On the damage tolerance of 3-D printed Mg-Ti interpenetrating-phase composites with bioinspired architectures

Bioinspired architectures are effective in enhancing the mechanical properties of materials, yet are difficult to construct in metallic systems. The structure-property relationships of bioinspired metallic composites also remain unclear. Here, Mg-Ti composites were fabricated by pressureless infiltrating pure Mg melt into three-dimensional (3-D) printed Ti-6Al-4V scaffolds. The result was composite materials where the constituents are continuous, mutually interpenetrated in 3-D space and exhibit specific spatial arrangements with bioinspired brick-and-mortar, Bouligand, and crossed-lamellar architectures. These architectures promote effective stress transfer, delocalize damage and arrest cracking, thereby bestowing improved strength and ductility than composites with discrete reinforcements. Additionally, they activate a series of extrinsic toughening mechanisms, including crack deflection/twist and uncracked-ligament bridging, which enable crack-tip shielding from the applied stress and lead to “Γ”-shaped rising fracture resistance R-curves. Quantitative relationships were established for the stiffness and strengths of the composites by adapting classical laminate theory to incorporate their architectural characteristics.

Tensile stress-strain curves of the 3-D printed a dense Ti-6Al-4V alloy and b Ti-6Al-4V scaffolds with bioinspired brick-and-mortar, Bouligand, and crossed-lamellar architectures without infiltration of Mg. The insets show the dog-bone shaped tensile specimens of them. and red arrows indicate the loading direction and twisting direction of laminae, respectively. a As β increases from 6° to 9° and then to 12°, the overall deviation of fibers in the laminate firstly increases and then decreases, leading a wavy behavior in the modulus/strength diagrams for the Bouligand architecture. b At specific angles that are divisible by 90° (such as β=30°), the fibers in some of the layers become perpendicular to the loading direction and thereby play a minimal role in stiffening/strengthening the composite, leading to the minimum values on the diagrams. c At specific angles especially those are divisible by 180° but not divisible by 90° (such as β=36°), the fibers at some of the layers are aligned right along the loading direction and none of them are perpendicular to the loading direction, leading to the peak points on the diagrams. The variation in the J-integral as a function of the crack extension Δ is manifested by the fitted R-curve for the 3-D printed dense Ti-6Al-4V alloy. The dashed line for determining the fracture toughness has a slope of ( YS + UTS ) and corresponds to a 0.2 mm offset strain. The inset shows the sample for compact tension testing.

Supplementary Note 1. Young's modulus of the structural element
Under plane-strain conditions, the elastic constants of the basic structural element where the reinforcements are unidirectionally aligned can be accessed from those of the constituents according to the rule-of-mixtures. Specifically, the Young's moduli along the 1 and 2 axes, 1 and 2 , can be obtained in line with, respectively, the Voigt and Reuss models (because the stresses and strains are identical between constituents respectively in these configurations) as [1,2]: Poisson's ratio of the element along the 1-2 direction, 12 , can be obtained based on the Voigt model as: The corresponding Poisson's ratio along the 2-1 direction 21 can then be obtained following the relationship 21 = 12 The stiffness matrix of the element in the local coordinate system (1, 2), 1,2 , can be described using the above elastic constants as [2,3]:  ].
The compliance matrix in the (L1, L2) coordinate system, , can be obtained by inversing as = − . On this basis, the Young's moduli along the L1 and L2 axes, L1 and L2 , in the (L1, L2) coordinate system can be obtained from the compliance matrix as: The distance of the k th lamina from the mid-plane can be described as: Then, the principal forces and moments of the laminate, both normalized by the area, can be correlated to the in-plane strains and curvatures following the relationship [2,4]: Vectors are used here for all the parameters to encompass the different components along different directions. , and are the stiffness matrices for the stress states of extension, extension-bending coupling and bending, which are given by [2,4]: is half the total number of laminae in the laminate which equals 12. is the stiffness matrix of the k th lamina which can be obtained by transforming its stiffness matrix in the local coordinate system (1,2) to that of the mid-plane (l, t) using the rotation angle , as detailed in Supplementary Note 1.
The above relationship of Eq. (10) can then be converted to: When the laminate is subject to in-plane stress, i.e., with ≠ and = , the plane-strain vector of the laminate can be simplified as: The compliance matrix of the laminate in the global coordinate system (L1, L2), , can be obtained by transforming the compliance matrix in the mid-plane coordinate system (l, t), , by taking into account their inclination angle = ( − 1 2 ) , as [4]:

Supplementary Note 3. Failure stress of the structural element
The Tsai-Hill failure criterion has proven to be applicable to a range of composites, including those with bioinspired architectures [3][4][5][6][7][8]. The critical stresses causing the failure (either yielding or fracture under tension) of the structural element can be determined by such a criterion following the relationship [5][6][7]: where 1 and 2 are the principal stresses along the axes 1 and 2, and 12 is the shear stress along the 1-2 direction in the local coordinate system (1, 2). X and Y represent the critical tensile failure stresses along the principal axes 1 and 2, respectively, and S is the shear failure stress along the 1-2 direction. X can be approximated to be the weighted average of the tensile failure stresses of the Ti-6Al-4V and Mg phases by their volume fractions. Y and S can be approximated using those of the coarse-grained pure Mg considering that the failure of the composite in these loading configurations is principally governed by the weak matrix [4,7]. The values of Y and S were set to be identical for different architectures and were determined by fitting the experimental data.
Considering the inclination of the local coordinate system (1, 2) with respect to the global one (L1, L2), 1 , 2 and 12 can be described using the critical failure stress of the structural element along the loading direction, f , as: where is the inclination angle between the local and global coordinate systems. As such, the failure stress of the element, which represents either the yield strength or the ultimate tensile strength, can be obtained by converting Eq. (21) to the following form: where X I , X II , Y and S can be determined following the methods as detailed in Supplementary Note 3.

Bouligand architecture
For the Bouligand architecture, the plane-stress matrix of the laminate in the global coordinate system (L1, L2) can be expressed as [3,4]: with = [1 ] T . f,B is the critical stress along the loading direction L1, and and are the ratios of, respectively, the stress along the L2 direction and the shear stress along the L1-L2 direction in the global coordinate system with respect to f,B . As the laminate is free of bending or distortion deformation, the total plane-strain matrix of the laminate can be given according to the stress-strain relationship as: where is the compliance matrix in the global coordinate system (L1, L2). The plane strain in the k th lamina in its local coordinate system (1, 2) can be obtained by transforming L as [3,4]: where = ( − 1) is the inclination angle between the local and global coordinate systems for the k th lamina. The plane stress in the k th lamina in the local coordinate system can then be given by: where is the stiff matrix of the lamina in the local coordinate system and can be obtained in line with the methods described in Supplementary where X B , Y B and S B can be determined following the methods as detailed in Supplementary Note 3. Accordingly, the failure stress of the laminate can be obtained by incorporating Eqs. (28)-(30) into Eq. (31).
On the above basis, the yield strength of the laminate can be accessed according to the conservative first-ply failure criterion which assumes that the laminate yields when the first lamina (i.e., the lamina with the lowest strength) fails [7,8]. As such, the yield strength of the laminate can be determined using the specific failure stress corresponding to the failure of the first lamina. Similarly, the ultimate tensile strength of the laminate can be approximated using the critical stress at which all the laminae fail [4,6], i.e., corresponding to the failure of the final lamina.