Peridynamic (PD) models of bodies without pre -cracks, based on a single fracture parameter (associated with the critical fracture energy ), produce different strengths when different horizon sizes are used to simulate crack nucleation under quasi -static conditions. To maintain the same strength and fracture energy under different horizon sizes, extra parameters h ave to be introduced in the failure model. B ilinear and trilinear bond force -strain relationship s have been proposed in the literature for crack propagation in quasi -brittle materials . In this paper w e study crack nucleation in a plate with a hole under quasi -static loading using bilinear and trilinear PD model s. We provide analytical formulas to calibrat e the models to measurable material properties. We show convergence for both strength and fracture toughness. The bilinear PD constitutive model works well for both brittle (e.g. ceramics) and quasi -brittle (e.g. concrete) systems , while the trilinear version is more suited for quasi -brittle fracture behavior. We also find that for quasi -brittle fracture, a model that accounts, stochastically, for the p resence of small -scale pores/ defec ts performs better than a homogen ized model . A wedge -splitting test in concrete and crack nucleation in a quasi -isotropic composite plate with a circular hole are used to demonstrate the model’s performance . In contrast with other models, the current formulation does not depend on the sample geometry. Keywords: Crack nucleation ; peridynamics ; failure model; crack growth ; strength ; fracture energy . 1. Introduction Brittle and quasi-brittle crack propagation under static or dynamic loading ha ve been successfully analyzed using peridynamics (PD) (e.g. see [1-6]). PD models converge in terms of crack path and strength (maximum load before failure, in the load -displacement curve) displayed as the nonlocal region (horizon) size goes to zero , when pre-cracks are present . In certain problems, where material or physical length -scale effects are significant in the material’s mechanical behavior , an upper bound for the horizon size can be determined based on experimental results (see [7, 8] ). As we explain in Section 3, existing PD models based on a single fracture parameter (calibrated to match a material’s critical fracture energy ), lead to strength values that do not converge as the horizon decreases, for problems without pre-cracks or other defects . The reason for this behavior is the absence of a parameter linked to crack nucleation. The prototype microelastic brittle ( PMB ) model (see [9]) contain s only the critical bond strain that is related to the energy release rate for a growing 2 crack, not nucleation of a crack. To address this issue, one can introduce strength -dependent parameter s into the peridynamic b ond behavior, leading to, for example, bilinear or trilinear bond force -strain relationships . Recent works based on such bilinear/trilinear behavior of PD bonds focused on aspects of crack nucleation from a material stability point of view and cra ck propagation [10-14]. Reference [10] presented a bilinear law for PD bond force -strain curve, alongside the original linear bond force -strain curve presented in [15] for the PMB model . In [10], a condition associated with material stability was suggested for spontaneous emergence of a discontinuity (“nucleation” of a crack ) in a peridynamic body, and a numerical example of a plate with a centered hole was tested . A study on strength as the horizon size decreases was not provide d. A bilinear model was also employed in [11, 12] in PD modeling of static crack growth from a pre -crack in brittle and quasi -brittle materials, such as concrete . The focus was only on crack propagation from existing pre -cracks, not on crack nucleation or the issue of convergence for strength as the horizon decreases . We also note that in [10-12], the extra parameter in the bilinear model is determined via curve fitting to an experimental for ce-displacement curve, with a goal of match ing the location where the load-displacement curve departs from linearity as the moment for crack nucleation . In [13], the bilinear model is used to study quasi -brittle behavior , and the model parameters are calibrated to match the experimental force -displacement curves . An alternative approach to computing the model parameters by curve -fitting experimental data, as pursued in [10-13], is to directly connect these to some measurable quantities (e.g. ultimate strength) . In [14], the extra parameter in the bilinear model is set to the ultimate tension strain, i.e. 𝜎𝑢𝐸⁄, in which 𝜎𝑢 is the ultimate strength, and E is the elastic modulus. The other parameter in the bilinear model is then evaluated from the critical fracture energy (similar to the calculation of the critical bond strength in the PMB model in [9]). A PD model with a trilinear bond force -strain relationship for quasi -brittle behavior was proposed in [14] with the softening part of the bond force -strain curve being calibrated to the correspondin g experimental softening curve. Note that PD models with bilinear force -bond strain behavior have also been used to simulate elasto -plastic - type behavior ( [16, 17] ). In these