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The paper attempts to analyze the hypoelastic stress responses of isotropic solids to the right pure shear deformation. After a concise description of the right pure shear deformation, a new expression of the deformation described in forms of trigonometric functions is developed. Thereafter, the necessary deformation and rate-form tensors are formulated. In addition, a detailed derivation of the eigenvectors of the left and right Cauchy–Green deformation tensors and their eigenprojection tensors is presented. On the basis of the representation principle of the material spin tensors, a set of concise representation equations of the spin tensors for the right pure shear deformation is derived. From the derived deformation and rate-form tensors, the hypoelastic equations based on eight representative stress rates are formulated and solved, and multiple sets of analytical solutions are derived. The analytical stress solutions are plotted vs. the shear strain and compared by graphs. A unique and interesting result is that the spins for the Jaumann, Green–Naghdi, logarithmic and Ω E -based stress rates are the same under the right pure shear deformation; hence, the analytical stress responses rendered by the hypoelastic equations based on these spins are the same and consistent with the response shown by the linear isotropic elastic equation. Under the right pure shear deformation, all the corotational hypoelastic equations can render the pure shear stress state, only the pure shear stress components are not observed under the reference Cartesian frame. The hypoelastic equations related to the spatial velocity gradient cannot present the pure shear stress state. [ABSTRACT FROM AUTHOR]

For a one dimensional 2-lattice we write down the momentum equation and the equation ruling the shift vector for the dynamic case as a first order system and characterize it in terms of its hyperbolicity. We use similar arguments for problems of increasing difficulty and tackle: one dimensional 3-lattices, three dimensional 2-lattices, three dimensional 3-lattices and finally the most general case of three dimensional (n + 1) -lattices. Our approach is confined to the geometrically and materially linear elastodynamic theory and is valid for generic anisotropic materials that have the multilattice structure. The main finding is that, with the assumptions adopted, the presence of each shift vector is related with zero eigenvalues when the system is written as a first order system. [ABSTRACT FROM AUTHOR]

Motivated from standard procedures in linear wave equations, we write the equations of classical elastodynamics as a linear symmetric hyperbolic system in terms of the displacement gradient ( u x ) and the velocity ( u t ); this is in contrast with common practice, where the stress tensor and the velocity are used as the basic variables. We accomplish our goal by a judicious use of the compatibility equations. The approach using the stress tensor and the velocity requires use of the time differentiated constitutive law as a field equation; the present approach is devoid of this need. The symmetric form presented here is based on a Cartesian decomposition of the variables and the differential operators that does not alter the Hamiltonian structure of classical elastodynamics. We comment on the differences of our approach with that using the stress tensor in terms of the differentiability of the coefficients and the differentiability of the solution. Our analysis is confined to classical elastodynamics, namely geometrically and materially linear anisotropic elasticity which we treat as a linear theory per se and not as the linearization of the nonlinear theory. We, nevertheless, comment on the symmetrization processes of the nonlinear theories and the potential relation of them with the present approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider isotropic Mindlin–Toupin strain gradient elasticity theory, in which the equilibrium equations contain two additional length-scale parameters and have the fourth order. For this theory, we developed an extended form of Boussinesq–Galerkin (BG) and Papkovich–Neuber (PN) general solutions. The obtained form of BG solution allows to define the displacement field through a single vector function that obeys the eight-order bi-harmonic/bi-Helmholtz equation. The developed PN form of the solution provides an additive decomposition of the displacement field into the classical and gradient parts that are defined through the standard Papkovich stress functions and modified Helmholtz decomposition, respectively. Relations between different stress functions and the completeness theorem for the derived general solutions are established. As an example, it is shown that a previously known fundamental solution within the strain gradient elasticity can be derived by using the developed PN general solution. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a factorization of the equilibrium equation in the theory of Toupin–Mindlin strain-gradient elasticity in the framework of quaternion analysis by using the inframonogenic operator F u = D u D and the Dirac operator of mixed sides D a u = cos a D u + sin a u D , where D is the Dirac operator in R 3 . We construct some integral representation formulae for the displacement fields. In application, we obtain a decomposition formula of the displacement fields. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose an experimental procedure for determining the characteristic length of a strain gradient model. The identification problem is studied through a virtual pull-out test of a rigid bar along the symmetry axis of a cylindrical strain gradient elastic domain. To allow an accurate parameter identification based on measured data, we investigate the effect of the characteristic length on the mechanical fields for this problem. We see a significant sensitivity of the inflection point of the displacement profile evaluated on the cross section of the cylinder, with respect to the characteristic length. By adjusting the characteristic length of the strain gradient such that the theoretical models match best with experimental measurements of the surface displacement fields, the characteristic length of the strain gradient can be estimated. In order to allow for more efficient analysis and an almost real-time parameter identification, the initial three-dimensional (3D) problem is reduced to a one-dimensional (1D) problem by exploiting the cylindrical symmetry of the problem. As will be shown, an accurate 1D finite element method (FEM) strain gradient solution can be obtained for this simplified problem. Since the cylindrical symmetry is only true in an infinitely long cylinder, specific boundary conditions are constructed on a cylinder of finite length, which is then used for the comparison of the 1D and 3D problems. Results show, however, that the structural response at the inflection point is insensitive to whether the specific boundary conditions are considered or not, which is why the 1D model can be used for parameter identification. Since the proposed approach is methodological, it can be applied to any material. As a prototype problem in this paper, we consider the case of a bar embedded in Portland cement concrete. [ABSTRACT FROM AUTHOR]

The exact solutions for the propagation of Love waves in one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) nanoplates with surface effects are derived. An electro-elastic model is developed to investigate the anti-plane strain problem of Love wave propagation. By introducing three shape functions, the wave equations and electric balance equations are decoupled into three uncorrelated problems. Satisfying the boundary conditions of the top surface on the covering layer, the interlayer interface, and the matrix, a dispersive equation with the influence of multi-physical field coupling is provided. A surface PQC model is developed to investigate the surface effects on the propagation behaviors of Love waves in quasicrystal (QC) multilayered structures with nanoscale thicknesses. A novel dispersion relation for the PQC structure is derived in an explicit closed form according to the non-classical mechanical and electric boundary conditions. Numerical examples are given to reveal the effects of the boundary conditions, stacking sequence, characteristic scale, and phason fluctuation characteristics on the dispersion curves of Love waves propagating in PQC nanoplates with surface effects. [ABSTRACT FROM AUTHOR]

We present an embedded-hybridizable discontinuous Galerkin finite element method for the total pressure formulation of the quasi-static poroelasticity model. Although the displacement and the Darcy velocity are approximated by discontinuous piece-wise polynomials, H (div) -conformity of these unknowns is enforced by Lagrange multipliers. The semi-discrete problem is shown to be stable and the fully discrete problem is shown to be well-posed. Additionally, space-time a priori error estimates are derived, and confirmed by numerical examples, that show that the proposed discretization is free of volumetric locking. [ABSTRACT FROM AUTHOR]

Moiré patterns result from setting a 2D material such as graphene on another 2D material with a small twist angle or from the lattice mismatch of 2D heterostructures. We present a continuum model for the elastic energy of these bilayer moiré structures that includes an intralayer elastic energy and an interlayer misfit energy that is minimized at two stackings (disregistries). We show by theory and computation that the displacement field that minimizes the global elastic energy subject to a global boundary constraint gives large alternating regions of one of the two energy-minimizing stackings separated by domain walls. We derive a model for the domain wall structure from the continuum bilayer energy and give a rigorous asymptotic estimate for the structure. We also give an improved estimate for the L 2 -norm of the gradient on the moiré unit cell for twisted bilayers that scales at most inversely linearly with the twist angle, a result which is consistent with the formation of one-dimensional domain walls with a fixed width around triangular domains at very small twist angles. [ABSTRACT FROM AUTHOR]

The validity of Korn’s first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn’s first inequality holds in the case ps>1ps\gt 1 for fractional W0s,p(Ω){W}_{0}^{s,p}\left(\Omega ) Sobolev fields in open and bounded C1{C}^{1}-regular domains Ω⊂Rn\Omega \subset {{\mathbb{R}}}^{n}. Also, in the case ps1ps\gt 1 follows a standard compactness approach adopted in the classical case, combined with a Hardy inequality, and a recently proven Korn second inequality by Mengesha and Scott [A Fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations, Commun. Math. Sci. 20 (2022), no. 2, 405–423]. The counterexamples constructed in the case ps

We classify all exactly stress-free solutions to the cubic-to-trigonal phase transformation within the geometrically linearized theory of elasticity, showing that only simple laminates and crossing-twin structures can occur. In particular, we prove that although this transformation is closely related to the cubic-to-orthorhombic phase transformation, all its solutions are rigid. The argument relies on a combination of the Saint-Venant compatibility conditions together with the underlying nonlinear relations and non-convexity conditions satisfied by the strain components. [ABSTRACT FROM AUTHOR]

It is well known that the conventional couple-stress theory leaves the spherical part of the couple-stress indeterminate. This indeterminacy problem is recently resolved for fibrous composites subjected to either small or large deformations and containing a single family of fibres resistant in bending (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595). However, the problem remains still unsolved in simpler cases where the implied preference material direction is not related to fibre bending resistance, and even in the simplest possible case where the polar material of interest is linearly elastic and isotropic. This communication aims (i) to show that a relevant virtual spin concept employed (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595) is further applicable in the latter case of polar linear isotropic elasticity, (ii) to demonstrate the process in which that concept thus leads to determination of the spherical part of the couple-stress, (iii) to exemplify this process by providing a couple of simple illustrative examples, (iv) to specify and discuss the reason that the outlined method meets a hurdle in cases of linear anisotropic elasticity that is due to one or more preferential material directions, and, hence, (v) to further discuss the manner in which that newly identified difficulty is currently confronted, and may thus be handled successfully. [ABSTRACT FROM AUTHOR]

A new mathematical formulation for the constitutive laws governing elastic perfectly plastic materials is proposed here. In particular, it is shown that the elastic strain rate and the plastic strain rate form an orthogonal decomposition with respect to the tangent cone and the normal cone of the yield domain. It is also shown that the stress rate can be seen as the projection on the tangent cone of the elastic stress tensor. This approach leads to a coherent mathematical formulation of the elasto-plastic laws and simplifies the resulting system for the associated flow evolution equations. The cases of one or two yield functions are treated in detail. The practical examples of the von Mises and Tresca yield criteria are worked out in detail to demonstrate the usefulness of the new formalism in applications. [ABSTRACT FROM AUTHOR]

Mechanical models of residually stressed fibre-reinforced solids, which do not resist bending, have been developed in the literature. However, in some residually stressed fibre-reinforced elastic solids, resistance to fibre bending is significant, and the mechanical behavior of such solids should be investigated. Hence, in this paper, we model the mechanical aspect of residually stressed elastic solids with bending stiffness due to fibre curvature, which up to the authors' knowledge has not been mechanically modeled in the past. The proposed constitutive equation involves a nonsymmetric stress and a couple-stress tensor. Spectral invariants are used in the constitutive equation, where each spectral invariant has an intelligible physical meaning, and hence they are useful in experiment and analysis. A prototype strain energy function is proposed. Moreover, we use this prototype to give results for some cylindrical boundary value problems. [ABSTRACT FROM AUTHOR]

The Flamant problem of a cubic quasicrystal half-plane is solved when its surface is loaded by normal and tangential concentrated forces. Its solution serves as a fundamental solution since its superposition forms other solutions. The Flamant problem is converted to an associated boundary value problem. The Fourier transform method is applied to solve the Flamant problem. Explicit expressions for the fundamental solution of the phonon and phason stresses are obtained. A comparison between the fundamental solutions of the Flamant problem for cubic quasicrystals and conventional cubic crystals is made. The influence of the presence of phason field on the phonon stress and deformation is shown in graph. [ABSTRACT FROM AUTHOR]

The paper deals with the Free Material Design (FMD) problem aimed at constructing the least compliant structures from an elastic material, the constitutive field of which plays the role of design variable in the form of a tensor valued measure λ supported in the design domain. Point-wise the constitutive tensor is referred to a given anisotropy class H , while the integral of a cost c (λ) is bounded from above. The convex p-homogeneous elastic potential j is parameterized by the constitutive tensor. The work puts forward the existence result and shows that the original problem can be reduced to the Linear Constrained Problem (LCP) known from the theory of optimal mass distribution by G. Bouchitté and G. Buttazzo. A theorem linking solutions of (FMD) and (LCP) allows to effectively solve the original problem. The developed theory encompasses several optimal anisotropy design problems known in the literature as well as it unlocks new ones. By employing the derived optimality conditions we give several analytical examples of optimal designs. [ABSTRACT FROM AUTHOR]

In this paper, we consider a conformal mapping function with complex constants coefficients to solve the problem of infinite plate weakened by a curvilinear hole. The conformal mapped is used outside the unit circle ϖ in the presence of an initial heat flowing perpendicular to the plate. All previous works, in this field, considered this problem without time and most of them regarded the conformal mapping with real constants. Here, we use complex variable method by new rational mapping to give convenient expression of Gaursat functions in applications with time effect. The hole takes different shapes that make this study applicable to many cases, such as caves, tunnels, excavations in soils or rocks. Stress components are obtained and plotted to investigate their physical meaning. Using maple 18, the different forms a hole are received and distribution of stresses in 3-D are obtained. [ABSTRACT FROM AUTHOR]

This article considers the influence of incompressibility on the compliance and stiffness constants that appear in the weakly nonlinear theory of elasticity. The formulation first considers the incompressibility constraint applied to compliances, which gives explicit finite limits for the second-, third-, and fourth-order compliance constants. The stiffness/compliance relationships for each order are derived and used to determine the incompressible behavior of the second-, third-, and fourth-order stiffness constants. Unlike the compressible case, the fourth-order compliances are not found to be dependent on the fourth-order stiffnesses. [ABSTRACT FROM AUTHOR]

In this paper, a new continuous model capable of representing a class of elastic metamaterials with strong local rotational and translational coupling is developed. This material model differs from that used in classical elasticity, as well as the micropolar continuum model. One-dimensional harmonic wave propagation is then studied based on this newly developed model, and the results are compared with those obtained from a corresponding discrete metamaterial model. It is interesting to observe that the new continuum model captures the main dynamic characteristics of the original metamaterial system and yields a dispersion relation close to that of the discrete model near the resonant frequency. This work provides a model for a class of new continuous metamaterials whose salient features cannot be captured by traditional models of continuum materials, as well as highlights the feasibility of using such continuum models to accurately predict the behaviour of discrete elastic metamaterials. [ABSTRACT FROM AUTHOR]

Our modelling of brain mechanics is based on observations of Budday and colleagues [6], who analyzed the elastic properties of human brain tissue samples under multiple loading modes. Using these data, Budday et al. determined a realistic constitutive model for brain tissue mechanics. In these studies, they found that compression and shear responses were best modelled by a non-linear one-term Ogden elasticity model, although other elasticity models are possible as well. Here we analyze the role of the elasticity model of brain tissue on the invasion speed of glioma and the resulting tissue deformation (mass effect). We present a one dimensional continuum model that couples cell dynamics to tissue mechanics. Since the mechanics of glioma-compromised brain tissue is not clear, for comprehensive studies, we incorporate both elastic and viscoelastic versions of two brain tissue elasticity models - the commonly employed linear model and the experimentally determined one-term Ogden model. For each elasticity model we identify travelling wave solutions in one dimension and calculate the corresponding invasion speeds. We find that the invasion speed is, in fact, independent of the chosen elasticity model. However, the deformations of the brain tissue, and resulting stress, between the linear and one-term Ogden models are drastically different: the Ogden model shows two orders of magnitude less deformation and three orders of magnitude less stress as compared to the linear model. Such a discrepancy might be relevant when looking at glioma-induced health complications. Statement of significance: Cancers arising from glial cells, known as gliomas, form in the spine and the brain. The spread of glioma is not fully understood, although recent studies have highlighted the role of tissue mechanics as a main factor in the invasion process. We present a one dimensional continuum model framework of glioma invasion that incorporates proliferation and invasion of glioma cells, as well as mass effects by coupling cell dynamics to tissue mechanics. We explore both elastic and viscoelastic versions of two brain tissue elasticity models - the commonly employed linear model and the experimentally determined one-term Ogden model. This is the first time the one-term Ogden model has been incorporated into a model of glioma invasion. We show that although the choice of elasticity model does not affect the invasion speed, the deformation and stress generated in the tissue are significantly different with the Ogden model producing three orders of magnitude less deformation and stress as compared to the linear model. Such a discrepancy might be relevant when looking at glioma-induced health complications.

Third- and fourth-order weakly nonlinear theories of elasticity are widely used by applied mathematicians, physicists and acousticians. Although their introduction is traced back to a paper by Signorini dated 1930, some aspects related to the rigorous mathematical derivation of the strain energy for quasi-incompressible isotropic elastic solids are not satisfactorily clear. In this paper we address these aspects and use our results to discuss the incompressible limit of some fourth-order nonlinear constitutive parameters. [ABSTRACT FROM AUTHOR]

The unusual properties of quasicrystals (QCs) have attracted tremendous attention from researchers. In this paper, a semi-analytical solution is presented for the static response of a functionally graded (FG) multilayered two-dimensional (2D) decagonal QC rectangular plate with mixed boundary conditions. Based on the elastic theory of FG 2D QCs, the state-space method is used to derive the state equations composed of partial differential along the thickness direction. Besides, the Fourier series expansion and the differential quadrature technique are utilized to simulate the simply supported boundary conditions and the mixed boundary conditions, respectively. Then, the propagator matrix which connects the field variables at the upper interface to those at the lower interface of any homogeneous layer can be derived based on the state equations. Combined with the interface continuity condition, the static response can be obtained by imposing the sinusoidal load on the top surfaces of laminates. Finally, the numerical examples are presented to verify the effectiveness of this method, and the results are very useful for the design and understanding of the characterization of FG QC materials in their applications to multilayered systems. [ABSTRACT FROM AUTHOR]

An instantaneous sub-surface disturbance in a two-dimensional elastic half-space is considered. The disturbance propagates through the elastic material until it reaches the free surface, after which it propagates out along the surface. In conventional theory, the free-surface conditions on the unknown surface are projected onto the flat plane y = 0 , so that a linear model may be used. Here, however, we present a formulation that takes explicit account of the fact that the location of the free surface is unknown a priori, and we show how to solve this more difficult problem numerically. This reveals that, while conventional linearized theory gives an accurate account of the decaying waves that travel outwards along the surface, it can under-estimate the strength of the elastic rebound above the location of the disturbance. In some circumstances, the non-linear solution fails in finite time, due to the formation of a "peakon" at the free surface. We suggest that brittle failure of the elastic material might in practice be initiated at those times and locations. [ABSTRACT FROM AUTHOR]

In this paper, we develop a method for solving planar nonstationary problems on the influence of surface pressure on an elastic-porous half-space. Using surface influence functions, we obtain an integral relation, which expresses the desired displacements with the surface pressure. A numericalanalytical algorithm is constructed. For calculating singular integrals, we derive special quadrature formulas based on the canonical regularization of integrands. [ABSTRACT FROM AUTHOR]

The eye grows during childhood to position the retina at the correct distance behind the lens to enable focused vision, a process called emmetropization. Animal studies have demonstrated that this growth process is dependent upon visual stimuli, but dependent on genetic and environmental factors that affect the likelihood of developing myopia. The coupling between optical signal, growth, remodeling, and elastic response in the eye is particularly challenging to understand. To analyse this coupling, we develop a minimal morphoelastic model of an eye growing under intraocular pressure in response to visual stimuli. Distinct to existing three-dimensional finite-element models of the eye, we treat the sclera as a thin axisymmetric hyperelastic shell which undergoes local growth in response to external stimulus. This simplified analytic morphoelastic model provides a tractable framework in which we can evaluate various emmetropization hypotheses and understand different types of growth feedback. As an example, we demonstrate that local growth laws are sufficient to tune the global size and shape of the eye for focused vision across a wide range of parameter values. [ABSTRACT FROM AUTHOR]

We consider a quasi-static fluid–structure interaction problem where the fluid is modeled by the Stokes equations and the structure is an active elastic material. More precisely, the displacement of the structure satisfies the equations of elasticity with an additional active stress, which models the presence of internal biological motors in the structure. Under smallness assumptions on the data, we prove the existence of a solution for this strongly coupled system. [ABSTRACT FROM AUTHOR]

This paper shows how the classical representation techniques for the solution of elasticity problems, based on the Green's functions, can be generalized to second-gradient continua focusing on the specific case of pantographic lattices. As these last are strongly anisotropic, the fundamental solutions of isotropic second-gradient continua involving bi-Helmholtz-type operators are not applicable. More specifically we establish the analytical fundamental solution for the linearized equations governing the equilibrium of pantographic 2D continua in the neighbourhood of the reference configuration. Moreover, by means of found novel Green's functions, it is shown that it is possible to solve aforesaid equilibrium equations by using Fredholm integral equations. It is seen that an approximated analytical solution for the standard bias test for pantographic 2D continua can be found by using judiciously the found analytical fundamental solutions. The micro-macro-asymptotic identification allows for a clear and satisfactory physical interpretation of the obtained analytical results. [ABSTRACT FROM AUTHOR]

We consider the problem of waves propagating in a viscoelastic solid. For the material properties of the solid we consider both classical and fractional differentiation in time versions of the Zener, Maxwell, and Voigt models, where the coupling of different models within the same solid are covered as well. Stability of each model is investigated in the Laplace domain, and these are then translated to time-domain estimates. With the use of semigroup theory, some time-domain results are also given which avoid using the Laplace transform and give sharper estimates. We take the time to develop and explain the theory necessary to understand the relation between the equations we solve in the Laplace domain and those in the time-domain which are written using the language of causal tempered distributions. Finally we offer some numerical experiments that highlight some of the differences between the models and how different parameters effect the results.

H-matrices play an important role in applied sciences such as numerical analysis and optimization theory. An attractive question is to identify whether a given matrix is an H-matrix. In this paper, we propose a new iterative algorithm for identifying H-matrices. We show that the proposed algorithm has linear convergence and can determine the H-matrix characterization for any given matrix. Its performance is illustrated in a set of numerical tests. [ABSTRACT FROM AUTHOR]

The peridynamic theory has been used to model and simulate numerically various kinds of mechanical behavior of solids. This work is devoted to analytical solutions of the elastodynamic behavior of linearized isotropic state-based peridynamic materials. First, we present the solutions of the dispersion relations, group velocities, and phase velocities of longitudinal and transverse waves, and examine in detail the effects of the Poisson's ratio on these properties. It is shown that the elastodynamic behavior of the state-based peridynamic material with a negative Poisson's ratio is remarkably different from that of the material with a positive Poisson's ratio. We then derive the general solutions of initial-value problems, and obtain the Green's function in a closed form. Finally, we study the evolution of a displacement discontinuity in the state-based peridynamic medium, and find that each component of the discontinuity in the three-dimensional theory varies independently according to the same vibrational mode. The results may have implications in investigations of wave propagations, including discontinuities such as phase transitions and kink propagations. [ABSTRACT FROM AUTHOR]

In this paper, first we give the definition of standard tensor. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensor. And we prove that the singular values of rectangular tensors are the special cases of the eigen-values of standard tensors related to rectangular tensors. Based on standard tensor, we present a generalized version of the weak Perron-Frobenius Theorem of nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors, we get some new results of rectangular tensors. Besides, by using the special structure of standard tensors corresponding to nonnegative rectangular tensors, we show that the largest singular value is really geometrically simple under some weaker conditions. [ABSTRACT FROM AUTHOR]

In this study, Mindlin's second strain gradient elasticity is employed to extract an estimate of the effective shear modulus of a functionally graded (FG) unidirectional fibrous composite subjected to an anti-plane mode of loading. The composite is supposed to be a periodically microstructured medium reinforced with infinitely long fibers, each of them surrounded by a coating made of a FG material. To determine the elastic field of the composite, an extended version of the equivalent inclusion method adapted with second strain gradient elasticity is utilized and, then, a unified procedure for the solution to the associated consistency equations is implemented that is applicable, in a similar manner, to any kind of two-phase, multi-phase, or FG composites. Subsequently, within a homogenization scheme, the effective shear modulus of the composite is calculated. The obtained results reflect well the size effect on the overall properties of the composite and demonstrate that the displacement field predicted by second strain gradient elasticity turns out to be a smooth function over the entire medium. [ABSTRACT FROM AUTHOR]

We study the conditioning and error analysis of novel nonlocal operators in 1D with local boundary conditions. These operators are used, for instance, in peridynamics (PD) and nonlocal diffusion. The original PD operator uses nonlocal boundary conditions (BC). The novel operators agree with the original PD operator in the bulk of the domain and simultaneously enforce local periodic, antiperiodic, Neumann, or Dirichlet BC. We prove sharp bounds for their condition numbers in the parameter δ only, the size of nonlocality. We accomplish sharpness both rigorously and numerically. We also present an error analysis in which we use the Nyström method with the trapezoidal rule for discretization. Using the sharp bounds, we prove that the error bound scales like O ( h 2 δ − 2 ) and verify the bound numerically. The conditioning analysis of the original PD operator was studied by Aksoylu and Unlu (2014). For that operator, we had to resort to a discretized form because we did not have access to the eigenvalues of the analytic operator. Due to analytical construction, we now have direct access to the explicit expression of the eigenvalues of the novel operators in terms of δ . This gives us a big advantage in finding sharp bounds for the condition number without going to a discretized form and makes our analysis easily accessible. We prove that the novel operators have ill-conditioning indicated by δ − 2 sharp bounds. For the original PD operator, we had proved the similar δ − 2 ill-conditioning when the mesh size approaches 0. From the conditioning perspective, we conclude that the modification made to the original PD operator to obtain the novel operators that accommodate local BC is minor. Furthermore, the sharp δ − 2 bounds shed light on the role of δ in nonlocal problems. [ABSTRACT FROM AUTHOR]

Using Mindlin’s general microstructural elasticity theory we derive models pertaining to a gradient elasticity based on both Laplacians of stress and strain. We prove that such Laplacian-based gradient elasticity models can also be derived on the basis of a thermodynamically consistent gradient elasticity. The proof relies upon a non-conventional thermodynamic framework. It is shown that a general analogy between linear viscoelastic solids based on spring and dashpot elements and specific gradient elasticity models can be established. The governing differential equations of motion of resulting gradient elasticity models are then formulated. By employing energy related arguments, conditions on the material parameters are derived and the appropriate concomitant boundary conditions are specified for both approaches. Finally, the considered models are compared to each other with reference to one-dimensional dispersion relations. [ABSTRACT FROM AUTHOR]

We study the equilibrium of a three-dimensional solid having a uniform thickness 2ε along a middle surface which satisfies the usual assumptions of shell theory. The solid is linearly elastic at small strains and is submitted to unilateral contact conditions with an obstacle on a part of its boundary. When ε tends to zero, the three-dimensional domain tends to a two-dimensional one, so that the contact conditions pass from a part of the boundary to the interior of the domain. We restrict our attention to the so-called bending case, that is when the shell undergoes only inextensional deformations. As a major difference with the case of a shallow shell, we get in general a coupling between the three components of the displacement in the contact conditions. The work is closed by explicit examples showing the corresponding variation of the non-penetrability condition along the surface of the shell and by comments about the model and the remaining difficulties. [ABSTRACT FROM AUTHOR]

We study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition. [ABSTRACT FROM AUTHOR]

We study nonlocal equations from the area of peridynamics on bounded domains. We present four main results. In our recent paper, we have discovered that, on R, the governing operator in peridynamics, which involves a convolution, is a bounded function of the classical (local) governing operator. Building on this, as main result 1, we construct an abstract convolution operator on bounded domains which is a generalization of the standard convolution based on integrals. The abstract convolution operator is a function of the classical operator, defined by a Hilbert basis available due to the purely discrete spectrum of the latter. As governing operator of the nonlocal equation we use a function of the classical operator, this allows us to incorporate local boundary conditions into nonlocal theories. As main result 2, we prove that the solution operator can be uniquely decomposed into a Hilbert—Schmidt operator and a multiple of the identity operator. As main result 3, we prove that Hilbert—Schmidt operators provide a smoothing of the input data in the sense a square integrable function is mapped into a function that is smooth up to boundary of the domain. As main result 4, for the homogeneous nonlocal wave equation, we prove that continuity is preserved by time evolution. Namely, the solution is discontinuous if and only if the initial data is discontinuous. As a consequence, discontinuities remain stationary. [ABSTRACT FROM AUTHOR]

We consider the interaction of a screw dislocation with a thin film-substrate interface in the anti-plane deformations of a couple stress elastic solid. We formulate and solve the corresponding boundary value problem in three main cases: when the screw dislocation is located inside the substrate; when the screw dislocation is located inside the film; and when the screw dislocation is located inside an unconfined (isolated) thin film in the absence of the substrate. In each case, we discuss the contribution of couple stresses to the interaction force acting on the dislocation. Moreover, we examine the changes in the interaction force resulting from the mismatch caused by the corresponding couple stress parameters. [ABSTRACT FROM AUTHOR]