An adhesion model for plane-strain shearable hyperelastic beams

Mechanics Research Communications 90 (2018) 42–46

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An adhesion model for plane-strain shearable hyperelastic beams Liwen He a, Jia Lou a, Jianbin Chen a, Aibing Zhang a, Jie Yang b,∗ a b

Department of Mechanics and Engineering Science, Ningbo University, Ningbo, Zhejiang 315211, China School of Engineering, RMIT University, Bundoora, VIC 3083 Australia

a r t i c l e

i n f o

Article history: Received 26 January 2018 Revised 27 April 2018 Accepted 27 April 2018 Available online 30 April 2018 Keywords: Adhesion Hyperelastic Soft materials Peeling criterion First integral

a b s t r a c t In the present work, a new adhesion model is proposed to analyze the peeling behavior of plane-strain shearable hyperelastic beams from a rigid flat substrate. The large strain effect, the bending effect and the transverse shear effect are all taken into account in the model. The variational method is utilized to derive the equilibrium equations and associated boundary conditions, including one that physically means the local peeling (fracture) criterion. A first integral is found for such kind of beams and is also used to derive an equivalent global peeling criterion. It is proven that the critical peeling force for the steady peeling of such shearable hyperelastic beams is the same as that for hyperelastic thin films with membrane approximation. The effect of pre-stretch on the peeling behavior is further considered. The developed model will contribute to the modeling and understanding of the adhesion and fracture behaviors of soft structures and biomimetic adhesives. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Detachment of thin, flexible films by peeling is a ubiquitous phenomenon of practical importance to a wide range of problems. Examples include the fabrication and reliability of multifunctional layered components [5,6], adhesive tapes used to fix objects in place [11,39,40], transfer printing of micro/nano-scale materials and devices from one substrate to another [9,37,41], the ability of plants and animals to cling to surfaces [4,24,30,34], and the achievement of physiological functions of tissues involving cell contact, adhesion and mechanotransduction [10,31,35]. Peeling mechanics of thin films has been extensively studied [2–4,7,8,12,18,19,21,26–30,33,34]. Many theoretical works were conducted via employing the inextensible elastica model [8,18,26] or extensible elastica model [17,19,27,28]. The finite rotation of the peeled film can be described by such kind of models, while inextensibility or small strain assumption is adopted in these models. For the case of small-angle peeling with a moderate interfacial adhesion energy or large-angle peeling with a strong interfacial adhesion energy, large strain probably occurs in the peeled film, and thus the elastica-based adhesion models fail to accurately describe the peeling behavior. The detachment of hyperelastic membranes from a flat substrate has also been studied by adopting the membrane approximation [2,7,12,38]. These adhesion models adopt hyperelastic constitutive relations and account for the large strain ef-

∗

Corresponding author. E-mail addresses: [email protected] (L. He), [email protected] (J. Yang).

https://doi.org/10.1016/j.mechrescom.2018.04.010 0093-6413/© 2018 Elsevier Ltd. All rights reserved.

fect. However, the bending effect (bending resistance) is neglected in such types of models, which may be of great importance in some cases [18,19,27,34]. As far as we know, the bending effect and the large strain effect have not been simultaneously considered in a single theoretical adhesion model. It is known that hyperelastic beam models account for the bending deformation, while most hyperelastic beam models do not consider the variation of the cross-section of the beam under large strain [1,36]. Thus the large strain effect is not accurately described in these models. In a recent work [15], we proposed a new finite strain beam model which accounts for the thickness stretchability (with the plane strain assumption). Hence, both the bending effect and the large strain effect are captured in this model. Based on this model, an adhesion model was developed to describe the peeling behavior of Euler-type hyperelastic beams [13]. However, the transverse shear effect is neglected in the adhesion model. It is known that for a moderately thick beam, the shear effect has a significant effect on the mechanical behavior of the beam. Motivated by such a gap, we will incorporate the shear effect into the adhesion model, and consequently, develop a new adhesion model for shearable hyperelastic beams within the plane-strain context. The remainder of this paper is structured as follows. In Sections 2.1 and 2.2, the kinematics and constitutive relations for shearable hyperelastic beams are briefly presented. Based on these results and by using the variational method, the equilibrium equations and associated boundary conditions are derived in Section 2.3. In the subsequent subsection, a first integral is found for the equilibrium equation and it is then utilized to derive the global peeling criterion and also the critical force for steady peel-

L. He et al. / Mechanics Research Communications 90 (2018) 42–46

43

where ( ) represents derivative with respect to the coordinate X. Due to the shear deformation, the slanted angle θ of the tangent plane of the deformed geometrical mid plane is not the same as the cross-sectional rotation angle ϕ . The difference between them, denoted by α = θ − ϕ , is obviously the shear angle. The derivative ϕ  physically means the nominal bending curvature of the beam (not equal to the nominal curvature θ  of the deformed geometrical mid line) and it is denoted by a new symbol κ. Through detailed kinematic analysis [14], it can be found for incompressible materials that:

λX =



λ2 − 2κ Z .

(3)

 −1/2 λZ = λ2 cos2 α − 2κ Z . Fig. 1. Schematic figures for the reference and current configurations of a shearable hyperelastic beam lying on a rigid flat substrate and subjected to a peeling force.

ing. The pre-stretch effect is further considered in Section 2.5. At last, some conclusions are presented in Section 3. 2. Theoretical modeling 2.1. Kinematics A finite strain beam model was proposed for plane-strain shearable hyperelastic beams in a previous work [14]. We will use the model to establish a new adhesion model to describe the peeling behavior of a plane-strain shearable hyperelastic beam. It is assumed in that model that any planar cross-section of the beam remains planar after deformation, and the beam is transversely shearable. However, the rigid cross-section hypothesis usually adopted in the classical Timoshenko beam model is relaxed by considering the thickness stretchability. Moreover, for the sake of simplicity, the plane strain assumption is adopted. Thus for a initially straight hyperelastic beam with rectangular cross-section (the width and thickness denoted by B and H, respectively) as shown in Fig. 1, deformation only occurs in the O-XZ plane. The plane strain assumption is applicable to beams with stiff fiber constraint in the width direction [16]. The deformation of a plane-strain hyperelastic beam from an initial stress-free configuration, which is referred to as the reference configuration, can be described by a mapping x = χ(X), i.e., any material point denoted by its initial position X in the reference configuration is moved to a new position x. In a Cartesian coordinate system, it can be written as x = X + u(X, Z), y = Y, z = Z + w(X, Z), where u and w are the horizontal and vertical components of the displacement of any material point in the beam, respectively. According to the aforementioned deformation hypothesis, we have the following expressions for the two displacement components:

u(X, Z ) = u0 (X ) − z∗ (X, Z )sin[ϕ (X )], w(X, Z ) = w0 (X ) + z∗ (X, Z ) cos [ϕ (X )] − Z,

(1)

where u0 and w0 are the displacement components of any point on Z the geometrical mid plane, z∗ = 0 λZ dZ , in which λZ is the stretch of any line element dZ and the absolute value of z∗ means the deformed distance between the material point (X, Y, Z) to the corresponding one (X, Y, 0) on the geometrical mid plane, and ϕ is the rotation angle of the cross-section. The slanted angle and stretch of any line element dX on the deformed geometrical mid plane are denoted by θ (X) and λ(X), respectively. According to the geometric relation as shown in Fig. 1, it is easy to find that:

θ = arctan

w0 ,λ = 1 + u 0



1 + u0

2

+ w02 ,

(2)

I1 = trC = λ2 − 2κ Z +



λ2 cos2 α − 2κ Z

(4)

−1

+1

(5)

where λX is the stretch of any (initially horizontal) line element dX. It is noted that for the present homogeneous plane strain beam, we also have I2 = I1 , I3 = 1, where Ik (k = 1, 2, 3) are the principal invariants of the right Cauchy-Green deformation tensor C = FT F with F = ∂ x/∂ X the deformation gradient. 2.2. Constitutive equations For the studied finite strain beam, the strain energy per unit reference length is defined by

φ (λ, α , κ ) =



A

W dA,

(6)

where W is the strain energy per unit reference volume of the beam, and the area integral is over the referential (undeformed) cross-section of the beam. It is noted that Simo [36] derived the constitutive relations for spatial rods based on the 3D finite deformation theory and the assumed rigid planar cross-section hypothesis. Following Simo’s method, and neglecting some additional strain energies due to thickness stretching, which are very small compared with the dominant stretching, shear and bending energies, we obtain the energy formula for shearable and thickness stretchable beams,

δφ = Nn δλn + Ns δγ + Mδκ ,

(7)

which is the same with that given by Simo [36] and also by Ressiner [32]. Eq. (7) is equivalent to the following constitutive equations:

∂φ (λ, α , κ ), ∂ λn ∂φ Ns = (λ, α , κ ), ∂γ ∂φ M= (λ, α , κ ), ∂κ

Nn =

(8)

where λn = λcos α − 1 and γ = λsin α are the normal strain and shear strain on the deformed planar cross-section (whose rotation angle is ϕ ), and Nn , Ns and M are the normal stress resultant, shear stress resultant and bending moment, respectively, on the same cross-section. Eq. (8)1 – 3 shows that the generalized forces Nn , Ns and M on the cross-section are respectively work conjugated to the corresponding generalized strains λn , γ and κ . With the substitution of the expressions for λn and γ into Eq. (7), we also have:

δφ = T δλ + Sλδα + Mδκ ,

(9)

44

L. He et al. / Mechanics Research Communications 90 (2018) 42–46

where T = Nn cos α + Ns sin α and S = Ns cos α − Nn sin α comes from another kind of decomposition of the stress resultant vector on the deformed cross-section, and they are the tangent and normal components relative to the deformed geometrical mid plane, respectively. Therefore, we have another form of constitutive equations for shearable hyperelastic beams:

∂φ T = (λ, α , κ ), ∂λ ∂φ S= (λ, α , κ ), λ∂α ∂φ M= (λ, α , κ ), ∂κ

δ Etot =



L a

+

(T − Fx cos θ − Fy sin θ )δλdX



+

(10)

 1 κ 2H2   φ 1 2 1 λ + 2 2 −2 + = + O ξ4 μBH 2 6 λ6 cos6 α λ cos α   + O ξ4 , 7 6 λ cos α 

  S 1 κ 2H2 = tan α + + O ξ4 , μBH λ3 cos2 α λ7 cos6 α   M 1 κ = + O ξ4 , 3 6 6 3 λ cos α μB H 1 λ− 3 2 − λ cos α

L

(S + Fx sin θ − Fy cos θ )λδα dX

a



which state that the tangent force T is work conjugated to the stretch λ, and the quantity λS is work conjugated to the shear angle α . By using the general constitutive relations (6) and (10) and the kinematic relation (5), the specific constitutive equations for shearable neoHookean beams can be approximately written as [14]:

T = μBH

where ω is the adhesion work per unit reference length at the interface between the beam and the substrate. It is assumed that the attached part does not slide relative to the substrate, i.e., the no-slipping assumption is adopted. Substituting Eqs. (14), (15) and (16) into Eq. (13), we obtain:

L

[−M + λ(Fx sin θ − Fy cos θ )]δϕ dX + Mδϕ |a L

a

− [Fx (1 − λ cos θ ) − Fy λ sin θ + ω + φ ]δ a

(17)

Considering the arbitrariness of independent kinematic variables λ, α , ϕ and a, and using the relation δϕ |a = δϕ a − ϕ  δ a (i.e., the variation of ϕ a has two contributions, the one δϕ |a is from the variation of the function itself at an assumed fixed boundary a and the other ϕ  δ a is due to the variation δ a of the boundary), we derive the following Euler-Lagrangian equations:

(11)

T = Fx cos θ + Fy sin θ , S = Fy cos θ − Fx sin θ ,

κ 2H2

M + λS = 0

(18)

and associated boundary conditions:

θ = θr , at X = a, Mκ + T λ − φ − Fx = ω, at X = a, M = 0, at X = L,

(12)

(19)

The principle of minimum energy states that the total free energy of the beam-substrate system (as shown in Fig. 1), consisting of the elastic energy Eela , the external potential energy Eext and the adhesion energy Ead , reaches the minimum value when the system is at equilibrium. By the principle, we have:

where θ r is the root rotation of the beam at the peeling front (X = a). From the assumed boundary condition ϕ a = 0, it can be inferred that θ r = α (a), i.e., the root rotation θ r at the peeling front is due to the shear deformation at the peeling front. It is also noted that from a fracture mechanics point of view, Eq. (19) 2 can be interpreted as a local peeling criterion (or fracture criterion), i.e., the energy release rate G (the left-hand side of Eq. (19)2 ) is equal to the adhesion work ω (or fracture energy) when the peeling front (or crack front) at X = a neither propagates nor recedes. If the beam is assumed to be unshearable, then we have θ r = α (a) = 0, and ϕ = θ . Hence, the governing Eqs. (18) and associated boundary conditions (19) reduce to the following form:

δ Etot = δ Eela + δ Eext + δ Ead = 0.

T = Fx cos θ + Fy sin θ , M + λS = 0,

κ H , which is usually much smaller than 1 even in where ξ = λ2 cos 2α the case of large rotation.

2.3. Variational derivation of governing equations and boundary conditions

(13)

The variation of the elastic energy of the beam can be written as:



δ Eela = δ  =

a

L

a L

θ = 0, at X = a, Mθ  + Fx (λ − 1 ) − φ = ω, at X = a,

φ dX

M = 0, at X = L,

(T δλ + Sλδα + Mδκ )dX − φδ a.

(14)

where a is the referential length of the adherent part of the beam and L is the total referential length of the whole hyperelastic beam. Here, it is assumed that the attached part is undeformed. The effect of possible pre-stretch in the attached part will be discussed in Section 2.5. The variation of external potential energy of the beam is:



δ Eext = −Fx δ a +



L a



λ cos θ dX − Fy δ



a

L



λ sin θ dX ,

(15)

where Fx and Fy are the horizontal and vertical components of the  peeling force at the end X = L, respectively. We also use F = Fx2 + Fy2 , θF = arctan(Fy /Fx ) to denote the magnitude and direction of the peeling force. The adhesion energy of the beam-substrate system is [20]:

δ Es = ωδ (L − a ),

(16)

(20)

(21)

where S = Fy cos θ − Fx sin θ . It is found that the present adhesion model for shearable hyperelastic beams can be degenerated to that for the unshearable case [13]. As discussed in that work, the adhesion model for unshearable hyperelastic beams can be further degenerated to a corresponding adhesion model based on the extensible elastica theory [19,27] in the infinitesimal strain limit or an adhesion model based on membrane approximation for hyperelastic thin films [7]. 2.4. A first integral and a global peeling criterion Integrating the moment equilibrium Eq. (18)3 multiplied by ϕ  and using integration by parts, we have



X2

X1



ϕ  dX = (Mκ + T λ) XX21



M + λS

 −

X2 X1

(Mdκ + T dλ + λSdα ) = 0

(22)

L. He et al. / Mechanics Research Communications 90 (2018) 42–46

where ϕ = θ − α , S = Fy cos θ − Fx sin θ and T = Fx cos θ + Fy sin θ have been used. With the work conjugation relations (10)1 – 3 , Eq. (22) can be simplified to be:

Mκ + T λ − φ = constant.

(23)

Thus the expression (Mκ + Tλ − φ ) is a first integral of the moment equilibrium equation for hyperelastic Timoshenko beams. It is noted that the first integral physically means the complementary strain energy density per unit reference length of the beam. In fact, such a conservation law is a general form of the well-known conservation of material momentum in rod theories, which is usually based on the rigid cross-section assumption [25]. The first integral has been extensively used to obtain solutions of integral form for inextensible elastica [18] and extensible elastica [19,23,25,27] within the context of inextensibility or small strain. The present work clearly shows that the first integral also exists for plane-strain shearable hyperelastic beams besides plane-strain unshearable hyperelastic beams as demonstrated in a previous work [13]. By the first integral and the boundary condition (19)3 , the local peeling criterion (19)2 is equivalent to the following “global” peeling criterion:

T λ − φs (λ, α ) − F cos θF = ω, at X = L.

(24)

where φ s (λ,α ) = φ (λ, α , 0). In the special case of steady peeling, θ (L) = θ F , T(L) = F, and consequently the critical condition for steady peeling is derived:

F (λ − cos θF ) − φm (λ ) = ω,

(25)

where φ m (λ) = φ s (λ,0) = φ (λ, 0, 0) and λ depends on the peeling force through dφm /dλ = F . This is the same with the corresponding condition for plane-strain unshearable hyperelastic beams [13] and also the one for hyperelastic thin films with membrane approximation [7], i.e., neither the bending effect nor transverse shear effect affects the steady peeling condition of a shearable hyperelastic beam subjected to plane strain. 2.5. Pre-stretch effect When pre-stretch exists in the attached part of the beam, by using the variational method as well as the no-slipping condition, we obtain the same governing equation and boundary conditions for the peeled part of the beam, except that the local peeling condition (19)2 should be replaced by the following equation:

[[Mκ + T λ − φ ]] = ω,

(26)

where [[]] = ( )|a+ −( )|a− denotes the jump of the bracketed quantity at the peeling front, with “a + ” and “a − ” representing the limits towards the peeling front from the peeled and attached parts, respectively. Thus, the well-known “jump condition” in rod theories based on the rigid cross-section assumption [25] still holds true in the present more general context. It is also found that the energy release rate as given on the left-hand side of Eq. (26) has a similar form to that presented by Srivastava and Hui [38]. The main differences stem from two aspects. One is that the large strain effect was considered in their work by using the membrane approximation, while the bending and shearing effects are further accounted for in the present work. The other is that the energy release rate and the adhesion work in the present work are measured per unit reference length rather than per unit current length in their work. Similarly, by using the first integral, a global peeling condition can be easily obtained:

[T λ − φs (λ, α )]X=L − [Mκ + T λ − φ (λ, α , κ )]X=a− = ω.

(27)

45

For the steady peeling of a pre-stretched hyperelastic beam from a rigid flat substrate, we have θ (L) = θ F , T(L) = F and consequently,

F (λL − λ pre cos θF ) − [φm (λL ) − φm (λ pre )] = ω

(28)

= λ(a − ),

where λpre λL = λ(L) satisfies dφm /dλ = F , and it is assumed here that κ (a − ) = 0 and α (a − ) = θ (a − ) = 0. It is evident that the critical stretch λL and thus the critical peeling force for the no-slipping steady peeling of hyperelastic beams from a rigid flat substrate can be determined from Eq. (28). This peeling condition is completely the same with that given by Begley et al., who derived the condition with the membrane approximation [2]. It is also easy to check that Eq. (28) can be degenerated to Eq. (25) for the case without pre-stretch by simply letting λpre be 1 and considering that φ m (1) = 0. At last, it is pointed out that not only the analytical formulae (25) and (28) can be used to predict the steady peeling behavior of plane-strain shearable hyperelastic beams, the boundary value problem formulated in the present work, i.e., Eqs. (18) and (19) (or Eqs. (18) and (19) with Eq. (19) 2 replaced by Eq. (26)) could be applied to analyze the adhesion behavior, such as unsteady peeling, of plane-strain shearable hyperelastic beams. For instance, the finite difference method can be adopted to solve such kind of boundary value problems as demonstrated in our previous works ([14,15]) and thus to clarify the stability of adhesion. 3. Conclusion A novel adhesion model is developed for plane-strain shearable hyperelastic beams in the present work by using a recently developed transversely shearable and thickness stretchable finite strain beam model and adopting the adhesion energy concept. Via the standard variational method, the equilibrium equations and boundary conditions, including one physically means the local peeling criterion, are derived. By inspecting the moment equilibrium equation, a first integral (also called conservation law of material momentum) is found in the present context and it is used to derive an equivalent global peeling criterion. Moreover, the condition for the steady peeling of such kind of beams is easily obtained. The effect of pre-stretch on the peeling behavior is further considered. The present adhesion model is expected to be useful for the analyses and characterization of the adhesion and debonding behaviors of soft structures and adhesives. It should also be noted that there are several limitations for the present adhesion model. Firstly, dissipative processes, such as viscoelastic [8] and plastic effects are not taken into account in the model, and actually a further extension to account for anyone of these effects is possible. Secondly, the adhesion energy concept and a Timoshenko-type kinematic hypothesis with a relaxation of the rigid cross-section assumption is adopted in the present model, which may limit its applications in some cases. In a more general context, the finite deformation elasticity theory [22] augmented with a cohesion–separation law may be needed to accurately clarify the adhesion and fracture behaviors of soft materials and structures. Acknowledgment The present work is supported by National Natural Science Foundation of China (Grant Nos. 11602118 and 11602117) and also sponsored by K.C. Wong Magna Fund in Ningbo University. References [1] M.M. Attard, Finite strain–beam theory, IJSS 40 (2003) 4563–4584. [2] M.R. Begley, R.R. Collino, J.N. Israelachvili, R.M. McMeeking, Peeling of a tape with large deformations and frictional sliding, J. Mech. Phys. Solids 61 (2013) 1265–1279.

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