Cohesive-strength homogenisation model of porous and non-porous materials using linear comparison composites and application

An estimation of the strength of composite materials with different strength behaviours of the matrix and inclusion is of great interest in science and engineering disciplines. Linear comparison composite (LCC) is an approach introduced for estimating the macroscopic strength of matrix-inclusion composites. The LCC approach has however not been expanded to model non-porous composites. Therefore, this paper is to fill this gap by developing a cohesive-strength method for modelling frictional composite materials, which can be porous and non-porous, using the LCC approach. The developed cohesive-strength homogenisation model represents the matrix and inclusion as a two-phase composite containing solids and pores. The model is then implemented in a multiscaling model in which porous cohesive-frictional solids intermix with each other at different scale levels classified as micro, meso and macro. The developed model satisfies an upscaling scheme and is suitable for investigating the effects of the microstructure, the composition, and the interface condition of the materials at micro scales on the macroscopic strength of the composites. To further demonstrate the application of the developed cohesive-strength homogenisation model, the cohesive-strength properties of very high strength concrete are determined using instrumented indentation, nonlinear limit analysis and second-order cone programming to obtain material properties at different scale levels.

The rest of the paper is organised as follows: Section 2 describes the cohesive-strength homogenisation model based on a second-order homogenisation and LCC approach. Section 3 proposes the concept of a multiscale-link approach for determining properties of materials at micro, meso and macro scales. Section 4 demonstrates the application of the proposed approach on very high strength concrete to investigate the cohesive-strength properties at different scales.

cohesive-Strength Homogenisation Model
A second-order homogenisation approach and strength homogenisation method have been introduced by Castaneda 6,7 and Ortega et al. 9 , respectively. In order to develop a cohesive-strength homogenisation model of porous or non-porous composite, we recall briefly concepts of strength homogenisation method. Consider a representative volume element (RVE) of a composite material, Ω = Ω 1 ∪ Ω 2 where the Ω 1 and Ω 2 are the domains occupied by the two-phases, the subscripts 1 and 2 denote each phase domain. The material behaviour of a generic material phase i in the composite material is assumed to be characterised by local convex failure criterion where G i is a local convex strength domain. At the plastic collapse, each material phase no longer stores external work thus the external work is dissipated through plastic flow. This leads to the maximum dissipation capacity at the plastic collapse of each material phase which can be defined by the support , where d is the microscopic strain rate corresponding to the velocity field v 9,10 . According to Hill lemma 11 , the upper bound theorem and plastic flow rule with relevant kinematic admissible (K.A) field, the microscopic dissipation function π i is linked within the counterpart at the scale of RVE. It leads to the macroscopic dissipation function Π hom with respect to the macroscopic stress Σ and the strain rate D as:

hom G hom
where G hom is the boundary of the macroscopic domain 10 . According to the dual definition, the stress located at the intersection of G hom can be expressed by: The position vector of the macroscopic scale in Ω is denoted by x that leads to the local state equation as: For an isotropic material, the elastic stiffness  and prestress tensor τ of each phase can be expressed as: where k i and g i are the bulk and shear modulus of the corresponding phase. With classic linear micromechanics, the macroscopic stress equation can be represented as: where  hom and τ hom are macroscopic elastic stiffness tensor and prestress, respectively. Considering two-phase composite without discontinuity (variables associated with subscript 1 and 2), the strain rate energy function of an isotropic material can be expressed by: , respectively. The Mori-Tanaka estimation of the effective behaviour of a two-phase composite material can be illustrated implicitly with a matrix-inclusion scheme 12 . In the present work, the microstructure of the two-phase composite will be represented by a spherical inclusion. The results from the use of linear micromechanics 13 to obtain the homogenised bulk and shear modulus can be found in Part A of the Supplementary Information. The classic yield function of Drucker-Prager is defined by the mean stress σ σ = 1/3tr( ) m and stress invariance σ = s s (1/2) : d with s = σ − σ m 1 as: where α and c are the Drucker-Prager friction coefficient and cohesion describing the intrinsic strength of each phase, respectively, i.e., α < 3 /2 which is corresponding to Morh-Coulomb friction angle of 90°. With the application of stationary conditions through the material dissipation function of each phase, the degree of nonlinearity for the Drucker-Prager condition of each phase can be 9 : www.nature.com/scientificreports www.nature.com/scientificreports/ Therefore, the macroscopic strain rate energy density Π hom for the upper bound solution can be found by employing the generated expression of the strain rate energy function in Eq. (6) and nonlinearity function Eq. (8) with the stationary conditions as: where f i is the volume fraction of each phase. The macroscopic strain rate energy density function can be obtained by rearranging Eqs. 1 to (8), which yields: The constraint in Eq. 10 ensures the validity of the macroscopic strain rate energy density function, and the shape of the yield function will depend on the ρ sign, hyperbolic when ρ > 0, and elliptical when ρ < 0, as shown in Fig. 1. In order to derive the dual definition of the strength domain represented by Eq. 2, the macroscopic yield strength criterion can be obtained as:

Multiscaling cohesive-Strength
A general class of heterogeneous materials such as cementitious or geological materials is composed of particles which forms a porous composite material at the micrometre scale 8 . Let's consider a composite material at three scale levels: a cohesive-strength solid (Level 0), porous solid (Level I), and porous inclusion-matrix (Level II), as shown in Fig. 2. Implementation of the proposed cohesive-strength yield criterion with LCC approach requires multiscale-link modelling in the strength homogenisation. The yield strength domain of the porous solid (Level I) is considered a cohesive-strength porous material in which the existence of elementary solid build block (Level 0) is present. In Level II, an RVE of Ω of the two-phase composite material has the inclusion phase surrounding the matrix phase as sub-index I and II, respectively. The systematic multiscale-link structure of the composite material as presented in Fig. 2 serves as a reference for implementing the cohesive-strength model in Eq. 11.
Level ii: porous inclusion-matrix composite. Consider a two-phase composite material with perfect adherence between interface at Level II as shown in Fig. 2, the first phase and second phase are cohesive frictional porous inclusion and matrix, respectively. In turn, the volume fraction of the matrix phase is characterised by f 1 while the volume fraction of the inclusion phase f 2 = 1 − f 1 . The scaling relation of the macroscopic strain rate energy density function between Level I and Level II can be expressed as: www.nature.com/scientificreports www.nature.com/scientificreports/ I   I  I   I  I   I  I   I   II  II  I  I I  I I I  I  I  g  I  I  I  I I  I  II  g  I  I  II  g  I  I   II  g  I  I I  I I   I  I  II  g  I  I I  I I   II  g  I  I  I  I I  g  I  I  I  I  II  II  II  I   g  I  I I  g  I  I I  I I  II  g  I  I  II  II  g  I  I   g  I   I   I   II  II  I   I  I   II   I   I  I II where  δ δ = (1/3) ij kl and K I J = − with fourth-order identity tensor . With the dual definition in Eq. 2, the cohesive-strength yield criterion in Level II can be expressed by: Similarly, the macroscopic frictional coefficient α II ( ) and cohesion c II ( ) in Level II can be expressed by: Level i: porous solid. In this section, the mesoscopic cohesive-strength yield criterion of a two-phase composite material formed by pores (voids) and solid frictional matrix phase is established. The volumetric description at this level involves the solid packing density η, i.e., pore void φ = 1 − η. According to Eq. (4), the distribution of the elastic stiffness and prestress of porous solid can be described as: Therefore, by substituting the limits, α 2 → 0, c 2 → 0, r g → 0, f 2 → 0, and f 1 → η in Eq. 13, the cohesive-strength yield criterion of porous solid becomes: In other words, Eq. 13 is reduced to Eq. (17) to present the solution for the porous solid support function Π hom I ( ) , which is found to be the same as that of Ortega et al. 9 . It should be noted that corresponds to an elliptical cohesive-strength yield criterion. Based on the dual definition of the strength domain presented in Eq. 2, the macroscopic yield strength criterion can then be derived as: To retrieve the Drucker-Prager yield criterion, the macroscopic frictional coefficient α II ( ) and cohesion c II ( ) in in Level I can then be obtained as:

Application of cohesive-Strength Homogenisation
The application of the proposed cohesive-strength homogenisation model is demonstrated in this section with a focus on the investigation of the properties of very high strength concrete (VHSC) (>100 MPa). The two main phases of VHSC are aggregate and matrix phases. High strength concrete (HSC) (50-100 MPa) and VHSC have been used increasingly in the construction industry due to its inherent performance characteristics. High compressive strength, high elastic modulus, very low permeability, low deformations are some of the contributing factors to the increasing uses 14 . SC and VHSC are very brittle in nature compared with the normal strength concrete (NSC), and the damage and fracture behaviour of HSC and VHSC are considerably different from the fracture behaviour of NSC [14][15][16] . Two mixtures, M1 and M2, are used to achieve VHSC with a water to binder ratio of 0.22. It is assumed that nano-silica is able to fill nano-sized gaps between the materials and therefore enhances the strength and performance of the mixtures. The high-range water reducing, slump retention, and viscosity modifying admixture have been used to retain the slump, workability, and avoid bleeding of the self-compacting and segregation. Two different types of coarse aggregate, basalt (M1) and granite (M2), are used in order to determine cohesive-strength properties with the mixture contents as shown in In this section, the methodology for evaluating the materials properties based on the proposed cohesive-strength homogenisation model is demonstrated. Firstly, the cohesive-strength homogenisation model in Level I is applied for the continuum discretisation of instrumented indentation solution. The instrumented indentation is a well-known technique that has emerged for determining mechanical properties 17,18 . Recently, the instrumented indentation makes it possible to test the structure of porous materials with the characteristic size of porosity much smaller than the maximum indentation depth 19 . Bobko et al. 20 developed an indentation hardness response with the developed strength homogenisation model by Ortega et al. 9 . They used a limit analysis solver to predict the indentation hardness which was nominalised by solid cohesion as a function of varying solid www.nature.com/scientificreports www.nature.com/scientificreports/ friction coefficient. In this paper, a similar approach is adopted to investigate the applicability of the developed cohesive-strength homogenisation model, based on the second-order cone programming (SOCP) 11 . Consider the indentation test of a rigid contact into a cylindrical composite material with polar coordinates r, θ, and z as shown in Fig. 3.
The work rate provides a rigid contact to composite material (assumed frictionless contact) during testing process as: With the combination of Eq. 10, the dimensional function in the form of the indentation hardness H to cohesion c ratio, with the upper bound solution of the cohesive-strength homogenisation can be expressed by: where U is the velocity field, π = A h c m ax 2 θ tan i 2 is the contact area of the indentation test when using an equivalent conical Berkovich tip with θ i = 70.3 degree. The displacement rate  h of the rigid indenter tip in the z direction is fixed as one. The dimensional function in Eq. (21) consists of the key unknowns, S hom , A hom , and B hom developing the relation between the cohesive-strength responses of the composite material. In determining the dimensional function, a design of experimental (DOE) approach 18 is adopted considering variables such as α and η (more details are provided in Part B of the Supplementary Information). An advanced implementation of a continuum discretisation into finite elements together with nonlinear composites and SOCP techniques have been recognised for the upper bound theorem 6,7 . In particular, the approach used in this study is the numerical implementation of the discretised dissipation of the cohesive-strength criterion (α < 3/2) in the form of a second-order cone optimisation problem which can be solved by MOSEK optimisation algorithm 21 available in MATLAB 22 . In the implementation of the finite element discretisation, the triangle shape function in the axisymmetric condition shown in Fig. 3 has been modelled. The radius of the sample to contact ratio, r max /a, and the thickness to the indentation ratio, z max /h, are set to 5; and the radius of the sample to element size ratio, r max /L i = 100; the equivalent conical Berkovich tip apex angle, θ i , of 70.3 degree is set with the frictionless contact between the rigid contact and the sample. Eq. (21) can be solved as a SOCP optimisation problem, in which the dimensional functions, as presented in Eq. (17), are calculated for all the combination of the parameters outlined in the DOE, Table B, given the Supplementary Information. Figures 4 and 5 show a typical series of the velocity field and the relationship between various α (0) and the dimensional function H c / It should be noted that the efficiency of the minimisation process depends on computational resources. The first application of the cohesive-strength homogenisation model in Level I focuses on the investigation of solid properties of aggregate and cementitious matrix of VHSC. Each specimen was cut using diamond saw to obtain 10mm cube core part after 90 days curing ages. It is important to reduce the surface roughness of specimens to obtain an accurate nanoindentation results because the specimen's surface has a significant influence on the test results. In order to apply the continuum indentation model of an infinite half-space, the indentation depth must be larger than the surface roughness. Therefore, fine emery paper was used to grind all specimens to reduce the surface roughness then the specimens were polished using a suspension solution ranging 9.0 μm to 0.1 μm for 15 mins. This is achieved with the maximum indentation depth 500 nm of VHSC samples. Nanoindentation test was conducted on the three sets of cube specimens with XP BASIC testing mode with Keysight Nanoindentation G200; each set was specified around 300 indentation points on the cementitious matrix, fine sand and coarse aggregate using the Berkovich indenter. The hardness values of each indentation point then determined using our inverse algorithm 18    695 GPa, respectively. he present results show that even though M1 and M2 matrices have the same mix design, Level 0 (solid) properties are not consistent to each other. It is believed that the development of calcium silicate hydrated (CSH) which is one of major hydration products of cementitious matrix contributing to the primary source of nanometre-scale elastic modulus degradation 13,18 . In fact that the nanostructure of CSH is still not known, forms poorly crystalline during the hydration of the cementitious matrix and characterised by extensive atomic imperfection and structural variations at

conclusion
The proposed cohesive-strength homogenisation model enables predicting the behaviour of porous and non-porous two-phase composites. The fundamental idea behind the developed approach presented in this paper is that it is possible to assess the cohesive-strength behaviour of composite materials at different scale levels. The novelty of the present model is the incorporation of the two-phase nonlinear relationship of cohesive-strength composites based on a linear comparison composite (LCC) approach. The developed cohesive-strength model represents a multiscale-link relation of composite materials which incorporates the contribution of different phases for determining the effective mechanical properties. An application of the proposed model has been demonstrated on very high strength concrete (VHSC) materials which have two main phases, aggregate and  www.nature.com/scientificreports www.nature.com/scientificreports/ matrix. A three-level multiscale conceptual model is formulated which makes it possible to determine microscale and macroscale properties of the materials. The proposed cohesive-strength model approach combines an instrumented indentation technique, a nonlinear limit analysis and a second-order cone programming method to quantitatively predict the strength of VHSC from micro to macro level. Overall, the proposed cohesive-strength model can be used to predict the mechanical properties of material phases at different scale levels.