Theory and computation of higher gradient elasticity theories based on action principles

In continuum mechanics, there exists a unique theory for elasticity, which includes the first gradient of displacement. The corresponding generalization of elasticity is referred to as strain gradient elasticity or higher gradient theories, where the second and higher gradients of displacement are involved. Unfortunately, there is a lack of consensus among scientists how to achieve the generalization. Various suggestions were made, in order to compare or even verify these, we need a generic computational tool. In this paper, we follow an unusual but quite convenient way of formulation based on action principles. First, in order to present its benefits, we start with the action principle leading to the well-known form of elasticity theory and present a variational formulation in order to obtain a weak form. Second, we generalize elasticity and point out, in which term the suggested formalism differs. By using the same approach, we obtain a weak form for strain gradient elasticity. The weak forms for elasticity and for strain gradient elasticity are solved numerically by using open-source packages—by using the finite element method in space and finite difference method in time. We present some applications from elasticity as well as strain gradient elasticity and simulate the so-called size effect

Verification of strain gradient elasticity computation by analytical solutions

As there are different computational methods for simulating problems in generalized mechanics, we present simple applications and their closed-form solutions for verifying a numerical implementation. For such a benchmark, we utilize these analytical solutions and examine three-dimensional numerical simulations by the finite element method (FEM) using IsoGeometric Analysis (IGA) with the aid of open source codes, called tIGAr, developed within the FEniCS platform. A study for the so-called wedge forces and double tractions help to comprehend their roles in the displacement solution as well as examine the significance by comparing to the closed form solutions for given boundary conditions. It is found that numerical results are in a good agreement with the analytical solutions if wedge forces and double tractions are considered. It is also presented how the wedge forces become necessary in order to maintain equilibrium in strain gradient materials

Verification of asymptotic homogenization method developed for periodic architected materials in strain gradient continuum

Strain gradient theory is an accurate model for capturing the size effect and localization phenomena. However, the challenge in identification of corresponding constitutive parameters limits the practical application of the theory. We present and utilize asymptotic homogenization herein. All parameters in rank four, five, and six tensors are determined with the demonstrated computational approach. Examples for epoxy carbon fiber composite, metal matrix composite, and aluminum foam illustrate the effectiveness and versatility of the proposed method. The influences of volume fraction of matrix, the stack of RVEs, and the varying unit cell lengths on the identified parameters are investigated. The homogenization computational tool is applicable to a wide class materials and makes use of open-source codes in FEniCS. We make all of the codes publicly available in order to encourage a transparent scientific exchange

Strain gradient elasticity with geometric nonlinearities and its computational evaluation

The theory of linear elasticity is insufficient at small length scales, e.g., when dealing with micro-devices. In particular, it cannot predict the “size effect” observed at the micro- and nanometer scales. In order to design at such small scales an improvement of the theory of elasticity is necessary, which is referred to as strain gradient elasticity

Material characterization and computations of a polymeric metamaterial with a pantographic substructure

The development of additive manufacturing methods, such as 3D printing, allows the design of more complex architectured materials. Indeed, the main structure can be obtained by means of periodically (or quasi-periodically) arranged substructures which are properly conceived to provide unconventional deformation patterns. These kinds of materials which are ‘substructure depending’ are called metamaterials. Detailed simulations of a metamaterial is challenging but accurately possible by means of the elasticity theory. In this study, we present the steps taken for analyzing and simulating a particular type of metamaterial composed of a pantographic substructure which is periodic in space—it is simply a grid. Nevertheless, it shows an unexpected type of deformation under a uniaxial shear test. This particular behavior is investigated in this work with the aid of direct numerical simulations by using the finite element method. In other words, a detailed mesh is generated to properly describe the substructure. The metamaterial is additively manufactured using a common polymer showing nonlinear elastic deformation. Experiments are undertaken, and several hyperelastic material models are examined by using an inverse analysis. Moreover, a direct numerical simulation is repeated for all studied material models. We show that a good agreement between numerical simulations and experimental data can be attained

Investigating infill density and pattern effects in additive manufacturing by characterizing metamaterials along the strain-gradient theory

Infill density used in additive manufacturing incorporates a structural response change in the structure. Infill pattern creates a microstructure that affects the mechanical performance as well. Whenever the length ratio of microstructure to geometry converges to one, metamaterials emerge and the strain-gradient theory is an adequate model to predict metamaterials response. All metamaterial parameters are determined by an asymptotic homogenization, and we investigate the effects of infill density and pattern on these parameters. In order to illuminate the role of infill characteristics on the strain-gradient parameters, an in-depth numerical investigation is presented for one, widely used case in three-dimensional (3D) printers, rectangular grid

Verification of asymptotic homogenization method developed for periodic architected materials in strain gradient continuum

Strain gradient theory is an accurate model for capturing the size effect and localization phenomena. However, the challenge in identification of corresponding constitutive parameters limits the practical application of the theory. We present and utilize asymptotic homogenization herein. All parameters in rank four, five, and six tensors are determined with the demonstrated computational approach. Examples for epoxy carbon fiber composite, metal matrix composite, and aluminum foam illustrate the effectiveness and versatility of the proposed method. The influences of volume fraction of matrix, the stack of RVEs, and the varying unit cell lengths on the identified parameters are investigated. The homogenization computational tool is applicable to a wide class materials and makes use of open-source codes in FEniCS. We make all of the codes publicly available in order to encourage a transparent scientific exchange