Linear Strain Theory Research Articles

Effect of creep on high-order shear deformable beams

A powerful and new theoretical approach is used to obtain an expression for the effect of creep on reinforced concrete shear deformable beams. First, a method for Euler-Bernulli beam is proposed to represent long-term behavior of concrete beams based on linear strain theory. Secondly, a formulation is developed for analyzing the strain distribution in shear deformable concrete beams. Finally, three numerical examples are included in order to compare well-known codes with the proposed method. Comparison between proposed method, FEM, codes and experimental works demonstrate that the proposed analytical procedure can effectively simulate creep behavior in reinforced concrete beams.

Large-displacement strain theory and its application to graphene

Under the application of a force a material will deform and, hence, the crystal lattice will experience strain. This induced strain will alter the electronic properties of the material. In particular, strain in graphene generates an artificial vector potential that, if spatially varying, admits a pseudomagnetic field. Current theories for spatially varying strain use linear or finite strain theory, whose derivation is based on small displacements of infinitesimal length vectors. Here we apply a differential geometry method to derive a strain theory for large displacements of finite length vectors. This method gives a finite displacement term whose contribution is comparable to that of the linear strain term. Furthermore, we show that a ``domain-wall''-like pseudomagnetic-field profile can be generated when a wide graphene ribbon is subjected to a pair of opposing point forces (point stretch). The resulting field is a function of the new finite displacement term only and displays a maximum strength of over three times that which is predicted by the linear strain theory. These results extend the current theories of strain, which are based on the transformation of infinitesimal length vectors, to finite length vectors, thus providing an accurate description of pseudomagnetic-field structures in strained materials.

Heterogeneity and phase transformation in materials: Energy minimization, iterative methods and geometric nonlinearity

We consider the relationship between methods used to solve problems involving heterogeneities with a focus on phase transformations. We compare methods that solve for mechanical equilibrium based on iterative spectral techniques with those evolving a free energy using inertial and relaxational dynamics in terms of microstructure, convergence and efficiency for the two-dimensional versions of the cubic-to-tetragonal and hexagonal-to-orthorhombic transformations. We generalize the strain-based approach using kinematic compatibility, in conjunction with a Landau-based nonconvex energy functional, to the geometrically nonlinear case for computation using the iterative spectral as well energy-minimizing methods. Our approach uses the strain compatibility equations for geometrically nonlinear elasticity in two dimensions, the geometrically linear strain compatibility equations being recovered in the small strain limit. We show how for the two-dimensional version of the hexagonal-to-orthorhombic transformation with three variant distortions in the microstructure, the linear compatibility equations capture relatively large rotations with small to moderate stains. We propose a reconstruction that uses the mapping between displacement gradients and Lagrange strains to approximate the geometrically nonlinear result within the scope of the linear strain theory if the linear strains are substituted for the Lagrange strains. We illustrate this by evaluating the disclination angle associated with the unique microstructure for this transformation, which otherwise requires geometric nonlinear theory to correctly capture.

Analysis of Large Deformation Wound Roll Models

Almost all winding models incorporate the assumption of small linear deformations and strain in their development. These models treat the addition of a layer of web to a winding roll with linear analysis using linear strain theory. Very few winding models have been developed that incorporate large deformation theory although many models treat material nonlinearity. Tissue and nonwoven webs are highly extensible in-plane and highly compressible in the thickness dimension when compared to paper, plastic film, and metal foil webs. Winding models that embody large deformation theory should apply to all web materials. Such models may be wasteful in computation time for web materials such as paper, film, and foils where models that employ small deformation theory may provide sufficient accuracy. This would appear deterministic based upon the extensibility and compressibility of a web material, but the issue becomes more complex due to limitations in tension that can be exerted on the webs. Herein, a large deformation winding model will be developed. Results from this model will be used to benchmark results from other small and large deformation models, and with laboratory test data, a review of all results will be used to determine when or if large deformation winding models are required.

Application of a photogrammetric technique to a model tunnel

Close range photogrammetric technique (digital imaging process) is a useful non-intrusive method for studying deformation patterns imposing no constraints on the material. In this study, the close range photogrammetric technique for a laboratory two-dimensional model tunnel test using the multi-sized rod material was introduced and consequently, the displacement and strain data were provided to compare with the finite element analysis. The close photogrammetric technique with a linear strain theory showed a remarkable agreement with the physical model data.

Relating crevassing to non-linear strain in the floating part of Jakobshavn Isbræ, West Greenland

AbstractJakobshavn Isbræ is a major ice stream that drains the west-central Greenland ice sheet and becomes afloat in Jakobshavn Isfjord (69° N, 49° W), where it has maintained the world’s fastest-known sustained velocity and calving rate (7 kma−1) for at least four decades. The floating portion is approximately 12 km long and 6 km wide. Surface elevations and motion vectors were determined photogrammetrically for about 500 crevasses on the floating ice, and adjacent grounded ice, using aerial photographs obtained 2 weeks apart in July l985. Surface strain rates were computed from a mesh of 399 quadrilateral elements having velocity measurements at each corner. It is shown that heavy crevassing of floating ice invalidates the assumptions of linear strain theory that (i) surface strain in the floating ice is homogeneous in both space and time, (ii) the squares and products of strain components are nil, and (iii) first- and second-order rotation components are small compared to strain components. Therefore, strain rates and rotation rates were also computed using non-linear strain theory. The percentage difference between computed linear and non-linear second invariants of strain rate per element were greatest (mostly in the range 40–70%) where crevassing is greatest. Isopleths of strain rate parallel and transverse to flow and elevation isopleths relate crevassing to known and inferred pinning points.

Magnetoelastic second-order strain effects in paramagnetic rare earth systems

The qualitative and quantitative importance of second-order strain interactions in paramagnetic rare earth systems is demonstrated within a recently developed theory. In an applied magnetic field second-order strain interactions contribute measurable anisotropy effects to transverse sound velocities, which are as large as rotational effects and which are not described by the conventional linear-strain theory. For the example of SmSb, the magnetic-field-dependent and field-independent second-order strain effects turn out to be of the same order of magnitude as the corresponding first-order strain effects.