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imold_folder:E:top_assembly:10prefix:10project_unit:Millimeteradd_to_std:TRUE在大块金属玻璃钢板中建模和测量残余应力a 凯克实验室。材料科学学院，加州理工学院，加州1200年东大道，M/C138-78 ， 帕萨迪纳加州,CA91125 ，美国b 工程科学与应用部门，洛斯阿拉莫斯国家实验室，洛斯阿拉莫斯国家实验室，新墨西哥州87545 ，美国c cLiquidmetal Technologies, Lake Forest, CA 92630,美国摘 要 最近出现的多组分特殊合金玻璃形成能力，可以处理大量的金属样品的非晶态结构.在研究加工这些大块金属玻璃(BMG)样本期间用的两种模式有可能形成热回火压力: () 即时冻结模型，（ ii ）粘弹性模型。第一种承担了突然之间的过渡和弹性固体液体在玻璃化转变温度. 第二种模式考虑了BMG的平衡粘度. 两种模式虽然使用大量不同的方法但取得了类似的结果. 结果表明， 等金属对流冷却具有高传热系数可能在表面平衡和中平面张力中潜在的产生重大应力。裂纹遵从（纵剪）方法，然后在镶铸铜模BMG板中用来测量剖面图的应力.这些剖面图大致表明，热回火确实是占主导地位的残余应力产生机制。然而，巨大的压力测量（与峰值只有约1.5 屈服强度）明显低于模型的预测.可能原因是所描述的与实际铸造工艺和材料特性的差异. 极低的残余应力测量在这些BMG样本上，加上其高强度和韧性，有助于进一步提高BMGs的优势,超过对应的结晶金属。1 引言具有极好的玻璃形成能力的多组分金属合金最近已制出，第一次，处理大量的标本与无定形结构, 称为大块金属玻璃（ BMGs ）, 这些材料已被证明有令人印象深刻性能，如很高的弹性应变极限（2%）和屈服强度（2千兆）, 良好的破裂韧性（高达55兆帕货币供应量M1=2） ，优良的耐蚀 电阻等, 产生的一个重要的问题是大量生产的性质和重大处理残余应力. 在BMG的数据处理通常涉及合金铸造成模具其次是严重的淬火. 由于 BMG低导热系数,这一步骤可能会导致大的热梯度. 此外，在玻璃转变的合金展览品期间,在一个小的温度范围内大量改变其粘度。所有这些参数可能会导致热回火,这产生压缩表面残余应力与中平面张力的平衡。类似的现象，有人曾使用硅酸盐玻璃 1 , 热回火在这些玻璃中主要是研究在一定初始边值问题：无限金属板在一个统一的初始温度从两边传递冷却.合成应力剖面粗略成抛物线,那里压缩的 表面（s）的内部是平衡的. 首先理论利用即时冻结的假设（见，例如 2 ），推测物质行为作为一个非粘性流体高于其玻璃化转变温度和线性弹性体下方. 这种方法只需要简单的玻璃化转变温度为流变投入和忽视的细节，玻璃化转变的范围和它的冷却速度的依赖以及任何应力松弛低于该范围。最终进化的理论提供必要的材料 功能（例如，放松和结构模 3 ）等任何几何可处理有限元模型 4,5 然而，大多数粘弹性能的BMGs ，特别是所需要的先进的建模热回火，还没有研究系统。出于这个原因，即时冻结的设想先被调用，首先研究获得的估计残余应力产生的这种现象。下一步，唯一可用的在BMG合金上的粘弹数据用在这里，其作为一个平衡粘度随温度的变化，被雇用于在BMGs中发展的第一个粘弹性模型的热回火。该模型预测后来被用于和从BMG金属中用裂缝柔度法收集的残余应力数据相比，按作者的理解，这是第一次广泛理论/试验研究BMGs中的热回火。2 热回火的建模2.1 即时冻结模型热回火是由一个过程引起的，其中包括供热玻璃高于其玻璃化转变温度（Tg），然后迅速冷却它。在冷却期间，首先表面的玻璃板凝固和缩小，导致表面扩张，中平面缩小。这一过程同样将发生在弹性固体.然而，这一阶段的淬火玻璃，其核心仍然是低粘度和中期的平面压应力同时放宽粘性流动。因此，产生的压力在整个截面凝固时低于在其他部位相同的值，但弹性材料与它具有同样的温度分布。然而，当中平面最后凝固，粘带作用停止且材料的弹性实际上跟玻璃板一样，这时施加的温度分布线。所有在应力上进一步的改变仅仅确定温度分布线的变化。当在室温条件下温度梯度衰减到常数，应力产生相反的意思：中平面扩张，表面缩小。对于总是弹性固体的来说，当温度梯度增长时，这些将是平等的，与那些在第一阶段生产的相反。在玻璃中，其中包括开始时应力松弛，两个类型的应力不相等。换句话说，应力衰变过程中产生的温度梯度会超过那些最初产生的。由此产生的残余应力剖面的函数的距离来自于的金属板的中心，由即间冻结模型给出了，如下式： （1）其中是线性热膨胀系数，E是杨氏模量，是泊松比率 ，Tg是玻璃化转变温度是环境温度，衡量冷却速度，更明确来说，被作为的第一个根其中是有 h，l，k的Biot数，定义为传热系数，分别为半厚度钢板和热 电导率。图1显示了在平面残余应力剖面的板厚度。X和Y是平面坐标系， 当Z是不共平面坐标。对于这种无限平面问题热回火的残余应力是分布在X-Y平面上，目前，确切的试验热值转移系数中发现BMG加工通常无法处理。为了取得一个粗略的估计，最近的一项处理方法称为熔体渗透铸造，从文献 7 指出。在这种技术的一个类型中，BMG合金铸造不锈钢钢管，然后熄火在水中。这个方式的冷却是通过对流层的或者湍流层的流动方式。传热系数为自由对流水（即积水）在室温下的约900 W/(m2 K) 8 。当水被迫流动，这个值可以很容易到达2000 -3500 W/(m2 K). 如果水沸腾（如有时发现熔体渗透处理） ， 这个值高达35 000W/(m2 K)是可能的。为了保守，选择。h的这个值也屈服于一个时间比例，它考虑到BMG金属板从熔体冷却到室温， 图1：一种典型的由于热回火残余应力剖面厚度的大金属板：表面压缩与中平面张力平衡 ，内平面压力是等双轴的，仅仅作用于坐标（z）的厚度如第3节所述时提到铸造工艺用于此项研究。Eq.（1) 预测，表面压缩和中平面扩张，金属板厚度（t）和传热系数（h）分别如图2或者图3所示，材料参数的计算采用表1列出的。即间冻结模型预测提出有效的残余应力由于热回火效应可产生BMGs。另一个即时冻结模型预测是：当表面压缩值增加，表面层的厚度被局限而减少而中和中平面张力饱和烃（参见图2 ），这意味着， 薄表面层逐步增加应力梯度可形成一定程度的回火增加图2，在表面等双轴（绝对）应力值和中平面大量的Zr41:2Ti13:8Cu12:5Ni10Be22:5(Vit.1)金属板，作为金属板厚度的一部分。在这个研究中即时冻结模型的应力预测是8.25 mm厚金属板，以短划线为记号，这些预测的粘弹性模型通过和指出，粘弹性模型计算在图4中以进行图3：表面和中平面应力的绝对值预测通过即时冻结（IFM）和粘弹性VFT模式热回火的8.25 mm厚Vit.1金属板作为（对流）传热系数的一部分。该VFT数据显示在图4中使用和1是两种不同的计算.表1 用于模拟计算 6 的材料参数2.2 粘弹性模型2.2.1介绍从历史上看，这是第二种模式制定量化热回火硅酸盐玻璃9 。粘弹模型在每个温度中采用了流变性能的测定平衡液。因此，它取代了笨拙的从非粘性流体跳到线性弹性固体，前者为即时冻结模式的平稳变换。粘弹性模型适用于热流变简单的一类材料。这样的材料，蠕变函数，松弛模量和任何其他典型粘弹性描绘的函数相对比,时间显示的对数当温度改变时只是一简单的转变后更改为另一个值。平衡液体的硅酸盐玻璃（称为稳定玻璃）被证明是事先应用热流变简单的分析9.对回火问题，适当的粘弹性功能是剪切松弛模量和一定的温度T,记为.通常散装放松比剪切松弛更迟钝,Lee等人让假设弹性散装反应更合理. 因此，就这个问题，粘弹性行为的材料，仅仅在整个温度范围内得到使用两种材料职能. 首先是剪切松弛模量测量参考温度,记为第二个是改变函数,负有的温度依赖性.它在时间算法中将参考函数改变为即放松模量的利益温度.2.2.2. 粘弹性模型的大块金属玻璃作者不知道任何在等温测量剪切松弛模量上不同温度的的确切数据.因此，该模型认为，平衡合金粘度在玻璃化转变和旋转杯实验周围的熔点已被彻底蠕变试验研究 10-12 。粘度数据包涵了一套数量为14的命令，并成功地符合VogelFulcherTammann(VFT)的关系，如下 (2)其中叫做脆弱性参数是VFT动力学的冻结温度=，在最适合的实验粘度生产数据是和.自稳态流粘度在温度范围内的利益用蠕变试验来测量 下面在粘度和松弛模量的关系靠温度来保持 13 （3）从标量粘度数据中确定松弛模量是不可能的。因此，对松弛时间范围作为一个参数研究的分析是必要的。换言之，就要考虑一组具有不同光谱松弛模量对残余应力的计算，这样他们满足方便涉及范围广泛的松弛时间谱，为是松弛模量假定的 (4) 其中是伸展指数，是在温度下的即时剪切模量，对应于Debye 松弛，当下降时光谱拓宽 。如下代入和计算. (5) 在范围内，在量的有组织的递增0.05对值的进行计算首先，对剪切模量的温度依赖是被忽视的，同时它的室温值是被利用的。这是一个由Narayanaswamy3在他对硅酸盐玻璃钢化的分析中提出的假设，后来Kurkjian，s在玻璃剪切模量作为其温度接近玻璃过渡区的数据14中注释中显示15%的增长。研究残余应力对这种变化的剪切模的灵敏度，第二个组计算是假设一个线性变化的剪切模量从在室温转变为在下的。这些为了传热系数进行计算。结果表明，就最后的所关心的残余应力值剪切模量随随温度的变化几乎可以忽略不计（变化小于0.25%）。因此，这一影响在下面的讨论被排除了.粘弹性模型计算用有限元方法计算和细节描述在下一节。从即时冻结和粘弹性模型展示的比较结果在图3 。可以预测残余应力的对 Kohlrausch 因子，考虑的范围（）值是相当敏感的。在中平面张力和表面压缩的变化范围分别低于1.6%和1.3%（当从0.5变化到1，计算的应力增长.注意，在上面参数的研究，依赖于Kohlrausch因子的任何可能的温度都是不加以考虑的。这样做的目的是维护简单热流变学假设。简单热流变学允许的温度变化是无效的。过分复杂的计算阻碍现有的计算方法的使用。此外，在上解具有极其微弱的依赖性; 用它的温度变化调整这个缺点。最后，上述提出的BMG合金热稳定性的所有参数遍及整淬火工艺，相位分离10,11的现象不加以考虑。此外，非线性粘性反应（例如，剪切减薄）材料 16 由于高剪切率也被忽视了.2.2.3.启用大块金属玻璃粘弹模型用ABAQUS有限元软件对粘弹性模型计算。一维模型的建立（见 图4 ），代表一个无限板。原理是：沿板厚方向（Z）性的两个垂直方向X和Y对无限板执行相容性条件，首先，正方向节点受到限制变形为一条线。其次，与正规的平面变形原理不同的是在Y向的位移仅设置为零，一般的平面变形原理是在出平面方向Y许使用均匀形变。沿Z方向长度的原理是有偏颇的，以致网格朝有较高的温度梯度的表面变细。在最表面通过应用对流换热产生了冷却钢板。在ABAQUS中简单热流变学被定义为松弛函数。即在KWW形式中通过瞬时值标准化松弛模量为。这样一来，使用温度依赖性剪切模量使得简单热流变学对松弛模量而不是松弛函数是无效的，不构成任何额外困难。计算松弛函数和在参考温度中输入程序而且转移下面给定的函数都是用户定义的子程序执行的 (6)在表1的材料数据显示对于Vit.1金属板的值（厚=8.25mm）应用在下，用这个模型去获得下面表面缩小和中平面扩张的值：和，这些结果同样在图2显示。另外，在保持金属板的厚度在8.25mm（图3）下，传热系数(h)是不同的。3 试样制备在这里研究金属板是衡量回火引起的残余应力。金属板名义上要长为150mm，宽为100mm，厚为8.25mm。这是在室温下利用大型造铜模铸塑。由于加工的自然专利，这里只讨论与出版物有关的详细资料。通过真空援助在低压下将合金熔体输入铜模。在母模表面，当熔融合金流入到模具时Vit.1的固体皮肤很可能形成，皮肤厚度像预期那样与金属板厚度成反比的，厚度的热质馈入模具，填满模具估计花2-3s的时间。考虑到金属板的有效凝固，进刀压力再保持10s，以致它能在铸模型腔中保留。当这个过程由于BMG高于其 玻璃化转变和模具较低的初始平均温度的热收缩，金属板表面可能从母模表面脱离。最后通过在水中的淬火模具冷却到室温。由于铜热膨胀系数大于Vit.1（ 低于它的玻璃化转变17），模具可能在淬火期间可以压在金属板上。图4。有限元模型中使用的粘弹性模型计算的图解。当金属板半厚度一直伸延到Z时通过对称原理（在X和Y中）描绘无限金属板，在右侧元素（或者节点）要求沿X方向均匀移动。4.用裂纹柔度法测量残余应力在BMGs中, 残余应力通过非破坏性方法是不能便利的确定的. 加工过程中光弹性可用于硅酸盐玻璃衡量残留和剖面的现场应力 1 ，BMGs的不透明度阻止使用这种方法。由于BMGs是无定形的,散射同样不适用. 因此，为了获取应力剖面内所需的空间分辨率,机械松弛方法仍然是唯一的选择，这些方法用特定的方式消除材料，在残余应力下依靠的样本干扰力矩平衡。样本的形变是当它达到一个新的平衡。然后监测并且用这个信息反过来计算残余应力。在这篇文章中选择裂纹遵守方法 19,20 作研究，因为在测量方向上它可以准确地测定具有良好的空间分辨率的完全穿厚侧面图。在此方法中，压力测量的狭缝是逐步穿过一个样本。假设应力松弛切割狭缝是弹性的，从所测的压力中计算原始侧面图的残余应力。图5说明了裂纹柔度测量和定义术语。裂缝的介绍，和它在Z方向的深度是逐步扩大。该测试常常通过样本的厚度来确定。采用两个应变仪。顶端应变仪，放近切口，用来确定近表面区域的应力。反向行程限位器放在与切口相反方向，通过样本的残余部分计算应力.图5。裂缝柔度法术语（从文章19中改编）用级数展开的方法测量张力来确定原来的残余应力 19,21 ，这是在实验应变数据中能容忍的噪声和错误。它首先是假定未知的应力变化为一个函数，穿厚坐标可以表示为一个级数展开。 (7)其中代表未知级数系数，这项研究，因为对于Legendre多项式扩展超过金属板厚度，通过排除第0个和第一个多项式，由此产生的应力分布能保证满足压力和力矩的平衡。在级数中，每一项计算切口深度时测量压力。这叫做屈从函数。利用叠加，通过级数展开的压力可写成： (8)用最小二乘法拟合，尽量减少Eq.(8)式的压力误差，使压力测量屈服于（通过Eq.(7)加强压力）并且记为 （9）在这项研究中，顶端应变仪柔度函数是用数值计算，整体力法解是插槽在半无限固体22中用一些改进的数值解23。后退应变仪的柔度函数的计算是通过在金属板中的二维有限元裂缝计算。在最后的张力预测的不确定性是基于标准误差传播公式应用于上述方程在衡量和计算之间的使用差异（Eq.（ 8 ），当作张力的测量不确定度 24 。在Eq.(7)中n的顺序利用率太低，扩展的结果不适合很好的测量压力，因此，导致在压力中的大量不确定性。相反，一个顺序使用过高的导致更多的不可逆矩阵求逆，Eq.(9)，同样在压力在有更大的不确同样在压力在有更大的不确定性。因此，最理想的适合顺序是选择是装将不确定性减到最小。 切口是使用钢丝电火花加工（EDM）而成的优于机械切割的。将机器调到skim cut，在切割中设置最小感应应力。必须指出的是样本的尺寸精度不是完美的，因为他们是从一大块金属板中提取的。在不同切割平面中样本厚度大多数样本都小于0.1mm切割深度和厚度值在精密量度的基础上用光学显微镜从样本两侧测量.图6,是大概的规模,显示从金属板中提取样本的原来定位件.在Y方向穿厚度应力测试的7个样本是,同样的在X方向测试4个样本是.金属板的一个表面当作顶端被除选择并且除之外的所有样本都被保留,和从检查的一方到对称性应力剖面故意留下狭缝. 退火样样本A1是在290 C下进行2小时解除残余应力。这温度高得足以让快速应力松弛 26 无显著改变Vit.1的结构. 非晶结构的A1热处理后证实与X -射线衍射, 对样品要求是长为25.4mm，宽为12.7mm。样本的长要足够长以便根本上保留在测量方向剖面上的原始残余应力27.同样宽b的尺寸也要足够大以符合平面应变使用计算中的假设 27 。虽然早些时候的测试是采用0.006英寸(0.150mm)直复径线,在后来的测试使用0.004英寸（ 0.100mm）直径线(见图6关于每个样本的分布). 插槽被切入误差是在0.254mm. 在前几个步骤这个值减少到0.127mm，以获得在附近的上表面更高的分辨率图6.金属板被切割之前的定位件.金属板尺寸规格是:150mm100mm8.25mm. 在每个样品中切割的钢丝直径注明英寸,在铸造期间BMG熔化的流动方向也显示.样本是12.7mm25.4mm.通过X指定的方向被用于确定沿X方向的内平面应力,其它的命名为等.用于测量.5.实验结果在一些早期的测试, 顶端指标没有取得可靠的数据，这是由于穷尽依附和涂料的估计阻止其的防水安装。电火花加工机用于这些试验采用了水射流对保持线工件地区在介质流体中吞没。幸运的是，来自回衡量大部分数据（从10 至90 的厚度样本）和最高指标不是绝对必要的。因此，几乎完整的应力剖面，可只用回衡量数据。缺点是在这种情况下在近表面压力的精确度降低和顶端计量器的稳定作用下解丢失。然而，不确定性分析正确解决这些问题。在这项研究中,可以准确的看到中平面应力仍然十分好并且应力剖面的扩展也得到了充分的解决另一方面,从数据的估计和从前四个样本中获得结果类似于所有其他铸标本说明样本和.的测试是成功的。从所有样本中的后应变剖面力距在图7中显示了. 值得考虑的是所有样本的剖面,在X和Y方向的测试(除应力解除格A1), 即使其中的一些测试, 表现显着相似的形状和规模, 例如和 取得一些低质量的数据. 这表明,首先, 该应力剖面具有弱的坐标(空间)的信赖,同时也表明这种应力状态大约是等双轴的.注意,如果样本的应力剖面仅仅是相互之间的倍数才考虑,应变边距和标准化数据相比将成用应力剖面幅度的直线规格. 另一方面，最高估计数据无法直接比较, 其价值会取决于插槽估计的距离。典型的应力剖面图从图8(a)所示样本的回衡量数据中取得. 这些试验产生了特别清晰的数据和直截了当的缩小应力剖面图.在图8(b), 的测试和退火样本A1相比较. 自从在测试,最上表面应力的顶端估计工作也被呈现了.在图8的误差被认为远小于观察应力剖面.更重要的是,目前在退火样本(A1)中的应力变化表明在内裂纹柔度法的应力解。 注意到仅仅应力值被重要的描绘。这包括不包括若干在上表面附近点，其中回估计反应太弱（顶端估计的测试并不能正常工作）和一些点关闭整个样本厚度使得应力不再提供良好的残余应力估计。当残余的韧带（在图5中的）变小，几个因素（例如，样本的重量和在引入线的张力）通过回应变仪测量应力造成残余韧带的弯曲的扭力的贡献越来越大。在小样本这类从估计涂层中附加的刚度当残余韧带变小时也可以影响应变指数。残余韧带变小的净影响是当切口接近背割面实验测量的应力变得奇异。见图7，而当反表面在切口附近没有这些影响的接近应变有限值（表面应力除以平面应变弹性模量）。在这些样本中，在之后奇数影响变得重要；因些，应力超过它的厚度是不记录的。图7 后应力与样本（a）标准化厚度数据在Y方面，（b）在X方向的与退火样本A1比较。见图6在金属板上的原始样本位置。图8 （a）仅仅从边距压力数据计算应力剖面图与标准化厚度数据的比较。（b）从顶端的后测量仪器获得应力剖面图。正如图7所见,特别是样本和不遵循一般的光滑剖面. 他们的数据显示尴尬变化斜率和一些特别糟糕的数据点被删除。后来人们认识到，变化的原因是流动的电介质，并估计表面形成泡沫时作为EDM线进入样本。这仅仅发生在几个样本中,因为蜡涂层压力表的样本显示的是粗糙的表面,这就造成复杂的流态.如在图8的一些样本通常是非常稳定的.当从未定的应变数据中转化应力应当小心谨慎,因为拟合的噪音将导致波形剖面图,而不是简单的表面缩小,中平面扩展分配.下一个误差原因适用于所有的样本,叫做”EDM影响”. 在EDM切割，一层薄薄的材料（有一个新的应力状态）可以改写切割表面, 这尤其影响到顶端估计数据，而边距相对不敏感.这些数据来自测试的退火样本，通过边距非常小的压力测量和大小不一的大测试值从顶部的估计，确认EDM影响的可能性。底部的一致性测试估计结果都取得了0.004和0.006 英寸,直径电线也证实说距数据没有受到影响。在25,一个详细的EDM校正被用于顶端压力估计, 以获得应力分布图,如图8(b)所示.剖面图的应变是一个切割厚度的函数,这是由EDM影响引起的,是由残余应力引起的完全不同的剖面图. 它可以分析的估计和从应变测试中分离出来.在这些测试中几个因素结合起来使得EDM影响成一个争论点,但它通常是微不足道的 25 。主要的因素是残余应力在测试样本中是很低的.当实施轻微切割时,由应变引起的EDM影响通常是独立于的残余应力规模的一个固定小值, 因此,它们通常包括一些测试应变的微不足道的部分,并且没有明显的结果影响.当应力非常低时,在记录方面,EDM应变能在测量数据方面做出重大贡献 ,因此,影响结果. 其他EDM影响因素是，其他参数相同, 更大的材料具有较高的屈服应力和低导热系数,这两个条件被BMG样本采用. 即便如此, 如果有较高的残余应力，EDM的影响将是微不足道的。6.讨论图2和3呈现的是即时冻结和粘弹性模型两个预测。尽管方法大不相同和每一个使用独立的数据，他们的相似性还是显著的。比如，应力的形状和图3的传热系数几乎是相同的。但该模型的预测明显不同于实验结果。最初假定的对流冷却传热系数是，从自由牵引表面遍及整个BMGs的生产过程，显示一个合理的估计6。因为猜测Vit.1的对流冷却从熔体到室温持续一段时间，大约接近冷却期的实际铸造工艺（ 10s）。所以选择了这个值。正如图2，即时冻结模型预测金属板的厚度是8.25mm，表面缩小是-217MPa，中平面扩展是+85MPa，同时使用粘弹性模型，同样应力在表面上的估计是-230MPa，内部是+90MPa。注意，尤其是表面压缩值高度依赖于h，而，实验结果在中平面扩展是+10到+13MPa，表面压缩是-25到-30MPa，这和预测模型相比都是非常小的。这种差异可能是因为界定热回火问题的两个成分：试样制备工艺和材料特性。上述论点目前的想法是：虽然实际的铸造工艺并没有类似在热回火情况下考虑模具，大多数残余应力仍然由于热回火。然而，尽管压力产生的原理是一样的，但热的问题并不是跟这个原理一样。换言之，在模型中的温度曲线图进化计算和在实际样本中建立的可能不同。对流冷却问题所考虑的生产指数下降，余弦高阶项迅速衰变的计算 6 。实际淬火已实现通过最初在室温下传导到大型铜块是更复杂的。了解新一代热回火残余应力驱逐热梯度的温度变化是至关重要的，或更简单来说，表面和中期平面的温度差距。低程度压力测量指出了在铸造Vit.1关于玻璃转变过程相对平稳的温度曲线。这样的温度分布有两种可能的来源，第一，金属板和模具表面的接触过程中可能损耗一些时间，这和可能保持在相对较低的温度的大铜模相比将造成较大的热收缩（尤其是上面的玻璃化转变）。在这之后的分离，传热从板到真空模腔将大大减少。第二，更加明显，在金属板中，平坦温度分布曲线的温度提高的一种可能是来源于铜块。这将导致妨碍传热，特别是在低温条件下。这些结果指出，实验仪器铸造过程中有必要的知道确切BMG的温度分布可在原位监测，这项工作目前正在进行中。即时冻结模型和粘弹性模型这两种模型忽视玻璃行为的一个的重要组成部分：其在热力学平衡中从稳非晶态固体到过冷液体的结构松弛 10,11 。在此过程中，过多的自由体积通过短程原子运动导致的密度，粘性固体被困在松懈材料中. 粘弹性理论，平衡物质的数据粘弹行为在每个温度被采用. 这意味着，当达到力平衡时,结构松弛已经完成，因此，仅仅从粘度的角度来看，这项研究中粘弹性模型的采用超过更先进的结构模型回火应力的预期估计，这可以在结构松弛期间准确地考虑暂态效应。不幸的是，在BMGs中没有存在可靠的数据应用于结构模型，并获得一个更实际的估计回火引起的残余应力。目前在加州理工学院的研究涉及这里具体BMGs研究的这些数据的产生。最后，值得一提的事实是，在BMGs中缺乏高回火应力不一定是不利的。常规（结晶）金属通常有残余应力的大约是25-50 的屈变力 27,30 。这种应力有助于断裂故障，软化，应力腐蚀分解，和其他应力驱动程序 31 。这些压力也是一个昂贵的变形加工过程到最后确定中造成的问题的重要来源32，在常规金属中通过热处理或冷加工 18,33 消除应力通常是昂贵的，而且往往不可能的，因为它可以大幅度降低机械性能。相比之下，在本研究中测量BMGs残余应力峰值只相当于约1.5 的屈变力。从失真和机械故障的角度来看，如此低的压力可以有效地被忽视。7.结论这项热回火引起的残余应力在一个厚BMG金属板中研究的有两个前提：（ i ）在BMG金属板中样本精确地测量残余应力，（ ii ）用理论对结果进行适当的比较，并获得处理方法和材料性能的洞察力。首先是通过使用裂纹柔度方法可以完成，它是能够屈从穿厚应力分布的。该方法适用于在公平电导率BMGs有优势的 EDM线切割。虽然测量应力程度比较低，但是产生了良好的技术解决。即时冻结模型和粘弹性模型在这项研究中都被采用了估计过高的残余应力。这种差异是由于缺乏关于细节处理条件的信息（例如，在表面传热系数的确切值） ，以及缺乏适当关于BMG合金的法定的结构关系。BMGs的粘弹性和结构性能都必须考虑到一个热回火更准确的描述。在明确界定的条件和材料试验下，铸造的仪表化到产生粘弹性的BMG数据适用于更先进的热回火模型所必需的是全面了解这一BMGs现象。谢词这项研究得到美国加洲理工学院的结构非晶态金属中心的支持（陆军研究办公室授予号.DAAD19 - 01 - 0525 ）参考文献1 R.Gardon,in:D.R.Uhlmann,N.J.Kreidl(Eds.),Glass Science and Technology,Elasticity and Strength in Glasses,vol.5,Academic 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Forum 347349(2001)235.17Section 4. Bulk metallic glassesModeling and measurement of residual stressesin a bulk metallic glass plateC. Can Aydinera, ErsanUUst u undaga,*, Michael B. Primeb, Atakan PekercaKeck Lab., Department of Materials Science, California Institute of Technology, 1200 E. California Boulevard, M/C 138-78,Pasadena, CA 91125, USAbEngineering Sciences and Applications Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USAcLiquidmetal Technologies, Lake Forest, CA 92630, USAAbstractThe recent advent of multi-component alloys with exceptional glass forming ability has allowed the processing oflarge metallic specimens with amorphous structure. The possibility of formation of thermal tempering stresses duringthe processing of these bulk metallic glass (BMG) specimens was investigated using two models: (i) instant freezingmodel, and (ii) viscoelastic model. The first one assumed a sudden transition between liquid and elastic solid at the glasstransition temperature. The second model considered the equilibrium viscosity of BMG. Both models yielded similarresults although from vastly different approaches. It was shown that convective cooling of Zr41:2Ti13:8Cu12:5Ni10Be22:5plates with high heat transfer coefficients could potentially generate significant compressive stresses on the surfacesbalanced with mid-plane tension. The crack compliance (slitting) method was then employed to measure the stressprofiles in a BMG plate that was cast in a copper mold. These profiles were roughly parabolic suggesting that thermaltempering was indeed the dominant residual stress generation mechanism. However, the magnitude of the measuredstresses (with peak values of only about 1.5% of the yield strength) was significantly lower than the modeling pre-dictions. Possible reasons for this discrepancy are described in relation to the actual casting process and materialproperties. The extremely low residual stresses measured in these BMG specimens, combined with their high strengthand toughness, serve to further increase the advantages of BMGs over their crystalline metal counterparts.? 2003 Elsevier Science B.V. All rights reserved.PACS: 61.43.Dq; 81.70.Bt; 83.60.Bc; 62.20.)x1. IntroductionMulti-component metallic alloys with superbglass formation ability have recently been devel-oped allowing, for the first time, the processing oflarge specimens with amorphous structure. Re-ferred to as bulk metallic glasses (BMGs), thesematerials have been shown to have impressiveproperties such as very high elastic strain limit(2%) and yield strength (?2 GPa), good fracturetoughness (up to 55 MPam1=2), excellent corrosionresistance, etc. An important question that ariseswith bulk production is the nature and magnitude*Corresponding author. Tel.: +1-626 395 2329; fax: +1-626395 3933.E-mail address: ersan (E.U Ust u undag).0022-3093/03/$ - see front matter ? 2003 Elsevier Science B.V. All rights reserved.PII: S0022-3093(02)01940-3Journal of Non-Crystalline Solids 316 (2003) 8295/locate/jnoncrysolof processing-induced residual stresses. The BMGprocessing typically involves casting an alloy into amold followed by severe quenching. This proce-dure can lead to large thermal gradients due to thelow thermal conductivity of BMG. In addition,during glass transition the alloy exhibits largechanges in its viscosity within a small temperaturerange. All of these parameters can lead to ?thermaltempering? which generates compressive surfaceresidual stresses balanced with mid-plane tension.A similar phenomenon was observed previouslyin silicate glasses 1. Thermal tempering in theseglasses was mostly studied for a certain initialboundary value problem: an infinite plate at auniform initial temperature convectively cooledfrom both sides. The resulting stress profile wasroughly parabolic where compression on the sur-face(s) was balanced by tension in the interior.First theories made use of the instant freezing as-sumption (see, e.g. 2) that presumed the materialbehaves as a non-viscous fluid above its glasstransition temperature and as a linear elastic solidbelow it. This simplistic approach required onlythe glass transition temperature as the rheologicalinput and ignored the details of the glass transitionrange and its cooling rate dependence as well asany stress relaxation below that range. The theoryeventually evolved with the necessary materialfunctions (e.g., relaxation and structural moduli)3 such that any geometry could be handled withfinite element modeling 4,5.Unfortunately, most viscoelastic properties ofBMGs, especially the ones needed for advancedmodeling of thermal tempering, have not yet beeninvestigated systematically. For this reason, theinstant freezing assumption was invoked first inthe present study to obtain estimates of the resid-ual stresses generated by this phenomenon. Next,the only available viscoelastic data on the BMGalloy used here, its equilibrium viscosity as afunction of temperature, were employed to de-velop the first viscoelastic model of thermal tem-pering in BMGs. The model predictions were thencompared to residual stress data collected from acast BMG plate using the crack compliancemethod. To the authors? knowledge, this is the firstextensive theoretical/experimental investigation ofthermal tempering in BMGs.2. Modeling of thermal tempering2.1. Instant freezing modelThermal tempering is a result of a processwhich involves heating glass above its glass tran-sition temperature (Tg) and then rapidly cooling it.During cooling, the surfaces of a glass plate so-lidify and contract first, leading to tension on thesurface and compression in the mid-plane. Thesame would occur in an elastic solid. However, inthis phase of quenching a glass, the core is still oflowviscosity andthemid-planecompressivestresses are simultaneously relaxed by viscous flow.Therefore, stresses developed before the entirecross-section solidifies are lower than those in anotherwise identical but elastic material with thesame temperature distribution. However, when themid-plane finally solidifies, the viscous effects ceaseand the material becomes practically an elasticglass plate, imposed with the temperature profileat that instant. All further changes in stresses aredetermined by changes in the temperature profileonly. As the temperature gradients decay to aconstant value at room temperature, stresses in theopposite sense are produced: tension in the mid-plane and compression on the surface. For analways elastic solid these would be equal and op-posite to those produced in the first phase whentemperature gradients grew. In a glass, which in-volves stress relaxation at the beginning, the twotypes of stresses are not equal. In other words, thestresses produced during the decay of temperaturegradients have an excess over the ones producedinitially. This leads to residual tension in the mid-plane and compression on the surface.The resulting residual stress profile as a functionof distance from the center of the plate is given bythe instant freezing model as follows 6:rZ aETg? Ta1 ? m1?sindd?ZZ=L0sind ? sindfsindf1 ? fcos2dfdf?;1where a is the coefficient of linear thermal expan-sion, E is the Young?s modulus, m is the Poisson?sratio, Tgis the glass transition temperature, TaisC. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 829583the ambient temperature and d is a measure of thecooling rate. More explicitly d is given as the firstroot of dtand Bi where Bi hL=k is the Biotnumber with h, L, k denoting heat transfer coeffi-cient, half-thickness of the plate and thermalconductivity, respectively. Fig. 1 shows the in-planeresidual stress profile acrosstheplatethickness. In-plane coordinates are X and Y whileZ is the out-of-plane coordinate. Thermal-tem-pering-induced residual stresses are equibiaxial inthe XY plane for this infinite plate problem.Presently, the exact values of experimental heattransfer coefficients typically found in BMG pro-cessing are unavailable. To obtain a rough esti-mate, a recent processing method called meltinfiltration casting 7 is noted. In one version ofthis technique, BMG alloys are cast in stainlesssteel tubes and then quenched in water. Thismeans cooling is via convection by water whichflows in either laminar or turbulent manner. Theheat transfer coefficient for free convection ofwater (i.e., stagnant water) at room temperature isabout 900 W/(m2K) 8. When water is forced toflow laminarly, this value can easily reach 20003500 W/(m2K). If water boils (as is sometimesobserved during melt infiltration processing), thenvalues up to 35000 W/(m2K) are possible. To beconservative, h 2000 W/(m2K) was chosen. Thisvalue for h also yields the same time scale (?10 s) ittakes to cool a BMG plate from melt to roomtemperature as described in Section 3 when thecasting procedure used in this study is mentioned.The predictions of Eq. (1) for surface compressionand mid-plane tension as a function of platethickness (t) and heat transfer coefficient (h) areshown in Figs. 2 and 3, respectively. The materialparameters employed in the calculation are listedin Table 1.The instant freezing model predictions suggestsignificant residual stresses can be generated inBMGs due to the thermal tempering effect. An-other prediction of the instant freezing model isthat as the value of surface compression increases,the thickness of the surface layer it is confined todecreases since the balancing mid-plane tensionsaturates (see Fig. 2). This means, a progressivelythinner surface layer with an increasingly highstress gradient can be formed as the degree oftemper increases.ZX (Z)0 SmLLtFig. 1. A typical residual stress profile across the thickness of alarge plate due to thermal tempering: surface compression (rs)is balanced with mid-plane tension (rm). The in-plane stressesare equibiaxial and function of the thickness coordinate (Z)only.01002003004005006000510152025303540Plate Thickness, t (mm)Stress MPasurface compressionmid-plane tensionFig. 2. Equibiaxial (absolute) stress values at the surface andmid-plane of a large Zr41:2Ti13:8Cu12:5Ni10Be22:5(Vit.1) plate forh 2000 W/(m2K) as a function of plate thickness 6. Thestresses predicted by the instant freezing model for the 8.25 mmthick plate used in this study are marked with dashed lines whilethose predicted by the viscoelastic model are indicated by (?)and (?). The viscoelastic model calculations were performed forb 1 in Eq. (4).84C. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 82952.2. Viscoelastic model2.2.1. IntroductionHistorically, this was the second model devel-oped to quantify the thermal tempering of silicateglasses 9. The viscoelastic model employed themeasured rheological properties of the equilibriumliquid at each temperature. Hence, it replaced theawkward jump from a non-viscous fluid to a linearelastic solid in the former instant freezing modelswith a smooth transition. The viscoelastic modelwas applied to a class of materials with thermo-rheological simplicity. For such a material, thecreep function, relaxation modulus and any otherTable 1Material parameters used in the modeling calculations 6Thermal conductivityk4 W/(mK)Densityq5.9 g/cm3Linear thermal expansion coefficienta10:1 ? 10?6K?1Specific heatCp666 J/(kgK) at 25 ?C850 J/(kgK) at 900 ?CGlass transition temperatureTg352 ?CInitial (melt) temperatureTi900 ?CQuenching temperatureTa25 ?CYoung?s modulusE90 GPaPoisson?s ratiom0.354Shear modulus (at room temperature)lRT33.2 GPa0501001502002500400800120016002000Heat Transfer Coefficient, h W/(m2 K)Stress MPaIFM (surface)IFM (mid-plane)VFT (surface, beta=0.5)VFT (mid-plane, beta=0.5)VFT (surface, beta=1)VFT (mid-plane, beta=1)Fig. 3. Absolute values of surface and mid-plane stresses predicted by both instant freezing (IFM) and viscoelastic VFT models ofthermal tempering of an 8.25 mm thick Vit.1 plate as a function of (convection) heat transfer coefficient. The VFT data are shown fortwo different calculations using b 0:5 and 1 in Eq. (4).C. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 829585characteristic viscoelastic function plotted vs. thelogarithm of time exhibit only a simple shift whenthe temperature is changed to another constantvalue. Equilibrium liquids of silicate glasses (calledstabilized glass) were shown to be thermorhe-ologically simple prior to the analysis in 9.For the tempering problem, the appropriateviscoelastic function is the shear relaxation mod-ulus and at a certain temperature T, it will be de-noted by GTt. Bulk relaxation is typically muchmore sluggish than shear relaxation and Lee et al.made the reasonable assumption of elastic bulkresponse. Hence in this problem, the viscoelasticbehavior of the material over the entire tempera-ture range is obtained using two material functionsonly. The first is the shear relaxation modulusmeasured at a reference temperature, TR, denotedby GTRt and the second is the ?shift function?,aT, which bears the temperature dependence. Itshifts the reference function GTRt in logarithmof time to obtain GTt, namely, the relaxationmodulus at the temperature of interest.2.2.2. Viscoelastic model for bulk metallic glassThe authors are not aware of any definitive dataon isothermal measurements of shear relaxationmoduli for Zr41:2Ti13:8Cu12:5Ni10Be22:5(Vit.1e) atvarious temperatures. Therefore, this model con-siders the equilibrium viscosity of this alloy whichhas been thoroughly studied with creep testsaround the glass transition and rotating cup ex-periments about the melting point 1012. Theviscosity data covers a range of 14 orders ofmagnitude and is successfully fit with the VogelFulcherTammann (VFT) relation as follows:gT g0expD?T0T ? T0?;2where D?is called the fragility parameter and T0isthe VFT kinetic freezing temperature. The value ofg0is 4 ? 10?5Pas and the best fits to experimentalviscosity data yield D? 18:5 and T0 412:5 K10.Since the steady-flow viscosities were measuredwith creep tests in the temperature range of inter-est, the following relation between viscosity andrelaxation modulus at temperature T holds 13:gT Z10GTtdt:3The determination of the relaxation modulus fromthe scalar viscosity data is not possible. There-fore, the analysis was pursued as a parametricstudy for the relaxation time spectrum. In otherwords, a set of relaxation moduli with varyingspectra were considered for residual stress calcu-lations such that they satisfied Eq. (3). To conve-niently cover a broad range of relaxation spectra,the KohlrauschWilliamsWatts (KWW) formwas assumed for the relaxation modulus:GTt lTexp?t=sTb?;4where b is the stretching exponent and lT is theinstantaneous shear modulus at T. b 1 corre-sponds to Debye relaxation and the spectrumbroadens as b decreases. sT is calculated bysubstituting Eq. (4) in Eq. (3) as follows:sT gTlT1C1 1=b:5Calculations were carried out for b values thatsystematically varied in increments of 0.05 in the0:56b61 range. First, the temperature depen-dence of the shear modulus was neglected and itsroom temperature value (lRT 33:2 GPa) wasused. This was one of the assumptions made byNarayanaswamy 3 in his analysis of silicate glasstempering after noting Kurkjian?s data 14 thatshowed ?15% decrease in the shear modulus ofglass as its temperature approached the glasstransition region. Recently, a similar decrease (1020%) in the elastic modulus of some metallicglasses has been measured as the temperaturereached Tg15. To investigate the sensitivity of theresidual stresses to such changes in the shearmodulus, a second set of computations was madeby assuming a linear variation of the shear mod-ulus from lRT 33:2 GPa at room temperature to0:85lRT( 28:2 GPa) at Tg. These calculationswere performed for a heat transfer coefficient ofh 2000 W/(m2K). The results showed that theeffect of shear modulus variation with temperaturewas negligible as far as the final residual stressvalues were concerned (the change was less than86C. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 82950.25%). Therefore, this effect is excluded from thediscussion that follows.The viscoelastic model calculations were per-formed using the finite element method and thedetails are described in the next section. Thecomparison of results obtained from both the in-stant freezing and viscoelastic models is exhibitedin Fig. 3. It is seen that the predicted residualstresses are rather insensitive to the value of theKohlrausch factor, b, for the range of b valuesconsidered (0:56b61). The variation in mid-plane tension and surface compression with b inthis range is under 1.6% and 1.3%, respectively(the calculated stresses increase when b changesfrom 0.5 to 1).Note that any possible temperature dependenceof the Kohlrausch factor b was not considered inthe above parametric study. This was done topreservethethermorheologicalsimplicityas-sumption. Allowing for the temperature variationof b invalidates thermorheological simplicity andexcessively complicates the calculations impedingthe use of the current numerical method. Fur-thermore, the extremely weak dependence of thesolution on b justifies this negligence of its varia-tion with temperature.Finally, all the arguments presented above as-sume the thermal stability of the BMG alloythroughout the quenching process, and phenom-ena such as phase separation 10,11 were notconsidered. In addition, the non-linear viscousresponse (e.g., shear thinning) of the material 16due to high shear rates was also ignored.2.2.3. Implementation of the viscoelastic model forbulk metallic glassThe calculations for the viscoelastic model wereconducted using the ABAQUS finite elementsoftware. A one-dimensional model was built (seeFig. 4) to represent an infinite plate. The elementswere arranged along the plate thickness direction(Z) with symmetry imposed on both perpendiculardirections (X and Y). To implement the compati-bility condition for the infinite plate, first, the rightside nodes were constrained to deform as a line.Second, unlike regular plane strain elements wherethe displacements in the Y-direction would merelybe set to zero, generalized plane strain elementswere used that allowed uniform deformation in theout-of-plane direction Y. The lengths of the ele-ments along the Z-direction were biased such thatthe mesh became finer towards the surface wherethere are higher temperature gradients. The cool-ing of the plate was carried out via convection heattransfer applied on the top surface. Thermorhe-ological simplicity in ABAQUS was defined for therelaxation function, namely, relaxation modulusnormalized by its instantaneous value was given byexp?t=sbin the KWW formalism. This way, theuse of temperature-dependent shear modulus lT,which invalidates thermorheological simplicity forFig. 4. Schematic of the finite element model used in visco-elastic model calculations. An infinite plate is represented viasymmetric elements (in X and Y) while the plate half-thicknessextends along Z. The elements (or nodes) on the right-hand sideare required to move uniformly along the X-direction.C. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 829587relaxation modulus but not relaxation function,did not pose any additional difficulties. The re-laxation function was calculated and input to theprogram at a reference temperature TRand theshift function given below was implemented with auser-defined subroutine:aT sTsTRlTRlTexp D?T01TR? T0?1T ? T0?:6The material data shown in Table 1 were appliedfor the Vit.1 plate of interest (thickness 8:25mm) using this model to obtain the followingvalues for surface compression and mid-planetension at h 2000 W/(m2K): rs? ?230 MPaand rm? 90 MPa. These results are also shownin Fig. 2. In addition, the heat transfer coefficient(h) was varied while keeping the plate thicknessfixed at 8.25 mm (Fig. 3).3. Specimen preparationA Zr41:2Ti13:8Cu12:5Ni10Be22:5(Vit.1e) plate wasused in this study to measure the tempering-induced residual stresses. This plate was nominally150 mm long, 100 mm wide and 8.25 mm thick. Itwas cast using a large copper mold that was ini-tially at room temperature. Due to the proprietarynature of the process, only details relevant to thispublication are discussed here. The alloy melt wasfed into the copper mold at low pressure by vac-uum assistance. A solid skin of Vit.1 likely formedat the cavity surfaces as the molten alloy flowedinto the mold. The skin thickness was expected tobe inversely proportional to the thickness of theplate, i.e., the thickness of the thermal mass fedinto the mold. Filling up the mold was estimatedto take 23 s. The feeding pressure was kept foranother 10 s to allow sufficient solidification of theplate such that it could be retained in the moldcavity. The plate surfaces possibly separated fromthe cavity surfaces during this process due to thelarger thermal contraction of the BMG above itsglass transition (aVit:1 17:7 ? 10?6K?117) andthe low initial average temperature of the mold.Final cooling to room temperature was achievedby quenching the mold in water. Since the coeffi-cient of thermal expansion of copper (aCu16:5 ? 10?6K?118) is larger than that of Vit.1(aVit:1ffi10 ? 10?6K?1below its glass transition17), the mold likely pressed on the plate duringquenching.4. Residual stress measurement using crack compli-ance methodResidual stresses in BMGs cannot be conve-niently determined by non-destructive methods.While photoelasticity could be used in silicateglasses to measure both residual and in situ stressprofiles during processing 1, the opacity ofBMGs prevents the use of this method. Diffractionis similarly not applicable since BMGs are amor-phous. Therefore, mechanical relaxation methodsremain as the only options to obtain the stressprofile within the desired spatial resolution. Thesemethods rely on disturbing the equilibrium of asample under residual stress by removing materialin a particular way. The deformation of the sampleas it reaches a new equilibrium is then monitoredand this information is used to back-calculate theoriginal residual stresses. The crack compliancemethod 19,20 was chosen in this study since itallows accurate determination of the completethrough-thickness profile with good spatial reso-lution in the measurement direction.In this method, strains are measured as a slit isincrementally cut through a specimen. Assumingthat the stress relaxation from cutting the slit iselastic, the original profile of residual stress iscalculated from the measured strains. Fig. 5 illus-trates a crack compliance measurement and de-fines the terminology. A slit is introduced, and itsdepth a in the Z-direction is extended incremen-tally. The test is used to determine rXZ throughthe thickness of the specimen. Two strain gaugesare employed. The top strain gauge, placed verynear the cut, is used for determining the stresses inthe near surface region (a=t 0:05). The backgauge is placed directly opposite to the cut and isused to calculate the stresses through the remain-ing portion of the specimen.The original residual stresses are determinedfrom the measured strains using the series expan-88C. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 8295sion approach 19,21, which is very tolerant ofnoise and errors in the experimental strain data. Itis first assumed that the unknown stress variationas a function of the through-thickness coordinatecan be expressed as a series expansion:rxZ Xni1AiPiZ P?fAg;7where the Airepresent unknown series coefficients.For this study, Legendre polynomials expandedover the thickness of the plates were chosen for thePibecause, by excluding the 0th and 1st orderpolynomials, the resulting stress distribution isguaranteed to satisfy force and moment equilib-rium. The strains that would be measured at thecut depths ajare calculated for each term in theseries. These are called the compliance functionsCij. Using superposition, the strains given by theseries expansion can be written asexaj Xni1AiCaj;Pi C?fAg:8A least squares fit to minimize the error betweenthe strains given by Eq. (8) and the measuredstrains yields the Ai(and hence the stresses by Eq.(7) and can be written asfAg C?TC?1C?Tfemeasuredg:9In this study, the compliance functions for thetop strain gauge were calculated using a numerical,body-force method solution for a slot in a semi-infinite solid 22 with some improvements made tothe numerical solution 23. The compliance func-tions for the back strain gauge were calculated bytwo-dimensional finite element calculations of acrack in a plate. Uncertainties in the final stressprediction were based on standard error propa-gation formulas applied to the above equationsusing the difference between the measured andcalculated (Eq. (8) strains as the measurementuncertainty 24. Using too low an order, n in Eq.(7), of an expansion results in not fitting themeasured strains well and, therefore, leads to largeuncertainties in the stresses. Inversely, using toohigh of an order results in a more singular matrixinverse, Eq. (9), and also large uncertainties in thestresses. Thus, an optimal fit order was chosen tominimize uncertainty.The cuts were made using wire electric dis-charge machining (EDM), which is preferred overmechanical cutting. The machine was set to ?skimcut? settings to minimize stresses induced duringcutting 25. It must be mentioned that the di-mensional accuracy of the samples was not perfectsince they were extracted from a cast plate. Thethickness of the sample in the cut plane variedwithin 0.1 mm for most samples. The cut depthand thickness values were based on accuratemeasurements performed with an optical micro-scope from both sides of the sample.Fig. 6, which is roughly to scale, shows theoriginal locations of samples extracted from theplate. The seven samples tested for through-thickness stress profiles in the Y-direction werenamed Y1;.;Y7, and similarly the four samplestested in the X-direction were called X1;.;X4.One face of the plate was selected as the top andwas kept as such for all samples except Y7, X2 andX4 that were intentionally slit from the other sideto check the symmetry of the stress profile. SampleA1 was annealed at 290 ?C for 2 h to relieve theresidual stresses. This temperature is high enoughto allow rapid stress relaxation 26 without sig-nificantly changing the structure of Vit.1. Theamorphous structure of A1 after heat treatmentwas confirmed with X-ray diffraction. The sampleswere nominally 25.4 mm long by 12.7 mm wide.The sample length, l, was large enough (l=t ? 3) tosubstantially preserve the original residual stressesin the measurement direction on the cut plane 27.Also the width dimension, b, was large enough tozyslot width, wdepth, astrain gaugeback facestrain gaugethickness, txspecimen width, bspecimen length, lFig. 5. Crack compliance method terminology (adapted from19).C. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 829589conform to the plane strain assumption used in thecalculations 27. While the earlier tests employed a0.006 in. (0.150 mm) diameter wire, the later onesused a 0.004 in. (0.100 mm) diameter wire (see Fig.6 about a distribution of samples for each). Theslot was cut in 0.254 mm increments. This valuewas reduced to 0.127 mm during the first few stepsto obtain a higher resolution near the top surface.5. Experimental resultsThe top gauge did not yield reliable data insome early tests. This was attributed to the poorattachment and coating of the gauge which pre-vented its waterproof installation. The EDM ma-chine used in these tests employed a water jet tokeep the wire-workpiece region engulfed in a di-electric fluid. Fortunately, most of the data comefrom the back gauge (from 10% to 90% of thethickness of the sample) and a top gauge is notstrictly necessary. Therefore, almost the completestress profile can be obtained using only backgauge data. The disadvantages in this case are thatthe precision in the near-surface stresses is low andthe stabilizing effect of the top gauge on the solu-tion is lost. Nevertheless, the uncertainty analysiscorrectly accounts for such issues. In this study, itwas seen that the accuracy of the mid-planestresses was still quite good and the trend of thestress profile was adequately resolved. On theother hand, the tests for samples X1, X2, X4, Y7and A1 were successful in yielding data from bothgauges and the results obtained from the first fourof these samples are similar to those obtained fromall other as-cast specimens.The back gauge strain profiles from all speci-mens are shown in Fig. 7. It is worth noting thatthe profiles for all samples, tested both in X- andY-directions (except the stress-relieved A1), ex-hibited remarkable similarity in shape and mag-nitude even though a few of the tests, e.g., that ofY6 and X4, yielded somewhat lower quality data.This shows, first, that the stress profile has weakXY coordinate (spatial) dependence and it alsosuggests that the stress state is approximatelyequibiaxial. Note that if samples with stress pro-files that are merely multiples of each other areconsidered, back gauge strain vs. normalized depthdata will scale linearly with the amplitude of thesestress profiles. On the other hand, top gauge dataare not directly comparable for their values willalso depend on the gauge-to-slot distance.Typical stress profiles obtained from the backgauge data of specimens Y2, Y3, Y4 are shown inFig. 8(a). These tests yielded particularly cleandata and reduction of the stress profile wasstraightforward. In Fig. 8(b), the test of Y7 iscompared with that of the annealed sample, A1.Since the top gauge worked in both tests, thestresses on the top surface are also exhibited. Theerror bars in Fig. 8 are seen to be much smallerthan the observed stress profile. More significantly,the apparent stress variation in the annealedspecimen (A1) is within 5 MPa indicating the stressresolution of the crack compliance method.Note that only stress values that are significantare plotted. This involves excluding a number ofpoints near the top surface where the back gaugeresponse is too weak (for the tests where the topgauge did not function properly) and a few pointsclose the full thickness of the sample where theFig. 6. Locations of samples in the plate before cutting. Platedimensions are: 150 mm by 100 mm by 8.25 mm. Wire dia-meters used in cutting each sample are indicated in inches. Theflow direction of the molten BMG during casting is also shown.The samples are 12.7 mm by 25.4 mm. The ones designated byX were used to determine in-plane stresses along the X-direction(rX), while those named Y1, etc. were used to measure rY.90C. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 8295strains no longer give a good estimate of the re-sidual stress. As the remaining ligament (ta inFig. 5) becomes small, several factors (e.g., theweight of the specimen and tension in the leadwires) increasingly contribute to the strain mea-sured by the back strain gauge by causing bendingand torsion of the remaining ligament. In smallsamples such as these, additional stiffness from thegauge coating can also affect the strain readings asthe remaining ligament becomes small. The neteffect of the remaining ligament becoming small isthat the experimentally measured strains become(a)010020030040050060000.81a/tBack gauge strain (x10-6)Y1Y2Y3Y4Y5Y6Y7A1(b)010020030040050060000.81a/tBack gauge strain (x10-6)X1X2X3X4A1Fig. 7. Back strain vs. normalized depth data for samples (a) in the Y-direction, and (b) in the X-direction, in comparison to theannealed sample A1. See Fig. 6 for original specimen locations on the plate.(a)-30-20-100102000.81a/tStress (MPa)Y2Y3Y4(b)-40-30-20-100102000.81a/tStress (MPa)Y7A1Fig. 8. (a) Calculated stress profile vs. normalized depth reduced from the back gauge strain data only. (b) The stress profile obtainedfrom both top and back gauges.C. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 829591singular as the cut approaches the back face, seeFig. 7, whereas without these effects the strainwould approach a finite value (surface stress di-vided by plane strain elastic modulus) as the cutneared the opposite surface. In these specimens,the singular effects became significant after abouta=t 0:92; therefore, stresses are not reportedbeyond that depth.As seen in Fig. 7, especially samples Y6 and X4do not follow the generally observed smoothprofile. Their data show awkward changes in slopeand some especially bad data points were deleted.Later it was realized that the reason was variationsin the flow of the dielectric fluid and formation ofbubbles on the gauge surfaces as the EDM wiremoved into the sample. This happened only for afew samples because the wax coating on the gaugesof these samples had apparently a rougher surfacethat caused complicated flow patterns. Somesamples such as the ones in Fig. 8(a) were usuallyvery stable. Caution should be exercised when in-verting the stresses from problematic strain data asfitting the noise will result in a wavy profile insteadof the simple ?compression on the surface, tensionin the middle? distribution.The next source of error that was valid for allsamples is called the ?EDM effect?. During EDMcutting, a thin material layer (with a new stressstate) can be recast on the cut surfaces. This es-pecially affects the top gauge data while the backgauge is relatively insensitive. The data from thetest on the annealed specimen very small strainsmeasured by the back gauge and inconsistentlylarge values from the top gauge confirmed thelikelihood of an EDM effect. The consistency ofbottom gauge results for tests made with both0.004 and 0.006 in. diameter wires also corrobo-rates that the back gauge data were unaffected. AnEDM correction detailed in 25 was performed tothe top gauge strains in order to obtain the stressprofiles shown in Fig. 8(b). The profile of strain asa function of cut depth that is caused by an EDMeffect is quite different from the profile caused byresidual stresses. It can be estimated analyticallyand separated out from the measured strains.Several factors combined to make the EDMeffect an issue in these tests, whereas it is usuallyinsignificant 25. The major factor is that the re-sidual stresses are so low in the tested specimens.The strains caused by the EDM effect are generallya fixed small value, independent of the residualstress magnitudes, when cutting is performedgently. Thus, they generally comprise an insignifi-cant portion of the measured strains and do nothave a noticeable effect on the results. When thestresses are extremely low, as in the tests reportedhere, the EDM strains can contribute significantlyto the measured data and, therefore, affect theresults. The other factor is that the EDM effect is,other parameters equal, greater for materials withhigher yield stress and low thermal conductivity,both of which apply to the BMG specimens. Evenso, if the residual stresses were higher, the EDMeffect would have been negligible.6. DiscussionFigs. 2 and 3 exhibit the predictions of both theinstant freezing and viscoelastic models. Despitevastly different approaches and the use of inde-pendent data in each, their similarity is remark-able. For instance, the shapes of the stress vs. heattransfer coefficient plots in Fig. 3 are almostidentical. The model predictions, however, aresignificantly different than the experimental re-sults. It was initially assumed that convectivecooling with a heat transfer coefficient of h 2000W/(m2K) from traction-free surfaces throughoutthe production process of BMGs represented areasonable estimate 6. This value was chosensuch that the imagined convective cooling of Vit.1from melt to room temperature lasted a time pe-riod that roughly approximates the cooling dura-tion of the actual casting process (?10 s). Asobserved in Fig. 2, the prediction of the instantfreezing model for this 8.25 mm thick plate is )217MPa surface compression and 85 MPa mid-plane tension, while the same stresses are estimatedto be about )230 MPa on the surface and 90MPa in the interior using the viscoelastic model.Note that especially the value of surface com-pression is highly dependent on h. However, theexperimental results with 10 to 13 MPa mid-plane tension and )25 to )30 MPa surface com-pression are much smaller than model predictions.92C. Can Aydiner et al. / Journal of Non-Crystalline Solids 316 (2003) 8295This discrepancy is likely due to the two constit-uents that define the thermal tempering problem:specimenpreparationprocedureandmaterialproperties.The casting process described above does notinvolve convective cooling on traction-free sur-faces. Actually, it is more similar to the injectionmolding process used in the polymer industry28,29. For polymers, the residual stresses areclassified as flow-induced stresses and residualthermal stresses. Flow-induced stresses are thestresses generated during the filling stage of themold whereas the residual thermal stresses are dueto the differential cooling of the material, which isthe only case for the thermal tempering problem.It is noted in 29 that the flow stresses are an orderof magnitude lower than the thermal stresses inabsolute value. This is not surprising since theexistence of flow at considerable rates indicatesthat most of the material still has very short shearstress relaxation times, and thus, is not able toaccumulate residual stresses. Based on this argu-ment, it is likely that the residual stresses that formin the BMG plate are also mostly thermal residualstresses. The fact that the measured stress profileshave no significant spatial and directional depen-dence within the plate gives credence to thisconclusion. Furthermore, the observed ?compres-siontensioncompression? stress profile is typicalof thermal tempering 1,6. The thin layer of sur-face tension that forms in injection molding ofpolymers superimposed on this type of stressprofile was not observed in this study.The above arguments present the idea that al-though the actual casting process did not resemblethe thermal tempering case considered by themodels, the majority of the residual stresses werestill due to thermal tempering. However, althoughthe stress generation mechanism was the same, thethermal problem was not. In other words, thetemperature profile evolution in the modelingcalculation and that found in the actual samplewerelikelydifferent.Theconvectivecoolingproblem considered by the calculations yields anexponentially decreasing cosine profile after higherordertermsdecayquickly6.Theactualquenching realized by conduction to a large cop-per block initially at room temperature is morecomplicated. An understanding of the tempera-ture evolution is crucial for the generation of re-sidual stresses as thermal tempering is driven bythermal gradients, or in more simple terms, thedifferential between the surface and mid-planetemperatures. The low magnitude of measuredstressessuggestsa relativelyflattemperatureprofile about the glass transition of Vit.1 existedduring the casting. There are two possible sourcesof such a temperature profile. First, the contactbetween the plate and the mold surface was likelylost some time during the process. This wouldresult from the larger thermal contraction (espe-cially above the glass transition) of the platecompared to the large copper mold which prob-ably remained at relatively low temperatures.After this separation, the heat transfer from theplate to the vacuumed mold cavity would be sig-nificantly reduced. The second, and more obvious,source of a flat temperature profile in the plate is apossible increase of the temperature of the copperblock. This would result in hindered heat transferespecially at low temperatures. These results pointto the need for an instrumented casting experi-ment during which the exact temperature profileof BMG can be monitored in situ. This work iscurrently in progress.Both instant freezing and viscoelastic modelsdisregard a critical component of glass behavior:its structural relaxation from a metastable amor-phous solid to a supercooled liquid in thermody-namic equilibrium 10,11. During this process, theexc