Stiffness linear elasticity

If an compressive normal force Fn S acting on a sojl contact of two isotropic, stiff, linear elastic, mono-disperse spherical particles the previous contact point is deformed to a small contact area and the adhesion force between these two partners is increasing, see Rumpf et al. [22] and Molerus [8]. 

Fiber-reinforced composite materials such as boron-epoxy and graphite-epoxy are usually treated as linear elastic materials because the essentially linear elastic fibers provide the majority of the strength and stiffness. Refinement of that approximation requires consideration of some form of plasticity, viscoelasticity, or both (viscoplasticity). Very little work has been done to implement those models or idealizations of composite material behavior in structural applications. 

The purpose of this chapter is to remind the reader of the basis of the theory of elasticity, to outline some of its principal results and to discuss to what extent the classical theory can be applied to polymeric systems. We shall begin by reviewing the definitions of stress and strain and the compliance and stiffness matrices for linear elastic bodies at small strains. We shall then state several important exact solutions of these equations under idealised loading conditions and briefly discuss the changes introduced if realistic loading conditions are considered. We shall then move on to a discussion of viscoelasticity and its application to real materials. 

Some researchers have used approximate microscopic descriptions to develop more rigorous macroscopic constitutive laws. A microstructural model of AC [5] linked the directionality of mechanical stiffness of cartilage to the orientation of its microstructure. The biphasic composite model of [6] uses an isotropic fiber network described by a simple linear-elastic equation. A homogenization method based on a unit cell containing a single fiber and a surrounding matrix was used to predict the variations in AC properties with fiber orientation and fiber-matrix adhesion. A recent model of heart valve mechanics [8] accounts for fiber orientation and predicts a wide range of behavior but does not account for fiber-fiber interactions. 

If the solid is linear elastic (stiffness tensor C = S,v ), the potential energy Fj 1 takes the form (Deude et al., 2002) ... 

The commercial finite element program, Abaqus [17], was used to calculate the stress distribution in an edge delamination sample. A fully three-dimensional model of the combinatorial edge delamination specimen was constructed for the finite element analyses (FEA). For clarity, some of the FEA results and schematics are presented as two-dimensional configurations in this paper (e.g.. Fig. 1). The film and substrate were assumed to be linearly elastic. The ratio of the film stiffness to the substrate stiffness was assumed to be 1/100 to reflect the relative rigidity of the substrate. This ratio also represents a typical organic... 

Reiterer et al. (1999) observed that their 200 p,m thick specimens stretched considerably once the elastic limit had been exceeded, but only where the MFA is quite large (> 15-20°). Then it stretches even further for each increment of strain - in this region, beyond the linear elastic limit, the material deforms irrecoverably by viscoelastic or plastic flow. Finally the sample breaks in tension. The strength of the material, i.e. the failure stress, is read from the y-axis. The stiffness of all woods ranges from 0.5-20 GPa and strength ranges from 1-40 MPa, from the corewood of low density species to the outerwood of very dense species. 

What this equation tells us is that a particular state of stress is nothing more than a linear combination (albeit perhaps a tedious one) of the entirety of components of the strain tensor. The tensor Cijn is known as the elastic modulus tensor or stiffness and for a linear elastic material provides nearly a complete description of the material properties related to deformation under mechanical loads. Eqn (2.52) is our first example of a constitutive equation and, as claimed earlier, provides an explicit statement of material response that allows for the emergence of material specificity in the equations of continuum dynamics as embodied in eqn (2.32). In particular, if we substitute the constitutive statement of eqn (2.52) into eqn (2.32) for the equilibrium case in which there are no accelerations, the resulting equilibrium equations for a linear elastic medium are given by... 

Assuming linear elastic behavior, the body can be viewed as a linear spring. The stored elastic strain energy U is given by the applied load (P) and the load-point displacement (A), or in terms of the compliance (C) of the body, or the inverse of its stiffness or spring constant i.e.. 

Finally, high porosity and reactive shales are strongly anisotropic, and are likely to be neither linear elastic nor with confining stress- independent stiffness. These factors are affect analysis and interpretation, yet they remain ill understood. 

The second characterisation (b) is more often used in practice. In linear elastic multi-layer calculations, for instance, the absolute value of the complex modulus, lE" , which is equivalent to the stiffness modulus, , is used as input. 

FWD provides the ability to estimate the stiffness moduli of each individual layer after back-calculation analysis of the deflection data. This is carried out using specially developed computer programs, almost all using the principles and assumptions of the multi-layer system linear elastic analysis. Some guidance for calculating in situ equivalent elastic moduli of pavement materials using layered elastic theory is provided by ASTM D 5858 (2008). 

Ziegert and Lewis [9] measured the in vivo indentation properties of the soft tissues covering the anterior-medial tibiae. A preload of 22.4 N was used with indentors of 6 to 25 mm in diameter. The observed load displacement relationship were essentially linearly elastic. The structural stiffness was noted to vary by up to 70% between sites in one individual and up to 300% between individuals. Unfortunately, the thicknesses of overlying tissues were not determined at the different sites for the individuals studied. 

Lanir et al. [3] measured the in vivo indentation behavior of human forehead skin with pressures up to 5 kPa. The observed behavior was linearly elastic and calculated stiffnesses were 4 to 12 kPa. 

A, B, C and D are coefficients determined by the relevant boundary conditions, k is the foundation stiffness, and E and I are the Young s modulus and second moment of area of the beam, respectively. To cast this solution in terms of adhesive bonds, the adhesive, of thickness h, is assumed to be linearly elastic with a modulus of E, unaffected by deformations in the surrounding material, and acting over a bond width, w, on the beam. Under these conditions, the foundation stiffness becomes... 

For single crystals with transverse dimensions large enough to permit a plane wave condition to be attained, the results are unambiguous and virtually free from theoretical assumptions. Five independent elastic constants (stiffnesses or compliances) are required to describe the linear elastic stress-strain relations for hexagonal materials. Only three independent constants are required for cubic (y-Ce, Eu, Yb) materials. Since there are no single crystal elastic constant data for the cubic rare earth metals, this discussion will concentrate on the relationships for hexagonal symmetry. 

The Geogrid was modeled as linearly elastic-perfectly plastic material with Mohr-Coulomb failure criterion. The axial stiffness (J) and the tensile strength (T) were needed in the program and taken J = 620 kN/m and T = 70 kN/m, respectively. The interaction coefficient between the geogrid and cohesive backfill was taken 0.7. 

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