An ex vivo heart perfusion device preserves the donor heart in a warm beating state during transfer between extraction and implantation surgeries. One of the current challenges includes the use of rigid and noncompliant plastic tubes, which causes injuries to the heart at the junction between the tissue and the tube. The compliant and rapidly strain-stiffening mechanical property that generates a “J-shaped” stress–strain behavior is necessary for producing the Windkessel effect, which ensures continuous flow of blood through the aorta. In this study, we mimic the J-shaped and anisotropic stress–strain behavior of human aorta in synthetic elastomers to replace the problematic noncompliant plastic tube. First, we assess the mechanical properties of human (n = 1) and porcine aorta (n = 14) to quantify the nonlinear and anisotropic behavior under uniaxial tensile stress from five different regions of the aorta. Second, fabric-reinforced elastomer composites were prepared by reinforcing silicone elastomers with embedded fabrics in a trilayer geometry. The knitted structures of the fabric provide strain-stiffening as well as anisotropic mechanical properties of the resulting composite in a deterministic manner. By optimizing the combination between different elastomers and fabrics, the resulting composites matched the J-shaped and anisotropic stress–strain behavior of natural human and porcine aorta. Finally, improved analytical constitutive models based on Gent’s and Mooney–Rivlin’s constitutive model (to describe the elastomer matrix) combined with Holzapfel–Gasser–Ogden’s model (to represent the stiffer fabrics) were developed to describe the J-shaped behavior of the natural aortas and the fabric-reinforced composites. We anticipate that the suggested fabric-reinforced silicone elastomer composite design concept can be used to develop complex soft biomaterials, as well as in emerging engineering fields such as soft robotics and microfluidics, where the Windkessel effect can be useful in regulating the flow of fluids.