Question

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2.59 The Brinell hardness test is used to measure the indentation hardness of materials. It involves penetration of an indenter, in the form of a ball of diameter D (mm), under a load P (kg) as shown in Fig. 2.13a. The Brinell hardness number (BHN) is defined as 2P where A (in mm2) is the spherical surface area and d (in mm) is the diameter of the crater or indentation formed. The diameter d and the depth h of indentation are related by (Fig. 2.13b) It is desired to find the size of indentation, in terms of the values of d and h, when a tungsten carbide ball indenter of diameter 10 mm is used under a load of P 3000 kgr on a stainless steel test specimen of BHN 1250. Find the values of d and h by formulating and solving the problem as an unconstrained minimization problem. Hint: Consider the objective function as the sum of squares of the equations implied by Eqs. (1) and (2) Spherical (ball) indenter of diameter D th Indentation or crater dーグ//-of diameter d and depth h Figure 2.13 Brinell hardness test.

Answer 1

Page No. Date: 2 Mecd tn ind vee ha 2.P Ia 2 23.03243 d3 03243 14139 2JhCi-) Alo-)08756

Page No Date: 2 h lo 9.84221 2 h9.92361mm is not ussible hecause its RD 2.

First of all , find d by plugging in values in the formula of BHN. Then use the 2nd relation to relate d with h and eventually find h