Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams are studied through a finite element approach. The exact shape functions for uniform homogeneous Timoshenko beam elements are used to formulate the proposed element. The accuracy of the present element is considerably improved by considering the exact variations of cross-sectional profile and mechanical properties in the evaluation of the structural matrices. Carrying out several numerical examples, the convergence of the method is verified and the effects of taper ratio, elastic constraint, attached mass and the material non-homogeneity on the natural frequencies and critical buckling load are investigated.

Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions

This paper presents a new approach for investigating the vibration behaviors of axially functionally graded Timoshenko beams with non-uniform cross-section. By introducing an auxiliary function, we can change the coupled governing equations with variable coefficients for the deflection and rotation to a single governing equation. Moreover, all physical quantities can be expressed in terms of the solution of the resulting equation. Making use of power series for unknown function, we can transform the single equation to a system of linear algebraic equations and will get a characteristic equation in natural frequencies for different boundary conditions. An advantage of the suggested approach is that the derived characteristic equation is a polynomial equation, where the lower and higher-order natural frequencies can be determined simultaneously from the multi-roots. Several examples of estimating natural frequencies for axially grade beams and non-uniform beams are presented, which show that our method has fast convergence and obtained numerical results have high accuracy.

Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section

https://academiccommons.columbia.edu/doi/10.7916/D8K661SC/download

A mixed-mode phase field fracture model in anisotropic rocks with consistent kinematics

Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams

A nonlinear parallel-bonded stress corrosion (NPSC) model is proposed to simulate the fatigue characteristics of artificial rock (concrete) during cyclic loading. Numerical simulations of fatigue tests replicate the main mechanical features of concrete specimens subjected to cyclic loading observed in the laboratory. A nonlinear reduction speed of the bond diameter between two bonded particles represents the damage rate induced by the fatigue load. The damage rate is proportional to the maximum cyclic load level when the minimum cyclic load level is fixed. Compared with laboratory data, a logarithmic function of bond diameter in the NPSC model resulted in the best fit to simulate the fatigue behaviour of concrete. The simulation includes acoustic emission (AE) monitoring during fatigue tests. The axial strain of the assembly is governed by the evolution of bond breakages. The sum of released bond strain energy is documented as value proportional to cumulative AE energy. The simulation results show very similar evolution compared with laboratory data, which verifies the effectiveness of AE energy simulation.

Bonded-particle model-based simulation of artificial rock subjected to cyclic loading

Geomaterials such as soils and rocks are inherently anisotropic and sensitive to temperature changes caused by various internal and external processes. They are also susceptible to strain localization in the form of shear bands when subjected to critical loads. We present a thermoplastic framework for modeling coupled thermomechanical response and for predicting the inception of a shear band in a transversely isotropic material using the general framework of critical state plasticity and the specific framework of an anisotropic modified Cam–Clay model. The formulation incorporates anisotropy in both elastic and plastic responses under the assumption of infinitesimal deformation. The model is first calibrated using experimental data from triaxial tests to demonstrate its capability in capturing anisotropy in the mechanical response. Subsequently, stresspoint simulations of strain localization are carried out under two different conditions, namely, isothermal localization and adiabatic localization. The adiabatic formulation investigates the effect of temperature on localization via thermomechanical coupling. Numerical simulations are presented to demonstrate the important role of anisotropy, hardening, and thermal softening on strain localization inception and orientation. Copyright © 2016 John Wiley & Sons, Ltd.

Thermoplasticity and strain localization in transversely isotropic materials based on anisotropic critical state plasticity

Correspondence Ronaldo I. Borja, Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USA. Email: borja@stanford.edu Summary Accurate prediction of strength in rocks with distinct bedding planes requires knowledge of the bedding plane orientation relative to the load direction. Thermal softening adds complexity to the problem since it is known to have significant influence on the strength and strain localization properties of rocks. In this paper, we use a recently proposed thermoplastic constitutive model appropriate for rocks exhibiting transverse isotropy in both the elastic and plastic responses to predict their strength and strain localization properties. Recognizing that laboratory-derived strengths can be influenced by material and geometric inhomogeneities of the rock samples, we consider both stress-point and boundary-value problem simulations of rock strength behavior. Both plane strain and 3D loading conditions are considered. Results of the simulations of the strength of a natural Tournemire shale and a synthetic transversely isotropic rock suggest that the mechanical model can reproduce the general U-shaped variation of rock strength with bedding plane orientation quite well. We show that this variation could depend on many factors, including the stress loading condition (plane strain versus 3D), degree of anisotropy, temperature, shear-induced dilation versus shear-induced compaction, specimen imperfections, and boundary restraints.

On the strength of transversely isotropic rocks

Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams

Shale is a highly heterogeneous material across multiple scales. A typical shale consists of nanometer-scale pores and minerals mixed with macroscale fractures and particles of varying size. High-resolution imaging is crucial for characterizing the composition and microstructure of this rock. However, it is generally not feasible to image a large sample of shale at a high resolution over a large field of view (FOV), thus limiting a full characterization of the microstructure of this material. We present a stochastic framework based on multiple-point statistics that uses high-resolution training images to enhance low-resolution images obtained over a large FOV. We demonstrate the approach using X-ray micro-tomography images of organic-rich Woodford shale obtained at two different resolutions and FOV. Results show that the proposed technique can generate realistic high-resolution images of the microstructure of shale over a large FOV.

Quantifying the heterogeneity of shale through statistical combination of imaging across scales

Introducing the concept of Basic Displacement Functions (BDFs), an innovative method is presented which yields a mechanical based approach rather than a mathematical one for exact static analysis of arbitrarily tapered Timoshenko beams. Holding pure mechanical interpretations, BDFs are obtained using energy methods such as unit load method. It is shown that exact shape functions and consequently structural matrices could be derived in terms of BDFs. Unlike the most of finite element formulations which are based on prescribed displacement fields, the present finite element takes advantage of a flexibility basis which guarantees the exact interpolation of displacement field along the element and leads to fast convergence of the method in static analysis even with few elements. The method could be easily implemented into any standard displacement-based program. Carrying out numerical examples, the competency of the method in both static and free vibration is verified.

Shabnam J. Semnani

Papers

Thermoplasticity and strain localization in transversely isotropic materials based on anisotropic critical state plasticity

On the strength of transversely isotropic rocks

Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams

Quantifying the heterogeneity of shale through statistical combination of imaging across scales

Analysis of Non-Prismatic Timoshenko Beams Using Basic Displacement Functions