中值定理在圆盘锻造应变矢量内积中的应用

Application of Integral Mean Value Theorem to Inner-Product of Strain Rate Vector During Disk Forging

提出了一种以积分中值定理简化应变速率矢量内积的积分方法．首先将有鼓形圆盘锻造等效应变速率表示成二维应变速率矢量，然后采用积分中值定理确定应变速率比值函数及该矢量的方向余弦，再对其内积进行了逐项积分；其次，将逐项积分结果求和并给出相应的鼓形参数b的计算公式及应力影响因子的解析解．最后经压缩试验将应力状态系数与总压力计算结果与Avitzur公式的相应计算结果及压力机实测值进行了比较，表明计算结果与Avitzur上界近似基本一致，但高于实测结果．道次压下率在10％～33％范围内相对误差为1．9％～9％．

A new integration method proposed i.e., the integral mean value theorem, is applied to simplifying the inner product of strain rate vector. The equivalent strain rate during barrel-shaped disk forging is expressed in terms of twodimensional strain rate vector, then the strain rate ratio function and direction cosine of this vector are detemined according to the integral mean value theorem, and its inner-product is integrated term by term. The integration results are summed up to obtain the formula of relevant barreling parameter b and analytical solution of the stress influence factor. A compression test was done to compare the values of the stress influence factor and total pressure thus calculated with those calculated according to Avitzur＇s approxirnate solution and the measured values given by the test. The comparison results show that the values thus calculated conform basically to Avitzur＇ s upper bound approximation but are higher than measured ones, among which the relative error of pass reduction from 10% to 33% is 1.9%-9%.