MATERIALS. DIFFERENT MODELS OF ELASTIC DEFORMATION
Note 8.21. Where the subject of interest is wave propagation in granular powders such that the wavelengths are sufficiently close to the characteristic size of the microstructure
(for example, the major diameter of the granule), a good approximation yields another microstructural model, the model of a mixture. This model gives a good approach for the linear and nonlinear wave investigation and will be discussed later.
Pobedrya’s micropolar structural theory
The classic micropolar approach arises when the modern procedure of averaging is applied to the compo- site with the regularly repeated elementary cell. Such an approach was proposed by Pobedrya. The written in displacements initial inhomogeneous problem of the linear theory of elasticity
Focus on composite materials
, , ,
( ) ;
ijkl k l j i i tt
C x u X Uu
ª º
¬ ¼ (8.44)
Focus on composite materials
, ;
ij jkln l n k ij j i
a C u n b u6 6 S6 u xi( ,0) U xi( ); ( ,0)u xi t, V xi( )
is reduced to the recurrent sequence of written in displacements problems of the linear theory of elasticity for an anisotropic homogeneous medium with some effective elastic moduli
Focus on composite materials
( ) ( ) ( )
, , ;
k k k
ijmn m nj i i tt
h w X Uw (8.45)
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
FOCUS ON COMPOSITE MATERIALS. DIFFERENT MODELS OF ELASTIC DEFORMATION
( ) ( ) ( )
, ;
k k k
ij jlmn m n i ij j i
a h w n b w6 6 S6 w xi( )k ( ,0) Ui( )k ( ); x w xi t( ),k ( ,0) Vi( )k ( )x .
Besides, the solution u x ti( , ) of initial problem (8.44) is linked in sufficiently complicate form with the solution w x ti( )k ( , ) of new problem
1 1
, ...
( ) 0 ...
( , )
( , ) q( ) q ;
p j k k
p q p
i ijk k p
p q
v x t
u x t D N [
W
f
w
¦
w ( )0
k k
i i
v w
D fD
¦
.Here, the small parameter D is equal to the ratio of the characteristic size of microstructure and the construction, and [ xD.
The feature of this model is that in the zero approximation the local displacement within the cell can be evaluated as
Focus on composite materials
( ) , ( )
i i ijk j k
u v DN [ v x .
In the notation of equations (8.44) and (8.45) the constitutive equations are absent.
Therefore, the system of equations is not complete. In other notation, when the problem is written in stre-ses, this problem is reduced to the problem of moment homogeneous theory of elasticity.
In one’s turn, the moment problem is reduced to a recurrent sequence of problems of the theory of elasticity in stresses for an anisotropic homogeneous medium with averaged moduli of mechanical compliances. One considers that the procedure of evaluation of effective moduli of elasticity and compliance is well developed.
The scheme of the microstructural theory permits, in Pobedrya opinion, to use successfully the modern numerical methods.
Thus, some fundamental formulations and sentences relative the composite materials are expounded in this chapter. Also, the basic facts from the theory of effective (averaged) constants and the main microstructural theories of composite materials are presented here.
They together form only some basis for understanding the theory of composite materials that is developed in the structural mechanics of materials.
Comments
Comment 8.1. The developed in the mechanics approximate models and structural theories of elastic deformation of the composite materials are mainly sufficiently simple. The mechanical tests showed that these models and theories predict some mechanical phenomena quite
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
FOCUS ON COMPOSITE MATERIALS. DIFFERENT MODELS OF ELASTIC DEFORMATION
who once said: “We should be obliged to God, that He founded the world in such a way that all that is simple is the truth, and all that is complicated, is not the truth”.
Comment 8.2. It seems to rational to repeat here the basic facts from the structural mechanics of materials. The structural mechanics of materials is meant as a part of investigations on the mechanics of materials, in which the internal structure of materials is taken into account in a quantitative and qualitative sense when the models of materials being constructed and corresponding problems being studied.
When the structural mechanics of materials is defined in such a way, then the object of its study is coming to the large class of modern materials, including:
natural composites, like the true wood from trees as well as the woody plants, internal structure of which is defined by the presence of layers of different properties;
reinforced concrete, the internal structure of which is defined by the presence of armature;
metals, alloys, and ceramics, the internal structure of which is defined by the presence of grains and other structural components;
composite materials, the internal structure of which is defined by the presence of granules, fibers, and layers and which include last time nanocomposites, the internal structure of which is defined by the presence of nanogranules, nanofibers, and nanolayers.
Thus, the science on the mechanical behavior of composite materials is only one big part of the huge volume of knowledge on materials with the internal structure. Therefore, it is constructed on the rich prior experience of studying the materials.
Comment 8.3. The John Bernal’s closing remarks at a 1963 science conference on new material show the philosophical essence of mechanics of materials as the theoretical science.
So, he said:
We, academic scientists, are in a certain sphere. We can continue to be useless to some extent in the certainty that sooner or later our works will be found some use. Of course, the mathematician prides himself on his complete futility, but he is usually the most useful of all. He finds a solution, but he doesn’t care what the problem is. Sooner or later someone will find a problem, the answer to which is his solution.
Here we must rethink our goals. We’re talking about new materials, but ultimately we’re not so much interested in the materials themselves as in the structures in which they should function. Our theme is primarily the practical requirements and
their satisfaction. But how do we know what these requirements are if we do not know what they can be satisfied? We often like all kinds of things we can’t get.
On the other hand, how do we know that someone needs the substance we got.
… We should be able to provide the requests that we will have, and fulfill them, so to speak, upfront, invent all sorts of crazy materials, and then find their use.
Further reading
8.1. Achenbach, JD 1976, Generalized continuum theories for directionally reinforced solids.
Arch. of Mech., 35, N4, 257-278.
8.2. Achenbach, JD & Sun, CT 1972, The directionally reinforced composite as a homogeneous continuum with microstructure. In: Dynamics of Composite Materials. Lee, EH (ed).
ASME, New York, 48-69.
8.3. Atkin, RJ & Crain, RE 1976, Continuum theory of mixtures: basic theory and historical developments. Quart. J. Mech. and Appl. Math., 29, N2, 209-244.
8.4. Bedford, A & Drumheller, DS 1983, Theories of immiscible and structured mixtures.
Int. J. Engng. Sci., 21, N8, 863-960.
8.5. Bedford, A, Drumheller, DS & Sutherland, HJ 1976, On modelling the dynamics of composite materials. In: Nemat-Nasser, S, Mechanics Today, Vol.3, Pergamon Press, New York, 1-54.
8.6. Bedford, A & Sutherland, HJ 1973, A lattice model for stress wave propagation in composite materials. Trans. ASME. J. Appl. Mech., 40, N1, 157-164.
8.7. Ben-Amoz, M 1975, On wave propagation in laminated composites. - I. Propagation parallel to the laminates. Int. J. Engng. Sci., 14, N1, 43-56.
8.8. Ben-Amoz, M 1975, On wave propagation in laminated composites. - II. Propagation normal to the laminates. Int. J. Engng. Sci., 14, N1, 57-67.
8.9. Bhushan, B (ed) 2004, Springer Handbook on Nanotechnology. Springer, Berlin.
8.10. Bolotin, VV & Novichkov, YN 1980, Mechanics of Multilayer Structures. Mashinostroenie, Moscow. (in Russian).
8.11. Bowen, PM 1976, Mixtures and EM Field Theories. In: Continuum Physics, Vol III, Eringen, AC (ed). Academic Press, New York, 1-127.
8.12. Broutman, LJ & Krock, RH (eds) 1974-1975, Composite Materials, In 8 vols., Academic Press, New York.
8.13. Buryachenko, VA, Roy, A, Lafdi, K, Anderson, KL & Chellapilla, S 2005, Multiscale mechanics of nanocomposites including interface: Experimental and numerical investigation. Composites Science and Technology, 65, 2435-2465.
8.14. Cattani, C & Rushchitsky, JJ 2007, Wavelet and Wave Analysis as applied to Materials with Micro- and Nano- structure. World Scientific, London-Singapore.
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
FOCUS ON COMPOSITE MATERIALS. DIFFERENT MODELS OF ELASTIC DEFORMATION
8.15. Christensen, RM 1979, Mechanics of Composite Materials. John Wiley & Sons, New York.
8.16. Courtney, TH 2000, Mechanical Behavior of Materials. (2nd ed). McGraw Hill, Boston.
8.17. Daniel, IM & Ishai, O 1994, Engineering Mechanics of Composite Materials. Oxford University Press, Oxford.
8.18. Drumheller, DS & Bedford, A 1974, Wave propagation in elastic laminates using a second order microstruc ture theory. Int. J. Solids Struct., N10, 61-76.
8.19. Drumheller, DS & Sutherland, HJ 1973, Lattice model of a composite material for the investigation of stress waves propagation. Trans. of ASME, J. Appl. Mech., 40, N1,157-164.
8.20. Elhajjar, R, La Saponara, V & Muliana, A (eds) 2017, Smart Composites: Mechanics and Design (Composite Materials). CRC Press, Boca Raton.
8.21. Eringen, AC & Suhubi, ES 1964, Nonlinear theory of simple microelastic solids, Int.
J. Engng. Sci., 2, N2, 189-203.
8.22. Eringen, AC 1972, Theory of micromorphic materials with memory, Int. J. Engng.
Sci., 10, N7, 623-641.
8.23. Green, AE & Steel, TR1966, Constitutive equations for interacting continua, Int. J.
Engng.Sci., 4, N4, 483-500.
8.24. Guz, AN (ed) 1982, Mechanics of Composite Materials and Structural Elements. In 3 vols. Naukova Dumka, Kiev. (In Russian).
8.25. Guz, AN (ed) 1993-2003, Mechanics of Composites. In 12 vols, Naukova Dumka, Kiev. (In Russian).
8.26. Guz, AN, Markus, S, Kabelka, J, Ehrenstein, G, Rushchitsky, JJ and others 1993, Dynamics and Stability of Layered Composite Materials. Naukova Dumka, Kiev. (In Russian).
8.27. Guz, AN, Shulga, NA, Kosmodamiansky, AS, Rushchitsky, JJ and others 1993, Dynamics and Stability of Materials, Naukova Dumka, Kiev. (In Russian).
8.28. Guz, AN & Rushchitsky, JJ 2004, Nanomaterials. On mechanics of nanomaterials.
Int. Appl. Mech., 39, N11, 1271-1293.
8.29. Guz, AN & Rushchitsky, JJ 2013, Short Introduction to Mechanics of Nanocomposites.
Scientific&Academic Publishing, Rosemead, CA.
8.30. Jones, RM. 1999, Mechanics of Composite Materials. (2nd ed). Taylor & Francis, London.
8.31. Hegemier, GA & Nayfeh, AN 1973, A continuum theory for wave propagation in composites.- Case 1: propagation normal to the laminate, Trans. ASME. J. Appl.
Mech., 40, N2, 503-510.
8.32. Hegemier, GA & Bache, TC 1973, A continuum theory for wave propagation in composites.- Case 2: propagation parallel the laminates. J. of Elasticity, 3, N2, 125-140.
8.33. Hegemier, GA & Bache, TC 1974, A general continuum theory with the microstructure for the wave propagation in elastic laminated composites. Trans. ASME. J. Appl. Mech., 41, N1, 101-105.
8.34. Herrmann, G, Kaul, RK & Delph, TG 1978, On continuum modelling of the dynamic behaviour of layered composites. Archives of Mechanics, 28, N3, 405-421.
8.35. Hollaway, L (ed) 1994, Handbook of Polymer Composites for Engineers. Woodhead Publishing, New York.
8.36. Hull, D 1981, Introduction to Composite Materials. Cambridge University Press, Cambridge.
8.37. Katz, HS & Milewski, JV (eds) 1978, Handbook of Fillers and Reinforcements for Plastics. Van Nostrand Reinhold Company, New York.
8.38. Kaw, AK 2005, Mechanics of Composite Materials. (2nd ed.). CRC, Boca Raton.
8.39. Kelly, A 1985, Composites in context. Composites Science and Technology. 23, 171-199.
8.40. Kelly, A & Zweben, C (eds) 2000, Comprehensive Composite Materials. In 6 vols, Pergamon Press, Amsterdam.
8.41. Kravchuk, AS, Mayboroda, VP & Urzhumtsev, YS 1985, Mechanics of polymeric and composite materials. Experimental and numerical methods. Nauka, Moscow. (In Russian) 8.42. Lau, KT & Hui, D 2002, The revolutionary creating of new advanced carbon nanotube
composite. Composites. Part B: Engineering, 33, 263-277.
8.43. Lee, EH 1972, A survey of variational methods for elastic wave propagation analysis in composites with periodical structures. In: Lee, EH (ed), Dynamic of Composite Materials, ASME, New York, 1-10.
8.43. Lempriere, B 1969, On practicability of analyzing waves in composites by the theory of mixtures. Lockheed Palo Alto Research Laboratory, Report No LMSC-6-78-69-21, 76-90.
8.44. Lubin, G (ed) 1982, Handbook of Composites. Van Nostrand Reinhold Company, New York.
8.45. Matthews, FL & Rawlings, RD 1999, Composite Materials: Engineering and Science.
CRC Press, Boca Raton.
8.46. Maugin, GA & Eringen, AC 1977, On the equations of the electrodynamics of deformable bodies of finite ex- Tent. Journal de Mechanique, 16, N1, 101-147.
8.47. McNiven, HD & Mengi, Y 1979, A mathematical model for the linear dynamic behavior of two-phase periodic materials. Int. J. Solids Struct., 15, N1, 271-280.
8.48. McNiven, HD & Mengi, Y 1979, A mixture theory for elastic laminated composites.
Int. J. Solids Struct., 15, N1, 281-302.
8.49. Milne, I, Ritchie, RO & Karihaloo, B (eds) 2003, Comprehensive Structural Integrity.
In 10 vols, Elsevier, New York.
8.50. Mindlin, RD 1964, Microstructure in linear elasticity. Arch. Rat. Mech. Anal.,16, N1, 51-78.
8.51. Nalwa, HS 2000, Handbook of Nanostructured Materials and Nanotechnology. Academic Press, San Diego.
8.52. Nemat-Nasser, S & Hori, M 1993, Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, Amsterdam.
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
FOCUS ON COMPOSITE MATERIALS. DIFFERENT MODELS OF ELASTIC DEFORMATION
8.53. Nigmatulin, RI 1978, Foundations of Mechanics of Heterogeneous Media. Nauka, Moscow. (In Russian)
8.54. Nigmatulin, RI 1987, Dynamics of Multi-Phase Media. In 2 parts, Nauka, Moscow.
(In Russian)
8.55. Pobedrya, BE 1984, Mechanics of Composite Materials. Moscow University Publishing House, Moscow. (In Russian)
8.56. Rushchitsky, JJ (1991), Elements of the theory of mixtures. Naukova Dumka, Kyiv. (in Russian)
8.57. Rushchitsky, JJ & Tsurpal, S.I. (1998), Waves in Materials with the Microstructure.
S.P.Timoshenko Institute of Mechanics, Kiev. (In Ukrainian)
8.58. Rushchitsky, JJ (1999), Interaction of waves in solid mixtures. App. Mech. Rev., 52, N2, 35-74.
8.59. Sahimi, M 2003, Heterogeneous Materials. Nonlinear and Breakdown Properties and Atomistic Modeling. Springer, New York.
8.60. Skudra, AM & Bulavs, FY (1978), Structural Theory of Reinforced Plastics. Zinatne, Riga. (In Russian)
8.61. Steel, TR (1967), Applications of a theory of interacting continua. Quart. J. Mech.
and Appl. Math., 20, N1, 57-72.
8.62. Steel, TR (1968), Determination of the constitutive coefficients for a mixture of two solids. Int. J. Solids and Struct., 4, N12, 1149-1160.
8.63. Sutherland, HJ (1979), Dispersion of acoustic waves by an alumina-epoxy mixture.
J. Compos. Mater., 13, N1, 35-47.
8.64. Tiersten, TR & Jahanmir, M (1977), A theory of composites modeled as interpenetrating solid continua. Arch. Rat. Mech. Anal., 54, N2, 153-163.
8.65. Thostenson, E, Chunyu, L. & Chou, TW (2005), Nano-composites in context (review).
Composites Science and Technology, 65, 491-516.
8.66. Torquato, S (2003), Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer, New York.
8.67. Tsai, SW & Hahn, HT 1980, Introduction to Composite Materials. Technomic, Wesport, CT.
8.68. Vaia RA & Wagner, HD 2004, Framework for nanocomposites, Materials Today, N10, 32-37.
8.69. Van Fo Fy, GA 1971, Theory of Armed Materials with Coatings, Naukova Dumka, Kiev. (In Russian)
8.70. Vanin, GA 1985, Micromechanics of Composite Materials. Naukova Dumka, Kiev. (In Russian)
8.71. Wagner, HD & Vaia, RA 2004, Nanocomposites: issues the interface. Materials Today, N10, 38-42.
8.72. Yakobson, BI & Avouris, P 2001, Mechanical properties of carbon nanotubes. In:
Topics in Advanced Physics. Vol. 80. Carbon nanotubes: synthesis, structure, properties,
and applications. Dresselhaus, MS, Dresselhaus, G & Avouris, P (eds) 2001. Springer, Berlin, 287-329.
Questions
8.1. Enlarge on the list of the natural composite materials which are used in private life.
8.2. Which artificial composite materials were used in ancient Rome and China empires?
8.3. Write three types of industrial composites most widespread over the world.
8.4. Comment on the situation when macro-, meso-, micro-, and nanomechanics of composite materials are developed basing on the continuum models. This fact testifies the conservatism or the power of the continuum approach in mechanics?
8.5. Comment the diffuseness of the border between the macro-, meso-, micro-, and nanomechanics of composite materials by the criterion of the characteristic size of internal structure. Why the ranges of these parts of mechanics are so overlapped?
8.6. Why the theoretical prediction of the mechanical properties of the artificial composite materials is more important as compared with other types of engineering materials, where the experimental tests are the main tool in the determination of mechanical properties?
8.7. Try to find the formulas for averaged properties of composite materials which are proposed for the not classical (isotropic, transversely isotropic, orthotropic) cases of symmetry.
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
FOCUS ON COMPOSITE MATERIALS. STRUCTURAL MODEL OF ELASTIC MIXTURES