2, B and C can be solved:

[0182]

[0183] where

[0184]

[0185] Then the maximum stress is:

[0186]

[0187]

[0188] Step four, judge the interface maximum shear stress Whether the shear strength τ c , and further evaluate the interface failure

[0189] When the interface maximum shear stress With the increase of the system compression strain to the shear strength τ c , the composite film will slip, and the corresponding compression strain is the critical strain of slip If The composite film-substrate structure will not slip when the compression strain is loaded If The composite film-substrate structure will slip when the compression strain is loaded .

[0190] First, the constitutive relation of the composite film based on the traction-free model is established, and the balance equation and control equation of the composite beam-soft substrate structure before slip are derived.

[0191] Then, the stress distribution and the maximum interface stress before slip are solved, different modeling parameter values are input into the expression, and the stress distribution τ 0 and the maximum interface stress

[0192] Finally, referring to the interface shear strength τ c , when the interface maximum shear stress With the increase of the system compression strain to the shear strength τ c , the composite film will slip, and further the device slip failure can be judged.

[0193] I. Obtain the constitutive relation of the composite film based on the traction-free model

[0194] Figure 1 The corresponding thickness, plane strain modulus and Poisson's ratio of the three layers in the composite film-soft substrate structure are represented as: h f , ν f ; h s , ν s and ha , ν a The subscripts "f", "s", and "a" represent the thin film, substrate, and adhesive layer, respectively, and the torque, shear stress, and axial stress are represented by M. i Q i and N i (i = f or s) indicates that the lengths of the thin film and the substrate are l and L, respectively (l << L).

[0195] Figure 1 The medium-hard film is a composite film, as shown in the schematic diagram. Figure 2 The composite film can be viewed as a perfectly bonded n-layer composite beam, subjected to a uniformly distributed axial force f at its boundaries. x and f z and bending moment m x and m z Thickness t of each layer i With coordinate y i The relationship is t i =y i -y i-1 Under tensile and bending loads, the principal strains are along the coordinate directions. The strain can be obtained using Kirchhoff's assumptions:

[0196]

[0197] Where, ε x0 and ε z0 It is the mid-plane of the composite beam along the x and z directions. Membrane strain at κ x and κ z These are the curvatures. For laminates containing multiple materials, the intermediate plane is not necessarily a neutral plane, hence ε x0 and ε z0 It is not always zero. The normal stress given by the linear elastic constitutive relation is:

[0198]

[0199] Where E is Young's modulus and ν is Poisson's ratio. Therefore, the axial force and bending moment per unit length can be obtained by integrating over the entire thickness of the composite beam, as shown below:

[0200]

[0201] Combining equations (1), (2), and (3), the relationship between axial force / bending moment and membrane strain / curvature can be given as follows:

[0202]

[0203] The values ​​of each element in the coefficient matrix are...

[0204]

[0205] Here, E i , v i and t i are the Young's modulus, Poisson's ratio and thickness of the i-th layer of the composite beam, respectively.

[0206] For the bending load, the no-drag condition is imposed at the boundary normal to the x-axis:

[0207] f x = m x = 0 (50)

[0208] The axial force and bending moment are imposed at the boundary normal to the z-axis:

[0209]

[0210] Substituting eqs. (6) and (7) into eq. (4), we have:

[0211]

[0212] Further, we have:

[0213]

[0214] where the coefficient matrix is the element of the inverse matrix of the coefficient matrix of eq. (4). According to the knowledge of elasticity mechanics, we have:

[0215]

[0216] By combining eqs. (9) and (10), the constitutive relation of the composite beam under the typical bending load is given as follows

[0217]

[0218] II. Obtain the equilibrium equation and control equation of the composite beam-soft substrate structure before sliding:

[0219] First, consider the stress distribution under the perfect bonding state (i.e., the interface without sliding failure). The stress analysis of the composite thin film-soft substrate structure flexible device is as follows Figure 1 . By symmetry, only half of the system needs to be analyzed, and at this time the equilibrium equations of the soft substrate and hard thin film are written as:

[0220]

[0221]

[0222] where σ and τ are the interfacial normal and shear stresses, respectively. The constitutive relations for the axial elongation and bending of the soft substrate can be written as

[0223]

[0224] where u s and w s are the axial displacement and deflection components, respectively. Now consider the balance of the adhesive layer, which is modeled as a discrete distribution of springs with given normal and shear stiffnesses. In the normal direction, the interfacial normal stress is given by

[0225]

[0226] The shear stress is also related to the displacement components of the film and substrate:

[0227]

[0228] where the first and second parts of the interlayer are the axial displacements of the film bottom and substrate top, respectively, is the shear modulus of the adhesive layer. Combining the fourth and third order derivatives of equations (58) and (59) with equations (54)-(57) then leads to the coupled differential equations for the interfacial stresses, as follows:

[0229]

[0230]

[0231] Substituting the above into equations (62) and (63) and eliminating the normal stress gives the differential expression for the shear stress:

[0232]

[0233]

[0234] where the coefficient values are

[0235]

[0236] Solving equations (62) and (63) together eliminates the normal stress and gives the differential expression for the shear stress:

[0237]

[0238] where ξ = x / h a , and the coefficients are

[0239]

[0240] For a general composite film-soft substrate structure of a flexible electronic device, it is concluded that k1 » k2, k3, so the last term in equation (65) can be neglected.

[0241] III. Solving the pre-slip stress distribution and maximum interfacial stress

[0242] Since the shear stress is known to decay rapidly from the free edge x = l / 2, the approximate solution can be expressed as

[0243]

[0244] where the superscript "0" represents the stress field before slip occurs, A1and A2are constants determined by the boundary conditions, and

[0245]

[0246]

[0247] In practical applications, we are mainly interested in the region near the edge of the film, which is a few times the thickness of the adhesive layer. Therefore, by neglecting the higher order infinitesimal terms, we can approximate the expression for the shear stress differential in equation (24) as

[0248]

[0249] where A = λ1A2+ λ2A1. Substituting equation (70) into equation (60), the positive interfacial stress is

[0250]

[0251] where B and C are constants determined by the boundary conditions, and

[0252]

[0253]

[0254] In the composite film-soft substrate structure, one end of the substrate is subjected to a compressive displacement load dL, while the other end is fixed, and the system loading compressive strain is defined as |ε| = dL / L. The elastic stability theory gives the critical buckling strain: which is the minimum applied strain that causes buckling. At the same time, because the length of the film can be neglected compared to the bending curvature, the moment and axial force of the delaminated part can be considered constant, which are

[0255]

[0256]

[0257] where K(·) is the first kind of complete elliptic integral, α is the maximum slope rotation angle (absolute value) determined by

[0258]

[0259] Therefore, the boundary conditions at the free edges of the thin film and the substrate can be obtained as follows:

[0260]

[0261]

[0262] Due to symmetry, the boundary conditions for normal equilibrium are:

[0263]

[0264] Using the above conditions, the following boundary conditions can be obtained.

[0265]

[0266]

[0267]

[0268] Substituting the boundary conditions (79)-(82) into the expressions for interface shear stress and normal stress (67), (71), and ignoring higher-order terms (O(h) s / L)) can be used to solve for the indeterminate constants A1, A2, B, and C:

[0269]

[0270] in

[0271]

[0272] Then the maximum stress is calculated as follows:

[0273]

[0274]

[0275] IV. Determining the maximum shear stress at the interface Has the shear strength τ been reached? c To assess whether the interface is malfunctioning.

[0276] When the maximum shear stress at the interface The shear strength τ increases with increasing compressive strain of the system until it reaches the experimentally obtained shear strength τ. c When the composite film slips, the corresponding compressive strain is the critical strain at which slip occurs. if Composite thin film-substrate structure under compressive strain At that time, slip failure will not occur; if Slip failure of composite film-substrate structures under loading compression strain Slip failure will occur.

[0277] V. Calculation results of the slip failure evaluation method of composite film-substrate structures

[0278] To verify the correctness of the new method, the results of the new and old methods are compared. By setting the number of layers of the composite beam to 1, the composite beam is degenerated into a single-layer beam, the material is set to silicon, and the thickness is set to 2.5 μm. The structural schematic diagram is as shown in Figure 3 The comparison results are as shown in Figure 4 , wherein the results of the relationship between the maximum interfacial shear stress and the pre-compression strain of the system and the typical distribution of the interfacial shear stress calculated by the new and old methods can be seen from Figure 4 (a), Figure 4 (b), which proves the correctness and universality of the newly established slip failure evaluation method of the composite film-substrate structure.

[0279] According to Figure 5 , Figure 6 , the slip failure evaluation method of the composite film-substrate structure can be further analyzed to obtain some qualitative conclusions. FIG. shows the relationship between the critical compression strain at which the interface of the PI / Si film, PZT / Si film, and Pt / Si film flexible device slips and fails and the thickness of the film. As can be seen from Figure 6 (a), the critical strain that can be borne by the three types of composite beams decreases with the increase of the film thickness; among the three types of composite beams, the PI / Si film flexible device can bear the largest critical strain under the condition of the same film thickness, the PZT / Si film flexible device is the second, and the Pt / Si film flexible device is the smallest. While the elastic modulus E PI of the PI material is 4.5 GPa, the elastic modulus E PZT of the PZT material is 76.5 GPa, and the elastic modulus E Pt of the Pt material is 169.0 GPa, it can be preliminarily obtained from Figure 6 (a) that under the condition of the same film thickness, the critical strain that can be borne by the system decreases with the increase of the elastic modulus. As can be seen from Figure 6 (b), the interfacial shear stress of the three types of composite beam systems is continuous in the length direction of the interface, all first points to the positive direction of the x-axis near the free end, and changes to the negative direction at a more intense speed with the decrease of the dimensionless axial coordinate, decays to zero at about 1 / 20 of the length of the silicon strip from the free end, and the integral sum along the length direction is not zero. However, the smaller the elastic modulus of the second layer material is, the smaller the value of the shear stress at the free end is, and the earlier the change direction occurs with the decrease of the dimensionless axial coordinate.

[0280] To further study the factors affecting the interface strength reliability, the layer number of the composite beam is set to 2, the first layer material is set to silicon, the thickness is set to 1.0 μm, the second layer material thickness is set to 1.5 μm, the Poisson's ratio v = 0.28, the elastic modulus E ∈ 0 ~ 200 GPa is changed, and the structural schematic diagram is as shown in Figure 5 (d). From Figure 6 (c) The relationship between the critical compressive strain of interface slip failure and the elastic modulus of the second layer thin film material. The solid line in the figure shows the critical strain predicted by the theory in this chapter. As can be seen from the figure, the smaller the critical strain that the system can withstand increases with the elastic modulus, and the structure is more prone to slip failure. The pentagram-shaped data points are the critical strains measured in the experiment about the interface slip failure of the 2.5 μm silicon thin film. The pentagram-shaped data points are located on the solid line, indicating that the theoretical prediction and the experiment are in good agreement, proving the correctness of the slip failure evaluation method of the composite thin film-substrate structure.

Claims

1. A method for assessing slip failure in a composite thin film-substrate structure, characterized in that, Includes the following steps: Step 1: Obtain the constitutive relation of the composite thin film based on the traction-free model; Step 1-1: Specify the geometric and mechanical parameters involved in the composite thin film-substrate structure: The thickness, plane strain modulus, and Poisson's ratio of the thin film, substrate, and adhesive layer are expressed as: h f , ν f h s , ν s and h a , ν a The subscripts f, s, and a represent the thin film, substrate, and adhesive layer, respectively; the torque, shear stress, and axial stress of the thin film and substrate are represented by M. i Q i and N i , i = f or s means that the lengths of the thin film and the substrate are l and L, respectively, l << L; Steps 1-2: The composite film is considered as an n-layer composite beam bonded together, which is subjected to a uniformly distributed axial force f at the boundary. x and f z and bending moment m x and m z Thickness t of each layer of the composite film j With coordinate y j The relationship is t j =y j -y j-1 Under tensile and bending loads, the principal strains are along the coordinate directions, and the strains are obtained through Kirchhoff's assumptions: Where, ε x0 and ε z0 It is the mid-plane of the composite beam along the x and z directions. Membrane strain at κ x and κ z These are the curvatures; the normal stress, given by the linear elastic constitutive relation, is: Where E is Young's modulus and ν is Poisson's ratio; therefore, the axial force and bending moment per unit length are obtained by integrating over the entire thickness of the composite beam, as shown in equation (3): Among them, y0, y n These represent the coordinates of the lower surface of the first-layer beam and the upper surface of the nth-layer beam, respectively. Steps 1-3: Combining equations (1), (2), and (3), the relationship between axial force / bending moment and membrane strain / curvature is given as follows: The values ​​of each element in the coefficient matrix are: In the formula, E i ,ν i and t i These are Young's modulus, Poisson's ratio, and thickness of the i-th layer material in the composite beam, respectively. Steps 1-4: For bending loads, there is no traction condition at the boundary perpendicular to the x-axis: f x =m x =0 (6) There is axial force and bending moment at the boundary perpendicular to the z-axis: Substituting equations (6) and (7) into equation (4), we get: Therefore, we get: The coefficient matrix is ​​the element of the inverse matrix of the coefficient matrix in equation (4); according to the knowledge of elasticity, we have: Combining equations (9) and (10), the constitutive relation of the composite beam under bending load is obtained as follows: Step 2: Obtain the equilibrium equations and governing equations of the composite thin film-substrate structure before slippage; Step 2-1: The equilibrium equation for the soft substrate and the hard film can be written as: Where σ and τ are the interface normal stress and shear stress, respectively; Step 2-2: The constitutive relations for axial elongation and bending of the soft substrate are written as follows: Among them, u s and w s These are the axial displacement and deflection components, respectively; Step 2-3: In the normal direction, the interfacial normal stress is expressed as: Shear stress is also related to the displacement components of the film and the substrate: The first and second parts of the intermediate layer represent the axial displacements of the bottom surface of the thin film and the top surface of the substrate, respectively. It is the shear modulus of the adhesive layer; by combining the fourth and third derivatives of equations (15) and (16) with equations (11)-(14), the coupled differential equations of the interface stress are established as follows: Substituting the above equation, we get Among them, the values ​​of each coefficient are Combining equations (19) and (20) to eliminate the normal stress, we obtain the differential expression for the shear stress: Where ξ=x / h a The coefficients are: For flexible electronic devices with composite thin film-soft substrate structure, k1 >> k2, k3 are satisfied, so the last term of equation (22) is ignored; Step 3: Solve for the stress distribution and maximum interface stress before slip; Step 3-1: Since the shear stress is known to decrease rapidly from the free edge x = l / 2, the approximate solution is expressed as follows: The superscript 0 represents the stress field before slip occurs, A1 and A2 are constants determined by the boundary conditions, and: Ignoring higher-order infinitesimal terms, the expression for the differential of shear stress in the approximate equation (24) is as follows: When x→l / 2 (27) Where A = λ1A2 + λ2A1; Substituting equation (27) into equation (17), the normal stress is: Where B and C are constants determined by the boundary conditions, and: Step 3-2: In the composite film-soft substrate structure, one end of the substrate is subjected to a compressive displacement load dL, while the other end is fixed. The compressive strain of the system is defined as |ε|=dL / L; the elastic stability theory gives the critical buckling strain: This represents the minimum applied strain that induces buckling; simultaneously, since the length of the film is negligible compared to the bending curvature, the moment and axial force of the delamination are considered constant, respectively: Where K(·) is the first kind of complete elliptic integral, α is the maximum ramp rotation angle determined by the following formula: The boundary conditions at the free edges of the thin film and the substrate are then obtained as follows: Due to symmetry, the boundary conditions for normal equilibrium are: The following boundary conditions are obtained: Step 3-3: Substitute the boundary conditions (36)-(39) into the expressions (24) and (28) for the interface shear stress and normal stress, and ignore the higher-order terms (O(h)). s Solve for the indeterminate constants A1, A2, B, and C: in Then the maximum stress is calculated as follows: Step 4: Determine the maximum shear stress at the interface Has the shear strength τ been reached? c This allows for an assessment of whether the interface is malfunctioning. When the maximum shear stress at the interface The shear strength τ increases with increasing compressive strain of the system. c When the composite film slips, the corresponding compressive strain is the critical strain at which slip occurs. if Composite thin film-substrate structure under compressive strain At that time, slip failure will not occur; if Composite thin film-substrate structure under compressive strain At this time, slip failure will occur.

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