Details

Autor(en) / Beteiligte

• Bogdanov, Viacheslav L
• Guz, Aleksander N
• Nazarenko, Vladimir M

Titel

Three-Dimensional Problems on Loading of Bodies Along Cracks

Ist Teil von

• Fracture of Materials Under Compression Along Cracks, 2020, Vol.138, p.249-439

Ort / Verlag

Switzerland: Springer International Publishing AG

Erscheinungsjahr

2020

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• In this chapter we examine spatial problems of the fracture mechanics of materials loaded along cracks located in a single plane or in parallel planes, relying on the approaches of the three-dimensional linearized mechanics of deformable bodies (TLMDB), whose principal ideas are presented in Chap. 10.1007/978-3-030-51814-1_2 and in monographs [22, 23, 28, 30, 31, 39, 40]. For the uniform initial stress-strain state, which is defined by relations (10.1007/978-3-030-51814-1_2), (10.1007/978-3-030-51814-1_2) and considering that relations (10.1007/978-3-030-51814-1_2), (10.1007/978-3-030-51814-1_2) and (10.1007/978-3-030-51814-1_2) are valid, in the case of spatial problems we have obtained the representations of the general solutions of the equilibrium equations in terms of potential harmonic functions, which allows the integral transformations methods to be used for the investigation of the problems. The general solutions are constructed in a common unified form for compressible and incompressible bodies for the theory of large (finite) initial strains and for two variants of the theory of small initial strains. The focus here is on the investigation of three-dimensional problems on the compression of bodies with internal cracks under the action of forces parallel to the planes of cracks location. We examine problems on biaxial uniform compression along an array of cracks located in one plane (in particular, for a single circular internal crack), a near-surface crack, two parallel coaxial cracks and a periodic array of parallel coaxial cracks. For different material models (highly elastic, elastoplastic, and composite ones) we have obtained the values of critical compression parameters that correspond to the local loss of stability of the material equilibrium near cracks, i.e., the fracture mechanism described in Sect. 10.1007/978-3-030-51814-1_1 has been investigated. In the case of elastoplastic materials we also employ the generalized concept of continuing loading (see Sect. 10.1007/978-3-030-51814-1_2), which permits changes in the unloading zones during stability loss to be neglected. In investigating composite materials we use the continuum approach when the composite is modeled as a homogeneous transversely isotropic body with the reduced (averaged) values of mechanical parameters (see Sect. 10.1007/978-3-030-51814-1_2). The dependence of those critical compression parameters on the physical and mechanical characteristics of materials and the geometric parameters of the problems has been analyzed. Besides, by using the example of the problems of fracture mechanics on unbounded bodies containing parallel interacting cracks, we have analyzed the employment of the combined (unified) approach to investigating spatial problems of the fracture mechanics of bodies with initial stresses and problems on the fracture of cracked bodies under compression along cracks [7, 12, 13, 16, 50]. Relying on the combined approach mentioned in Sect. 10.1007/978-3-030-51814-1_2, we have proposed a new method of determining critical (limit) loading parameters in problems on body compression along the cracks it contains when special investigations of eigenvalue problems within the three-dimensional linearized theory of stability are not necessary. Those parameters are calculated by solving the corresponding boundary value problems of the fracture mechanics of prestressed materials, when during the continuous change of loading parameters we determine such values of initial compressive loads which, when achieved, cause a dramatic “resonance-like” change of amplitude values (stresses and displacements) near crack tips. The values of initial loading parameters determined in that way correspond to characteristic values of eigenvalue problems on bodies compressed along cracks. The effect of initial stresses and geometric parameters of problems on stress intensity factors has been analyzed for highly elastic bodies and composites with elastic components. We discuss the abovementioned “resonance-like” phenomena that appear in prestressed bodies with parallel coaxial cracks when initial compressive stresses approach the values corresponding to the local loss of material stability in the vicinity of the cracks. In this chapter, we partially use the results presented in [5, 10, 11, 14–19, 24–27, 33, 37, 39–41, 46–49, 63].