Stress strain relationship polar coordinates worksheet

stress strain relationship polar coordinates worksheet

arity (Hookean stress-strain behavior) or linear relation between stress and strain . dial distance, J is the polar second moment of area and G is the shear modulus. speaking, the equations for bending stress and beam deflections are only. control variables conversions –42 Conyplex coordinate system products costing worksheet –15 costs –17 cowoven fabric 99 crack growth behavior crack propagation, schematics of crackle –91 constant stress Overview –6 stress-strain-time in vs. log. In equation form, Hooke's law is given by. F=k\Delta L,. where \Delta Figure a is a cylindrical rod standing on its end with a height of L sub. Experiments have . [ link] shows a stress-strain relationship for a human tendon. Some tendons have.

The tangent base vector is tangent to the coordinate lines, see the Figure below. The components of tensor expressed in these base vectors are its contravariant components, p. This is the space of tangent vectors Ch. They are written as described w. See pages in [1], page in [3], page 18 in [4]. The metric introduced later will make it possible to use an alternative representation of a tensor, covariant components.

This can be interpreted in our context as an alternative representation using cotangent base vectors. They belong to a dual space Ch. It is the space of tangent functions. The cotangent base vectors are defined as Eq. They are obtained from differentials of these functions thereby the space of tangent functions. Note that the base vectors are not unit vectors. The tangent base vector is defined as in the Figure above and the length and the direction of the normal base vector is given by Eq.

See page 18 in [4]. Contravariant and covariant tensor transforms differently under change of coordinates Ch. More specifically, the ratio of the principal cross-sectional moments e.

Studies by Carlson and co-workers Carlson, ; Carlson and Judex, provide good examples of how these measures of robusticity and ellipiticality are defined and used in anthropological studies that are aimed at deciphering load history.

What this means is that the greatest bending rigidity occurs when the direction of bending is across i. These definitions can be confusing in the anthropological literature because some prominent investigators have adhered to them e.

Material properties of bone. At the most basic level, bone is a composite of type 1 collagen and mineral typically a carbonated hydroxyapatiteand this composite is enriched with non-collagenous proteins that also have important biochemical and biomechanical functions.

Important mechanical and biophysical interactions of the collagen-mineral composite for elastic and plastic behaviors are described by Burstein et al. Bone material properties are the tissue-level mechanical properties that describe the constituent material and are independent of the size and shape of the bone.

Tensor Analysis of Deformation - Application Center

These material properties are determined by machining precise samples from the bone of interest and testing them in a particular loading mode. With respect to material properties, it is important for all students of bone adaptation to memorize all features of a typical stress-strain curve, which shows the results of a material test Fig. Beyond the yield point B is where permanent deformation of the specimen occurs. The strain at any point in the elastic region of the curve is proportional to the stress at that point.

Therefore the slope of the elastic region A-B is the material stiffness or elastic modulus; increased steepness of the A-B slope represents increased stiffness. The ultimate failure point C is the point beyond which failure of the specimen occurred.

Total energy absorbed is a sum of elastic and plastic energy. Brittle materials have shorter B-C regions than more ductile materials longer B-C regions.

App Preview:

This material heterogeneity and its directionality e. Bone tissue is also viscoelastic — it has stress-strain characteristics that are dependent upon the applied strain rate. In other words, a specimen of bone tissue that is exposed to very rapid loading will absorb more energy than a specimen that is loaded more slowly. Therefore, bone tissue is both anisotropic and viscoelastic. Because of these characteristics one must specify the strain rate and the direction of applied loading when discussing bone material behavior.

As noted, mechanical behaviors defined by stress and strain deal with material properties; the corollaries for structural properties are load and deformation, respectively. Stiffness and strength are the chief properties of a bone whether it is considered as a structure or as a material.

The modulus of elasticity shows how stiff the bone material is. Yield stress strength at initial failure and strain determine how much energy can be absorbed before irreversible changes take place in the material.

stress strain relationship polar coordinates worksheet

Ultimate stress is the strength at final failure. Post-yield stress and strain 1 see footnotes determine mainly how much energy can be absorbed after yielding but before the material fractures Fig. Irreversible changes occur at the yield point and are caused by accumulating microdamage. The total area under the stress-strain curve is equivalent to the work that must be done per unit volume of the bone specimen before it breaks. The total area under the stress-strain curve can be divided into two portions: As described below, regional CFO variations typically correlate more strongly with energy absorption than with stiffness elastic modulus or strength Skedros et al.

Fracture mechanical properties show the extent to which bone Footnote 1. Among various stimuli, available data suggests that strain is the mechanical parameter most directly involved in causally mediating bone adaptations Rubin and Lanyon, ; Lanyon, ; Ehrlich and Lanyon, Mechanical strain is the change in length of a loaded structure as a percentage of its initial unloaded length Note: This dimensionless ratio is a measure of material or tissue deformation.

In vivo strain data from a variety of animals suggests that physiological strains are generally between and 3, microstrain i. The upper limit may be only 1, microstrain in tension Fritton et al. Crack travel resistance is indicated by post-yield stress and strain.

The more compliant material at right requires more energy to break it. Energy absorption is proportional to the area under each curve; hence the less compliant material at left absorbs less energy than the more compliant material even though it has higher yield stress YS and ultimate stress US.

In addition to stiffness, strength, and toughness, fatigue resistance is one of the four most-important mechanical parameters that must be considered when interpreting the load history and adaptation of a bone or bone region.

Fatigue failure is when a structure is loaded repeatedly and breaks at a lower load than would cause it to fail if it were loaded only once. Fatigue resistance is when fatigue failure is prolonged or avoided altogether. Ritchie Ritchie et al. Using the stress-strain curve for considering mechanisms of bone adaptation produced by remodeling-induced affects on CFO, osteon morphotypes, and osteon population densities. The final portion of this section is an exercise that considers how primary modifications for pre-yield elastic behavior can secondarily have beneficial consequences for post-yield behavior.

Stress components in Rectangular Cartesian Coordinate system (3 mins)

Another way to think about this is to ask: This depiction is based on data and observations described by Bigley et al. Three drawings depicting an energy absorption toughness test in tension T of a machined specimen at top loaded at each end with a force F. Energy is absorbed as the osteons debond, pullout, and bridge the forming crack middle drawing.

stress strain relationship polar coordinates worksheet

Adapted from Martin et al. These primary adaptations can have beneficial perhaps serendipitous effects for post-yield behavior. Osteon size might also play a role in this context Hiller et al. This figure helps to conceptualize the putative shifts in the temporal importance of genetic, epigenetic, and extra-genetic influences, especially with respect to varying histocompositional characteristics within or between bones.

From the pre-natal phase and well into the attainment phase, the adaptive growth response of cartilage, or chondral modeling not discussed in this chapterhas a profound influence on the growth and form of limb bones, especially at their epiphyses where articulations are formed with adjacent bones Hammond et al.

Shown is also the second bone mass growth spurt that occurs in humans, which has not been demonstrated in any other amniote.

These images are reproduced from the original study of Martin et al. The numerical values of each of the six morphotypes are used to calculate the osteon morphotype score MTS of entire microscopic images that contain many osteons Skedros et al. Images are reproduced from Martin et al.

Images are reproduced with permission granted by Lanny Griffin and Elsevier Ltd.

stress strain relationship polar coordinates worksheet

This refers to a material with a grain whose properties or technical constants are different when measured in different directions, most having some degree of symmetry to their internal structure. Bone is anisotropic, but examples of limited anisotropy can also be found in bone orthotropic and transversely isotropic Martin et al.

If the properties are the same in different directions, then the material is isotropic. Adaptation in cortical bone commonly refers to either: This section primarily deals with identifying correlations between structure, material, function, and load history. These correlations might be produced by: We have suggested that the non-uniform strain distribution experienced in the early development of most limb bones is proximate to the historical origin i.

Fatigue failure is when a structure is loaded repeatedly and breaks at a load that would not cause it to fail if it were loaded only once. Consequently, modeling is a concept describing a combination of non-proximate, though coordinated, resorption and formation drifts whose net result is, typically, to change the distribution of bone.

Such drifts are called macro-modeling in cortical bone and mini-modeling in cancellous bone. The re-alignment of trabecular tracts along the lines of stress would be a consequence of mini-modeling. Tough materials resist damage propagation but do not necessarily resist damage formation.

Toughness tests typically involve propagating a crack in a controlled direction through a specimen machined into a specific shape for this test.

On the effect of X-ray irradiation on the deformation and fracture behavior of human cortical bone. Biol Rev Camb Philos Soc Mechanical loading and bone growth in vivo.

Bone, Volume 7, Bone Growth — B. Bone stress in the horse forelimb during locomotion at different gaits: A comparison of two experimental methods.

Biomechanics of Bone | Team Bone

Mechanics of locomotion and jumping in the forelimb of the horse Equus: In vivo stress developed in the radius and metacarpus. Osteonal interfacial strength and histomorphometry of equine cortical bone.

Contribution of collagen and mineral to the elastic-plastic properties of bone. J Bone Joint Surg Am Investigating the form-function interface in African apes: Relationships between principal moments of area and positional behaviors in femoral and humeral diaphyses.