An Experimentally Validated Combined Stiffness Formulation for a Finite Domain Considering Volume Fraction, Shape, Orientation, and Location of a Single Inclusion

Abstract

This work characterizes the stiffness of a finite domain containing one (biaxial ellipsoidal) void due to the combined effect of inclusion’s attributes: (1) size or volume fraction, VF, (2) shape or aspect ratio, AR, (3) angular orientation, and (4) location (position) within the matrix. The values and ranges of these ellipsoidal inclusion attributes are varied according to a matrix developed using design of experiments (DOE). Modified Mori–Tanaka method combined with dual-eigenstrain method (interior and exterior eigenstrain methods) is used to determine the effective stiffness tensor of the composite domain. Employing the numerically calculated normalized axial modulus [Formula: see text] values in SAS/STAT®, a nonlinear mathematical expression of [Formula: see text] as function of the void’s variables is arrived at Stiffness values found from the numerical homogenization scheme are experimentally corroborated using compression tests conducted on 3D-printed ABS cubes having a single ellipsoidal inclusion of various geometric attributes. In addition, finite element simulations were run of said uniaxial compression test cases to further validate the numerical homogenization results. Corroborated findings suggest that while the location of the inclusions in the matrix have no significant effect on normalized modulus [Formula: see text], the void’s volume fraction has the largest effect where it decreases with VF. The effect of the void’s orientation and elliptical aspect ratio are significant. [Formula: see text] increases with AR at angles ranging from 0–[Formula: see text]; at [Formula: see text][Formula: see text] are almost constant with AR, at angles of 60–[Formula: see text] values of [Formula: see text] decrease with AR. As AR approaches unity, the effect of orientation decreases significantly.

Experimentally validated combined stiffness expression for finite domain containing multiple inclusions

“…Also shown in the 3D RVE are inclusions distributed with multiple orientations with their corresponding rotation angles identified with respect to the RVE Cartesian coordinate. To move from each inclusion's local coordinate system to the global RVE coordinate system, a transformation matrix is utilized to account for the orientation of each individual inclusion within the RVE (Hage and Hamade, [9]). The composite is considered as domain Ωu having multiple inclusions Ω i I from i = 1, .…”

“…The coordinates (x (1) 1 , x (1) 2 , x (1) 3 ) for the origin of each inclusion are obtained with respect to the origin of the composite domain coordinate system. Using the transformation matrix and considering that r is the distance between point P in the domain and the origin of the considered inclusion, the coordinate position vector is reoriented to the inclusion coordinate system and reformulated in the composite domain (Hage and Hamade, [9])…”

“…In addition, Q i j is a transformation matrix is utilized to account for the orientation of the inclusion within the 3D RVE and is utilized to jump from the inclusion local coordinate system to the global RVE coordinate system [9]. This reformulated coordinate position distance r is used to determine the exterior Eshelby tensor and is function of the orientation and coordinates of the center of each inclusion w.r.t.…”

“…Translating the effective stiffness tensor into the global domain, the strain concentration tensors for each inclusion A E , and A I is calculated in the local domain and equivalent strain concentration tensors are calculated in the global system using a 4th rank transformation tenor, Q i j kl (Koay [16], this tensor is function of the transformation matrix Q i j [9] defined in the previous section). The "Bond transformation" [17] is a tensor that accounts for the inclusion's 3-dimensional attributes.…”

“…This work is devoted to validating a new homogenization method for linear elastic composite materials by comparing the resulting estimate of the overall stiffness to estimates produced by finite element computations and also to strain measurements utilizing 3D-printed microstructures. This homogenization scheme of multi inclusions is an extension of recent work by the authors (Hage and Hamade, [9]) that calculates the stiffness of a domain containing a single inclusion with its corresponding shape, volume fraction, and orientation. The work is based in part on the superposition of strain field principle as described by Li et al [10] that considers one inclusion in finite and infinite domains (without accounting for location) and also expands on the works of Ju and Sun [11][12][13].…”