Mathematics and Mechanics of Solids

We explore alternative writings of the equations of classical elastodynamics as a first-order symmetric system. In the one-dimensional case, we present symmetric writings with respect to (1) the velocity ( u t ) and the displacement gradient ( u x ), (2) the velocity and stress (σ), (3) all three quantities: the velocity, the displacement gradient and the stress, and finally, (4) the momentum ( ρ 0 u t ), the velocity, the displacement gradient and the stress. In the two-dimensional case, we present similar writings with respect to (1) the velocity ( u t ) and the strain tensor (e), (2) the velocity and the stress tensor ( σ ), (3) all three variables ( u t , e , σ ) , and finally, (4) one more writing utilizing the momentum as well, i.e., ( ρ 0 u t , u t , e , σ ) . We accomplish our goal by judiciously using the compatibility equations as well as the momentum equation and the time-differentiated constitutive law. This is done in an inverse way: we start by writing our initial equations as a first-order system of the form (q being the vector representing the variables in each writing)

A ∂ q ∂t + ∑ i = 1 n B i ∂ q ∂ x i = 0 ,

with n = 1 , 2 depending on whether we are in 1 or 2 dimensions. We then check what are the symmetric forms of matrices B i and which combinations of the compatibility equations, the momentum equations and the time-differentiated constitutive law should be used in order the symmetric form of matrices B i to appear into the system. This “symmetrization” process alters matrix A and if the resulting matrix A is symmetric our goal is accomplished. Our analysis is confined to classical elastodynamics, namely, geometrically and materially linear anisotropic elasticity.

Predicting buckling loads and optimizing stiffener configurations for composite stiffened panels present significant challenges due to nonlinear behavior and the computational demands of iterative simulations by traditional finite element analysis (FEA). This study addresses these challenges by integrating artificial neural networks (ANNs) with FEA to develop an efficient and accurate predictive framework. An in-plane shear load experiment was designed and conducted to validate the combined ANN-FEA model, which was further utilized to investigate buckling phenomena and provide initial predictions of critical buckling loads. The FEA results demonstrated that the stiffener configuration significantly affects load-carrying capacity, underscoring its critical role in structural performance. To reduce the computational intensity of FEA, ANN was trained on a subset of FEA-generated data, achieving high predictive accuracy for buckling loads with reduced modeling effort. The proposed hybrid approach successfully optimized stiffener parameters, offering a robust solution for improving the design and performance of composite stiffened panels under shear loading.

This article presents a sensitivity analysis aimed at examining the pressure–inflation relation and the axial force within a pressurized and elongated cylindrical tube. The material under consideration is an isotropic ground material that is reinforced by two families of fibers. Within these families, the fibers are taken to be dispersed. The natural configuration of these fibers may not coincide with the one of the ground substance, potentially due to a pre-stretch of the fibers or imperfect bonding between the constituents. The input parameters relevant to the mechanical system, including the fiber dispersion, the azimuthal and axial stretches of the cylinder, the stiffness parameters of the constituents, the fiber winding angle, and the natural configurations of the fibers are presumed to follow two distinct probability distribution functions. In the sensitivity analysis, we utilize the Sobol method to assess how variations of the input parameters influence the necessary inflation pressure and the corresponding axial force, which are the output variables. The application of the Sobol method enables us to consider the interactions among various parameters and to pinpoint the most significant factors affecting both the pressure–inflation relation and the axial force. This analysis highlights these elements, revealing a diverse array of results.

The present paper aims to develop an efficient fracture model allowing the prediction of the quasi-brittle material phase field damage coupled to cohesion. The adopted method is based on finite element simulations using COMSOL Multiphysics when considering the penalization approach as an irreversibility constraint. A comparative study is conducted applying linear and quadratic Ambrosio–Tortorelli’s models and the phase field cohesive zone model (PF-CZM), in order to evaluate their performance and assess the ability of the PF-CZM to generate a realistic crack path. Particularly, its effectiveness coupled with the penalized approach is applied to predict the French Pantheon crack. The originality of this work lies on different aspects. First, the study presents the penalization technique within the phase field framework as an alternative to Miehe history field, providing a robust enhancement to the irreversibility condition. Second, the implementation adopts a staggered algorithmic scheme which refines the stability and the robustness of computations. Third, this study gives a more in-depth numerical analysis for the optimal penalty parameter so that the optimal range is provided for solver stability and computational efficiency of the cohesive zone model. Also, the effectiveness of PF-CZM coupled with the penalization approach is investigated in predicting complex crack phenomena. The test of the French Pantheon structure response is validated with literature, and an extended solution is provided based on the developed cohesive penalized approach.

We use the reduced relaxed micromorphic model (RRMM) to capture the effective “bulk” dynamical response of finite-size metamaterial specimens made out of a Labyrinthine unit cell. We show that for small finite-size specimens, boundary effects can play a major role, so that the RRMM needs an enrichment to capture the metamaterial’s bulk response, as well as the boundary effects. A benchmark test is introduced to show that different metamaterial/ homogeneous material interfaces can drive completely different responses even if the bulk metamaterial remains the same. We show with no remaining doubts that the concept of “interface forces” must necessarily be introduced if one wants to model finite-size metamaterials in a homogenized framework.

AMS 2010 subject classification: 74A10, 74B05, 74J05, 74M25

This study establishes a three-dimensional frictionless contact model for the rigid indenter and the thermoelectric thin film under multiple physical fields. By combining the Navier–Cauchy equations, the Fourier heat conduction equation, and the thermoelectric transport equation, the governing equations of thermoelectric thin film are established. The frequency response functions of thermoelectric thin film are derived by using the double Fourier transform to convert the space domain equations into the frequency-domain equations. The discrete convolution fast Fourier transform algorithm combined with the conjugate gradient method and local mesh refinement technique is used to further improve the efficiency and precision of the numerical solution. This study solved the stress field, temperature field, and electric field at the contact interface, elucidating the synergistic mechanism through which multiple parameters (thermoelectric load, film thickness, probe radius, and probe spacing) regulate contact responses in single-indenter and arrayed-indenter thermoelectric thin film contact systems.

In this paper, the scattering of SH wave by a nano elliptical hole embedded the infinite inhomogeneous medium is studied. First, the displacement governing equation with variable coefficients is transformed into the complex Helmholtz equation by using complex variable function and conformal transformation. By the wave function expanding method, the analytical expressions of the displacement field and stress field in the inhomogeneous medium are obtained. Considering the surface effect and using the generalized Young–Laplace equation, we obtain the boundary conditions at nano elliptical hole; then the field equations satisfied by the boundary conditions are attributed to solving a set of infinite algebraic equations. The numerical results show that when the size of the elliptical hole reaches the nanometer level, the influences of surface energy, inhomogeneous coefficients, and wave number on the dynamic stress concentration factor near the hole are significant.