at Bioengineering the tissue level
Specialized organization of cells
Quote of the Day
Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. Albert Einstein
HMMM
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Learning Objectives
Define tissues List the four tissue types Define tissue engineering and how engineering is involved Define the term biomaterial Explain biocompatibility Provide examples of tissue engineering and biomaterials use
Tissue
Tissue: a group of cells and surrounding substances that function together to perform more specialized activities Four primary tissue types
Epithelial Connective Muscle Nervous
Epithelial Tissues
Arranged in sheets Organized into glands Free surface Functions:
Adapted for secretion One or more layers thick
Form the interfaces of your body Absorption (e.g. small intestine) Secretion (e.g. glands) Transport (e.g. kidney tubules) Excretion (e.g. sweat glands) Protection (e.g. skin) Sensory Reception (e.g. taste buds)
http://www.agen.ufl.edu/~chyn/age2062/lect/lect_19/lect_19.htm
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Connective Tissue
Most abundant and widely distributed
Loose woven fibers around and between organs Protective capsules around organs Ligaments and tendons Blood Bone Cartilage Adipose tissue
Specialized connective tissues include:
http://www.agen.ufl.edu/~chyn/age2062/lect/lect_19/lect_19.htm
Muscle Tissue
Muscles comprise 40 to 50 percent of body weight Generate forces by contraction Three muscle types
skeletal Smooth (found in walls of blood vessels) cardiac
http://www.agen.ufl.edu/~chyn/age2062/lect/lect_19/lect_19.htm
Nervous Tissue
Neurons Conduct electrical impulses Glial cells Protect, support, nourish neurons
http://www.agen.ufl.edu/~chyn/age2062/lect/lect_19/lect_19.htm
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Tissue Material Complexity Bone Example
Cortical Trabecular "Structure within a structure" Features at every scale
cm mm m nm
Trabecular Bone
Cortical Bone
Bone Scales
Trabecular Bone
Struts are circumferentially lamellar in form Blood vessel in center of strut Bone cells in the interlamellar spaces
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Bone Cells
Osteoblasts a bone forming cell Osteoclasts a specialized bone cell that enzymatically removes (excavates) bone Osteocytes retired osteoblasts that maintain the tissue. They are surrounded by bone and are interconnected Bone lining cell retired osteoblasts that regulate the passage of calcium in and out of the bone
Tissue Engineering
Tissue Engineering
Central idea: Inducing the body to develop new tissues <perhaps organ systems eventually>, instead of replacing the tissue with an engineered device... An biomedical engineering discipline integrated with biology to create tissues or cellular products outside the body Diverse knowledge of molecular and cellular biology as well as chemical engineering, materials science, and other engineering disciplines
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Examples
Blunt Trauma to an extremity Highly comminuted fracture Amputate
Tissue Engineering Solution
In some way, yet unknown, create an environment that would cause the body to restore the damaged tissue. Currently, the skin is one of the most successful applications of tissue engineering
Skin
An example of epithelial tissue Skin is one of the largest structures of the body Replacement necessitated by Sensory Protective
tramatic injury burn (12000 fatalities / year) cosmetic/vanity Microbe/disease Radiation (Sun/UV) Chemical
Thermoregulation Secretion
http://www.agen.ufl.edu/~chyn/age2062/lect/lect_19/lect_19.htm
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Artificial Skin
Scaffold
Artificial Allograft derived
Culture cells the scaffold Usually cultured in the presence of loads
Dermal Grafts for Burn Victims
Implant tissue
The circulating stem cells target damaged tissue Cells deposit and adhere to the matrix Cells differentiate into tissue-specific cell types Differentiated cells elaborate the new matrix to regenerate tissue
www.bvmed.de/bilderpool/Bilder_Medizinprodukt...
Tissue Engineering
Engineered tissue can be used in patients who are in need of transplantation, eliminating problems such as shortage of donors and immune rejection. Several tissue types are in the process of being engineered:
Pancreas Liver Skin Nerves Cornea Breast Bone Blood Muscle Cartilage Blood vessels
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Cartilage Tissue Engineering
More than one million patients every year suffer from some form of cartilage damage in the United States. The engineering of cartilage includes
Isolation of young chondrocytes (cartilage cells) Seeding on a bio-resorbable porous scaffold Culturing on a bioreactor under the appropriate conditions
Cells are seeded in a bioresorbable matrix to guide tissue development in a 3-dimensional way. The cell-matrix system is cultured for a period of time inside a bioreactor or culture vessel. Ulltimately, the new tissue is implanted into the patient. So far, none of the engineered cartilages produced around the world can guarantee to match the performance of natural articular cartilage.
http://www.coe.neu.edu/research/berl/BERL_research_tissueeng.html
Bioengineers and Materials
Bioengineers use materials to make things!
Cardiovascular/thoracic devices Orthopedic devices Dentistry devices and materials Environmental remediation Characterization of biologic tissue
Engineering Materials Classes
A material class is a set of materials with similar microstructural organization, and/or performance i.e. electronic configuration, bonding, structure, function, etc.
Material Classes
Metals Ceramics Polymers Semiconductors Composites Biomaterials
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Engineering Materials Classes
Material classes may include elements of other material classes. For example
A certain metal may be a biomaterial However, not all biomaterials are metals And not all metals are biomaterials
Material Properties
A material property is an intrinsic characteristic that can be used to describe a material behavior
Quality
Stiffness Strength Fracture Resistance Wear Resistance Conductance
Material Property
Elastic Modulus Tensile Yield Fracture Toughness Hardness Resistivity
Symbol
E Sy K Rc (example)
Stress
Tensile stress, :
Ft
A rea, A
Shear stress, :
Ft
A rea, A
F Fs
=
Ft Ao
Ft
=
original area before loading
Fs Ao
Fs F Ft
Stress has units: N/m2 or lb/in2
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Strain
Tensile strain:
= Lo
L/2 /2
Lateral strain:
- L = L wo
wo
Lo
/2 L/2
Shear strain:
/2
= tan
/2 - /2
Strain is always dimensionless.
/2
Simple Tensile Mechanical Properties
Obtained by a tensile test
Stress-Strain Testing
Typical tensile specimen Typical tensile test machine
Adapted from Fig. 6.2,
Callister 6e.
load cell
extensometer
gauge (portion of sample with = length reduced cross section)
specimen
moving cross head
Other types of tests:
--compression: brittle
shafts. materials (e.g., concrete)
--torsion: cylindrical tubes,
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Modulus of Elasticity, E
Slope of the linear portion of the stress-strain graph
F
1
E
Linearelastic
Hooke's Law =E
F
simple tension test
Units: E: [GPa] or [psi]
Poisson's ratio,
= - L
metals: ~ 0.33 ceramics: ~0.25 polymers: ~0.40
Units: : dimensionless
F
F
simple tension test
L - 1
Elastic Shear modulus, G
Slope of the linear portion Pressurevolume change curve =G Special relations for isotropic materials
G= E 2(1 + )
M
simple torsion test
M
1
G
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Yield Strength, y
Stress at which noticeable plastic deformation has occurred. when p = 0.002
tensile stress,
y
engineering strain,
p = 0.002
Tensile Strength, TS
Maximum possible engineering stress in tension
TS engineering stress
Typical response of a metal
strain
Metals: occurs when noticeable necking starts. Ceramics: occurs when crack propagation starts. Polymers: occurs when polymer backbones are aligned and about to break. Other symbols UTS, UTS
Biomaterials
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Biomaterials Experts?
Overview
Background Traditional biomaterials New generation biomaterials The Future Conclusions
Historical Uses of Biomaterials
Earliest examples are sutures. Circa 4000 BC. Linen and adhesive plaster Horse hair, cotton, leather (600 BC) Catgut (2nd century) Silk (11th century) Wire (16th century) Violin strings (19th century)
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Biomaterials?
What are they... living, dead, animate, both, neither, something else??? Is a biomaterial biological? Is a biomaterial organic? Could a biomaterial be inorganic?
Biomaterials: Definition
YES! A biomaterial is any non-native substance -- typically man-made, but can be derived from living tissue -- that is systemically and pharmacologically inert and designed for implantation or incorporation with the living system.
Clemson University Biomaterials Symposia Definition.
Biomaterial
A biomaterial is a nonviable material used in a medical device intended to interact with biologic systems
Williams, D.F. (1987) Definitions in Biomaterials. Proceedings of a Consensus Conference of the European Society of Biomaterials, Chester, England, Mar 3-5, 1986, Vol 4, Elsevier, NY
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Biomaterials Requirements
Must not evoke adverse reactions (toxic, carcinogenic) or worse (death) The material must not interfere with, or modify (in a negative fashion), life processes such as wound healing. The material must not be damaged by being in the bodily environment (such as corrosion). The purpose is generally for functional replacement or interventional devices
Uses of Biomaterials
Biomaterials usage constitutes approximately $100 billion per year from medical device industry Medical uses (partial list from A to Z...):
Arterial graft Heart valves Joint replacement Pacemakers Stents Zirconium knee joint
Interdisciplinary Field
Materials Science Surface chemistry Biochemists Biology Medicine Ethicists Lawyers
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What is Biocompatibility?
Materials that are biocompatible evoke minimal (at best no) adverse responses while in contact with the host, as well as maintaining material integrity over the intended service life.
Host
Human or animal Local tissue and distant systems/organs
Response
Dies, heals, carcinogenic, thrombogenic, etc.
Contact
Brief, <24 hours, <30 days, >30 days Implant, mucous membrane, skin, eye, etc.
What is Biocompatibility?
Foreign
Not originating from the host (including autograft)
Material
Polymer, metal alloys, ceramics, silicone, xenograft, splinter, microspheres, collagen, etc.
Affected
Corrosion, cracking, degradation, color change, fracture, fatigue
Long Term Implants
Devices intended to be left in the body Fracture fixation plates Pacemakers Stents Dental implants
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Soft Tissue Response to Implanted Biomaterials
Non-toxic reactionEssence of Biocompatibility
Typical Biological Response to Non-Toxic Implanted Materials (Cardiovascular or Soft Tissue)
1. 2. 3.
4. 5. 6.
Implant Protein Adsorption (seconds) Thrombosis and Complement Activation (seconds to hours) Cellular Recognition (minutes to hours) Inflammation (days) Resolution (weeks/months)
Foreign Body Reaction
Giant cell formation & cytokine release
1 sec to 1 hr 2 10 days
Protein adsorption
Cellular attack/recognition
30 min 2 days 3+ weeks
Fibrous Capsule
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Time Course of Inflammation
Thrombosis, Complement Activation
Protein Adsorption
J. Anderson, "Inflammation and the Foreign Body Response," Problems in General Surgery, 11, 147-160 (1994).
Foreign Body Reaction
Histology Slides of Silicone Implanted in SVC for 6 months
50 m 50 m
Macrophages and FBGCs
Acellular Fibrous Capsule
Silicone Implant Haematoxylin and Eosin
Nuclei - Purple/Black
Masson's Trichrome
Collagen - Blue Cytoplasm - Red
Blood Responses Foreign Body
Clotting system pathways activated Thrombus formation (clot)
Restricting flow
Thrombus embolization
Strokes, heart attacks
Blood cells damage (hemolysis) Surrounding tissues damage Bacterial colonization
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Foreign Body Reaction
Summary
The body is designed to detect foreign materials The basic strategy is to either digest, or encapsulate the foreign body if removal is not possible From an engineering perspective, it is hoped that the device evokes minimal reaction
Typical Implantable Materials
Metals Ceramics Polymers
Bioabsorbable
Metals
Strong (good) Wear resistant (good) Formable (good) Stiff (may not be so good) Chemically active (not so good)
Stainless Steel (316LV) Titanium and titanium alloys Zirconium Nitinol Tantalum MP35N Co-Cr-(Mo) Alloys Gold Platinum and Pt-Ir
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Ceramics
Most biocompatible class of materials Chemically inert due to atomic bonding Easily sterilized (good) Generally brittle (bad) Generally hard (good) Some ceramic materials are naturally degraded by the body Some ceramic coatings may enhance device biocompatibility
Zirconia Alumina Graphite Diamond Hydroxyapatite Bioglass
Polymeric Materials
Polymers represent a class of materials that are generally biocompatable Due primarily to atomic bonding stability Most biocompatable polymers are in pure form Generally polymers must be highly polymerized (mers may be carcinogenic) Polymers must not leach substances
Fillers, plasticizers, colorants, etc.
Polymers
Polymers are large molecules made up of large numbers of repeating units
H C H H H C C H H H H C C H H H H C C H H H C H
Mer unit
Classified as Thermoplastics, Thermosets, and Elastomers
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Polmer Microstructure
Polymer = many mers
mer H H H H H H C C C C C C H H H H H H
Polyethylene (PE)
mer H H H H H H C C C C C C H Cl H Cl H Cl
Polyvinyl (PVC)
mer chloride H H H H H H C C C C C C H CH3 H CH3 H CH3
Polypropylene (PP)
Adapted from Fig. 14.2, Callister 6e.
Covalent chain configurations and strength:
secondary bonding
Linear
Branched
Cross-Linked
Network
Direction of increasing strength
Adapted from Fig. 14.7, Callister 6e.
Molecular Weight & Crystallinity
Molecular weight, Mw: Mass of a mole of chains. Tensile strength (TS):
often increases with Mw. Longer chains are entangled (anchored) better.
smaller Mw larger Mw
% Crystallinity: % of material that is crystalline. and E often increase with % crystallinity. Annealing causes crystalline regions to grow. % crystallinity increases.(TS):
crystalline region amorphous region
Adapted from Fig. 14.11, Callister 6e. (Fig. 14.11 is from H.W. Hayden, W.G. Moffatt, and J. Wulff, The Structure and Properties of Materials, Vol. III, Mechanical Behavior, John Wiley and Sons, Inc., 1965.)
Tensile Response: Brittle & Plastic
Near Failure
(MPa)
60 x 40
brittle failure
onset of necking near failure
plastic failure
x
Initial
20 0 0
unload/reload
2
4
6
8
crystalline regions slide
aligned,networked case crosslinked case semicrystalline case amorphous regions elongate
crystalline regions align
Stress-strain curves adapted from Fig. 15.1, Callister 6e. Inset figures along plastic response curve (purple) adapted from Fig. 15.12, Callister 6e. (Fig. 15.12 is from J.M. Schultz, Polymer Materials Science, Prentice-Hall, Inc., 1974, pp. 500-501.)
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Tensile Response: Elastomer Case
Compare to responses of other polymers:
(MPa)
x 60 brittle failure 40
plastic failure
x
brittle response 20 (aligned, cross linked 0 2 0 & networked case) initial: amorphous chains are plastic response kinked, heavily cross-linked. (semi-crystalline case)
elastomer
x
4
6
8
final: chains are straight, still cross-linked
Deformation is reversible!
Stress-strain curves adapted from Fig. 15.1, Callister 6e. Inset figures along elastomer curve (green) adapted from Fig. 15.14, Callister 6e. (Fig. 15.14 is from Z.D. Jastrzebski, The Nature and Properties of Engineering Materials, 3rd ed., John Wiley and Sons, 1987.)
Thermoplastics Vs Thermosets
Thermoplastics:
little cross-linking ductile soften w/heating Polyethylene, polypropylene polycarbonate, polystyrene
Thermosets:
Extensive cross-linking (10 to 50% of mers) hard and brittle do NOT soften w/heating vulcanized rubber, epoxies,polyester resin, phenolic resin
Biodegradation & Erosion
Bio-as a prefix will be used to indicate processes that are influenced by biological agents
Cells Enzymes Microorganisms
Emphasizes the process is facilitated by biologic agents or physiologic conditions
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Biodegradation
Breakdown of a substance by biological means Chemical structure is broken down into smaller components Cleavage of covalent bonds Cannot strictly be considered as hydrolysis, such as poly(lactic acid) Sometimes called bioresorption or bioabsorption
Bioerodable Polymers
A water-insoluble polymer that is converted under physiologic conditions into water-soluble materials without regard to the specific mechanism involved in the erosion process The polymer backbone is not broken down
Degradable Materials
Variable life materials, may be short-, mid-, or long-life A degradable material is designed to function as a biomaterial for a pre-defined life, then be digested by the body, which evokes minimal foreign body reactions at any stage
The by-products must be inert The rate of decomposition must be sufficiently slow to allow the device to function Some by-products can be metabolized...
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Biodegradable Materials
Most commonly composed of polymers Often times are formed from block copolymers Degradation typically involves hydrolysable bonds
Esters Carbonyls Amides
Some Degradable Poly's
O
Poly(glycolic acid)
*
O C C H2
O H O C C CH3
n
*
Poly(lactic acid)
*
n
*
O
Poly(-caprolactone)
*
C
CH2 5 O
n
*
Inert Polymers
These polymers are intended for longterm use Highly stable Do not leach Few (if any) additives
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Some Stable (Inert) Poly's
CH3
Silicone Rubber
*
Si O CH3
F F
n
n
*
Teflon
*
C C F
H H
*
F
Polyethylene
*
C C H H
n
*
Polymers
Any polymer must be able to be sterilized Typical sterilization techniques
Exposure to ethylene oxide Gamma irradiation Autoclave (pressure and steam)
Polymers can be designed to selectively degrade with time
Biomaterials: Some Common Sterilization Methods
Ethylene oxide
Lethal Can be problematic for polymers
Pressure and Steam (Autoclave)
Effective Simple, inexpensive Many polymers cannot be treated
Gamma irradiation
Lethal Possibly the best method, but very expensive Can damage polymers
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Infection: Possible Strategies
Surface coatings
Hydroxyapatite Silver
Antibiotic coatings
Gentimicin
Cell seeding
Endothelialization
Surface modified materials to prevent colonization of materials
Dental Materials
Dental Materials
Dental biomaterials even include mercury (dental amalgams) Materials must not be susceptible to corrosion, fatigue, or fracture Microbiologically influenced corrosion can be problem Thermal expansion coefficients of material and tissue must be compatable Gold is still the best filling material available Composite materials are widely used
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Some Orthopaedic Applications
OUCH!
OUCH! Hey, we can fix that!
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IM Nails in the Femur
Some Mechanics Issues
Kinematics Stress Analysis Device Size MORE LATER...
Materials Issues in Joints
Low Friction High Hardness Fatigue Resistance Elastic Properties
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Friction...Less is Better
Responses to Foreign Objects
Bone loss leading to loosening 1. Caused by sub-micron size particles (wear debris) -Hardness 2. Stress shielding -Stiffness (Elastic Modulus) 3. Infection Fracture
Pain Development of an allergic response
Corrosion products
Metallosis
A Joint Gone Bad...
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Composite Femoral Stem
Composite EPOCH femoral stems are
75% less stiff than a comparable cobalt-chromium 50% less stiff than a comparable titanium alloy implant.
These implants have a cobalt-chromium alloy core surrounded by a flexible polymer material. The exterior surface is composed of a titanium fiber mesh that is designed to encourage bone in-growth Smaller sizes of the EPOCH Hip are manufactured from a titanium alloy and have a porous titanium fiber surface, but do not have the polymer middle layer found in the larger sizes.
Composite Stem
PolyaryletherKetone
CoCr Alloy
Porous Titanium
Carbon Fiber Composite Stem
Invibio Composite Hip Replacement system Carbon fiber reinforced PEEK Hydroxyapatite coated proximally Ceramic Femoral Head
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Cardiovascular Biomaterials Requirements
Must not evoke tissue and or blood responses Thrombosis - clotting Hemolysis - blood killing Device must not interfere with blood hemodynamics
Estimated use of cardiovascular devices in the US
Device
Heart valves Mechanical Bioprosthetic Pacemakers Vascular Grafts Oxygenators (Cardiopulminary Bypass) Heart Assist Systems Intraaortic balloon pumps Ventricular assist devices (pulsitile, non-pulsitile) Total Artificial Hearts
Implants
32000 20000 144000 160000 260000
31300 400 20+
Ball and Cage Valve
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Current Designs
Blood Vessels
Polymers such as ePTFE, and woven Dacron work very well for a tissue replacement Polymers have the advantage of being heparinized Polymers should not "look like" proteins. (Nylon is an example -NH2, amine terminal group)
long term implants will be resorbed
Modeling Biologic Tissue
Normal tissues are collagen rich All tissues are viscoelastic to some extent Viscoelastic solids can be represented using springs and dashpots Creep: Constant stress (strain increases over time) Relaxation: Constant strain (stress decreases over time)
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Examples of Tissue Response to Rate of Loading
Bone
Stiffness is rate dependent Strength is rate dependent Fracture pattern is rate dependent
Tendons
Stiffness is rate dependent Strength is rate dependent
Fracture Pattern Examples
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Summary of Traditional Biomaterials
Foreign body reaction is a natural course This reaction is not controlled The fibrous capsule can lead to device failure over time
New Generation Biomaterials
Foreign body reaction is surface driven Attempt to control foreign body reaction
Short term is possible now
Long-term solutions
Combination of materials and tissue engineering Appropriate scaffolds and extra-cellular matrix
Surface Modified Materials
Surface chemistry and texture Non-fouling surfaces (no protein adsorption)
Polyethylene glycol
HO-(CH2-CH2-O)n-H Short term, in-vitro Not so good for long term, in vivo
Heparin binding (non-thrombogenic)
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PEG Background
Molecular weights range from 200 to 10,000 Low MW PEG is clear, viscous liquid High MW PEG is soft, wax-like solid Solubility in polar solvents (water, ethanol, etc.) decreases with increasing MW Protein rejection excellent biocompatibility
Applications
Hydrogel PEG surface coatings
Implants Nanoparticles
Surface Coatings
Source: Alcantar N, Aydil E, Israelachvili J. Polyethylene glycol-coated biocompatible surfaces. Journal of Biomedical Materials Research. 2000; 51: 343-351. Available at: http://www3.interscience.wiley.com/cgibin/abstract/72509278/ABSTRACT.
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Surface Coatings
Benefits of excellent biocompatibility on virtually any substrate
Must be chemically bonded to surface, otherwise biofluid solvents remove PEG chains
Amorphous silica (SiO2) deposited via PECVD Water plasma treatment creates high density of surface silanol (Si-OH) bonds Exposure to low MW (~400) PEG vapor Silanol groups react with hydroxyl groups, form covalent bonds (Si-O-C)
Long-term non-fouling surfaces
Most surface modified materials work better in-vitro The physiologic milieu is more complex than we know Stealth biomaterials may be unobtainable
Controlling the Fibrous Capsule
Surface Microarchitecture Porosity
Non porous Thick capsule 5-15 microns Favorable to tissue ingrowth
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Synthesis of New Biomaterials
Hydrogels
Hydrophilic cross-linked polymer structure Can absorb up to 1,000 times their dry weight in water Bond character determines biodegradation
Chemically stable or biodegradable
Physical hydrogels molecular entanglements, secondary bonds Chemical hydrogels covalent crosslinking
Hydrogels
Saturated flexible, soft, elastic
Primary bound water Secondary bound water
Applications:
Contact lenses Tissue engineering matrices Wound closure implants
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Tunable Structure
Swelling behavior can be adjusted by various methods
Complexing pH Sensitivity Temperature sensitivity
Becomes a sort of "smart" material
Stimulus
Complex formation
Complexing
Insoluble macromolecular structure formed by polymers with affinity for one another Increased cross-links decrease in swelling
Drug release rate will decrease
Smart Material
Biomaterials Science: An introduction to materials in medicine, 2nd Ed. (2004). Elsevier, NY
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Cell Entrapment
Immobilizing cells in the hydrogel forms the basis of tissue
Islets of Langerhans Cartilage
Photopolymerization is a common method for developing hydrogel scaffolds
Allows for 3-D structures
Biodegradable Cell-sheet engineering
Biomimetics
Biomimetic biomaterials
Nacre is known to be osteoinductive Creating osteoinductive materials based on nacre Bioactive materials incorporate short oligopeptides (RGD) to emulate the ECM Acellular materials derived from tissues that include tissue growth factors
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Patterned Biomaterials
Photolithography and other fabrication techniques can be used for MEMs or patterned materials
Falconneta, D.; Csucsb, G.; Grandina, H.M.; Textora, M. (2006) Surface engineering approaches to micropattern surfaces for cell-based assays. Biomaterials 27:30443063
Summary
Current materials used in devices will persist for the next 10-20 years Deeper understanding of the foreign body reaction will minimize the foreign body reaction Controlling the reaction will be realized by surface chemistry, biology, and materials science collaboration
Elastic Behavior
Stress is proportional to strain Linear Constant of proportionality is called elastic modulus, E
900 800 700 Stress, MPa 600 500 400 300 200 100 0 0 0.01 0.02 0.03 0.04 0.05 Strain, mm/mm
= E
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Modeling of Tissue
We might do this...
Viscous Behavior
Newtonian Fluid Stress proportional to strain rate
800 700 600 S tress, M P a 500 400 300 200 100 0 0 500 1000 1500 Strain Rate mm/mm/s
& =
Maxwell Model
An elastic element A viscous element Series configuration Total strain is sum of individual elements
E
T = e +
& & & T = e +
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Maxwell Model
Substituting the constitutive models into the strain rate equation produces a first order ordinary differential equation with constant coefficients
& & & T = e +
& & E = E E & =
= e =
& & + = E = E 1
Maxwell Model
Relaxation response: Apply a constant strain, the stress will tend toward zero over time.
& + =0 B.C. (0) = o
Stress, MPa
1
50 40 30 20 10 0 0 20 40 60 Time, Days
0.006 Strain, mm/mm 0.005 0.004 0.003 0.002 0.001 0
= o exp - t
( )
Viscoelastic Summary
Given simple spring and dashpot model, find relaxation time Set up a simple viscoelastic model ODE Sketch creep or relaxation response
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