The stress model of the hybrid-Tre!tz "nite element formulation is applied to the linear elastostatic analysis of solids. The stresses are approximated in the domain of the element and displacements on its boundary. Complete, linearly independent, hierarchical polynomial approximation functions are used in both domain and boundary approximations. The displacement basis is de"ned independently on each inter-element surface. Continuity at the edges and on the corners of the elements is not enforced a priori. The stress basis is constrained to solve locally the Beltrami governing di!erential equation. It is derived from the associated Papkovitch}Neuber elastic displacement solution. Generalized variables are used to ensure that the approximations are independent of the geometric description of the elements. The solving system is derived directly from the fundamental relations of elastostatics. The solving system is symmetric, when the same property applies to the local elasticity condition, sparse, described by boundary integral arrays and well suited to p-re"nement and parallel processing. The numerical implementation of these equations is discussed and numerical tests are presented to illustrate the performance of the "nite element formulation. thus obtained can be recognized in the embracing works of Pian and Tong [3] and Brezzi and Fortin [4].The formulation used here was identi"ed later as the stress model of the hybrid-Tre!tz "nite element formulation and applied to di!erent two-dimensional structural analysis problems [5}8]. It is, therefore, directly related with the displacement frame Tre!tz element developed by Jirousek and Stein and their co-workers [9, 10]. The theoretical basis of the formulation is established in [11], where a comparative analysis with the related T-elements is also presented.In the terminology adopted here, a "nite element derived from the direct approximation of the stress "eld is termed a stress "nite element model. The stress "eld is directly related with two fundamental conditions, namely the equilibrium condition in the domain of the element and the di!usivity (Neumann) condition on the associated boundary tractions (Cauchy stresses). Three classes of formulations can be established depending on the equilibrium constraints enforced a priori on the stress basis, namely the hybrid-mixed, the hybrid and the hybrid-Tre!tz formulations of the "nite element stress model [12,13].The hybrid-mixed formulation is the most general. No constraints (besides linear independence and completeness) are placed on the stress approximation basis. Besides the stresses, the displacements are also approximated, and independently, in the domain and on the boundary of the element. These bases are used to enforce on average the equilibrium and the di!usivity conditions, respectively. The hybrid formulation is obtained by constraining a priori the stress approximation to satisfy locally the equilibrium condition. Consequently, the displacement approximation in the domain of the element becomes redundant an...

Additively manufactured composites have been demonstrating promising results with the development of new materials of high mechanical performance, which draws attention from several fields, for example, biomedical, electronics and aeronautics. However, as such materials are based on novel technologies, it is necessary to better understand their resulting characteristics and properties. For instance, evaluating the effect of environmental conditions on their mechanical performance is important, especially when moisture‐sensitive polymers such as polyamide (PA) are employed as matrix. This work aims to understand and to characterize the moisture effect on the mechanical properties of additively manufactured Nylon and continuous carbon fiber (CF)‐reinforced Nylon‐based thermoplastic. Tensile and compressive tests were carried out in accordance with ASTM standards for the printed samples at their maximum moisture content and for samples submitted to drying after the saturated condition. Moreover, moisture absorption and swelling behaviors were assessed and discussed. The experimental results showed that moisture significantly affects the fiber/matrix interface, as well as the adhesion between printed filaments. These changes led to a decrease in the general mechanical properties in saturated state, including those in the fiber direction. Furthermore, a permanent degradation was observed in some properties after drying. Thus, the importance of considering water content and aging effect on the characterization and engineering application of 3D printed CF/PA composite was evidenced.

This work presents the experimental determination of fracture mechanics parameters of composite specimens manufactured by fused filament fabrication (FFF) with continuous carbon fiber reinforced thermoplastic filaments, based on Linear Elastic Fracture Mechanics (LEFM). The critical mode I translaminar fracture toughness (KIc) and the critical energy release rate (GIc) are found for unidirectional and cross-ply laminates. The specimens were submitted to quasi-static tensile testing. Digital Image Correlation (DIC) is used to find the stress field. The stress fields around the crack tip are compared to linear elastic finite element simulations. The results demonstrate the magnitude of fracture toughness is in the same range as for polymers and some metals, depending on lay-up configuration. Besides, fractographic analyses show some typical features as river lines, fiber impression, fiber pulls-out and porosity aspects.

Summary Three‐dimensional hybrid‐Trefftz stress finite elements for plates and shells are proposed. Two independent fields are approximated: stresses within the element and displacement on their boundary. The required stress field derived from the Papkovitch‐Neuber solution of the Navier equation, which a priori satisfies the Trefftz constraint, is generated using homogeneous harmonic polynomials. Restriction on the polynomial degree in the coordinate measured along the thickness direction is imposed to reduce the number of independent terms. Explicit expressions of the corresponding independent polynomials are listed up to the fifth order. Illustrative applications to evaluate displacements and stresses are conducted by hexahedral hybrid‐Trefftz stress element models. The hierarchical p‐ and h‐refinement strategy are exploited in the numerical tests.