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Abstract This paper presents an asymptotically rigorous one-dimensional analytical formulation capable of accurately capturing the stress and strain distributions that develop within the evolving neck of bars and sheets of rate-dependent materials stretched in tension. The work is an extension of an earlier study by the authors on necking instabilities in rate-independent materials. The one-dimensional model accounts for the gradients of the stress and strain that develop as the necking instability grows. Material strain-rate dependence has a significant influence on the strain that can be imposed on a bar or sheet before necking becomes pronounced. The formulation in this paper enables a quantitative assessment of the interplay in necking retardation due to rate-dependence and that due to the development of hydrostatic tension in the neck. The connection with a much simpler long-wavelength approximation which does not account for curvature induced hydrostatic tension in the neck is also emphasized and extended. [ABSTRACT FROM AUTHOR]

Dimensional reduction is applied to derive a one-dimensional energy functional governing tensile necking localization in a family of initially uniform prismatic solids, including as particular cases rectilinear blocks in plane strain and cylindrical bars undergoing axisymmetric deformations. The energy functional depends on both the axial stretch and its gradient. The coefficient of the gradient term is derived in an exact and general form. The one-dimensional model is used to analyze necking localization for nonlinear elastic materials that experience a maximum load under tensile loading, and for a class of nonlinear materials that mimic elastic-plastic materials by displaying a linear incremental response when stretch switches from increasing to decreasing. Bifurcation predictions for the onset of necking from the simplified theory compared with exact results suggest the approach is highly accurate at least when the departures from uniformity are not too large. Post-bifurcation behavior is analyzed to the point where the neck is fully developed and localized to a region on the order of the thickness of the block or bar. Applications to the nonlinear elastic and elastic-plastic materials reveal the highly unstable nature of necking for the former and the stable behavior for the latter, except for geometries where the length of the block or bar is very large compared to its thickness. A formula for the effective stress reduction at the center of a neck is established based on the one-dimensional model, which is similar to that suggested by Bridgman (1952) . [ABSTRACT FROM AUTHOR]

Abstract: For crack growth along an interface between dissimilar materials the effect of combined modes I, II and III at the crack-tip is investigated. First, in order to highlight situations where crack growth is affected by a mode III contribution, examples of material configurations are discussed where mode III has an effect. Subsequently, the focus is on crack growth along an interface between an elastic-plastic solid and an elastic substrate. The analyses are carried out for conditions of small-scale yielding, with the fracture process at the interface represented by a cohesive zone model. Due to the mismatch of elastic properties across the interface the corresponding elastic solution has an oscillating stress singularity, and this solution is applied as boundary conditions on the outer edge of the region analyzed. For several combinations of modes I, II and III crack growth resistance curves are calculated numerically in order to determine the steady-state fracture toughness. For given values of and the minimum fracture toughness corresponds to in most of the range analyzed, but there is a range where the minimum occurs for a nonzero value of . [Copyright &y& Elsevier]

A continuum theory for elastic–plastic solids that accounts for the size-dependence of strain hardening is employed to analyze trends in the indentation hardness test. Strain gradient plasticity theory incorporates an elevation of flow stress when non-uniform plastic deformations occur at the micron scale. Extensive experimental data exists for size-dependence of indentation hardness in the micron range for conical (pyramidal) indenters, and recent data delineates trends for spherical indenters. Deformation induced by rigid conical and spherical indenters is analyzed in two ways: by exploiting an approximation based on spherically symmetric void expansion and by finite element computations. Trends are presented for hardness as a function of the most important variables in the indentation test, including the size of the indent relative to the material length parameters, the strain hardening exponent, the ratio of initial yield stress to Young''s modulus, and the geometry of the indenter. The theory rationalizes seemingly different trends for conical and spherical indenters and accurately simulates hardness data presented recently for iridium, a low yield strain/high hardening material. The dominant role of one of the material length parameters is revealed, and it is suggested that the indentation test may the best means of measuring this parameter. [Copyright &y& Elsevier]

Foreign object damage (FOD) occurs when hard, millimeter-sized objects such as gravel or sand are ingested into aircraft jet engines. Particles impacting turbine blades at velocities up to about 300 m/s produce small indentation craters which can become sites for fatigue crack initiation, severely limiting the lifetime of the blade. A framework for analyzing FOD and its effect on fatigue cracking is established in this paper. Finite element analysis is used to determine the residual stresses and geometric stress concentration resulting from FOD. The roles of material rate sensitivity and inertia are delineated. The most important non-dimensional parameters governing impact indents are identified, significantly reducing the set of independent parameters. The second step in the analysis focuses on the potency of cracks emerging from critical locations at the indents. The results have been used to address the question: When and to what extent do the residual stresses and stress concentration caused by FOD reduce the critical crack size associated with threshold fatigue crack growth? For deep indents, it is found that elastic stress concentration is the dominant factor in reducing critical crack threshold when the applied cyclic load ratio, R, is large, otherwise the residual stresses are also important. Comparisons with a set of experiments conducted in parallel with the theory show that the numerical approach can account for various phenomena observed in practice. [Copyright &y& Elsevier]

Surface instabilities on the top surface of a soft elastomeric dielectric layer bonded on its bottom to a stiff substrate are investigated for two electro-static conditions: (I) a voltage difference imposed across the top and bottom surfaces of the layer which are both conducting, and (II) a voltage difference imposed across a rigid planar electrode above the layer and the conducting top surface of the layer. In the absence of surface energy, the critical voltage associated with the onset of instability of the planar state is shown to be a surface mode with any wavelength that is short compared to the layer thickness. With no layer pre-stretch and no surface energy, the uniform state in (I) has hydrostatic compression while that of (II) has hydrostatic tension, nevertheless the critical voltage is the same for both problems if the permittivity is the same for both systems. The role of equi-biaxial pre-stretch and surface energy is presented for both problems for neo-Hookean materials and for the Gent generalization. The thrust of the study is the investigation of the post-bifurcation behavior involving nonlinear interactions among the simultaneous surface modes associated with the critical voltage. The post-bifurcation response of Problem I is a crease-like mode with the top surface probing downward towards the lower surface. The opposite occurs for Problem II with the formation of a sharp ridge protruding towards the upper electrode. The bifurcation behavior is highly unstable leading to an extreme sensitivity to small imperfections. The study complements earlier work and highlights unresolved issues. [ABSTRACT FROM AUTHOR]

The axial buckling behavior is determined for an elastic beam or rod which has a uniform curvature in its natural state, is straightened by pure bending, and clamped at its ends. Buckling can be either identical to the classical two-dimensional behavior determined by Euler, or it can be three-dimensional involving twist and deflection out of the plane of natural curvature depending on the bending and torsional stiffnesses and the natural curvature. While the classical two-dimensional buckling behavior of Euler's elastica is stable under applied load, the three-dimensional buckling behavior can be stable or unstable. Theoretical and experimental examples are presented illustrating the full range of possibilities. [ABSTRACT FROM AUTHOR]

This paper addresses localization of the deformation due to buckling that occurs immediately following the onset of bifurcation in the axisymmetric buckling of a perfect spherical elastic shell subject to external pressure. The localization process is so abrupt that the buckling mode of the classical eigenvalue analysis, which undulates over the entire shell, becomes modified immediately after bifurcation transitioning to an isolated dimple surrounded by an unbuckled expanse of the shell. The paper begins by revisiting earlier attempts to analyze the initial post-buckling behavior of the spherical shell, illustrating their severely limited range of validity. The unsuccessful attempts are followed by an approximate Rayleigh-Ritz solution which captures the essence of the localization process. The approximate solution reveals the pathway that begins at bifurcation from the classical mode shape to the localized dimple buckle. The second part of the paper presents an exact asymptotic expansion of the initial post-buckling behavior which accounts for localization and which further exposes the analytic details of the abruptness of the transition. [ABSTRACT FROM AUTHOR]

Abstract When an elastomer is compressed, its surface forms creases at a critical strain about 0.35. When a plastically deformable material is compressed, however, its surface remains smooth at much larger strains. As the smooth surface folds locally into a crease, the material around the crease loads and unloads. We show that the hysteresis of plasticity retards the formation of the crease. For a crease growing in an infinite body, the stress field is self-similar as the crease grows, and the critical strain ec is the applied strain needed to maintain the self-similar growth. We simulate the formation of creases using an elastic-plastic model with linear hardening, and characterize the degree of plasticity of the model by the ratio of the tangent modulus to elastic modulus, E t / E. A small value of E t / E leads to a large critical strain for the onset of creases. We further show experimentally that creases can form at a strain of 0.49 for high-density polyethylene (HDPE) with E t / E ∼ 0.025, but cannot form for metals (aluminum, copper, and stainless steel) with E t / E ∼ 0.001. [ABSTRACT FROM AUTHOR]

Viscoelastic spherical shells exhibit a wide range of time/rate-dependent buckling behaviors when subjected to pressure. For certain loadings, buckling can even occur after a significant time delay, termed creep buckling. To gain a thorough understanding of the nonlinear time-dependent buckling behavior of viscoelastic spherical shells, this work develops an analytical model employing the small-strain, moderate-rotation shell theory combined with a linearly viscoelastic material law. Numerical results are presented for axisymmetric spherical shells with geometric imperfections for two types of loading: a prescribed rate of volume change and a prescribed pressure that remains constant after it is applied. The first type reveals the rate-dependent behavior of viscoelastic buckling while the constant pressure loading is used to quantify creep buckling phenomena. The results show that viscoelasticity and loading rates play important roles in the load-carrying behavior of these shells, and the results for the constant pressure loading reveal an unexpected and important connection between the short-time elastic buckling limit and the long-time creep buckling limit. An imperfection sensitivity map is constructed for the constant pressure loading showing three regimes with qualitatively different behaviors: near-instantaneous buckling, creep buckling and no buckling. [ABSTRACT FROM AUTHOR]

This paper investigates issues that have arisen in experimental and theoretical studies of the stability of a dielectric elastomeric layer bonded to a stiff substrate and subject to a voltage difference across the top and bottom conducting surfaces of the layer. The role of equi-biaxial pre-stretch of the layer prior to bonding to the substrate is a central factor in the investigation. The focus is the competition between wrinkling and creasing and how this competition is affected by pre-stretch. A finite element model of the system is employed to generate wrinkling bifurcation and advanced post-bifurcation solutions in the form of localized modes, either crease-like or groove-like depending on the pre-stretch. The constitutive model includes elastic compressibility, but the formulation produces accurate solutions for nearly incompressible materials which coincide with the neo-Hookean solid in the incompressible limit. The numerical simulations reveal that localized crease solutions exist at voltages below the critical voltage for wrinkling bifurcation for equi-biaxial pre- stretches below about 2.4. In this range, the wrinkling bifurcation is highly unstable, creasing rather than wrinkling can be expected, and a discontinuous transition is predicted with a finite energy barrier that may be overcome due to the presence of surface defects. With an equi-biaxial pre-stretch greater than 2.4, a continuous transition is predicted with no energy barrier, forming a localized groove due to nonlinear interactions among the unstable wrinkling modes. [ABSTRACT FROM AUTHOR]

The mechanics of the formation and propagation of ridges on compressed stiff film/compliant substrate systems is studied theoretically and experimentally. Ridges form on bilayer systems where the elastomeric substrate is subject to a significant pre-stretch prior to attachment of the film. When the bilayer is then subject to increasing overall compressive strain, sinusoidal wrinkles first form and subsequently become unstable giving way to localized ridges with relatively large amplitudes. Two-dimensional plane strain simulations for neo-Hookean film/substrate systems reveal the transition from wrinkles to ridges under increasing compression and the reverse transition from ridges to wrinkles when the overall compression is subsequently reduced. For a significant range of pre-stretch, the two transition strains differ, and a significant hysteresis response is observed in a complete cycle of loading and unloading. The Maxwell equal-energy condition has been identified associated with co-existence of wrinkles and ridges and with the three-dimensional steady-state propagation condition for the ridges. Experiments conducted with a specially designed film/substrate loading system have been performed that confirm the essential features of ridge formation and the hysteretic behavior in loading/unloading cycles that span the two transitions. [ABSTRACT FROM AUTHOR]

Curved-sided hexagrams with multiple equilibrium states have great potential in engineering applications such as foldable architectures, deployable aerospace structures, and shape-morphing soft robots. In Part I, the classical stability criterion based on energy variation was used to study the elastic stability of the curved-sided hexagram and identify the natural curvature range for the stability of each state with circular and rectangular rod cross-sections. Here, we combine a multi-segment Kirchhoff rod model, finite element simulations, and experiments to investigate the transitions between four basic equilibrium states of the curved-sided hexagram. The four equilibrium states, namely the star hexagram, the daisy hexagram, the 3-loop line, and the 3-loop "8", carry uniform bending moments in their initial states, and the magnitudes of these moments depend on the natural curvatures and their initial curvatures. Transitions between these equilibrium states are triggered by applying bending loads at their corners or edges. It is found that transitions between the stable equilibrium states of the curved-sided hexagram are influenced by both the natural curvature and the loading position. Within a specific natural curvature range, the star hexagram, the daisy hexagram, and the 3-loop "8" can transform among one another by bending at different positions. Based on these findings, we identify the natural curvature range and loading conditions to achieve transition among these three equilibrium states plus a folded 3-loop line state for one specific ring having a rectangular cross-section with a height to thickness ratio of 4. The results obtained in this part also validate the elastic stability range of the four equilibrium states of the curved-sided hexagram in Part I. We envision that the present work could provide a new perspective for the design of multi-functional deployable and foldable structures. [ABSTRACT FROM AUTHOR]

The stability of the multiple equilibrium states of a hexagram ring with six curved sides is investigated. Each of the six segments is a rod having the same length and uniform natural curvature. These rods are bent uniformly in the plane of the hexagram into equal arcs of 120o or 240o and joined at a cusp where their ends meet to form a 1-loop planar ring. The 1-loop rings formed from 120o or 240o arcs are inversions of one another and they, in turn, can be folded into a 3-loop straight line configuration or a 3-loop ring with each loop in an "8" shape. Each of these four equilibrium states has a uniform bending moment. Two additional intriguing planar shapes, 6-circle hexagrams, with equilibrium states that are also uniform bending, are identified and analyzed for stability. Stability is lost when the natural curvature falls outside the upper and lower limits in the form of a bifurcation mode involving coupled out-of-plane deflection and torsion of the rod segments. Conditions for stability, or lack thereof, depend on the geometry of the rod cross-section as well as its natural curvature. Rods with circular and rectangular cross-sections will be analyzed using a specialized form of Kirchhoff rod theory, and properties will be detailed such that all four of the states of interest are mutually stable. Experimental demonstrations of the various states and their stability are presented. Part II presents numerical simulations of transitions between states using both rod theory and a three-dimensional finite element formulation, includes confirmation of the stability limits established in Part I, and presents additional experimental demonstrations and verifications. [ABSTRACT FROM AUTHOR]

Abstract: Wrinkling of thin stiff films on thick compliant elastomeric substrates subject to plane strain compression is considered for cases in which the substrate is pre-stretched prior to film attachment. Advanced wrinkling modes are investigated that evolve as the systems are compressed beyond the onset of the primary sinusoidal wrinkling mode. If the substrate pre-stretch is greater than about 40%, an advanced mode in the form of a series of well-spaced ridges separated by relatively flat film is observed in the simulations. Our experiments reveal a localization mode in the form of alternating packets of large and small amplitude wrinkles, but not ridges, while ridge formation has been observed in other recent experiments. Measurements of undulation amplitudes have been made for wrinkle fields of stiff films formed by oxidation of the surface of pre-stretched PDMS substrates. Simulations have been performed with a finite element model and an analytical film/substrate model. The formation of the ridge mode is a consequence of the altered nonlinearity of the substrate produced by the pre-stretch. The role of the tangential substrate stiffness in suppressing localization at the ridges is also highlighted. If there is no substrate pre-stretch, or if the substrate is pre-compressed, the primary sinusoidal mode gives way to an entirely different sequence of advanced modes usually entailing period doubling followed by folding. The nature of substrate nonlinearity that leads to ridges or folds is discussed. [Copyright &y& Elsevier]

Abstract: A family of robust stretch-dominated bimaterial lattices is introduced which combines low (or zero) thermal expansion with high stiffness, structural robustness over wide temperature ranges and manufacturing facility. This combination of properties is unavailable through any other material solution. The concept uses two constituents configured as adjoining sub-lattices. It accommodates the thermal expansion through rotation of the members of one sub-lattice. Moreover, the lattice exhibits large stiffness to weight because it is fully triangulated and does not rely on rotational resistance at the joints for structural rigidity. A wide range of constituents can be used to build the new lattices enabling many desirable properties to be incorporated, especially high strength and toughness. Examples of both planar and volumetric lattices are presented, and their thermo-mechanical properties derived. The results are verified by conducting experiments and finite element simulations on a lattice fabricated using aluminium and titanium alloy constituents. [Copyright &y& Elsevier]