Materials Science · Biological, bio-derived and biomedical materials

Hierarchy in materials made by living systems

Useful background: Atoms, bonding, structure and defects; Energy, stability and environmental context; Composites, fibres and interfaces. Each worked case explains the quantities it uses.

Back to the chapter

After reading, you can

Identify levels of organisation in a biological material.
Trace named structural levels in wood and bone without giving them the same chemistry or geometry.
Distinguish a tissue-level response from the apparent response of a porous structure and the stiffness of a complete specimen.
Compare orientation and hydration through matched observations, keeping other biological and testing context.
Use a bounded load-path model to show why a bulk response does not identify every lower-level property.

A material built at several scales

Bone, wood and other biological materials organise molecular constituents into fibres, layers, pores and larger structures. Different levels can perform different jobs: carrying load, transporting substances or limiting damage. A bulk response emerges from their interaction.

A wood beam and a thin wood cell wall are not the same test object. Hierarchy means that organised units at one level form part of another level: molecular chains form fibrils; fibrils and surrounding matter form a wall; connected walls enclose cell spaces; many cells form a tissue region and larger stem. Naming a level says what has been included in an observation and what has been averaged over.

A constituent is a material component. Architecture is its spatial arrangement and connections, including spaces. A change in either can alter response. A larger structure can also serve transport, storage or growth, so mechanical stiffness alone does not describe its whole function. Constituents and composite architecture develops that distinction.

The retained nested sketch is a qualitative reminder of organisation. Its circles and aligned strokes are not identified microscopy, a measured scale sequence or a drawing of every bone or wood structure. The named examples below provide the actual material meanings.

Trace a load path through wood rather than calling it one fibre

Cellulose consists of polymer chains gathered into microfibrils. Within a wood wall these fibrils occur with hemicelluloses and lignin-rich material. The primary wall and secondary layers S 1, S 2 and S 3 differ in arrangement; the thick S 2 layer is especially significant for the wall response. A microfibril direction is measured relative to the cell axis, not automatically to the axis of a cut board.

The next level contains walls and lumina, the spaces enclosed by cells. A lumen is not a soft polymer matrix: it is a space whose contents and transport role need to be identified. Many mature conducting and support cells no longer contain living protoplasts, while other wood cells remain living. A material made by an organism is not composed entirely of living cells.

Longitudinal arrangements provide paths along a stem; radial rays connect across it. In a straight-grained region, longitudinal, radial and tangential directions are different material directions. Radial points outward across growth rings; tangential runs along them. The shape of the board and the direction of its applied force must be stated relative to that organisation.

Pulling along connected walls can load their material axially. Loading across a cellular arrangement can instead involve wall bending, changes of cell shape and connections. The same chemical names therefore do not guarantee the same response at wall and tissue scales. Real wood also has species, position, growth and processing differences; this account supplies no universal modulus or density exponent.

Named organisation in wood; the levels are related descriptions, not a universal measured scale ladder.
Level being described	What is organised	Question it helps answer
Wall constituents	Cellulose microfibrils with surrounding hemicelluloses and lignin	What can carry force within a wall?
Layered wall	Primary wall and S 1/S 2/S 3 arrangements	In which directions can the wall resist deformation?
Cellular tissue	Connected walls, lumina, axial cells and radial rays	How do load and transport paths cross the region?
Larger stem or specimen	Tissue arrangement, growth regions and external shape	Which region, direction and geometry was tested?

Bone has a different mineral–organic hierarchy

Bone is not wood with different pore sizes. Its extracellular material includes collagen, a protein, with an apatite mineral phase, water and other components. Collagen molecules form fibrillar arrangements with mineral; lamellae organise mineralised material with local orientations. These names describe distinct organisation levels, not interchangeable words for a single chemical phase.

In much adult cortical bone, osteons contain layered material around central canals, alongside other cortical structures. Cortical does not mean pore-free. Trabecular bone instead contains a connected architecture of tissue struts and plates with spaces between them. The mineralised material making a trabecula and the porous region containing many trabeculae are different mechanical test objects. Not every bone region follows one identical concentric-osteon pattern.

The architecture can redirect force, allow struts to bend or stretch, and change which connections remain after local damage. A test of a small tissue region, an indentation contact, a porous block and a whole bone includes different geometry and boundary effects. Their stiffness numbers cannot be exchanged solely because all specimens contain collagen and mineral.

Cells maintain and remodel bone over time. The architecture is therefore also a record of growth, loading and biological context. A synthetic copy of one visible pattern does not inherit that maintenance process. These distinctions concern material structure and evidence; they do not provide a clinical interpretation or an implant choice.

Separate the solid response, architecture and specimen stiffness

A simple original model makes the level distinction calculable. Consider straight parallel rods of one fictional linear-elastic solid, all of length L. They span the complete gap between two end plates held parallel and guided to translate axially without rotation. The common small axial extension is δ. Empty space carries no force; there are no other members, contacts or fluid forces. Rods remain straight and elastic, and their connections and plates are ideal. This is not a model fitted to bone or wood.

Let E_s be the rods’ axial modulus, A_g the complete envelope area including gaps, and A_s the sum of rod areas. The common strain is δ/L. Each rod carries its own area times E_sδ/L, so summing gives F = E_sA_sδ/L. Specimen stiffness is K = F/δ = E_sA_s/L, in N/mm if these lengths use mm and E_s uses N/mm².

Define f = A_s/A_g. Because every rod spans the same length, f is also the solid volume fraction within this model envelope. Define an apparent envelope modulus E_app = (F/A_g)/(δ/L). Substitution gives E_app = fE_s. The apparent response includes the gaps even though their force is zero; E_s describes the solid carrying the force. Neither is the whole-specimen stiffness K.

For original E_s = 20.0 GPa, f = 0.200, A_g = 10.0 mm² and L = 20.0 mm, E_app = 4.00 GPa and K = 2000 N/mm. A 10.0 N force gives δ = 0.00500 mm and envelope strain 0.000250. The force divided by complete area is 1.00 MPa; divided by the 2.00 mm² solid area it is 5.00 MPa. Both force balances agree.

A second supplied solid has lower E_s = 16.0 GPa but f = 0.300, giving E_app = 4.80 GPa. The architecture fraction reverses a ranking based only on solid modulus. Alternatively f = 0.250 with that 16.0 GPa solid gives the same E_app = 4.00 GPa as the first model. One apparent modulus cannot distinguish those two lower-level accounts.

The formula requires its specified continuous axial paths and common strain. A trabecular network with angled or bending members, a wall with its own anisotropy, incomplete joints or a different loading direction needs its own model. The same f in a different geometry does not establish the same E_app. Use the model to identify included levels and missing information, rather than to assign an actual bone or wood property.

Three original ideal straight-rod accounts, not measured bone or wood. Each envelope is 100 pixels across, with four equal-depth rods spanning its complete height. Total drawn rod widths are 20, 25 and 30 pixels, representing solid fractions 0.200, 0.250 and 0.300 only for this extruded axial model. Rod modulus is 20.0 gigapascals in A and 16.0 gigapascals in B and C. Apparent envelope moduli are 4.00, 4.00 and 4.80 gigapascals. All cases stipulate envelope area 10.0 square millimetres, length 20.0 millimetres, identical guided nonrotating plates, continuous rods, common small axial strain and no force through gaps. Stiffnesses are 2000, 2000 and 2400 newtons per millimetre. Plate strokes and height are schematic, not measured dimensions; actual bending, joints and biological architecture are not solved.

The living context matters

Growth, remodelling, hydration and age can change structure. Sampling orientation and location therefore matter. A dry specimen removed from a living system does not automatically represent its hydrated in-service state.

Water can be associated with solid wall or matrix material as well as occupy larger spaces. Those locations have different meanings. Wood can change dimensions with water uptake or loss; a dry-mass moisture fraction is not the fraction of the present total mass. Measurements must retain both their mass basis and the dimensions/state used in a mechanical comparison.

For example, an invented piece with dry mass 5.00 g and current mass 5.50 g contains 0.500 g added water if no other mass changed. Its water/dry-solid mass ratio is 0.500/5.00 = 0.100, or 10.0%. Water/current-total mass is instead 0.500/5.50 ≈ 9.09%. Neither ratio specifies where the water is located or a swelling amount.

A primary collagen thin-film study reports changed fibril dimensions and increased contact stiffness during controlled dehydration in that particular system. Contact stiffness of a supported thin film is not automatically a whole-bone axial modulus, and this result does not establish that every dry biological material is mechanically better. The testing scale and measured response are part of the finding.

The next table is an original fictional matched observation set, not data from that study or a named species. It concerns four otherwise matched tissue coupons with current test geometry A = 10.0 mm² and L = 20.0 mm, at one temperature and small recoverable axial excursion. “Longitudinal” and “transverse” refer to a recorded tissue axis. Hydrated and dry designate independently established states; no drying history or universal effect is supplied. At 100 N, E_app = FL/(Aδ) for this stipulated uniform linear account.

Original fictional tissue-coupon observations. Current dimensions and testing method are matched; values are not clinical data, material specifications or a universal drying law.
Material direction	State	Extension at 100 N (mm)	E_app from this account (GPa)
Longitudinal	Hydrated	0.0166667	12.0
Transverse	Hydrated	0.0250000	8.00
Longitudinal	Dry	0.0125000	16.0
Transverse	Dry	0.0200000	10.0

A component contains a cellular region, which contains aligned structures and smaller constituents. The nested sketch illustrates hierarchy without a measured scale bar.

Match the comparison before assigning a mechanism

Within the hydrated state, the longitudinal/transverse apparent-modulus ratio is 12/8 = 1.50. Within the dry state it is 16/10 = 1.60. These matched directional comparisons differ. Comparing dry-longitudinal 16 with hydrated-transverse 8 gives 2.00 while changing both state and direction; it does not isolate either effect.

At fixed longitudinal direction the dry/hydrated ratio is 16/12 = 1.3333; at fixed transverse direction it is 10/8 = 1.25. The differences are 4.00 and 2.00 GPa. Thus this supplied record does not support adding one direction-independent hydration increment. Nor do four means establish a population distribution, mechanism or causal response to a particular drying process.

Geometry remains separate. K = EA/L for the stated uniform axial interpretation. A 12.0 GPa specimen with A = 10.0 mm² and L = 20.0 mm has K = 6000 N/mm. Doubling both area and length leaves that K unchanged. An 8.00 GPa specimen with A = 15.0 mm² and L = 20.0 mm also has K = 6000 N/mm. A matching component stiffness therefore need not establish a matching tissue modulus.

To explain a difference, identify the region and orientation, constituent/state measurements, architecture and dimensions, loading history and time scale, and actual observation method. A picture can resolve an arrangement without measuring its modulus or hydration. A force–extension record can establish a response without uniquely identifying its microscopic cause. Pair complementary observations at the level needed for the question.

Four original fictional matched tissue-coupon observations, not source data or a universal drying law. Horizontal bar length uses 25 pixels per gigapascal on a common zero-origin scale. Hydrated longitudinal and transverse apparent moduli are 12.0 and 8.00 gigapascals; dry longitudinal and transverse values are 16.0 and 10.0 gigapascals. Circular blue and square orange keys distinguish longitudinal and transverse directions beyond colour. Current tested area 10.0 square millimetres and length 20.0 millimetres, temperature and small recoverable axial method are stipulated matched. The corresponding extensions at 100 newtons are 0.0166667, 0.0250000, 0.0125000 and 0.0200000 millimetre. The two direction ratios are 1.50 and 1.60; the two dry over hydrated ratios at fixed direction are 1.3333 and 1.25. Comparing 16.0 with 8.00 changes both conditions. No uncertainty distribution, particular drying process, biological cause, strength or medical interpretation is supplied.

Worked examples

Work through the case

Two bone specimens come from different orientations. Should their stiffness be pooled without recording direction?

Identify the hierarchical orientation.
Compare testing and hydration conditions.
Retain directional information in the interpretation.
In the fictional matched record, compare 12.0 with 8.00 GPa while holding hydration fixed, giving a directional ratio 1.50. The 16.0/8.00 comparison changes two recorded conditions.

No. Organisation can produce directional response. Matched state and geometry are also needed to isolate that directional comparison.

Recover the two stress bases in one load path

For the straight-rod model, E_s = 20.0 GPa, f = 0.200, A_g = 10.0 mm² and L = 20.0 mm. At 10.0 N find the solid area, apparent modulus, specimen stiffness and extension. Explain why using the apparent modulus with solid-area stress would be inconsistent.

A_s = fA_g = 2.00 mm². In matching units E_s = 20000 N/mm².
E_app = fE_s = 4000 N/mm² = 4.00 GPa. K = E_sA_s/L = 2000 N/mm.
The extension is 10.0/2000 = 0.00500 mm. Strain 0.00500/20.0 = 0.000250 gives solid stress 20000×0.000250 = 5.00 MPa and envelope stress 4000×0.000250 = 1.00 MPa.
Solid stress times 2.00 mm² and envelope stress times 10.0 mm² both return 10.0 N. Combining 5.00 MPa with 4.00 GPa silently mixes two different area bases.

A_s = 2.00 mm², E_app = 4.00 GPa, K = 2000 N/mm and δ = 0.00500 mm. The response bases must stay paired.

Separate a directional contrast from a state contrast

Use all four fictional tissue coupons. Determine the directional ratios in each state and the state ratios in each direction. Does dry-longitudinal divided by hydrated-transverse isolate a mechanism?

Hydrated direction ratio 12/8 = 1.50; dry direction ratio 16/10 = 1.60.
Longitudinal state ratio 16/12 = 1.3333; transverse state ratio 10/8 = 1.25.
The cross-condition ratio 16/8 = 2.00 changes both factors. Its value is reproducible arithmetic, but it does not isolate a single direction or state contrast.
The table supplies illustrative conditional means without a population or a specimen evolution history. Use matched further observations before identifying a cause.

The two direction ratios are 1.50 and 1.60; the two state ratios are 1.3333 and 1.25. The 2.00 cross-condition ratio is confounded.

Distinguish equal component stiffness from equal material response

Compare the 12.0 GPa tissue account at A = 10.0 mm², L = 20.0 mm with an 8.00 GPa account at A = 15.0 mm², L = 20.0 mm. At 100 N what extensions follow, and what does equality identify?

First stiffness is 12000×10.0/20.0 = 6000 N/mm.
Second is 8000×15.0/20.0 = 6000 N/mm.
Both give 100/6000 = 0.0166667 mm extension. Their engineering stresses differ because areas differ: 10.0 and 6.66667 MPa.
Equal force–extension response at this excursion is consistent with different moduli. It does not identify their chemistry, hierarchy, strength or long-duration response.

Both extend 0.0166667 mm under the stipulated linear account. Equal component stiffness does not prove equal tissue modulus.

Try the reasoning

Does a hierarchy mean every level has the same composition?
Show the answer and reasoning
No.
Composition and architecture can vary with scale. A lumen and a cellulose-rich wall are different parts of one cellular region; a mineralised trabecula and a porous block describe different inclusions.
Why record hydration?
Show the answer and reasoning
Water can affect structure and response.
The measured state must match the question. Keep its measured basis and geometry; do not transfer a supported-film contact response directly to a whole tissue.
In the same ideal straight-rod model, E_s = 15.0 GPa and f = 0.400, with A_g = 12.0 mm², L = 30.0 mm. Find E_app and K. Does f alone supply a formula for a bending network?
Show the answer and reasoning
E_app = 6.00 GPa and K = 2400 N/mm under this model. A bending network is not determined by f alone.
A_s = 4.80 mm², so 15000×4.80/30.0 = 2400 N/mm. The derivation needs complete straight axial rods, common small strain and ideal guided ends; these conditions were not established for an arbitrary porous architecture.
A measured apparent modulus is 4.00 GPa. In the ideal rod model, compare (f,E_s) = (0.200,20.0 GPa) and (0.250,16.0 GPa). What additional information is missing?
Show the answer and reasoning
Both give 4.00 GPa. Independent architecture/solid-property evidence is needed to select between them, as well as evidence that this model applies.
Their products coincide. The same apparent modulus does not recover two unknown factors or identify a biological mechanism.
A report takes the fictional 16.0 GPa dry-longitudinal value divided by 8.00 GPa hydrated-transverse and calls it a drying factor of 2.00. Give two correctly matched state comparisons.
Show the answer and reasoning
At fixed longitudinal direction, 16.0/12.0 = 1.3333. At fixed transverse direction, 10.0/8.00 = 1.25.
The 2.00 quotient changed direction as well as state. The differing matched ratios also prevent treating one state multiplier as established for both directions.
An invented specimen has dry mass 4.00 g and current mass 5.00 g, with no other mass changes. Find added-water mass, water/dry mass percentage and water/current-total percentage. Does this identify where the water is?
Show the answer and reasoning
Added water 1.00 g; dry-mass basis 25.0%; current-total basis 20.0%. Its location is not identified.
Subtract masses, then retain the denominator. Bound or wall-associated water and water in larger spaces are not distinguished by this whole mass account alone; neither is a swelling amount supplied.
A board test is described only as “across the grain.” Which wood directions still need separation, and can the cellulose name alone choose the measured response?
Show the answer and reasoning
Radial and tangential directions remain distinct; composition alone does not choose either response.
Record direction relative to growth-ring and cell arrangement, specimen geometry and state. Wall response, cellular bending and connections can contribute differently.
A supported collagen thin film becomes harder to indent after dehydration in one experiment. Does that establish the change in axial stiffness of a complete hydrated bone?
Show the answer and reasoning
No. Specimen, support, response quantity, scale and state differ.
The actual film finding can support a role for hydration in that system. A whole-bone response additionally contains tissue distribution, architecture, geometry and boundary conditions. Keep the narrow observation without inventing a cross-scale conversion.

What to carry forward

Describe organisation at relevant scales.
Retain biological and specimen context.
Name constituents, structural units, spaces and connections at the level actually observed.
Keep solid/tissue response, architecture-dependent apparent response and complete-specimen stiffness separate.
Match orientation, hydration, geometry and method before attributing a response contrast.
One bulk measurement need not identify the lower-level constituents or architecture, and biological origin does not guarantee a particular use performance.

References

Teaching resources and research supporting this lesson’s explanations and examples.