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2023 | OriginalPaper | Buchkapitel

2. The Effect of Residual Hygrothermal Stresses on the Energy Release Rate and Mode Mixity of Interfacial Cracks in Beams with Bending–Extension Coupling

verfasst von : Panayiotis Tsokanas

Erschienen in: Fracture Analysis of Layered Beams With an Elastically Coupled Behavior and Hygrothermal Stresses

Verlag: Springer International Publishing

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Abstract

The present chapter proposes an analytical framework for calculating the mode I and mode II components of the ERR of an interfacial crack between two generally layered sublaminates while considering the effects of BEC and RHTS. In this model, both sublaminates are modeled as Timoshenko beams, the crack tip is assumed to be semi-rigid (rotationally flexible), and mode partitioning is performed according to the so-called global partitioning method. ERR and MM are obtained for a generally loaded cracked beam specimen and are then reduced for some typical interfacial fracture test configurations. Next, an asymmetrically disbonded FML is analyzed using the DCB and ENF configurations. Notably, the inherent contact problem in the case of the ENF configuration is also approached analytically. It is demonstrated that the effect of RTS on the ERR and MM of the FML, which has largely been ignored in the relevant literature, is non-negligible and might even be significant. The analytical results are validated through a comparison against numerical ones via the finite element method. Lastly, the proposed analytical solution is well-suited for future experimental data reduction purposes since it is provided as closed-form expressions and does not require complicated calculations.

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Vorheriges Kapitel Introduction

Nächstes Kapitel Fracture Toughness of Metal-to-Composite Adhesive Joints with Bending–Extension Coupling and Residual Thermal Stresses

An illustrative example of temperature and moisture exchange comes from the aerospace industry. During a typical flight, an aircraft is subjected to a wide spectrum of temperatures and relative humidities. While the aircraft is on the runway, temperatures can rise up to 60 °C with 100% relative humidity. During take-off, there is significant temperature and moisture release. While cruising at an altitude of some 37,000 ft, the temperature is usually –55 °C, and the relative humidity is normally lower than 30%. Conversely, during landing and parking, the temperature increases again, and moisture uptake is also observed.

Typical examples of such structures include FMLs, dissimilar adhesive joints, and, of course, composite laminates consisting of layers of different materials or different fiber orientations.

Usually, such stresses are induced after a manufacturing process at a high temperature or in cases where the difference between the service temperature and manufacturing temperature is high.

For example, many symmetric and nonsymmetric cross-ply and angle-ply composite laminates can be analyzed by our model.

Common analytical expressions and DRSs (see, for example, those cited in Sect. 2.1.1) are based on 2D theories, which assume a straight crack front and a uniform ERR distribution along the width of the beam. Even though this assumption is logical for \(b \ll \ell\), it has been established (see, for instance, [35]) as only an idealization of the reality.

In the present work, the term sublaminate is used to be consistent with the pre-existing literature that prefers this term, even though the term sub-beam is probably more accurate.

In the literature, the beam is sometimes modeled as the assemblage of three beams: two beams (the upper and lower ones) in the cracked region and a third beam in the uncracked region [13, 45]. Nevertheless, Bruno and Greco [8] showed that modeling the entire uncracked region using a single beam fails to capture satisfactorily the shear deformation of the beam.

When the slenderness ratio (i.e., the ratio of free length to thickness) is small, approaches based on Euler beam theory are inappropriate for interface fracture analysis. This is because, in moderately thick beams, the effect of shear deformation is obviously significant and, therefore, should be considered.

The wedge forces applied are generic; they will be reduced in the following sections to meet specific test configurations of interest.

These assumptions remain valid in all theoretical models described in the present thesis.

It is called pseudo because, for Timoshenko beam theory, it does not coincide with the geometric curvature of the mid-thickness plane.

Equations (2.22) and (2.23) may require further explanation. We respectively define as \(\Delta u\left( x \right) = \overline{u}_{2} \left( x \right) - \underline {u}_{1} \left( x \right)\) and \(\Delta w\left( x \right) = \overline{w}_{2} \left( x \right) - \underline {w}_{1} \left( x \right)\) the axial and transverse relative displacements along the interface. \(\underline {u}_{1} \left( x \right)\) and \(\underline {w}_{1} \left( x \right)\) are the displacements at the bottom surface of sublaminate 1 \(\left( {z_{1} = - h_{1} /2} \right)\). Correspondingly, \(\overline{u}_{2} \left( x \right)\) and \(\overline{w}_{2} \left( x \right)\) are the displacements at the top surface of sublaminate 2 \(\left( {z_{2} = h_{2} /2} \right)\). Based on Eq. (2.1), the axial displacements of both sublaminates are linear functions of the thickness coordinate, \(z_{{\text{i}}}\), with \(i \in \left\{ {1,{ }2} \right\}\). As a result, \(\underline {u}_{1} \left( x \right) = u_{1} \left( x \right) - \left( {h_{1} /2} \right) \cdot \varphi_{1} \left( x \right)\) and \(\overline{u}_{2} \left( x \right) = u_{2} \left( x \right) + \left( {h_{2} /2} \right) \cdot \varphi_{2} \left( x \right)\). In addition, according to Eq. (2.2), the transverse displacements of the sublaminates are constant throughout the thickness such that \(\underline {w}_{1} \left( x \right) = w_{1} \left( x \right)\) and \(\overline{w}_{2} \left( x \right) = w_{2} \left( x \right)\). Consequently, Eqs. (2.22) and (2.23) are obtained from these expressions.

There are various ways to present the analytical expressions presented in the rest of the thesis. In the present work, the expressions are presented to be as consistent as possible with expressions in the main body of the relevant literature. This will help the reader to directly compare and clearly observe the differences between the previous expressions and the ones proposed here.

Also recall that \(\frac{{d^{2} {\mathcal{M}}_{{\text{T}}} \left( x \right)}}{{dx^{2} }} = 0\) and \(\frac{{d^{2} {\mathcal{N}}_{{\text{T}}} }}{{dx^{2} }} = 0\).

\(c_{0}\) is finally given in Eq. (2.45).

Therefore, the obvious relation of \({\mathcal{Q}}_{{\text{T}}} = \frac{{d{\mathcal{M}}_{{\text{T}}} \left( x \right)}}{dx}\) is fulfilled.

Regarding sublaminate 2, we do not report the respective magnitudes here because we do not need to use them in the solution. We will define them in Chap. 6.

The flexibility coefficients of the semi-rigid model may also be referred to as root-rotation coefficients [3], given that the deformability of the CTE is attributed to the rotations only. The higher these coefficients, the higher the root rotation.

In Chap. 5, an additional test configuration of interest (namely, the DCB-UBM test) will be studied. The respective expressions will be introduced in that chapter, while the analytical formulation to be used is the one presented in this chapter.

The external load, \(P_{{{\text{MMB}}}}\), as well as the loads \(P_{{\text{e}}}\) and \(P_{{\text{c}}}\) and the crack-tip forces, \({\mathcal{N}}_{{\text{c}}}\) and \({\mathcal{Q}}_{{\text{c}}}\), are reduced to load over width, \(b\), of the specimen. Thus, they are measured in N/mm in the present thesis. The measurement units for all magnitudes are listed in the Nomenclature of the book.

This is why we adopt the notation of Yokozeki [52] for the loads \({P}_{\mathrm{e}}\) and \(P_{{\text{c}}}\) in the present chapter. The same notation for both these loads is also adopted in the rest of the thesis.

More information on the process of determining this equation is given in Chap. 6.

Thus, the layup of the entire FML can be notated as [0°_Al./0°_GFRP/90°_GFRP/0°_GFRP/0°_Al.//0°_GFRP/90° _GFRP/0°_GFRP/0°_Al.], where the single slash, “/,” denotes the interface between two successive layers and the double slash, “//,” denotes the position of the crack plane.

However, contact may also occur in other test configurations if strong curvatures are induced due to thermal stresses.

This temperature difference represents the difference between a reference temperature (operational temperature) and the stress-free temperature (typically, the maximum temperature of the manufacturing process).

This was the case even though we verified that the material properties of the interfacial layer do not significantly affect the results.

Such an approach is also supported by fractographic evidence following experimental works in delamination or interfacial disbonding studies; in most cases, the crack grows within a (thin) interfacial layer, whereas it is very uncommon for the crack to propagate along a bimaterial interface.

In contrast, this technique introduces the difficulty of having to carefully create the mesh inside the thin artificial layer to prevent elements with a distorted geometry or a high aspect ratio from being created.

These elements were chosen due to the advantages they offer in terms of both accuracy and fast convergence. Plane-strain elements are commonly used to model structures with high length-to-thickness ratios since the displacement of all nodes in the direction vertical to the modeling plane is zero. In addition, the eight-node elements have shape functions of the same order as the bending equation of beam theory; therefore, they are suitable for modeling bending problems. Another advantage of these elements is the avoidance of shear locking, due to the reduced integration, in contrast to the full-integration elements. Lastly, quadrilateral elements offer a uniform grid formation, as opposed to triangular ones.

This classification is also reported in [5].

Appendix F refers to the topic of mode decoupling.

In contrast, Valvo’s [45] approach, which was also tested when developing the present solution, does not seem to be able to perform mode partitioning in the presence of RHTS due to the existence of the crack-tip moment, \({\mathcal{M}}_{{\text{c}}}\). This issue may concern us in a future work.

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Titel: The Effect of Residual Hygrothermal Stresses on the Energy Release Rate and Mode Mixity of Interfacial Cracks in Beams with Bending–Extension Coupling
verfasst von: Panayiotis Tsokanas
Verlag: Springer International Publishing
Buch: Fracture Analysis of Layered Beams With an Elastically Coupled Behavior and Hygrothermal Stresses
Print ISBN: 978-3-031-17620-3

Electronic ISBN: 978-3-031-17621-0

Copyright-Jahr: 2023
DOI: https://doi.org/10.1007/978-3-031-17621-0_2

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