This paper examines the size eﬀect in quasi-brittle materials using a three-dimensional bond-based peridynamic model. This is the ﬁrst time that the capability of a peridynamic model to capture the size eﬀect in quasi-brittle materials has been examined. Correctly reproducing the size eﬀect is an essential check on the validity of any computational model and it is demonstrated that a bond-based peridynamic model can accurately capture the failure stress of geometrically identical structures over a range of sizes. A systematic examination of geometrically similar notched and unnotched beams provides new insights into the predictive capabilities of the model, and mode I and mixed-mode problems are considered to examine the generality of the model. The model is validated using published experimental data, and the predictive accuracy is equivalent, if not superior, to well-established numerical methods whilst oﬀering several beneﬁts that justify further research and development. This study provides evidence that the length scale in a peridynamic model is a numerical parameter and not a material property. Keywords: Bond-based peridynamics, Size eﬀect, Model validation, Three-dimensional fracture propagation 1. Introduction Based on the strength-of-materials theory, structural failure is assumed to occur when the maximum stress in a structure exceeds some limiting value of stress that can be determined from small scale tests on representative material samples. Simple fundamental tests such as uniaxial tension, uniaxial compression, and ﬂexural tests are used to establish the limiting stress for diﬀerent loading conditions. This simplistic view does not suﬃce for quasi-brittle materials [1]. Quasi-brittle materials exhibit a size eﬀect where large elements fail at lower stresses than Corresponding author Email addresses: mhobbs@turing.ac.uk (Mark Hobbs), tdodwell@turing.ac.uk (Tim Dodwell), gh465@cam.ac.uk (Gabriel Hattori), jjo33@cam.ac.uk (John Orr) Preprint submitted to Engineering Structures November 10, 2021 geometrically identical but smaller-scale elements. Accurately capturing the structural size eﬀect is essential for safe predictions of load capacity. According to the strength-of-materials theory, the maximum stress that a material can resist is independent of size when geometrically similar specimens are considered. Any deviation from predictions made using stress failure criteria is known as the structural size eﬀect. The maximum stress that a quasi-brittle material can resist deviates signiﬁcantly from predictions made using stress failure criteria, and large specimens fail at signiﬁcantly lower stresses than small specimens. There are two primary sources of size eﬀect in quasi-brittle materials [ 2]: (1) release of stored energy (deterministic size eﬀect), and (2) statistical variability in material properties (statistical size eﬀect). The release of stored energy is by far the most important factor inﬂuencing the size eﬀect on structural strength, and the statistical size eﬀect is of secondary importance. The deterministic size eﬀect is governed by the size of the fracture process zone (zone of energy dissipation) relative to the size of the structure. The statistical size eﬀect is a result of the randomness of material properties and defects. The probability that a specimen contains a defect from which failure will initiate increases as the size of the specimen increases. Understanding the structural size eﬀect is essential for safe predictions of load capacity. The design of large scale structures relies on the properties of quasi-brittle materials measured in small scale laboratory tests. These properties must be extrapolated to structures that are one or two orders of magnitude larger than laboratory tests. Full scale tests are prohibitively expensive and reliable models that can capture structural size eﬀect are needed. Accurately capturing the size eﬀect is of utmost importance for the design of eﬃcient structural forms with small margins of safety. Existing analytical formulas for predicting size eﬀect in geometrically similar structures are extremely limited, and they generally employ empirical parameters that must be determined through best-ﬁt procedures [ 3]. Numerical methods oﬀer the only practicable approach for addressing complex industry motivated problems. This paper examines the size eﬀect in quasi-brittle materials using a deterministic three- dimensional bond-based peridynamic model. The capability of a peridynamic model to capture the size eﬀect in quasi-brittle materials has not been investigated yet and remains a major question. Understanding the predictive capabilities of the peridynamic model is essential for further application to more complex industry-motivated problems. As stated by Bažant [4], the capability to correctly reproduce the size eﬀect is an important check on the validity of any computational model. There are relatively few papers in the peridynamic literature that compare numerical results 2 against experimental data, and when a comparison is made, the majority of papers calibrate model parameters against a single experiment, and only a few provide further validation. Diehl et al.[5]provided a review of benchmark experiments for the validation of peridynamic models and noted the same issue. Hobbs [6]also discusses this issue at length and expands on the key elements of model validation. The lack of rigorous validation studies is a concern that is limiting the application of peridynamic models. This study quantiﬁes the predictive capabilities of the peridynamic model against carefully selected benchmark tests. Geometrically similar beams of diﬀerent sizes are modelled to examine the size eﬀect on structural strength. Notched and unnotched beams are modelled to examine the eﬀect of boundary types on the predictive accuracy of the peridynamic model, and mode I and mixed-mode problems are considered to evaluate the generality of the model. The aims of this paper are: (1) quantify the predictive accuracy of the bond-based peridy- namic model in capturing the size eﬀect; (2) validate the model against experimental data from the literature; and (3) identify future research directions. The paper is organised as follows: Section 2 provides a review of structural size eﬀect and the mechanisms behind size eﬀect are examined. Section 3 introduces the peridynamic theory and discusses the advantages of using a bond-based peridynamic model. Section 4 brieﬂy outlines the numerical framework. Mode I problems are examined in section 5 and the eﬀect of boundary types is explored. Mixed-mode problems are examined in section 6. Section 7 provides a discussion of the results and areas of future work are identiﬁed. Section 8 concludes the paper. 2. Structural size eﬀect 2.1. What is size eﬀect? According to the classical failure theories, such as elasticity with a material strength limit and plasticity theory, the nominal strength nof a structure is independent of size when geometrically similar structures are considered. The classical theories express the material failure criterion in terms of stress and strain and the nominal strength nis deﬁned as the maximum stress that a structure can resist [ 4]. For a concrete beam in bending, the limiting nominal strength n is given by the ultimate tensile strength ft. Any deviation from predictions made using stress failure criteria is known as the structural size eﬀect. A size eﬀect is observed in many materials. To clarify further, the basic theory of strength of materials predicts that a small and large beam made of the same material will fail at the same value of stress. In reality, the larger beam will fail at a lower stress. The post-peak behaviour of quasi-brittle materials also exhibits a 3 size eﬀect. At a small scale, concrete behaves like a ductile material and the post-peak load deﬂection curve descends slowly. At a suﬃciently large scale, concrete becomes perfectly brittle, and the load-deﬂection curve descends rapidly and may exhibit snap-back behaviour [ 7]. Clearly, a proper understanding of structural size eﬀect is essential for safe predictions of load capacity. For a detailed examination of size eﬀect in quasi-brittle materials, see the work of Bažant and Planas [2] and Bažant [7, 4]. 2.2. Size eﬀect mechanisms In brittle and quasi-brittle materials, the size eﬀect can primarily be explained by two mechanisms [ 2,4]: (1) release of stored energy, and (2) statistical variability in material properties. These eﬀects are often referred to as the deterministic and statistical size eﬀect. Secondary factors that inﬂuence the size eﬀect include the boundary layer eﬀect, diﬀusion phenomena, and hydration heat [ 8]. The inﬂuence of these secondary factors is insigniﬁcant and will not be considered any further. The release of stored energy is by far the most important factor inﬂuencing the size eﬀect on structural strength. The inﬂuence of statistical variability in material properties is less signiﬁcant. But until the late 1980’s, the primary cause of size eﬀect was generally believed to be of statistical origin [9]. When a fracture initiates and begins to propagate, the elastic strain energy stored in the structure is consumed by the fracture propagation. In quasi-brittle materials, energy is dissipated in a non-linear zone of micro-cracking formed at the crack tip. This damage zone is known as the fracture process zone (FPZ). The ratio between the elastic energy stored in the structure and the energy consumed by the fracture process varies with the size of the structure. This is the main mechanism responsible for structural size eﬀect. The mechanism by which the statistical size eﬀect occurs is simpler to comprehend. As the size of a structure increases, so does the probability that a defect will be present from which a fracture will initiate. Studying the mechanisms that govern size eﬀect is limited by the physical size of the experiments that can be performed within a reasonable timescale and budget. Many structures of interest, such as dams and reactor containment buildings, are far beyond the scale of laboratory experiments. Numerical methods provide new ways to explore the structural size eﬀect without the aforementioned limitations. 2.3. Size eﬀect laws Numerous analytical expressions have been developed to describe the size eﬀect of quasi- brittle materials. Size eﬀect laws provide predictions of nominal strength as a function of 4 the characteristic size of the structure for geometrically similar structures, with and without notches. These expressions have been developed from experimental observations and theoretical considerations. The central characteristic of quasi-brittle materials is that there exists a sizeable fracture process zone and area of softening damage. The size eﬀect is governed by the size of the fracture process zone relative to the size of the structure and depending on the structure size, diﬀerent theories are applicable for predicting strength. At a suﬃciently small scale, the theory of plasticity, which assumes no size eﬀect, is applicable. The theory of plasticity is valid when the size of the fracture process zone is comparable to the structural size. At a suﬃciently large scale, the theory of linear elastic fracture mechanics, which predicts the strongest possible deterministic size eﬀect, is applicable. The theory of linear elastic fracture mechanics is valid when the size of the fracture process zone is negligible with respect to the structural size. Many attempts have been made to bridge the gap between the theory of plasticity and the theory of linear elastic fracture mechanics. Numerous expressions have been proposed to combine the diﬀerent theories and provide a general size eﬀect law. The analytical expressions deﬁned by Eq. 1 were proposed by Bažant [10]and Bažant and Planas [2]to describe the deterministic size eﬀect. Two types of size eﬀect law are deﬁned: Type 1 - structures with no notches or pre-existing cracks (fracture initiates from a smooth surface); Type 2 - structures with a notch. Notched problems are representative of reinforced concrete structures, where due to the reinforcement, the stable growth of cracks occurs prior to the ultimate load. N(D) =8 >>>>>>< >>>>>>:f1 r 1 +rDb D+lp1=r Type 1 (unnotched case) Bftr 1 +D D0Type 2 (notched case)(1) Nis the nominal strength and Dis the characteristic size of the structure. For Type 1 structures,f1 r,Db,lp, andrare positive constants representing unknown empirical parameters. f1 rrepresents the elastic-brittle strength reached for large structures, Dbis a deterministic characteristic length set principally by the material inhomogeneity size (aggregate size in concrete),lpis a second deterministic characteristic length controlling the center of transition to a horizontal asymptote of the log Nversus logDcurve, andrcontrols the curvature and shape of the law. For Type 2 structures, ftrepresents the tensile strength, and BandD0are positive constants representing unknown empirical parameters. Bis a dimensionless constant, and D0 5 represents the transitional size at which the power laws of plasticity and LEFM intersect. It should be noted these deﬁnitions are not exact and diﬀerent interpretations can be found within the literature. The size eﬀect laws are illustrated in Fig. 1 and it becomes clear that Type 1 and Type 2 structures have diﬀerent asymptotic properties when D!1. Figure 1: Size eﬀect laws in a log-log scale: (a) Type 1 (unnotched case); (b) Type 2 (notched case). Adapted from Fig. 1 in [11]. AnextendedanalyticalexpressionwasproposedbyBažantetal. [12]todescribethecombined deterministic-stochastic size eﬀect for Type 1 structures. N(D) =f1 r ls D+lsrn m +rDb D+lp!1=r (2) lsis a stochastic scaling length, mis the dimensionless Weibull modulus (shape parameter of the Weibull distribution), and nis the number of spatial dimensions in which the structure is scaled. The proposed expression accounts for the stochastic and deterministic component of size eﬀect. The practical applicability of size eﬀect laws is limited. This has been noted by Moallemi et al.[3]. Many of the parameters are empirical and there is no systematic method to determine the parameter values. A precise deﬁnition for the characteristic length is lacking and this ambiguity further undermines the general applicability of the size eﬀect laws. This is discussed further in the following section. The nominal strength is inﬂuenced by multiple factors including the geometry of the structure, boundary conditions, and material properties. It is unlikely that it will be possible to develop a general size eﬀect law that can be reliably applied to a wide range of non-trivial problems. Numerical methods oﬀer the only practicable approach for addressing complex industry motivated problems. 6 2.4. Numerical methods A signiﬁcant body of work has been published on the numerical simulation of size eﬀect in quasi-brittle materials, and models can be broadly categorised as deterministic or stochastic. In a deterministic model, the material properties are constant over the spatial domain. Deterministic models have been employed by many researchers including Moallemi et al. [3], Feng and Wu [13] and Barbat et al. [8]. In a stochastic model, the spatial variability of the material properties is considered. Stochastic models have been used by Gutiérrez and De Borst [14], Vorechovský [15], Bažant et al. [12], Yang and Frank Xu [16], Bobiński et al. [17], Grassl et al. [18], Syroka- Korol et al. [11], Syroka-Korol et al. [19], Zhou and Chen [20], Eliáš et al. [21]and Eliáš and Vorechovský [22]. These