Physics of Life 2. Biomechanics

Introduction to Lecture 6

What biomechanics is. An introduction to basic concepts of force work and power, and the important distinction between units and dimensions.

KW:work; force; SI units; units; dimensions; dimensional analysis;

What stress and strain are. How to quantify them using the stress-strain curve. How to define the elastic modulus. What elasticity is. Energetics of stress and strain. Compliance versus stiffness. Hookean versus non-Hookean materials.

What elasticity is. What elastic energy storage is. Composite materials. The aorta as an example of a composite material.

What hysteresis is. Energetics of deformation in nonelastic materials. Tendons as shock absorbers for articulated skeletons. What shock absorbers are, and how they work. Why tendons should serve as shock absorbers.-

How materials fail. Ductile failure and plastic deformation. Analysis of ductile failure using the stress-strain curve. Brittle failure. Thermodynamics of brittle failure. Crack stability and energetics of crack propagation.

The precise meaning of these words. How they are measured. Defining strength, resiliency and toughness using the stress-strain curve. Tough, weak and brittle materials.

KW: strength; toughness; resiliency; elastic modulus; stiffness; ductile failure; brittleness;

Imposed strain by wave surge on the kelp stipe. The stipe as a brittle material, compared to bone and wood. Kelp does not fail by ductile failure, but by brittle failure. Stress concentration in the kelp stipe. Self healing of damage to the stipe, and stress relaxation. Kelp as an adaptive material.

A survey of strength and toughness among common materials. Laminate materials and crack stopping mechanisms. Lamination in tree trunks. Lamination in bone. Strain homeostasis in bone and wood. Bone and wood as adaptive materials.

How tough materials can be made to fail. Nacre as a tough material in seashells. How nacre fails. Sheer fracture versus tensile fracture. Strain hardening of mussel shells by crabs.

KW: nacre; chitin; aragonite; shear stress; fracture stress; strain hardening; crabs; mussels;

How to quantify changes of shape. How to evaluate changes of bone shape with increasing body size. The mathematics of scaling. Understanding and working with the power equation.

KW: allometry; allometric scaling; isomorphic scaling; hypermorphic scaling; hypomorphic scaling; bone diameter; bone length; body size;

What is a cantilever, and how does it differ from a column? Why bones are not columns. How do cantilevers support loads? Are the bones of skeletons cantilevers?

KW: cantilever; skeleton; skeletal mass; elephant; wrapped; body mass; allometry; allometric scaling; power equation;

How cantilever beams support loads. Parasite material and what is. Second moment of area and flexural stiffness. Second moment of area and design of the I-beam, box beam, and tube.

KW: cantilever; second moment of area; flexural stiffness; elastic modulus; I-beam; box beam; tube;

Different models of scaling for bone size and bone shape. The difference between statics stress similarity and dynamic stress similarity. How bones fail under a buckling loads. Mitigating risk of buckling failure with natural selection. Do bones scale according to elastic similarity criteria?

KW: cantilever; geometric similarity; static stress similarity; dynamic stress similarity; elastic similarity; bone diameter; bone length; Henry Ford;

Articulated skeletons are based upon levers of various types. These levers act to enhance the limited performance of muscle itself, acting as transmissions to either magnify force, speed, or power of a muscle. There are three types of levers based upon the relative placement of forces and the fulcrum.

KW: lever; work; force; arc; lever arm; fulcrum;

The operating principle of a lever is the conservation of energy. Levers are machines for transmitting work. Because work is the product of force and distance, these two quantities can be altered depending upon the relative placement of forces and fulcrums on lever arms. Depending on the placement of the fulcrum, levers can trade-off either magnify force, or magnifying distance, or alternatively, magnifying power, or magnifying speed. These are known generically as mechanical advantages. Ultimately, the principle of conservation of energy is the fundamental operating principle of any lever system.

KW: lever; force; power; mechanical advantage; speed; length advantage; force advantage; speed advantage; power advantage;

In the body, articulated skeletons act as levers. The joints around which skeletal elements rotate represent the fulcrum, while the insertion of the various muscles that control the joints position act as the in force of the lever. All three lever types are represented in a typical skeleton. By altering the placement of the insertion of a muscle, the various mechanical advantages can be altered, rendering skeletal muscular systems is relatively high gear systems, which advantage speed over power, or low gear systems, which advantage power over speed.

KW: joints; articulated skeleton; high gear levers; low gear levers; lion; cheetah; muscle function; muscle work; mechanical advantage; power; speed;

Muscle functions well over a fairly limited range of shortening distances and rates, and generation of force. Often, there is a mismatch between a muscles’ functional capabilities, and what the muscle is expected to do. In the case of running, and animal must accelerate from a standing stop up to running speed. Physical acceleration of the body is frequently much lower compared to the muscles that was power it. The mismatch is bridged in two ways. First, there may be multiple systems that control the joint. For example one muscle might act as a low gear muscle to power acceleration, shifting off to another muscle that may act as a high gear system to power running speed. The other bridge involves storage of muscle work into elastic elements in the muscles and tendons. This energy can be stored temporarily in these elastic elements, to be tapped later to power actual movement of the joint.

KW: acceleration; muscle function; running; vicuna; elastic energy storage; gluteus medius; semimembranosus; mechanical advantage; force advantage; speed advantage;

Ultimately, locomotion amounts to a system of energy transfers between muscles and the external environment. The functional limitations of muscles are illustrated by the energetics of jumping, that is elevating the body’s center of mass to some height above the ground. The energetics of jumping lead to the surprising conclusion that jumping height is independent of body size, that is no matter how large or small an animal is, the muscle-powered jump will be the same no matter how large or small the animal is.

KW: jumping; muscle power; jumping height; potential energy; muscle work; dimensional analysis; body size; scaling;

The unusual prediction of a jumping height that is independent of body size is based upon the assumption that it is muscle alone that powers the jump. An analysis of the energetics of jumping of the flea shows that muscle alone cannot power the fleas observed accelerations and impulse times. The flea’s muscles are both too weak, and to slow to be able to explain the jumping of the flea. What fills the gap is the transient storage of elastic energy in elastic elements within the flea’s body. The flea uses its muscles to store energy in a pad of elastic resilin, which is held in place through a catch mechanism. Upon release of the catch mechanism this stored energy can be released very fast. In this instance, the use of elastic energy storage converts work done by relatively weak and low-power muscles into an engine for powering a very rapid and high power jump.

Skeletons are devices that transmit forces from muscles to the external environment. Skeletons are commonly thought of as articulated skeletons, that is rigid elements like bones joined together at joints. In fact, a skeleton can be anything which transmits work. Incompressible water servers that function in animals that do not have articulated skeletons.

KW: skeleton; bone; chitin; articulated skeleton; incompressible water; hydrostatic skeleton; cantilevers; mechanical advantage;

Articulated skeletons work by transmitting muscle work through the lever system of rigid elements rotating about joints. Hydrostatic skeletons can do the same thing, that is, water contained within a flexible container can transmit muscle work through various kinds of speed and force advantages. This makes hydrostatic skeletons inherently more versatile than articulated skeletons.

KW: hydrostatic skeleton; mechanical advantage; hoop muscle; circumferential muscle; longitudinal muscle; earthworm;

Earthworms provide a beautiful example of a hydrostatic skeleton in action. Earthworms are segmented worms, and each segment can serve as a hydrostatic skeleton, independently controllable from all the other earthworms segments. This type of organization as increasing versatility to the use of a hydrostatic skeleton, because different segments can be made to do different things, either pushing radially, or pushing along the worm’s longitudinal axis.

KW: earthworm; segment; hoop muscle; circumferential muscle; longitudinal muscle; radial force; longitudinal force; mechanical advantage;

The chameleon tongue is an impressive example of a hydrostatic skeleton in action. The tongue of the chameleon is a muscular organ that can project at impressive speeds in acceleration, all without an articulated skeleton. The action of the tongue illustrates dramatically the remarkable versatility of the hydrostatic skeleton.

KW: chameleon; tongue; hyoid apparatus; acceleration; speed; mechanical advantage;

Many hydrostatic skeletons are fiber wound cylinders, that is structures that are enveloped in a helical winding of connective tissue. This winding of fibers as to the versatility of hydrostatic skeletons, enabling cylinders to do a variety of things with only a single structural design element, namely the fiber angle of the wound cylinder. Depending upon the fiber angle, a fiber wound cylinder can either elongate, shorten and widen, or bend without kinking.

KW: fiber wound cylinder; helical winding; fiber angle; pitch; hoop stress; longitudinal stress;

The tube feet of echinoderms represents one of the simplest of the fiber wound cylinders. These structures are water-filled cylinders, wrapped with a helical winding of fibers, and pressurized by muscles located elsewhere in the water vascular system of the animal. The tube feet are capable of elongation, forceful retraction, and bending, all without any hard skeletal elements.

KW: tube feet; echinoderm; starfish; water vascular system; ampullary muscle; longitudinal muscle; helical winding; bending; elongation; retraction; hoop stress; longitudinal stress;

The helical winding in a fiber wound cylinder need not be a passive tension very element like collagen. The fiber winding can also be muscle, which can be thought of as a variable stiffness of fiber. Muscle wound hydrostats add further to the remarkable versatility of the hydrostatic skeleton, because it can balance hoop stress and longitudinal stress in some remarkable ways.

KW: hydrostatic skeleton; muscle wound cylinder; hoop stress; longitudinal stress; modulus of elasticity;

The tentacle of the squid is capable of a remarkable versatility of motion. This versatility of motion comes about through a combination of differently oriented muscle fibers, including helically wound layers of muscles. All operate to transmit work through the hydrostatic skeleton that is the tentacle, and this makes the tentacle capable of remarkable dexterity, with motions including elongation and retraction, but also twisting, and helical bending of the tentacle.

KW: squid; tentacle; longitudinal muscle; transverse muscle; helical muscle; bending; elongation; retraction; twisting; torque; helical bending; hydrostatic skeleton;

Locomotion in articulated skeletons involves the repetitive back-and-forth motion of limbs. In this, limbs act like pendulums, and the physics of pendulums can be used to predict a number of interesting properties of locomotion.

KW: biomechanics; locomotion; pendulum; frequency; period; Natural frequency; walking; energy;

The back-and-forth motion of the limbs during walking can be considered analogous to the back-and-forth motion of a pendulum and it swing. This leads to a model of walking known as pendulum walking, which enables us to predict the energetics of walking and how to optimize it. Pendulums are driven most efficiently at their natural frequency, which is proportional to the length of the limb. Pendulum walking supposes that walking is powered most efficiently when the stride frequency corresponds to the legs natural frequency, which is related to the length of the leg.

KW: biomechanics; pendulum walking; leg length; natural frequency; energy cost; optimum walking speed;

Forward locomotion at speeds other than the optimum walking speed elected from pendulum walking involves transitions to different patterns of oscillatory motions of the limbs known as gaits. Gates enable an animal to circumvent the limitations of pendulum walking, including transitioning increases of locomotion from changes of stride frequency to changes of stride length. In this way, animals can extend the range of locomotory speeds beyond what pendulum walking would allow.

KW: biomechanics; walking; trotting; galloping; running; Froude number; pendulum walking elastic similarity; scaling; stride frequency; stride length;

Transitions between different gaits occur at a ticket speeds. Some of this can be explainable by the theory of pendulum walking, but ultimately the transition from one gait to another is a problem of physiological control. Transition from one gait to another appears to be regulated through the sensing of strain in bones as they are pulled back and forth during locomotion. When the strains get too high, these are sense, and translated into variation in gait pattern and frequency, modulated by the central nervous system.

KW: biomechanics; gait transitions; strain homeostasis; optimum stride frequency; energetics of locomotion; bone strain; pendulum walking;