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In the bending diagrams, one can see a beneficial type of failure, similar to (or even slightly milder than) that for the 3D-W laminate. However, at the initial stage of the basic failure (from about 13 mm of deflection), the main mechanism is matrix fracture. Fiber fracture takes place only at the moment of a partial load capacity loss of the beam. Such behavior proves a significant contribution of the translaminar interweave strands in the work of the 3D laminate, during bending in direction W. The lack of this action in direction P results in a milder model of failure, yet, at the same time, laminate weakening (lower maximal strain); a similar regularity was observed for stitched laminates [55]. The lattice structure and the presence of relatively large cracks between the fiber bands (Figure 8) result in a locally-facilitated bending of the laminate in direction P (local reduction of the section height). Figure 11 shows practically no delamination in the upper part of the beam, whereas the main failure is the extensive fracture in the lower part.

Part of the research presented in this study has been financed by Silesian University of Technology within the scope of the statutory project BK-220/RM3/2015. Part of the research presented in this study has been financed by the Polish National Science Centre within the scope of the project No. N N508 628640.

Abstract Optimal fiber designs for the maximal pull-off force have been indispensable for increasing the attachment performance of recently introduced gecko-inspired reversible micro/nanofibrillar adhesives. There are several theoretical studies on such optimal designs; however, due to the lack of three-dimensional (3D) fabrication techniques that can fabricate such optimal designs in 3D, there have not been many experimental investigations on this challenge. In this study, we benefitted from recent advances in two-photon lithography techniques to fabricate mushroomlike polyurethane elastomer fibers with different aspect ratios of tip to stalk diameter (β) and tip wedge angles (θ) to investigate the effect of these two parameters on the pull-off force. We found similar trends to those predicted theoretically. We found that β has an impact on the slope of the force-displacement curve while both β and θ play a role in the stress distribution and crack propagation. We found that these effects are coupled and the optimal set of parameters also depends on the fiber material. This is the first experimental verification of such optimal designs proposed for mushroomlike microfibers. This experimental approach could be used to evaluate a wide range of complex microstructured adhesive designs suggested in the literature and optimize them.

The concept of crack tip opening angle (CTOA) was probably introduced by Anderson in 1972 [1] to simulate stable crack growth by the finite element method. In this method, crack growth is obtained by successive relaxation of the nodal forces at the node representing the crack tip. The y angle between two element sides representing the crack tip was chosen as the criterion for the crack growth. However, crack growth dependence of this angle was expected, and a constant value was used for all of the stages of growth. The value at the first increment of crack growth called y 0 was determined from the experimentally determined criti cal value of the J integral. Anderson assumed that after large crack growth, steady state conditions prevail and y stabilizes to the y stab value with y stab < y0. Rice [2] has pointed out the difficulty of using this parameter for the simulation of crack growth in an elastic-plastic material. In this case, displacement exhibits a profile proportional to the product r (where r is the distance measured from the crack tip). Therefore the tangent at the crack tip corresponds to an ambiguous definition of the CTOA.

One of the first experimental determinations of the CTOA was mentioned by Luxmoore et al. [7]. Using centre notched and double-edge notched specimens of aluminium alloy, they measured the CTOA during stable crack growth. They noted that the crack tip opening 8 varied linearly with increases in crack length. This linear behaviour indicated that the CTOA remains constant and equal to 2.1°. This was also previously noticed by de Koning et al. [8].

Conditions of stable crack growth require that the rate of change of the crack driving force with increasing crack length Aa is smaller than the increase of crack growth resistance expressed in terms of crack opening displacement:

During the last decade, extensive studies applying the CTOA concept to ductile fracture have been undertaken. As a result, the CTOA-based fracture mechanics method has become mature, and a standard test method for critical CTOA testing was developed recently by ASTM with the designation E2472-06e 1 [11]. The recommended specimens are the compact-tension C(T) and middle-crack-tension M(T) specimens made from thin-sheet materials in order to achieve low constraint conditions at the crack tip. The standard was validated by Heerens and Schodel [12] using a comprehensive dataset on the stable crack extension in an aluminium sheet material with a thickness of 3 mm.

The CTOA fracture criterion has now become one of the most promising fracture criteria used for characterizing stable tearing in thin metallic materials. Initially, fracture resistance to crack extension was given by Charpy energy, as in the Battelle two-curves method (BTCM) [13]. The Charpy test is related to crack initiation, bending of the specimen, and plastic deformation at the load points. It is necessary that tests performed to characterize the propagation resistance be able to isolate and quantify the propagation energy with respect to incremental crack advance. For this reason and due to the development of higher strength steels with increased toughness and lower transition temperatures by using controlled rolling techniques, Charpy energy was replaced by drop-weight tear test (DWTT) energy in the HLP two-curve method [14]. DWTT tests are also related to crack initiation, bending of the specimen, and plastic deformation. However, notched DWTT specimens are larger

than Charpy ones and therefore relatively less of the total fracture energy is related to initiation. The statically pre-cracked DWTT showed the best compromise between isolating the propagation energy and ease of specimen preparation. Another step was the development of a test methodology to indirectly measure CTOA, derived on the basis of the difference in energy between two modified DWTT specimens with different initial crack lengths [15].

The use of CTOA for prediction of dynamic ductile fracture arrest in pipelines began in the late 1980s with a model which calculates the crack driving force in terms of CTOA as a function of crack speed.

The CTOA is defined as the angle between the crack faces of a growing crack. The practical realization of this definition is not possible because the crack faces are not straight but curved with a curvature which depends on the specimen and loading type. A definition based on crack tip opening displacement 8 at a distance d of the order of 1 mm is consistent for both experimental and numerical determination (Fig. 3)

In order to overcome the zigzag pattern of real crack faces or the influence of mesh size, it is more convenient to determine CTOAs at several distinct complementary positions on the upper and lower crack surfaces Y, and to

In microtopography [17], the fracture surfaces are topographically analysed post-mortem. This method is based on the assumption that the CTOA is preserved in the plastic deformations of the fracture surface. After both surfaces have been scanned, they are recombined in the computer. The reconstructed crack contours at fracture allow the determination of the CTOA. This method is of interest because it allows the CTOA to be identified in positions of different thicknesses, but it is time consuming.

The calculation of the critical CTOA can be made using the dynamic fracture results from two specimens with different notch depths [18]. These specimens are three-point bend specimens similar to the DWTT specimen but with a straight notch machined in the place of the standard pressed notch. Evaluating the fracture energy difference makes it possible to suppress the crack initiation energy and give the propagation energy proportional to ligament area. CTOA is obtained by the following formula:

(E/A)shallow is the energy per area for a shallow-notch specimen (a = 10 mm, a/W = 0.13), (E/A)deep is the energy per area for a deep-notch specimen (a = 38 mm, a/W = 0.5), Rc d is the dynamic flow stress (Rcd = Rcs), Rcs is the static flow stress, a is the crack length, and W is the specimen width.

Optical microscopy is one of the most common methods of measuring CTOA [18]. The crack contour close to the crack tip is investigated at the polished surface using a light microscope. A special case of optical measurement is the digital image correlation (DIC) method. For our measures, a commercial DIC camera (Gom FASTCAM SA. 1 Photron) and a software analysis package with integrated length and angle measurement tools (ARAMIS V6.3) has been used to measure CTOA and crack extension Aa. The recording time was automatically available from the videotapes, where a digital stopwatch was used to synchronize the still images. All of this allowed test parameters such as load, displacement, and crack length Aa are correlated with CTOA. One example of such a digital image and the corresponding CTOA values is given in Fig. 5. 2b1af7f3a8