Linear theory of elasticity

Godunov SK, Romenskii El (1972) Non-steady state equation of non-linear theory of elasticity in Euler co-ordinates. Zh Prikl Mech Technich Phys 6 124-144 (in Russian) Golovicheva IE, Sinovich SA, Pyshnograi GV (2000) Effect of molecular weight on shear and elongational viscosity of linear polymers. Prikl Mech Tekhnich Fiz 41(2) 154-160 (in Russian)... 

As we will see in subsequent chapters, for many purposes (e.g. the linear theory of elasticity) the small-strain tensor suffices to characterize the deformation of the medium. 

If a crystal is subjected to small strain elastic deformation it is convenient to imagine the energetics of the strained solid in terms of the linear theory of elasticity. As we noted in chap. 2, the stored strain energy may be captured via the elastic strain energy density which in this context is a strictly local quantity of the form... 

For the moment, we will restrict our attention to the general statements that can be made concerning such forces in the context of the linear theory of elasticity. In this case, the results simplify considerably in that we can think of such energetic... 

The elastic solution upon which many approaches to the consideration of microstructure are built is that of the so-called Eshelby inclusion in which one considers a single ellipsoidal inclusion in an otherwise unperturbed material. From the standpoint of the linear theory of elasticity, this problem is analytically tractable... 

Dislocation Screening at a Crack Tip From the standpoint of the linear theory of elasticity, the problem of crack tip shielding may be seen as a question of solving the boundary value problem of a crack in the presence of a dislocation in its vicinity. As indicated schematically in fig. 11.15, the perspective we will adopt here is that of the geometrically sterilized two-dimensional problem in which both the crack front and the dislocation line are infinite in extent and perfectly straight. As we will see, even this case places mathematical demands of some sophistication. The basic question we... 

From a multiple scale modeling perspective, the presence of phenomenological parameters in various effective theories provides an opportunity for information passage in which one theory s phenomenological parameters are seen as derived quantities of another. We have already seen that although the linear theory of elasticity is silent on the particular values adopted by the elastic moduli (except for important thermodynamic inequalities), these parameters may be deduced on the basis of microscopic analysis. The advent of reliable models of material behavior makes it possible to directly calculate these parameters, complementing the more traditional approach which is to determine them experimentally. 

Gurtin, M.E. The Linear Theory of Elasticity Handbuch der Physik, vol. VIa/2. Springer, Berlin (1972)... 

As the last variable in this list of materials, the tear strength has to be mentioned. First the basic task for this test has to be defined A membrane has a tear-type damage of a defined length. Required is the load at which the tear will be subject to unstable spread so that the membrane finally fails. The theory here is that of fracture mechanics, in which the stress concentration factor is defined as the decisive variable. This material variable is defined on the basis of the linear theory of elasticity and can be applied to anisotropic materials without any problems. It shows that this theory is also valid as a good approximation for coated fabrics. 

The linear theory of elasticity provides relationships between the modulae ... 

This step is called kinematic or geometrical linearization. For the following development we will employ the coordinates Su of this infinitesimal strain tensor as a measure of the local strain of the body in the immediate vicinity of material point X at position x in the actual configuration. Usually they will simply be referred to as strain coordinates. As a further important consequence of the geometrical linearization the fact should be noted that derivatives with respect to material coordinates Xk may be replaced by the corresponding derivatives with respect to spatial coordinates Xk. Within the limits of applicabiUty of the linear theory of elasticity relations (3.26) and (3.28) are also simplified. The former reduces to... 

For the special case of the sphere, an analysis based on the classical linear theory of elasticity (Lame formulae) yields the following expression for the incremental modulus E namely , = 3l2)Vd oJdV. This same result is obtained from the expression for on setting a = b, and y = 1. 

The confusion surrounding the terms ventricular or chamber stiffness (dP/dT), myocardial stiffness ,nc (do/de) and factors affecting the diastolic P-V relation are best explained by a simple mathematical model which is based on the classical linear theory of elasticity. 

The absorption of ultrasound in smectic phases is significantly more anisotropic than that in nematics, and even the velocity has a measurable anisotropy of about 5%. Details of the behaviour of SmA, SmB, SmC and SmE phases can be found in the literature [14—16, 18, 86, 94-97]. The usual approach to the analysis of smectic phases, based on the linear theory of elasticity and hydrodynamics, results in the relationship a—f, which does not agree with the experimental data. In the low-frequency range the coefficients a, o, 4 and c% demonstrate singularity, induced by nonlinear effects, in the form of oT. This results in a linear frequency dependence of the ultrasound absorption. The corrections for the coefficients of elasticity B and K, taking into account the nonlinear fluctuation effects in smectic phases, depend on the wavevector of the smectic phase layer structure B=(ln9,)- [96, 97]. In... 

Most of the work in mechanism deformation analysis is based on the linear theory of elasticity [1-5]. In most of these, a kinematics and rigid body analysis is used to solve for the gross body motion and the inertial forces. By employing different techniques, the elastic deformations are then found by applying these inertial forces as externally applied forces to the linear elasticity problem. The small elastic deformations are superimposed onto the gross motion in order to predict the total motion of each link in the system. 

Gross, G. (1953) Mathematical Structures of the Theories of Viscoelasticity (Hermann C, Paris) Gurtin, M.E. (1972) The Linear Theory of Elasticity , in Encyclopedia of Physics, Vol. VIa/2 Mechanics of Solids If ed. by C. Truesdell (Springer-Verlag, Berlin, Heidelberg)... 

The linear theory of elasticity provides relationships between the modulae P = 2G(l + v) = 3jfeb(l-2v) (7)... 

---