A quantity expressing the strength of the stress field at a crack tip. Given by K = Yσ√(πa), it depends on both the stress and the crack length, and the crack grows unstably when it reaches a critical value.
Japanese name: 応力拡大係数
Stress intensity factor appears in 4 practice questions across 2 topics, including Fracture mechanics and fatigue, Computational materials science.
Near the tip of a crack in an elastic body, the stress grows in inverse proportion to the square root of the distance r from the tip, taking the form σ ∝ K/√(2πr). The shape of this distribution is the same regardless of the shape of the crack or the part, and only its strength is set by the stress intensity factor K. So even for a part and a test specimen of different shapes, the crack tips can be regarded as being in the same state if K is the same. This is why fracture toughness measured on small specimens can be used to design real structures. In reality the region near the tip yields and becomes a plastic zone, so this idea holds when the plastic zone is small enough.
The stress intensity factor can be written K = Yσ√(πa), where the geometry factor Y depends on the shape and position of the crack, the width of the part and how the load is applied. For a central crack (length 2a) in an infinitely wide plate, Y = 1; for a short crack entering from the edge of a plate (an edge crack) it is about 1.12, and Y rises as the crack becomes large compared with the width of the part. Y for many geometries is compiled in handbooks as formulas and tables. Within the elastic range, when several loads such as tension and bending act together, K values of the same mode can be added, and the K from residual stress is also included in the assessment.
The elastic energy released when a crack advances by a unit area is called the energy release rate G, related to K by G = K²/E in plane stress and G = K²(1 − ν²)/E in plane strain. Griffith's condition, that G equals the energy needed to create surfaces, is equivalent when written with K to K reaching a critical value K_c. The critical value for Mode I is the fracture toughness K_IC, and under repeated loading the range of K, ΔK, determines the fatigue crack growth rate. In this way, K is the quantity that links the assessment of both static fracture and fatigue.
Answer: B. About 0.57 mm
K/σy = 30/500 = 0.060 √m, which squared gives 3.6×10⁻³ m; dividing by 2π ≈ 6.28 gives r_p ≈ 5.7×10⁻⁴ m, about 0.57 mm. In the elastic solution, the stress at the crack tip grows without limit in proportion to 1/√r, but in reality the region where it exceeds the yield stress becomes a plastic zone, whose size is proportional to (K/σy)². About 3.6 mm forgets to divide by 2π, and about 9.5 mm skips the squaring and treats 0.060/2π as a length. Linear elastic fracture mechanics, which judges fracture by K, can be used when the plastic zone is much smaller than the crack length and the plate thickness. Under plane strain, the plastic zone is about one third of this value.
Answer: A. The middle region is a straight line given by the Paris law da/dN = C(ΔK)ᵐ, and below ΔK_th the crack hardly grows
The curve is usually divided into three regions. In the low region where ΔK approaches the threshold ΔK_th, the growth rate drops sharply, and below ΔK_th the crack effectively does not grow. The intermediate region is a straight line on log-log axes and can be described by the Paris law da/dN = C(ΔK)ᵐ, with an exponent m of roughly 2–4 for metals. In the high region where the maximum stress intensity factor K_max approaches the fracture toughness K_IC, the rate rises sharply and leads to unstable fracture, so the claim that growth stops is backwards. ΔK = YΔσ√(πa) depends on both the stress range and the crack length, so the rate is not determined by crack length alone.
The stress concentration factor Kt is a dimensionless number giving how many times the average stress the maximum stress at a rounded hole or notch root is. When the tip is sharp, as at a crack, the maximum stress is in theory infinite and cannot be expressed by Kt, so the strength of the stress field at the tip is expressed by the stress intensity factor K (in MPa·√m).
In K = Yσ√(πa), σ is the stress (MPa) and a is the crack length (m), and since its square root is multiplied in, the units become MPa·√m. Even at the same stress, a crack 4 times longer doubles K. The square root of a length appears in the units because the effect of crack size is included.
When the plastic zone at the crack tip is large compared with the crack length, the plate thickness or the remaining cross-section. In highly ductile materials, thin plates or under high loads, the plastic zone spreads and K alone cannot describe the state. In that case, elastic-plastic fracture mechanics quantities that include plasticity, such as the J-integral or the crack tip opening displacement (CTOD), are used.
In the high region where the maximum stress intensity factor K_max approaches the fracture toughness K_IC, the rate rises sharply and leads to unstable fracture, so the claim that growth stops is backwards.
Which statement correctly describes the curve of fatigue cr…The stress intensity factors are likewise written K_I, K_II and K_III.
Among the modes of crack deformation, which statement corre…Cracks are therefore evaluated not by stress but by the stress intensity factor or the J-integral, with singular elements or similar used at the tip.
Which is the correct way of thinking when using the finite …Fracture toughness Paris law Finite element method Energy release rate Fatigue striations Griffith theory Miner's rule Modified Goodman diagram Stress concentration factor