## Abstract

This paper investigates the effects of large deflections on the energy release rate and mode partitioning of face/core debonds for the single cantilever beam sandwich composite testing configuration, which is loaded with an applied shear force and/or a bending moment. Studies in this topic have been done by employing geometrically linear theories (either Euler–Bernoulli or Timoshenko beam theory). This assumes that the deflection at the tip of the loaded debonded part is small, which is not always the case. To address this effect, we employ the elastica theory, which is a nonlinear theory, for the debonded part. An elastic foundation analysis and the linear Euler Bernoulli theory are employed for the “joined” section where a series of springs exists between the face and the substrate (core and bottom face). A closed form expression for the energy release rate is derived by the use of J-integral. Another closed-form expression for the energy release rate is derived from the energy released by a differential spring as the debond propagates. Furthermore, a mode partitioning angle is defined based on the displacement field solution. Results for a range of core materials are in very good agreement with the corresponding ones from a finite element analysis. The results show that large deflection effects reduce the energy release rate but do not have a noteworthy effect on the mode partitioning. A small deflection assumption can significantly overestimate the energy release rate for relatively large applied loads and/or relatively long debonds.

## 1 Introduction

In a sandwich composite, the onset of an interface crack indicates an imperfection/defect that may compromise the integrity of a structure. The initiation of such a crack, referred to herein as “debond,” is caused by weakening adhesion between dissimilar materials (i.e., face and core). Once a crack forms, the resultant stresses and moments redistribute unevenly across the interface, leading to possible crack propagation once critical conditions are attained. This process continues until the sandwich composite completely delaminates.

A sandwich panel is one type of a composite structure that consists of two thin faces (skins) separated by a relatively thick core, i.e., it is a tri-material. Compared to a bulk material, a sandwich panel offers the benefits of high stiffness-to-weight ratio, impact absorption, etc., making it a very good structural concept for aerospace, military, automotive, and renewable energy applications. The weak link is its vulnerability to mechanical failure caused by an interface debond. Such debonds can be caused by subpar manufacturing quality (i.e., trapped air bubbles, adhesive irregularities, or insufficient cure time) [1] or in-service conditions [2]. Other failure mechanisms include shear yielding/cracking of the core, face wrinkling (local buckling), and in some cases, global buckling [3].

For a monolithic composite, Yin and Wang [4] derived a closed-form expression for the energy release rate of a delamination in the homogeneous solid. In contrast, Suo and Hutchinson [5] did the same for a bi-material interface crack. The study also analyzed the homogeneous material as a limiting case. In a separate study, Kardomateas et al. [6] expanded Suo and Hutchinson’s framework for a tri-material, since this is what a sandwich structure is. The general consideration for Kardomateas’s study was an asymmetric sandwich composite in which the thickness and material of the top and bottom face sheets are not necessarily the same. The limiting solution for the case of delaminated homogeneous bi-material was also retrieved and discussed. All these solutions were based on the assumption of vanishing shear loads. Note that the complexity of the analysis increases significantly with the inclusion of shear effects. Subsequently, Li et al. [7] included shear by expanding Suo’s framework with constants obtained from finite elements. In a more recent study, Barbieri et al. [8] developed a model for a crack at a face/core interface with shear effects. The configuration was that of a single cantilever beam (SCB) and the fracture parameters were expressed in hybrid forms as a combination of analytical and numerical components.

All previous studies provided solutions that depended on one or more constants that are determined numerically (typically from finite elements). For a fully closed-form solution, a beam theory combined with an elastic foundation (EF) approach has been pursued in the past. Kanninen [9] first applied the elastic foundation approach, without considering shear effects, to analyze crack propagation. In the study, the bonded section of a double cantilever beam (DCB) configuration was treated as resting on elastic springs. The crack was given a finite length and placed at a homogeneous material’s mid-thickness. An Euler–Bernoulli beam theory was then applied to describe its deformation mathematically. Because of the crack placement and the homogeneous material assumption, this work only derived the energy release rate and the mode mixity was not of concern.

In a similar test configuration, Kanninen [10] performed a dynamic analysis to investigate the rate of crack growth and the mechanisms of crack arrest. The effort led to the development of a generalized elastic foundation formula that also accounted for shear effects. To this extent, Kanninen applied the Timoshenko beam theory to the equation of motion that describes crack behavior and the energy analysis of the entire structure. Subsequently, Williams [11] emulated Kanninen’s model and used the Timoshenko beam theory to study the crack root rotation in an orthotropic homogeneous material. Results from the studies show that an elastic foundation model can yield accurate solutions for the energy release rate and underscored the importance of including shear effects. However, the mid-thickness placement of the crack in a homogeneous solid was also a limitation to the study. Even so, later studies by Li and Carlsson [12] and Saseendran et al. [13] demonstrated the use of the elastic foundation analysis in a bi- and tri-material of an SCB sandwich structure.

In all these previous studies with the elastic foundation approach, the mode mixity was obtained numerically and the foundation modulus was fitted to numerical data. In a recent study, Kardomateas et al. [14] addressed this void by deriving a closed-form solution for the foundation modulus in a sandwich composite from the theory of elasticity. This solution was also compared to that from the extended high-order sandwich panel theory [15]. These formulas for the elastic foundation constants were then applied to study the debond in a DCB configuration, in which the debonded layer and the substrate part are both loaded in bending and shear [16], and, subsequently, in a SCB configuration, in which only the debonded face is loaded [17]. In addition to the energy release rate, a closed-form expression for the mode partitioning was also derived in these two papers, by making use of the normal and shear displacements at the beginning of the elastic foundation. The analysis was done based on the Euler–Bernoulli beam theory and the effect of transverse shear was accounted for using an approximate shear correction angle. More recently, Niranjan Babu and Kardomateas [18] proposed a model that captured the transverse shear effect in these elastic foundation approaches by use of the Timoshenko beam theory; in this paper, a DCB configuration was employed.

Despite the robustness of the beam theories, their accuracy is limited to the linear regime, which means sufficiently low applied loads. It is an open question how high loads, which cause large deflections of the debonded face, would affect the energy release rate and mode partitioning of the debond. As such, the material behavior is then characterized by a nonlinear elastic deflection, thereby making the linear beam theories inadequate. By contrast, a large deflection theory can accurately capture a beam’s deformation if the applied loads are within the beam’s load-bearing capacity. On this topic, most works have been in connection with the post-buckling behavior of structures. Indeed, Kadormateas [19] applied a large deflection theory to investigate the large deformation effect on the post-buckling behavior of thin delaminations in homogeneous composites. Results revealed that large deformation effects lead to a more growth resistant system, in that the energy release rate is lower than the corresponding linear theory.

In this paper, the effect of a large deflection of the debonded face in a sandwich composite is studied by use of the elastica theory. An elastic foundation approach is used for the intact part of the beam. The displacements in the latter part are still small and, therefore, the Euler–Bernoulli theory is used. The continuity conditions at the common section are used to formulate the problem. The energy release rate is derived from the J-integral and the mode partitioning from the displacement field. Results for a range of core stiffnesses and load combinations are tabulated and compared to finite element results obtained from both small and large deflection analyses. Notice that this formulation is applicable to orthotropic faces and core, not only isotropic.

## 2 Formulation

The single cantilever beam specimen (SCB) for sandwich structural testing is shown in Fig. 1 and consists of a sandwich beam with a top face/core debond (crack) of length *a*. The total length of the specimen is *L*, thus the ligament (uncracked) length is *l* = *L* − *a*. The sandwich construction consists of top and bottom faces of thickness *f*_{t} and *f*_{b}, with extensional Young’s modulus (along *x*) *E*_{ft} and *E*_{fb}, respectively (assumed to be equal in tension and compression), a core of thickness 2*c* and of Young’s modulus in the transverse direction (along *z*), *E*_{c}, again assumed to be equal in tension and compression, and shear modulus *G*_{c}. The width of the sandwich beam is *b*. The analysis that follows employs the nonlinear elastica theory for the debonded portion of the top face and the linear Euler–Bernoulli theory for the joined (intact) portion of the top face whereas the substrate’s contribution is represented by the elastic foundation, which mainly depends on *E*_{c}. Therefore, the formulation is such that both faces and core can be either isotropic or orthotropic. Also, it should be noted that, regarding the core, the transverse compressibility and the shear compliance are the properties that are of importance. The specimen can be generally loaded with a shear force, *P*, and a moment, *M*_{0}, which are applied at the left end of the debond. The specimen is supported so that the transverse displacement of the bottom edge is zero. Notice that in this paper we focus on the overall behavior of a sandwich beam with an interfacial debond, thus the debond length can be arbitrary. In the following, we shall use “*d*” to denote the debonded part of the face (of thickness *f*_{t}). We shall also use “*f*” when referring to the top face properties.

*a*)

*s*gives

*b*)

*dθ*and integrating gives

*c*)

*d*)

*C*being a constant of integration.

*M*

_{0}and the angle

*β*so

*e*)

*f*)

*d*)

*a*)

*θ*decreases with increasing

*s*, i.e.,

*dθ*/

*ds*is negative (Fig. 2), we have

*b*)

*θ*

_{0}the slope at the point where the debond ends (and the elastic foundation begins). Assuming incompressibility, integrating (3

*b*) over the entire debond, would give the length of the debond: $a=\u222b\beta \theta 0ds$, i.e.,

*θ*

_{0}is the slope at the end of the debond.

Thus, the debonded part deflection is expressed in terms of two unknowns, namely, the slopes at the loaded tip and the end of the debond, *β* and *θ*_{0}, respectively.

For the jointed (intact) part of the top face, the coordinate system is set so that *x* = 0 is at the end of the debond, i.e., the debond is for negative *x* and the intact part is for positive *x* (Fig. 1). We denote by *w*_{f}, *u*_{f} the transverse and axial displacements, respectively, in this region.

*a*)

*k*

_{n}is the modulus of the elastic foundation.

*b*)

*x*> 0 in the form

*c*)

*x*, 2 ≡

*y*, and 3 ≡

*z*:

*d*)

*e*)

*x*, 2 ≡

*y*, and 3 ≡

*z*; also, note that

*E*

_{c}is the transverse (along

*z*) extensional modulus of the core and the

*ν*’s are the Poisson’s ratios of the core.

*x*= 0, which is the common section for the debonded and the intact part, we have the following conditions:

- Slope continuity at the beginning of the elastic foundation, $\u2212\theta 0=wf\u2032\u2223x=0$, which gives from (6
*c*)(7$\u2212\theta 0=\lambda (C2+C3)$*a*) - Shear force equality at the beginning of the elastic foundation, $\u2212P=\u2212(EI)fwf\u2033\u2032\u2223x=0$, which gives from (6
*c*)(7$P=(EI)f2\lambda 3(C2\u2212C3)$*c*)Further conditions are at the right end of the face. Specifically, the conditions at

*x*=*l*are of zero moment and shear. - The condition of zero moment at
*x*=*l*iswhich, from (6(8$(EI)fd2wfdx2\u2223x=l=0$*a*)*c*), becomes(8$C1sinh\lambda lsin\lambda l\u2212C2sinh\lambda lcos\lambda l+C3cosh\lambda lsin\lambda l\u2212C4cosh\lambda lcos\lambda l=0$*b*)The above equation contains hyperbolic sin and cos functions, which can quickly become very large numbers, and thus lead the numerical calculations to failure. Therefore, one important step is to divide (8*b*) by cosh*λl*, which would result in hyperbolic tan functions, which are bounded; thus we obtain(8$C1tanh\lambda lsin\lambda l\u2212C2tanh\lambda lcos\lambda l+C3sin\lambda l\u2212C4cos\lambda l=0$*c*) - The condition of zero shear at
*x*=*l*isand, from (6(9$(EI)fd3wfdx3\u2223x=l=0$*a*)*c*), it leads to(9$C1(cosh\lambda lsin\lambda l+sinh\lambda lcos\lambda l)\u2212C2(cosh\lambda lcos\lambda l\u2212sinh\lambda lsin\lambda l)+C3(sinh\lambda lsin\lambda l+cosh\lambda lcos\lambda l)\u2212C4(sinh\lambda lcos\lambda l\u2212cosh\lambda lsin\lambda l)=0$*b*)Again, we divide (9*b*) by cosh*λl*, in order to avoid numerical calculation failure; thus we obtainEquations (4), (7(9$C1(sin\lambda l+tanh\lambda lcos\lambda l)\u2212C2(cos\lambda l\u2212tanh\lambda lsin\lambda l)+C3(tanh\lambda lsin\lambda l+cos\lambda l)\u2212C4(tanh\lambda lcos\lambda l\u2212sin\lambda l)=0$*d*)*a*)–(7*c*), (8*c*), and (9*c*) are six equations that can be solved for the six unknowns*β*,*θ*_{0},*C*_{1},*C*_{2},*C*_{3}, and*C*_{4}. Equations (4) and (7*b*) are nonlinear. We can reduce the problem to solving just the nonlinear equations as follows:

*c*) and (9

*c*) and solving for

*C*

_{1}and

*C*

_{4}, we can obtain

*C*

_{1}and

*C*

_{4}in terms of

*θ*

_{0}:

*a*)

*b*)

*b*) into (7

*b*) gives a nonlinear equation in terms of only

*β*and

*θ*

_{0}:

### 2.1 Special Case of Only Applied Moment.

*P*, is applied at the tip, i.e., only a moment,

*M*

_{0}, then from (2

*a*),

*a*)

*a*) over the entire debond would give the length of the debond: $a=\u222b\beta \theta 0ds$, i.e.,

*b*)

*c*) into (8

*c*) and (9

*c*), we can solve for

*θ*

_{0}and

*C*

_{1}:

*a*)

*b*)

*β*, can be determined from (13

*b*):

*c*)

## 3 *J*-Integral

*a*)

*T*

_{i}and

*u*

_{i}are the components of the traction vector and the displacement vector, respectively. We will choose the integration path Γ =

*BAA*′

*FED*′

*DCB*, as in Fig. 3, that follows the outer boundary of the structure on the three sides and on the fourth side it is infnitesimally close to the debond tip. We chose this path because on all sides we have small deflections and strains, thus the validity of the J-integral is assured.

On both the top and bottom horizontal segments of the path, *dz* = 0; therefore, the first term in the *J*-integral expression (15*a*) is zero. Furthermore, on the top horizontal segment of the path, $T\u2192=0\u2192$. Thus, at the top horizontal segment *J*_{CB} = 0.

*b*)

*w*is the transverse displacement and we use “,” to denote the partial derivation.

We have made the assumption of a fixed support (zero transverse displacement, *w*, throughout the segment *FE*), which also means that *w*_{,x} = 0. Moreover, since the reaction at the bottom edge is along the transverse direction only and no additional shear loads are applied, *τ*_{xz} is zero. Thus, on the bottom horizontal segment of the path, we also have *J*_{FE} = 0.

We thus only have the vertical sides of the path that could contribute to *J*.

*c*)

On the right vertical side, $T\u2192=0$, which means *σ*_{xx} = *τ*_{xz} = 0, and moreover for a plane stress assumption, *σ*_{zz} = 0, and for a plane strain assumption ($\epsilon zz=0$), *σ*_{zz} = *ν*_{xz}*σ*_{xx} = 0. Therefore, on these sides, it is also *W* = 0. As a result, on the vertical sides *ED*′ and *D*′*C*, we have *J* = 0.

*A*′

*F*does not contribute to the J-integral. Therefore, only the segment

*BA*(debonded part) contributes. On this segment,

*d*)

In the above equation, the normal stress and strain are created by the bending moment and the shear stress by the shear load. The corresponding shear strain is *γ*_{xz} = *κτ*_{xz}/*G*_{eq}, where *G*_{eq} is the equivalent shear modulus of the section and *κ* is the shear correction factor, which takes into account the non-uniform distribution of shear stresses. Since the top face is homogeneous, the equivalent shear modulus is the shear modulus of the top face, *G*_{eq} = *G*_{ft}, and the shear correction factor [21] is *κ*_{d} = 6/5.

*a*)

*A*

_{d}=

*bf*

_{t}is the cross-sectional area of the debonded section.

*BA*segment, on which

*dz*= −

*ds*(Fig. 3):

*b*)

*σ*

_{zz}= 0 and $\epsilon xx=\sigma xx/Eft$, thus

*c*)

*σ*

_{zz}=

*ν*

_{xz}

*σ*

_{xx}, therefore, $\epsilon xx=(\sigma xx\u2212\nu zx\sigma zz)/$$Eft=(1\u2212\nu zx\nu xz)\sigma xx/Eft$, and the first term in (16

*c*) should be multiplied by (1 −

*ν*

_{zx}

*ν*

_{xz}).

*BA*, gives (for a plane stress assumption)

*x*= 0 denotes the debond tip.

The above equation contains the *dw*_{f}/*dx* at the point where the elastic foundation starts (debond tip), which is the rotation of the beam on the elastic foundation at its beginning; considering the coordinate systems in Fig. 2, this is (−*θ*_{0}).

*M*

_{0}is applied,

*P*= 0 and

*M*

_{BA}=

*M*

_{0}, thus

*b*)

### 3.1 Rate of Energy Released by the Springs.

*da*, means that the corresponding spring,

*k*

_{n}

*da*, breaks, thus releasing the energy that was stored in it. The energy stored in the differential spring length,

*da*, is

*a*)

*b*)

*C*

_{1}is given by (11

*a*).

We shall compare this simple formula to the energy release rate, as determined previously by the *J*-integral.

## 4 Mode Partitioning

So far, the large deflection analysis of a composite sandwich debond has been limited to determining the energy release rate. Prediction of the debond growth necessitates determining another parameter, namely, the relative amounts of mode I and mode II. This is quantified by the mode mixity angle, which ranges from zero for mode I to ±90 for mode II. To this extent, for the case of no shear loading, Suo and Hutchinson [5] and Kardomateas et al. [6] used the complex stress intensity factor approach to obtain the mode mixity in terms of a single parameter (*ω*), which was left to be determined numerically. However, with shear loading, an expression in terms of a single parameter is not possible. Nevertheless, it is possible to use a superposition approach and express the mode mixity in terms of multiple parameters, dependent on the geometry and materials, which are determined numerically; this approach was pursued in Ref. [8]. Our goal in this paper is to provide a closed-form solution and employ an elastic foundation model. Thus, an alternative approach is pursued, which makes use of the displacements. This new approach was introduced by Kardomateas et al. [16], and it is based on the relative amounts of normal and shear displacements where the elastic foundation begins (i.e., the debond tip). Since this displacement-based approach for apportioning the modes I and II is different than the one based on the stress intensity factors, we shall use the name “mode partitioning” and we will reserve the name “mode mixity” for the fracture mechanics, stress intensity factor-based approach. Notice that displacements as an alternative approach to determine mode mixity have been used in bi-material fracture mechanics by Berggreen et al. [22]. However, the latter is based on the fracture mechanics singular field and thus it is different conceptually than our measure of mode partitioning, which is based on the elastic foundation model.

*w*

_{f0}, and axial,

*u*

_{f0}, displacements of the debonded part at the beginning of the elastic foundation, which is the end of the debond (debond tip) in the limit, are from Eq. (6

*c*) following the Euler–Bernoulli beam theory (notice that the positive slope is counter-clockwise):

*a*)

*b*)

*γ*=

*κV*/(

*G*

_{ft}

*A*

_{d}). Notice that according to Fig. 1, a positive shear would create a clockwise slope. Thus, the axial displacements at the face/core interface due to the shear (to be added to the

*u*

_{f0}) are

*c*)

*x*= 0) and for pure mode II, only axial (shearing) displacement occurs at

*x*= 0, a mode partitioning phase angle,

*ψ*

_{EF}, can be defined from the relative crack flank opening and shearing displacements,

*δ*

_{w}and

*δ*

_{u}, respectively, at the tip, i.e.,

*d*)

*e*)

*ψ*

_{EF}= 0 for pure mode I and

*ψ*

_{EF}= ±90 deg for pure mode II.

*a*)–(20

*c*) results in a closed-form expression for the mode partitioning angle, written as $\psi EFL$, to denote that it is from the elastic foundation approach (EF) and for large deflections (

*L*) as

*C*

_{2}and

*C*

_{3}are given in Eq. (10).

## 5 Results and Discussion

The large deflection analysis is applied to a symmetric sandwich SCB configuration with faces made out of isotropic aluminum with Young’s modulus *E*_{f} = 70 GPa and Poisson’s ratio *ν*_{f} = 0.3. Two cores are examined: (i) isotropic aluminum foam with Young’s modulus *E*_{c} = 7 GPa and Poisson’s ratio *ν*_{c} = 0.32 and (ii) isotropic H100 with *E*_{c} = 0.13 GPa and *ν*_{c} = 0.30. The results from this semi-closed-form analysis will be compared with the commercial finite element code abaqus [23], which, however, can only produce results for the modes I and II stress intensity factors of a bi-material crack when both materials are isotropic; this is the reason we have opted for isotropic faces and core.

Regarding the geometry, we shall examine a symmetric sandwich of length *L* = 500 mm, with top and bottom face thicknesses *f*_{t} = *f*_{b} = 2 mm, and core thickness 2*c* = 20 mm. We shall examine two debond lengths, a relatively long debond of length *a* = 200 mm and a small one of length *a* = 20 mm.

Regarding the finite element analysis (FEA), isoparametric eight-node bi-quadratic plane stress elements (CPS8R) were used to model the sandwich beam. A mesh base discretization used for the linear and nonlinear simulation are shown in Figs. 4(a) and 4(b). For the linear analysis, a coarse mesh is applied to both the faces and core. A finer mesh of quadratic plane stress triangle elements (CPS6R) is applied around the crack tip. The nonlinear analysis requires mesh refinement near the crack tip and also along the debonded top face. Also, the nonlinear analysis must be performed by activating the geometric nonlinearity feature in abaqus. abaqus uses the interaction integral method [24] to calculate the J-integral and the stress intensity factors *K*_{I} and *K*_{II} via a contour integral. Accordingly, the mode mixity from the FEA analysis is further calculated from

Care must be taken when assigning the incremental and total time to prevent non-convergences or inaccuracies in results. For a linear elastic analysis, a total time of even 1 s can yield good results. On the contrary, a nonlinear elastic analysis requires an extended total time ( ≥ 30 s) with smaller step size (0.1 s) for an accurate result and to overcome any non-convergence issues.

The solution of the nonlinear equations (4) and (12) for *β* and *θ*_{0} was achieved using the Newton’s numerical approach with starting values of zero.

Tables 1 and 3 show the energy release rate results for several loading cases. These loads include pure shear, pure moment, and a combination of both. Specifically, Table 1 is for an aluminum foam core (with Young’s modulus of *E*_{c} = 7 GPa), while Table 2 is for an H100 core (with Young’s modulus of *E*_{c} = 0.13 GPa).

Applied load | Small deflection | Large deflection | |||||
---|---|---|---|---|---|---|---|

P | M_{0} | J_{EF} | J_{FEA} | G_{spring} | $JEFL$ (N/mm) | $JFEAL$ | $GspringL$ |

(N) | (N mm) | (N/mm) [17] | (N/mm) | (N/mm) [17] | Eqs. (18a) and (18b) | (N/mm) | (N/mm) Eq. (19b) |

0.25 | 0.0 | 0.0277 | 0.0278 | 0.0275 | 0.0267 | 0.0273 | 0.0267 |

0.50 | 0.0 | 0.111 | 0.110 | 0.111 | 0.109 | 0.107 | 0.109 |

0.0 | 100 | 0.107 | 0.107 | 0.107 | 0.107 | 0.107 | 0.107 |

0.20 | 20 | 0.0395 | 0.0392 | 0.0395 | 0.0391 | 0.0388 | 0.0391 |

0.50 | 100 | 0.436 | 0.434 | 0.436 | 0.403 | 0.400 | 0.403 |

1.0 | 200 | 1.750 | 1.740 | 1.750 | 1.346 | 1.340 | 1.346 |

2.0 | 200 | 3.953 | 3.918 | 3.953 | 2.376 | 2.354 | 2.376 |

2.0 | 300 | 5.361 | 5.315 | 5.361 | 2.960 | 2.936 | 2.960 |

3.0 | 300 | 8.894 | 8.815 | 8.894 | 3.937 | 3.902 | 3.937 |

5.0 | 300 | 18.62 | 18.16 | 18.62 | 5.850 | 5.800 | 5.850 |

Applied load | Small deflection | Large deflection | |||||
---|---|---|---|---|---|---|---|

P | M_{0} | J_{EF} | J_{FEA} | G_{spring} | $JEFL$ (N/mm) | $JFEAL$ | $GspringL$ |

(N) | (N mm) | (N/mm) [17] | (N/mm) | (N/mm) [17] | Eqs. (18a) and (18b) | (N/mm) | (N/mm) Eq. (19b) |

0.25 | 0.0 | 0.0277 | 0.0278 | 0.0275 | 0.0267 | 0.0273 | 0.0267 |

0.50 | 0.0 | 0.111 | 0.110 | 0.111 | 0.109 | 0.107 | 0.109 |

0.0 | 100 | 0.107 | 0.107 | 0.107 | 0.107 | 0.107 | 0.107 |

0.20 | 20 | 0.0395 | 0.0392 | 0.0395 | 0.0391 | 0.0388 | 0.0391 |

0.50 | 100 | 0.436 | 0.434 | 0.436 | 0.403 | 0.400 | 0.403 |

1.0 | 200 | 1.750 | 1.740 | 1.750 | 1.346 | 1.340 | 1.346 |

2.0 | 200 | 3.953 | 3.918 | 3.953 | 2.376 | 2.354 | 2.376 |

2.0 | 300 | 5.361 | 5.315 | 5.361 | 2.960 | 2.936 | 2.960 |

3.0 | 300 | 8.894 | 8.815 | 8.894 | 3.937 | 3.902 | 3.937 |

5.0 | 300 | 18.62 | 18.16 | 18.62 | 5.850 | 5.800 | 5.850 |

Applied load | Small deflection | Large deflection | |||
---|---|---|---|---|---|

P | M_{0} | ψ_{EF} | ψ_{FEA} | $\psi EFL$ | $\psi FEAL$ |

(N) | (N mm) | (deg) [17] | (deg) | (deg) Eq. (21) | (deg) |

0.25 | 0.0 | −28.24 | −31.10 | −28.24 | −31.03 |

0.50 | 0.0 | −28.24 | −31.10 | −28.24 | −30.97 |

0.0 | 100 | −28.43 | −31.40 | −28.43 | −31.30 |

0.20 | 20 | −28.30 | −31.20 | −28.30 | −31.13 |

0.50 | 100 | −28.33 | −31.30 | −28.33 | −31.00 |

1.0 | 200 | −28.33 | −31.30 | −28.32 | −29.80 |

2.0 | 200.0 | −28.30 | −31.22 | −28.27 | −30.56 |

2.0 | 300.0 | −28.32 | −31.27 | −28.28 | −30.51 |

3.0 | 300.0 | −28.30 | −32.61 | −28.24 | −30.32 |

5.0 | 300 | −28.28 | −31.20 | −28.19 | −30.31 |

Applied load | Small deflection | Large deflection | |||
---|---|---|---|---|---|

P | M_{0} | ψ_{EF} | ψ_{FEA} | $\psi EFL$ | $\psi FEAL$ |

(N) | (N mm) | (deg) [17] | (deg) | (deg) Eq. (21) | (deg) |

0.25 | 0.0 | −28.24 | −31.10 | −28.24 | −31.03 |

0.50 | 0.0 | −28.24 | −31.10 | −28.24 | −30.97 |

0.0 | 100 | −28.43 | −31.40 | −28.43 | −31.30 |

0.20 | 20 | −28.30 | −31.20 | −28.30 | −31.13 |

0.50 | 100 | −28.33 | −31.30 | −28.33 | −31.00 |

1.0 | 200 | −28.33 | −31.30 | −28.32 | −29.80 |

2.0 | 200.0 | −28.30 | −31.22 | −28.27 | −30.56 |

2.0 | 300.0 | −28.32 | −31.27 | −28.28 | −30.51 |

3.0 | 300.0 | −28.30 | −32.61 | −28.24 | −30.32 |

5.0 | 300 | −28.28 | −31.20 | −28.19 | −30.31 |

Applied load | Small deflection | Large deflection | |||||
---|---|---|---|---|---|---|---|

P | M_{0} | J_{EF} | J_{FEA} | G_{spring} | $JEFL$ (N/mm) | $JFEAL$ | $GspringL$ |

(N) | (N mm) | (N/mm) [17] | (N/mm) | (N/mm) [17] | Eqs. (18a) and (18b) | (N/mm) | (N/mm) Eq. (19b) |

0.25 | 0.0 | 0.0295 | 0.0294 | 0.0295 | 0.0294 | 0.0298 | 0.0294 |

0.50 | 0.0 | 0.118 | 0.118 | 0.118 | 0.116 | 0.117 | 0.116 |

0.0 | 100 | 0.107 | 0.107 | 0.107 | 0.107 | 0.109 | 0.107 |

0.20 | 20 | 0.041 | 0.041 | 0.041 | 0.041 | 0.041 | 0.041 |

0.50 | 100 | 0.450 | 0.449 | 0.450 | 0.413 | 0.420 | 0.413 |

1.0 | 200 | 1.80 | 1.80 | 1.80 | 1.36 | 1.38 | 1.36 |

2.0 | 200 | 4.123 | 4.192 | 4.123 | 2.391 | 2.426 | 2.391 |

2.0 | 300 | 5.559 | 5.533 | 5.559 | 2.955 | 2.998 | 2.955 |

3.0 | 300 | 9.276 | 9.431 | 9.276 | 3.922 | 3.979 | 3.923 |

5.0 | 300 | 19.55 | 19.45 | 19.55 | 5.85 | 5.91 | 5.85 |

Applied load | Small deflection | Large deflection | |||||
---|---|---|---|---|---|---|---|

P | M_{0} | J_{EF} | J_{FEA} | G_{spring} | $JEFL$ (N/mm) | $JFEAL$ | $GspringL$ |

(N) | (N mm) | (N/mm) [17] | (N/mm) | (N/mm) [17] | Eqs. (18a) and (18b) | (N/mm) | (N/mm) Eq. (19b) |

0.25 | 0.0 | 0.0295 | 0.0294 | 0.0295 | 0.0294 | 0.0298 | 0.0294 |

0.50 | 0.0 | 0.118 | 0.118 | 0.118 | 0.116 | 0.117 | 0.116 |

0.0 | 100 | 0.107 | 0.107 | 0.107 | 0.107 | 0.109 | 0.107 |

0.20 | 20 | 0.041 | 0.041 | 0.041 | 0.041 | 0.041 | 0.041 |

0.50 | 100 | 0.450 | 0.449 | 0.450 | 0.413 | 0.420 | 0.413 |

1.0 | 200 | 1.80 | 1.80 | 1.80 | 1.36 | 1.38 | 1.36 |

2.0 | 200 | 4.123 | 4.192 | 4.123 | 2.391 | 2.426 | 2.391 |

2.0 | 300 | 5.559 | 5.533 | 5.559 | 2.955 | 2.998 | 2.955 |

3.0 | 300 | 9.276 | 9.431 | 9.276 | 3.922 | 3.979 | 3.923 |

5.0 | 300 | 19.55 | 19.45 | 19.55 | 5.85 | 5.91 | 5.85 |

The energy release rate results in Tables 1 and 3 are segmented by results produced using the small or large deflection analysis. We use the superscript “*L*” to denote the results from the large deflection analysis. For the small deflection, *J*_{EF} is for values obtained from the closed-form analytical model [17], which combines the Euler–Bernoulli theory with the elastic foundation analysis. Another small deflection energy release rate result presented is the formula based on the energy stored in the springs, *G*_{spring} [17]. For the large deflection, $JEFL$ is the energy release rate value obtained using the present large deflection expression, Eqs. (18*a*) and (18*b*). The corresponding energy release rate values predicted by the springs are listed under $GspringL$ and are calculated from Eq. (19*b*). All calculated results are validated using the abaqus linear (small deflection) and nonlinear (large deflection) results, which are presented under *J*_{FEA} and $JFEAL$, respectively.

The energy release rate values from the elastic foundation analysis and the corresponding FEA are within about 2% of each other for both the small deflection and large deflection analyses. Another very interesting result from these tables is that the values of *G*_{spring} (i.e., the energy released by the springs) are exactly the same as the *J*-integral values, *J*_{EF}. Thus, Eq. (19*b*) can be used as an alternative estimate of the energy release rate. There is a marked difference between the values of the energy release rates from small deflections and large deflections as the loads become larger and the last entry of Tables 1 and 3 shows that the energy release rate from the large deflections analysis is only about one-third of the one from the small deflections analysis.

Supplementary to Tables 1 and 3, Fig. 5 shows a plot of the energy release rate results for an applied force, *P*, and no moment. In the low load range (for *P* less than about 0.75 N), the small and large deflection frameworks yield remarkable results. Beyond the low load range, the results begin to diverge. These trends are also confirmed by the corresponding FEA linear and nonlinear results.

We now shift focus to discussing the large deflection effect on the mode partitioning. Tables 2 and 4 show results for an aluminum foam and H100 core sandwich configuration, respectively. The tabulated values are segmented by small deflection solutions obtained from Ref. [17] and large deflection solutions obtained from Eq. (21). The small deflection mode partitioning angles are listed under *ψ*_{EF}, and the large deflection mode partitions are listed under $\psi EFL$. Both results are compared to the FEA results listed under *ψ*_{FEA} and $\psi FEAL$, respectively. In all cases, the difference between the elastic foundation and the corresponding FEA results is within about 3 deg. The agreement between elastic foundation and finite element results is actually better for the softer H100 core. The mode partitioning results are larger by absolute value for the stiffer aluminum foam core, indicating a higher mode II component than the weaker H100 core. The most important observation, however, is that there does not seem to be a noteworthy effect of large deflections on the mode partitioning angles, i.e., both the small and large deflection analyses yield similar values.

Applied load | Small deflection | Large deflection | |||
---|---|---|---|---|---|

P | M_{0} | ψ_{EF} | ψ_{FEA} | $\psi EFL$ | $\psi FEAL$ |

(N) | (N mm) | (deg) [17] | (deg) | (deg) Eq. (21) | (deg) |

0.25 | 0.0 | −10.88 | −9.00 | −10.88 | −9.00 |

0.50 | 0.0 | −10.88 | −9.00 | −10.88 | −9.00 |

0.0 | 100 | −11.13 | −9.52 | −11.13 | −9.40 |

0.20 | 20 | −10.96 | −9.22 | −10.99 | −9.23 |

0.50 | 100 | −11.00 | −9.29 | −10.99 | −8.97 |

1.0 | 200 | −11.00 | −9.30 | −10.98 | −8.95 |

2.0 | 200 | −10.96 | −9.06 | −10.90 | −9.22 |

2.0 | 300 | −10.98 | −9.26 | −10.93 | −8.91 |

3.0 | 300 | −10.96 | −9.06 | −10.87 | −9.96 |

5.0 | 300 | −10.93 | −9.17 | −10.77 | −10.20 |

Applied load | Small deflection | Large deflection | |||
---|---|---|---|---|---|

P | M_{0} | ψ_{EF} | ψ_{FEA} | $\psi EFL$ | $\psi FEAL$ |

(N) | (N mm) | (deg) [17] | (deg) | (deg) Eq. (21) | (deg) |

0.25 | 0.0 | −10.88 | −9.00 | −10.88 | −9.00 |

0.50 | 0.0 | −10.88 | −9.00 | −10.88 | −9.00 |

0.0 | 100 | −11.13 | −9.52 | −11.13 | −9.40 |

0.20 | 20 | −10.96 | −9.22 | −10.99 | −9.23 |

0.50 | 100 | −11.00 | −9.29 | −10.99 | −8.97 |

1.0 | 200 | −11.00 | −9.30 | −10.98 | −8.95 |

2.0 | 200 | −10.96 | −9.06 | −10.90 | −9.22 |

2.0 | 300 | −10.98 | −9.26 | −10.93 | −8.91 |

3.0 | 300 | −10.96 | −9.06 | −10.87 | −9.96 |

5.0 | 300 | −10.93 | −9.17 | −10.77 | −10.20 |

Next, a parametric study is performed to examine the large deflection effects for various core stiffnesses. Results from the small and large deflection analyses are compared for a face-to-core stiffness ratio ranging from 10 to 1000. Tabulated in Table 5 are the energy release rates and mode partitioning angles for *P* = 3 N and *M*_{0} = 300 N mm, along with the corresponding small and large deflection FEA results. It can be seen that there is good agreement between the analytical and the FEA results. Overall, the small deflection *J*_{EF} and *J*_{FEA} results are within 1% of each other for all *E*_{f}/*E*_{c} ratios, while the large deflection $JEFL$ and $JFEAL$ results are within 1% of each other up to a stiffness ratio of 500 and show higher differences for the very high *E*_{f}/*E*_{c} ratios of 700 and 1000, i.e., for the very weak cores. This is due to the fact that for the very high stiffness ratios, transverse shear effects are expected to be significant, and these are not accounted for in the Euler–Bernoulli theory employed herein. Another observation is that the energy release rate increases as the core becomes weaker, but this increase is small, i.e., the *E*_{f}/*E*_{c} ratio does not have a noteworthy effect on the energy release rate. However, there is a marked effect of large deflections on the energy release rate, with the large deflection values being less than half of the corresponding small deflection ones.

Small deflection | Large deflection | |||||||
---|---|---|---|---|---|---|---|---|

J_{EF} | J_{FEA} | ψ_{EF} | ψ_{FEA} | $JEFL$ (N/mm) | $JFEAL$ | $\psi EFL$ | $\psi FEAL$ | |

E_{f}/E_{c} | (N/mm) [17] | (N/mm) | (deg) [17] | (deg) | Eqs. (18a) and (18b) | (N/mm) | (deg) Eq. (21) | (deg) |

10 | 8.894 | 8.821 | −28.30 | −31.24 | 3.937 | 3.905 | −28.24 | −30.32 |

50 | 9.001 | 8.925 | −19.75 | −21.28 | 3.933 | 3.902 | −19.67 | −18.81 |

100 | 9.063 | 8.986 | −16.77 | −17.26 | 3.931 | 3.898 | −16.68 | −15.47 |

500 | 9.256 | 9.214 | −11.33 | −9.47 | 3.923 | 3.963 | −11.24 | −9.93 |

700 | 9.308 | 9.279 | −10.42 | −8.11 | 3.921 | 4.070 | −10.33 | −9.65 |

1000 | 9.368 | 9.359 | −9.54 | −6.77 | 3.919 | 4.282 | −9.44 | −6.16 |

Small deflection | Large deflection | |||||||
---|---|---|---|---|---|---|---|---|

J_{EF} | J_{FEA} | ψ_{EF} | ψ_{FEA} | $JEFL$ (N/mm) | $JFEAL$ | $\psi EFL$ | $\psi FEAL$ | |

E_{f}/E_{c} | (N/mm) [17] | (N/mm) | (deg) [17] | (deg) | Eqs. (18a) and (18b) | (N/mm) | (deg) Eq. (21) | (deg) |

10 | 8.894 | 8.821 | −28.30 | −31.24 | 3.937 | 3.905 | −28.24 | −30.32 |

50 | 9.001 | 8.925 | −19.75 | −21.28 | 3.933 | 3.902 | −19.67 | −18.81 |

100 | 9.063 | 8.986 | −16.77 | −17.26 | 3.931 | 3.898 | −16.68 | −15.47 |

500 | 9.256 | 9.214 | −11.33 | −9.47 | 3.923 | 3.963 | −11.24 | −9.93 |

700 | 9.308 | 9.279 | −10.42 | −8.11 | 3.921 | 4.070 | −10.33 | −9.65 |

1000 | 9.368 | 9.359 | −9.54 | −6.77 | 3.919 | 4.282 | −9.44 | −6.16 |

The mode partitioning results shown in Table 5 are also compared for the same range of core stiffnesses. The table includes the small and large deflection mode partitioning angles, which are validated against the corresponding FEA results. It can be observed that for all stiffness ratios, the results show essentially no effect of large deflections on the mode partitioning angle and this is observed in both the analytical and the FEA results. However, the mode partitioning angles become smaller by absolute value, as the stiffness ratio increases, indicating a smaller mode II component for the weaker cores. In addition, comparing the analytical with the corresponding FEA results, the predicted mode partitioning angles from the two models deviate by at most about 3 deg in absolute value, i.e., for all stiffness ratios, the analytical and FEA results agree well with each other.

Finally, a study of the effect of the debond length is performed. Tables 6–9 show the energy release rate and mode partitioning angles for the geometry and materials in Tables 1–4, except that the debond length now is *a* = 20 mm, i.e., one-tenth of the debond length in Tables 1–4. A few select high and low loads of pure shear, bending, and a combination of both are considered. Tables 6 and 7 are for an aluminum foam core, whereas Tables 8 and 9 are for an H100 core. Results between the two cores re-iterate trends observed in Ref. [17] that the small and large deflection closed-form solutions are just as accurate (by comparison to FEA) with the smaller debond lengths. More importantly, the study underscores the capability of the large deflection closed-form solution in producing results for various debond lengths, load ranges, and core stiffnesses, making it a very robust semi-closed-form framework and more accurate than the small deflection framework; however, notice that the solution for small deflections in Ref. [17] is in fully closed form, whereas the present large deflection solution is semi-closed form in that it still requires the numerical solution of the two nonlinear equations (4) and (12). The total specimen length is still *L* = 500 mm. The results show clearly that the elastic foundation analysis is just as accurate as it was with the smaller debond lengths. One observation that can be made is that the energy release rates are substantially smaller for the smaller debond length, as expected, but the mode partitioning angles are only slightly affected by the smaller debond length. More importantly, the large deflection effects are essentially not observed in the smaller debond length cases, i.e., the energy release rates are similar whether from a small deflection or a large deflection analysis. This demonstrates that the large deflection effects depend strongly not only on the loading but also on the debond length. All these trends are very well captured by the present elastic foundation model and confirmed by the corresponding FEA results.

Applied load | Small deflection [17] | Large deflection | |||||
---|---|---|---|---|---|---|---|

P | M_{0} | J_{EF} | J_{FEA} | G_{spring} | $JEFL$ (N/mm) | $JFEAL$ | $GspringL$ |

(N) | (N mm) | (N/mm) | (N/mm) | (N/mm) | Eqs. (18a) and (18b) | (N/mm) | (N/mm) Eq. (19b) |

0.0 | 100 | 0.107 | 0.107 | 0.107 | 0.107 | 0.107 | 0.107 |

1.0 | 200 | 0.536 | 0.527 | 0.536 | 0.536 | 0.527 | 0.536 |

2.0 | 200 | 0.656 | 0.640 | 0.656 | 0.655 | 0.639 | 0.655 |

2.0 | 300 | 1.293 | 1.270 | 1.293 | 1.292 | 1.267 | 1.292 |

3.0 | 300 | 1.475 | 1.439 | 1.475 | 1.472 | 1.436 | 1.473 |

5.0 | 300 | 1.876 | 1.813 | 1.876 | 1.870 | 1.808 | 1.870 |

Applied load | Small deflection [17] | Large deflection | |||||
---|---|---|---|---|---|---|---|

P | M_{0} | J_{EF} | J_{FEA} | G_{spring} | $JEFL$ (N/mm) | $JFEAL$ | $GspringL$ |

(N) | (N mm) | (N/mm) | (N/mm) | (N/mm) | Eqs. (18a) and (18b) | (N/mm) | (N/mm) Eq. (19b) |

0.0 | 100 | 0.107 | 0.107 | 0.107 | 0.107 | 0.107 | 0.107 |

1.0 | 200 | 0.536 | 0.527 | 0.536 | 0.536 | 0.527 | 0.536 |

2.0 | 200 | 0.656 | 0.640 | 0.656 | 0.655 | 0.639 | 0.655 |

2.0 | 300 | 1.293 | 1.270 | 1.293 | 1.292 | 1.267 | 1.292 |

3.0 | 300 | 1.475 | 1.439 | 1.475 | 1.472 | 1.436 | 1.473 |

5.0 | 300 | 1.876 | 1.813 | 1.876 | 1.870 | 1.808 | 1.870 |

Applied load | Small deflection | Large deflection | |||
---|---|---|---|---|---|

P | M_{0} | ψ_{EF} | ψ_{FEA} | $\psi EFL$ | $\psi FEAL$ |

(N) | (N mm) | (deg) [17] | (deg) | (deg) Eq. (21) | (deg) |

0.0 | 100 | −28.43 | −31.25 | −28.43 | −31.13 |

1.0 | 200 | −28.26 | −30.94 | −28.26 | −30.67 |

2.0 | 200 | −28.12 | −30.69 | −28.12 | −30.34 |

2.0 | 300 | −28.28 | −30.85 | −28.21 | −30.31 |

3.0 | 300 | −28.12 | −30.69 | −28.12 | −30.02 |

5.0 | 300 | −27.98 | −30.42 | −27.98 | −29.65 |

Applied load | Small deflection | Large deflection | |||
---|---|---|---|---|---|

P | M_{0} | ψ_{EF} | ψ_{FEA} | $\psi EFL$ | $\psi FEAL$ |

(N) | (N mm) | (deg) [17] | (deg) | (deg) Eq. (21) | (deg) |

0.0 | 100 | −28.43 | −31.25 | −28.43 | −31.13 |

1.0 | 200 | −28.26 | −30.94 | −28.26 | −30.67 |

2.0 | 200 | −28.12 | −30.69 | −28.12 | −30.34 |

2.0 | 300 | −28.28 | −30.85 | −28.21 | −30.31 |

3.0 | 300 | −28.12 | −30.69 | −28.12 | −30.02 |

5.0 | 300 | −27.98 | −30.42 | −27.98 | −29.65 |

Applied load | Small deflection [17] | Large deflection | |||||
---|---|---|---|---|---|---|---|

P | M_{0} | J_{EF} | J_{FEA} | G_{spring} | $JEFL$ | $JFEAL$ | $GspringL$ |

(N) | (N mm) | (N/mm) | (N/mm) | (N/mm) | (N/mm) Eqs. (18a) and (18b) | (N/mm) | (N/mm) Eq. (19b) |

0.0 | 100 | 0.107 | 0.106 | 0.107 | 0.107 | 0.106 | 0.107 |

1.0 | 200 | 0.568 | 0.556 | 0.568 | 0.567 | 0.556 | 0.567 |

2.0 | 200 | 0.726 | 0.709 | 0.726 | 0.725 | 0.708 | 0.725 |

2.0 | 300 | 1.391 | 1.362 | 1.391 | 1.390 | 1.359 | 1.390 |

3.0 | 300 | 1.634 | 1.595 | 1.634 | 1.627 | 1.591 | 1.627 |

5.0 | 300 | 2.177 | 2.117 | 2.177 | 2.162 | 2.107 | 2.163 |

Applied load | Small deflection [17] | Large deflection | |||||
---|---|---|---|---|---|---|---|

P | M_{0} | J_{EF} | J_{FEA} | G_{spring} | $JEFL$ | $JFEAL$ | $GspringL$ |

(N) | (N mm) | (N/mm) | (N/mm) | (N/mm) | (N/mm) Eqs. (18a) and (18b) | (N/mm) | (N/mm) Eq. (19b) |

0.0 | 100 | 0.107 | 0.106 | 0.107 | 0.107 | 0.106 | 0.107 |

1.0 | 200 | 0.568 | 0.556 | 0.568 | 0.567 | 0.556 | 0.567 |

2.0 | 200 | 0.726 | 0.709 | 0.726 | 0.725 | 0.708 | 0.725 |

2.0 | 300 | 1.391 | 1.362 | 1.391 | 1.390 | 1.359 | 1.390 |

3.0 | 300 | 1.634 | 1.595 | 1.634 | 1.627 | 1.591 | 1.627 |

5.0 | 300 | 2.177 | 2.117 | 2.177 | 2.162 | 2.107 | 2.163 |

## 6 Conclusions

The effect of large deflections on the energy release rate and mode partitioning of a face/core debond in a sandwich composite was investigated by considering a SCB configuration, which consists of a partially debonded sandwich beam loaded in shear and/or bending applied at the debond end. The beam is fixed at the bottom bounding surface. The study is done by use of the elastica theory for the debonded face and an elastic foundation approach for the rest of the structure. The solution derived is in semi-closed form, in that it is reduced to the solution of two nonlinear algebraic equations for two unknowns: the rotation at the loaded end and the rotation at the debond tip (where the elastic foundation begins). A J-integral approach is subsequently applied to derive a formula for the energy release rate. In addition, a simple formula for the energy release rate is derived by considering the differential energy stored in the springs at the beginning of the elastic foundation, i.e., the energy released by the “broken” differential spring element as the debond propagates. The mode partitioning angle derivation in the context of the elastic foundation approach is based on the transverse and axial displacements at the beginning of the elastic foundation (“debond tip”). Results from this solution are compared with finite element data for a wide range of loadings, face/core moduli ratios, and debond lengths. It is shown that the energy release rate from this semi-closed-form solution is in very good agreement with the finite element data. Moreover, the results from the J-integral and the simple formula based on the energy released by the broken differential spring as the debond propagates are coincident. It is also seen that the mode partitioning angles are in good agreement with the corresponding finite element mode mixity angles, being within about 3 deg of each other. In general, large deflections result in a reduced energy release rate but do not affect the mode partitioning. For relatively large loads and debond lengths, it is seen that the large deflection effects can significantly reduce the energy release rate. The large deflection solution converges to the small deflection solution for smaller loads. As expected, for the same load, large deflection effects are absent for a small debond length. The solution presented in this paper applies to orthotropic faces and core and also to finite length beams and all formulas are given explicitly, including the value of the elastic foundation modulus, in addition to the formulas for the energy release rate and the mode partitioning angle.

## Acknowledgment

The financial support of the Office of Naval Research, Grant N00014-20-1-2605, and the interest and encouragement of the Grant Monitors, Dr. Jessica Dibelka, Dr. Paul Hess, and Dr. Y. D. S. Rajapakse, are both gratefully acknowledged. The authors would also like to acknowledge Dr. Siddarth Niranjian Babu’s help with the finite element analysis.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.