Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave

The interactions between linear elastic waves and a nonlinear crack with finite compressibility are studied in the one-dimensional context. Numerical studies on a hyperbolic model of contact with sinusoidal forcing have shown that the mean values of the scattered elastic displacements are discontinuous across the crack. The mean dilatation of the crack also increases with the amplitude of the forcing levels. The aim of the present theoretical study is to analyse these nonlinear processes under a larger range of nonlinear jump conditions. For this purpose, the problem is reduced to a nonlinear differential equation. The dependence of the periodic solution on the forcing amplitude is quantified under sinusoidal forcing conditions. Bounds for the mean, maximum and minimum values of the solution are presented. Lastly, periodic forcing with a null mean value is addressed. In that case, a result about the mean dilatation of the crack is obtained.Comment: submitted to the SIAM J. App. Mat

Stability of neutral delay differential equations modeling wave propagation in cracked media

International audiencePropagation of elastic waves is studied in a 1D medium containing N cracks modeled by nonlinear jump conditions. The case N = 1 is fully understood. When N > 1, the evolution equations are written as a system of nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. In the case N = 2, some mathematical results about the existence, uniqueness and attractivity of periodic solutions have been obtained in 2012 by the authors, under the assumption of small sources. The difficulty of analysis follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. Here we propose a new result of stability in the homogeneous case, based on an energy method. One deduces the asymptotic stability of the zero steady-state. Extension to N = 3 cracks is also considered, leading to new results in particular configurations. 1. Introduction. Understanding the interactions between ultrasonic waves and contact defects have crucial applications in the field of mechanics, especially as far as the non-destructive testing of materials is concerned. When the cracks are much smaller than the wavelengths, they are usually replaced by interfaces with appropriate jump conditions. Here we consider realistic models describing cracks with finite compressibility, in a 1D geometry (section 2). The case of N = 1 crack, which involves a nonlinear ordinary differential equation, has been completely analysed in [7]. When tackling with N > 1 cracks, the analysis becomes much more intricate. The successive reflections of waves between the cracks are described mathematically by a system of N nonlinear neutral-delay differential equations (NDDE) with forcing [5]. The main features of such systems are already contained in the following scalar NDDE:

Double scale analysis of periodic solutions of some non linear vibrating systems

We consider {\it small solutions} of a vibrating system with smooth non-linearities for which we provide an approximate solution by using a double scale analysis; a rigorous proof of convergence of a double scale expansion is included; for the forced response, a stability result is needed in order to prove convergence in a neighbourhood of a primary resonance.Comment: 36 page

Interaction between periodic elastic waves and two contact nonlinearities

International audiencePropagation of elastic waves is studied in a 1D medium containing two cracks. The latter are modeled by smooth nonlinear jump conditions accounting for the finite, non-null compressibility of real cracks. The evolution equations are written in the form of a system of two nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. Perturbation analysis indicates that, under periodic excitation, the periodic solutions oscillate around positive mean values, which increase with the forcing level. This typically nonlinear phenomenon offers non-destructive means to evaluate the cracks. Existence, uniqueness and attractivity of periodic solutions is then examined. At some particular values of the ratio between the wave travel time and the period of the source, results are obtained whatever the forcing level. With a much larger set of ratios but at small forcing levels, results are obtained under a Diophantine condition. Lastly, numerical experiments are proposed to illustrate the behavior of the periodic diffracted waves

The Method of Strained Coordinates for Vibrations with Weak Unilateral Springs

We study some spring mass models for a structure having a unilateral spring of small rigidity $\epsilon$. We obtain and justify an asymptotic expansion with the method of strained coordinates with new tools to handle such defects, including a non negligible cumulative effect over a long time: T_\eps \sim \eps^{-1} as usual; or, for a new critical case, we can only expect: T_\eps \sim \eps^{-1/2}. We check numerically these results and present a purely numerical algorithm to compute "Non linear Normal Modes" (NNM); this algorithm provides results close to the asymptotic expansions but enables to compute NNM even when $\epsilon$ becomes larger

Software for evaluating probability-based integrity of reinforced concrete structures

In recent years, much research work has been carried out in order to obtain a more controlled durability and long-term performance of concrete structures in chloride containing environment. In particular, the development of new procedures for probability-based durability design has proved to give a more realistic basis for the analysis. Although there is still a lack of relevant data, this approach has been successfully applied to several new concrete structures, where requirements to a more controlled durability and service life have been specified. A probability-based durability analysis has also become an important and integral part of condition assessment of existing concrete structures in chloride containing environment. In order to facilitate the probability-based durability analysis, a software named DURACON has been developed, where the probabilistic approach is based on a Monte Carlo simulation. In the present paper, the software for the probability-based durability analysis is briefly described and used in order to demonstrate the importance of the various durability parameters affecting the durability of concrete structures in chloride containing environment

Potential field theory and its applications to classical mechanical problems

Advances in many scientific fields are expected to come from work in nanotechnology. Engineering at nano-scales presents novel problems that classical mechanics cannot solve. Many engineers are uncomfortable designing at this level because classical or continuum mechanics does not apply and quantum mechanics is said to apply in a tangible way. There are unique opportunities to contribute to the design, controls, and analysis of systems that are particularly suited to mechanical engineering. Within the derivations of classical mechanics are assumptions that limit its use to bulk engineering. These assumptions are examined to determine what principles can be extended to smaller scales. To allow engineers to do their job at these scales, it is necessary to understand strength and how changing scales affects the strength of material this leads directly to sets of variables necessary for engineering at any scale. Potential field theory is an old method that is experiencing a resurgence of interest. Potential fields are used to study quantum mechanics at the atomic scale, crack and dislocation mobility at the micro-scale, and even bulk analysis. It encompasses many problems that can be formulated using partial differential equations. These series solutions are well suited for computerized numerical approximation. Because of recent advances in computational abilities, potential field theory deserves a fresh look as a candidate for multiscale modeling and as the math that binds each level together

Multiscale analysis and damage tolerance of carbon fibre biaxial non-crimp-fabric composites

Composite materials are attractive in aerospace and automotive applications due to their high stiffness-to-weight ratios. Non-crimp-fabric (NCF) reinforced composites have been receiving attention in the composite market due to their cost-effectiveness and excellent mechanical performance. However, compared to traditional unidirectional laminates (UD), NCF composites have different mechanical behaviours due to their heterogeneous internal structures. The ability to predict the mechanical performance of NCF composites is necessary for a robust and reliable design. The objective of this PhD research was to develop numerical methods to predict the in-plane mechanical behaviour and damage tolerance of the carbon fibre reinforced biaxial NCF composites. The in-plane mechanical behaviours of NCF composites were investigated at different scales by conducting multiscale analyses in the LS-DYNA finite element (FE) software. The macroscopical FE modelling results were validated by a series of in-plane characterisation tests of biaxial NCF composites. The compression-after-impact (CAI) test was adopted to assess the damage tolerance of the composite laminates. The complex failure mechanisms of NCF composites involved in a CAI failure process were comprehensively studied by experimental methods. The experimental results contributed to the validation of FE models to predict the low-velocity impact (LVI) and CAI behaviours of NCF composites in LS-DYNA. Furthermore, different laminate designs were employed to change the CAI behaviour of NCF composites by altering layup sequence and ply-level hybridisation. An optimised scheme was proposed to enhance the CAI behaviour of NCF composites, providing a practical guide to damage tolerance design.Open Acces

Multi-Scale Traction Dynamics in Obliquely Impacted Polymer-Metal Targets

The characterization of polymer behavior at high strain-rates is a criticalarea of research driven by their use as adhesives, structural components, or evenas binders for energetic systems. Current experimentation has been limited to eitherlow strain-rate mechanical testing or plane strain (uniaxial) impact testing.As such, more complicated loading conditions at high strain-rate have remainedunexplored. Particularly of interest include the rate dependencies of polymerstrength as well as interface traction behaviors like adhesion and dynamic frictionphenomena. To investigate these characteristics, fully dense, high durometer,polyurethane (PUR) and epoxy were subjected to combined pressure-shear loadingvia oblique impact experiments. Two distinct configurations of oblique impactexperiments were used to investigate both shear strength and friction behaviors.Oblique impact resulted in high strain-rate (105s��1) combined normal (pressure)and shear stress loading of polymers with magnitudes of approximately 800(pressure) and 120 MPa (shear) respectively, depending on impact velocity and angle.Material response was inferred from free-surface particle velocities measuredusing transverse photon Doppler velocimetry techniques. The impact of a 7075-T6aluminum projectile onto a target consisting of a thin polymer specimen confinedbetween two anvils of the same aluminum allowed for the measurement of polymershear strengths. Strength was found to increase with higher confining normalstress, increasing strength by almost an order of magnitude. This normal stress(or pressure) dependence was observed to have a greater effect on strengtheningthan that of strain-rate alone. The oblique impact of a polymer projectile against a7075-T6 aluminum target provided both a quantification for coefficient of friction,m and additional material strength information. Polyurethane and epoxy werefound to have m values of approximately 0.11 and 0.26 respectively, though withviscoelastic effects distorting the latter. Results are in agreement with previous experimentation.The role of adhesion is discussed and, in agreement with literature,it is speculated that the strength of adhesion is greater than or equal to that of thebulk polymer