Predicting the behaviour of structures under impact loads using geometrically distorted scaled models

When a scaled structure (model or replica) is used to predict the response of a full-size compound (prototype), the model geometric dimensions should relate to the corresponding prototype dimensions by a single scaling factor. However, owing to manufacturing technical restrictions, this condition cannot be accomplished for some of the dimensions in real structures. Accordingly, the distorted geometry will not comply with the overall geometric scaling factor, infringing the Π theorem requirements for complete dynamic similarity. In the present study, a method which takes geometrical distortions into account is introduced, leading to a model similar to the prototype. As a means to infer the performance of this method, three analytical problems of structures subjected to dynamic loads are analysed. It is shown that the replica developed applying this technique is able to accurately predict the full-size structure behaviour even when the studied models have some of their dimensions severely distorted.     

Scaling of structures subject to impact loads when using a power law constitutive equation

It is well known that structures subjected to dynamic loads do not follow the usual similarity laws when the material is strain rate sensitive. As a consequence, it is not possible to use a scaled model to predict the prototype behaviour. In the present study, this problem is overcome by changing the impact velocity so that the model behaves exactly as the prototype. This exact solution is generated thanks to the use of an exponential constitutive law to infer the dynamic flow stress. Furthermore, it is shown that the adopted procedure does not rely on any previous knowledge of the structure response. Three analytical models are used to analyze the performance of the technique. It is shown that perfect similarity is achieved, regardless of the magnitude of the scaling factor. For the class of material used, the solution outlined has long been sought, inasmuch as it allows perfect similarity for strain rate sensitive structures subject to impact loads.     

Role of strain-rate sensitivity in the crystal plasticity of hexagonal structures

In the first part of this paper the stress and strain-rate response of hexagonal crystal structures are examined when slip is viscoplastic according to a power law. The stress and strain-rate equi-potential surfaces are constructed and discussed as a function of the strain-rate sensitivity index m. The second part of this paper deals with the case of linear viscous slip; i.e., for the case when m is equal to one. A simple analytic solution is presented to obtain the deviatoric stress state for a given strain-rate. It is shown that the plastic spin is not zero for m = 1 in hexagonal crystal structures, contrary to the cubic case where the plastic spin vanishes. In addition, the rate of texture evolution in simple shear of a magnesium polycrystal is examined as a function of m.     

Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects

Recently it has been demonstrated that, on the basis of the separation D=D^e+D^p arising from the split of the stress power and two consistency criteria for objective Eulerian rate formulations, it is possible to establish a consistent Eulerian rate formulation of finite elastoplasticity in terms of the Kirchhoff stress and the stretching, without involving additional deformation-like variables labelled “elastic” or “plastic”. It has further been demonstrated that this consistent formulation leads to a simple essential structure implied by the work postulate, namely, both the normality rule for plastic flow D^p and the convexity of the yield surface in Kirchhoff stress space. Here, we attempt to place such an Eulerian formulation on the thermodynamic grounds by extending it to a general case with thermal effects, where the consistency requirements are treated in a twofold sense. First, we propose a general constitutive formulation based on the foregoing separation as well as the two consistency criteria. This is accomplished by employing the corotational logarithmic rate and by incorporating an exactly integrable Eulerian rate equation for D^e for thermo-elastic behaviour. Then, we study the consistency of the formulation with thermodynamic laws. Towards this goal, simple forms of restrictions are derived, and consequences are discussed. It is shown that the proposed Eulerian formulation is free in the sense of thermodynamic consistency. Namely, a Helmholtz free energy function in explicit form may be found such that the restrictions from the thermodynamic laws can be fulfilled with positive internal dissipation for arbitrary forms of constitutive functions included in the constitutive formulation. In particular, that is the case for the foregoing essential constitutive structure in the purely mechanical case. These results eventually lead to a complete, explicit constitutive theory for coupled fields of deformation, stress and temperature in thermo-elastoplastic solids at finite deformations.     

Theoretical properties of cubic crystals at arbitrary pressure—I. Density and bulk modulus

The theoretical elastic behaviour of simple monatomic cubic crystals at arbitrary pressure is to be presented in a series of papers. In the present paper, general expressions are derived for calculating the pressure P and the bulk modulus κ as a function of the all-round stretch λ for crystals in which the interatomic interaction energies are modelled by pairwise functions φ. With the aid of a particular family of functions φ, calculations are carried out for the three cubic structures, and a detailed study is made of the influence of crystal structure and explicit nature of φ upon the theoretical elastic behaviour under pressure loading. Calculations are also made of higher order derivatives of P(λ) and gk(λ) in the reference state (i.e. at λ = 1) and the “higher order moduli” thus calculated are used to formulate series expansion approximations to the functions P(λ) and κ(λ). Values of P(λ) and κ(λ) in the series approximations, based upon successively higher order moduli (evaluated in the reference state), are compared with the corresponding “exact” values evaluated in the current state. The theoretical results are useful as empirical relationships modelling the elastic behaviour of crystals at arbitrary pressure.     

Inelastic seismic demand of low-rise buildings with soil-flexibility

Relying on the ductile behaviour of structures during earthquake, building codes introduce response reduction factors (R) to reduce design forces in earthquake resistant design. However, applicability of such factors has not been systematically explored for low-rise buildings with stiff periods. Present study is an attempt to address this issue. Both elasto-plastic and degrading hysteresis behaviour for lateral load-resisting structural elements are considered herein, while sub-soil is idealized as linear and elasto-plastic in parallel. The study recognizes that inelastic response for short period systems is very sensitive to R and may be phenomenally amplified even for small R due to soil-structure interaction implying restrictive applicability of dual-design philosophy. Limited study on the plan-asymmetric low-rise buildings depicts that inelastic response of the asymmetric structure relative to its symmetric counterpart is not appreciably influenced due to soil-structure interaction (SSI). The study also confirms that equivalent single story model characterized by the lowest period rather than the fundamental one of the real system tends to yield conservative estimation of inelastic demand at least for the short-period systems.     

Finite deformation and stability behaviour of cylindrical membranes subjected to symmetric line loads2

An ‘exact’ analysis of the complete non-linear load-deflection and stability behaviour of cylindrical membranes without end ‘shear walls’ subjected to longitudinal Symmetric line loads is presented. The analysis includes low as well as high profile structures. In order to determine the lateral stability behaviour, infinitesimal lateral displacements are superimposed on the symmetric finite deflection field.Results indicate that such structures may experience vertical and/or lateral instability depending on their initial geometry. Membranes with initial central angles, θ₀ ? 90° are stable, both vertically and laterally, for all load values. For 42.23° < θ₀ < 90° the structure is laterally stable but becomes vertically unstable at a certain ‘limit load’ W?_V while for θ₀< 42.23°, the structure becomes laterally unstable at a load value w?_l<W?_V.The analysis admits contact between vertical segments of the membrane to either side of the line load as well as between the membrane and the horizontal surface next to the supports.     

箱型结构内部爆炸破坏研究进展

结构内部爆炸破坏机理和规律是常规武器毁伤效能预测与评估、建筑物和舰船抗爆防护设计的重要支撑。基于结构内爆炸载荷、内部爆炸作用下结构塑性响应、内部爆炸作用下箱壁结构破坏模式、内部爆炸作用下多箱型结构破坏模式和分布四个方面详细论述了箱型结构内部爆炸破坏的研究现状及存在的问题,并对内部爆炸后续研究给出了建议。建议研究并建立更加复杂的结构内部爆炸载荷和破坏效应描述模型、内部爆炸作用下箱壁的动力响应机理、多箱型结构与内部爆炸波产生的耦合效应、内部爆炸作用下结构的破坏模式和破坏范围的快速准确预测方法等。     

Polar plasticity of thin-walled composites made of ideal fibre-reinforced material

The present study investigates the influence that polar material response has on the plastic behaviour of thin-walled structures made of ideal fibre-reinforced materials (Spencer, 1972); or, equivalently, on the response of thin-walled fibrous composites within the first branch of the matrix dominated form (MDM) of the bimodal theory of plasticity (Soldatos, 2011, Dvorak and Bahei-El-Din, 1987). The plasticity studies mentioned above assume that fibres are infinitely thin and, therefore, perfectly flexible. They possess no bending stiffness and, hence, their negligible bending resistance cannot influence the developed stress state, which is accordingly described by a symmetric stress tensor. In contrast, the present study considers that if fibres resistant in bending are embedded in a material at high volume concentrations, their flexure produces couple-stress and, as a result of this kind of polar material behaviour, the stress tensor becomes non-symmetric. Under plane stress conditions that dominate behaviour of thin-walled structures, the stress-space and, therefore, conditions of plastic yield and relevant yield surfaces are thus four-dimensional. However, shapes and properties of initial yield surfaces relevant to the f₁-branch of MDM are studied comprehensively by considering their projection on particular planes of such a four-dimensional stress-space. It then becomes easier understood that, in the regime of polar material response, a thin-walled structure made of ideal fibre-reinforced material deforms plastically when suitable combinations of shear stress values are reached simultaneously, rather than when only one of two unequal shear stress components reaches some maximum absolute value. Thus, polar material plasticity dismisses the conventional concept of material yield stress in shear and replaces it with a pair of two independent yield moduli. Existence of the latter is perceived as a theoretical justification of the expectation that, due to the presence of fibres, two rather than one shear yield parameters of the composite should be present and accountable for. The non-zero values of those parameters are shown to exert paramount influence on the form of the yield surface of the ideal fibre-reinforced material of interest.     

Pseudo-bistable pre-stressed morphing composite panels

The pseudo-bistable phenomenon already shown to exist in the case of spherical domes is demonstrated in pre-stressed composite panels. This new concept for morphing structures uses intrinsic material viscoelasticity to actuate the structure passively between its different states. A pseudo-bistable structure is first snapped into a buckled state and allowed to relax under a constant strain. Once the actuation is removed, the structure remains in its buckled configuration for a period of time, before quickly returning to its initial state. In this paper, the principles of the pseudo-bistable behaviour are first outlined using a discrete truss model. An equivalent numerical model is then used to show how the time-dependent behaviour imparted to the structure can be controlled by the choice of the pre-straining boundary conditions. Next, the effect of a composite layup on the pseudo-bistable behaviour is shown, and a volume fraction limit is given. Finally, preliminary experimental results confirm the numerical simulations.     

Application of the NSCD method to analyse the dynamic behaviour of stone arched structures

In this paper, a masonry arch is simulated in order to assess both its structural and seismic vulnerability. The non-smooth contact dynamics (NSCD) computational method is used to simulate this type of structure as a collection of bodies under the hypothesis of unilateral constraints and frictional contact, with or without cohesion. Sinusoidal oscillations in three dimensions and real earthquake data have been applied to the supporting base element of the arch model.The primary aim of this study is to better understand the dynamic behaviour of the masonry arch, a typical component of historic unreinforced masonry (URM) structures. This study also assesses the influences of the input parameters on the mechanical and dynamic behaviour of the arch structure. Its collapse mechanism is studied for both cohesive and non-cohesive contact.In addition, we examine the behaviour under seismic loading of the Arles aqueduct, a real historical arched structure located in the south-east of France. Significant information can be obtained from the comparison of the results of advanced numerical analysis, taking into account the precise geometry of the model, the mechanical characteristics of the materials and the observation of the in situ monuments after their collapse.     

Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture

Because the uncertainty in current empirical safety factors for structural strength is far larger than the relative errors of structural analysis, improvements in statistics offer great promise. One improvement, proposed here, is that, for quasibrittle structures of positive geometry, the understrength factors for structural safety cannot be constant but must be increased with structures size. The statistics of safety factors has so far been generally regarded as independent of mechanics, but further progress requires the cumulative distribution function (cdf) to be derived from the mechanics and physics of failure. To predict failure loads of extremely low probability (such as 10^-6 to 10^-7) on which structural design must be based, the cdf of strength of quasibrittle structures of positive geometry is modelled as a chain (or series coupling) of representative volume elements (RVE), each of which is statistically represented by a hierarchical model consisting of bundles (or parallel couplings) of only two long sub-chains, each of them consisting of sub-bundles of two or three long sub-sub-chains of sub-sub-bundles, etc., until the nano-scale of atomic lattice is reached. Based on Maxwell-Boltzmann distribution of thermal energies of atoms, the cdf of strength of a nano-scale connection is deduced from the stress dependence of the interatomic activation energy barriers, and is expressed as a function of absolute temperature T and stress-duration τ (or loading rate 1/τ). A salient property of this cdf is a power-law tail of exponent 1. It is shown how the exponent and the length of the power-law tail of cdf of strength is changed by series couplings in chains and by parallel couplings in bundles consisting of elements with either elastic-brittle or elastic-plastic behaviors, bracketing the softening behavior which is more realistic, albeit more difficult to analyze. The power-law tail exponent, which is 1 on the atomistic scale, is raised by the hierarchical statistical model to an exponent of m=10 to 50, representing the Weibull modulus on the structural scale. Its physical meaning is the minimum number of cuts needed to separate the hierarchical model into two separate parts, which should be equal to the number of dominant cracks needed to break the RVE. Thus, the model indicates the Weibull modulus to be governed by the packing of inhomogeneities within an RVE. On the RVE scale, the model yields a broad core of Gaussian cdf (i.e., error function), onto which a short power-law tail of exponent m is grafted at the failure probability of about 0.0001-0.01. The model predicts how the grafting point moves to higher failure probabilities as structure size increases, and also how the grafted cdf depends on T and τ. The model provides a physical proof that, on a large enough scale (equivalent to at least 500 RVEs), quasibrittle structures must follow Weibull distribution with a zero threshold. The experimental histograms with kinks, which have so far been believed to require the use of a finite threshold, are shown to be fitted much better by the present chain-of-RVEs model. For not too small structures, the model is shown to be essentially a discrete equivalent of the previously developed nonlocal Weibull theory, and to match the Type 1 size effect law previously obtained from this theory by asymptotic matching. The mean stochastic response must agree with the cohesive crack model, crack band model and nonlocal damage models. The chain-of-RVEs model can be verified and calibrated from the mean size effect curve, as well as from the kink locations on experimental strength histograms for sufficiently different specimen sizes.     

The effective prediction of wave-in-deck loads

The present paper concerns the extreme wave loads acting on an offshore structure; specifically the wave-in-deck loading component that arises when the height of an incident wave crest exceeds the elevation of the topside structure. In this case wave inundation occurs, the resulting loads on the topside structure represent a significant part of the total wave load. A new model for the effective prediction of this important loading component is presented. This is based upon the conservation of momentum, is formulated in a Lagrangian frame of reference, can incorporate any incident wave form, and takes due account of the porosity (or openness) of the topside structure. Comparisons between the model predictions and wide-ranging laboratory observations are shown to be in good agreement; the latter based upon deterministic focused wave events that are known to be representative of the largest waves arising in realistic sea-states. In addition, comparisons are also made with independent cfd calculations. Taken together, the proposed model is shown to accommodate changes in the spectral shape, the spectral peak period, the incident crest elevation (and hence the level of inundation), the directional spread of the incident waves, and the porosity of the topside structure. Importantly, this agreement applies to both non-breaking and breaking waves, involves no empirical calibration, and can be achieved with limited computational resources. As such, the model is ideally suited to design/re-assessment calculations in which the reliability of any offshore structure must be based upon a rigorous assessment of the long-term distribution of the total wave loads, including any wave-in-deck loading component.     

First applications of a novel unified model for global and local buckling of sandwich columns

Due to the intrinsic heterogeneity of sandwich structures, phenomena at various scales can co-exist in these layered-like assembly of thick-soft and thin-stiff materials. Especially under in-plane compression loadings, geometrical instabilities can occur at both global (structure) and local (skins) scales. Therefore, the in-plane compressive response of sandwich structures is of major concern in designing structural applications. In the present paper, the first applications of a novel unified model for sandwiches are presented, with closed-form solutions for both global and local buckling. For the perfect structure, analytical critical loads are extracted for a simply supported beam, through the calculation of two eigenvalues leading to three buckling modes: it appears that the eigenvalue associated with the antisymmetrical mode can correspond to the occurrence of either global or local (wrinkling) buckling. These global and local loads from the present unified model are shown to compare very well with the predictions given by the most complete specific models from the literature. Moreover, it is shown that conversely to the classical models, our approach yields critical loads that depend only on rigorous well-founded mechanical hypotheses. The simple but general analytical expressions from the unified model permit to select quickly configurations against local and global buckling. In this simplified framework, conclusions can be drawn from this unified model capable of properly predicting the phenomena at both scales. This simplified study is essential in getting an insight in the role played by each geometrical and material parameter, the combination of which is of importance for subsequent non-linear interactive post-buckling analyses (Léotoing et al., 2001).     

On fatigue crack growth in ductile materials by crack-tip blunting

One of the basic mechanisms for fatigue crack growth in ductile metals is that depending on crack-tip blunting under tensile loads and re-sharpening of the crack-tip during unloading. In a standard numerical analysis accounting for finite strains it is not possible to follow this process during many cycles, as severe mesh distortion at the crack-tip results from the huge geometry changes developing during the cyclic plastic straining. In the present numerical studies, based on an elastic-perfectly plastic material model, crack growth computations are continued up to 200 full cycles by using remeshing at several stages of the plastic deformation. Three different values of the load ratio R=K_min/K_max are considered. It is shown that the crack-tip opening displacement, CTOD, typically undergoes a transient behaviour, with no crack closure during many cycles, before a steady-state cycling with crack closure at the tip starts to gradually develop.     

Mechanics of fish skin: A computational approach for bio-inspired flexible composites

Natural materials and structures are increasingly becoming a source of inspiration for the design novel of engineering systems. In this context, the structure of fish skin, made of an intricate arrangement of flexible plates growing out of the dermis of a majority of fish, can be of particular interest for materials such as protective layers or flexible electronics. To better understand the mechanics of these composite shells, we introduce here a general computational framework that aims at establishing a relationship between their structure and their overall mechanical response. Taking advantage of the periodicity of the scale arrangement, it is shown that a representative periodic cell can be introduced as the basic element to carry out a homogenization procedure based on the Hill-Mendel condition. The proposed procedure is applied to the specific case of the fish skin structure of the Morone saxatilis, using a computational finite element approach. Our numerical study shows that fish skin possesses a highly anisotropic response, with a softer bending stiffness in the longitudinal direction of the fish. This softer response arises from significant scale rotations during bending, which induce a stiffening of the response under large bending curvature. Interestingly, this mechanism can be suppressed or magnified by tuning the rotational stiffness of the scale-dermis attachment but is not activated in the lateral direction. These results are not only valuable to the engineering design of flexible and protective shells, but also have implications on the mechanics of fish swimming.     

Numerical investigation of the slope discontinuities in large deformation finite element formulations

In many multibody system applications, the system components are made of structural elements that can have different orientations, leading to slope discontinuities. In this paper, a numerical investigation of a new procedure that can be used to model structures with slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is presented. This procedure can be applied to model slope discontinuities in the case of commutative rotations of gradient deficient elements that are used for modeling thin beam and plate structures. An important special case to which the proposed procedure can be applied is the case of all planar gradient deficient ANCF finite elements. The use of the proposed method leads to a constant orthogonal element transformation that describes an arbitrary initial configuration. As a consequence, one obtains, in the case of large commutative rotations and large deformations, a constant mass matrix for structures which have complex geometry. The procedure used in this investigation to model slope discontinuities requires the use of the concept of the intermediate finite element coordinate system. For each finite element, a new set of gradient coordinates that define, at the discontinuity node, the element deformation with respect to the intermediate element coordinate system is introduced. These new gradient coordinates are assumed to be equal for the two finite elements at the point of intersection. That is, the change of the gradients of two elements at the intersection point from their respective intermediate initial reference configuration is assumed to be the same. This procedure leads to a set of linear algebraic equations that define the orthogonal transformation matrix for the finite element. Numerical examples are presented in order to demonstrate the use of the proposed procedure for modeling slope discontinuities.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.