TY - JOUR

T1 - Multi-scale micromorphic theory for hierarchical materials

AU - Vernerey, Franck

AU - Liu, Wing Kam

AU - Moran, Brian

N1 - Funding Information:
The authors gratefully acknowledge the support of the ONR D3D Digital Structure Consortium (award N00014-05-C-0241) and the National Science Foundation. The authors also acknowledge Cahal McVeigh for reviewing this paper.

PY - 2007/12

Y1 - 2007/12

N2 - For the design of materials, it is important to faithfully model macroscopic materials response together with mechanisms and interactions occurring at the microstructural scales. While brute-force modeling of all the details of the microstructure is too costly, many of the current homogenized continuum models suffer from their inability to capture the correct underlying deformation mechanisms-especially when localization and failure are concerned. To overcome this limitation, a multi-scale continuum theory is proposed so that kinematic variables representing the deformation at various scales are incorporated. The method of virtual power is then used to derive a system of coupled governing equations, each representing a particular scale and its interactions with the macro-scale. A constitutive relation is then introduced to preserve the underlying physics associated with each scale. The inelastic behavior is represented by multiple yield functions, each representing a particular scale of microstructure, but collectively coupled through the same set of internal variables. The theory is illustrated by two applications. First, a one-dimensional example of a three-scale material is presented. After the onset of softening, the model shows that the localization zone is distributed according to two distinct length scale determined by the model. Second, a two-scale continuum model is introduced for the failure of porous metals. By comparing the theory to a direct numerical simulation (DNS) of the microstructure for a specimen in tension, we show that the model capture the main physics, and at the same time, remains computationally affordable.

AB - For the design of materials, it is important to faithfully model macroscopic materials response together with mechanisms and interactions occurring at the microstructural scales. While brute-force modeling of all the details of the microstructure is too costly, many of the current homogenized continuum models suffer from their inability to capture the correct underlying deformation mechanisms-especially when localization and failure are concerned. To overcome this limitation, a multi-scale continuum theory is proposed so that kinematic variables representing the deformation at various scales are incorporated. The method of virtual power is then used to derive a system of coupled governing equations, each representing a particular scale and its interactions with the macro-scale. A constitutive relation is then introduced to preserve the underlying physics associated with each scale. The inelastic behavior is represented by multiple yield functions, each representing a particular scale of microstructure, but collectively coupled through the same set of internal variables. The theory is illustrated by two applications. First, a one-dimensional example of a three-scale material is presented. After the onset of softening, the model shows that the localization zone is distributed according to two distinct length scale determined by the model. Second, a two-scale continuum model is introduced for the failure of porous metals. By comparing the theory to a direct numerical simulation (DNS) of the microstructure for a specimen in tension, we show that the model capture the main physics, and at the same time, remains computationally affordable.

KW - Finite elements

KW - Inhomogeneous material

KW - Microstructures

KW - Multi-scale micromorphic theory

KW - Plastic collapse

UR - http://www.scopus.com/inward/record.url?scp=36048952147&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36048952147&partnerID=8YFLogxK

U2 - 10.1016/j.jmps.2007.04.008

DO - 10.1016/j.jmps.2007.04.008

M3 - Article

AN - SCOPUS:36048952147

SN - 0022-5096

VL - 55

SP - 2603

EP - 2651

JO - Journal of the Mechanics and Physics of Solids

JF - Journal of the Mechanics and Physics of Solids

IS - 12

ER -