Showing papers in "Annals of Solid and Structural Mechanics in 2012"

Bending relationships between the modified couple stress-based functionally graded Timoshenko beams and homogeneous Bernoulli–Euler beams

Abstract: The Bernoulli–Euler and Timoshenko beam theories are reformulated using a modified couple stress theory and through-thickness power-law variation of a two-constituent material [functionally graded material (FGM)]. The model contains a material length scale parameter that can capture the size effect in a FGM. The equations are then used to develop algebraic relationships for the deflections, slopes, stress resultants of the Timoshenko beam theory (TBT) for microstructure-dependent FGM beams in terms of the same quantities of the conventional Bernoulli–Euler beam theory (BET). The relationships allow determination of the solutions of the TBT for microstructure-dependent FGM beams whenever solutions based on the BET are available. Examples of the use of the relationships are presented using straight beams with simply supported and clamped boundary conditions.

Comparison and critical analysis of invariant-based models with respect to their ability in fitting human aortic valve data

Abstract: With the increase of life expectancy and population average age, heart valve diseases have become more frequent, representing an always increasing percentage among cardiovascular diseases, which are the predominant cause of death in the western country. For this reason, research activities within such a context and, in particular, computer-based predictions of valve behavior are strongly motivated. Consequently, the study of the tissue mechanical response and the constitutive relationships for modeling material behavior represent crucial a aspect to be investigated in order to perform realistic simulations. The mechanical response of the aortic valve tissue depends on the contribution, composition, and interaction of different constituents, such as collagen fibers and elastin network. Accordingly, constitutive laws including non-linearity and anisotropy are necessary. Clearly, the complexity of a constitutive model increases more as it takes into account the histological structure of the tissue. Numerous constitutive models have been developed to describe arterial tissue, but relatively few models have been calibrated specifically for the aortic valve. This study focuses on the investigation of constitutive models so far proposed in the literature which could be suitable to capture the mechanical behavior of the aortic valvular tissue. To make the right choice, the comparison between these constitutive models is done in terms of the fitting quality achieved with respect to human aortic valve data proposed in the literature. For this purpose, an optimization technique based on the nonlinear least square method is used. The obtained material parameters could be later used in finite element analysis adopted, in this last decade, as an innovative approach to support the operation planning procedure and the design of artificial grafts.

Rate constitutive theory for ordered thermoelastic solids

Abstract: When the mathematical models for the deforming solids are constructed using the Eulerian description, the material particle displacements and hence the strain measures are not known. In such cases the constitutive theory must utilize convected time derivatives of the strain measures. The entropy inequality provides a mechanism for determining constitutive equations for the equilibrium stress with the additional requirement that the work expanded due to the deviatoric part of the Cauchy stress tensor be positive, but provides no mechanism for establishing the constitutive theory for it. In the development of the constitutive theory in the Eulerian description for thermoelastic solids, one must consider a coordinate system in the current configuration in which the deformed material lines can be identified. Thus the covariant, contravariant and Jaumann convected coordinate systems are natural choices for the development of the constitutive theory. The compatible conjugate pairs of convected time derivatives of the stress and strain measures in these bases in conjunction with the theory of generators and invariants provide a general mathematical framework for the development of the constitutive theory for thermoelastic solids. This framework has a foundation based on the basic principles and axioms of continuum mechanics but the resulting constitutive theory must satisfy the conditions resulting from the entropy inequality to ensure thermodynamic equilibrium of the deforming matter. This paper presents development of rate constitutive theories for compressible as well as incompressible, homogeneous, isotropic solids. The density, temperature, and temperature gradient in the current configuration and the convected time derivatives of the strain tensor up to any desired order in the chosen basis are considered as the argument tensors of the first convected time derivative of the deviatoric Cauchy stress tensor and heat vector. The thermoelastic solids described by these constitutive theories are termed ordered thermoelastic solids due to the fact that the constitutive theories for the deviatoric Cauchy stress tensor and heat vector are dependent on the convected time derivatives of the strain tensor up to any desired order, the highest order defining the order of the solid.

Lazy zone bone remodeling theory and its relation to topology optimization

Abstract: The connection between apparent density-type bone remodeling theories and density formulations of topology optimization is well known from a large number of publications and its theoretical basis has recently been discussed by making use of a dynamical systems approach to optimization. The present paper takes this connection one step further by showing how the Coleman–Noll procedure of rational thermodynamics can be used to derive general dynamical systems, where a special case includes the lazy zone concept of bone remodeling theory. It is also shown how a numerical solution method for the dynamical system can be developed by using the sequential convex approximation idea. The method is employed to obtain a series of solutions that show the influence of modeling parameters representing elements of plasticity and viscosity in the growth process.

Nonlinear dynamics of a rotating shaft with a breathing crack

Abstract: In this paper, the effects of a breathing crack on the vibratory characteristics of a rotating shaft are investigated. A new, simple and robust model composed of two rigid bars connected with a nonlinear flexural spring is proposed. The nonlinear spring, located at the cracked transverse section position, concentrates the global stiffness of the cracked shaft. The breathing mechanism of the crack is described by a more realistic periodic variation of the global stiffness depending not only but substantially on the system vibratory response. It is based on an energy formulation of the problem of 3D elasticity with unilateral contact conditions on the crack lips. A possible partial opening and closing of the crack is considered which makes the approach more appropriate for deep cracks modeling. The harmonic balance method, direct time-integration schemes and nonlinear dynamics tools are used to characterize the global dynamics of the system. The effects of the crack depth and rotating frequency have been meticulously examined and it was found that the cracked shaft never exhibits chaotic or quasi-periodic vibratory response.

Frequency-locking in nonlinear forced oscillators near 3:1 and 4:1 resonances

Abstract: In this paper, frequency-locking phenomenon in self-excited nonlinear oscillators subjected to harmonic excitation is investigated near the 3:1 and 4:1 subharmonic resonances. Analytical treatment based on perturbation techniques is performed to study the quasiperiodic modulation domain and the frequency-locking area in the vicinity of these resonances. It is shown that this analytical method, based on a double averaging procedure, is efficient to capture the modulation domain of the quasiperiodic response as well as the threshold of synchronization area near the considered subharmonic resonances.

On a simple cyclic plasticity modeling with implicit kinematic hardening restoration

Abstract: This paper presents a fully three-dimensional plastic constitutive modeling framework suitable for the prediction of cyclic loading at large number of cycles. It can require only one yield surface and it is motivated by a simple rheological model where a restoration of the kinematic hardening is introduced. The classical kinematic hardening rules are then simply adapted leading to time-dependent evolution laws that are consistent with continuum thermodynamics requirements. The resulting behavior is physically motivated by many man-made materials of engineering interest such as bituminous material. This framework allows all types of yield functions to be easily implemented numerically. This is first illustrated with algorithmic details through a simple associative pressure-insensitive model example of the von Mises type. Then a more elaborated model is given where the present framework is applied to the description of bituminous materials submitted to triaxial static creep and to large number of cyclic loadings. Of particular interest is the ratcheting and the mean stress relaxation. The responses agree well with some experimental test results found in the literature.