Fracture mechanics is an emerging discipline that has only developed in recent decades. It mainly studies the conditions under which a bearing body fails due to the expansion of a main crack (including expansion under static load and fatigue load). Fracture mechanics is applied to the analysis of various complex structures, and the process from crack initiation and expansion to instability is within its analysis scope. Since it is directly related to the safety issues of materials or structures, although it started late, both experiments and theories have developed rapidly and have been widely used in engineering. The method of fracture mechanics research is: starting from the elastic mechanics equation or elastic-plastic mechanics equation, taking the crack as a boundary condition, examining the stress field, strain field and displacement field at the top of the crack, and trying to establish the relationship between these fields and the physical parameters that control the fracture and the local fracture conditions near the crack tip.
Current status of related research at home and abroad
At present, the overall research trend of fracture mechanics is: from linear elasticity to elastic-plasticity; from static fracture to dynamic fracture; from macroscopic and microscopic separation to macroscopic and microscopic combination; from deterministic methods to probabilistic and statistical methods. Therefore, as far as fracture mechanics itself is concerned, according to the specific content and scope of the research, it is divided into macroscopic fracture mechanics (engineering fracture mechanics) and microscopic fracture mechanics (belonging to the category of metal physics). Macroscopic fracture mechanics can be divided into elastic fracture mechanics (which includes linear elastic fracture mechanics and nonlinear elastic fracture mechanics) and elastoplastic fracture mechanics (including small-scale yield fracture mechanics and large-scale yield fracture mechanics and comprehensive yield fracture mechanics). Engineering fracture mechanics also includes important aspects of engineering such as fatigue fracture, creep fracture, corrosion fracture, corrosion fatigue fracture and creep fatigue fracture. Nowadays, reliability theory is introduced in the research methods of fracture mechanics, which is called probabilistic fracture mechanics, enriching the research content of fracture mechanics, and further developing and improving the theory of fracture mechanics, and playing an increasingly important guiding role in engineering practice.
1. Griffith theory
In order to study the influence of cracks inside the material on the strength of the material, Griffith in the 1920s first studied the strength of glass containing cracks and derived the relationship of fracture energy:
This is the famous Griffith fracture criterion, in which G is the energy release rate at the crack tip and γs is the surface free energy (the energy required for the material to form a unit crack area). From this relationship, the relationship between Griffith crack stress and crack size can be obtained:
In the formula, a is the crack length. If G>2γs, the crack will expand; if G<2γs, the crack will not expand; if G=2γs, it is a limit state. In addition, if the crack expands and dG/da>0, it can be determined as unstable expansion; if the crack expands and dG/da<0, the crack stops.
2. Stress intensity factor K
The abbreviation of the elastic stress field intensity factor in the crack tip area is a mechanical parameter in linear elastic mechanics that reflects the strength of the elastic stress field in the crack tip area, represented by the symbol KI. From the study of stress field near the crack tip, we know that the stress near the crack tip tends to infinity in some way, that is, it has singularity. Therefore, the stress here cannot be used to measure its strength. The KI value can reflect the strength of the elastic stress field in the crack tip area. Its value is related to the load, crack size and geometry. The mathematical expression of Griffith crack is:
Where σ is stress, a is crack length, and there are three forms of crack extension: KI, KII, and KIII, which represent the stress intensity factors of type I, type II and type III cracks respectively. Among them, for type I crack:
Where E is plane stress.
Note: The stress intensity factor is applicable to the plastic zone at the crack tip that is several times smaller than the K field zone and several times smaller than the crack length, such as ductile materials.
3. J integral
Proposed by Rice (J.R.Rice) in 1968. It reflects the concentration of stress and strain at the crack tip due to large-scale yielding. The definition of J integral is:
It is used to study plane problems and represents the energy related to crack extension. The first term on the right side of the formula is the energy related to strain energy, where W is the density of strain energy (i.e., strain energy per unit volume). In the case of elastic-plasticity, it is the stress-deformation work density (including elastic strain energy and plastic deformation work) received by each volume element of the specimen during monotonic loading. The second term is the force component on ds; ds is the arc element on the path Γ.
J integral has the following properties:
J integral is independent of the path;
J integral can determine the elastic-plastic stress-strain field at the crack tip;
J integral has the following relationship with deformation work power:
Where B is the specimen thickness, U is the deformation work of the specimen, and ▽ is a given position. The above formula is the basis for the experimental determination of J integral.
4. Resistance curve
In fracture mechanics, a curve that represents the stable expansion behavior of a crack in a material (as shown in the figure below). The ordinate is the resistance to crack extension, expressed by J integral, δ of CTOD or stress intensity factor K, and the abscissa is the crack extension amount △a. When the crack does not extend, the curve coincides with the ordinate. Once extended, △a≠0, the curve deviates from the ordinate, and the inflection point is the crack initiation point. The following represents the stable extension process. When the tangent of a point on the curve can pass through the point on the horizontal negative axis representing the crack length, it means that unstable extension will occur. When instability occurs, the crack extension driving force and crack extension resistance have the same rate of change with the crack size. The crack will expand rapidly and break without loading. The resistance curve can be tested with a specimen, which can be used to determine the crack initiation value (δi or JIC) or the conditional crack initiation value (δ0.005 or J0.005, etc.), and can also be used to predict the process of subcritical crack extension in a component.
5. Numerical calculation methods
With the deepening of fracture mechanics research, the problems that need to be solved are becoming more and more complex and diversified, making how to establish efficient and high-precision calculation methods a hot topic for scholars. Due to the continuous development of disciplines such as computer science, computational mathematics and mechanics, numerical calculation methods for solving fracture mechanics problems continue to emerge, from the early finite difference method, finite element method, boundary element method to the current meshless method, numerical manifold method, wavelet numerical method, discontinuous deformation analysis, etc., they are becoming important tools to promote the continuous development of fracture mechanics research.
Finite element method:
In the case of finite element solution, stress recovery, error estimation and automatic division of new grids are used to perform finite element solution, and this process is repeated until a satisfactory finite element solution is obtained. In addition, stochastic analysis is an important direction for the development of fracture mechanics and the basis for structural reliability assessment. On the basis of finite element method, stochastic finite element method uses random parameters to describe practical engineering problems. The main research contents include random variation principle, establishment of random finite element control equations and their solutions.
Boundary Element Method:
This is a numerical method for solving mechanical problems developed after the finite element method. Its composition includes the following three main parts:
The characteristics of the basic solution and its application;
The selection of discretization and boundary elements;
The superposition method and solution technology.
The advantage of this method is that the Guass theorem is used to reduce the problem order, converting the three-dimensional problem into a two-dimensional problem, and converting the two-dimensional problem into a one-dimensional problem, which greatly simplifies the data preparation, makes the grid division and readjustment more convenient, and the size of the final algebraic equation group is much smaller.
Meshless method:
Also called elementless method. This method discretizes the entire solution domain into independent nodes without connecting the nodes into units. It does not need to divide the grid, thus overcoming the defect of the finite element method that the grid must be continuously updated during the calculation process. During the calculation process, the crack tip area can be tracked in real time for local refinement, and the continuous crack extension process is regarded as multiple linear increments. The crack extension angle in each increment is determined according to the stress intensity factor. The calculation accuracy is improved by introducing external basis functions at the crack tip refinement node.
Numerical manifold method:
The basic idea of this method is to introduce the manifold principle of differential geometry into material analysis, based on topological manifolds and differential manifolds, while absorbing the advantages of the interpolation function construction method in finite elements and the block kinematics theory in discontinuous deformation analysis, unifying the problems of continuous and discontinuous deformation mechanics.
Wavelet numerical method:
This method takes advantage of the good localization characteristics of wavelets, approximates the displacement field with wavelet functions, establishes a wavelet numerical calculation format, simulates the singularity problem at the crack tip and solves the stress intensity factor at the crack tip.
Existing problems and technical key
The above methods or theories are all derived from Griffith's fracture theory and are based on singularity, that is, they are all based on the model where the stress and strain at the crack tip are infinite. The elastic mechanics explanation of the fracture theory of the Inglis mathematical tip crack model is the basis of the mathematical tip crack model. The distance between the upper and lower surfaces is zero, and the radius of curvature of the crack tip is also zero. Therefore, the stress component obtained by elastic mechanics is infinite at the crack tip. This phenomenon is called singularity.
The singularity theory has been continued to this day, but the singularity fracture mechanics has essential defects in physics, which are mainly manifested in two aspects:
First, the upper and lower surface spacing and the radius of curvature of the crack tip found in practice are finite values and not equal to zero;
Second, in actual cracks, even at the crack tip, the stress and strain are finite values, and there is no so-called singularity of stress and strain.
In this way, the physical quantities based on mathematical tip cracks and stress singularities lack a solid physical foundation. In order to improve the theory and present non-singularity, a blunt crack (or cut) model with a semicircular tip that is more in line with the actual situation can be used, but the measurement of the curvature radius of the blunt crack needs to be measured by metallographic methods, which requires the development of metallographic fracture mechanics.
Future development trends
Although some progress has been made in elastic-plastic fracture mechanics, there are still many issues that need to be studied in depth. It is one of the main research directions of fracture mechanics at present. Fracture dynamics, for linear materials, needs to be improved; for nonlinear materials, it is still in the early stages of research and is another main research direction of fracture mechanics. With the in-depth study of fracture problems and the convenient use of mathematical tools, fracture mechanics theory will become increasingly mature and fracture mechanics applications will become increasingly widespread.
For numerical calculation methods, the future development trends are: cross-scale fracture mechanics numerical calculation methods, parallel numerical calculation methods, the combination of analytical methods and numerical methods, the organic combination and fusion of multiple calculation methods, and data processing automation.
