Hi! Answer is quite long but I believe that the best way to understand something is to do it! Hope it works!
Intro.
First of all, scale homogenization is a mathematical technique which was developed mainly by French and Russian mathematicians in the mid 70's (see Benssousan et al., 1975 and 1978).
From a physical point of view, this concept deals with what we understand by physical properties and how can we define a composite material as a mixture/aggregate of homogeneus constituents.
As a practical example, think about wood or steel. In a "macroscopical" scale, namely X, we can see with our eyes a continuum and "practically homogeneus material". However, as you may already been noticed, we know that internally these kind of materials are heterogeneous.
A natural question is, how we can even think about in material properties when we know that internally, in a "different scale", namely Y, they are heterogeneus?
After some reflections, you may notice that this theory relies on the assumption, until some extent, that material heterogenities may exhibit a sort of periodicity.
To clarify the idea, think about crystals. They are often modeled as "periodic arrangement of atoms in a crystal-lattice".
For the effective properties of a composite material, analysis is limited to a "periodic model", "representative volume of analysis" or "periodic element" in the Y-scale.
The key here is to choose this "periodic object" in manner that ALL the physical and geometrical (distribution) characteristics of each constituent are present.
Such periodic object will contain, as in a crystal, all the information needed to compute their physical properties (elastic, thermal, electrical and magnetic).
With the above mentioned elements, it is possible to state a connection (both, geometrical and physical) between the scales X and Y. This is:
(1) Y = X/e
where e is a small (e g1 only depends on the Y-scale.
(7) u2(x,y) = [partial u1(x,y)/partial x] * [g2(y)] --> g2 depends only on y.
u3(x,y) = [partial u2(x,y)/partial x] * [g3(y)] --> g3 depends only on y.
and so on.
Now the physics.
Do you remember the periodic assumption? Well, if periodicity is present at the Y-scale, it is possible, until some extent, to assume that functions g1, g2, g3, etc. are all related as follows:
(8) g2 = dg1/dy
g3 = dg2/dy = -g1
This idea is clarified if you think that g1 is a periodic function like the sine. From calculus, g2 will be similar to the cosine, hence, their derivative (g3) is the negative of the first and so on.
With this assumption it is feasible to truncate the system to the first two or three equations. A stronger assumption than the previously exposed relies on the fact that, for any periodical function their integral inside their period is zero.
Classical homogenization only uses the equations related to the terms 1/e, e^0 and e^1 (an additional term, 1/e^2, appears if a0 is set as function of both, x and y in (6) just to figure out that a0 only depends on x).
The integral of the equation related to e^1 is strongly related to divergence theorem for the stresses (you may need to rewrite a "local" version of Hooke's law using one of the first equations of the system and the definition of “traction”).
From this, you can set that equation related to this power of "e" can be neglected in your calculations. Note that this is only valid for linear elasticity, I don't know the details of what you pretend to do.
Caution: This argument is stronger than the stunning asseveration, let's make e = 0. Don't forget that you still have an equation related to 1/e.
Now the effective property.
At the end of the "physics" step, you must have (in linear elasticity) a 2x2 system of partial differential equations for coefficients a0 and a1.
Since we are depicting effective properties, which can be only seen in the X-scale, we need to make an "average" of this system. For this reason, this kind of method is also called "an averaging technique".
Since g1 is periodic (a1 doesn't) and we are modeling the properties at the Y-scale level, typically, an integral on the defined periods of your model should be enough to average your model. Since this integral is performed in the Y-scale, all the functions dependent only on x can be treated as "constants" inside the integral.
Formally, the average of function h(y) can be defined as:
(9) = int h(y) dy with limits in the "periodic model boundaries".
Now, just do the average!
Take a look in the equation containing the second derivative of a0(x) with respect x.
What's the meaning of the term in brackets? The effective Young's module!!! :D
Hence, you don't need to solve anything for a0(x)! Remember! We are interested in the "effective properties".
For linear elasticity, if you did all the steps in a 1D bar with two constituents “perfectly bonded” (you will need to introduce an expression to this!) you will get Reuss estimate for the effective properties!!!
Concluding remarks.
1) This is a summary! A detailed and formal exposition of the method can be found in Bensoussan et al. (1978). Use of more and more equations of this system in the "heat equation" can be found in Amar & Gianni (2005).
2) In the case of linear elasticity, it is possible to use Finite Element Method tools like ANSYS, ABACUS and so on... This is possible, if and only if, adequate constraints are imposed to the periodic Y-model and a "good averaging schema" is properly defined.
A detail of this technique with ANSYS can be found in a set of papers published by Dr. Harald Berger of Magdeburg University (Berger et al, 2005(a to c)). Another good example is depicted in Segurado & Llorca (2002) paper. You may need to sleep together with your favorite FORTRAN or MatLab book!
3) We are working with Newton's movement laws only! If you want to mix physical laws (say, classical mechanics + molecular dynamics), I reccommed the papers from Fish et al. (2007) and Chung et al. (2003).
4) A clear and formal insight about the limits of this approximation can be found in Blanc's et al. paper (2007).
5) Real 3D models makes a huge use of cartesian tensors. Don't be afraid! Just put “i”'s , “j”'s , “k”'s and “l”'s at the beginning (using generalized Hooke's law) of this human-readable recipe. For equations (7) and (8) you will need additional indexes (“p”'s, “q”'s, “r”'s and “s”'s). Abuse of “index switching” like “change i's for j's” is highly recommended to visualize symmetries in those “non-understandable” steps! ] :D A good introduction to tensors can be found in ... Platicity's theory book and in ... book too! And, of course!, the remaining equation is a “closed-system of partial differential equations”.
References.
Amar M., Gianni R. 2005. A Brief Survey on Homogenization with a Physical Application. MAT A Series (9), Rosario, Argentina.
Bensoussan A., Lions J. L., Papanicolau G. 1975. Sur quelques phénomènes asymptotiques stationnaires (I). Comptes Rendus Acad. Sci. Paris, (281) 89--94.
Bensoussan A., Lions J. L., Papanicolau G. 1978. Asymptotic Methods in Periodic
Structures. Amsterdam. North Holland.
Berger H., Kari S., Gabbert U., Rodríguez-Ramos R., Bravo-Castillero J., Guinovart-Díaz R. 2005 (a). A comprehensive numerical homogenisation technique for calculating effective coefficients of uniaxial piezoelectric fibre composites. Materials Science and Engineering A (412), 53–60.
Berger H., Kari S., Gabbert U., Rodríguez-Ramos R., Bravo-Castillero J., Guinovart-Díaz R. 2005 (b). Calculation of effective coefficients for piezoelectric fiber composites based on a general numerical homogenization technique. Composite Structures (71), 397–400.
Berger H., Kari S., Gabbert U., Rodríguez-Ramos R., Bravo-Castillero J., Guinovart-Díaz R. 2005 (c). An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. International Journal of Solids and Structures (45), 5692–5714.
Blanc X., Le Bris C., Lions P. L. 2007. Atomistic to Continuum limits for computational materials science. Mathematical Modelling and Numerical Analysis (41), 391–426.
Chung P W., Namburu. R. R., 2003. On a formulation for a multiscale atomistic–continuum homogenization method. International Journal of Solids and Structures (40), 2563–2588.
Fish J., Chen W., Li R. 2007. Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions.Comput. Methods Appl. Mech. Eng. 196, 908–922.
Segurado J., Llorca J. 2002. A numerical approximation to the elastic properties of sphere–reinforced composites. Journal of the mechanics of physics and solids (50), 2107–2121.
For scale homogenization, you mean size effect predicted by homogenization techniques? You should check my last publications about it.
Is for linear elasticity but it may give you some inspiration for non-linear materials.
Good luck!
Mat
Hello, you can linearize numerically and provide it in step-by-step mode. What we need is a precise knowledge about stress-strain relationship for the original components of the composite.
Dear Jawad,
We might be able to give you an answer that is more specific, but we might need some information about the problem you're interested in. Regarding the "how" part of your question, most standard commercial and free FEM packages can deal with homogenization problems, but since you, I assume, want to study the nonlinear mechanical behaviour of materials, you may want to look into Abaqus (industry-oriented) or Z-Set (academy-oriented, www.zset-software.com).
Cheers,
Justin
Hello everyone,
about the "what is" question, I think Brett gave a good simple definition. Regarding the "how" to non.linear material I also think we need more information. First, do you mean non-linear material or structures with non-linear behaviours? Because I have been working on corrugated core sandwich structures for the last 3 years and I am applying homogenization techniques to the core. The homogenized material has a non-linear plastic behaviour. So, I think we need a bit more details.
CIAO!
Giorgio
Hi! Answer is quite long but I believe that the best way to understand something is to do it! Hope it works!
Intro.
First of all, scale homogenization is a mathematical technique which was developed mainly by French and Russian mathematicians in the mid 70's (see Benssousan et al., 1975 and 1978).
From a physical point of view, this concept deals with what we understand by physical properties and how can we define a composite material as a mixture/aggregate of homogeneus constituents.
As a practical example, think about wood or steel. In a "macroscopical" scale, namely X, we can see with our eyes a continuum and "practically homogeneus material". However, as you may already been noticed, we know that internally these kind of materials are heterogeneous.
A natural question is, how we can even think about in material properties when we know that internally, in a "different scale", namely Y, they are heterogeneus?
After some reflections, you may notice that this theory relies on the assumption, until some extent, that material heterogenities may exhibit a sort of periodicity.
To clarify the idea, think about crystals. They are often modeled as "periodic arrangement of atoms in a crystal-lattice".
For the effective properties of a composite material, analysis is limited to a "periodic model", "representative volume of analysis" or "periodic element" in the Y-scale.
The key here is to choose this "periodic object" in manner that ALL the physical and geometrical (distribution) characteristics of each constituent are present.
Such periodic object will contain, as in a crystal, all the information needed to compute their physical properties (elastic, thermal, electrical and magnetic).
With the above mentioned elements, it is possible to state a connection (both, geometrical and physical) between the scales X and Y. This is:
(1) Y = X/e
where e is a small (e g1 only depends on the Y-scale.
(7) u2(x,y) = [partial u1(x,y)/partial x] * [g2(y)] --> g2 depends only on y.
u3(x,y) = [partial u2(x,y)/partial x] * [g3(y)] --> g3 depends only on y.
and so on.
Now the physics.
Do you remember the periodic assumption? Well, if periodicity is present at the Y-scale, it is possible, until some extent, to assume that functions g1, g2, g3, etc. are all related as follows:
(8) g2 = dg1/dy
g3 = dg2/dy = -g1
This idea is clarified if you think that g1 is a periodic function like the sine. From calculus, g2 will be similar to the cosine, hence, their derivative (g3) is the negative of the first and so on.
With this assumption it is feasible to truncate the system to the first two or three equations. A stronger assumption than the previously exposed relies on the fact that, for any periodical function their integral inside their period is zero.
Classical homogenization only uses the equations related to the terms 1/e, e^0 and e^1 (an additional term, 1/e^2, appears if a0 is set as function of both, x and y in (6) just to figure out that a0 only depends on x).
The integral of the equation related to e^1 is strongly related to divergence theorem for the stresses (you may need to rewrite a "local" version of Hooke's law using one of the first equations of the system and the definition of “traction”).
From this, you can set that equation related to this power of "e" can be neglected in your calculations. Note that this is only valid for linear elasticity, I don't know the details of what you pretend to do.
Caution: This argument is stronger than the stunning asseveration, let's make e = 0. Don't forget that you still have an equation related to 1/e.
Now the effective property.
At the end of the "physics" step, you must have (in linear elasticity) a 2x2 system of partial differential equations for coefficients a0 and a1.
Since we are depicting effective properties, which can be only seen in the X-scale, we need to make an "average" of this system. For this reason, this kind of method is also called "an averaging technique".
Since g1 is periodic (a1 doesn't) and we are modeling the properties at the Y-scale level, typically, an integral on the defined periods of your model should be enough to average your model. Since this integral is performed in the Y-scale, all the functions dependent only on x can be treated as "constants" inside the integral.
Formally, the average of function h(y) can be defined as:
(9) = int h(y) dy with limits in the "periodic model boundaries".
Now, just do the average!
Take a look in the equation containing the second derivative of a0(x) with respect x.
What's the meaning of the term in brackets? The effective Young's module!!! :D
Hence, you don't need to solve anything for a0(x)! Remember! We are interested in the "effective properties".
For linear elasticity, if you did all the steps in a 1D bar with two constituents “perfectly bonded” (you will need to introduce an expression to this!) you will get Reuss estimate for the effective properties!!!
Concluding remarks.
1) This is a summary! A detailed and formal exposition of the method can be found in Bensoussan et al. (1978). Use of more and more equations of this system in the "heat equation" can be found in Amar & Gianni (2005).
2) In the case of linear elasticity, it is possible to use Finite Element Method tools like ANSYS, ABACUS and so on... This is possible, if and only if, adequate constraints are imposed to the periodic Y-model and a "good averaging schema" is properly defined.
A detail of this technique with ANSYS can be found in a set of papers published by Dr. Harald Berger of Magdeburg University (Berger et al, 2005(a to c)). Another good example is depicted in Segurado & Llorca (2002) paper. You may need to sleep together with your favorite FORTRAN or MatLab book!
3) We are working with Newton's movement laws only! If you want to mix physical laws (say, classical mechanics + molecular dynamics), I reccommed the papers from Fish et al. (2007) and Chung et al. (2003).
4) A clear and formal insight about the limits of this approximation can be found in Blanc's et al. paper (2007).
5) Real 3D models makes a huge use of cartesian tensors. Don't be afraid! Just put “i”'s , “j”'s , “k”'s and “l”'s at the beginning (using generalized Hooke's law) of this human-readable recipe. For equations (7) and (8) you will need additional indexes (“p”'s, “q”'s, “r”'s and “s”'s). Abuse of “index switching” like “change i's for j's” is highly recommended to visualize symmetries in those “non-understandable” steps! ] :D A good introduction to tensors can be found in ... Platicity's theory book and in ... book too! And, of course!, the remaining equation is a “closed-system of partial differential equations”.
References.
Amar M., Gianni R. 2005. A Brief Survey on Homogenization with a Physical Application. MAT A Series (9), Rosario, Argentina.
Bensoussan A., Lions J. L., Papanicolau G. 1975. Sur quelques phénomènes asymptotiques stationnaires (I). Comptes Rendus Acad. Sci. Paris, (281) 89--94.
Bensoussan A., Lions J. L., Papanicolau G. 1978. Asymptotic Methods in Periodic
Structures. Amsterdam. North Holland.
Berger H., Kari S., Gabbert U., Rodríguez-Ramos R., Bravo-Castillero J., Guinovart-Díaz R. 2005 (a). A comprehensive numerical homogenisation technique for calculating effective coefficients of uniaxial piezoelectric fibre composites. Materials Science and Engineering A (412), 53–60.
Berger H., Kari S., Gabbert U., Rodríguez-Ramos R., Bravo-Castillero J., Guinovart-Díaz R. 2005 (b). Calculation of effective coefficients for piezoelectric fiber composites based on a general numerical homogenization technique. Composite Structures (71), 397–400.
Berger H., Kari S., Gabbert U., Rodríguez-Ramos R., Bravo-Castillero J., Guinovart-Díaz R. 2005 (c). An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. International Journal of Solids and Structures (45), 5692–5714.
Blanc X., Le Bris C., Lions P. L. 2007. Atomistic to Continuum limits for computational materials science. Mathematical Modelling and Numerical Analysis (41), 391–426.
Chung P W., Namburu. R. R., 2003. On a formulation for a multiscale atomistic–continuum homogenization method. International Journal of Solids and Structures (40), 2563–2588.
Fish J., Chen W., Li R. 2007. Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions.Comput. Methods Appl. Mech. Eng. 196, 908–922.
Segurado J., Llorca J. 2002. A numerical approximation to the elastic properties of sphere–reinforced composites. Journal of the mechanics of physics and solids (50), 2107–2121.
Thanks everyone.
I am looking into dynamic analysis of wire rope (non linear material with hysteresis).
Hello,
I suggest you to read my dissertation on the ULB site (ulb.ac.be). I discussed largely the situation of (pseudo) homogeneous situations in biotechnical systems. It may help you.
Good luck.
Hi Brett & Georgio,
I am returning back after a while to refresh this discussion again. I am interested in a combination of material and structural non-linearity. I am interested in modeling and analyzing the pseudoelastic wire covered with the rubber. This makes this application more complex as there is two nonlinear material interaction. To make the problem more interesting the wire is passing through the high-speed steel pulleys. I would like to study the dynamic behavior of the system to check at which torque and speed the wire will fail. For more questions let me know.
Thanks!!
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