On the Generalization of Reissner Plate Theory to Laminated Plates,Part II: Comparison with the Bending-Gradient Theory

In the first part of this two-part paper (Lebée and Sab in On the generalization of Reissner plate theory to laminated plates, Part I: theory, doi: 10.1007/s10659-016-9581-6, 2015), the original thick plate theory derived by Reissner (J. Math. Phys. 23:184–191, 1944) was rigorously extended to the case of laminated plates. This led to a new plate theory called Generalized-Reissner theory which involves the bending moment, its first and second gradients as static variables. In this second paper, the Bending-Gradient theory (Lebée and Sab in Int. J. Solids Struct. 48(20):2878–2888, 2011 and 2889–2901, 2011) is obtained from the Generalized-Reissner theory and several projections as a Reissner–Mindlin theory are introduced. A comparison with an exact solution for the cylindrical bending of laminated plates is presented. It is observed that the Generalized-Reissner theory converges faster than the Kirchhoff theory for thin plates in terms of deflection. The Bending-Gradient theory does not converge faster but improves considerably the error estimate.     

Two-subsystem thermodynamics for the mechanics of aging amorphous solids

The effect of physical aging on the mechanics of amorphous solids as well as mechanical rejuvenation is modeled with nonequilibrium thermodynamics, using the concept of two thermal subsystems, namely a kinetic one and a configurational one. Earlier work (Semkiv and Hütter in J Non-Equilib Thermodyn 41(2):79–88, 2016) is extended to account for a fully general coupling of the two thermal subsystems. This coupling gives rise to hypoelastic-type contributions in the expression for the Cauchy stress tensor, that reduces to the more common hyperelastic case for sufficiently long aging. The general model, particularly the reversible and irreversible couplings between the thermal subsystems, is compared in detail with models in the literature (Boyce et al. in Mech Mater 7:15–33, 1988; Buckley et al. in J Mech Phys Solids 52:2355–2377, 2004; Klompen et al. in Macromolecules 38:6997–7008, 2005; Kamrin and Bouchbinder in J Mech Phys Solids 73:269–288 2014; Xiao and Nguyen in J Mech Phys Solids 82:62–81, 2015). It is found that only for the case of Kamrin and Bouchbinder (J Mech Phys Solids 73:269–288, 2014) there is a nontrivial coupling between the thermal subsystems in the reversible dynamics, for which the Jacobi identity is automatically satisfied. Moreover, in their work as well as in Boyce et al. (Mech Mater 7:15–33, 1988), viscoplastic deformation is driven by the deviatoric part of the Cauchy stress tensor, while for Buckley et al. (J Mech Phys Solids 52:2355–2377, 2004) and Xiao and Nguyen (J Mech Phys Solids 82:62–81, 2015) this is not the case.     

A Bending-Gradient model for thick plates,Part II: Closed-form solutions for cylindrical bending of laminates

In the first part (Lebée and Sab, 2010a) of this two-part paper we have presented a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called Bending-Gradient plate theory is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case when the plate is homogeneous. Moreover, we demonstrated that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In this paper, the Bending-Gradient theory is applied to laminated plates and its predictions are compared to those of Reissner–Mindlin theory and to full 3D (Pagano, 1969) exact solutions. The main conclusion is that the Bending-Gradient gives good predictions of deflection, shear stress distributions and in-plane displacement distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

A Dugdale-Barenblatt Crack Between Dissimilar Media

The problem of a Dugdale-Barenblatt crack between dissimilar media is treated. The corresponding singular integral equation of the second kind is solved numerically. The crack propagation criteria is deduced using the revisited Griffith theory (Francfort and Marigo in J. Mech. Phys. Solids 46(8), 1319–1342, 1998). A parametric study is performed. An important result is the absence of the non physical phenomenon of overlapping of the crack faces near the ends observed in (England in J. Appl. Mech. 32(2), 400–402, 1965).     

Spontaneous bending of pre-stretched bilayers

We discuss spontaneously bent configurations of pre-stretched bilayer sheets that can be obtained by tuning the pre-stretches in the two layers. The two-dimensional nonlinear plate model we use for this purpose is an adaptation of the one recently obtained for thin sheets of nematic elastomers, by means of a rigorous dimensional reduction argument based on the theory of Gamma-convergence (Agostiniani and DeSimone in Meccanica. doi: 10.1007/s11012-017-0630-4, 2017, Math Mech Solids. doi: 10.1177/1081286517699991, arXiv:1509.07003, 2017). We argue that pre-stretched bilayer sheets provide us with an interesting model system to study shape programming and morphing of surfaces in other, more complex systems, where spontaneous deformations are induced by swelling due to the absorption of a liquid, phase transformations, thermal or electro-magnetic stimuli. These include bio-mimetic structures inspired by biological systems from both the plant and the animal kingdoms.     

Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Thermodynamic models for viscoplastic solids are often formulated in the context of continuum thermodynamics and the dissipation principle. The purpose of the current work is to show that models for such material behavior can also be formulated in the form of a General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC), see, e.g., Grmela and Öttinger (Phys Rev E, 56:6620–6632, 1997), Öttinger and Grmela (Phys Rev E, 56:6633–6655, 1997), Grmela (J Non-Newtonian Fluid Mech, 165:980–986, 2010). A GENERIC combines Hamiltonian-dynamics-based modeling of time-reversible processes with Onsager–Casimir-based modeling of time-irreversible processes. The result is a model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to model complex fluids, it has recently been applied to anisotropic inelastic solids in Eulerian (Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:45–52, 2008; Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:53–65, 2008; Hütter and Tervoort, in Adv Appl Mech, 42:254–317, 2008) and Lagrangian (Hütter and Svendsen, in J Elast 104:357–368, 2011) settings, as well as to damage mechanics. For simplicity, attention is focused in the current work on the case of thermoelastic viscoplasticity. Central to this formulation is a GENERIC-based form for the viscoplastic flow rule. A detailed comparison with the formulation based on continuum thermodynamics and the dissipation principle is given.     

A Bending-Gradient model for thick plates. Part I: Theory

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In part two (Lebée and Sab, 2011), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner–Mindlin theory and to full 3D Pagano’s exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

Topological and Measure-Theoretical Entropies of Nonautonomous Dynamical Systems

We consider the dynamics of a nonautonomous dynamical system determined by a sequence of continuous self-maps \(f_n:X \rightarrow X,\) where \( n \in {\mathbb {N}},\) defined on a compact metric space X. Applying the theory of the Carathéodory structures, elaborated by Pesin (Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago, 1997), we construct a Carathéodory structure whose capacity coincides with the topological entropy of the considered system. Generalizing the notion of local measure entropy, introduced by Brin and Katok (in: Palis (ed) Geometric Dynamics, Lecture Notes in Mathematics. Springer, Berlin 1983) for a single map, to a nonautonomous dynamical system we provide a lower and upper estimations of the topological entropy by local measure entropies. The theorems of the paper generalize results of Kawan (Nonautonomous Stoch Dyn Syst 1:26–52, 2013) and of Feng and Huang (J Funct Anal 263:2228–2254, 2012). Also, we construct a new entropy-like invariant such the entropy of a sequence \(\{f_n:X \rightarrow X\}_{n=1}^{\infty }\) of Lipschitz continuous maps with the same Lipschitz constant \(L >1,\) restricted to a subset \(Y\subset X,\) is upper bounded by Hausdorff dimension of Y multiplied by the logarithm of the Lipschitz constant L. This gives a generalizations of results of Dai et al. (Sci China Ser A 41:1068–1075, 1998) and Misiurewicz (Discret Contin Dyn Syst 10:827–833, 2004).     

Strichartz Estimates and Moment Bounds for the Relativistic Vlasov–Maxwell System

We consider the relativistic Vlasov–Maxwell system with data of unrestricted size and without compact support in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, Glassey–Schaeffer proved (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:331–354, 1998; Arch Ration Mech Anal. 141:355–374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the \({L^\infty_x}\) norms of the electromagnetic fields compared to Glassey and Schaeffer (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:355–374, 1998). In the three-dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as \({{\|p_0^{\theta} f \|_{L^{q}_{x}L^1_{p}}}}\) remains bounded for \({\theta > \frac{2}{q}}\), \({2 < q \leqq \infty}\). This improves previous results of Pallard (Indiana Univ Math J 54(5):1395–1409, 2005; Commun Math Sci 13(2):347–354, 2015).     

On a Nonlinear Model for Tumor Growth in a Cellular Medium

We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the “tumor” is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the “waste” and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum \(\Omega \) with boundary \(\partial \Omega \) both of which evolve in time. In particular, the evolution of the boundary \(\partial \Omega \) is prescibed by a given velocity \({{{\varvec{V}}}.}\) The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles (Enault in Mathematical study of models of tumor growth, 2010; Li and Lowengrub in J Theor Biol, 343:79–91, 2014) where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions (Mathematical topics in fluid dynamics. Compressible models, 1998) [see also Donatelli and Trivisa (J Math Fluid Mech 16: 787–803, 2004), Feireisl (Dynamics of viscous compressible fluids, 2014)].     

Statistical and Numerical Considerations of Backus-Average Product Approximation

In this paper, we examine the applicability of the approximation, \(\overline{f g}\approx \overline{f}\,\overline{g}\), within Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging. This approximation is a crucial step in the method proposed by Backus (J. Geophys. Res. 67(11):4427–4440, 1962), which is widely used in studying wave propagation in layered Hookean solids. According to this approximation, the average of the product of a rapidly varying function and a slowly varying function is approximately equal to the product of the averages of those two functions.Considering that the rapidly varying function represents the mechanical properties of layers, we express it as a step function. The slowly varying function is continuous, since it represents the components of the stress or strain tensors. In this paper, beyond the upper bound of the error for that approximation, which is formulated by Bos et al. (J. Elast. 127:179–196, 2017), we provide a statistical analysis of the approximation by allowing the function values to be sampled from general distributions.Even though, according to the upper bound, Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging might not appear as a viable approach, we show that—for cases representative of physical scenarios modelled by such an averaging—the approximation is typically quite good. We identify the cases for which there can be a deterioration in its efficacy.In particular, we examine a special case for which the approximation results in spurious values. However, such a case—though physically realizable—is not likely to appear in seismology, where Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging is commonly used. Yet, such values might occur in material sciences, in general, for which Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging is also considered.     

Bail-Down Test Simulation at Laboratory Scale

This paper presents a comparison of hydraulic oil conductivity obtained from interpreting bail-down test data to values calculated from theory. The bail-down tests were performed at laboratory scale, on a radial portion of a circular domain filled with calibrated sand allowing hydraulic oil conductivity to be calculated using Parker’s theoretical model (Parker et al. in Water Resour Res 23(4):618–624, 1987). The bail-down tests were interpreted using the modified Bouwer and Rice (Huntley in Ground Water 38(1):46–52, 2000) and the modified Cooper methods (Beckett and Lyverse in API Interact LNAPL Guide 2:1–27, 2002). The results show that (1) both interpretation methods from bail-down test data give similar hydraulic oil conductivities, and (2) the hydraulic oil conductivities estimated from bail-down test data agree well with the hydraulic oil conductivity predicted when using the Parker theoretical model. Overall, this paper confirms that the modified Bouwer and Rice (Huntley 2000) and the modified Cooper methods (Beckett and Lyverse 2002) are valid to estimate hydraulic oil conductivity, giving realistic values despite test conditions not meeting all the assumptions and boundary conditions of each analytical solution.     

A Rigidity Result for a Reduced Model of a Cubic-to-Orthorhombic Phase Transition in the Geometrically Linear Theory of Elasticity

In this article we study a simplified two-dimensional model for a cubic-to-orthorhombic phase transition occurring in certain shape-memory-alloys. In the low temperature regime the linear theory of elasticity predicts various possible patterns of martensite arrangements: Apart from the well known laminates this phase transition displays additional structures involving four martensitic variants—so called crossing twins.Introducing a variational model including surface energy, we show that these structures are rigid under small energy perturbations. Combined with an upper bound construction this gives the optimal scaling behavior of incompatible microstructures. These results are related to papers by Capella and Otto (Commun. Pure Appl. Math. 62(12):1632–1669, 2009; Proc. R. Soc. Edinb., Sect. A, Math. 142:273–327, 2012) as well as to a paper by Dolzmann and Müller (Meccanica 30:527–539, 1995).     

Far-Field Boundary Conditions for Calculation of Hole-Drilling Residual Stress Calibration Coefficients

The Hole-Drilling method for residual stress measurement, both in its standard version based on strain gauge rosettes (ASTM E837-08e1 2008) and its derivative using optical methods for estimating the displacement field around the hole (Baldi (2005) J Eng Mater Technol 127(2):165–169; Schajer and Steinzig (2005) Exp Mech 45(6):526–532; Schajer (2010) Exp Mech 50(2):159–168), relies on numerical calibrated coefficients (A and B) to correlate the experimentally acquired strains (displacements) with residual stress components. To estimate the A and B coefficients, two FEM (Finite Element Method) computations are required, the former related to a hydrostatic stress state, the latter to a pure shear case. Both can be implemented using either a semi-analytical approach (i.e. an axis-symmetric mesh expanded in the tangential direction using a Fourier series) or a tri-dimensional mesh, usually exploiting the double symmetry of the problem. Whatever the approach selected, the definition of constraints to be applied to the outer boundary is critical because the hole-drilling method assumes an infinite plate, thus both the usual solutions—fully constrained or free boundaries—are unable to correctly describe the theoretical situation. In the following, the problem of correct simulation of the infinite domain will be discussed and two simple and effective solutions will be proposed.     

Micromorphic approach for gradient-extended thermo-elastic–plastic solids in the logarithmic strain space

The coupled thermo-mechanical strain gradient plasticity theory that accounts for microstructure-based size effects is outlined within this work. It extends the recent work of Miehe et al. (Comput Methods Appl Mech Eng 268:704–734, 2014) to account for thermal effects at finite strains. From the computational viewpoint, the finite element design of the coupled problem is not straightforward and requires additional strategies due to the difficulties near the elastic–plastic boundaries. To simplify the finite element formulation, we extend it toward the micromorphic approach to gradient thermo-plasticity model in the logarithmic strain space. The key point is the introduction of dual local–global field variables via a penalty method, where only the global fields are restricted by boundary conditions. Hence, the problem of restricting the gradient variable to the plastic domain is relaxed, which makes the formulation very attractive for finite element implementation as discussed in Forest (J Eng Mech 135:117–131, 2009) and Miehe et al. (Philos Trans R Soc A Math Phys Eng Sci 374:20150170, 2016).     

The peculiar elongational viscosity of concentrated solutions of monodisperse PMMA in oligomeric MMA

Concentrated solutions of nearly monodisperse poly(methyl methacrylate), PMMA-270k and PMMA-86k, in oligo(methyl methacrylate), MMA o-4k and MMA o-2k, investigated by Wingstrand et al. (Phys Rev Lett 115:078302, 2015) and Wingstrand (2015) do not follow the linear-viscoelastic scaling relations of monodisperse polystyrenes (PS) dissolved in oligomeric styrene (Wagner in Rheol Acta 53:765–777, 2014a, in Non-Newtonian Fluid Mech 222:121–131, 2014b; Wagner et al. in J Rheol 59:1113–1130, 2015). Rather, PMMA-270k shows an attractive interaction with MMA, in contrast to the interaction of PMMA-86k and MMA. This different behavior can be traced back to different tacticities of the two polymers. The attractive interaction of PMMA-270k with o-4k creates pseudo entanglements, which increase the interchain tube pressure, and therefore, the solution PMMA-270k/o-4k shows, as reported by Wingstrand et al. (Phys Rev Lett 115:078302, 2015), qualitatively a similar scaling of the elongational viscosity with \( {\left(\dot{\varepsilon}{\tau}_R\right)}^{-1/2} \) as observed for polystyrene melts. For the solution PMMA-270/o-2k, this effect is only seen at the highest elongation rates investigated. The elongational viscosity of PMMA-86k dissolved in oligomeric MMA is determined by the Rouse time of the melt, as in the case of polystyrene solutions.

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Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model

Loladze et al. (Bull Math Biol 62:1137–1162, 2000) proposed a highly cited stoichiometric predator–prey system, which is nonsmooth, and thus it is extremely difficult to analyze its global dynamics. The main challenge comes from the phase plane fragmentation and parameter space partitioning in order to perform a detailed and complete global stability and bifurcation analysis. Li et al. (J Math Biol 63:901–932, 2011) firstly discussed its global dynamical behavior with Holling type I functional response and found that the system has no limit cycles, and the internal equilibrium is globally asymptotically stable if it exists. Secondly, for the system with Holling type II functional response, Li et al. (2011) fixed all parameters (with realistic values) except K to perform the bifurcation analysis and obtained some interesting phenomena, for instance, the appearance of bistability and many bifurcation types. The aim of this paper is to provide a complete global analysis for the system with Holling type II functional response without fixing any parameter. Our analysis shows that the model has far richer dynamics than those found in the previous paper (Li et al. 2011), for example, four types of bistability appear: besides the bistability between an internal equilibrium and a limit cycle as shown in Li et al. (2011), the other three bistabilities occur between an internal equilibrium and a boundary equilibrium, between two internal equilibria, or between a boundary equilibrium and a limit cycle. In addition, this paper rigorously provides all possible bifurcation passways of this stoichiometric model with Holling type II functional response.     

Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations

We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L ² norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.     

Homogenization of thick periodic plates: Application of the Bending-Gradient plate theory to a folded core sandwich panel

In a previous paper from the authors, the bounds from Kelsey et al. (1958) were applied to a sandwich panel including a folded core in order to estimate its shear forces stiffness (Lebée and Sab, 2010b). The main outcome was the large discrepancy of the bounds. Recently, Lebée and Sab (2011a) suggested a new plate theory for thick plates – the Bending-Gradient plate theory – which is the extension to heterogeneous plates of the well-known Reissner–Mindlin theory. In the present work, we provide the Bending-Gradient homogenization scheme and apply it to a sandwich panel including the chevron pattern. It turns out that the shear forces stiffness of the sandwich panel is strongly influenced by a skin distortion phenomenon which cannot be neglected in conventional design. Detailed analysis of this effect is provided.     

Numerical Study of Smoothly Perturbed Shocks in the Newtonian Limit

Previous weakly nonlinear analyses of strong shocks in the Newtonian limit have shown that the main characteristics of the cellular pattern of detonations, namely the network of triple points propagating in the transverse direction, are associated with nonlinear mechanisms which are inherent to the leading shock (Clavin and Denet, Phys. Rev. Lett. 88(4), 044,502, 2002; Clavin, J. Fluid Mech. 721, 324–339, 2013). Motivated by this theoretical analysis, experimental and numerical studies have been conducted on a smoothly perturbed Mach 1.5 shock in air, reflected from a sinusoidal wall of small amplitude (Jourdan et al., Shock Waves 13(6), 501–504, 2004; Denet et al., Combust Sci. Technol. 187, 296–323, 2015; Lodato et al., J. Fluid Mech. 789, 221–258, 2016). Under such flow conditions, the reflected shock is relatively weak and the Newtonian limit, used in the above mentioned analysis, is rather far from being met. Despite of this, the theoretical results concerning the nonlinear dynamics of the shock front were, for the most part, confirmed. In an effort to get closer to the conditions of the theoretical analysis, namely strong shocks in the Newtonian limit, a similar numerical analysis is performed in the present study where the incident Mach number is increased up to 5 and the specific heat ratio is decreased down to 1.15, leading to reflected shocks Mach numbers of about 3.2. This provides additional evidence about the main driving mechanism behind the structure of cellular detonations. Theoretical predictions regarding the spontaneous formation and transverse velocity of the triple points are further confirmed. In particular, significant improvements are observed in reproducing the theoretically predicted trajectories of the triple points. As a result of the increased Mach number of the reflected shock, stronger vortex sheets are formed within the shocked gases. This enables to better assess the impact of the molecular viscosity—a previously left open question—but also to highlight similarities with cellular detonations on a wider range of heat releases.