Influence of anisotropic elasticity on the mechanical properties of fivefold twinned nanowires

Previous atomistic simulations and experiments have shown an increased Young's modulus and yield strength of fivefold twinned (FT) face-centered cubic metal nanowires (NWs) when compared to single crystalline (SC) NWs of the same orientation. Here we report the results of atomistic simulations of SC and FT Ag, Al, Au, Cu and Ni NWs with diameters between 2 and 50 nm under tension and compression. The simulations show that the differences in Young's modulus between SC and FT NWs are correlated with the elastic anisotropy of the metal, with Al showing a decreased Young's modulus. We develop a simple analytical model based on disclination theory and constraint anisotropic elasticity to explain the trend in the difference of Young's modulus between SC and FT NWs. Taking into account the role of surface stresses and the elastic properties of twin boundaries allows to account for the observed size effect in Young's modulus. The model furthermore explains the different relative yield strengths in tension and compression as well as the material and loading dependent failure mechanisms in FTNWs.     

Coaxiality of strain and stress in anisotropic linear elasticity

For a linearly elastic anisotropic body there are at least two rotations of the principal axes of strain such that the stress and strain tensors become coaxial. These rotations correspond to critical points for the stored energy, viewed as a function of the relative orientation between the body and the strain tensor.Supported by Gruppo Nazionale per la Fisica Matematica of C.N.R. (Italy).     

Optimization of the strain energy density in linear anisotropic elasticity

The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.     

On the directional dependence of turbulence properties in anisotropic turbulent flows

 An examination is made of the consequences of a zero correlation between fluctuations in velocity level and flow direction in a stationary anisotropic turbulent flow. This zero correlation results from the fact that in a stationary turbulent flow fluctuations in both velocity level and flow direction are entirely random processes. In this paper this is considered to be an inherent property of stationary flow and utilised to simplify both the computation and the measurement of anisotropic turbulence. This new method shows several advantages compared to earlier methods. First, the spatial distribution of the flow turbulence has been shown to be a trigonometric function of the spatial parameter. Second, the relationship between Reynolds normal and shear stresses has been established. Third, when a two-dimensional flow field is concerned, two measurements using a one-component LDA system are sufficient to identify the turbulence. Until now it has been considered that three such measurements were necessary. The feasibility of the method, known as zero correlation method, has been analysed and demonstrated by experiment. Received: 29 January 1997/Accepted: 30 July 1997     

The completeness and a new derivation of the Stroh formalism of anisotropic linear elasticity

In this paper we present a new, simpler and unified derivation of the Stroh formalism of anisotropic linear elasticity, for both nondegenerate and degenerate cases. It is based on the potential representation and Jordan canonical representation theorems. The completeness of the Stroh formalism is proved in the derivation process itself. This new approach is also extended to piezoelastic problems. Besides, we show that the eigenvalues of the fundamental elastic matrix in planar anisotropic elasticity are always distinct, except for the case of isotropy. The project supported by the National Natural Science Foundation of China (19525207 and 19891180).     

On the failure of hyperbolicity in elasticity

The failure of hyperbolicity leading to the instability as an ill-posedness of the Cauchy problem is investigated for elasticity. The criteria for that instability are derived in terms of the potential energy as a function of the strain invariants. The theory is illustrated by examples.     

Equations of the linear theory of elasticity of anisotropic materials,reduced to three independent wave equations

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 6, pp. 143–150, November–December, 1994.     

On the convergence of the semi-discrete incremental method in nonlinear,three-dimensional,elasticity

The effect of the application of an incremental method is the approximation of the three-dimensional nonlinear equations of finite elasticity by a sequence of linear problems. We give here sufficient conditions which guarantee the convergence of such a method.     

On the calculation of some definite integrals in antenna theory

Summary The mutual radiation resistance of two identical linear antennae is usually calculated by using the e.m.f. method. However, in a recent paper it has been demonstrated by the author that sometimes a simple expression for the mutual radiation resistance of two antennae, derived on the basis of the Poynting vector method, may be used with advantage. In most cases, however, the results obtained by using this modified Poynting vector method contain integrals which are difficult to express by known functions. By equating the expression for the mutual radiation resistance derived on the basis of the Poynting vector method with that obtained by the e.m.f. method we may therefore obtain a useful mathematical formula expressing a definite integral by known functions. This heuristic procedure for generating formulae for some definite integrals has been used in several simple cases of pairs of antennae. Only an outline of the rather bulky calculations has been given here, details being reserved for separate papers on some new antenna formulae. Although the paper is mainly concerned with demonstrating the above mentioned method for generating integral formulae, the compilation of formulae for mutual radiation resistance of antennae, mostly not published before, should be of interest for those working in antenna theory.     

Symmetrization of the tensor operator of the compatibility equations in stresses in the anisotropic theory of elasticity

The general form of the term whose addition to the left-hand side of the compatibility equation in stresses in anisotropic elasticity symmetrizes the rank four differential tensor operator of these equations is obtained. In the case of an arbitrary type of anisotropy, this term contains two arbitrary parameters of dimension of elastic compliances. The symmetrized compatibility equations themselves contain only one of these parameters, and the combination of the terms with this parameter can be separated from the terms containing the tensor of elastic compliances.     

On the completeness of solutions of Boussinesq,Timpe, Love,and Michell in axisymmetric elasticity

The completeness of Love's solution was proved by Noll, Gurtin, and Carlson if the meridional half-section is z-convex. In this paper it is pointed out that the condition of z-convexity is unnecessary for the completeness of Love's solution.     

On the (u, p) problem in the theory of elasticity

A problem of the theory of elasticity is considered for a body with vectors of displacements u and loads p simultaneously defined on one part of the body and with undefined conditions on the remaining part of the body. For a doubly connected domain, where the vectors u and p are set on one of its boundaries (inner or outer), an iterative method based on reduction of the initial problem to a sequence of mixed problems is justified. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 100–103, May–June, 2006.     

Some properties of the set of fourth-order tensors,with application to elasticity

In this paper the algebraic structure of the set of fourth-order tensors is examined. Special emphasis is given to the two different tensor products which can be defined over this set, when it is regarded as a vector space and as the set of all linear transformations over a vector space, respectively.The new formalism introduced here makes it possible to write in a simple form the restrictions imposed on the elasticity tensors, in finite elasticity, by balance of angular momentum and by the principle of material frame-indifference.A reassessment of the various elasticity tensors used in the literature is presented, and some classical results are re-stated in a simple and natural way.

---

Riassunto Questo articolo tratta della struttura algebrica dell' insieme dei tensori del quarto ordine. Particolare rilievo è dato alla definizione di prodotto tensoriale tra tensori del secondo ordine, che è diversa a seconda che essi siano considerati transformazioni lineari su di uno spazio vettoriale, oppure essi stessi elementi di uno spazio vettoriale.Il nuovo formalismo introdotto rende semplice la determinazione delle restrizioni imposte ai tensori elastici, in Elasticità finita, dal bilancio della quantità di moto e dal principio di indifferenza materiale.Diventa anche possibile un riordinamento dei tensori elastici piò frequentemente usati nella letteratura. Ciò permette di ritrovare, in modo semplice e naturale, alcuni risultati classici.

---

---

Dedicated to Guido Stampacchia,in mortem