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**Multiscale
Materials and Design Optimization Lab**

This lab aims to create and optimize
the microstructure of multifunctional materials using computational and
experimental means. The objective is to optimally design materials and
structures that meet certain functional requirements. Following are some of the
research projects pursued in our lab:

- Design of functionally graded materials with enhanced load carrying capabilities
- Design of scaffolds and implants for bone tissue engineering
- Optimization of applications catering
to the field of bio-mechanics and bio-ergonomics

The long term goal of this laboratory is
to integrate computer aided engineering tools together with 3D printing technologies
to build useful real world applications. Graduate and undergraduate students in
our lab get trained in using CAD/CAM/CAE tools as well as in conducting hands
on experimental tests. Our lab is located in the South Jersey Technology Park
and collaborates with medical professionals at Cooper University Hospital and
researchers at U.S. Army Armament Research, Development and Engineering Center
(ARDEC). Some of the past research projects are summarized below:

1. Multiscale mechanics of heterogeneous
materials

The
framework of stochastic mechanics is employed to obtain scale-dependent bounds
on the response of heterogeneous media (random composites, polycrystals). In
doing so, one infers the approach to the Representative Volume Element (RVE),
the corner-stone of the separation of scales in continuum mechanics. The RVE is
approached by setting up and solving stochastic Dirichlet and Neumann boundary
value problems consistent with the Hill(-Mandel) macrohomogeneity condition.
Further, the concept of a scaling function is introduced to establish unifying
scaling laws. It turns out that the scaling function depends on a mesoscale
(scale of observation relative to grain size) and an appropriate metric
quantifying the single crystal anisotropy. Based on the scaling function, a
material selection diagram is constructed that clearly separates the microscale
from the macroscale [Figs. 1(a), 1(b)]. Using such a diagram, one can determine
the size of RVE for a whole range of polycrystals made of various crystal
classes. Application problems include the scaling of the fourth-rank elasticity
and the second-rank thermal/electrical conductivity tensors. The trends in
approaching the RVE for planar conductivity, linear/non-linear
(thermo)elasticity, plasticity, and Darcy permeability is also established.

(a)
(b)

FIG.
1: (a) Homogenization methodology; (b) Material scaling diagram

The impact of this research lies in the fact that one can establish scaling laws that seamlessly integrate the microscale to the macroscale. Such a methodology can be used to understand and predict the properties of a variety of materials. In turn, this will enable the design of novel microstructures with target properties (enhanced strength, stiffness, conductivity and buckling resistance). Such microstructures can then be engineered to build low cost applications that are light in weight and safer.

Related Publications- Dalaq, A.S.,
__Ranganathan, S.I.__, 2015. Invariants of mesoscale thermal conductivity and resistivity tensors in random checkerboards, Engineering Computations (accepted)

- Raghavan, B.V.,
__Ranganathan, S.I.__, Ostoja-Starzewski, M., 2015. , Electrical properties of random checkerboards at finite scales, AIP Advances 5 (1), 017131

- Ostoja-Starzewski,
M., Costa, L.,
__Ranganathan, S.I.__, 2015, Scale dependent homogenization of random hyperbolic thermoelastic solids,*Journal of Elasticity*118, 243-250 (External Link)

- Abed,
F. H.,
__Ranganathan, S.I.__, Serry, M.A., 2014. Constitutive modeling of nitrogen-alloyed austenitic stainless steel at low and high strain rates and temperatures.*Mechanics of Materials*77, 142157

- Aldadah,
M.G.,
__Ranganathan, S.I.__, Abed, F.H., 2014, Buckling of two phase inhomogeneous columns at arbitrary phase contrasts and volume fractions,*Journal of Mechanics of Materials and Structures*9, 465-474

- Raghavan,
B.V.,
__Ranganathan, S.I.__, 2014, “Bounds and scaling laws at finite scales in planar elasticity,”*Acta Mechanica*(External Link)

- Dalaq,
A.S.,
__Ranganathan, S.I.__, Ostoja-Starzewski, M., 2013, “Scaling Function in Conductivity of Planar Random Checkerboards,”*Computational Materials Science*79, 252-261 (External Link)

- Ostoja-Starzewski,
M.,
__Ranganathan, S.I.__, 2013, “Scaling and Homogenization in Spatially Random Composites,” in*Mathematical Methods and Models in Composites*(Ed. Vladislav Mantič), World Scientific, to appear (Book Chapter)

__Ranganathan, S.I.__, Ostoja-Starzewski, M., 2009, “Towards scaling laws in random polycrystals,”*International Journal of Engineering Science*47, 1322-1330 (External Link)

__Ranganathan, S.I.__, Ostoja-Starzewski, M., 2008, “Mesoscale conductivity and scaling function in aggregates of cubic, trigonal, hexagonal, and tetragonal crystals,”*Physical Review B*77, 214308-1-10 (External Link)

__Ranganathan, S.I.__, Ostoja-Starzewski, M., 2008, “Scaling function, anisotropy and the size of RVE in elastic random polycrystals,”*Journal of the Mechanics and Physics of Solids*56, 2773-2791 (Ranked 07/25 in "Top 25 Hottest Articles" between Jul - Sep 2008-External Link)

__Ranganathan, S.I.__, Ostoja-Starzewski, M., 2008, “Scale-dependent homogenization of inelastic random polycrystals,”*ASME Journal of Applied Mechanics*75, 051008-1-9 (External Link)

2. Smart Biomaterials and Biomechanics

a) Smart Biomaterials:

The focus of this project is on the design and development of a novel BioNanoScaffold (biologically active composite) for Post-Traumatic OsteoRegeneration. The objective is to design a `smart fracture putty' to help repair serious bone injuries [see Fig. 2(a)]. The fracture putty is a complex biomaterial made up of individual components with characteristic dimensions that span across a variety of length scales (nano to macroscales). The putty has both mechanical and biological functions to fulfill and is essentially made up of a biodegradable polymeric matrix with stiffer reinforcements, pores and other functional materials [see Fig. 2(b)]. The putty is designed to take up the normal mechanical loads acting on the bone and is administered with stem cells and growth factors to facilitate the growth of new bone. Eventually, as the healthy bone grows, the putty starts degrading until the wound is completely healed. From a mechanistic perspective, the successful design of the putty requires a careful control of key design parameters such as: i) volume fraction of particles; ii) particle orientation distribution; iii) the shape; and iv) the size of particles. We have also developed mathematical models and design maps [see Fig. 2(c)] to predict bone growth and the degradation of the polymeric scaffold.

(a)
(b)
(c)

FIG. 2: (a) A scaffold
is implanted or injected into the fracture site; (b) Schematic of a possible design
for the scaffold construct; (c) Admissible design region favoring bone growth

b) Biomechanics:

Motorcyclists are
road-users with a higher risk than the car occupants. In crash safety research,
there is a significant increase in the use of computer simulations over the past
two decades. This is due to the advances in computer hardware/software and due
to the development of
reliable models describing the human body in a crash situations. Mathematical
models offer a very economical and versatile method for the analysis of the
crash scenarios and substantially bring down the need for experimental studies.
One of the most important task in such studies is to position the MATD model
close to the experimentally measured co-ordinates. A computer program was
developed to generate the necessary geometrical transformations to achieve this
objective [see Figs. 3(a), 3(b)].

(a) (b)

FIG. 3: (a) Side view
of the initial and final positions of the dummy; (b) Front view of the initial and
final positions of the dummy

Another
area of interest in biomechanics is the dynamics of thin elastic rods and
helices such as the DNA. We have developed a finite element model to include the
effects of rotary inertia in isotropic circular rods. In developing this model,
the angular velocities are first expressed in terms of the time derivatives of
the Frenet frame. This definition is then used to determine the contributions of
rotary inertia. A salient feature of this model is that only six variables
representing the absolute position vectors are used as nodal coordinates.
Galerkin’s weighted residual approach is used to complete the discretization.

Related Publications

__Ranganathan, S.I.__, Ferrari, M., Decuzzi, P., 2013, “Design maps for scaffold constructs in bone regeneration,”*Biomedical Microdevices*15, 1005-1013 (External Link)

__Ranganathan, S.I.__, 2013, “A novel finite element model incorporating rotary inertia in thin circular rods,”*Journal of Mechanical Engineering Science (IMechE, Part C)*227 (10), 2339-2343 (External Link)

__Ranganathan, S.I.__, Yoon, D.M., Henslee, A., Nair, M.B., Smid, C., Kasper, K.F., Tasciotti, E., Mikos, A.G., Decuzzi, P., Ferrari, M., 2010, “Shaping the micromechanical behavior of multi-phase composites for bone tissue engineering,”*Acta Biomaterialia*6, 3448-3456 (External Link)

- Sakamoto et al., 2010,
“Enabling individualized therapy through nanotechnology,”
*Pharmocological Research*62, 57-89 (Ranked 01/25 in "Top 25 Hottest Articles" between Jan - Mar 2010-External Link)

- Mukherjee, S., Chawla,
A.,
__Ranganathan, S.I.*__, 2006, “Positioning of motorcyclist dummies in crash simulations,”*International Journal of Crashworthiness*11, 337-344 (External Link)

3. Material and Geometrical Anisotropy

Etymologically, the
word anisotropy is a combination of `an' (not), `isos' (equal) and `tropos'
(turn) implying `not equal turn'. In the context of this research, anisotropy implies
geometric and material properties are direction dependent. As a simple example,
an ellipse (ellipsoid) can be thought of being anisotropic whereas a circle
(sphere) could be considered isotropic when viewed from the center. There are
several situations in which there is an inherent need to use a measure that
quantifies the extent of anisotropy (material and geometric). Basing on sound
mathematical principles, we came up with a definition of an anisotropy index
that was universally applicable. The impact of this research lies in its
simplicity, yet a very fundamental and useful definition that is applicable to
a variety of fields.

a) Material anisotropy--application to
elastic crystals:

Practically all
elastic single crystals are anisotropic, which calls for an appropriate universal
measure to quantify the extent of anisotropy. A review of the existing anisotropy
measures in the literature leads to a conclusion that they lack universality in
the sense that they are non-unique and ignore contributions from the bulk part
of the elastic stiffness (or compliance) tensor. Proceeding from extremal
principles of elasticity, a new universal anisotropy index is introduced that
overcomes the above limitations. Furthermore, special relationships between the
proposed anisotropy index and the existing anisotropy measures for special cases
are established. A new elastic anisotropy diagram is constructed for over 100
different crystals (from cubic through triclinic), demonstrating that the
proposed anisotropy measure is applicable to all types of elastic single
crystals, and thus fills an important void in the existing literature.

b) Material anisotropy--application to
biological materials:

Anisotropy is an
essential attribute exhibited by most biological materials. Based on the recent
work on anisotropy of a wide range of crystals and polycrystals, we propose an
appropriate measure (*A*) to quantify
the extent of elastic anisotropy in biomaterials by accounting the tensorial
nature (both stiffness-based and compliance-based) of their elastic properties.
Next, we derive a relationship between A and an empirically defined existing
measure. Also, the preceding measure is used to quantify the extent of
anisotropy in select biological materials that include bone, dentitional
tissues, and a variety of woods. Results indicate that woods are an order of
magnitude more anisotropic than hard tissues and apatites. Finally, based on
the available data, it is found that the anisotropy in human femur increases by
over 40 % when measured between 30 % and 70 % of the total femur length.

c) Geometrical anisotropy:

Particle shape plays
a crucial role in the design of novel reinforced composites. The notion of a
geometrical anisotropy index `*A*' is
introduced to characterize the particle shape and its relationship with the effective
elastic constants of biphase composite materials is established. The analysis
identifies three distinct regions of `*A*':
i) by using ovoidal particles with small `*A*',
the effective stiffness scales linearly with `*A*' for a given volume fraction *a*;
ii) for intermediate values of `*A*',
the use of prolate particles yield better elastic properties; and iii) for
large `*A*', the use of oblate
particles result in higher effective stiffness. Interestingly, the transition
from ii) to iii) occurs at a critical anisotropy `*A _{cr}*' and is independent of

Both
forms of anisotropy (material and geometric) have applications in a wide range
of disciplines that include geophysics, biological materials [see Fig. 4(a)],
diffusion tensor imaging, phase transformations, dislocation dynamics,
condensed matter physics and composite materials. For instance, using stiffer
ellipsoidal fillers as reinforcement in a softer matrix generates a composite
material with properties superior than a similar composite reinforced with spherical
fillers [see Figs. 4(b), 4(c)]. Further, material anisotropy plays a very
important role in seismology with applications to petroleum
exploration/production and in the understanding of lower mantle composition and
behavior [see Fig. 4(d)].

(a) |
(b) |

(c) |
(d) |

FIG.
4: (a) Anisotropy of trees; (b) Contours of geometrical anisotropy; (c)
Contours of elastic properties; (d) Lowermost mantle anisotropy [see
Walker & Wookey, Computers and Geosciences (2012)]

Related Publications

__Ranganathan, S.I.__, Ostoja-Starzewski, M., Ferrari, M., 2011, “Quantifying the anisotropy in biological materials,”*ASME Journal of Applied Mechanics*78, 064501-4 (External Link)

__Ranganathan, S.I.__, Decuzzi, P., Wheeler, L.T., Ferrari, M., 2010, “Geometrical anisotropy in biphase particle reinforced composites,”*ASME Journal of Applied Mechanics*77, 041017-1-4 (External Link)

__Ranganathan, S.I.__, Ostoja-Starzewski, M., 2008, “Universal elastic anisotropy index,”*Physical Review Letters*101, 055504-1-4 (External Link)

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